file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Order/Filter/SmallSets.lean
|
Filter.hasBasis_smallSets
|
[] |
[
55,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean
|
algebraMap_isometry
|
[
{
"state_after": "α : Type ?u.578231\nβ : Type ?u.578234\nγ : Type ?u.578237\nι : Type ?u.578240\n𝕜 : Type u_2\n𝕜' : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\nx y : 𝕜\n⊢ dist (↑(algebraMap 𝕜 𝕜') x) (↑(algebraMap 𝕜 𝕜') y) = dist x y",
"state_before": "α : Type ?u.578231\nβ : Type ?u.578234\nγ : Type ?u.578237\nι : Type ?u.578240\n𝕜 : Type u_2\n𝕜' : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\n⊢ Isometry ↑(algebraMap 𝕜 𝕜')",
"tactic": "refine' Isometry.of_dist_eq fun x y => _"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.578231\nβ : Type ?u.578234\nγ : Type ?u.578237\nι : Type ?u.578240\n𝕜 : Type u_2\n𝕜' : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedRing 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : NormOneClass 𝕜'\nx y : 𝕜\n⊢ dist (↑(algebraMap 𝕜 𝕜') x) (↑(algebraMap 𝕜 𝕜') y) = dist x y",
"tactic": "rw [dist_eq_norm, dist_eq_norm, ← RingHom.map_sub, norm_algebraMap']"
}
] |
[
541,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/Data/List/Basic.lean
|
List.reduceOption_concat_of_some
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.369210\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nl : List (Option α)\nx : α\n⊢ reduceOption (concat l (some x)) = concat (reduceOption l) x",
"tactic": "simp only [reduceOption_nil, concat_eq_append, reduceOption_append, reduceOption_cons_of_some]"
}
] |
[
3493,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3491,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
tsum_ne_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nκ : Type ?u.55666\nα : Type u_1\ninst✝² : CanonicallyOrderedAddMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na : α\nhf : Summable f\n⊢ (∑' (i : ι), f i) ≠ 0 ↔ ∃ x, f x ≠ 0",
"tactic": "rw [Ne.def, tsum_eq_zero_iff hf, not_forall]"
}
] |
[
218,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/Topology/Algebra/Monoid.lean
|
continuousOn_finset_prod
|
[] |
[
772,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.mapFun.neg
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_2\nS : Type u_1\nT : Type ?u.310095\nhp : Fact (Nat.Prime p)\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nα : Type ?u.310110\nβ : Type ?u.310113\nf : R →+* S\nx y : 𝕎 R\n⊢ mapFun (↑f) (-x) = -mapFun (↑f) x",
"tactic": "map_fun_tac"
}
] |
[
124,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.int_cast_comp_cast
|
[
{
"state_after": "case zero\nR : Type u_1\ninst✝ : Ring R\n⊢ Int.cast ∘ cast = cast\n\ncase succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\n⊢ Int.cast ∘ cast = cast",
"state_before": "n : ℕ\nR : Type u_1\ninst✝ : Ring R\n⊢ Int.cast ∘ cast = cast",
"tactic": "cases n"
},
{
"state_after": "case succ.h\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nx✝ : ZMod (Nat.succ n✝)\n⊢ (Int.cast ∘ cast) x✝ = ↑x✝",
"state_before": "case succ\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\n⊢ Int.cast ∘ cast = cast",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case succ.h\nR : Type u_1\ninst✝ : Ring R\nn✝ : ℕ\nx✝ : ZMod (Nat.succ n✝)\n⊢ (Int.cast ∘ cast) x✝ = ↑x✝",
"tactic": "simp [ZMod, ZMod.cast]"
}
] |
[
253,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
249,
1
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
|
le_iff_exists_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : CanonicallyOrderedMonoid α\na b c d : α\n⊢ a ≤ b ↔ ∃ c, b = c * a",
"tactic": "simp only [mul_comm _ a, le_iff_exists_mul]"
}
] |
[
214,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.filter_parts_partition_eq_cycleType
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\na : ℕ\nh : a ∈ replicate (Fintype.card α - Finset.card (support σ)) 1\n⊢ ¬2 ≤ a",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\n⊢ filter (fun n => 2 ≤ n) (partition σ).parts = cycleType σ",
"tactic": "rw [parts_partition, filter_add, Multiset.filter_eq_self.2 fun _ => two_le_of_mem_cycleType,\n Multiset.filter_eq_nil.2 fun a h => ?_, add_zero]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\na : ℕ\nh : a ∈ replicate (Fintype.card α - Finset.card (support σ)) 1\n⊢ ¬2 ≤ 1",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\na : ℕ\nh : a ∈ replicate (Fintype.card α - Finset.card (support σ)) 1\n⊢ ¬2 ≤ a",
"tactic": "rw [Multiset.eq_of_mem_replicate h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\na : ℕ\nh : a ∈ replicate (Fintype.card α - Finset.card (support σ)) 1\n⊢ ¬2 ≤ 1",
"tactic": "decide"
}
] |
[
540,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Topology/MetricSpace/Completion.lean
|
LipschitzWith.completion_extension
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : MetricSpace β\ninst✝ : CompleteSpace β\nf : α → β\nK : ℝ≥0\nh : LipschitzWith K f\nx y : Completion α\n⊢ Continuous fun x => dist (Completion.extension f x.fst) (Completion.extension f x.snd)",
"tactic": "continuity"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : MetricSpace β\ninst✝ : CompleteSpace β\nf : α → β\nK : ℝ≥0\nh : LipschitzWith K f\nx y : Completion α\n⊢ Continuous fun x => ↑K * dist x.fst x.snd",
"tactic": "continuity"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : MetricSpace β\ninst✝ : CompleteSpace β\nf : α → β\nK : ℝ≥0\nh : LipschitzWith K f\nx y : Completion α\n⊢ ∀ (a b : α), dist (Completion.extension f (↑α a)) (Completion.extension f (↑α b)) ≤ ↑K * dist (↑α a) (↑α b)",
"tactic": "simpa only [extension_coe h.uniformContinuous, Completion.dist_eq] using h.dist_le_mul"
}
] |
[
201,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Continuous.measurable2
|
[] |
[
951,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
948,
1
] |
Mathlib/Data/Multiset/Functor.lean
|
Multiset.traverse_map
|
[
{
"state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\n⊢ ∀ (a : List α), traverse h (map g (Quotient.mk (isSetoid α) a)) = traverse (h ∘ g) (Quotient.mk (isSetoid α) a)",
"state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\n⊢ traverse h (map g x) = traverse (h ∘ g) x",
"tactic": "refine' Quotient.inductionOn x _"
},
{
"state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\na✝ : List α\n⊢ traverse h (map g (Quotient.mk (isSetoid α) a✝)) = traverse (h ∘ g) (Quotient.mk (isSetoid α) a✝)",
"state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\n⊢ ∀ (a : List α), traverse h (map g (Quotient.mk (isSetoid α) a)) = traverse (h ∘ g) (Quotient.mk (isSetoid α) a)",
"tactic": "intro"
},
{
"state_after": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\na✝ : List α\n⊢ Coe.coe <$> Traversable.traverse h (List.map g a✝) = Coe.coe <$> Traversable.traverse (h ∘ g) a✝",
"state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\na✝ : List α\n⊢ traverse h (map g (Quotient.mk (isSetoid α) a✝)) = traverse (h ∘ g) (Quotient.mk (isSetoid α) a✝)",
"tactic": "simp only [traverse, quot_mk_to_coe, coe_map, lift_coe, Function.comp_apply]"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝³ : Applicative F\ninst✝² : CommApplicative F\nα' β' : Type u\nf : α' → F β'\nG : Type u_1 → Type u_1\ninst✝¹ : Applicative G\ninst✝ : CommApplicative G\nα β γ : Type u_1\ng : α → β\nh : β → G γ\nx : Multiset α\na✝ : List α\n⊢ Coe.coe <$> Traversable.traverse h (List.map g a✝) = Coe.coe <$> Traversable.traverse (h ∘ g) a✝",
"tactic": "rw [← Traversable.traverse_map h g, List.map_eq_map]"
}
] |
[
135,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Topology/Algebra/WithZeroTopology.lean
|
WithZeroTopology.Iio_mem_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.83617\nΓ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nγ γ₁ γ₂ : Γ₀\nl : Filter α\nf : α → Γ₀\nh : γ₁ < γ₂\n⊢ Iio γ₂ ∈ 𝓝 γ₁",
"tactic": "rcases eq_or_ne γ₁ 0 with (rfl | h₀) <;> simp [*, h.ne', Iio_mem_nhds_zero]"
}
] |
[
132,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
Polynomial.algebraMap_hahnSeries_apply
|
[] |
[
1325,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1323,
1
] |
Mathlib/Data/Part.lean
|
Part.some_mul_some
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.57768\nγ : Type ?u.57771\ninst✝ : Mul α\na b : α\n⊢ some a * some b = some (a * b)",
"tactic": "simp [mul_def]"
}
] |
[
740,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
740,
1
] |
Mathlib/LinearAlgebra/Pi.lean
|
LinearMap.pi_zero
|
[
{
"state_after": "case h.h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nx✝¹ : M₂\nx✝ : ι\n⊢ ↑(pi fun i => 0) x✝¹ x✝ = ↑0 x✝¹ x✝",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\n⊢ (pi fun i => 0) = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M₃\nφ : ι → Type i\ninst✝¹ : (i : ι) → AddCommMonoid (φ i)\ninst✝ : (i : ι) → Module R (φ i)\nx✝¹ : M₂\nx✝ : ι\n⊢ ↑(pi fun i => 0) x✝¹ x✝ = ↑0 x✝¹ x✝",
"tactic": "rfl"
}
] |
[
72,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Topology/Separation.lean
|
compl_singleton_mem_nhds
|
[] |
[
622,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
div_one
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.20357\nβ : Type ?u.20360\nG : Type u_1\ninst✝ : DivInvOneMonoid G\na : G\n⊢ a / 1 = a",
"tactic": "simp [div_eq_mul_inv]"
}
] |
[
341,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Data/Real/Sign.lean
|
Real.sign_mul_pos_of_ne_zero
|
[
{
"state_after": "r : ℝ\nhr : r ≠ 0\nh : 0 = sign r * r\n⊢ r = 0",
"state_before": "r : ℝ\nhr : r ≠ 0\n⊢ 0 < sign r * r",
"tactic": "refine' lt_of_le_of_ne (sign_mul_nonneg r) fun h => hr _"
},
{
"state_after": "r : ℝ\nhr : r ≠ 0\nh : 0 = sign r * r\nhs0 : sign r = 0\n⊢ r = 0",
"state_before": "r : ℝ\nhr : r ≠ 0\nh : 0 = sign r * r\n⊢ r = 0",
"tactic": "have hs0 := (zero_eq_mul.mp h).resolve_right hr"
},
{
"state_after": "no goals",
"state_before": "r : ℝ\nhr : r ≠ 0\nh : 0 = sign r * r\nhs0 : sign r = 0\n⊢ r = 0",
"tactic": "exact sign_eq_zero_iff.mp hs0"
}
] |
[
104,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.continuousAt
|
[] |
[
64,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
11
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.nonempty_of_mem
|
[] |
[
754,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.uIcc_subset_uIcc_iff_mem
|
[] |
[
944,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
943,
1
] |
Mathlib/RingTheory/Polynomial/ScaleRoots.lean
|
Polynomial.coeff_scaleRoots
|
[
{
"state_after": "no goals",
"state_before": "A : Type ?u.2391\nK : Type ?u.2394\nR : Type u_1\nS : Type ?u.2400\ninst✝⁴ : CommRing A\ninst✝³ : IsDomain A\ninst✝² : Field K\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nM : Submonoid A\np : R[X]\ns : R\ni : ℕ\n⊢ coeff (scaleRoots p s) i = coeff p i * s ^ (natDegree p - i)",
"tactic": "simp (config := { contextual := true }) [scaleRoots, coeff_monomial]"
}
] |
[
38,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/CategoryTheory/Monoidal/Braided.lean
|
CategoryTheory.tensor_associativity
|
[
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "have :\n (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv :=\n by pure_coherence"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "rw [this]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "clear this"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (((((((((((α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫ (𝟙 (X₁ ⊗ Y₁) ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 2 4 => rw [tensor_μ_def₁]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n ((((((((((α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫ (𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (((((((((((α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫ (𝟙 (X₁ ⊗ Y₁) ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 4 5 => rw [← tensor_id, associator_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (((((((((𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (𝟙 X₁ ⊗ (𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n ((((((((((α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫ (𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 5 6 => rw [← tensor_comp, associator_inv_naturality, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫\n ((((((((α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (((((((((𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (𝟙 X₁ ⊗ (𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 6 7 => rw [associator_inv_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =\n ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫\n ((((((((α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫\n ((((((((α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "have :\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =\n ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂) :=\n by pure_coherence"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =\n ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((((((((((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =\n ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫\n ((((((((α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 2 6 => rw [this]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((((((((((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =\n ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((((((((((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "clear this"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (((((((((((((((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂))) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv) ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((((((((((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 1 3 => rw [← tensor_comp, ← tensor_comp, tensor_μ_def₁, tensor_comp, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (((((((((((((α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫ (((𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₁) ⊗ 𝟙 Z₂)) ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (((((((((((((((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂))) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv) ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 3 4 => rw [← tensor_id, associator_inv_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n (((((((((((((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ ((β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₁) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (((((((((((((α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫ (((𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₁) ⊗ 𝟙 Z₂)) ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 4 5 => rw [← tensor_comp, associator_naturality, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((((((((((((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂ ⊗ 𝟙 Z₁) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n (((((((((((((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ ((β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₁) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 5 6 =>\n rw [← tensor_comp, ← tensor_comp, associator_naturality, tensor_comp, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((((((((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((((((((((((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂ ⊗ 𝟙 Z₁) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (β_ (X₂ ⊗ Y₂) Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 6 10 =>\n rw [← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ←\n tensor_comp, ← tensor_comp, tensor_id, tensor_associativity_aux, ← tensor_id, ←\n id_comp (𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁), ← id_comp (𝟙 Z₂ ≫ 𝟙 Z₂ ≫ 𝟙 Z₂ ≫ 𝟙 Z₂ ≫ 𝟙 Z₂),\n tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp,\n tensor_comp, tensor_comp, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((((𝟙 ((X₁ ⊗ (Y₁ ⊗ Z₁ ⊗ X₂) ⊗ Y₂) ⊗ Z₂) ≫ ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((((((((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).hom) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 11 12 =>\n rw [← tensor_comp, ← tensor_comp, Iso.hom_inv_id]\n simp"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((((𝟙 ((X₁ ⊗ (Y₁ ⊗ Z₁ ⊗ X₂) ⊗ Y₂) ⊗ Z₂) ≫ ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "simp only [assoc, id_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (((𝟙 ((X₁ ⊗ ((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) ⊗ Z₂) ≫ (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 10 11 =>\n rw [← tensor_comp, ← tensor_comp, ← tensor_comp, Iso.hom_inv_id]\n simp"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (((𝟙 ((X₁ ⊗ ((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) ⊗ Z₂) ≫ (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "simp only [assoc, id_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((((α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫ (𝟙 X₁ ⊗ ((β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 9 10 => rw [associator_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫ (𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂ ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n ((((α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫ (𝟙 X₁ ⊗ ((β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 10 11 => rw [← tensor_comp, associator_naturality, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫ (𝟙 X₁ ⊗ (β_ X₂ (Y₁ ⊗ Z₁)).hom ⊗ 𝟙 Y₂ ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom)) ≫\n (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 11 13 => rw [tensor_id, ← tensor_μ_def₂]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "have :\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv) :=\n by pure_coherence"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 7 12 => rw [this]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\nthis :\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "clear this"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (((((((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫ (𝟙 X₁ ⊗ ((𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom)) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 6 7 => rw [associator_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n ((((((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫ (𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom)) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (((((((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫ (𝟙 X₁ ⊗ ((𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom) ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom)) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom)) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 7 8 => rw [← tensor_comp, associator_naturality, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (((((𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫ (𝟙 X₁ ⊗ 𝟙 X₂ ⊗ 𝟙 Y₁ ⊗ (β_ Y₂ Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n ((((((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫ (𝟙 X₁ ⊗ (𝟙 X₂ ⊗ 𝟙 Y₁) ⊗ (β_ Y₂ Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom)) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 8 9 => rw [← tensor_comp, associator_naturality, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫\n ((((α_ X₁ X₂ (Y₁ ⊗ (Y₂ ⊗ Z₁) ⊗ Z₂)).inv ≫ ((𝟙 X₁ ⊗ 𝟙 X₂) ⊗ 𝟙 Y₁ ⊗ (β_ Y₂ Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (((((𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫ (𝟙 X₁ ⊗ 𝟙 X₂ ⊗ 𝟙 Y₁ ⊗ (β_ Y₂ Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 9 10 => rw [associator_inv_naturality]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Y₂ ⊗ Z₁) ⊗ Z₂)).inv ≫\n (((𝟙 X₁ ⊗ 𝟙 X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Y₂ Z₁ Z₂).hom) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Y₂ (Z₁ ⊗ Z₂)).inv) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂))) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫\n ((((α_ X₁ X₂ (Y₁ ⊗ (Y₂ ⊗ Z₁) ⊗ Z₂)).inv ≫ ((𝟙 X₁ ⊗ 𝟙 X₂) ⊗ 𝟙 Y₁ ⊗ (β_ Y₂ Z₁).hom ⊗ 𝟙 Z₂)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom)) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "slice_lhs 10 12 => rw [← tensor_comp, ← tensor_comp, ← tensor_μ_def₂, tensor_comp, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Y₂ ⊗ Z₁) ⊗ Z₂)).inv ≫\n (((𝟙 X₁ ⊗ 𝟙 X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Y₂ Z₁ Z₂).hom) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Y₂ (Z₁ ⊗ Z₂)).inv) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂))) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Y₂ ⊗ Z₁) ⊗ Z₂)).inv ≫\n (((𝟙 X₁ ⊗ 𝟙 X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Y₂ Z₁ Z₂).hom) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Y₂ (Z₁ ⊗ Z₂)).inv) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂))) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂ ⊗ Z₁) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Y₂ ⊗ Z₁) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Y₂ ⊗ Z₁) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Y₂ ⊗ Z₁) ⊗ Z₂)).inv ≫\n (((𝟙 X₁ ⊗ 𝟙 X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Y₂ Z₁ Z₂).hom) ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Y₂ (Z₁ ⊗ Z₂)).inv) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂))) ≫\n tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) =\n (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂)",
"tactic": "coherence"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =\n (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫\n (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫\n ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv",
"tactic": "pure_coherence"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫\n (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫\n (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫\n (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =\n ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫\n (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫\n ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫ ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂)",
"tactic": "pure_coherence"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁸ : Category C\ninst✝⁷ : MonoidalCategory C\ninst✝⁶ : BraidedCategory C\nD : Type u₂\ninst✝⁵ : Category D\ninst✝⁴ : MonoidalCategory D\ninst✝³ : BraidedCategory D\nE : Type u₃\ninst✝² : Category E\ninst✝¹ : MonoidalCategory E\ninst✝ : BraidedCategory E\nX₁ X₂ Y₁ Y₂ Z₁ Z₂ : C\n⊢ ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫\n ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫\n (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =\n (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫\n (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫\n (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫\n (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫\n (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv)",
"tactic": "pure_coherence"
}
] |
[
555,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/Data/Complex/ExponentialBounds.lean
|
Real.log_two_gt_d9
|
[
{
"state_after": "no goals",
"state_before": "⊢ 0.6931471803 < 287209 / 414355 - 1 / 10 ^ 10",
"tactic": "norm_num1"
}
] |
[
75,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/RingTheory/Algebraic.lean
|
Polynomial.algebraMap_pi_self_eq_eval
|
[] |
[
477,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
475,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_neg_one
|
[
{
"state_after": "x✝ y z x : ℝ\nH : x ^ ↑(-1) = x⁻¹\n⊢ x ^ (-1) = x⁻¹\n\ncase H\nx✝ y z x : ℝ\n⊢ x ^ ↑(-1) = x⁻¹",
"state_before": "x✝ y z x : ℝ\n⊢ x ^ (-1) = x⁻¹",
"tactic": "suffices H : x ^ ((-1 : ℤ) : ℝ) = x⁻¹"
},
{
"state_after": "no goals",
"state_before": "case H\nx✝ y z x : ℝ\n⊢ x ^ ↑(-1) = x⁻¹",
"tactic": "simp only [rpow_int_cast, zpow_one, zpow_neg]"
},
{
"state_after": "no goals",
"state_before": "x✝ y z x : ℝ\nH : x ^ ↑(-1) = x⁻¹\n⊢ x ^ (-1) = x⁻¹",
"tactic": "rwa [Int.cast_neg, Int.cast_one] at H"
}
] |
[
372,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.im_le_norm
|
[] |
[
737,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
736,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubmodule.nontrivial_iff
|
[] |
[
539,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
MulDistribMulAction.toMonoidHom_apply
|
[] |
[
1036,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1034,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_mono_null
|
[] |
[
200,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.ext
|
[] |
[
562,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
561,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Set.iUnion_eq_iUnion_finset
|
[] |
[
1855,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1854,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.sum_sub_index
|
[
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddGroup (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommGroup γ\nf g : Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂\nthis :\n ↑(↑liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) (f - g) =\n ↑(↑liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) f -\n ↑(↑liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\n⊢ sum (f - g) h = sum f h - sum g h",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddGroup (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommGroup γ\nf g : Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂\n⊢ sum (f - g) h = sum f h - sum g h",
"tactic": "have := (liftAddHom (β := β) fun a => AddMonoidHom.ofMapSub (h a) (h_sub a)).map_sub f g"
},
{
"state_after": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddGroup (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommGroup γ\nf g : Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂\nthis✝² :\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) (f - g) =\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) f -\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\nthis✝¹ :\n (sum (f - g) fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) =\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) f -\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\nthis✝ :\n (sum (f - g) fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) =\n (sum f fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) -\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\nthis :\n (sum (f - g) fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) =\n (sum f fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) -\n sum g fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))\n⊢ sum (f - g) h = sum f h - sum g h",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddGroup (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommGroup γ\nf g : Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂\nthis :\n ↑(↑liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) (f - g) =\n ↑(↑liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) f -\n ↑(↑liftAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\n⊢ sum (f - g) h = sum f h - sum g h",
"tactic": "rw [liftAddHom_apply, sumAddHom_apply, sumAddHom_apply, sumAddHom_apply] at this"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ndec : DecidableEq ι\ninst✝² : (i : ι) → AddGroup (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommGroup γ\nf g : Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂\nthis✝² :\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) (f - g) =\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) f -\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\nthis✝¹ :\n (sum (f - g) fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) =\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) f -\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\nthis✝ :\n (sum (f - g) fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) =\n (sum f fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) -\n ↑(sumAddHom fun a => AddMonoidHom.ofMapSub (h a) (_ : ∀ (b₁ b₂ : β a), h a (b₁ - b₂) = h a b₁ - h a b₂)) g\nthis :\n (sum (f - g) fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) =\n (sum f fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))) -\n sum g fun x => ↑(AddMonoidHom.ofMapSub (h x) (_ : ∀ (b₁ b₂ : β x), h x (b₁ - b₂) = h x b₁ - h x b₂))\n⊢ sum (f - g) h = sum f h - sum g h",
"tactic": "exact this"
}
] |
[
2081,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2076,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.sub_self_im
|
[] |
[
823,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
16
] |
Mathlib/GroupTheory/Sylow.lean
|
Sylow.pow_dvd_card_of_pow_dvd_card
|
[] |
[
665,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Cofork.condition
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf g : X ⟶ Y\nt : Cofork f g\n⊢ f ≫ π t = g ≫ π t",
"tactic": "rw [← t.app_zero_eq_comp_π_left, ← t.app_zero_eq_comp_π_right]"
}
] |
[
398,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Order/Basic.lean
|
eq_of_forall_le_iff
|
[] |
[
525,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
524,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.trans_right_ne
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.323287\nP : Type u_3\nP' : Type ?u.323293\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nw x z : P\nh₁ : Wbtw R w x z\nh : w ≠ x\nh₂ : Wbtw R x w z\n⊢ False",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.323287\nP : Type u_3\nP' : Type ?u.323293\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nw x y z : P\nh₁ : Wbtw R w x z\nh₂ : Wbtw R x y z\nh : w ≠ x\n⊢ w ≠ y",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.323287\nP : Type u_3\nP' : Type ?u.323293\ninst✝⁷ : OrderedRing R\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : Module R V\ninst✝⁴ : AddTorsor V P\ninst✝³ : AddCommGroup V'\ninst✝² : Module R V'\ninst✝¹ : AddTorsor V' P'\ninst✝ : NoZeroSMulDivisors R V\nw x z : P\nh₁ : Wbtw R w x z\nh : w ≠ x\nh₂ : Wbtw R x w z\n⊢ False",
"tactic": "exact h (h₁.swap_left_iff.1 h₂)"
}
] |
[
514,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.smul
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.724295\nα : Type ?u.724298\nβ : Type u_3\nι : Type ?u.724304\ninst✝⁵ : OrderedCommSemiring 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : OrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf : E → β\nc : 𝕜\nhc : 0 ≤ c\nhf : ConvexOn 𝕜 s f\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ c • (a • f x + b • f y) = a • c • f x + b • c • f y",
"tactic": "rw [smul_add, smul_comm c, smul_comm c]"
}
] |
[
983,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
978,
1
] |
Mathlib/GroupTheory/Subgroup/Finite.lean
|
Subgroup.list_prod_mem
|
[] |
[
54,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
11
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.bot_or_exists_ne_one
|
[] |
[
1389,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1388,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
Homotopy.nullHomotopicMap'_f
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ Hom.f (nullHomotopicMap fun i j => dite (ComplexShape.Rel c j i) (h i j) fun x => 0) k₁ =\n d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ Hom.f (nullHomotopicMap' h) k₁ = d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"tactic": "simp only [nullHomotopicMap']"
},
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ (d C k₁ k₀ ≫ dite (ComplexShape.Rel c k₁ k₀) (h k₀ k₁) fun x => 0) +\n (dite (ComplexShape.Rel c k₂ k₁) (h k₁ k₂) fun x => 0) ≫ d D k₂ k₁ =\n d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ Hom.f (nullHomotopicMap fun i j => dite (ComplexShape.Rel c j i) (h i j) fun x => 0) k₁ =\n d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"tactic": "rw [nullHomotopicMap_f r₂₁ r₁₀]"
},
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ (d C k₁ k₀ ≫ dite (ComplexShape.Rel c k₁ k₀) (h k₀ k₁) fun x => 0) +\n (dite (ComplexShape.Rel c k₂ k₁) (h k₁ k₂) fun x => 0) ≫ d D k₂ k₁ =\n d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ (d C k₁ k₀ ≫ dite (ComplexShape.Rel c k₁ k₀) (h k₀ k₁) fun x => 0) +\n (dite (ComplexShape.Rel c k₂ k₁) (h k₁ k₂) fun x => 0) ≫ d D k₂ k₁ =\n d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"tactic": "dsimp"
},
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁ = d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ (d C k₁ k₀ ≫ dite (ComplexShape.Rel c k₁ k₀) (h k₀ k₁) fun x => 0) +\n (dite (ComplexShape.Rel c k₂ k₁) (h k₁ k₂) fun x => 0) ≫ d D k₂ k₁ =\n d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni k₂ k₁ k₀ : ι\nr₂₁ : ComplexShape.Rel c k₂ k₁\nr₁₀ : ComplexShape.Rel c k₁ k₀\nh : (i j : ι) → ComplexShape.Rel c j i → (X C i ⟶ X D j)\n⊢ d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁ = d C k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ d D k₂ k₁",
"tactic": "rfl"
}
] |
[
383,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.coe_coe
|
[] |
[
153,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.mem_divisors_prime_pow
|
[
{
"state_after": "n p : ℕ\npp : Prime p\nk x : ℕ\n⊢ (∃ k_1, k_1 ≤ k ∧ x = p ^ k_1) ↔ ∃ j x_1, x = p ^ j",
"state_before": "n p : ℕ\npp : Prime p\nk x : ℕ\n⊢ x ∈ divisors (p ^ k) ↔ ∃ j x_1, x = p ^ j",
"tactic": "rw [mem_divisors, Nat.dvd_prime_pow pp, and_iff_left (ne_of_gt (pow_pos pp.pos k))]"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\npp : Prime p\nk x : ℕ\n⊢ (∃ k_1, k_1 ≤ k ∧ x = p ^ k_1) ↔ ∃ j x_1, x = p ^ j",
"tactic": "simp"
}
] |
[
313,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.pure_injective
|
[] |
[
2599,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2598,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
AlgHom.FinitePresentation.of_finiteType
|
[] |
[
526,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean
|
MeasureTheory.TendstoInMeasure.congr_right
|
[] |
[
99,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Std/Data/List/Lemmas.lean
|
List.extractP_eq_find?_eraseP
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\n⊢ extractP p l = (find? p l, eraseP p l)",
"tactic": "exact go #[] _ rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nh : l = acc.data ++ []\n⊢ extractP.go p l [] acc = (find? p [], acc.data ++ eraseP p [])",
"tactic": "simp [extractP.go, find?, eraseP, h]"
},
{
"state_after": "α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs →\n (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (Array.push acc x)) =\n (match p x with\n | true => some x\n | false => find? p xs,\n acc.data ++ bif p x then xs else x :: eraseP p xs)",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs → extractP.go p l (x :: xs) acc = (find? p (x :: xs), acc.data ++ eraseP p (x :: xs))",
"tactic": "simp [extractP.go, find?, eraseP]"
},
{
"state_after": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs → extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs →\n (bif p x then (some x, acc.data ++ xs) else extractP.go p l xs (Array.push acc x)) =\n (match p x with\n | true => some x\n | false => find? p xs,\n acc.data ++ bif p x then xs else x :: eraseP p xs)",
"tactic": "cases p x <;> simp"
},
{
"state_after": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)",
"state_before": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\n⊢ l = acc.data ++ x :: xs → extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)",
"tactic": "intro h"
},
{
"state_after": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ (find? p xs, (Array.push acc x).data ++ eraseP p xs) = (find? p xs, acc.data ++ x :: eraseP p xs)\n\ncase false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"state_before": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ extractP.go p l xs (Array.push acc x) = (find? p xs, acc.data ++ x :: eraseP p xs)",
"tactic": "rw [go _ xs]"
},
{
"state_after": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"state_before": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ (find? p xs, (Array.push acc x).data ++ eraseP p xs) = (find? p xs, acc.data ++ x :: eraseP p xs)\n\ncase false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"tactic": "{simp}"
},
{
"state_after": "no goals",
"state_before": "case false\nα : Type u_1\np : α → Bool\nl : List α\nacc : Array α\nx : α\nxs : List α\nh : l = acc.data ++ x :: xs\n⊢ l = (Array.push acc x).data ++ xs",
"tactic": "simp [h]"
}
] |
[
1021,
21
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1013,
9
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrix'_symm
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
1
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.degree_div_lt
|
[
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ degree (p / q) < degree p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\n⊢ degree (p / q) < degree p",
"tactic": "have hq0 : q ≠ 0 := fun hq0 => by simp [hq0] at hq"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ degree (p /ₘ (q * ↑C (leadingCoeff q)⁻¹)) < degree p",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ degree (p / q) < degree p",
"tactic": "rw [div_def, mul_comm, degree_mul_leadingCoeff_inv _ hq0]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ degree (p /ₘ (q * ↑C (leadingCoeff q)⁻¹)) < degree p",
"tactic": "exact\n degree_divByMonic_lt _ (monic_mul_leadingCoeff_inv hq0) hp\n (by rw [degree_mul_leadingCoeff_inv _ hq0]; exact hq)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q = 0\n⊢ False",
"tactic": "simp [hq0] at hq"
},
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ 0 < degree q",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ 0 < degree (q * ↑C (leadingCoeff q)⁻¹)",
"tactic": "rw [degree_mul_leadingCoeff_inv _ hq0]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np q : R[X]\nhp : p ≠ 0\nhq : 0 < degree q\nhq0 : q ≠ 0\n⊢ 0 < degree q",
"tactic": "exact hq"
}
] |
[
274,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.ncard_eq_three
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.164615\ns t : Set α\na b x y : α\nf : α → β\nh : ncard s = 3\n⊢ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.164615\ns t : Set α\na b x y : α\nf : α → β\n⊢ (∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}) → ncard s = 3",
"state_before": "α : Type u_1\nβ : Type ?u.164615\ns t : Set α\na b x y : α\nf : α → β\n⊢ ncard s = 3 ↔ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}",
"tactic": "refine' ⟨fun h ↦ _, _⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ncard {x, y, z} = 3",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.164615\ns t : Set α\na b x y : α\nf : α → β\n⊢ (∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}) → ncard s = 3",
"tactic": "rintro ⟨x, y, z, xy, xz, yz, rfl⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ¬y ∈ {z}\n\ncase refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ¬x ∈ {y, z}",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ncard {x, y, z} = 3",
"tactic": "rw [ncard_insert_of_not_mem, ncard_insert_of_not_mem, ncard_singleton]"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ¬(x = y ∨ x = z)",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ¬x ∈ {y, z}",
"tactic": "rw [mem_insert_iff, mem_singleton_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ¬(x = y ∨ x = z)",
"tactic": "tauto"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt✝ : Set α\na b x✝ y : α\nf : α → β\nx : α\nt : Set α\nhxt : ¬x ∈ t\nht : ncard t = 2\nh : ncard (insert x t) = 3\n⊢ ∃ x_1 y z, x_1 ≠ y ∧ x_1 ≠ z ∧ y ≠ z ∧ insert x t = {x_1, y, z}",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.164615\ns t : Set α\na b x y : α\nf : α → β\nh : ncard s = 3\n⊢ ∃ x y z, x ≠ y ∧ x ≠ z ∧ y ≠ z ∧ s = {x, y, z}",
"tactic": "obtain ⟨x, t, hxt, rfl, ht⟩ := eq_insert_of_ncard_eq_succ h"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nhyz : y ≠ z\nhxt : ¬x ∈ {y, z}\nht : ncard {y, z} = 2\nh : ncard {x, y, z} = 3\n⊢ ∃ x_1 y_1 z_1, x_1 ≠ y_1 ∧ x_1 ≠ z_1 ∧ y_1 ≠ z_1 ∧ {x, y, z} = {x_1, y_1, z_1}",
"state_before": "case refine'_1.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt✝ : Set α\na b x✝ y : α\nf : α → β\nx : α\nt : Set α\nhxt : ¬x ∈ t\nht : ncard t = 2\nh : ncard (insert x t) = 3\n⊢ ∃ x_1 y z, x_1 ≠ y ∧ x_1 ≠ z ∧ y ≠ z ∧ insert x t = {x_1, y, z}",
"tactic": "obtain ⟨y, z, hyz, rfl⟩ := ncard_eq_two.mp ht"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nhyz : y ≠ z\nhxt : ¬x = y ∧ ¬x = z\nht : ncard {y, z} = 2\nh : ncard {x, y, z} = 3\n⊢ ∃ x_1 y_1 z_1, x_1 ≠ y_1 ∧ x_1 ≠ z_1 ∧ y_1 ≠ z_1 ∧ {x, y, z} = {x_1, y_1, z_1}",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nhyz : y ≠ z\nhxt : ¬x ∈ {y, z}\nht : ncard {y, z} = 2\nh : ncard {x, y, z} = 3\n⊢ ∃ x_1 y_1 z_1, x_1 ≠ y_1 ∧ x_1 ≠ z_1 ∧ y_1 ≠ z_1 ∧ {x, y, z} = {x_1, y_1, z_1}",
"tactic": "rw [mem_insert_iff, mem_singleton_iff, not_or] at hxt"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nhyz : y ≠ z\nhxt : ¬x = y ∧ ¬x = z\nht : ncard {y, z} = 2\nh : ncard {x, y, z} = 3\n⊢ ∃ x_1 y_1 z_1, x_1 ≠ y_1 ∧ x_1 ≠ z_1 ∧ y_1 ≠ z_1 ∧ {x, y, z} = {x_1, y_1, z_1}",
"tactic": "exact ⟨x, y, z, hxt.1, hxt.2, hyz, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.164615\nt : Set α\na b x✝ y✝ : α\nf : α → β\nx y z : α\nxy : x ≠ y\nxz : x ≠ z\nyz : y ≠ z\n⊢ ¬y ∈ {z}",
"tactic": "rwa [mem_singleton_iff]"
}
] |
[
767,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
757,
1
] |
Mathlib/Control/Traversable/Basic.lean
|
ApplicativeTransformation.preserves_map'
|
[
{
"state_after": "case h\nF : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β : Type u\nx : α → β\ny : F α\n⊢ ((fun {α} => app η α) ∘ Functor.map x) y = (Functor.map x ∘ fun {α} => app η α) y",
"state_before": "F : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β : Type u\nx : α → β\n⊢ (fun {α} => app η α) ∘ Functor.map x = Functor.map x ∘ fun {α} => app η α",
"tactic": "ext y"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type u → Type v\ninst✝³ : Applicative F\ninst✝² : LawfulApplicative F\nG : Type u → Type w\ninst✝¹ : Applicative G\ninst✝ : LawfulApplicative G\nη : ApplicativeTransformation F G\nα β : Type u\nx : α → β\ny : F α\n⊢ ((fun {α} => app η α) ∘ Functor.map x) y = (Functor.map x ∘ fun {α} => app η α) y",
"tactic": "exact preserves_map η x y"
}
] |
[
159,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
LipschitzWith.comp_lipschitzOnWith
|
[] |
[
236,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Order/Iterate.lean
|
Monotone.antitone_iterate_of_map_le
|
[] |
[
244,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/CategoryTheory/Abelian/Exact.lean
|
CategoryTheory.Abelian.exact_iff_exact_coimage_π
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f (coimage.π g ≫ Abelian.factorThruCoimage g) ↔ Exact f (coimage.π g)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f g ↔ Exact f (coimage.π g)",
"tactic": "conv_lhs => rw [← Abelian.coimage.fac g]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ Exact f (coimage.π g ≫ Abelian.factorThruCoimage g) ↔ Exact f (coimage.π g)",
"tactic": "rw [exact_comp_mono_iff]"
}
] |
[
249,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean
|
Rel.interedges_disjoint_left
|
[
{
"state_after": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : ∀ ⦃a : α⦄, a ∈ s → ¬a ∈ s'\nt : Finset β\n⊢ ∀ ⦃a : α × β⦄, a ∈ interedges r s t → ¬a ∈ interedges r s' t",
"state_before": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : Disjoint s s'\nt : Finset β\n⊢ Disjoint (interedges r s t) (interedges r s' t)",
"tactic": "rw [Finset.disjoint_left] at hs⊢"
},
{
"state_after": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : ∀ ⦃a : α⦄, a ∈ s → ¬a ∈ s'\nt : Finset β\na✝ : α × β\nhx : a✝ ∈ interedges r s t\nhy : a✝ ∈ interedges r s' t\n⊢ False",
"state_before": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : ∀ ⦃a : α⦄, a ∈ s → ¬a ∈ s'\nt : Finset β\n⊢ ∀ ⦃a : α × β⦄, a ∈ interedges r s t → ¬a ∈ interedges r s' t",
"tactic": "intro _ hx hy"
},
{
"state_after": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : ∀ ⦃a : α⦄, a ∈ s → ¬a ∈ s'\nt : Finset β\na✝ : α × β\nhx : a✝.fst ∈ s ∧ a✝.snd ∈ t ∧ r a✝.fst a✝.snd\nhy : a✝.fst ∈ s' ∧ a✝.snd ∈ t ∧ r a✝.fst a✝.snd\n⊢ False",
"state_before": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : ∀ ⦃a : α⦄, a ∈ s → ¬a ∈ s'\nt : Finset β\na✝ : α × β\nhx : a✝ ∈ interedges r s t\nhy : a✝ ∈ interedges r s' t\n⊢ False",
"tactic": "rw [mem_interedges_iff] at hx hy"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.5508\nι : Type ?u.5511\nκ : Type ?u.5514\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrderedField 𝕜\nr : α → β → Prop\ninst✝ : (a : α) → DecidablePred (r a)\ns✝ s₁ s₂ : Finset α\nt✝ t₁ t₂ : Finset β\na : α\nb : β\nδ : 𝕜\ns s' : Finset α\nhs : ∀ ⦃a : α⦄, a ∈ s → ¬a ∈ s'\nt : Finset β\na✝ : α × β\nhx : a✝.fst ∈ s ∧ a✝.snd ∈ t ∧ r a✝.fst a✝.snd\nhy : a✝.fst ∈ s' ∧ a✝.snd ∈ t ∧ r a✝.fst a✝.snd\n⊢ False",
"tactic": "exact hs hx.1 hy.1"
}
] |
[
93,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean
|
finrank_vectorSpan_le_iff_not_affineIndependent
|
[] |
[
207,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_lt_iff
|
[] |
[
921,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
920,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherianRing_iff
|
[] |
[
493,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
492,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.countp_eq_card_filter
|
[] |
[
2226,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2225,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.coe_ker
|
[] |
[
771,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
770,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_Ico
|
[] |
[
161,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/BigOperators/NatAntidiagonal.lean
|
Finset.Nat.prod_antidiagonal_succ
|
[
{
"state_after": "M : Type u_1\nN : Type ?u.17\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (0, n + 1) *\n ∏ x in antidiagonal n,\n f\n (↑(Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl ℕ))\n x) =\n f (0, n + 1) * ∏ p in antidiagonal n, f (p.fst + 1, p.snd)",
"state_before": "M : Type u_1\nN : Type ?u.17\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ ∏ p in antidiagonal (n + 1), f p = f (0, n + 1) * ∏ p in antidiagonal n, f (p.fst + 1, p.snd)",
"tactic": "rw [antidiagonal_succ, prod_cons, prod_map]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.17\ninst✝¹ : CommMonoid M\ninst✝ : AddCommMonoid N\nn : ℕ\nf : ℕ × ℕ → M\n⊢ f (0, n + 1) *\n ∏ x in antidiagonal n,\n f\n (↑(Function.Embedding.prodMap { toFun := Nat.succ, inj' := Nat.succ_injective } (Function.Embedding.refl ℕ))\n x) =\n f (0, n + 1) * ∏ p in antidiagonal n, f (p.fst + 1, p.snd)",
"tactic": "rfl"
}
] |
[
30,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.conjTranspose_list_prod
|
[] |
[
2331,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2329,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.monotone_lift'
|
[] |
[
341,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.linear_injective_iff
|
[
{
"state_after": "case intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\n⊢ Function.Injective ↑f.linear ↔ Function.Injective ↑f",
"state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\n⊢ Function.Injective ↑f.linear ↔ Function.Injective ↑f",
"tactic": "obtain ⟨p⟩ := (inferInstance : Nonempty P1)"
},
{
"state_after": "case intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\nh : ↑f.linear = ↑(Equiv.vaddConst (↑f p)).symm ∘ ↑f ∘ ↑(Equiv.vaddConst p)\n⊢ Function.Injective ↑f.linear ↔ Function.Injective ↑f",
"state_before": "case intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\n⊢ Function.Injective ↑f.linear ↔ Function.Injective ↑f",
"tactic": "have h : ⇑f.linear = (Equiv.vaddConst (f p)).symm ∘ f ∘ Equiv.vaddConst p := by\n ext v\n simp [f.map_vadd, vadd_vsub_assoc]"
},
{
"state_after": "no goals",
"state_before": "case intro\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\nh : ↑f.linear = ↑(Equiv.vaddConst (↑f p)).symm ∘ ↑f ∘ ↑(Equiv.vaddConst p)\n⊢ Function.Injective ↑f.linear ↔ Function.Injective ↑f",
"tactic": "rw [h, Equiv.comp_injective, Equiv.injective_comp]"
},
{
"state_after": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\nv : V1\n⊢ ↑f.linear v = (↑(Equiv.vaddConst (↑f p)).symm ∘ ↑f ∘ ↑(Equiv.vaddConst p)) v",
"state_before": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\n⊢ ↑f.linear = ↑(Equiv.vaddConst (↑f p)).symm ∘ ↑f ∘ ↑(Equiv.vaddConst p)",
"tactic": "ext v"
},
{
"state_after": "no goals",
"state_before": "case h\nk : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type ?u.361436\nP3 : Type ?u.361439\nV4 : Type ?u.361442\nP4 : Type ?u.361445\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\nf : P1 →ᵃ[k] P2\np : P1\nv : V1\n⊢ ↑f.linear v = (↑(Equiv.vaddConst (↑f p)).symm ∘ ↑f ∘ ↑(Equiv.vaddConst p)) v",
"tactic": "simp [f.map_vadd, vadd_vsub_assoc]"
}
] |
[
474,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
finrank_span_singleton
|
[
{
"state_after": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ finrank K { x // x ∈ span K {v} } ≤ 1\n\ncase a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ 1 ≤ finrank K { x // x ∈ span K {v} }",
"state_before": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ finrank K { x // x ∈ span K {v} } = 1",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ finrank K { x // x ∈ span K {v} } ≤ 1",
"tactic": "exact finrank_span_le_card ({v} : Set V)"
},
{
"state_after": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ Nontrivial { x // x ∈ span K {v} }",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ 1 ≤ finrank K { x // x ∈ span K {v} }",
"tactic": "rw [Nat.succ_le_iff, finrank_pos_iff]"
},
{
"state_after": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ { val := v, property := (_ : v ∈ span K {v}) } ≠ 0",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ Nontrivial { x // x ∈ span K {v} }",
"tactic": "use ⟨v, mem_span_singleton_self v⟩, 0"
},
{
"state_after": "no goals",
"state_before": "case a\nK : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nhv : v ≠ 0\n⊢ { val := v, property := (_ : v ∈ span K {v}) } ≠ 0",
"tactic": "simp [hv]"
}
] |
[
1155,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1150,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.eq_of_subset_of_card_le
|
[] |
[
1247,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1245,
1
] |
Mathlib/Data/Int/Units.lean
|
Int.eq_of_mul_eq_one
|
[] |
[
66,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Data/Rat/Order.lean
|
Rat.le_refl
|
[
{
"state_after": "a b c : ℚ\n⊢ Rat.Nonneg (a - a)",
"state_before": "a b c : ℚ\n⊢ a ≤ a",
"tactic": "rw [Rat.le_iff_Nonneg]"
},
{
"state_after": "a b c : ℚ\n⊢ Rat.Nonneg (a - a)",
"state_before": "a b c : ℚ\n⊢ Rat.Nonneg (a - a)",
"tactic": "show Rat.Nonneg (a - a)"
},
{
"state_after": "a b c : ℚ\n⊢ Rat.Nonneg 0",
"state_before": "a b c : ℚ\n⊢ Rat.Nonneg (a - a)",
"tactic": "rw [sub_self]"
},
{
"state_after": "no goals",
"state_before": "a b c : ℚ\n⊢ Rat.Nonneg 0",
"tactic": "exact le_refl (0 : ℤ)"
}
] |
[
165,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
11
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.mkSpanSingleton'_apply_self
|
[
{
"state_after": "R : Type u_2\ninst✝⁶ : Ring R\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.87505\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → c • y = 0\nh : x ∈ (mkSpanSingleton' x y H).domain\nthis : ↑(mkSpanSingleton' x y H) { val := 1 • x, property := (_ : 1 • x ∈ (mkSpanSingleton' x y H).domain) } = 1 • y\n⊢ ↑(mkSpanSingleton' x y H) { val := x, property := h } = y",
"state_before": "R : Type u_2\ninst✝⁶ : Ring R\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.87505\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → c • y = 0\nh : x ∈ (mkSpanSingleton' x y H).domain\n⊢ ↑(mkSpanSingleton' x y H) { val := x, property := h } = y",
"tactic": "have := by refine mkSpanSingleton'_apply x y H 1 ?_; rwa [one_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : Ring R\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.87505\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → c • y = 0\nh : x ∈ (mkSpanSingleton' x y H).domain\nthis : ↑(mkSpanSingleton' x y H) { val := 1 • x, property := (_ : 1 • x ∈ (mkSpanSingleton' x y H).domain) } = 1 • y\n⊢ ↑(mkSpanSingleton' x y H) { val := x, property := h } = y",
"tactic": "convert this <;> rw [one_smul]"
},
{
"state_after": "R : Type u_2\ninst✝⁶ : Ring R\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.87505\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → c • y = 0\nh : x ∈ (mkSpanSingleton' x y H).domain\n⊢ 1 • x ∈ (mkSpanSingleton' x y H).domain",
"state_before": "R : Type u_2\ninst✝⁶ : Ring R\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.87505\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → c • y = 0\nh : x ∈ (mkSpanSingleton' x y H).domain\n⊢ ?m.91161",
"tactic": "refine mkSpanSingleton'_apply x y H 1 ?_"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : Ring R\nE : Type u_1\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_3\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.87505\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\nx : E\ny : F\nH : ∀ (c : R), c • x = 0 → c • y = 0\nh : x ∈ (mkSpanSingleton' x y H).domain\n⊢ 1 • x ∈ (mkSpanSingleton' x y H).domain",
"tactic": "rwa [one_smul]"
}
] |
[
171,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Analysis/Calculus/Monotone.lean
|
StieltjesFunction.ae_hasDerivAt
|
[
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"state_before": "f : StieltjesFunction\n⊢ ∀ᵐ (x : ℝ), HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"tactic": "filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (vitaliFamily (volume : Measure ℝ) 1) f.measure,\n rnDeriv_lt_top f.measure volume, f.countable_leftLim_ne.ae_not_mem volume] with x hx h'x h''x"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"tactic": "have L1 :\n Tendsto (fun y => (f y - f x) / (y - x)) (𝓝[>] x) (𝓝 (rnDeriv f.measure volume x).toReal) := by\n apply Tendsto.congr' _\n ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))\n filter_upwards [self_mem_nhdsWithin]\n rintro y (hxy : x < y)\n simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]\n rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]\n exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"tactic": "have L2 : Tendsto (fun y => (leftLim f y - f x) / (y - x)) (𝓝[<] x)\n (𝓝 (rnDeriv f.measure volume x).toReal) := by\n apply Tendsto.congr' _\n ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_left x)))\n filter_upwards [self_mem_nhdsWithin]\n rintro y (hxy : y < x)\n simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]\n rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal, ← neg_neg (y - x),\n div_neg, neg_div', neg_sub, neg_sub]\n exact div_nonneg (sub_nonneg.2 (f.mono.leftLim_le hxy.le)) (sub_pos.2 hxy).le"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL4 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun b => (↑f b - ↑f x) / (b - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x))) ∧\n Tendsto (fun b => (↑f b - ↑f x) / (b - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL4 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ HasDerivAt (↑f) (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)) x",
"tactic": "rw [hasDerivAt_iff_tendsto_slope, slope_fun_def_field, ← nhds_left'_sup_nhds_right', tendsto_sup]"
},
{
"state_after": "no goals",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL4 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun b => (↑f b - ↑f x) / (b - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x))) ∧\n Tendsto (fun b => (↑f b - ↑f x) / (b - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))",
"tactic": "exact ⟨L4, L1⟩"
},
{
"state_after": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc x y) =ᶠ[𝓝[Ioi x] x]\n fun y => (↑f y - ↑f x) / (y - x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))",
"tactic": "apply Tendsto.congr' _\n ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_right x)))"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ ∀ (a : ℝ),\n a ∈ Ioi x →\n (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc x y) a =\n (↑f a - ↑f x) / (a - x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc x y) =ᶠ[𝓝[Ioi x] x]\n fun y => (↑f y - ↑f x) / (y - x)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\ny : ℝ\nhxy : x < y\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc x y) y =\n (↑f y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\n⊢ ∀ (a : ℝ),\n a ∈ Ioi x →\n (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc x y) a =\n (↑f a - ↑f x) / (a - x)",
"tactic": "rintro y (hxy : x < y)"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\ny : ℝ\nhxy : x < y\n⊢ ENNReal.toReal (ENNReal.ofReal (↑f y - ↑f x) / ENNReal.ofReal (y - x)) = (↑f y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\ny : ℝ\nhxy : x < y\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc x y) y =\n (↑f y - ↑f x) / (y - x)",
"tactic": "simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc, Classical.not_not.1 h''x]"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\ny : ℝ\nhxy : x < y\n⊢ 0 ≤ (↑f y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\ny : ℝ\nhxy : x < y\n⊢ ENNReal.toReal (ENNReal.ofReal (↑f y - ↑f x) / ENNReal.ofReal (y - x)) = (↑f y - ↑f x) / (y - x)",
"tactic": "rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal]"
},
{
"state_after": "no goals",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\ny : ℝ\nhxy : x < y\n⊢ 0 ≤ (↑f y - ↑f x) / (y - x)",
"tactic": "exact div_nonneg (sub_nonneg.2 (f.mono hxy.le)) (sub_pos.2 hxy).le"
},
{
"state_after": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc y x) =ᶠ[𝓝[Iio x] x]\n fun y => (leftLim (↑f) y - ↑f x) / (y - x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))",
"tactic": "apply Tendsto.congr' _\n ((ENNReal.tendsto_toReal h'x.ne).comp (hx.comp (Real.tendsto_Icc_vitaliFamily_left x)))"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ (a : ℝ),\n a ∈ Iio x →\n (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc y x) a =\n (leftLim (↑f) a - ↑f x) / (a - x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc y x) =ᶠ[𝓝[Iio x] x]\n fun y => (leftLim (↑f) y - ↑f x) / (y - x)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhxy : y < x\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc y x) y =\n (leftLim (↑f) y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ (a : ℝ),\n a ∈ Iio x →\n (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc y x) a =\n (leftLim (↑f) a - ↑f x) / (a - x)",
"tactic": "rintro y (hxy : y < x)"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhxy : y < x\n⊢ ENNReal.toReal (ENNReal.ofReal (↑f x - leftLim (↑f) y) / ENNReal.ofReal (x - y)) = (leftLim (↑f) y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhxy : y < x\n⊢ (ENNReal.toReal ∘ (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) ∘ fun y => Icc y x) y =\n (leftLim (↑f) y - ↑f x) / (y - x)",
"tactic": "simp only [comp_apply, StieltjesFunction.measure_Icc, Real.volume_Icc]"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhxy : y < x\n⊢ 0 ≤ (↑f x - leftLim (↑f) y) / (x - y)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhxy : y < x\n⊢ ENNReal.toReal (ENNReal.ofReal (↑f x - leftLim (↑f) y) / ENNReal.ofReal (x - y)) = (leftLim (↑f) y - ↑f x) / (y - x)",
"tactic": "rw [← ENNReal.ofReal_div_of_pos (sub_pos.2 hxy), ENNReal.toReal_ofReal, ← neg_neg (y - x),\n div_neg, neg_div', neg_sub, neg_sub]"
},
{
"state_after": "no goals",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhxy : y < x\n⊢ 0 ≤ (↑f x - leftLim (↑f) y) / (x - y)",
"tactic": "exact div_nonneg (sub_nonneg.2 (f.mono.leftLim_le hxy.le)) (sub_pos.2 hxy).le"
},
{
"state_after": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝[Iio x] x) (𝓝[Iio x] x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))",
"tactic": "apply tendsto_apply_add_mul_sq_div_sub (nhds_left'_le_nhds_ne x) L2"
},
{
"state_after": "case h1\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝[Iio x] x) (𝓝 x)\n\ncase h2\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[Iio x] x, x_1 + ↑1 * (x_1 - x) ^ 2 ∈ Iio x",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝[Iio x] x) (𝓝[Iio x] x)",
"tactic": "apply tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within"
},
{
"state_after": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 x)",
"state_before": "case h1\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝[Iio x] x) (𝓝 x)",
"tactic": "apply Tendsto.mono_left _ nhdsWithin_le_nhds"
},
{
"state_after": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Tendsto (fun y => y + 1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + 1 * (x - x) ^ 2))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 x)",
"tactic": "have : Tendsto (fun y : ℝ => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + ↑1 * (x - x) ^ 2)) :=\n tendsto_id.add (((tendsto_id.sub_const x).pow 2).const_mul ↑1)"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Tendsto (fun y => y + 1 * (y - x) ^ 2) (𝓝 x) (𝓝 (x + 1 * (x - x) ^ 2))\n⊢ Tendsto (fun y => y + ↑1 * (y - x) ^ 2) (𝓝 x) (𝓝 x)",
"tactic": "simpa using this"
},
{
"state_after": "case h2\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\n⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[Iio x] x, x_1 + ↑1 * (x_1 - x) ^ 2 ∈ Iio x",
"state_before": "case h2\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[Iio x] x, x_1 + ↑1 * (x_1 - x) ^ 2 ∈ Iio x",
"tactic": "have : Ioo (x - 1) x ∈ 𝓝[<] x := by\n apply Ioo_mem_nhdsWithin_Iio; exact ⟨by linarith, le_refl _⟩"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\n⊢ ∀ (a : ℝ), a ∈ Ioo (x - 1) x → a + ↑1 * (a - x) ^ 2 ∈ Iio x",
"state_before": "case h2\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\n⊢ ∀ᶠ (x_1 : ℝ) in 𝓝[Iio x] x, x_1 + ↑1 * (x_1 - x) ^ 2 ∈ Iio x",
"tactic": "filter_upwards [this]"
},
{
"state_after": "case h.intro\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\ny : ℝ\nhy : x - 1 < y\nh'y : y < x\n⊢ y + ↑1 * (y - x) ^ 2 ∈ Iio x",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\n⊢ ∀ (a : ℝ), a ∈ Ioo (x - 1) x → a + ↑1 * (a - x) ^ 2 ∈ Iio x",
"tactic": "rintro y ⟨hy : x - 1 < y, h'y : y < x⟩"
},
{
"state_after": "case h.intro\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\ny : ℝ\nhy : x - 1 < y\nh'y : y < x\n⊢ y + ↑1 * (y - x) ^ 2 < x",
"state_before": "case h.intro\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\ny : ℝ\nhy : x - 1 < y\nh'y : y < x\n⊢ y + ↑1 * (y - x) ^ 2 ∈ Iio x",
"tactic": "rw [mem_Iio]"
},
{
"state_after": "case h.intro\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\ny : ℝ\nhy : x - 1 < y\nh'y : y < x\n⊢ y + (y - x) ^ 2 < x",
"state_before": "case h.intro\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\ny : ℝ\nhy : x - 1 < y\nh'y : y < x\n⊢ y + ↑1 * (y - x) ^ 2 < x",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case h.intro\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nthis : Ioo (x - 1) x ∈ 𝓝[Iio x] x\ny : ℝ\nhy : x - 1 < y\nh'y : y < x\n⊢ y + (y - x) ^ 2 < x",
"tactic": "nlinarith"
},
{
"state_after": "case H\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ x ∈ Ioc (x - 1) x",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Ioo (x - 1) x ∈ 𝓝[Iio x] x",
"tactic": "apply Ioo_mem_nhdsWithin_Iio"
},
{
"state_after": "no goals",
"state_before": "case H\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ x ∈ Ioc (x - 1) x",
"tactic": "exact ⟨by linarith, le_refl _⟩"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ x - 1 < x",
"tactic": "linarith"
},
{
"state_after": "case hgf\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ᶠ (b : ℝ) in 𝓝[Iio x] x, (leftLim (↑f) (b + ↑1 * (b - x) ^ 2) - ↑f x) / (b - x) ≤ (↑f b - ↑f x) / (b - x)\n\ncase hfh\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ᶠ (b : ℝ) in 𝓝[Iio x] x, (↑f b - ↑f x) / (b - x) ≤ (leftLim (↑f) b - ↑f x) / (b - x)",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))",
"tactic": "apply tendsto_of_tendsto_of_tendsto_of_le_of_le' L3 L2"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ (a : ℝ), a ∈ Iio x → (leftLim (↑f) (a + ↑1 * (a - x) ^ 2) - ↑f x) / (a - x) ≤ (↑f a - ↑f x) / (a - x)",
"state_before": "case hgf\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ᶠ (b : ℝ) in 𝓝[Iio x] x, (leftLim (↑f) (b + ↑1 * (b - x) ^ 2) - ↑f x) / (b - x) ≤ (↑f b - ↑f x) / (b - x)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x) ≤ (↑f y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ (a : ℝ), a ∈ Iio x → (leftLim (↑f) (a + ↑1 * (a - x) ^ 2) - ↑f x) / (a - x) ≤ (↑f a - ↑f x) / (a - x)",
"tactic": "rintro y (hy : y < x)"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ ↑f y ≤ leftLim (↑f) (y + ↑1 * (y - x) ^ 2)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x) ≤ (↑f y - ↑f x) / (y - x)",
"tactic": "refine' div_le_div_of_nonpos_of_le (by linarith) ((sub_le_sub_iff_right _).2 _)"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ y < y + ↑1 * (y - x) ^ 2",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ ↑f y ≤ leftLim (↑f) (y + ↑1 * (y - x) ^ 2)",
"tactic": "apply f.mono.le_leftLim"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\nthis : 0 < (x - y) ^ 2\n⊢ y < y + ↑1 * (y - x) ^ 2",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ y < y + ↑1 * (y - x) ^ 2",
"tactic": "have : ↑0 < (x - y) ^ 2 := sq_pos_of_pos (sub_pos.2 hy)"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\nthis : 0 < (x - y) ^ 2\n⊢ 0 < (y - x) ^ 2",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\nthis : 0 < (x - y) ^ 2\n⊢ y < y + ↑1 * (y - x) ^ 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\nthis : 0 < (x - y) ^ 2\n⊢ 0 < (y - x) ^ 2",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ y - x ≤ 0",
"tactic": "linarith"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ (a : ℝ), a ∈ Iio x → (↑f a - ↑f x) / (a - x) ≤ (leftLim (↑f) a - ↑f x) / (a - x)",
"state_before": "case hfh\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ᶠ (b : ℝ) in 𝓝[Iio x] x, (↑f b - ↑f x) / (b - x) ≤ (leftLim (↑f) b - ↑f x) / (b - x)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ (↑f y - ↑f x) / (y - x) ≤ (leftLim (↑f) y - ↑f x) / (y - x)",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\n⊢ ∀ (a : ℝ), a ∈ Iio x → (↑f a - ↑f x) / (a - x) ≤ (leftLim (↑f) a - ↑f x) / (a - x)",
"tactic": "rintro y (hy : y < x)"
},
{
"state_after": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ leftLim (↑f) y - ↑f x ≤ ↑f y - ↑f x",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ (↑f y - ↑f x) / (y - x) ≤ (leftLim (↑f) y - ↑f x) / (y - x)",
"tactic": "refine' div_le_div_of_nonpos_of_le (by linarith) _"
},
{
"state_after": "no goals",
"state_before": "case h\nf : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ leftLim (↑f) y - ↑f x ≤ ↑f y - ↑f x",
"tactic": "simpa only [sub_le_sub_iff_right] using f.mono.leftLim_le (le_refl y)"
},
{
"state_after": "no goals",
"state_before": "f : StieltjesFunction\nx : ℝ\nhx :\n Tendsto (fun a => ↑↑(StieltjesFunction.measure f) a / ↑↑volume a) (VitaliFamily.filterAt (vitaliFamily volume 1) x)\n (𝓝 (rnDeriv (StieltjesFunction.measure f) volume x))\nh'x : rnDeriv (StieltjesFunction.measure f) volume x < ⊤\nh''x : ¬leftLim (↑f) x ≠ ↑f x\nL1 :\n Tendsto (fun y => (↑f y - ↑f x) / (y - x)) (𝓝[Ioi x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL2 :\n Tendsto (fun y => (leftLim (↑f) y - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\nL3 :\n Tendsto (fun y => (leftLim (↑f) (y + ↑1 * (y - x) ^ 2) - ↑f x) / (y - x)) (𝓝[Iio x] x)\n (𝓝 (ENNReal.toReal (rnDeriv (StieltjesFunction.measure f) volume x)))\ny : ℝ\nhy : y < x\n⊢ y - x ≤ 0",
"tactic": "linarith"
}
] |
[
136,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.cases_succ'
|
[
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\nC : Fin (n + 1) → Sort u_1\nH0 : C 0\nHs : (i : Fin n) → C (succ i)\ni : ℕ\nh : i + 1 < n + 1\n⊢ cases H0 Hs { val := Nat.succ i, isLt := h } = Hs { val := i, isLt := (_ : i < n) }",
"tactic": "cases i <;> rfl"
}
] |
[
1747,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1746,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.det_conj'
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.411237\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA B : Matrix n n α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m m α\nh : IsUnit M\nN : Matrix m m α\n⊢ det (M⁻¹ ⬝ N ⬝ M) = det N",
"tactic": "rw [← h.unit_spec, ← coe_units_inv, det_units_conj']"
}
] |
[
710,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Data/List/Perm.lean
|
List.perm_replicate
|
[] |
[
199,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.attach_ndinsert
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht : ndinsert a s = t\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\n⊢ ∀ (t : Multiset α) (eq : ndinsert a s = t),\n attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"tactic": "intro t ht"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht : ndinsert a s = t\nh : a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))\n\ncase neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht : ndinsert a s = t\nh : ¬a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht : ndinsert a s = t\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"tactic": "by_cases h : a ∈ s"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht✝ : ndinsert a s = t\nht : s = t\nh : a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht : ndinsert a s = t\nh : a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"tactic": "rw [ndinsert_of_mem h] at ht"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nh : a ∈ s\nht : ndinsert a s = s\n⊢ attach s =\n ndinsert { val := a, property := (_ : a ∈ s) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) (attach s))",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht✝ : ndinsert a s = t\nht : s = t\nh : a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"tactic": "subst ht"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nh : a ∈ s\nht : ndinsert a s = s\n⊢ attach s =\n ndinsert { val := a, property := (_ : a ∈ s) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) (attach s))",
"tactic": "rw [eq, map_id, ndinsert_of_mem (mem_attach _ _)]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht✝ : ndinsert a s = t\nht : a ::ₘ s = t\nh : ¬a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht : ndinsert a s = t\nh : ¬a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"tactic": "rw [ndinsert_of_not_mem h] at ht"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nh : ¬a ∈ s\nht : ndinsert a s = a ::ₘ s\n⊢ attach (a ::ₘ s) =\n ndinsert { val := a, property := (_ : a ∈ a ::ₘ s) }\n (map (fun p => { val := ↑p, property := (_ : ↑p ∈ a ::ₘ s) }) (attach s))",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nt : Multiset α\nht✝ : ndinsert a s = t\nht : a ::ₘ s = t\nh : ¬a ∈ s\n⊢ attach t =\n ndinsert { val := a, property := (_ : a ∈ t) } (map (fun p => { val := ↑p, property := (_ : ↑p ∈ t) }) (attach s))",
"tactic": "subst ht"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\neq : ∀ (h : ∀ (p : { x // x ∈ s }), ↑p ∈ s), (fun p => { val := ↑p, property := (_ : ↑p ∈ s) }) = id\nh : ¬a ∈ s\nht : ndinsert a s = a ::ₘ s\n⊢ attach (a ::ₘ s) =\n ndinsert { val := a, property := (_ : a ∈ a ::ₘ s) }\n (map (fun p => { val := ↑p, property := (_ : ↑p ∈ a ::ₘ s) }) (attach s))",
"tactic": "simp [attach_cons, h]"
}
] |
[
120,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean
|
Qpf.recF_eq
|
[
{
"state_after": "case mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\na✝ : (P F).A\nf✝ : PFunctor.B (P F) a✝ → WType (P F).B\n⊢ recF g (WType.mk a✝ f✝) = g (abs (recF g <$> PFunctor.W.dest (WType.mk a✝ f✝)))",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\nx : PFunctor.W (P F)\n⊢ recF g x = g (abs (recF g <$> PFunctor.W.dest x))",
"tactic": "cases x"
},
{
"state_after": "no goals",
"state_before": "case mk\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\ng : F α → α\na✝ : (P F).A\nf✝ : PFunctor.B (P F) a✝ → WType (P F).B\n⊢ recF g (WType.mk a✝ f✝) = g (abs (recF g <$> PFunctor.W.dest (WType.mk a✝ f✝)))",
"tactic": "rfl"
}
] |
[
175,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.id_comp
|
[] |
[
719,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
718,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean
|
WithTop.mul_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Zero α\ninst✝ : Mul α\na : WithTop α\nh : a ≠ 0\n⊢ a * ⊤ = ⊤",
"tactic": "rw [mul_top', if_neg h]"
}
] |
[
53,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
9
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.LEquiv.onFormula_apply
|
[] |
[
976,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
974,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.restrict_lintegral_eq_lintegral_restrict
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.889169\nγ : Type ?u.889172\nδ : Type ?u.889175\nm : MeasurableSpace α\nμ ν : Measure α\nf : α →ₛ ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ lintegral (restrict f s) μ = lintegral f (Measure.restrict μ s)",
"tactic": "rw [f.restrict_lintegral hs, lintegral_restrict]"
}
] |
[
1087,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1085,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.le_prod_of_mem
|
[
{
"state_after": "case intro\nι : Type ?u.137543\nα : Type u_1\nβ : Type ?u.137549\nγ : Type ?u.137552\ninst✝ : CanonicallyOrderedMonoid α\na : α\nm' : Multiset α\nh : a ∈ a ::ₘ m'\n⊢ a ≤ prod (a ::ₘ m')",
"state_before": "ι : Type ?u.137543\nα : Type u_1\nβ : Type ?u.137549\nγ : Type ?u.137552\ninst✝ : CanonicallyOrderedMonoid α\nm : Multiset α\na : α\nh : a ∈ m\n⊢ a ≤ prod m",
"tactic": "obtain ⟨m', rfl⟩ := exists_cons_of_mem h"
},
{
"state_after": "case intro\nι : Type ?u.137543\nα : Type u_1\nβ : Type ?u.137549\nγ : Type ?u.137552\ninst✝ : CanonicallyOrderedMonoid α\na : α\nm' : Multiset α\nh : a ∈ a ::ₘ m'\n⊢ a ≤ a * prod m'",
"state_before": "case intro\nι : Type ?u.137543\nα : Type u_1\nβ : Type ?u.137549\nγ : Type ?u.137552\ninst✝ : CanonicallyOrderedMonoid α\na : α\nm' : Multiset α\nh : a ∈ a ::ₘ m'\n⊢ a ≤ prod (a ::ₘ m')",
"tactic": "rw [prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case intro\nι : Type ?u.137543\nα : Type u_1\nβ : Type ?u.137549\nγ : Type ?u.137552\ninst✝ : CanonicallyOrderedMonoid α\na : α\nm' : Multiset α\nh : a ∈ a ::ₘ m'\n⊢ a ≤ a * prod m'",
"tactic": "exact _root_.le_mul_right (le_refl a)"
}
] |
[
458,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
454,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.isPartitionDisjUnionOfEqDiff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nh : Prepartition.iUnion π₂ = ↑I \\ Prepartition.iUnion π₁\n⊢ Prepartition.iUnion π₁ ∪ Prepartition.iUnion π₂ = ↑I",
"tactic": "simp [h, π₁.iUnion_subset]"
}
] |
[
791,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
789,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
orthogonalProjection_tendsto_self
|
[
{
"state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ Tendsto (fun t => ↑(↑(orthogonalProjection (U t)) x)) atTop (𝓝 x)",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : ⊤ ≤ Submodule.topologicalClosure (⨆ (t : ι), U t)\n⊢ Tendsto (fun t => ↑(↑(orthogonalProjection (U t)) x)) atTop (𝓝 x)",
"tactic": "rw [← eq_top_iff] at hU'"
},
{
"state_after": "case h.e'_5.h.e'_3\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ x = ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ Tendsto (fun t => ↑(↑(orthogonalProjection (U t)) x)) atTop (𝓝 x)",
"tactic": "convert orthogonalProjection_tendsto_closure_iSup U hU x"
},
{
"state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ x ∈ Submodule.topologicalClosure (⨆ (i : ι), U i)",
"state_before": "case h.e'_5.h.e'_3\n𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ x = ↑(↑(orthogonalProjection (Submodule.topologicalClosure (⨆ (i : ι), U i))) x)",
"tactic": "rw [orthogonalProjection_eq_self_iff.mpr _]"
},
{
"state_after": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ x ∈ ⊤",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ x ∈ Submodule.topologicalClosure (⨆ (i : ι), U i)",
"tactic": "rw [hU']"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type ?u.871455\ninst✝⁷ : IsROrC 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝² : CompleteSpace E\nι : Type u_2\ninst✝¹ : SemilatticeSup ι\nU : ι → Submodule 𝕜 E\ninst✝ : ∀ (t : ι), CompleteSpace { x // x ∈ U t }\nhU : Monotone U\nx : E\nhU' : Submodule.topologicalClosure (⨆ (t : ι), U t) = ⊤\n⊢ x ∈ ⊤",
"tactic": "trivial"
}
] |
[
921,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
913,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
GroupCat.SurjectiveOfEpiAuxs.agree
|
[
{
"state_after": "case refine'_1\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\n⊢ b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier → b ∈ {x | ↑h x = ↑g x}\n\ncase refine'_2\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ False",
"state_before": "A B : GroupCat\nf : A ⟶ B\n⊢ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier = {x | ↑h x = ↑g x}",
"tactic": "refine' Set.ext fun b => ⟨_, fun hb : h b = g b => by_contradiction fun r => _⟩"
},
{
"state_after": "case refine'_1.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\n⊢ ↑f a ∈ {x | ↑h x = ↑g x}",
"state_before": "case refine'_1\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\n⊢ b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier → b ∈ {x | ↑h x = ↑g x}",
"tactic": "rintro ⟨a, rfl⟩"
},
{
"state_after": "case refine'_1.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\n⊢ ↑h (↑f a) = ↑g (↑f a)",
"state_before": "case refine'_1.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\n⊢ ↑f a ∈ {x | ↑h x = ↑g x}",
"tactic": "change h (f a) = g (f a)"
},
{
"state_after": "case refine'_1.intro.H.fromCoset.mk.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n ↑(↑g (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) })\n\ncase refine'_1.intro.H.infinity\nA B : GroupCat\nf : A ⟶ B\na : ↑A\n⊢ ↑(↑h (↑f a)) ∞ = ↑(↑g (↑f a)) ∞",
"state_before": "case refine'_1.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\n⊢ ↑h (↑f a) = ↑g (↑f a)",
"tactic": "ext ⟨⟨_, ⟨y, rfl⟩⟩⟩"
},
{
"state_after": "case refine'_1.intro.H.fromCoset.mk.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"state_before": "case refine'_1.intro.H.fromCoset.mk.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n ↑(↑g (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) })",
"tactic": "rw [g_apply_fromCoset]"
},
{
"state_after": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }\n\ncase neg\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : ¬y ∈ MonoidHom.range f\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"state_before": "case refine'_1.intro.H.fromCoset.mk.intro\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "by_cases m : y ∈ f.range"
},
{
"state_after": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"state_before": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]"
},
{
"state_after": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := ↑f a *l (y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier),\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"state_before": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "change fromCoset _ = fromCoset ⟨f a *l (y *l _), _⟩"
},
{
"state_after": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := ↑f a *l (y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier),\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"state_before": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := ↑f a *l (y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier),\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "simp only [← fromCoset_eq_of_mem_range _ (Subgroup.mul_mem _ ⟨a, rfl⟩ m)]"
},
{
"state_after": "case pos.e_a.e_val\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier =\n ↑f a *l (y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)",
"state_before": "case pos\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := ↑f a *l (y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier),\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case pos.e_a.e_val\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : y ∈ MonoidHom.range f\n⊢ ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier =\n ↑f a *l (y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)",
"tactic": "rw [leftCoset_assoc _ (f a) y]"
},
{
"state_after": "case neg\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : ¬y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"state_before": "case neg\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : ¬y ∈ MonoidHom.range f\n⊢ ↑(↑h (↑f a))\n (fromCoset\n { val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) }) =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "rw [h_apply_fromCoset_nin_range f (f a) ⟨_, rfl⟩ _ m]"
},
{
"state_after": "no goals",
"state_before": "case neg\nA B : GroupCat\nf : A ⟶ B\na : ↑A\ny : ↑B\nm : ¬y ∈ MonoidHom.range f\n⊢ fromCoset\n { val := ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n ↑f a * y *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n {\n val :=\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) },\n property :=\n (_ :\n ↑f a *l\n ↑{ val := Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y,\n property :=\n (_ :\n ∃ y_1,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y_1 =\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y) } ∈\n Set.range (Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier)) }",
"tactic": "simp only [leftCoset_assoc]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.H.infinity\nA B : GroupCat\nf : A ⟶ B\na : ↑A\n⊢ ↑(↑h (↑f a)) ∞ = ↑(↑g (↑f a)) ∞",
"tactic": "rw [g_apply_infinity, h_apply_infinity f (f a) ⟨_, rfl⟩]"
},
{
"state_after": "case refine'_2\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\neq1 :\n ↑(↑h b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ False",
"state_before": "case refine'_2\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ False",
"tactic": "have eq1 : (h b) (fromCoset ⟨f.range.carrier, ⟨1, one_leftCoset _⟩⟩) =\n fromCoset ⟨f.range.carrier, ⟨1, one_leftCoset _⟩⟩ := by\n change ((τ).symm.trans (g b)).trans τ _ = _\n dsimp [tau]\n simp [g_apply_infinity f]"
},
{
"state_after": "case refine'_2\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\neq1 :\n ↑(↑h b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\neq2 :\n ↑(↑g b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ False",
"state_before": "case refine'_2\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\neq1 :\n ↑(↑h b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ False",
"tactic": "have eq2 :\n (g b) (fromCoset ⟨f.range.carrier, ⟨1, one_leftCoset _⟩⟩) =\n fromCoset ⟨b *l f.range.carrier, ⟨b, rfl⟩⟩ := rfl"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nA B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\neq1 :\n ↑(↑h b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\neq2 :\n ↑(↑g b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ False",
"tactic": "exact (fromCoset_ne_of_nin_range _ r).symm (by rw [← eq1, ← eq2, FunLike.congr_fun hb])"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ ↑((τ.symm.trans (↑g b)).trans τ)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ ↑(↑h b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"tactic": "change ((τ).symm.trans (g b)).trans τ _ = _"
},
{
"state_after": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ ↑(Equiv.swap\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) })\n ∞)\n (↑(↑g b)\n (↑(Equiv.swap\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) })\n ∞)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }))) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ ↑((τ.symm.trans (↑g b)).trans τ)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"tactic": "dsimp [tau]"
},
{
"state_after": "no goals",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\n⊢ ↑(Equiv.swap\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) })\n ∞)\n (↑(↑g b)\n (↑(Equiv.swap\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) })\n ∞)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }))) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"tactic": "simp [g_apply_infinity f]"
},
{
"state_after": "no goals",
"state_before": "A B : GroupCat\nf : A ⟶ B\nb : ↑B\nhb : ↑h b = ↑g b\nr : ¬b ∈ (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier\neq1 :\n ↑(↑h b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\neq2 :\n ↑(↑g b)\n (fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }) =\n fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }\n⊢ fromCoset\n { val := (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) } =\n fromCoset\n { val := b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier,\n property :=\n (_ :\n ∃ y,\n Function.swap leftCoset (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier y =\n b *l (MonoidHom.range f).toSubmonoid.toSubsemigroup.carrier) }",
"tactic": "rw [← eq1, ← eq2, FunLike.congr_fun hb]"
}
] |
[
317,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_univ
|
[] |
[
232,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.eval_mk
|
[] |
[
707,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
705,
1
] |
Mathlib/Analysis/NormedSpace/AffineIsometry.lean
|
AffineIsometry.nndist_map
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_5\nV : Type u_2\nV₁ : Type ?u.47368\nV₂ : Type u_3\nV₃ : Type ?u.47374\nV₄ : Type ?u.47377\nP₁ : Type ?u.47380\nP : Type u_4\nP₂ : Type u_1\nP₃ : Type ?u.47389\nP₄ : Type ?u.47392\ninst✝²⁰ : NormedField 𝕜\ninst✝¹⁹ : SeminormedAddCommGroup V\ninst✝¹⁸ : SeminormedAddCommGroup V₁\ninst✝¹⁷ : SeminormedAddCommGroup V₂\ninst✝¹⁶ : SeminormedAddCommGroup V₃\ninst✝¹⁵ : SeminormedAddCommGroup V₄\ninst✝¹⁴ : NormedSpace 𝕜 V\ninst✝¹³ : NormedSpace 𝕜 V₁\ninst✝¹² : NormedSpace 𝕜 V₂\ninst✝¹¹ : NormedSpace 𝕜 V₃\ninst✝¹⁰ : NormedSpace 𝕜 V₄\ninst✝⁹ : PseudoMetricSpace P\ninst✝⁸ : MetricSpace P₁\ninst✝⁷ : PseudoMetricSpace P₂\ninst✝⁶ : PseudoMetricSpace P₃\ninst✝⁵ : PseudoMetricSpace P₄\ninst✝⁴ : NormedAddTorsor V P\ninst✝³ : NormedAddTorsor V₁ P₁\ninst✝² : NormedAddTorsor V₂ P₂\ninst✝¹ : NormedAddTorsor V₃ P₃\ninst✝ : NormedAddTorsor V₄ P₄\nf : P →ᵃⁱ[𝕜] P₂\nf₁ : P₁ →ᵃⁱ[𝕜] P₂\nx y : P\n⊢ nndist (↑f x) (↑f y) = nndist x y",
"tactic": "simp [nndist_dist, (dist_map)]"
}
] |
[
151,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
derivWithin_cos
|
[] |
[
814,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.Nat.card_coe_set_eq
|
[
{
"state_after": "case h.e'_2\nα : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ Nat.card ↑s = ncard univ\n\ncase h.e'_3\nα : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ ncard s = ncard ((fun a => ↑a) '' univ)\n\ncase convert_1\nα : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ Type ?u.47955\n\ncase convert_2\nα : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ ?convert_1 → Prop",
"state_before": "α : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ Nat.card ↑s = ncard s",
"tactic": "convert (ncard_image_ofInjective univ Subtype.coe_injective).symm using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ ncard s = ncard ((fun a => ↑a) '' univ)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nα : Type u_1\nβ : Type ?u.47787\ns✝ t : Set α\na b x y : α\nf : α → β\ns : Set α\n⊢ Nat.card ↑s = ncard univ",
"tactic": "rw [ncard_univ]"
}
] |
[
301,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
9
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.neg_apply
|
[] |
[
315,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
RingHom.eq_of_eqOn_set_dense
|
[] |
[
1246,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1244,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.Trident.ι_eq_app_zero
|
[] |
[
213,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.revzip_powersetAux_perm
|
[
{
"state_after": "α : Type u_1\nl₁ l₂ : List α\np : l₁ ~ l₂\nthis : DecidableEq α\n⊢ revzip (powersetAux l₁) ~ revzip (powersetAux l₂)",
"state_before": "α : Type u_1\nl₁ l₂ : List α\np : l₁ ~ l₂\n⊢ revzip (powersetAux l₁) ~ revzip (powersetAux l₂)",
"tactic": "haveI := Classical.decEq α"
},
{
"state_after": "α : Type u_1\nl₁ l₂ : List α\np : l₁ ~ l₂\nthis : DecidableEq α\n⊢ List.map (fun x => (x, ↑l₂ - x)) (powersetAux l₁) ~ List.map (fun x => (x, ↑l₂ - x)) (powersetAux l₂)",
"state_before": "α : Type u_1\nl₁ l₂ : List α\np : l₁ ~ l₂\nthis : DecidableEq α\n⊢ revzip (powersetAux l₁) ~ revzip (powersetAux l₂)",
"tactic": "simp [fun l : List α => revzip_powersetAux_lemma l revzip_powersetAux, coe_eq_coe.2 p]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl₁ l₂ : List α\np : l₁ ~ l₂\nthis : DecidableEq α\n⊢ List.map (fun x => (x, ↑l₂ - x)) (powersetAux l₁) ~ List.map (fun x => (x, ↑l₂ - x)) (powersetAux l₂)",
"tactic": "exact (powersetAux_perm p).map _"
}
] |
[
168,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Order/Antichain.lean
|
Set.Subsingleton.isWeakAntichain
|
[] |
[
394,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Analysis/Calculus/Inverse.lean
|
ApproximatesLinearOn.to_inv
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nx : F\nhx : x ∈ f '' s\ny : F\nhy : y ∈ f '' s\nA : LocalEquiv E F := toLocalEquiv hf hc\n⊢ ‖↑(LocalEquiv.symm A) x - ↑(LocalEquiv.symm A) y - ↑↑(ContinuousLinearEquiv.symm f') (x - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖x - y‖",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nx : F\nhx : x ∈ f '' s\ny : F\nhy : y ∈ f '' s\n⊢ ‖↑(LocalEquiv.symm (toLocalEquiv hf hc)) x - ↑(LocalEquiv.symm (toLocalEquiv hf hc)) y -\n ↑↑(ContinuousLinearEquiv.symm f') (x - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖x - y‖",
"tactic": "set A := hf.toLocalEquiv hc"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nx : F\nhx : x ∈ f '' s\ny : F\nhy : y ∈ f '' s\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\n⊢ ‖↑(LocalEquiv.symm A) x - ↑(LocalEquiv.symm A) y - ↑↑(ContinuousLinearEquiv.symm f') (x - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖x - y‖",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nx : F\nhx : x ∈ f '' s\ny : F\nhy : y ∈ f '' s\nA : LocalEquiv E F := toLocalEquiv hf hc\n⊢ ‖↑(LocalEquiv.symm A) x - ↑(LocalEquiv.symm A) y - ↑↑(ContinuousLinearEquiv.symm f') (x - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖x - y‖",
"tactic": "have Af : ∀ z, A z = f z := fun z => rfl"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\ny : F\nhy : y ∈ f '' s\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\n⊢ ‖↑(LocalEquiv.symm A) (f x') - ↑(LocalEquiv.symm A) y - ↑↑(ContinuousLinearEquiv.symm f') (f x' - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖f x' - y‖",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nx : F\nhx : x ∈ f '' s\ny : F\nhy : y ∈ f '' s\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\n⊢ ‖↑(LocalEquiv.symm A) x - ↑(LocalEquiv.symm A) y - ↑↑(ContinuousLinearEquiv.symm f') (x - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖x - y‖",
"tactic": "rcases (mem_image _ _ _).1 hx with ⟨x', x's, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ‖↑(LocalEquiv.symm A) (f x') - ↑(LocalEquiv.symm A) (f y') - ↑↑(ContinuousLinearEquiv.symm f') (f x' - f y')‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖f x' - f y'‖",
"state_before": "case intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\ny : F\nhy : y ∈ f '' s\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\n⊢ ‖↑(LocalEquiv.symm A) (f x') - ↑(LocalEquiv.symm A) y - ↑↑(ContinuousLinearEquiv.symm f') (f x' - y)‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖f x' - y‖",
"tactic": "rcases (mem_image _ _ _).1 hy with ⟨y', y's, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ‖x' - y' - ↑↑(ContinuousLinearEquiv.symm f') (↑A x' - ↑A y')‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖↑A x' - ↑A y'‖",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ‖↑(LocalEquiv.symm A) (f x') - ↑(LocalEquiv.symm A) (f y') - ↑↑(ContinuousLinearEquiv.symm f') (f x' - f y')‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖f x' - f y'‖",
"tactic": "rw [← Af x', ← Af y', A.left_inv x's, A.left_inv y's]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ‖x' - y' - ↑↑(ContinuousLinearEquiv.symm f') (↑A x' - ↑A y')‖ ≤\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖↑A x' - ↑A y'‖",
"tactic": "calc\n ‖x' - y' - f'.symm (A x' - A y')‖ ≤ N * ‖f' (x' - y' - f'.symm (A x' - A y'))‖ :=\n (f' : E →L[𝕜] F).bound_of_antilipschitz f'.antilipschitz _\n _ = N * ‖A y' - A x' - f' (y' - x')‖ := by\n congr 2\n simp only [ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.map_sub]\n abel\n _ ≤ N * (c * ‖y' - x'‖) := (mul_le_mul_of_nonneg_left (hf _ y's _ x's) (NNReal.coe_nonneg _))\n _ ≤ N * (c * (((N⁻¹ - c)⁻¹ : ℝ≥0) * ‖A y' - A x'‖)) := by\n gcongr\n rw [← dist_eq_norm, ← dist_eq_norm]\n exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩\n _ = (N * (N⁻¹ - c)⁻¹ * c : ℝ≥0) * ‖A x' - A y'‖ := by\n simp only [norm_sub_rev, NNReal.coe_mul]; ring"
},
{
"state_after": "case e_a.e_a\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑f' (x' - y' - ↑(ContinuousLinearEquiv.symm f') (↑A x' - ↑A y')) = ↑A y' - ↑A x' - ↑f' (y' - x')",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * ‖↑f' (x' - y' - ↑(ContinuousLinearEquiv.symm f') (↑A x' - ↑A y'))‖ =\n ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * ‖↑A y' - ↑A x' - ↑f' (y' - x')‖",
"tactic": "congr 2"
},
{
"state_after": "case e_a.e_a\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑f' x' - ↑f' y' - (↑(toLocalEquiv hf hc) x' - ↑(toLocalEquiv hf hc) y') =\n ↑(toLocalEquiv hf hc) y' - ↑(toLocalEquiv hf hc) x' - (↑f' y' - ↑f' x')",
"state_before": "case e_a.e_a\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑f' (x' - y' - ↑(ContinuousLinearEquiv.symm f') (↑A x' - ↑A y')) = ↑A y' - ↑A x' - ↑f' (y' - x')",
"tactic": "simp only [ContinuousLinearEquiv.apply_symm_apply, ContinuousLinearEquiv.map_sub]"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_a\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑f' x' - ↑f' y' - (↑(toLocalEquiv hf hc) x' - ↑(toLocalEquiv hf hc) y') =\n ↑(toLocalEquiv hf hc) y' - ↑(toLocalEquiv hf hc) x' - (↑f' y' - ↑f' x')",
"tactic": "abel"
},
{
"state_after": "case h.h\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ‖y' - x'‖ ≤ ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ‖↑A y' - ↑A x'‖",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * (↑c * ‖y' - x'‖) ≤\n ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * (↑c * (↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ‖↑A y' - ↑A x'‖))",
"tactic": "gcongr"
},
{
"state_after": "case h.h\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ dist y' x' ≤ ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * dist (↑A y') (↑A x')",
"state_before": "case h.h\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ‖y' - x'‖ ≤ ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ‖↑A y' - ↑A x'‖",
"tactic": "rw [← dist_eq_norm, ← dist_eq_norm]"
},
{
"state_after": "no goals",
"state_before": "case h.h\n𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ dist y' x' ≤ ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * dist (↑A y') (↑A x')",
"tactic": "exact (hf.antilipschitz hc).le_mul_dist ⟨y', y's⟩ ⟨x', x's⟩"
},
{
"state_after": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ *\n (↑c * (↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ‖↑(toLocalEquiv hf hc) x' - ↑(toLocalEquiv hf hc) y'‖)) =\n ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ↑c *\n ‖↑(toLocalEquiv hf hc) x' - ↑(toLocalEquiv hf hc) y'‖",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * (↑c * (↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ‖↑A y' - ↑A x'‖)) =\n ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊ * (‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * c) * ‖↑A x' - ↑A y'‖",
"tactic": "simp only [norm_sub_rev, NNReal.coe_mul]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nG : Type ?u.404204\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nG' : Type ?u.404307\ninst✝² : NormedAddCommGroup G'\ninst✝¹ : NormedSpace 𝕜 G'\nε : ℝ\ninst✝ : CompleteSpace E\nf : E → F\nf' : E ≃L[𝕜] F\ns : Set E\nc : ℝ≥0\nhf : ApproximatesLinearOn f (↑f') s c\nhc : Subsingleton E ∨ c < ‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹\nA : LocalEquiv E F := toLocalEquiv hf hc\nAf : ∀ (z : E), ↑A z = f z\nx' : E\nx's : x' ∈ s\nhx : f x' ∈ f '' s\ny' : E\ny's : y' ∈ s\nhy : f y' ∈ f '' s\n⊢ ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ *\n (↑c * (↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ‖↑(toLocalEquiv hf hc) x' - ↑(toLocalEquiv hf hc) y'‖)) =\n ↑‖↑(ContinuousLinearEquiv.symm f')‖₊ * ↑(‖↑(ContinuousLinearEquiv.symm f')‖₊⁻¹ - c)⁻¹ * ↑c *\n ‖↑(toLocalEquiv hf hc) x' - ↑(toLocalEquiv hf hc) y'‖",
"tactic": "ring"
}
] |
[
443,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
422,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.restrictScalars_inj
|
[] |
[
636,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
634,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.snd
|
[] |
[
309,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_neg
|
[] |
[
1096,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1095,
1
] |
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