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/Data/Nat/Multiplicity.lean
|
Nat.Prime.multiplicity_choose_prime_pow
|
[] |
[
247,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.map_add_right_Ioo
|
[
{
"state_after": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x => c + x) (Ioo a b) = Ioo (c + a) (c + b)",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x => x + c) (Ioo a b) = Ioo (a + c) (b + c)",
"tactic": "simp_rw [add_comm _ c]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (fun x => c + x) (Ioo a b) = Ioo (c + a) (c + b)",
"tactic": "exact map_add_left_Ioo _ _ _"
}
] |
[
326,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Deprecated/Ring.lean
|
RingHom.to_isRingHom
|
[] |
[
171,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/CategoryTheory/Limits/FullSubcategory.lean
|
CategoryTheory.Limits.hasColimitsOfShape_of_closed_under_colimits
|
[] |
[
145,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Mathlib/Algebra/BigOperators/Pi.lean
|
MonoidHom.functions_ext
|
[
{
"state_after": "case intro\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\nval✝ : Fintype I\n⊢ g = h",
"state_before": "I : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\n⊢ g = h",
"tactic": "cases nonempty_fintype I"
},
{
"state_after": "case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ↑g k = ↑h k",
"state_before": "case intro\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\nval✝ : Fintype I\n⊢ g = h",
"tactic": "ext k"
},
{
"state_after": "case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ∏ x : I, ↑g (Pi.mulSingle x (k x)) = ∏ x : I, ↑h (Pi.mulSingle x (k x))",
"state_before": "case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ↑g k = ↑h k",
"tactic": "rw [← Finset.univ_prod_mulSingle k, g.map_prod, h.map_prod]"
},
{
"state_after": "no goals",
"state_before": "case intro.h\nI : Type u_1\ninst✝³ : DecidableEq I\nZ : I → Type u_3\ninst✝² : (i : I) → CommMonoid (Z i)\ninst✝¹ : Finite I\nG : Type u_2\ninst✝ : CommMonoid G\ng h : ((i : I) → Z i) →* G\nH : ∀ (i : I) (x : Z i), ↑g (Pi.mulSingle i x) = ↑h (Pi.mulSingle i x)\nval✝ : Fintype I\nk : (i : I) → Z i\n⊢ ∏ x : I, ↑g (Pi.mulSingle x (k x)) = ∏ x : I, ↑h (Pi.mulSingle x (k x))",
"tactic": "simp only [H]"
}
] |
[
94,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Std/Classes/Order.lean
|
Std.TransCmp.lt_trans
|
[] |
[
71,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
70,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
isCompact_accumulate
|
[] |
[
422,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Topology/Basic.lean
|
Continuous.range_subset_closure_image_dense
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.175018\nδ : Type ?u.175021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type ?u.175033\nι : Type ?u.175036\nf✝ : κ → β\ng : β → γ\nf : α → β\nhf : Continuous f\ns : Set α\nhs : Dense s\n⊢ f '' closure s ⊆ closure (f '' s)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.175018\nδ : Type ?u.175021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type ?u.175033\nι : Type ?u.175036\nf✝ : κ → β\ng : β → γ\nf : α → β\nhf : Continuous f\ns : Set α\nhs : Dense s\n⊢ range f ⊆ closure (f '' s)",
"tactic": "rw [← image_univ, ← hs.closure_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.175018\nδ : Type ?u.175021\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type ?u.175033\nι : Type ?u.175036\nf✝ : κ → β\ng : β → γ\nf : α → β\nhf : Continuous f\ns : Set α\nhs : Dense s\n⊢ f '' closure s ⊆ closure (f '' s)",
"tactic": "exact image_closure_subset_closure_image hf"
}
] |
[
1821,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1818,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.frequently_atTop'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.35304\nι' : Type ?u.35307\nα : Type u_1\nβ : Type ?u.35313\nγ : Type ?u.35316\ninst✝² : SemilatticeSup α\ninst✝¹ : Nonempty α\ninst✝ : NoMaxOrder α\np : α → Prop\n⊢ (∀ (i : α), True → ∃ x, x ∈ Ioi i ∧ p x) ↔ ∀ (a : α), ∃ b, b > a ∧ p b",
"tactic": "simp"
}
] |
[
335,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
333,
1
] |
Mathlib/CategoryTheory/Preadditive/Basic.lean
|
CategoryTheory.Preadditive.comp_neg
|
[] |
[
162,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.coe_nnnorm
|
[] |
[
272,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
11
] |
Mathlib/Data/Set/Finite.lean
|
Set.finite_or_infinite
|
[] |
[
148,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
11
] |
Mathlib/Topology/Algebra/Order/UpperLower.lean
|
Set.OrdConnected.interior
|
[
{
"state_after": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : OrdConnected s\n⊢ OrdConnected (interior ↑(upperClosure s) ∩ interior ↑(lowerClosure s))",
"state_before": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : OrdConnected s\n⊢ OrdConnected (interior s)",
"tactic": "rw [← h.upperClosure_inter_lowerClosure, interior_inter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : HasUpperLowerClosure α\ns : Set α\nh : OrdConnected s\n⊢ OrdConnected (interior ↑(upperClosure s) ∩ interior ↑(lowerClosure s))",
"tactic": "exact\n (upperClosure s).upper.interior.ordConnected.inter (lowerClosure s).lower.interior.ordConnected"
}
] |
[
116,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
11
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.trunc_one
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\n⊢ (if m < n then if m = 0 then 1 else 0 else 0) = MvPolynomial.coeff m 1",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\n⊢ MvPolynomial.coeff m (↑(trunc R n) 1) = MvPolynomial.coeff m 1",
"tactic": "rw [coeff_trunc, coeff_one]"
},
{
"state_after": "case inl.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : m = 0\n⊢ 1 = MvPolynomial.coeff m 1\n\ncase inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : ¬m = 0\n⊢ 0 = MvPolynomial.coeff m 1\n\ncase inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ 0 = MvPolynomial.coeff m 1",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\n⊢ (if m < n then if m = 0 then 1 else 0 else 0) = MvPolynomial.coeff m 1",
"tactic": "split_ifs with H H'"
},
{
"state_after": "case inl.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nH : 0 < n\n⊢ 1 = MvPolynomial.coeff 0 1",
"state_before": "case inl.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : m = 0\n⊢ 1 = MvPolynomial.coeff m 1",
"tactic": "subst m"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nH : 0 < n\n⊢ 1 = MvPolynomial.coeff 0 1",
"tactic": "simp"
},
{
"state_after": "case inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : ¬m = 0\n⊢ MvPolynomial.coeff m 1 = 0",
"state_before": "case inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : ¬m = 0\n⊢ 0 = MvPolynomial.coeff m 1",
"tactic": "symm"
},
{
"state_after": "case inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : ¬m = 0\n⊢ (if 0 = m then 1 else 0) = 0",
"state_before": "case inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : ¬m = 0\n⊢ MvPolynomial.coeff m 1 = 0",
"tactic": "rw [MvPolynomial.coeff_one]"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : m < n\nH' : ¬m = 0\n⊢ (if 0 = m then 1 else 0) = 0",
"tactic": "exact if_neg (Ne.symm H')"
},
{
"state_after": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ MvPolynomial.coeff m 1 = 0",
"state_before": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ 0 = MvPolynomial.coeff m 1",
"tactic": "symm"
},
{
"state_after": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ (if 0 = m then 1 else 0) = 0",
"state_before": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ MvPolynomial.coeff m 1 = 0",
"tactic": "rw [MvPolynomial.coeff_one]"
},
{
"state_after": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ ¬0 = m",
"state_before": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ (if 0 = m then 1 else 0) = 0",
"tactic": "refine' if_neg _"
},
{
"state_after": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nH : ¬0 < n\n⊢ False",
"state_before": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nm : σ →₀ ℕ\nH : ¬m < n\n⊢ ¬0 = m",
"tactic": "rintro rfl"
},
{
"state_after": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nH : ¬0 < n\n⊢ 0 < n",
"state_before": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nH : ¬0 < n\n⊢ False",
"tactic": "apply H"
},
{
"state_after": "no goals",
"state_before": "case inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\nH : ¬0 < n\n⊢ 0 < n",
"tactic": "exact Ne.bot_lt hnn"
}
] |
[
722,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.mem_closure_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.222978\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\n⊢ (∀ (i : ℝ≥0∞), 0 < i → ∃ y, y ∈ s ∧ y ∈ ball x i) ↔ ∀ (ε : ℝ≥0∞), ε > 0 → ∃ y, y ∈ s ∧ edist x y < ε",
"tactic": "simp only [mem_ball, edist_comm x]"
}
] |
[
720,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
719,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigOWith_congr
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhc : c₁ = c₂\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ (∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖) ↔ ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₂ * ‖g₂ x‖",
"state_before": "α : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhc : c₁ = c₂\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ IsBigOWith c₁ l f₁ g₁ ↔ IsBigOWith c₂ l f₂ g₂",
"tactic": "simp only [IsBigOWith_def]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ (∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖) ↔ ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₁ * ‖g₂ x‖",
"state_before": "α : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhc : c₁ = c₂\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ (∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖) ↔ ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₂ * ‖g₂ x‖",
"tactic": "subst c₂"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ ∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖ ↔ ‖f₂ x‖ ≤ c₁ * ‖g₂ x‖",
"state_before": "α : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ (∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖) ↔ ∀ᶠ (x : α) in l, ‖f₂ x‖ ≤ c₁ * ‖g₂ x‖",
"tactic": "apply Filter.eventually_congr"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\na✝ : α\ne₁ : f₁ a✝ = f₂ a✝\ne₂ : g₁ a✝ = g₂ a✝\n⊢ ‖f₁ a✝‖ ≤ c₁ * ‖g₁ a✝‖ ↔ ‖f₂ a✝‖ ≤ c₁ * ‖g₂ a✝‖",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\n⊢ ∀ᶠ (x : α) in l, ‖f₁ x‖ ≤ c₁ * ‖g₁ x‖ ↔ ‖f₂ x‖ ≤ c₁ * ‖g₂ x‖",
"tactic": "filter_upwards [hf, hg]with _ e₁ e₂"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.55712\nE : Type u_2\nF : Type u_3\nG : Type ?u.55721\nE' : Type ?u.55724\nF' : Type ?u.55727\nG' : Type ?u.55730\nE'' : Type ?u.55733\nF'' : Type ?u.55736\nG'' : Type ?u.55739\nR : Type ?u.55742\nR' : Type ?u.55745\n𝕜 : Type ?u.55748\n𝕜' : Type ?u.55751\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E\ng₁ g₂ : α → F\nhf : f₁ =ᶠ[l] f₂\nhg : g₁ =ᶠ[l] g₂\na✝ : α\ne₁ : f₁ a✝ = f₂ a✝\ne₂ : g₁ a✝ = g₂ a✝\n⊢ ‖f₁ a✝‖ ≤ c₁ * ‖g₁ a✝‖ ↔ ‖f₂ a✝‖ ≤ c₁ * ‖g₂ a✝‖",
"tactic": "rw [e₁, e₂]"
}
] |
[
299,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Data/Set/Intervals/OrdConnected.lean
|
Set.ordConnected_def
|
[] |
[
51,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.kerLiftAlg_injective
|
[] |
[
306,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/Algebra/ContinuedFractions/Basic.lean
|
GeneralizedContinuedFraction.Pair.coe_toPair
|
[] |
[
94,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/RingTheory/Int/Basic.lean
|
multiplicity.finite_int_iff_natAbs_finite
|
[
{
"state_after": "no goals",
"state_before": "a b : ℤ\n⊢ Finite a b ↔ Finite (_root_.Int.natAbs a) (_root_.Int.natAbs b)",
"tactic": "simp only [finite_def, ← Int.natAbs_dvd_natAbs, Int.natAbs_pow]"
}
] |
[
337,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.mk_def_of_ne
|
[] |
[
223,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.setToFun_smul_left'
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1373587\nG : Type ?u.1373590\n𝕜 : Type ?u.1373593\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nc : ℝ\nh_smul : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T' s = c • T s\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T' hT' f = c • setToFun μ T hT f\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1373587\nG : Type ?u.1373590\n𝕜 : Type ?u.1373593\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nc : ℝ\nh_smul : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T' s = c • T s\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T' hT' f = c • setToFun μ T hT f",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1373587\nG : Type ?u.1373590\n𝕜 : Type ?u.1373593\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nc : ℝ\nh_smul : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T' s = c • T s\nf : α → E\n⊢ setToFun μ T' hT' f = c • setToFun μ T hT f",
"tactic": "by_cases hf : Integrable f μ"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1373587\nG : Type ?u.1373590\n𝕜 : Type ?u.1373593\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nc : ℝ\nh_smul : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T' s = c • T s\nf : α → E\nhf : Integrable f\n⊢ setToFun μ T' hT' f = c • setToFun μ T hT f",
"tactic": "simp_rw [setToFun_eq _ hf, L1.setToL1_smul_left' hT hT' c h_smul]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1373587\nG : Type ?u.1373590\n𝕜 : Type ?u.1373593\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nhT : DominatedFinMeasAdditive μ T C\nhT' : DominatedFinMeasAdditive μ T' C'\nc : ℝ\nh_smul : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T' s = c • T s\nf : α → E\nhf : ¬Integrable f\n⊢ setToFun μ T' hT' f = c • setToFun μ T hT f",
"tactic": "simp_rw [setToFun_undef _ hf, smul_zero]"
}
] |
[
1348,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1342,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Functor.mapIso_symm
|
[] |
[
603,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.nodup_attach
|
[] |
[
159,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.deriv_eq_id_of_nfp_eq_id
|
[
{
"state_after": "no goals",
"state_before": "f✝ : Ordinal → Ordinal\nf : Ordinal → Ordinal\nh : nfp f = id\n⊢ deriv f 0 = id 0 ∧ ∀ (a : Ordinal), deriv f a = id a → deriv f (succ a) = id (succ a)",
"tactic": "simp [h]"
}
] |
[
559,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.closure_induction_right
|
[] |
[
413,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
ContinuousWithinAt.snd
|
[] |
[
1297,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1295,
1
] |
Mathlib/Computability/Partrec.lean
|
Nat.Partrec.of_primrec
|
[
{
"state_after": "no goals",
"state_before": "f : ℕ → ℕ\nhf : Nat.Primrec f\n⊢ Partrec ↑f",
"tactic": "induction hf with\n| zero => exact zero\n| succ => exact succ\n| left => exact left\n| right => exact right\n| pair _ _ pf pg =>\n refine' (pf.pair pg).of_eq_tot fun n => _\n simp [Seq.seq]\n| comp _ _ pf pg =>\n refine' (pf.comp pg).of_eq_tot fun n => _\n simp\n| prec _ _ pf pg =>\n refine' (pf.prec pg).of_eq_tot fun n => _\n simp only [unpaired, PFun.coe_val, bind_eq_bind]\n induction n.unpair.2 with\n | zero => simp\n | succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact ⟨_, IH, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case zero\nf : ℕ → ℕ\n⊢ Partrec ↑fun x => 0",
"tactic": "exact zero"
},
{
"state_after": "no goals",
"state_before": "case succ\nf : ℕ → ℕ\n⊢ Partrec ↑Nat.succ",
"tactic": "exact succ"
},
{
"state_after": "no goals",
"state_before": "case left\nf : ℕ → ℕ\n⊢ Partrec ↑fun n => (unpair n).fst",
"tactic": "exact left"
},
{
"state_after": "no goals",
"state_before": "case right\nf : ℕ → ℕ\n⊢ Partrec ↑fun n => (unpair n).snd",
"tactic": "exact right"
},
{
"state_after": "case pair\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ Nat.pair (f✝ n) (g✝ n) ∈ Seq.seq (Nat.pair <$> ↑f✝ n) fun x => ↑g✝ n",
"state_before": "case pair\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\n⊢ Partrec ↑fun n => Nat.pair (f✝ n) (g✝ n)",
"tactic": "refine' (pf.pair pg).of_eq_tot fun n => _"
},
{
"state_after": "no goals",
"state_before": "case pair\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ Nat.pair (f✝ n) (g✝ n) ∈ Seq.seq (Nat.pair <$> ↑f✝ n) fun x => ↑g✝ n",
"tactic": "simp [Seq.seq]"
},
{
"state_after": "case comp\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ f✝ (g✝ n) ∈ ↑g✝ n >>= ↑f✝",
"state_before": "case comp\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\n⊢ Partrec ↑fun n => f✝ (g✝ n)",
"tactic": "refine' (pf.comp pg).of_eq_tot fun n => _"
},
{
"state_after": "no goals",
"state_before": "case comp\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ f✝ (g✝ n) ∈ ↑g✝ n >>= ↑f✝",
"tactic": "simp"
},
{
"state_after": "case prec\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ unpaired (fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n) n ∈\n unpaired\n (fun a n =>\n Nat.rec (↑f✝ a)\n (fun y IH => do\n let i ← IH\n ↑g✝ (Nat.pair a (Nat.pair y i)))\n n)\n n",
"state_before": "case prec\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\n⊢ Partrec ↑(unpaired fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n)",
"tactic": "refine' (pf.prec pg).of_eq_tot fun n => _"
},
{
"state_after": "case prec\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) (unpair n).snd ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) (unpair n).snd",
"state_before": "case prec\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ unpaired (fun z n => Nat.rec (f✝ z) (fun y IH => g✝ (Nat.pair z (Nat.pair y IH))) n) n ∈\n unpaired\n (fun a n =>\n Nat.rec (↑f✝ a)\n (fun y IH => do\n let i ← IH\n ↑g✝ (Nat.pair a (Nat.pair y i)))\n n)\n n",
"tactic": "simp only [unpaired, PFun.coe_val, bind_eq_bind]"
},
{
"state_after": "no goals",
"state_before": "case prec\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) (unpair n).snd ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) (unpair n).snd",
"tactic": "induction n.unpair.2 with\n| zero => simp\n| succ m IH =>\n simp only [mem_bind_iff, mem_some_iff]\n exact ⟨_, IH, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case prec.zero\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn : ℕ\n⊢ Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) Nat.zero ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) Nat.zero",
"tactic": "simp"
},
{
"state_after": "case prec.succ\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn m : ℕ\nIH :\n Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) m ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) m\n⊢ ∃ a,\n a ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) m ∧\n g✝\n (Nat.pair (unpair n).fst\n (Nat.pair m (Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) m))) =\n g✝ (Nat.pair (unpair n).fst (Nat.pair m a))",
"state_before": "case prec.succ\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn m : ℕ\nIH :\n Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) m ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) m\n⊢ Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) (Nat.succ m) ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) (Nat.succ m)",
"tactic": "simp only [mem_bind_iff, mem_some_iff]"
},
{
"state_after": "no goals",
"state_before": "case prec.succ\nf f✝ g✝ : ℕ → ℕ\na✝¹ : Nat.Primrec f✝\na✝ : Nat.Primrec g✝\npf : Partrec ↑f✝\npg : Partrec ↑g✝\nn m : ℕ\nIH :\n Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) m ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) m\n⊢ ∃ a,\n a ∈\n Nat.rec (Part.some (f✝ (unpair n).fst))\n (fun y IH => Part.bind IH fun i => Part.some (g✝ (Nat.pair (unpair n).fst (Nat.pair y i)))) m ∧\n g✝\n (Nat.pair (unpair n).fst\n (Nat.pair m (Nat.rec (f✝ (unpair n).fst) (fun y IH => g✝ (Nat.pair (unpair n).fst (Nat.pair y IH))) m))) =\n g✝ (Nat.pair (unpair n).fst (Nat.pair m a))",
"tactic": "exact ⟨_, IH, rfl⟩"
}
] |
[
199,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Order/BoundedOrder.lean
|
bot_le
|
[] |
[
266,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.liftRel_destruct_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel\n (LiftRelO R fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t))\n (destruct s) (destruct t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\n⊢ Computation.LiftRel\n (LiftRelO R fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t))\n (destruct s) (destruct t)",
"tactic": "have h : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t) := by\n cases' h with h h\n exact liftRel_destruct h\n assumption"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ ∀ {a : Option (α × WSeq α)} {b : Option (β × WSeq β)},\n LiftRelO R (LiftRel R) a b →\n LiftRelO R (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) a b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel\n (LiftRelO R fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t))\n (destruct s) (destruct t)",
"tactic": "apply Computation.LiftRel.imp _ _ _ h"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\n⊢ LiftRelO R (LiftRel R) a b →\n LiftRelO R (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) a b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ ∀ {a : Option (α × WSeq α)} {b : Option (β × WSeq β)},\n LiftRelO R (LiftRel R) a b →\n LiftRelO R (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) a b",
"tactic": "intro a b"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\n⊢ ∀ (s : WSeq α) (t : WSeq β),\n LiftRel R s t → LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\n⊢ LiftRelO R (LiftRel R) a b →\n LiftRelO R (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) a b",
"tactic": "apply LiftRelO.imp_right"
},
{
"state_after": "case H\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝¹ : WSeq α\nt✝¹ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝¹) (destruct t✝¹)\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s✝ t✝\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\ns : WSeq α\nt : WSeq β\n⊢ LiftRel R s t → LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\n⊢ ∀ (s : WSeq α) (t : WSeq β),\n LiftRel R s t → LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"tactic": "intro s t"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝¹ : WSeq α\nt✝¹ : WSeq β\nh✝¹ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝¹) (destruct t✝¹)\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s✝ t✝\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\na : Option (α × WSeq α)\nb : Option (β × WSeq β)\ns : WSeq α\nt : WSeq β\n⊢ LiftRel R s t → LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"tactic": "apply Or.inl"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : LiftRel R s t\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n\ncase inr\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : (fun s t => LiftRel R s t ∨ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)) s t\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"tactic": "cases' h with h h"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : LiftRel R s t\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n\ncase inr\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"tactic": "exact liftRel_destruct h"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\ns✝ : WSeq α\nt✝ : WSeq β\nh✝ : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s✝) (destruct t✝)\ns : WSeq α\nt : WSeq β\nh : Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)\n⊢ Computation.LiftRel (LiftRelO R (LiftRel R)) (destruct s) (destruct t)",
"tactic": "assumption"
}
] |
[
520,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Analysis/Normed/Group/Pointwise.lean
|
closedBall_one_div_singleton
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nε δ : ℝ\ns t : Set E\nx y : E\n⊢ closedBall 1 δ / {x} = closedBall x⁻¹ δ",
"tactic": "simp"
}
] |
[
197,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/GroupTheory/SchurZassenhaus.lean
|
Subgroup.smul_diff_smul'
|
[
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\n⊢ diff (MonoidHom.id { x // x ∈ H }) (g • α) (g • β) =\n { val := (unop g)⁻¹ * ↑(diff (MonoidHom.id { x // x ∈ H }) α β) * unop g,\n property := (_ : (unop g)⁻¹ * ↑(diff (MonoidHom.id { x // x ∈ H }) α β) * unop g ∈ H) }",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\n⊢ diff (MonoidHom.id { x // x ∈ H }) (g • α) (g • β) =\n { val := (unop g)⁻¹ * ↑(diff (MonoidHom.id { x // x ∈ H }) α β) * unop g,\n property := (_ : (unop g)⁻¹ * ↑(diff (MonoidHom.id { x // x ∈ H }) α β) * unop g ∈ H) }",
"tactic": "letI := H.fintypeQuotientOfFiniteIndex"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nϕ : { x // x ∈ H } →* { x // x ∈ H } :=\n {\n toOneHom :=\n { toFun := fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) },\n map_one' :=\n (_ : (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (h₁ h₂ : { x // x ∈ H }),\n { val := (unop g)⁻¹ * ↑(h₁ * h₂) * unop g, property := (_ : (unop g)⁻¹ * ↑(h₁ * h₂) * unop g ∈ H) } =\n { val := (unop g)⁻¹ * ↑h₁ * unop g, property := (_ : (unop g)⁻¹ * ↑h₁ * unop g ∈ H) } *\n { val := (unop g)⁻¹ * ↑h₂ * unop g, property := (_ : (unop g)⁻¹ * ↑h₂ * unop g ∈ H) }) }\nq : G ⧸ H\nx✝ : q ∈ Finset.univ\n⊢ ↑(↑(MonoidHom.id { x // x ∈ H })\n {\n val :=\n (↑(↑(toEquiv (_ : ↑(g • α) ∈ leftTransversals ↑H)) q))⁻¹ *\n ↑(↑(toEquiv (_ : ↑(g • β) ∈ leftTransversals ↑H)) q),\n property :=\n (_ :\n (↑(↑(toEquiv (_ : ↑(g • α) ∈ leftTransversals ↑H)) q))⁻¹ *\n ↑(↑(toEquiv (_ : ↑(g • β) ∈ leftTransversals ↑H)) q) ∈\n H) }) =\n ↑(↑ϕ\n (↑(MonoidHom.id { x // x ∈ H })\n {\n val :=\n (↑(↑(toEquiv (_ : ↑α ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)))⁻¹ *\n ↑(↑(toEquiv (_ : ↑β ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)),\n property :=\n (_ :\n (↑(↑(toEquiv (_ : ↑α ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)))⁻¹ *\n ↑(↑(toEquiv (_ : ↑β ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)) ∈\n H) }))",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nϕ : { x // x ∈ H } →* { x // x ∈ H } :=\n {\n toOneHom :=\n { toFun := fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) },\n map_one' :=\n (_ : (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (h₁ h₂ : { x // x ∈ H }),\n { val := (unop g)⁻¹ * ↑(h₁ * h₂) * unop g, property := (_ : (unop g)⁻¹ * ↑(h₁ * h₂) * unop g ∈ H) } =\n { val := (unop g)⁻¹ * ↑h₁ * unop g, property := (_ : (unop g)⁻¹ * ↑h₁ * unop g ∈ H) } *\n { val := (unop g)⁻¹ * ↑h₂ * unop g, property := (_ : (unop g)⁻¹ * ↑h₂ * unop g ∈ H) }) }\n⊢ diff (MonoidHom.id { x // x ∈ H }) (g • α) (g • β) =\n { val := (unop g)⁻¹ * ↑(diff (MonoidHom.id { x // x ∈ H }) α β) * unop g,\n property := (_ : (unop g)⁻¹ * ↑(diff (MonoidHom.id { x // x ∈ H }) α β) * unop g ∈ H) }",
"tactic": "refine'\n Eq.trans\n (Finset.prod_bij' (fun q _ => g⁻¹ • q) (fun q _ => Finset.mem_univ _)\n (fun q _ => Subtype.ext _) (fun q _ => g • q) (fun q _ => Finset.mem_univ _)\n (fun q _ => smul_inv_smul g q) fun q _ => inv_smul_smul g q)\n (map_prod ϕ _ _).symm"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nϕ : { x // x ∈ H } →* { x // x ∈ H } :=\n {\n toOneHom :=\n { toFun := fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) },\n map_one' :=\n (_ : (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (h₁ h₂ : { x // x ∈ H }),\n { val := (unop g)⁻¹ * ↑(h₁ * h₂) * unop g, property := (_ : (unop g)⁻¹ * ↑(h₁ * h₂) * unop g ∈ H) } =\n { val := (unop g)⁻¹ * ↑h₁ * unop g, property := (_ : (unop g)⁻¹ * ↑h₁ * unop g ∈ H) } *\n { val := (unop g)⁻¹ * ↑h₂ * unop g, property := (_ : (unop g)⁻¹ * ↑h₂ * unop g ∈ H) }) }\nq : G ⧸ H\nx✝ : q ∈ Finset.univ\n⊢ ↑(↑(MonoidHom.id { x // x ∈ H })\n {\n val :=\n (↑(↑(toEquiv (_ : ↑(g • α) ∈ leftTransversals ↑H)) q))⁻¹ *\n ↑(↑(toEquiv (_ : ↑(g • β) ∈ leftTransversals ↑H)) q),\n property :=\n (_ :\n (↑(↑(toEquiv (_ : ↑(g • α) ∈ leftTransversals ↑H)) q))⁻¹ *\n ↑(↑(toEquiv (_ : ↑(g • β) ∈ leftTransversals ↑H)) q) ∈\n H) }) =\n ↑(↑ϕ\n (↑(MonoidHom.id { x // x ∈ H })\n {\n val :=\n (↑(↑(toEquiv (_ : ↑α ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)))⁻¹ *\n ↑(↑(toEquiv (_ : ↑β ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)),\n property :=\n (_ :\n (↑(↑(toEquiv (_ : ↑α ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)))⁻¹ *\n ↑(↑(toEquiv (_ : ↑β ∈ leftTransversals ↑H)) ((fun q x => g⁻¹ • q) q x✝)) ∈\n H) }))",
"tactic": "simp only [MonoidHom.id_apply, MonoidHom.coe_mk, OneHom.coe_mk,\n smul_apply_eq_smul_apply_inv_smul, smul_eq_mul_unop, mul_inv_rev, mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\n⊢ (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1",
"tactic": "rw [Subtype.ext_iff, coe_mk, coe_one, mul_one, inv_mul_self]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\ninst✝¹ : IsCommutative H\ninst✝ : FiniteIndex H\nα β : ↑(leftTransversals ↑H)\nhH : Normal H\ng : Gᵐᵒᵖ\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nh₁ h₂ : { x // x ∈ H }\n⊢ OneHom.toFun\n { toFun := fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) },\n map_one' :=\n (_ : (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1) }\n (h₁ * h₂) =\n OneHom.toFun\n { toFun := fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) },\n map_one' :=\n (_ : (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1) }\n h₁ *\n OneHom.toFun\n { toFun := fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) },\n map_one' :=\n (_ : (fun h => { val := (unop g)⁻¹ * ↑h * unop g, property := (_ : (unop g)⁻¹ * ↑h * unop g ∈ H) }) 1 = 1) }\n h₂",
"tactic": "simp only [Subtype.ext_iff, coe_mk, coe_mul, mul_assoc, mul_inv_cancel_left]"
}
] |
[
72,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\nh : c ⟶ d\ni : d ⟶ e\n⊢ inv ((α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv ≫ (α_ f g h).hom ▷ i) =\n inv (f ◁ (α_ g h i).inv ≫ (α_ f (g ≫ h) i).inv)",
"tactic": "simp"
}
] |
[
318,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/CategoryTheory/Extensive.lean
|
CategoryTheory.NatTrans.Equifibered.comp
|
[] |
[
77,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Algebra/Ring/Equiv.lean
|
RingEquiv.toNonUnitalRingHom_injective
|
[] |
[
604,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.SimpleFunc.integral_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.116296\nF : Type u_2\n𝕜 : Type ?u.116302\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.116404\nF' : Type ?u.116407\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\ny : F\n⊢ integral μ (const α y) = ENNReal.toReal (↑↑μ Set.univ) • y",
"tactic": "classical\ncalc\n (const α y).integral μ = ∑ z in {y}, (μ (const α y ⁻¹' {z})).toReal • z :=\n integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)\n _ = (μ univ).toReal • y := by simp [Set.preimage]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.116296\nF : Type u_2\n𝕜 : Type ?u.116302\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.116404\nF' : Type ?u.116407\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\ny : F\n⊢ integral μ (const α y) = ENNReal.toReal (↑↑μ Set.univ) • y",
"tactic": "calc\n (const α y).integral μ = ∑ z in {y}, (μ (const α y ⁻¹' {z})).toReal • z :=\n integral_eq_sum_of_subset <| (filter_subset _ _).trans (range_const_subset _ _)\n _ = (μ univ).toReal • y := by simp [Set.preimage]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.116296\nF : Type u_2\n𝕜 : Type ?u.116302\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\np : ℝ≥0∞\nG : Type ?u.116404\nF' : Type ?u.116407\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : NormedSpace ℝ F'\nm✝ : MeasurableSpace α\nμ✝ : Measure α\nm : MeasurableSpace α\nμ : Measure α\ny : F\n⊢ ∑ z in {y}, ENNReal.toReal (↑↑μ (↑(const α y) ⁻¹' {z})) • z = ENNReal.toReal (↑↑μ Set.univ) • y",
"tactic": "simp [Set.preimage]"
}
] |
[
353,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.coe_neg
|
[] |
[
401,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
11
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.card_image₂
|
[] |
[
70,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Std/Data/Int/DivMod.lean
|
Int.emod_self
|
[
{
"state_after": "a : Int\nthis : 1 * a % a = 0\n⊢ a % a = 0",
"state_before": "a : Int\n⊢ a % a = 0",
"tactic": "have := mul_emod_left 1 a"
},
{
"state_after": "no goals",
"state_before": "a : Int\nthis : 1 * a % a = 0\n⊢ a % a = 0",
"tactic": "rwa [Int.one_mul] at this"
}
] |
[
486,
55
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
485,
15
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.Measure.sum_prod
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.4494925\nβ : Type u_3\nβ' : Type ?u.4494931\nγ : Type ?u.4494934\nE : Type ?u.4494937\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace β'\ninst✝⁵ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SigmaFinite ν\ninst✝² : SigmaFinite μ✝\nι : Type u_1\ninst✝¹ : Finite ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\ns : Set α\nt : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑↑(sum fun i => Measure.prod (μ i) ν) (s ×ˢ t) = ↑↑(sum μ) s * ↑↑ν t",
"state_before": "α : Type u_2\nα' : Type ?u.4494925\nβ : Type u_3\nβ' : Type ?u.4494931\nγ : Type ?u.4494934\nE : Type ?u.4494937\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace β'\ninst✝⁵ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SigmaFinite ν\ninst✝² : SigmaFinite μ✝\nι : Type u_1\ninst✝¹ : Finite ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\n⊢ Measure.prod (sum μ) ν = sum fun i => Measure.prod (μ i) ν",
"tactic": "refine' prod_eq fun s t hs ht => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.4494925\nβ : Type u_3\nβ' : Type ?u.4494931\nγ : Type ?u.4494934\nE : Type ?u.4494937\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : MeasurableSpace α'\ninst✝⁷ : MeasurableSpace β\ninst✝⁶ : MeasurableSpace β'\ninst✝⁵ : MeasurableSpace γ\nμ✝ μ' : Measure α\nν ν' : Measure β\nτ : Measure γ\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SigmaFinite ν\ninst✝² : SigmaFinite μ✝\nι : Type u_1\ninst✝¹ : Finite ι\nμ : ι → Measure α\ninst✝ : ∀ (i : ι), SigmaFinite (μ i)\ns : Set α\nt : Set β\nhs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑↑(sum fun i => Measure.prod (μ i) ν) (s ×ˢ t) = ↑↑(sum μ) s * ↑↑ν t",
"tactic": "simp_rw [sum_apply _ (hs.prod ht), sum_apply _ hs, prod_prod, ENNReal.tsum_mul_right]"
}
] |
[
580,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
MeasureTheory.lintegral_prod_symm'
|
[] |
[
777,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
775,
1
] |
Mathlib/Algebra/Order/Kleene.lean
|
mul_kstar_le
|
[] |
[
223,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.map_val_atTop_of_Ici_subset
|
[
{
"state_after": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\n⊢ map Subtype.val atTop = atTop",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\n⊢ map Subtype.val atTop = atTop",
"tactic": "haveI : Nonempty s := ⟨⟨a, h le_rfl⟩⟩"
},
{
"state_after": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ (∀ (i : α), Ici i ∈ ⨅ (i : ↑s), 𝓟 (Subtype.val '' Ici i)) ∧ ∀ (i : ↑s), Subtype.val '' Ici i ∈ ⨅ (a : α), 𝓟 (Ici a)",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ map Subtype.val atTop = atTop",
"tactic": "simp only [le_antisymm_iff, atTop, le_iInf_iff, le_principal_iff, mem_map, mem_setOf_eq,\n map_iInf_eq this, map_principal]"
},
{
"state_after": "case left\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ ∀ (i : α), Ici i ∈ ⨅ (i : ↑s), 𝓟 (Subtype.val '' Ici i)\n\ncase right\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ ∀ (i : ↑s), Subtype.val '' Ici i ∈ ⨅ (a : α), 𝓟 (Ici a)",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ (∀ (i : α), Ici i ∈ ⨅ (i : ↑s), 𝓟 (Subtype.val '' Ici i)) ∧ ∀ (i : ↑s), Subtype.val '' Ici i ∈ ⨅ (a : α), 𝓟 (Ici a)",
"tactic": "constructor"
},
{
"state_after": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\nx y : ↑s\n⊢ ∃ z,\n (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) x) ((fun x => 𝓟 (Ici x)) z) ∧\n (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) y) ((fun x => 𝓟 (Ici x)) z)",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\n⊢ Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)",
"tactic": "intro x y"
},
{
"state_after": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\nx y : ↑s\n⊢ (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) x)\n ((fun x => 𝓟 (Ici x)) { val := ↑x ⊔ ↑y ⊔ a, property := (_ : ↑x ⊔ ↑y ⊔ a ∈ s) }) ∧\n (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) y)\n ((fun x => 𝓟 (Ici x)) { val := ↑x ⊔ ↑y ⊔ a, property := (_ : ↑x ⊔ ↑y ⊔ a ∈ s) })",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\nx y : ↑s\n⊢ ∃ z,\n (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) x) ((fun x => 𝓟 (Ici x)) z) ∧\n (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) y) ((fun x => 𝓟 (Ici x)) z)",
"tactic": "use ⟨x ⊔ y ⊔ a, h le_sup_right⟩"
},
{
"state_after": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\nx y : ↑s\n⊢ ↑x ≤ ↑x ⊔ ↑y ⊔ a ∧ ↑y ≤ ↑x ⊔ ↑y ⊔ a",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\nx y : ↑s\n⊢ (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) x)\n ((fun x => 𝓟 (Ici x)) { val := ↑x ⊔ ↑y ⊔ a, property := (_ : ↑x ⊔ ↑y ⊔ a ∈ s) }) ∧\n (fun x x_1 => x ≥ x_1) ((fun x => 𝓟 (Ici x)) y)\n ((fun x => 𝓟 (Ici x)) { val := ↑x ⊔ ↑y ⊔ a, property := (_ : ↑x ⊔ ↑y ⊔ a ∈ s) })",
"tactic": "simp only [ge_iff_le, principal_mono, Ici_subset_Ici, ← Subtype.coe_le_coe, Subtype.coe_mk]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis : Nonempty ↑s\nx y : ↑s\n⊢ ↑x ≤ ↑x ⊔ ↑y ⊔ a ∧ ↑y ≤ ↑x ⊔ ↑y ⊔ a",
"tactic": "exact ⟨le_sup_left.trans le_sup_left, le_sup_right.trans le_sup_left⟩"
},
{
"state_after": "case left\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : α\n⊢ Ici x ∈ ⨅ (i : ↑s), 𝓟 (Subtype.val '' Ici i)",
"state_before": "case left\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ ∀ (i : α), Ici i ∈ ⨅ (i : ↑s), 𝓟 (Subtype.val '' Ici i)",
"tactic": "intro x"
},
{
"state_after": "case left\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : α\n⊢ Subtype.val '' Ici { val := x ⊔ a, property := (_ : x ⊔ a ∈ s) } ⊆ Ici x",
"state_before": "case left\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : α\n⊢ Ici x ∈ ⨅ (i : ↑s), 𝓟 (Subtype.val '' Ici i)",
"tactic": "refine' mem_of_superset (mem_iInf_of_mem ⟨x ⊔ a, h le_sup_right⟩ (mem_principal_self _)) _"
},
{
"state_after": "case left.intro.intro\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : α\ny : ↑s\nhy : y ∈ Ici { val := x ⊔ a, property := (_ : x ⊔ a ∈ s) }\n⊢ ↑y ∈ Ici x",
"state_before": "case left\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : α\n⊢ Subtype.val '' Ici { val := x ⊔ a, property := (_ : x ⊔ a ∈ s) } ⊆ Ici x",
"tactic": "rintro _ ⟨y, hy, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case left.intro.intro\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : α\ny : ↑s\nhy : y ∈ Ici { val := x ⊔ a, property := (_ : x ⊔ a ∈ s) }\n⊢ ↑y ∈ Ici x",
"tactic": "exact le_trans le_sup_left (Subtype.coe_le_coe.2 hy)"
},
{
"state_after": "case right\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : ↑s\n⊢ Subtype.val '' Ici x ∈ ⨅ (a : α), 𝓟 (Ici a)",
"state_before": "case right\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\n⊢ ∀ (i : ↑s), Subtype.val '' Ici i ∈ ⨅ (a : α), 𝓟 (Ici a)",
"tactic": "intro x"
},
{
"state_after": "case h\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : ↑s\nb : α\nhb : ↑x ⊔ a ≤ b\n⊢ b ∈ Subtype.val '' Ici x",
"state_before": "case right\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : ↑s\n⊢ Subtype.val '' Ici x ∈ ⨅ (a : α), 𝓟 (Ici a)",
"tactic": "filter_upwards [mem_atTop (↑x ⊔ a)]with b hb"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type ?u.314713\nι' : Type ?u.314716\nα : Type u_1\nβ : Type ?u.314722\nγ : Type ?u.314725\ninst✝ : SemilatticeSup α\na : α\ns : Set α\nh : Ici a ⊆ s\nthis✝ : Nonempty ↑s\nthis : Directed (fun x x_1 => x ≥ x_1) fun x => 𝓟 (Ici x)\nx : ↑s\nb : α\nhb : ↑x ⊔ a ≤ b\n⊢ b ∈ Subtype.val '' Ici x",
"tactic": "exact ⟨⟨b, h <| le_sup_right.trans hb⟩, Subtype.coe_le_coe.1 (le_sup_left.trans hb), rfl⟩"
}
] |
[
1537,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1520,
1
] |
Mathlib/Data/Polynomial/IntegralNormalization.lean
|
Polynomial.integralNormalization_coeff_natDegree
|
[] |
[
71,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Analysis/Calculus/Series.lean
|
tendstoUniformly_tsum
|
[
{
"state_after": "α : Type u_1\nβ : Type u_3\n𝕜 : Type ?u.6502\nE : Type ?u.6505\nF : Type u_2\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\nhfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n\n⊢ TendstoUniformlyOn (fun t x => ∑ n in t, f n x) (fun x => ∑' (n : α), f n x) atTop univ",
"state_before": "α : Type u_1\nβ : Type u_3\n𝕜 : Type ?u.6502\nE : Type ?u.6505\nF : Type u_2\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\nhfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n\n⊢ TendstoUniformly (fun t x => ∑ n in t, f n x) (fun x => ∑' (n : α), f n x) atTop",
"tactic": "rw [← tendstoUniformlyOn_univ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\n𝕜 : Type ?u.6502\nE : Type ?u.6505\nF : Type u_2\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : CompleteSpace F\nu : α → ℝ\nf : α → β → F\nhu : Summable u\nhfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n\n⊢ TendstoUniformlyOn (fun t x => ∑ n in t, f n x) (fun x => ∑' (n : α), f n x) atTop univ",
"tactic": "exact tendstoUniformlyOn_tsum hu fun n x _ => hfu n x"
}
] |
[
69,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.image_symm
|
[] |
[
224,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.IsRegularOfDegree.top
|
[
{
"state_after": "ι : Sort ?u.287669\n𝕜 : Type ?u.287672\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableEq V\nv : V\n⊢ degree ⊤ v = Fintype.card V - 1",
"state_before": "ι : Sort ?u.287669\n𝕜 : Type ?u.287672\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableEq V\n⊢ IsRegularOfDegree ⊤ (Fintype.card V - 1)",
"tactic": "intro v"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.287669\n𝕜 : Type ?u.287672\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\ninst✝¹ : Fintype V\ninst✝ : DecidableEq V\nv : V\n⊢ degree ⊤ v = Fintype.card V - 1",
"tactic": "simp"
}
] |
[
1502,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1499,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasurableEquiv.quasiMeasurePreserving_symm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.3211462\nδ : Type ?u.3211465\nι : Type ?u.3211468\nR : Type ?u.3211471\nR' : Type ?u.3211474\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ✝ : MeasureTheory.Measure α\nν : MeasureTheory.Measure β\nμ : MeasureTheory.Measure α\ne : α ≃ᵐ β\n⊢ Measure.map (↑(symm e)) (Measure.map (↑e) μ) ≪ μ",
"tactic": "rw [Measure.map_map, e.symm_comp_self, Measure.map_id] <;> measurability"
}
] |
[
4328,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4326,
1
] |
Std/Control/ForInStep/Lemmas.lean
|
ForInStep.bind_done
|
[] |
[
11,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
10,
9
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.le_map_of_map_zero
|
[] |
[
493,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_nonneg
|
[] |
[
174,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
Equiv.Perm.smul_def
|
[] |
[
101,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
11
] |
Mathlib/Algebra/Order/Group/MinMax.lean
|
abs_max_sub_max_le_abs
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\na✝ b✝ c✝ a b c : α\n⊢ abs (max a c - max b c) ≤ abs (a - b)",
"tactic": "simpa only [sub_self, abs_zero, max_eq_left (abs_nonneg (a - b))]\n using abs_max_sub_max_le_max a c b c"
}
] |
[
108,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.pi_empty
|
[
{
"state_after": "no goals",
"state_before": "G : Type ?u.318819\nG' : Type ?u.318822\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\nA : Type ?u.318831\ninst✝³ : AddGroup A\nH✝ K : Subgroup G\nk : Set G\nN : Type ?u.318852\ninst✝² : Group N\nP : Type ?u.318858\ninst✝¹ : Group P\nη : Type u_2\nf : η → Type u_1\ninst✝ : (i : η) → Group (f i)\nH : (i : η) → Subgroup (f i)\nx : (i : η) → f i\n⊢ x ∈ pi ∅ H ↔ x ∈ ⊤",
"tactic": "simp [mem_pi]"
}
] |
[
1848,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1847,
1
] |
Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean
|
MeasureTheory.integral_divergence_of_hasFDerivWithinAt_off_countable
|
[
{
"state_after": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)\n\ncase inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nhne : ¬∃ i, a i = b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"state_before": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "rcases em (∃ i, a i = b i) with (⟨i, hi⟩ | hne)"
},
{
"state_after": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n ∑ i : Fin (n + 1), ↑(f' x) (e i) i ∂Measure.pi fun x => volume) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"state_before": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "rw [volume_pi, ← set_integral_congr_set_ae Measure.univ_pi_Ioc_ae_eq_Icc]"
},
{
"state_after": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n ∑ i : Fin (n + 1), ↑(f' x) (e i) i ∂Measure.pi fun x => volume) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"state_before": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n ∑ i : Fin (n + 1), ↑(f' x) (e i) i ∂Measure.pi fun x => volume) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "have hi' : Ioc (a i) (b i) = ∅ := Ioc_eq_empty hi.not_lt"
},
{
"state_after": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n ∑ i : Fin (n + 1), ↑(f' x) (e i) i ∂Measure.pi fun x => volume) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"state_before": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n ∑ i : Fin (n + 1), ↑(f' x) (e i) i ∂Measure.pi fun x => volume) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "have : (pi Set.univ fun j => Ioc (a j) (b j)) = ∅ := univ_pi_eq_empty hi'"
},
{
"state_after": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\n⊢ ∀ (x : Fin (n + 1)),\n x ∈ Finset.univ →\n ((∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove x)) (b ∘ ↑(Fin.succAbove x)),\n f (Fin.insertNth x (b x) x_1) x) -\n ∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove x)) (b ∘ ↑(Fin.succAbove x)),\n f (Fin.insertNth x (a x) x_1) x) =\n 0",
"state_before": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.pi Set.univ fun i => Set.Ioc (a i) (b i),\n ∑ i : Fin (n + 1), ↑(f' x) (e i) i ∂Measure.pi fun x => volume) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "rw [this, integral_empty, sum_eq_zero]"
},
{
"state_after": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"state_before": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\n⊢ ∀ (x : Fin (n + 1)),\n x ∈ Finset.univ →\n ((∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove x)) (b ∘ ↑(Fin.succAbove x)),\n f (Fin.insertNth x (b x) x_1) x) -\n ∫ (x_1 : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove x)) (b ∘ ↑(Fin.succAbove x)),\n f (Fin.insertNth x (a x) x_1) x) =\n 0",
"tactic": "rintro j -"
},
{
"state_after": "case inl.intro.inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i) =\n 0\n\ncase inl.intro.inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\nhne : i ≠ j\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"state_before": "case inl.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"tactic": "rcases eq_or_ne i j with (rfl | hne)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.inl\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i) =\n 0",
"tactic": "simp [hi]"
},
{
"state_after": "case inl.intro.inr.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"state_before": "case inl.intro.inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\ni : Fin (n + 1)\nhi : a i = b i\nhi' : Set.Ioc (a i) (b i) = ∅\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\nhne : i ≠ j\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"tactic": "rcases Fin.exists_succAbove_eq hne with ⟨i, rfl⟩"
},
{
"state_after": "case this\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\n⊢ Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)) =ᵐ[volume] ∅\n\ncase inl.intro.inr.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis✝ : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\nthis : Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)) =ᵐ[volume] ∅\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"state_before": "case inl.intro.inr.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"tactic": "have : Icc (a ∘ j.succAbove) (b ∘ j.succAbove) =ᵐ[volume] (∅ : Set ℝⁿ)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.inr.intro\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis✝ : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\nthis : Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)) =ᵐ[volume] ∅\n⊢ ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (b j) x) j) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)), f (Fin.insertNth j (a j) x) j) =\n 0",
"tactic": "rw [set_integral_congr_set_ae this, set_integral_congr_set_ae this, integral_empty,\n integral_empty, sub_self]"
},
{
"state_after": "case this\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\n⊢ ENNReal.ofReal ((b ∘ ↑(Fin.succAbove j)) i - (a ∘ ↑(Fin.succAbove j)) i) = 0",
"state_before": "case this\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\n⊢ Set.Icc (a ∘ ↑(Fin.succAbove j)) (b ∘ ↑(Fin.succAbove j)) =ᵐ[volume] ∅",
"tactic": "rw [ae_eq_empty, Real.volume_Icc_pi, prod_eq_zero (Finset.mem_univ i)]"
},
{
"state_after": "no goals",
"state_before": "case this\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nthis : (Set.pi Set.univ fun j => Set.Ioc (a j) (b j)) = ∅\nj : Fin (n + 1)\ni : Fin n\nhi : a (↑(Fin.succAbove j) i) = b (↑(Fin.succAbove j) i)\nhi' : Set.Ioc (a (↑(Fin.succAbove j) i)) (b (↑(Fin.succAbove j) i)) = ∅\nhne : ↑(Fin.succAbove j) i ≠ j\n⊢ ENNReal.ofReal ((b ∘ ↑(Fin.succAbove j)) i - (a ∘ ↑(Fin.succAbove j)) i) = 0",
"tactic": "simp [hi]"
},
{
"state_after": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nhne : ¬∃ i, a i = b i\nhlt : ∀ (i : Fin (n + 1)), a i < b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nhne : ¬∃ i, a i = b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "have hlt : ∀ i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne ⟨i, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : Set.Countable s\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ (x : Fin (n + 1) → ℝ), x ∈ (Set.pi Set.univ fun i => Set.Ioo (a i) (b i)) \\ s → HasFDerivAt f (f' x) x\nHi : IntegrableOn (fun x => ∑ i : Fin (n + 1), ↑(f' x) (e i) i) (Set.Icc a b)\nhne : ¬∃ i, a i = b i\nhlt : ∀ (i : Fin (n + 1)), a i < b i\n⊢ (∫ (x : Fin (n + 1) → ℝ) in Set.Icc a b, ∑ i : Fin (n + 1), ↑(f' x) (e i) i) =\n ∑ i : Fin (n + 1),\n ((∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (b i) x) i) -\n ∫ (x : Fin n → ℝ) in Set.Icc (a ∘ ↑(Fin.succAbove i)) (b ∘ ↑(Fin.succAbove i)), f (Fin.insertNth i (a i) x) i)",
"tactic": "exact integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ ⟨a, b, hlt⟩ f f' s hs Hc\n Hd Hi"
}
] |
[
298,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
midpoint_unique
|
[] |
[
195,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.sin_add_pi_div_two
|
[
{
"state_after": "case h\nx✝ : ℝ\n⊢ sin (↑x✝ + ↑(π / 2)) = cos ↑x✝",
"state_before": "θ : Angle\n⊢ sin (θ + ↑(π / 2)) = cos θ",
"tactic": "induction θ using Real.Angle.induction_on"
},
{
"state_after": "no goals",
"state_before": "case h\nx✝ : ℝ\n⊢ sin (↑x✝ + ↑(π / 2)) = cos ↑x✝",
"tactic": "exact Real.sin_add_pi_div_two _"
}
] |
[
451,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.continuousAt_angle
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y : V\nx : V × V\nhx1 : x.fst ≠ 0\nhx2 : x.snd ≠ 0\n⊢ ‖x.fst‖ * ‖x.snd‖ ≠ 0",
"tactic": "simp [hx1, hx2]"
}
] |
[
56,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorBundleCore.localTrivAt_def
|
[] |
[
743,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
742,
1
] |
Mathlib/Computability/RegularExpressions.lean
|
RegularExpression.star_rmatch_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "have A : ∀ m n : ℕ, n < m + n + 1 := by\n intro m n\n convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one\n simp"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "have IH := fun t (_h : List.length t < List.length x) => star_rmatch_iff P t"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "clear star_rmatch_iff"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) x = true → ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\n\ncase mpr\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ (∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true) → rmatch (star P) x = true",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) x = true ↔ ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "constructor"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nm n : ℕ\n⊢ n < m + n + 1",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\n⊢ ∀ (m n : ℕ), n < m + n + 1",
"tactic": "intro m n"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nm n : ℕ\n⊢ n = 0 + n + 0",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nm n : ℕ\n⊢ n < m + n + 1",
"tactic": "convert add_lt_add_of_le_of_lt (add_le_add (zero_le m) (le_refl n)) zero_lt_one"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nm n : ℕ\n⊢ n = 0 + n + 0",
"tactic": "simp"
},
{
"state_after": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) [] = true → ∃ S, [] = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\n\ncase mp.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) (a :: x) = true → ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) x = true → ∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "cases' x with a x"
},
{
"state_after": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n_h : rmatch (star P) [] = true\n⊢ ∃ S, [] = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) [] = true → ∃ S, [] = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "intro _h"
},
{
"state_after": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n_h : rmatch (star P) [] = true\n⊢ [] = join [] ∧ ∀ (t : List α), t ∈ [] → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n_h : rmatch (star P) [] = true\n⊢ ∃ S, [] = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "use []"
},
{
"state_after": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n_h : rmatch (star P) [] = true\n⊢ [] = [] ∧ ∀ (t : List α), t ∈ [] → ¬t = [] ∧ rmatch P t = true",
"state_before": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n_h : rmatch (star P) [] = true\n⊢ [] = join [] ∧ ∀ (t : List α), t ∈ [] → t ≠ [] ∧ rmatch P t = true",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case mp.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n_h : rmatch (star P) [] = true\n⊢ [] = [] ∧ ∀ (t : List α), t ∈ [] → ¬t = [] ∧ rmatch P t = true",
"tactic": "tauto"
},
{
"state_after": "case mp.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ (∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true) →\n ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ rmatch (star P) (a :: x) = true → ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "rw [rmatch, deriv, mul_rmatch_iff]"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ (∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true) →\n ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "rintro ⟨t, u, hs, ht, hu⟩"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\nhwf : length u < length (a :: x)\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "have hwf : u.length < (List.cons a x).length := by\n rw [hs, List.length_cons, List.length_append]\n apply A"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : ∃ S, u = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\nhwf : length u < length (a :: x)\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\nhwf : length u < length (a :: x)\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "rw [IH _ hwf] at hu"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mp.cons.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : ∃ S, u = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\nhwf : length u < length (a :: x)\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "rcases hu with ⟨S', hsum, helem⟩"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: x = join ((a :: t) :: S') ∧ ∀ (t_1 : List α), t_1 ∈ (a :: t) :: S' → t_1 ≠ [] ∧ rmatch P t_1 = true",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ ∃ S, a :: x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "use (a :: t) :: S'"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro.intro.intro.left\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: x = join ((a :: t) :: S')\n\ncase mp.cons.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ ∀ (t_1 : List α), t_1 ∈ (a :: t) :: S' → t_1 ≠ [] ∧ rmatch P t_1 = true",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: x = join ((a :: t) :: S') ∧ ∀ (t_1 : List α), t_1 ∈ (a :: t) :: S' → t_1 ≠ [] ∧ rmatch P t_1 = true",
"tactic": "constructor"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\n⊢ length u < Nat.succ (length t + length u)",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\n⊢ length u < length (a :: x)",
"tactic": "rw [hs, List.length_cons, List.length_append]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhu : rmatch (star P) u = true\n⊢ length u < Nat.succ (length t + length u)",
"tactic": "apply A"
},
{
"state_after": "no goals",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro.left\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: x = join ((a :: t) :: S')",
"tactic": "simp [hs, hsum]"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\nht' : t' ∈ (a :: t) :: S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ ∀ (t_1 : List α), t_1 ∈ (a :: t) :: S' → t_1 ≠ [] ∧ rmatch P t_1 = true",
"tactic": "intro t' ht'"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro.intro.intro.right.head\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: t ≠ [] ∧ rmatch P (a :: t) = true\n\ncase mp.cons.intro.intro.intro.intro.intro.intro.right.tail\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝¹ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\na✝ : Mem t' S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro.right\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\nht' : t' ∈ (a :: t) :: S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"tactic": "cases ht'"
},
{
"state_after": "case mp.cons.intro.intro.intro.intro.intro.intro.right.tail\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝¹ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\na✝ : Mem t' S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro.right.head\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: t ≠ [] ∧ rmatch P (a :: t) = true\n\ncase mp.cons.intro.intro.intro.intro.intro.intro.right.tail\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝¹ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\na✝ : Mem t' S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"tactic": "case head ht' =>\n simp only [ne_eq, not_false_iff, true_and, rmatch]\n exact ht"
},
{
"state_after": "no goals",
"state_before": "case mp.cons.intro.intro.intro.intro.intro.intro.right.tail\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝¹ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\na✝ : Mem t' S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"tactic": "case tail ht' => exact helem t' ht'"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\nht' b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ rmatch (deriv P a) t = true",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\nht' b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ a :: t ≠ [] ∧ rmatch P (a :: t) = true",
"tactic": "simp only [ne_eq, not_false_iff, true_and, rmatch]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\nht' b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\n⊢ rmatch (deriv P a) t = true",
"tactic": "exact ht"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt u : List α\nhs : x = t ++ u\nht : rmatch (deriv P a) t = true\nhwf : length u < length (a :: x)\nS' : List (List α)\nhsum : u = join S'\nhelem : ∀ (t : List α), t ∈ S' → t ≠ [] ∧ rmatch P t = true\nt' : List α\nht' : Mem t' S'\n⊢ t' ≠ [] ∧ rmatch P t' = true",
"tactic": "exact helem t' ht'"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nS : List (List α)\nhsum : x = join S\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\n⊢ rmatch (star P) x = true",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\n⊢ (∃ S, x = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true) → rmatch (star P) x = true",
"tactic": "rintro ⟨S, hsum, helem⟩"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nS : List (List α)\nhsum : x = join S\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\n⊢ rmatch (star P) x = true",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nS : List (List α)\nhsum : x = join S\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\n⊢ rmatch (star P) x = true",
"tactic": "dsimp"
},
{
"state_after": "case mpr.intro.intro.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nS : List (List α)\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhsum : [] = join S\n⊢ rmatch (star P) [] = true\n\ncase mpr.intro.intro.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nS : List (List α)\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhsum : a :: x = join S\n⊢ rmatch (star P) (a :: x) = true",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nx : List α\nA : ∀ (m n : ℕ), n < m + n + 1\nIH :\n ∀ (t : List α),\n length t < length x →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nS : List (List α)\nhsum : x = join S\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\n⊢ rmatch (star P) x = true",
"tactic": "cases' x with a x"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nS : List (List α)\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\nIH :\n ∀ (t : List α),\n length t < length [] →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhsum : [] = join S\n⊢ rmatch (star P) [] = true",
"tactic": "rfl"
},
{
"state_after": "case mpr.intro.intro.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nS : List (List α)\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhsum : a :: x = join S\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"state_before": "case mpr.intro.intro.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nS : List (List α)\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhsum : a :: x = join S\n⊢ rmatch (star P) (a :: x) = true",
"tactic": "rw [rmatch, deriv, mul_rmatch_iff]"
},
{
"state_after": "case mpr.intro.intro.cons.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhelem : ∀ (t : List α), t ∈ [] → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join []\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true\n\ncase mpr.intro.intro.cons.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt' : List α\nU : List (List α)\nhelem : ∀ (t : List α), t ∈ t' :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join (t' :: U)\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"state_before": "case mpr.intro.intro.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\nS : List (List α)\nhelem : ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhsum : a :: x = join S\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "cases' S with t' U"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.cons.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhelem : ∀ (t : List α), t ∈ [] → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join []\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "exact ⟨[], [], by tauto⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nhelem : ∀ (t : List α), t ∈ [] → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join []\n⊢ x = [] ++ [] ∧ rmatch (deriv P a) [] = true ∧ rmatch (star P) [] = true",
"tactic": "tauto"
},
{
"state_after": "case mpr.intro.intro.cons.cons.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nhelem : ∀ (t : List α), t ∈ [] :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join ([] :: U)\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true\n\ncase mpr.intro.intro.cons.cons.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a :: x = join ((b :: t) :: U)\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"state_before": "case mpr.intro.intro.cons.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nt' : List α\nU : List (List α)\nhelem : ∀ (t : List α), t ∈ t' :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join (t' :: U)\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "cases' t' with b t"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"state_before": "case mpr.intro.intro.cons.cons.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a :: x = join ((b :: t) :: U)\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "simp only [List.join, List.cons_append, List.cons_eq_cons] at hsum"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_1\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ rmatch (deriv P a) t = true\n\ncase mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ rmatch (star P) (join U) = true",
"state_before": "case mpr.intro.intro.cons.cons.cons\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "refine' ⟨t, U.join, hsum.2, _, _⟩"
},
{
"state_after": "case mpr.intro.intro.cons.cons.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nhsum : a :: x = join ([] :: U)\nhelem : ([] ≠ [] ∧ rmatch P [] = true) ∧ ∀ (a : List α), a ∈ U → a ≠ [] ∧ rmatch P a = true\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"state_before": "case mpr.intro.intro.cons.cons.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nhelem : ∀ (t : List α), t ∈ [] :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a :: x = join ([] :: U)\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "simp only [forall_eq_or_imp, List.mem_cons] at helem"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.cons.cons.nil\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nhsum : a :: x = join ([] :: U)\nhelem : ([] ≠ [] ∧ rmatch P [] = true) ∧ ∀ (a : List α), a ∈ U → a ≠ [] ∧ rmatch P a = true\n⊢ ∃ t u, x = t ++ u ∧ rmatch (deriv P a) t = true ∧ rmatch (star P) u = true",
"tactic": "simp only [eq_self_iff_true, not_true, Ne.def, false_and_iff] at helem"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_1\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : a = b ∧ x = t ++ join U\nhelem : b :: t ≠ [] ∧ rmatch P (b :: t) = true\n⊢ rmatch (deriv P a) t = true",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_1\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ rmatch (deriv P a) t = true",
"tactic": "specialize helem (b :: t) (by simp)"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_1\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : a = b ∧ x = t ++ join U\nhelem : b :: t ≠ [] ∧ rmatch (deriv P b) t = true\n⊢ rmatch (deriv P a) t = true",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_1\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : a = b ∧ x = t ++ join U\nhelem : b :: t ≠ [] ∧ rmatch P (b :: t) = true\n⊢ rmatch (deriv P a) t = true",
"tactic": "rw [rmatch] at helem"
},
{
"state_after": "case h.e'_2.h.e'_3.h.e'_4\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : a = b ∧ x = t ++ join U\nhelem : b :: t ≠ [] ∧ rmatch (deriv P b) t = true\n⊢ a = b",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_1\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : a = b ∧ x = t ++ join U\nhelem : b :: t ≠ [] ∧ rmatch (deriv P b) t = true\n⊢ rmatch (deriv P a) t = true",
"tactic": "convert helem.2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_3.h.e'_4\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhsum : a = b ∧ x = t ++ join U\nhelem : b :: t ≠ [] ∧ rmatch (deriv P b) t = true\n⊢ a = b",
"tactic": "exact hsum.1"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ b :: t ∈ (b :: t) :: U",
"tactic": "simp"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\nhwf : length (join U) < length (a :: x)\n⊢ rmatch (star P) (join U) = true",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ rmatch (star P) (join U) = true",
"tactic": "have hwf : U.join.length < (List.cons a x).length := by\n rw [hsum.1, hsum.2]\n simp only [List.length_append, List.length_join, List.length]\n apply A"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\nhwf : length (join U) < length (a :: x)\n⊢ ∃ S, join U = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\nhwf : length (join U) < length (a :: x)\n⊢ rmatch (star P) (join U) = true",
"tactic": "rw [IH _ hwf]"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt✝ : List α\nhelem : ∀ (t : List α), t ∈ (b :: t✝) :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a = b ∧ x = t✝ ++ join U\nhwf : length (join U) < length (a :: x)\nt : List α\nh : t ∈ U\n⊢ t ∈ (b :: t✝) :: U",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\nhwf : length (join U) < length (a :: x)\n⊢ ∃ S, join U = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true",
"tactic": "refine' ⟨U, rfl, fun t h => helem t _⟩"
},
{
"state_after": "case mpr.intro.intro.cons.cons.cons.refine'_2.a\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt✝ : List α\nhelem : ∀ (t : List α), t ∈ (b :: t✝) :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a = b ∧ x = t✝ ++ join U\nhwf : length (join U) < length (a :: x)\nt : List α\nh : t ∈ U\n⊢ Mem t U",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_2\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt✝ : List α\nhelem : ∀ (t : List α), t ∈ (b :: t✝) :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a = b ∧ x = t✝ ++ join U\nhwf : length (join U) < length (a :: x)\nt : List α\nh : t ∈ U\n⊢ t ∈ (b :: t✝) :: U",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.cons.cons.cons.refine'_2.a\nα : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt✝ : List α\nhelem : ∀ (t : List α), t ∈ (b :: t✝) :: U → t ≠ [] ∧ rmatch P t = true\nhsum : a = b ∧ x = t✝ ++ join U\nhwf : length (join U) < length (a :: x)\nt : List α\nh : t ∈ U\n⊢ Mem t U",
"tactic": "assumption"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ length (join U) < length (b :: (t ++ join U))",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ length (join U) < length (a :: x)",
"tactic": "rw [hsum.1, hsum.2]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ sum (map length U) < length t + sum (map length U) + 1",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ length (join U) < length (b :: (t ++ join U))",
"tactic": "simp only [List.length_append, List.length_join, List.length]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.53385\nγ : Type ?u.53388\ndec : DecidableEq α\na✝ b✝ : α\nP : RegularExpression α\nA : ∀ (m n : ℕ), n < m + n + 1\na : α\nx : List α\nIH :\n ∀ (t : List α),\n length t < length (a :: x) →\n (rmatch (star P) t = true ↔ ∃ S, t = join S ∧ ∀ (t : List α), t ∈ S → t ≠ [] ∧ rmatch P t = true)\nU : List (List α)\nb : α\nt : List α\nhelem : ∀ (t_1 : List α), t_1 ∈ (b :: t) :: U → t_1 ≠ [] ∧ rmatch P t_1 = true\nhsum : a = b ∧ x = t ++ join U\n⊢ sum (map length U) < length t + sum (map length U) + 1",
"tactic": "apply A"
}
] |
[
349,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Algebra/GroupPower/Order.lean
|
pow_le_one
|
[] |
[
426,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Icc_union_Ioo_eq_Ico
|
[] |
[
1576,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1573,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Algebra.toMatrix_lmul_eq
|
[] |
[
906,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/Analysis/NormedSpace/WeakDual.lean
|
NormedSpace.Dual.dual_norm_topology_le_weak_dual_topology
|
[
{
"state_after": "case h.e'_4\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ WeakDual.instTopologicalSpace = TopologicalSpace.induced (fun x' => ↑toWeakDual x') WeakDual.instTopologicalSpace",
"state_before": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ UniformSpace.toTopologicalSpace ≤ WeakDual.instTopologicalSpace",
"tactic": "convert (@toWeakDual_continuous _ _ _ _ (by assumption)).le_induced"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ WeakDual.instTopologicalSpace = TopologicalSpace.induced (fun x' => ↑toWeakDual x') WeakDual.instTopologicalSpace",
"tactic": "exact induced_id.symm"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\n⊢ NormedSpace 𝕜 E",
"tactic": "assumption"
}
] |
[
150,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/RingTheory/JacobsonIdeal.lean
|
Ideal.le_jacobson
|
[] |
[
67,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
UpperSet.one_mul
|
[
{
"state_after": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns✝ t : Set α\na : α\ns : UpperSet α\n⊢ (⋃ (a : α) (_ : a ∈ Set.Ici 1), a • ↑s) ⊆ ↑s",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns✝ t : Set α\na : α\ns : UpperSet α\n⊢ Set.Ici 1 * ↑s ⊆ ↑s",
"tactic": "rw [← smul_eq_mul, ← Set.iUnion_smul_set]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns✝ t : Set α\na : α\ns : UpperSet α\n⊢ (⋃ (a : α) (_ : a ∈ Set.Ici 1), a • ↑s) ⊆ ↑s",
"tactic": "exact Set.iUnion₂_subset fun _ ↦ s.upper.smul_subset"
}
] |
[
185,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
9
] |
Mathlib/Data/Int/Units.lean
|
Int.ofNat_isUnit
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ IsUnit ↑n ↔ IsUnit n",
"tactic": "simp [isUnit_iff_natAbs_eq]"
}
] |
[
101,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.lookupAll_sublist
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nl l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\n⊢ map (Sigma.mk a) (lookupAll a []) <+ []",
"tactic": "simp"
},
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : a = a'\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l\n\ncase neg\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : ¬a = a'\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l",
"state_before": "α : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l",
"tactic": "by_cases h : a = a'"
},
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nb' : β a\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a, snd := b' } :: l)) <+ { fst := a, snd := b' } :: l",
"state_before": "case pos\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : a = a'\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l",
"tactic": "subst h"
},
{
"state_after": "case pos\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nb' : β a\n⊢ { fst := a, snd := b' } :: map (Sigma.mk a) (lookupAll a l) <+ { fst := a, snd := b' } :: l",
"state_before": "case pos\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nb' : β a\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a, snd := b' } :: l)) <+ { fst := a, snd := b' } :: l",
"tactic": "simp only [ne_eq, not_true, lookupAll_cons_eq, List.map]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na : α\nl : List (Sigma β)\nb' : β a\n⊢ { fst := a, snd := b' } :: map (Sigma.mk a) (lookupAll a l) <+ { fst := a, snd := b' } :: l",
"tactic": "exact (lookupAll_sublist a l).cons₂ _"
},
{
"state_after": "case neg\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : ¬a = a'\n⊢ map (Sigma.mk a) (lookupAll a l) <+ { fst := a', snd := b' } :: l",
"state_before": "case neg\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : ¬a = a'\n⊢ map (Sigma.mk a) (lookupAll a ({ fst := a', snd := b' } :: l)) <+ { fst := a', snd := b' } :: l",
"tactic": "simp only [ne_eq, h, not_false_iff, lookupAll_cons_ne]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u\nβ : α → Type v\nl✝ l₁ l₂ : List (Sigma β)\ninst✝ : DecidableEq α\na a' : α\nb' : β a'\nl : List (Sigma β)\nh : ¬a = a'\n⊢ map (Sigma.mk a) (lookupAll a l) <+ { fst := a', snd := b' } :: l",
"tactic": "exact (lookupAll_sublist a l).cons _"
}
] |
[
310,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Algebra/Ring/Units.lean
|
Units.sub_divp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : Type x\ninst✝ : Ring α\na✝ b✝ a b : α\nu : αˣ\n⊢ a - b /ₚ u = (a * ↑u - b) /ₚ u",
"tactic": "simp only [divp, sub_mul, Units.mul_inv_cancel_right]"
}
] |
[
80,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.transpose_zpow
|
[
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nn : ℕ\n⊢ (A ^ ↑n)ᵀ = Aᵀ ^ ↑n",
"tactic": "rw [zpow_ofNat, zpow_ofNat, transpose_pow]"
},
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nn : ℕ\n⊢ (A ^ -[n+1])ᵀ = Aᵀ ^ -[n+1]",
"tactic": "rw [zpow_negSucc, zpow_negSucc, transpose_nonsing_inv, transpose_pow]"
}
] |
[
340,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Topology/Basic.lean
|
nhds_basis_opens'
|
[
{
"state_after": "case h.e'_4.h.a\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nx✝ : Set α\n⊢ x✝ ∈ 𝓝 a ∧ IsOpen x✝ ↔ a ∈ x✝ ∧ IsOpen x✝",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\n⊢ HasBasis (𝓝 a) (fun s => s ∈ 𝓝 a ∧ IsOpen s) fun x => x",
"tactic": "convert nhds_basis_opens a using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.a\nα : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\na : α\nx✝ : Set α\n⊢ x✝ ∈ 𝓝 a ∧ IsOpen x✝ ↔ a ∈ x✝ ∧ IsOpen x✝",
"tactic": "exact and_congr_left_iff.2 IsOpen.mem_nhds_iff"
}
] |
[
937,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
Algebra.FinitePresentation.mvPolynomial_of_finitePresentation
|
[
{
"state_after": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nhfp : ∃ ι x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\nι : Type v\ninst✝ : Finite ι\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nhfp : FinitePresentation R A\nι : Type v\ninst✝ : Finite ι\n⊢ FinitePresentation R (MvPolynomial ι A)",
"tactic": "rw [iff_quotient_mvPolynomial'] at hfp ⊢"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nhfp : ∃ ι x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)\nι : Type v\ninst✝ : Finite ι\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"tactic": "obtain ⟨(ι' : Type v), _, f, hf_surj, hf_ker⟩ := hfp"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"tactic": "let g := (MvPolynomial.mapAlgHom f).comp (MvPolynomial.sumAlgEquiv R ι ι').toAlgHom"
},
{
"state_after": "case intro.intro.intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"state_before": "case intro.intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"tactic": "cases nonempty_fintype (Sum ι ι')"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG (RingHom.ker ↑↑(MvPolynomial.sumAlgEquiv R ι ι'))\n\ncase intro.intro.intro.intro.intro.refine_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG (RingHom.ker ↑(MvPolynomial.mapAlgHom f))",
"state_before": "case intro.intro.intro.intro.intro\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ ∃ ι_1 x f, Surjective ↑f ∧ Ideal.FG (RingHom.ker ↑f)",
"tactic": "refine\n ⟨Sum ι ι', by infer_instance, g,\n (MvPolynomial.map_surjective f.toRingHom hf_surj).comp (AlgEquiv.surjective _),\n Ideal.fg_ker_comp _ _ ?_ ?_ (AlgEquiv.surjective _)⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Fintype (ι ⊕ ι')",
"tactic": "infer_instance"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG ⊥",
"state_before": "case intro.intro.intro.intro.intro.refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG (RingHom.ker ↑↑(MvPolynomial.sumAlgEquiv R ι ι'))",
"tactic": "erw [RingHom.ker_coe_equiv (MvPolynomial.sumAlgEquiv R ι ι').toRingEquiv]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.refine_1\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG ⊥",
"tactic": "exact Submodule.fg_bot"
},
{
"state_after": "case intro.intro.intro.intro.intro.refine_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG (Ideal.map MvPolynomial.C (RingHom.ker ↑f))",
"state_before": "case intro.intro.intro.intro.intro.refine_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG (RingHom.ker ↑(MvPolynomial.mapAlgHom f))",
"tactic": "rw [AlgHom.toRingHom_eq_coe, MvPolynomial.mapAlgHom_coe_ringHom, MvPolynomial.ker_map]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.refine_2\nR : Type w₁\nA : Type w₂\nB : Type w₃\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nι : Type v\ninst✝ : Finite ι\nι' : Type v\nw✝ : Fintype ι'\nf : MvPolynomial ι' R →ₐ[R] A\nhf_surj : Surjective ↑f\nhf_ker : Ideal.FG (RingHom.ker ↑f)\ng : MvPolynomial (ι ⊕ ι') R →ₐ[R] MvPolynomial ι A :=\n AlgHom.comp (MvPolynomial.mapAlgHom f) ↑(MvPolynomial.sumAlgEquiv R ι ι')\nval✝ : Fintype (ι ⊕ ι')\n⊢ Ideal.FG (Ideal.map MvPolynomial.C (RingHom.ker ↑f))",
"tactic": "exact hf_ker.map MvPolynomial.C"
}
] |
[
208,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
189,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.star_fuzzy_zero
|
[
{
"state_after": "⊢ ∃ j, ∀ (i : LeftMoves (moveRight star j)), moveLeft (moveRight star j) i ⧏ 0",
"state_before": "⊢ star ⧏ 0",
"tactic": "rw [lf_zero]"
},
{
"state_after": "⊢ ∀ (i : LeftMoves (moveRight star default)), moveLeft (moveRight star default) i ⧏ 0",
"state_before": "⊢ ∃ j, ∀ (i : LeftMoves (moveRight star j)), moveLeft (moveRight star j) i ⧏ 0",
"tactic": "use default"
},
{
"state_after": "no goals",
"state_before": "⊢ ∀ (i : LeftMoves (moveRight star default)), moveLeft (moveRight star default) i ⧏ 0",
"tactic": "rintro ⟨⟩"
},
{
"state_after": "⊢ ∃ i, ∀ (j : RightMoves (moveLeft star i)), 0 ⧏ moveRight (moveLeft star i) j",
"state_before": "⊢ 0 ⧏ star",
"tactic": "rw [zero_lf]"
},
{
"state_after": "⊢ ∀ (j : RightMoves (moveLeft star default)), 0 ⧏ moveRight (moveLeft star default) j",
"state_before": "⊢ ∃ i, ∀ (j : RightMoves (moveLeft star i)), 0 ⧏ moveRight (moveLeft star i) j",
"tactic": "use default"
},
{
"state_after": "no goals",
"state_before": "⊢ ∀ (j : RightMoves (moveLeft star default)), 0 ⧏ moveRight (moveLeft star default) j",
"tactic": "rintro ⟨⟩"
}
] |
[
1873,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1865,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.getElem?_eq_getElem
|
[] |
[
48,
18
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
47,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.coeff_mul_monomial
|
[] |
[
708,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.shiftl_sub
|
[] |
[
430,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.quot_mk_to_coe'
|
[] |
[
50,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/RingTheory/DiscreteValuationRing/Basic.lean
|
DiscreteValuationRing.addVal_mul
|
[] |
[
439,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isBigO_const_const_iff
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.276396\nE : Type ?u.276399\nF : Type ?u.276402\nG : Type ?u.276405\nE' : Type ?u.276408\nF' : Type ?u.276411\nG' : Type ?u.276414\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.276423\nR : Type ?u.276426\nR' : Type ?u.276429\n𝕜 : Type ?u.276432\n𝕜' : Type ?u.276435\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nc : E''\nl : Filter α\ninst✝ : NeBot l\n⊢ ((fun _x => c) =O[l] fun _x => 0) ↔ 0 = 0 → c = 0\n\ncase inr\nα : Type u_1\nβ : Type ?u.276396\nE : Type ?u.276399\nF : Type ?u.276402\nG : Type ?u.276405\nE' : Type ?u.276408\nF' : Type ?u.276411\nG' : Type ?u.276414\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.276423\nR : Type ?u.276426\nR' : Type ?u.276429\n𝕜 : Type ?u.276432\n𝕜' : Type ?u.276435\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc✝ c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nc : E''\nc' : F''\nl : Filter α\ninst✝ : NeBot l\nhc' : c' ≠ 0\n⊢ ((fun _x => c) =O[l] fun _x => c') ↔ c' = 0 → c = 0",
"state_before": "α : Type u_1\nβ : Type ?u.276396\nE : Type ?u.276399\nF : Type ?u.276402\nG : Type ?u.276405\nE' : Type ?u.276408\nF' : Type ?u.276411\nG' : Type ?u.276414\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.276423\nR : Type ?u.276426\nR' : Type ?u.276429\n𝕜 : Type ?u.276432\n𝕜' : Type ?u.276435\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc✝ c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nc : E''\nc' : F''\nl : Filter α\ninst✝ : NeBot l\n⊢ ((fun _x => c) =O[l] fun _x => c') ↔ c' = 0 → c = 0",
"tactic": "rcases eq_or_ne c' 0 with (rfl | hc')"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.276396\nE : Type ?u.276399\nF : Type ?u.276402\nG : Type ?u.276405\nE' : Type ?u.276408\nF' : Type ?u.276411\nG' : Type ?u.276414\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.276423\nR : Type ?u.276426\nR' : Type ?u.276429\n𝕜 : Type ?u.276432\n𝕜' : Type ?u.276435\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nc : E''\nl : Filter α\ninst✝ : NeBot l\n⊢ ((fun _x => c) =O[l] fun _x => 0) ↔ 0 = 0 → c = 0",
"tactic": "simp [EventuallyEq]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.276396\nE : Type ?u.276399\nF : Type ?u.276402\nG : Type ?u.276405\nE' : Type ?u.276408\nF' : Type ?u.276411\nG' : Type ?u.276414\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.276423\nR : Type ?u.276426\nR' : Type ?u.276429\n𝕜 : Type ?u.276432\n𝕜' : Type ?u.276435\ninst✝¹³ : Norm E\ninst✝¹² : Norm F\ninst✝¹¹ : Norm G\ninst✝¹⁰ : SeminormedAddCommGroup E'\ninst✝⁹ : SeminormedAddCommGroup F'\ninst✝⁸ : SeminormedAddCommGroup G'\ninst✝⁷ : NormedAddCommGroup E''\ninst✝⁶ : NormedAddCommGroup F''\ninst✝⁵ : NormedAddCommGroup G''\ninst✝⁴ : SeminormedRing R\ninst✝³ : SeminormedRing R'\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedField 𝕜'\nc✝ c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl✝ l' : Filter α\nc : E''\nc' : F''\nl : Filter α\ninst✝ : NeBot l\nhc' : c' ≠ 0\n⊢ ((fun _x => c) =O[l] fun _x => c') ↔ c' = 0 → c = 0",
"tactic": "simp [hc', isBigO_const_const _ hc']"
}
] |
[
1261,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1257,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.symm_comp_self
|
[] |
[
838,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
837,
1
] |
Mathlib/CategoryTheory/Monoidal/Functor.lean
|
CategoryTheory.MonoidalFunctor.μ_inv_hom_id
|
[] |
[
246,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Control/Fold.lean
|
Traversable.foldl_toList
|
[
{
"state_after": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nf : α → β → α\nxs : t β\nx : α\n⊢ foldl f x xs = unop (↑(Foldl.ofFreeMonoid f) (↑FreeMonoid.ofList (toList xs))) x",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nf : α → β → α\nxs : t β\nx : α\n⊢ foldl f x xs = List.foldl f x (toList xs)",
"tactic": "rw [← FreeMonoid.toList_ofList (toList xs), ← foldl.unop_ofFreeMonoid]"
},
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nt : Type u → Type u\ninst✝¹ : Traversable t\ninst✝ : IsLawfulTraversable t\nf : α → β → α\nxs : t β\nx : α\n⊢ foldl f x xs = unop (↑(Foldl.ofFreeMonoid f) (↑FreeMonoid.ofList (toList xs))) x",
"tactic": "simp only [foldl, toList_spec, foldMap_hom_free, foldl.ofFreeMonoid_comp_of, Foldl.get,\n FreeMonoid.ofList_toList]"
}
] |
[
360,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean
|
MeasureTheory.SimpleFunc.integrable_approxOn
|
[
{
"state_after": "α : Type ?u.939616\nβ : Type u_2\nι : Type ?u.939622\nE : Type u_1\nF : Type ?u.939628\n𝕜 : Type ?u.939631\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : BorelSpace E\nf : β → E\nμ : Measure β\nfmeas : Measurable f\nhf : Memℒp f 1\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhi₀ : Memℒp (fun x => y₀) 1\nn : ℕ\n⊢ Memℒp (↑(approxOn f fmeas s y₀ h₀ n)) 1",
"state_before": "α : Type ?u.939616\nβ : Type u_2\nι : Type ?u.939622\nE : Type u_1\nF : Type ?u.939628\n𝕜 : Type ?u.939631\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : BorelSpace E\nf : β → E\nμ : Measure β\nfmeas : Measurable f\nhf : Integrable f\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhi₀ : Integrable fun x => y₀\nn : ℕ\n⊢ Integrable ↑(approxOn f fmeas s y₀ h₀ n)",
"tactic": "rw [← memℒp_one_iff_integrable] at hf hi₀⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.939616\nβ : Type u_2\nι : Type ?u.939622\nE : Type u_1\nF : Type ?u.939628\n𝕜 : Type ?u.939631\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : BorelSpace E\nf : β → E\nμ : Measure β\nfmeas : Measurable f\nhf : Memℒp f 1\ns : Set E\ny₀ : E\nh₀ : y₀ ∈ s\ninst✝ : SeparableSpace ↑s\nhi₀ : Memℒp (fun x => y₀) 1\nn : ℕ\n⊢ Memℒp (↑(approxOn f fmeas s y₀ h₀ n)) 1",
"tactic": "exact memℒp_approxOn fmeas hf h₀ hi₀ n"
}
] |
[
252,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean
|
CategoryTheory.FreeMonoidalCategory.normalizeIsoApp_unitor
|
[] |
[
207,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
derivWithin_subset
|
[] |
[
499,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.IsImage.union
|
[] |
[
444,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
11
] |
Mathlib/Data/Part.lean
|
Part.mem_some
|
[] |
[
168,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.continuousOn_sub_left
|
[
{
"state_after": "α : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ ContinuousOn ((fun p => p.fst - p.snd) ∘ fun x => (a, x)) {x | x ≠ ⊤}",
"state_before": "α : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ ContinuousOn (fun x => a - x) {x | x ≠ ⊤}",
"tactic": "rw [show (fun x => a - x) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨a, x⟩ by rfl]"
},
{
"state_after": "α : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ MapsTo (fun b => (a, b)) {x | x ≠ ⊤} {p | p ≠ (⊤, ⊤)}",
"state_before": "α : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ ContinuousOn ((fun p => p.fst - p.snd) ∘ fun x => (a, x)) {x | x ≠ ⊤}",
"tactic": "apply ContinuousOn.comp continuousOn_sub (Continuous.continuousOn (Continuous.Prod.mk a))"
},
{
"state_after": "case refl\nα : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nh : ⊤ ∈ {x | x ≠ ⊤}\n⊢ False",
"state_before": "α : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ MapsTo (fun b => (a, b)) {x | x ≠ ⊤} {p | p ≠ (⊤, ⊤)}",
"tactic": "rintro _ h (_ | _)"
},
{
"state_after": "no goals",
"state_before": "case refl\nα : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nh : ⊤ ∈ {x | x ≠ ⊤}\n⊢ False",
"tactic": "exact h none_eq_top"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.110099\nβ : Type ?u.110102\nγ : Type ?u.110105\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\na : ℝ≥0∞\n⊢ (fun x => a - x) = (fun p => p.fst - p.snd) ∘ fun x => (a, x)",
"tactic": "rfl"
}
] |
[
462,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.image_Ici
|
[] |
[
76,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.inductionOn₂
|
[] |
[
135,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/GroupTheory/Sylow.lean
|
not_dvd_index_sylow
|
[
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\n⊢ ¬p ∣ index ↑P",
"state_before": "p : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\n⊢ ¬p ∣ index ↑P",
"tactic": "cases nonempty_fintype (Sylow p G)"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\n⊢ ¬p ∣ index ↑P",
"tactic": "rw [← relindex_mul_index le_normalizer, ← card_sylow_eq_index_normalizer]"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\nthis : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"tactic": "haveI : (P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer).Normal :=\n Subgroup.normal_in_normalizer"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\nthis✝ : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\nthis : FiniteIndex ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\nthis : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"tactic": "haveI : FiniteIndex ↑(P.subtype le_normalizer : Subgroup (P : Subgroup G).normalizer) := ⟨hP⟩"
},
{
"state_after": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nval✝ : Fintype (Sylow p G)\nthis✝ : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\nthis : FiniteIndex ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\nhP : ¬p ∣ index ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nhP : relindex (↑P) (normalizer ↑P) ≠ 0\nval✝ : Fintype (Sylow p G)\nthis✝ : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\nthis : FiniteIndex ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"tactic": "replace hP := not_dvd_index_sylow' (P.subtype le_normalizer)"
},
{
"state_after": "no goals",
"state_before": "case intro\np : ℕ\nG : Type u_1\ninst✝¹ : Group G\nhp : Fact (Nat.Prime p)\ninst✝ : Finite (Sylow p G)\nP : Sylow p G\nval✝ : Fintype (Sylow p G)\nthis✝ : Normal ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\nthis : FiniteIndex ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\nhP : ¬p ∣ index ↑(Sylow.subtype P (_ : ↑P ≤ normalizer ↑P))\n⊢ ¬p ∣ relindex (↑P) (normalizer ↑P) * card (Sylow p G)",
"tactic": "exact hp.1.not_dvd_mul hP (not_dvd_card_sylow p G)"
}
] |
[
456,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Topology/Inseparable.lean
|
Inseparable.specializes'
|
[] |
[
279,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelEmbedding.isIrrefl
|
[] |
[
339,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
11
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
MulSalemSpencer.prod
|
[] |
[
105,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.mem_map
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.105727\nι : Sort ?u.105730\nκ : ι → Sort ?u.105735\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf✝ : α ≃o β\ns t : LowerSet α\na : α\nb✝ : β\nf : α ≃o β\nb : β\n⊢ b ∈ ↑(OrderIso.symm (map (OrderIso.symm f))) s ↔ ↑(OrderIso.symm f) b ∈ s",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.105727\nι : Sort ?u.105730\nκ : ι → Sort ?u.105735\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf✝ : α ≃o β\ns t : LowerSet α\na : α\nb✝ : β\nf : α ≃o β\nb : β\n⊢ b ∈ ↑(map f) s ↔ ↑(OrderIso.symm f) b ∈ s",
"tactic": "rw [← f.symm_symm, ← symm_map, f.symm_symm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.105727\nι : Sort ?u.105730\nκ : ι → Sort ?u.105735\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf✝ : α ≃o β\ns t : LowerSet α\na : α\nb✝ : β\nf : α ≃o β\nb : β\n⊢ b ∈ ↑(OrderIso.symm (map (OrderIso.symm f))) s ↔ ↑(OrderIso.symm f) b ∈ s",
"tactic": "rfl"
}
] |
[
1022,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1020,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.refl_trans
|
[] |
[
865,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/Order/CompleteLattice.lean
|
sSup_image2
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_1\nβ₂ : Type ?u.175570\nγ : Type u_2\nι : Sort ?u.175576\nι' : Sort ?u.175579\nκ : ι → Sort ?u.175584\nκ' : ι' → Sort ?u.175589\ninst✝ : CompleteLattice α\nf✝ g s✝ t✝ : ι → α\na b : α\nf : β → γ → α\ns : Set β\nt : Set γ\n⊢ sSup (image2 f s t) = ⨆ (a : β) (_ : a ∈ s) (b : γ) (_ : b ∈ t), f a b",
"tactic": "rw [← image_prod, sSup_image, biSup_prod]"
}
] |
[
1613,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1612,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.is_real_TFAE
|
[
{
"state_after": "case tfae_1_to_4\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\n⊢ ↑(starRingEnd K) z = z → ↑im z = 0\n\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"state_before": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "tfae_have 1 → 4"
},
{
"state_after": "case tfae_4_to_3\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\n⊢ ↑im z = 0 → ↑(↑re z) = z\n\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"state_before": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "tfae_have 4 → 3"
},
{
"state_after": "case tfae_3_to_2\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\n⊢ ↑(↑re z) = z → ∃ r, ↑r = z\n\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"state_before": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "tfae_have 3 → 2"
},
{
"state_after": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"state_before": "case tfae_3_to_2\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\n⊢ ↑(↑re z) = z → ∃ r, ↑r = z\n\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "exact fun h => ⟨_, h⟩"
},
{
"state_after": "case tfae_2_to_1\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\n⊢ (∃ r, ↑r = z) → ↑(starRingEnd K) z = z\n\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\ntfae_2_to_1 : (∃ r, ↑r = z) → ↑(starRingEnd K) z = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"state_before": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "tfae_have 2 → 1"
},
{
"state_after": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\ntfae_2_to_1 : (∃ r, ↑r = z) → ↑(starRingEnd K) z = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"state_before": "case tfae_2_to_1\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\n⊢ (∃ r, ↑r = z) → ↑(starRingEnd K) z = z\n\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\ntfae_2_to_1 : (∃ r, ↑r = z) → ↑(starRingEnd K) z = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "exact fun ⟨r, hr⟩ => hr ▸ conj_ofReal _"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\ntfae_4_to_3 : ↑im z = 0 → ↑(↑re z) = z\ntfae_3_to_2 : ↑(↑re z) = z → ∃ r, ↑r = z\ntfae_2_to_1 : (∃ r, ↑r = z) → ↑(starRingEnd K) z = z\n⊢ TFAE [↑(starRingEnd K) z = z, ∃ r, ↑r = z, ↑(↑re z) = z, ↑im z = 0]",
"tactic": "tfae_finish"
},
{
"state_after": "case tfae_1_to_4\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\nh : ↑(starRingEnd K) z = z\n⊢ ↑im z = 0",
"state_before": "case tfae_1_to_4\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\n⊢ ↑(starRingEnd K) z = z → ↑im z = 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case tfae_1_to_4\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\nh : ↑(starRingEnd K) z = z\n⊢ ↑im z = 0",
"tactic": "rw [← @ofReal_inj K, im_eq_conj_sub, h, sub_self, MulZeroClass.mul_zero, zero_div,\n ofReal_zero]"
},
{
"state_after": "case tfae_4_to_3\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\nh : ↑im z = 0\n⊢ ↑(↑re z) = z",
"state_before": "case tfae_4_to_3\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\n⊢ ↑im z = 0 → ↑(↑re z) = z",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case tfae_4_to_3\nK : Type u_1\nE : Type ?u.3632144\ninst✝ : IsROrC K\nz : K\ntfae_1_to_4 : ↑(starRingEnd K) z = z → ↑im z = 0\nh : ↑im z = 0\n⊢ ↑(↑re z) = z",
"tactic": "conv_rhs => rw [← re_add_im z, h, ofReal_zero, MulZeroClass.zero_mul, add_zero]"
}
] |
[
421,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
411,
1
] |
Mathlib/CategoryTheory/Monoidal/Braided.lean
|
CategoryTheory.braiding_leftUnitor_aux₂
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "coherence"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (𝟙 (𝟙_ C ⊗ X ⊗ 𝟙_ C) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv) ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).hom) ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "(slice_rhs 3 4 => rw [← id_tensor_comp, Iso.hom_inv_id, tensor_id])"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (𝟙 (𝟙_ C ⊗ X ⊗ 𝟙_ C) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv) ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "rw [id_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (𝟙 (𝟙_ C ⊗ X ⊗ 𝟙_ C) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv) ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ (𝟙_ C) X (𝟙_ C)).hom ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).hom) ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "slice_rhs 3 4 => rw [← id_tensor_comp, Iso.hom_inv_id, tensor_id]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((((α_ X (𝟙_ C) (𝟙_ C)).hom ≫ (β_ X (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (α_ (𝟙_ C) (𝟙_ C) X).hom) ≫ (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).inv) ≫\n ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫\n (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫\n (α_ tensorUnit' (𝟙_ C) X).hom ≫\n (𝟙 tensorUnit' ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ tensorUnit' X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).hom) ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫\n (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫\n (α_ tensorUnit' (𝟙_ C) X).hom ≫\n (𝟙 tensorUnit' ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ tensorUnit' X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "(slice_lhs 1 3 => rw [← hexagon_forward])"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((((α_ X (𝟙_ C) (𝟙_ C)).hom ≫ (β_ X (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (α_ (𝟙_ C) (𝟙_ C) X).hom) ≫ (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).inv) ≫\n ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫\n (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫\n (α_ tensorUnit' (𝟙_ C) X).hom ≫\n (𝟙 tensorUnit' ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ tensorUnit' X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "simp only [assoc]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((((α_ X (𝟙_ C) (𝟙_ C)).hom ≫ (β_ X (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (α_ (𝟙_ C) (𝟙_ C) X).hom) ≫ (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).inv) ≫\n ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫\n (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫\n (α_ tensorUnit' (𝟙_ C) X).hom ≫\n (𝟙 tensorUnit' ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ tensorUnit' X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫\n (α_ (𝟙_ C) X (𝟙_ C)).hom ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).hom) ≫\n (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ (𝟙_ C) X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫\n (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫\n (α_ tensorUnit' (𝟙_ C) X).hom ≫\n (𝟙 tensorUnit' ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ tensorUnit' X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C))",
"tactic": "slice_lhs 1 3 => rw [← hexagon_forward]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫\n (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫\n (α_ tensorUnit' (𝟙_ C) X).hom ≫\n (𝟙 tensorUnit' ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ tensorUnit' X (𝟙_ C)).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv",
"tactic": "rw [braiding_leftUnitor_aux₁]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ ((𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).inv =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom ≫ (β_ X (𝟙_ C)).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom ≫ (β_ X (𝟙_ C)).inv",
"tactic": "(slice_lhs 2 3 => rw [← braiding_naturality])"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ ((𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).inv =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom ≫ (β_ X (𝟙_ C)).inv",
"tactic": "simp only [assoc]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ ((𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).inv =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom ≫ (β_ X (𝟙_ C)).inv",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ (β_ X (tensorUnit' ⊗ 𝟙_ C)).hom ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom ≫ (β_ X (𝟙_ C)).inv",
"tactic": "slice_lhs 2 3 => rw [← braiding_naturality]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) ≫ (β_ X (𝟙_ C)).hom ≫ (β_ X (𝟙_ C)).inv =\n (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom)",
"tactic": "rw [Iso.hom_inv_id, comp_id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\n⊢ (α_ X tensorUnit' (𝟙_ C)).hom ≫ (𝟙 X ⊗ (λ_ (𝟙_ C)).hom) = (ρ_ X).hom ⊗ 𝟙 (𝟙_ C)",
"tactic": "rw [triangle]"
}
] |
[
167,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/MvPolynomial/Expand.lean
|
MvPolynomial.expand_one_apply
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type ?u.111976\nR : Type u_2\nS : Type ?u.111982\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : MvPolynomial σ R\n⊢ ↑(expand 1) f = f",
"tactic": "simp only [expand, pow_one, eval₂Hom_eq_bind₂, bind₂_C_left, RingHom.toMonoidHom_eq_coe,\n RingHom.coe_monoidHom_id, AlgHom.coe_mk, RingHom.coe_mk, MonoidHom.id_apply]"
}
] |
[
60,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.