file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
---|---|---|---|---|---|---|
Mathlib/Order/SuccPred/Basic.lean
|
exists_pred_iterate_or
|
[] |
[
1458,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1457,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.coind_bijective
|
[] |
[
198,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.sub_mem_ball
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.882649\nR' : Type ?u.882652\n𝕜 : Type u_1\n𝕜₂ : Type ?u.882658\n𝕜₃ : Type ?u.882661\n𝕝 : Type ?u.882664\nE : Type u_2\nE₂ : Type ?u.882670\nE₃ : Type ?u.882673\nF : Type ?u.882676\nG : Type ?u.882679\nι : Type ?u.882682\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : SMul 𝕜 E\np✝ : Seminorm 𝕜 E\nx y✝ : E\nr✝ : ℝ\np : Seminorm 𝕜 E\nx₁ x₂ y : E\nr : ℝ\n⊢ x₁ - x₂ ∈ ball p y r ↔ x₁ ∈ ball p (x₂ + y) r",
"tactic": "simp_rw [mem_ball, sub_sub]"
}
] |
[
768,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/MeasureTheory/Measure/Regular.lean
|
Set.exists_isOpen_lt_add
|
[] |
[
264,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Algebra/GCDMonoid/Multiset.lean
|
Multiset.lcm_dedup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns : Multiset α\n⊢ lcm (dedup 0) = lcm 0",
"tactic": "simp"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\nh : a ∈ s\n⊢ lcm s = GCDMonoid.lcm a (lcm s)",
"state_before": "α : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\n⊢ lcm (dedup (a ::ₘ s)) = lcm (a ::ₘ s)",
"tactic": "by_cases h : a ∈ s <;> simp [IH, h]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\nh : a ∈ s\n⊢ fold GCDMonoid.lcm 1 s = GCDMonoid.lcm a (fold GCDMonoid.lcm 1 s)",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\nh : a ∈ s\n⊢ lcm s = GCDMonoid.lcm a (lcm s)",
"tactic": "unfold lcm"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\nh : a ∈ s\n⊢ GCDMonoid.lcm a (fold GCDMonoid.lcm 1 (erase s a)) = GCDMonoid.lcm (↑normalize a) (fold GCDMonoid.lcm 1 (erase s a))",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\nh : a ∈ s\n⊢ fold GCDMonoid.lcm 1 s = GCDMonoid.lcm a (fold GCDMonoid.lcm 1 s)",
"tactic": "rw [← cons_erase h, fold_cons_left, ← lcm_assoc, lcm_same]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝² : CancelCommMonoidWithZero α\ninst✝¹ : NormalizedGCDMonoid α\ninst✝ : DecidableEq α\ns✝ : Multiset α\na : α\ns : Multiset α\nIH : lcm (dedup s) = lcm s\nh : a ∈ s\n⊢ GCDMonoid.lcm a (fold GCDMonoid.lcm 1 (erase s a)) = GCDMonoid.lcm (↑normalize a) (fold GCDMonoid.lcm 1 (erase s a))",
"tactic": "apply lcm_eq_of_associated_left (associated_normalize _)"
}
] |
[
103,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
IsLUB.ciSup_eq
|
[] |
[
537,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/NumberTheory/Fermat4.lean
|
not_fermat_4
|
[
{
"state_after": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 4\n⊢ False",
"state_before": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ a ^ 4 + b ^ 4 ≠ c ^ 4",
"tactic": "intro heq"
},
{
"state_after": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 4\n⊢ a ^ 4 + b ^ 4 = (c ^ 2) ^ 2",
"state_before": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 4\n⊢ False",
"tactic": "apply @not_fermat_42 _ _ (c ^ 2) ha hb"
},
{
"state_after": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 4\n⊢ c ^ 4 = (c ^ 2) ^ 2",
"state_before": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 4\n⊢ a ^ 4 + b ^ 4 = (c ^ 2) ^ 2",
"tactic": "rw [heq]"
},
{
"state_after": "no goals",
"state_before": "a b c : ℤ\nha : a ≠ 0\nhb : b ≠ 0\nheq : a ^ 4 + b ^ 4 = c ^ 4\n⊢ c ^ 4 = (c ^ 2) ^ 2",
"tactic": "ring"
}
] |
[
316,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.le_functorPushforward_pullback
|
[] |
[
681,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_eq_mul
|
[] |
[
934,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
933,
1
] |
Std/Data/List/Lemmas.lean
|
List.Fin.exists_iff
|
[] |
[
537,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
536,
1
] |
Mathlib/GroupTheory/Submonoid/Inverses.lean
|
Submonoid.fromLeftInv_mul
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nx : { x // x ∈ leftInv S }\n⊢ ↑(fromLeftInv S x) * ↑x = 1",
"tactic": "rw [mul_comm, mul_fromLeftInv]"
}
] |
[
126,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
|
BoxIntegral.IntegrationParams.hasBasis_toFilteriUnion_top
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁ π₂ : TaggedPrepartition I✝\nl✝ l₁ l₂ l : IntegrationParams\nI : Box ι\n⊢ HasBasis (toFilteriUnion l I ⊤) (fun r => ∀ (c : ℝ≥0), RCond l (r c)) fun r =>\n {π | ∃ c, MemBaseSet l I c (r c) π ∧ IsPartition π}",
"tactic": "simpa only [TaggedPrepartition.isPartition_iff_iUnion_eq, Prepartition.iUnion_top] using\n l.hasBasis_toFilteriUnion I ⊤"
}
] |
[
497,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
ChainComplex.of_d_ne
|
[
{
"state_after": "ι : Type ?u.77771\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X (n + 1) ⟶ X n\nsq : ∀ (n : α), d (n + 1) ≫ d n = 0\ni j : α\nh : i ≠ j + 1\n⊢ (if h : i = j + 1 then eqToHom (_ : X i = X (j + 1)) ≫ d j else 0) = 0",
"state_before": "ι : Type ?u.77771\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X (n + 1) ⟶ X n\nsq : ∀ (n : α), d (n + 1) ≫ d n = 0\ni j : α\nh : i ≠ j + 1\n⊢ HomologicalComplex.d (of X d sq) i j = 0",
"tactic": "dsimp [of]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.77771\nV : Type u\ninst✝⁴ : Category V\ninst✝³ : HasZeroMorphisms V\nα : Type u_1\ninst✝² : AddRightCancelSemigroup α\ninst✝¹ : One α\ninst✝ : DecidableEq α\nX : α → V\nd : (n : α) → X (n + 1) ⟶ X n\nsq : ∀ (n : α), d (n + 1) ≫ d n = 0\ni j : α\nh : i ≠ j + 1\n⊢ (if h : i = j + 1 then eqToHom (_ : X i = X (j + 1)) ≫ d j else 0) = 0",
"tactic": "rw [dif_neg h]"
}
] |
[
641,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/Init/Algebra/Order.lean
|
lt_trichotomy
|
[] |
[
322,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.order_single
|
[] |
[
249,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.lt_translationNumber_of_forall_add_lt
|
[
{
"state_after": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\n⊢ ∃ x, x ∈ Icc 0 1 ∧ ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n\ncase intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ z < τ f",
"state_before": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\n⊢ z < τ f",
"tactic": "obtain ⟨x, -, hx⟩ : ∃ x ∈ Icc (0 : ℝ) 1, ∀ y ∈ Icc (0 : ℝ) 1, f x - x ≤ f y - y"
},
{
"state_after": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ z < τ f",
"state_before": "f g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\n⊢ ∃ x, x ∈ Icc 0 1 ∧ ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n\ncase intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ z < τ f",
"tactic": "exact isCompact_Icc.exists_forall_le (nonempty_Icc.2 zero_le_one)\n (hf.sub continuous_id).continuousOn"
},
{
"state_after": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ ↑f x - x ≤ τ f",
"state_before": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ z < τ f",
"tactic": "refine' lt_of_lt_of_le (lt_sub_iff_add_lt'.2 <| hz x) _"
},
{
"state_after": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ ∀ (x_1 : ℝ), x_1 + (↑f x - x) ≤ ↑f x_1",
"state_before": "case intro.intro\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ ↑f x - x ≤ τ f",
"tactic": "apply le_translationNumber_of_add_le"
},
{
"state_after": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ ∀ (x_1 : ℝ), ↑f x - x ≤ ↑f x_1 - x_1",
"state_before": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ ∀ (x_1 : ℝ), x_1 + (↑f x - x) ≤ ↑f x_1",
"tactic": "simp only [← le_sub_iff_add_le']"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.hz\nf g : CircleDeg1Lift\nhf : Continuous ↑f\nz : ℝ\nhz : ∀ (x : ℝ), x + z < ↑f x\nx : ℝ\nhx : ∀ (y : ℝ), y ∈ Icc 0 1 → ↑f x - x ≤ ↑f y - y\n⊢ ∀ (x_1 : ℝ), ↑f x - x ≤ ↑f x_1 - x_1",
"tactic": "exact f.forall_map_sub_of_Icc _ hx"
}
] |
[
926,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
918,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec₂.nat_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : α → β → σ\nthis :\n ∀ (a : Option α) (b : Option β),\n Option.map (fun p => f p.fst p.snd) (Option.bind a fun a => Option.map (Prod.mk a) b) =\n Option.bind a fun a => Option.map (f a) b\n⊢ Primrec₂ f ↔\n Nat.Primrec (Nat.unpaired fun m n => encode (Option.bind (decode m) fun a => Option.map (f a) (decode n)))",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : α → β → σ\n⊢ Primrec₂ f ↔\n Nat.Primrec (Nat.unpaired fun m n => encode (Option.bind (decode m) fun a => Option.map (f a) (decode n)))",
"tactic": "have :\n ∀ (a : Option α) (b : Option β),\n Option.map (fun p : α × β => f p.1 p.2)\n (Option.bind a fun a : α => Option.map (Prod.mk a) b) =\n Option.bind a fun a => Option.map (f a) b := fun a b => by\n cases a <;> cases b <;> rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : α → β → σ\nthis :\n ∀ (a : Option α) (b : Option β),\n Option.map (fun p => f p.fst p.snd) (Option.bind a fun a => Option.map (Prod.mk a) b) =\n Option.bind a fun a => Option.map (f a) b\n⊢ Primrec₂ f ↔\n Nat.Primrec (Nat.unpaired fun m n => encode (Option.bind (decode m) fun a => Option.map (f a) (decode n)))",
"tactic": "simp [Primrec₂, Primrec, this]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Primcodable α\ninst✝¹ : Primcodable β\ninst✝ : Primcodable σ\nf : α → β → σ\na : Option α\nb : Option β\n⊢ Option.map (fun p => f p.fst p.snd) (Option.bind a fun a => Option.map (Prod.mk a) b) =\n Option.bind a fun a => Option.map (f a) b",
"tactic": "cases a <;> cases b <;> rfl"
}
] |
[
541,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_of_empty'
|
[] |
[
1503,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1502,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.iIndepSet.indep_generateFrom_le
|
[
{
"state_after": "case h.e'_2.h.e'_2\nΩ : Type u_2\nι : Type u_1\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝¹ : LinearOrder ι\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set Ω\nhsm : ∀ (n : ι), MeasurableSet (s n)\nhs : iIndepSet s\ni k : ι\nhk : i < k\n⊢ {s k} = {t | ∃ n, n ∈ {k} ∧ s n = t}",
"state_before": "Ω : Type u_2\nι : Type u_1\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝¹ : LinearOrder ι\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set Ω\nhsm : ∀ (n : ι), MeasurableSet (s n)\nhs : iIndepSet s\ni k : ι\nhk : i < k\n⊢ Indep (generateFrom {s k}) (generateFrom {t | ∃ j, j ≤ i ∧ s j = t})",
"tactic": "convert iIndepSet.indep_generateFrom_of_disjoint hsm hs {k} { j | j ≤ i }\n (Set.disjoint_singleton_left.mpr hk.not_le)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_2\nΩ : Type u_2\nι : Type u_1\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝¹ : LinearOrder ι\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set Ω\nhsm : ∀ (n : ι), MeasurableSet (s n)\nhs : iIndepSet s\ni k : ι\nhk : i < k\n⊢ {s k} = {t | ∃ n, n ∈ {k} ∧ s n = t}",
"tactic": "simp only [Set.mem_singleton_iff, exists_prop, exists_eq_left, Set.setOf_eq_eq_singleton']"
}
] |
[
494,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.tsum_eval
|
[] |
[
682,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
680,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.translationNumber_eq_of_semiconj
|
[] |
[
724,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
722,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
UniformCauchySeqOn.mono
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s'✝ : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ns' : Set α\nhf : UniformCauchySeqOnFilter F p (𝓟 s)\nhss' : s' ⊆ s\n⊢ UniformCauchySeqOnFilter F p (𝓟 s')",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s'✝ : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ns' : Set α\nhf : UniformCauchySeqOn F p s\nhss' : s' ⊆ s\n⊢ UniformCauchySeqOn F p s'",
"tactic": "rw [uniformCauchySeqOn_iff_uniformCauchySeqOnFilter] at hf⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s'✝ : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\ns' : Set α\nhf : UniformCauchySeqOnFilter F p (𝓟 s)\nhss' : s' ⊆ s\n⊢ UniformCauchySeqOnFilter F p (𝓟 s')",
"tactic": "exact hf.mono_right (le_principal_iff.mpr <| mem_principal.mpr hss')"
}
] |
[
493,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
lebesgue_number_lemma_sUnion
|
[
{
"state_after": "α✝ : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.183504\nα : Type u\ninst✝ : UniformSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ (t : Set α), t ∈ c → IsOpen t\nhc₂ : s ⊆ ⋃ (i : ↑c), ↑i\n⊢ ∃ n, n ∈ 𝓤 α ∧ ∀ (x : α), x ∈ s → ∃ t, t ∈ c ∧ ∀ (y : α), (x, y) ∈ n → y ∈ t",
"state_before": "α✝ : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.183504\nα : Type u\ninst✝ : UniformSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ (t : Set α), t ∈ c → IsOpen t\nhc₂ : s ⊆ ⋃₀ c\n⊢ ∃ n, n ∈ 𝓤 α ∧ ∀ (x : α), x ∈ s → ∃ t, t ∈ c ∧ ∀ (y : α), (x, y) ∈ n → y ∈ t",
"tactic": "rw [sUnion_eq_iUnion] at hc₂"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.183504\nα : Type u\ninst✝ : UniformSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ (t : Set α), t ∈ c → IsOpen t\nhc₂ : s ⊆ ⋃ (i : ↑c), ↑i\n⊢ ∃ n, n ∈ 𝓤 α ∧ ∀ (x : α), x ∈ s → ∃ t, t ∈ c ∧ ∀ (y : α), (x, y) ∈ n → y ∈ t",
"tactic": "simpa using lebesgue_number_lemma hs (by simpa) hc₂"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.183504\nα : Type u\ninst✝ : UniformSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ (t : Set α), t ∈ c → IsOpen t\nhc₂ : s ⊆ ⋃ (i : ↑c), ↑i\n⊢ ∀ (i : ↑c), IsOpen ↑i",
"tactic": "simpa"
}
] |
[
1865,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1862,
1
] |
Mathlib/Data/List/Basic.lean
|
List.zipLeft'_nil_left
|
[] |
[
4045,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4044,
1
] |
Mathlib/Analysis/Normed/Field/UnitBall.lean
|
coe_mul_unitSphere
|
[] |
[
167,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Rat.cast_finsupp_sum
|
[] |
[
423,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean
|
LocalizedModule.mul_smul'
|
[
{
"state_after": "case H\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ny : Localization S\nm : LocalizedModule S M\ndata : R × { x // x ∈ S }\n⊢ (Localization.mk data.fst data.snd * y) • m = Localization.mk data.fst data.snd • y • m",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : Localization S\nm : LocalizedModule S M\n⊢ (x * y) • m = x • y • m",
"tactic": "induction' x using Localization.induction_on with data"
},
{
"state_after": "case H.H\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : LocalizedModule S M\ndata data' : R × { x // x ∈ S }\n⊢ (Localization.mk data.fst data.snd * Localization.mk data'.fst data'.snd) • m =\n Localization.mk data.fst data.snd • Localization.mk data'.fst data'.snd • m",
"state_before": "case H\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ny : Localization S\nm : LocalizedModule S M\ndata : R × { x // x ∈ S }\n⊢ (Localization.mk data.fst data.snd * y) • m = Localization.mk data.fst data.snd • y • m",
"tactic": "induction' y using Localization.induction_on with data'"
},
{
"state_after": "case H.H.mk.mk\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : LocalizedModule S M\nr : R\ns : { x // x ∈ S }\nr' : R\ns' : { x // x ∈ S }\n⊢ (Localization.mk (r, s).fst (r, s).snd * Localization.mk (r', s').fst (r', s').snd) • m =\n Localization.mk (r, s).fst (r, s).snd • Localization.mk (r', s').fst (r', s').snd • m",
"state_before": "case H.H\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : LocalizedModule S M\ndata data' : R × { x // x ∈ S }\n⊢ (Localization.mk data.fst data.snd * Localization.mk data'.fst data'.snd) • m =\n Localization.mk data.fst data.snd • Localization.mk data'.fst data'.snd • m",
"tactic": "rcases data, data' with ⟨⟨r, s⟩, ⟨r', s'⟩⟩"
},
{
"state_after": "case H.H.mk.mk.h\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\ns : { x // x ∈ S }\nr' : R\ns' : { x // x ∈ S }\nm : M\nt : { x // x ∈ S }\n⊢ (Localization.mk (r, s).fst (r, s).snd * Localization.mk (r', s').fst (r', s').snd) • mk m t =\n Localization.mk (r, s).fst (r, s).snd • Localization.mk (r', s').fst (r', s').snd • mk m t",
"state_before": "case H.H.mk.mk\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm : LocalizedModule S M\nr : R\ns : { x // x ∈ S }\nr' : R\ns' : { x // x ∈ S }\n⊢ (Localization.mk (r, s).fst (r, s).snd * Localization.mk (r', s').fst (r', s').snd) • m =\n Localization.mk (r, s).fst (r, s).snd • Localization.mk (r', s').fst (r', s').snd • m",
"tactic": "induction' m using LocalizedModule.induction_on with m t"
},
{
"state_after": "no goals",
"state_before": "case H.H.mk.mk.h\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\ns : { x // x ∈ S }\nr' : R\ns' : { x // x ∈ S }\nm : M\nt : { x // x ∈ S }\n⊢ (Localization.mk (r, s).fst (r, s).snd * Localization.mk (r', s').fst (r', s').snd) • mk m t =\n Localization.mk (r, s).fst (r, s).snd • Localization.mk (r', s').fst (r', s').snd • mk m t",
"tactic": "rw [Localization.mk_mul, mk_smul_mk, mk_smul_mk, mk_smul_mk, mul_smul, mul_assoc]"
}
] |
[
351,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
9
] |
Mathlib/GroupTheory/Torsion.lean
|
Monoid.not_isTorsion_iff
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nH : Type ?u.104\ninst✝ : Monoid G\n⊢ ¬IsTorsion G ↔ ∃ g, ¬IsOfFinOrder g",
"tactic": "rw [IsTorsion, not_forall]"
}
] |
[
67,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_natCast
|
[] |
[
214,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_eq_one
|
[
{
"state_after": "no goals",
"state_before": "r : ℝ≥0\n⊢ ↑r = 1 ↔ r = 1",
"tactic": "rw [← NNReal.coe_one, NNReal.coe_eq]"
}
] |
[
221,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
220,
11
] |
Mathlib/FieldTheory/Fixed.lean
|
FixedPoints.minpoly.monic
|
[
{
"state_after": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\n⊢ Polynomial.Monic (prodXSubSmul G F x)",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\n⊢ Polynomial.Monic (minpoly G F x)",
"tactic": "simp only [minpoly, Polynomial.monic_toSubring]"
},
{
"state_after": "no goals",
"state_before": "M : Type u\ninst✝⁵ : Monoid M\nG : Type u\ninst✝⁴ : Group G\nF : Type v\ninst✝³ : Field F\ninst✝² : MulSemiringAction M F\ninst✝¹ : MulSemiringAction G F\nm : M\ninst✝ : Fintype G\nx : F\n⊢ Polynomial.Monic (prodXSubSmul G F x)",
"tactic": "exact prodXSubSmul.monic G F x"
}
] |
[
190,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/MeasureTheory/Constructions/Polish.lean
|
IsOpen.analyticSet_image
|
[
{
"state_after": "α : Type u_2\ninst✝² : TopologicalSpace α\nι : Type ?u.1515\nβ : Type u_1\ninst✝¹ : TopologicalSpace β\ninst✝ : PolishSpace β\ns : Set β\nhs : IsOpen s\nf : β → α\nf_cont : Continuous f\n⊢ AnalyticSet (range fun x => f ↑x)",
"state_before": "α : Type u_2\ninst✝² : TopologicalSpace α\nι : Type ?u.1515\nβ : Type u_1\ninst✝¹ : TopologicalSpace β\ninst✝ : PolishSpace β\ns : Set β\nhs : IsOpen s\nf : β → α\nf_cont : Continuous f\n⊢ AnalyticSet (f '' s)",
"tactic": "rw [image_eq_range]"
},
{
"state_after": "α : Type u_2\ninst✝² : TopologicalSpace α\nι : Type ?u.1515\nβ : Type u_1\ninst✝¹ : TopologicalSpace β\ninst✝ : PolishSpace β\ns : Set β\nhs : IsOpen s\nf : β → α\nf_cont : Continuous f\nthis : PolishSpace ↑s\n⊢ AnalyticSet (range fun x => f ↑x)",
"state_before": "α : Type u_2\ninst✝² : TopologicalSpace α\nι : Type ?u.1515\nβ : Type u_1\ninst✝¹ : TopologicalSpace β\ninst✝ : PolishSpace β\ns : Set β\nhs : IsOpen s\nf : β → α\nf_cont : Continuous f\n⊢ AnalyticSet (range fun x => f ↑x)",
"tactic": "haveI : PolishSpace s := hs.polishSpace"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\ninst✝² : TopologicalSpace α\nι : Type ?u.1515\nβ : Type u_1\ninst✝¹ : TopologicalSpace β\ninst✝ : PolishSpace β\ns : Set β\nhs : IsOpen s\nf : β → α\nf_cont : Continuous f\nthis : PolishSpace ↑s\n⊢ AnalyticSet (range fun x => f ↑x)",
"tactic": "exact analyticSet_range_of_polishSpace (f_cont.comp continuous_subtype_val)"
}
] |
[
106,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.IsDiag.sub
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.6998\nR : Type ?u.7001\nn : Type u_2\nm : Type ?u.7007\ninst✝ : AddGroup α\nA B : Matrix n n α\nha : IsDiag A\nhb : IsDiag B\ni j : n\nh : i ≠ j\n⊢ (A - B) i j = 0",
"state_before": "α : Type u_1\nβ : Type ?u.6998\nR : Type ?u.7001\nn : Type u_2\nm : Type ?u.7007\ninst✝ : AddGroup α\nA B : Matrix n n α\nha : IsDiag A\nhb : IsDiag B\n⊢ IsDiag (A - B)",
"tactic": "intro i j h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6998\nR : Type ?u.7001\nn : Type u_2\nm : Type ?u.7007\ninst✝ : AddGroup α\nA B : Matrix n n α\nha : IsDiag A\nhb : IsDiag B\ni j : n\nh : i ≠ j\n⊢ (A - B) i j = 0",
"tactic": "simp [ha h, hb h]"
}
] |
[
104,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Topology/DiscreteQuotient.lean
|
DiscreteQuotient.proj_bot_eq
|
[] |
[
268,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.coe_sInf
|
[] |
[
989,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
988,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
HasFDerivWithinAt.mono
|
[] |
[
364,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
8
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
continuousOn_zpow
|
[] |
[
516,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/Data/FunLike/Equiv.lean
|
EquivLike.bijective
|
[] |
[
174,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
11
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.inf_limZero
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : abs (↑f j) < ε\nH₂ : abs (↑g j) < ε\n⊢ abs (↑(f ⊓ g) j) < ε",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\n⊢ abs (↑(f ⊓ g) j) < ε",
"tactic": "let ⟨H₁, H₂⟩ := H _ ij"
},
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : -ε < ↑f j ∧ ↑f j < ε\nH₂ : -ε < ↑g j ∧ ↑g j < ε\n⊢ -ε < ↑(f ⊓ g) j ∧ ↑(f ⊓ g) j < ε",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : abs (↑f j) < ε\nH₂ : abs (↑g j) < ε\n⊢ abs (↑(f ⊓ g) j) < ε",
"tactic": "rw [abs_lt] at H₁ H₂⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedField α\nf g : CauSeq α abs\nhf : LimZero f\nhg : LimZero g\nε : α\nε0 : ε > 0\ni : ℕ\nH : ∀ (j : ℕ), j ≥ i → abs (↑f j) < ε ∧ abs (↑g j) < ε\nj : ℕ\nij : j ≥ i\nH₁ : -ε < ↑f j ∧ ↑f j < ε\nH₂ : -ε < ↑g j ∧ ↑g j < ε\n⊢ -ε < ↑(f ⊓ g) j ∧ ↑(f ⊓ g) j < ε",
"tactic": "exact ⟨lt_inf_iff.mpr ⟨H₁.1, H₂.1⟩, inf_lt_iff.mpr (Or.inl H₁.2)⟩"
}
] |
[
861,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
856,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.cpow_zero
|
[
{
"state_after": "no goals",
"state_before": "x : ℂ\n⊢ x ^ 0 = 1",
"tactic": "simp [cpow_def]"
}
] |
[
48,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
48,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.conj_im
|
[] |
[
513,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiff_id
|
[] |
[
174,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.ne_neg_self_iff
|
[
{
"state_after": "no goals",
"state_before": "θ : Angle\n⊢ θ ≠ -θ ↔ θ ≠ 0 ∧ θ ≠ ↑π",
"tactic": "rw [← not_or, ← eq_neg_self_iff.not]"
}
] |
[
219,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.list_tail
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.250075\nγ : Type ?u.250078\nσ : Type ?u.250081\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nl : List α\n⊢ (List.casesOn (id l) [] fun b l_1 => (l, b, l_1).snd.snd) = List.tail l",
"tactic": "cases l <;> rfl"
}
] |
[
1051,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1050,
1
] |
Mathlib/Analysis/SpecialFunctions/Arsinh.lean
|
Differentiable.arsinh
|
[] |
[
273,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_of_neg_pos
|
[
{
"state_after": "no goals",
"state_before": "a : Int\nh : 0 < -a\n⊢ -0 < -a",
"tactic": "rwa [Int.neg_zero]"
}
] |
[
896,
28
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
894,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.equalizer.condition
|
[] |
[
781,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
1
] |
Mathlib/Data/MvPolynomial/Monad.lean
|
MvPolynomial.bind₂_comp_bind₂
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_3\nτ : Type ?u.435135\nR : Type u_1\nS : Type u_2\nT : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : CommSemiring T\nf✝ : σ → MvPolynomial τ R\nf : R →+* MvPolynomial σ S\ng : S →+* MvPolynomial σ T\n⊢ RingHom.comp (bind₂ g) (bind₂ f) = bind₂ (RingHom.comp (bind₂ g) f)",
"tactic": "ext : 2 <;> simp"
}
] |
[
239,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
affineCombination_eq_centerMass
|
[
{
"state_after": "R : Type u_2\nE : Type u_3\nF : Type ?u.126038\nι✝ : Type ?u.126041\nι' : Type ?u.126044\nα : Type ?u.126047\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz : ι✝ → E\nι : Type u_1\nt : Finset ι\np : ι → E\nw : ι → R\nhw₂ : ∑ i in t, w i = 1\n⊢ ∑ i in t, w i • (p i -ᵥ 0) = ∑ i in t, w i • p i",
"state_before": "R : Type u_2\nE : Type u_3\nF : Type ?u.126038\nι✝ : Type ?u.126041\nι' : Type ?u.126044\nα : Type ?u.126047\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz : ι✝ → E\nι : Type u_1\nt : Finset ι\np : ι → E\nw : ι → R\nhw₂ : ∑ i in t, w i = 1\n⊢ ↑(affineCombination R t p) w = centerMass t w p",
"tactic": "rw [affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ w _ hw₂ (0 : E),\n Finset.weightedVSubOfPoint_apply, vadd_eq_add, add_zero, t.centerMass_eq_of_sum_1 _ hw₂]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nE : Type u_3\nF : Type ?u.126038\nι✝ : Type ?u.126041\nι' : Type ?u.126044\nα : Type ?u.126047\ninst✝⁷ : LinearOrderedField R\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : LinearOrderedAddCommGroup α\ninst✝³ : Module R E\ninst✝² : Module R F\ninst✝¹ : Module R α\ninst✝ : OrderedSMul R α\ns : Set E\ni j : ι✝\nc : R\nt✝ : Finset ι✝\nw✝ : ι✝ → R\nz : ι✝ → E\nι : Type u_1\nt : Finset ι\np : ι → E\nw : ι → R\nhw₂ : ∑ i in t, w i = 1\n⊢ ∑ i in t, w i • (p i -ᵥ 0) = ∑ i in t, w i • p i",
"tactic": "simp_rw [vsub_eq_sub, sub_zero]"
}
] |
[
240,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.cond
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.182461\nγ : Type ?u.182464\nσ : Type u_2\ninst✝³ : Primcodable α\ninst✝² : Primcodable β\ninst✝¹ : Primcodable γ\ninst✝ : Primcodable σ\nc : α → Bool\nf g : α → σ\nhc : Computable c\nhf : Computable f\nhg : Computable g\na : α\n⊢ (Nat.casesOn (encode (c a)) (g a) fun b => f (a, b).fst) = bif c a then f a else g a",
"tactic": "cases c a <;> rfl"
}
] |
[
681,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Combinatorics/SetFamily/Intersecting.lean
|
Set.Intersecting.not_mem
|
[] |
[
155,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
src/lean/Init/WF.lean
|
WellFounded.fix_eq
|
[] |
[
77,
27
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
75,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.nth_interleave_left
|
[
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\n⊢ nth (s₁ ⋈ s₂) (succ (succ (2 * n))) = nth s₁ (succ n)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\n⊢ nth (s₁ ⋈ s₂) (2 * (n + 1)) = nth s₁ (n + 1)",
"tactic": "change nth (s₁ ⋈ s₂) (succ (succ (2 * n))) = nth s₁ (succ n)"
},
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\n⊢ nth (tail s₁ ⋈ tail s₂) (2 * n) = nth s₁ (succ n)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\n⊢ nth (s₁ ⋈ s₂) (succ (succ (2 * n))) = nth s₁ (succ n)",
"tactic": "rw [nth_succ, nth_succ, interleave_eq, tail_cons, tail_cons]"
},
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\nthis : n < succ n\n⊢ nth (tail s₁ ⋈ tail s₂) (2 * n) = nth s₁ (succ n)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\n⊢ nth (tail s₁ ⋈ tail s₂) (2 * n) = nth s₁ (succ n)",
"tactic": "have : n < succ n := Nat.lt_succ_self n"
},
{
"state_after": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\nthis : n < succ n\n⊢ nth (tail s₁) n = nth s₁ (succ n)",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\nthis : n < succ n\n⊢ nth (tail s₁ ⋈ tail s₂) (2 * n) = nth s₁ (succ n)",
"tactic": "rw [nth_interleave_left n (tail s₁) (tail s₂)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nδ : Type w\nn : ℕ\ns₁ s₂ : Stream' α\nthis : n < succ n\n⊢ nth (tail s₁) n = nth s₁ (succ n)",
"tactic": "rfl"
}
] |
[
439,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
431,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.disjoint_atBot_principal_Ici
|
[] |
[
96,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Topology/CompactOpen.lean
|
ContinuousMap.gen_empty
|
[] |
[
60,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Std/Classes/Order.lean
|
Std.TransCmp.gt_trans
|
[
{
"state_after": "cmp✝ : ?m.2531 → ?m.2531 → Ordering\ninst✝¹ : TransCmp cmp✝\nx✝ : Sort ?u.2529\ncmp : x✝ → x✝ → Ordering\ninst✝ : TransCmp cmp\nx y z : x✝\nh₁ : cmp y x = Ordering.lt\nh₂ : cmp z y = Ordering.lt\n⊢ cmp z x = Ordering.lt",
"state_before": "cmp✝ : ?m.2531 → ?m.2531 → Ordering\ninst✝¹ : TransCmp cmp✝\nx✝ : Sort ?u.2529\ncmp : x✝ → x✝ → Ordering\ninst✝ : TransCmp cmp\nx y z : x✝\nh₁ : cmp x y = Ordering.gt\nh₂ : cmp y z = Ordering.gt\n⊢ cmp x z = Ordering.gt",
"tactic": "rw [cmp_eq_gt] at h₁ h₂ ⊢"
},
{
"state_after": "no goals",
"state_before": "cmp✝ : ?m.2531 → ?m.2531 → Ordering\ninst✝¹ : TransCmp cmp✝\nx✝ : Sort ?u.2529\ncmp : x✝ → x✝ → Ordering\ninst✝ : TransCmp cmp\nx y z : x✝\nh₁ : cmp y x = Ordering.lt\nh₂ : cmp z y = Ordering.lt\n⊢ cmp z x = Ordering.lt",
"tactic": "exact lt_trans h₂ h₁"
}
] |
[
74,
50
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
73,
1
] |
Mathlib/Data/List/Dedup.lean
|
List.dedup_cons_of_mem
|
[] |
[
56,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toNat_lt_iff_lt_of_lt_aleph0
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\nc d : Cardinal\nhc : c < ℵ₀\nhd : d < ℵ₀\n⊢ ↑toNat c < ↑toNat d ↔ c < d",
"tactic": "rw [← natCast_lt, cast_toNat_of_lt_aleph0 hc, cast_toNat_of_lt_aleph0 hd]"
}
] |
[
1700,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1698,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.mk_le_mk_of_dvd
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.307911\nγ : Type ?u.307914\nδ : Type ?u.307917\ninst✝ : CommMonoid α\na b : α\nx✝ : a ∣ b\nc : α\nhc : b = a * c\n⊢ Associates.mk (a * c) = Associates.mk a * Associates.mk c",
"state_before": "α : Type u_1\nβ : Type ?u.307911\nγ : Type ?u.307914\nδ : Type ?u.307917\ninst✝ : CommMonoid α\na b : α\nx✝ : a ∣ b\nc : α\nhc : b = a * c\n⊢ Associates.mk b = Associates.mk a * Associates.mk c",
"tactic": "simp [hc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.307911\nγ : Type ?u.307914\nδ : Type ?u.307917\ninst✝ : CommMonoid α\na b : α\nx✝ : a ∣ b\nc : α\nhc : b = a * c\n⊢ Associates.mk (a * c) = Associates.mk a * Associates.mk c",
"tactic": "rfl"
}
] |
[
940,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
939,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
isBoundedBilinearMap_compMultilinear
|
[] |
[
486,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.equivProdDfinsupp_smul
|
[] |
[
1693,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1691,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.insert
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List α\np : l₁ ~ l₂\nh : a ∈ l₁\n⊢ List.insert a l₁ ~ List.insert a l₂",
"tactic": "simpa [h, p.subset h] using p"
},
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁✝ l₂✝ : List α\ninst✝ : DecidableEq α\na : α\nl₁ l₂ : List α\np : l₁ ~ l₂\nh : ¬a ∈ l₁\n⊢ List.insert a l₁ ~ List.insert a l₂",
"tactic": "simpa [h, mt p.mem_iff.2 h] using p.cons a"
}
] |
[
965,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
963,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.IndepSets.indep_aux
|
[
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "let μ_inter := μ.restrict t1"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "let ν := μ t1 • μ"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "have h_univ : μ_inter Set.univ = ν Set.univ := by\n rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "haveI : IsFiniteMeasure μ_inter := @Restrict.isFiniteMeasure Ω _ t1 μ ⟨measure_lt_top μ t1⟩"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑(Measure.restrict μ t1) t2 = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "rw [Set.inter_comm, ← Measure.restrict_apply (h2 t2 ht2m)]"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ ↑↑μ_inter t = ↑↑ν t",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\n⊢ ↑↑(Measure.restrict μ t1) t2 = ↑↑μ t1 * ↑↑μ t2",
"tactic": "refine' ext_on_measurableSpace_of_generate_finite m p2 (fun t ht => _) h2 hpm2 hp2 h_univ ht2m"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\nht2 : MeasurableSet t\n⊢ ↑↑μ_inter t = ↑↑ν t",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ ↑↑μ_inter t = ↑↑ν t",
"tactic": "have ht2 : MeasurableSet[m] t := by\n refine' h2 _ _\n rw [hpm2]\n exact measurableSet_generateFrom ht"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\nht2 : MeasurableSet t\n⊢ ↑↑μ (t1 ∩ t) = ↑↑μ t1 • ↑↑μ t",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\nht2 : MeasurableSet t\n⊢ ↑↑μ_inter t = ↑↑ν t",
"tactic": "rw [Measure.restrict_apply ht2, Measure.smul_apply, Set.inter_comm]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\nht2 : MeasurableSet t\n⊢ ↑↑μ (t1 ∩ t) = ↑↑μ t1 • ↑↑μ t",
"tactic": "exact hyp t1 t ht1 ht"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\n⊢ ↑↑μ_inter Set.univ = ↑↑ν Set.univ",
"tactic": "rw [Measure.restrict_apply_univ, Measure.smul_apply, smul_eq_mul, measure_univ, mul_one]"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ MeasurableSet t",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ MeasurableSet t",
"tactic": "refine' h2 _ _"
},
{
"state_after": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ MeasurableSet t",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ MeasurableSet t",
"tactic": "rw [hpm2]"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type ?u.1680679\nm2 m : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\np1 p2 : Set (Set Ω)\nh2 : m2 ≤ m\nhp2 : IsPiSystem p2\nhpm2 : m2 = generateFrom p2\nhyp : IndepSets p1 p2\nt1 t2 : Set Ω\nht1 : t1 ∈ p1\nht2m : MeasurableSet t2\nμ_inter : MeasureTheory.Measure Ω := Measure.restrict μ t1\nν : MeasureTheory.Measure Ω := ↑↑μ t1 • μ\nh_univ : ↑↑μ_inter Set.univ = ↑↑ν Set.univ\nthis : IsFiniteMeasure μ_inter\nt : Set Ω\nht : t ∈ p2\n⊢ MeasurableSet t",
"tactic": "exact measurableSet_generateFrom ht"
}
] |
[
359,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
9
] |
Mathlib/InformationTheory/Hamming.lean
|
Hamming.ofHamming_zero
|
[] |
[
355,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.basisFun_map
|
[] |
[
992,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
990,
1
] |
Mathlib/Data/Option/NAry.lean
|
Option.map₂_none_left
|
[] |
[
61,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Analysis/Convex/Extreme.lean
|
IsExtreme.convex_diff
|
[] |
[
192,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
commutator_mem_commutatorSet
|
[] |
[
271,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.divMonomial_monomial
|
[] |
[
98,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.neg_fuzzy_neg_iff
|
[
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ -x ‖ -y ↔ x ‖ y",
"tactic": "rw [Fuzzy, Fuzzy, neg_lf_neg_iff, neg_lf_neg_iff, and_comm]"
}
] |
[
1342,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1341,
1
] |
Mathlib/Data/Nat/Sqrt.lean
|
Nat.sqrt_add_eq'
|
[] |
[
152,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.of_rat_sub
|
[
{
"state_after": "no goals",
"state_before": "q : ℚ\nx y : ℝ\nh : Irrational (↑q - x)\n⊢ Irrational (↑q + -x)",
"tactic": "simpa only [sub_eq_add_neg] using h"
}
] |
[
292,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Sbtw.trans_left_right
|
[] |
[
841,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean
|
Polynomial.multiset_prod_X_sub_C_coeff_card_pred
|
[
{
"state_after": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = -Multiset.sum t",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = -Multiset.sum t",
"tactic": "nontriviality R"
},
{
"state_after": "case h.e'_2\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) =\n nextCoeff (prod (Multiset.map (fun x => X - ↑C x) t))",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = -Multiset.sum t",
"tactic": "convert multiset_prod_X_sub_C_nextCoeff (by assumption)"
},
{
"state_after": "case h.e'_2\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) =\n if natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0 then 0\n else coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) - 1)",
"state_before": "case h.e'_2\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) =\n nextCoeff (prod (Multiset.map (fun x => X - ↑C x) t))",
"tactic": "rw [nextCoeff]"
},
{
"state_after": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0\n\ncase h.e'_2.inr\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : ¬natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) =\n coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) - 1)",
"state_before": "case h.e'_2\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) =\n if natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0 then 0\n else coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) - 1)",
"tactic": "split_ifs with h"
},
{
"state_after": "case h.e'_2.inr.e_a.e_a\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : ¬natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ ↑Multiset.card t = natDegree (prod (Multiset.map (fun x => X - ↑C x) t))",
"state_before": "case h.e'_2.inr\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : ¬natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) =\n coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) - 1)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\n⊢ Multiset ?m.820577",
"tactic": "assumption"
},
{
"state_after": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : Multiset.sum (Multiset.map (fun x => natDegree x) (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0\n\ncase h.e'_2.inl.h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ ∀ (f : R[X]), (∃ a, a ∈ t ∧ X - ↑C a = f) → Monic f",
"state_before": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0",
"tactic": "rw [natDegree_multiset_prod_of_monic] at h <;> simp only [Multiset.mem_map] at *"
},
{
"state_after": "case h.e'_2.inl.h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ ∀ (f : R[X]), (∃ a, a ∈ t ∧ X - ↑C a = f) → Monic f\n\ncase h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : Multiset.sum (Multiset.map (fun x => natDegree x) (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0",
"state_before": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : Multiset.sum (Multiset.map (fun x => natDegree x) (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0\n\ncase h.e'_2.inl.h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ ∀ (f : R[X]), (∃ a, a ∈ t ∧ X - ↑C a = f) → Monic f",
"tactic": "swap"
},
{
"state_after": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : ∀ (x : ℕ), (∃ a, (∃ a_1, a_1 ∈ t ∧ X - ↑C a_1 = a) ∧ natDegree a = x) → x = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0",
"state_before": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : Multiset.sum (Multiset.map (fun x => natDegree x) (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0",
"tactic": "simp_rw [Multiset.sum_eq_zero_iff, Multiset.mem_map] at h"
},
{
"state_after": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) ≠ 0\n⊢ ∃ x, (∃ a, (∃ a_1, a_1 ∈ t ∧ X - ↑C a_1 = a) ∧ natDegree a = x) ∧ x ≠ 0",
"state_before": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : ∀ (x : ℕ), (∃ a, (∃ a_1, a_1 ∈ t ∧ X - ↑C a_1 = a) ∧ natDegree a = x) → x = 0\n⊢ coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) = 0",
"tactic": "contrapose! h"
},
{
"state_after": "case h.e'_2.inl.intro\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) ≠ 0\nx : R\nhx : x ∈ t\n⊢ ∃ x, (∃ a, (∃ a_1, a_1 ∈ t ∧ X - ↑C a_1 = a) ∧ natDegree a = x) ∧ x ≠ 0",
"state_before": "case h.e'_2.inl\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) ≠ 0\n⊢ ∃ x, (∃ a, (∃ a_1, a_1 ∈ t ∧ X - ↑C a_1 = a) ∧ natDegree a = x) ∧ x ≠ 0",
"tactic": "obtain ⟨x, hx⟩ := card_pos_iff_exists_mem.mp ht"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.inl.intro\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : coeff (prod (Multiset.map (fun x => X - ↑C x) t)) (↑Multiset.card t - 1) ≠ 0\nx : R\nhx : x ∈ t\n⊢ ∃ x, (∃ a, (∃ a_1, a_1 ∈ t ∧ X - ↑C a_1 = a) ∧ natDegree a = x) ∧ x ≠ 0",
"tactic": "exact ⟨_, ⟨_, ⟨x, hx, rfl⟩, natDegree_X_sub_C _⟩, one_ne_zero⟩"
},
{
"state_after": "case h.e'_2.inl.h.intro.intro\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\nw✝ : R\nleft✝ : w✝ ∈ t\n⊢ Monic (X - ↑C w✝)",
"state_before": "case h.e'_2.inl.h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ ∀ (f : R[X]), (∃ a, a ∈ t ∧ X - ↑C a = f) → Monic f",
"tactic": "rintro _ ⟨_, _, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.inl.h.intro.intro\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\nw✝ : R\nleft✝ : w✝ ∈ t\n⊢ Monic (X - ↑C w✝)",
"tactic": "apply monic_X_sub_C"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.inr.e_a.e_a.h\nR : Type u\nι : Type w\ns : Finset ι\ninst✝ : CommRing R\nt : Multiset R\nht : 0 < ↑Multiset.card t\n✝ : Nontrivial R\nh : ¬natDegree (prod (Multiset.map (fun x => X - ↑C x) t)) = 0\n⊢ ∀ (f : R[X]), f ∈ Multiset.map (fun x => X - ↑C x) t → Monic f",
"tactic": "simp [natDegree_X_sub_C, monic_X_sub_C]"
}
] |
[
277,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Analysis/Calculus/Deriv/Pow.lean
|
deriv_pow
|
[] |
[
91,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean
|
Matrix.Represents.smul
|
[
{
"state_after": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_3\ninst✝³ : AddCommGroup M\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nA : Matrix ι ι R\nf : Module.End R M\nh : ↑(PiToModule.fromMatrix R b) A = ↑(PiToModule.fromEnd R b) f\nr : R\n⊢ ↑(PiToModule.fromMatrix R b) (r • A) = ↑(PiToModule.fromEnd R b) (r • f)",
"state_before": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_3\ninst✝³ : AddCommGroup M\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nA : Matrix ι ι R\nf : Module.End R M\nh : Represents b A f\nr : R\n⊢ Represents b (r • A) (r • f)",
"tactic": "delta Matrix.Represents at h ⊢"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝⁴ : Fintype ι\nM : Type u_3\ninst✝³ : AddCommGroup M\nR : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Module R M\nI : Ideal R\nb : ι → M\nhb : Submodule.span R (Set.range b) = ⊤\ninst✝ : DecidableEq ι\nA : Matrix ι ι R\nf : Module.End R M\nh : ↑(PiToModule.fromMatrix R b) A = ↑(PiToModule.fromEnd R b) f\nr : R\n⊢ ↑(PiToModule.fromMatrix R b) (r • A) = ↑(PiToModule.fromEnd R b) (r • f)",
"tactic": "rw [SMulHomClass.map_smul, SMulHomClass.map_smul, h]"
}
] |
[
149,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
Left.one_lt_mul_of_le_of_lt_of_pos
|
[] |
[
787,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
1
] |
Mathlib/CategoryTheory/Sites/Adjunction.lean
|
CategoryTheory.Sheaf.adjunctionToTypes_counit_app_val
|
[
{
"state_after": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ toSheafify J\n (((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫\n ((adjunctionToTypes J adj).counit.app X).val =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ ((adjunctionToTypes J adj).counit.app X).val =\n sheafifyLift J ((Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1)\n (_ : Presheaf.IsSheaf J X.val)",
"tactic": "apply J.sheafifyLift_unique"
},
{
"state_after": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ toSheafify J\n (((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫\n ((Functor.associator (sheafCompose J (forget D)) (Equivalence.symm (sheafEquivSheafOfTypes J)).inverse\n ((Equivalence.symm (sheafEquivSheafOfTypes J)).functor ⋙ composeAndSheafify J G)).hom.app\n X).val ≫\n ((whiskerLeft (sheafCompose J (forget D))\n (whiskerRight (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n (composeAndSheafify J G))).app\n X).val ≫\n ((adjunction J adj).counit.app X).val =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"state_before": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ toSheafify J\n (((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫\n ((adjunctionToTypes J adj).counit.app X).val =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"tactic": "dsimp only [adjunctionToTypes, Adjunction.comp, NatTrans.comp_app,\n instCategorySheaf_comp_val, instCategorySheaf_id_val]"
},
{
"state_after": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ toSheafify J\n (((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫\n ((Functor.associator (sheafCompose J (forget D)) (Equivalence.symm (sheafEquivSheafOfTypes J)).inverse\n ((Equivalence.symm (sheafEquivSheafOfTypes J)).functor ⋙ composeAndSheafify J G)).hom.app\n X).val ≫\n ((whiskerLeft (sheafCompose J (forget D))\n (whiskerRight (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n (composeAndSheafify J G))).app\n X).val ≫\n sheafifyLift J\n (↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) (X.val ⋙ forget D) X.val).symm\n (𝟙 (X.val ⋙ forget D)))\n (_ : Presheaf.IsSheaf J X.val) =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"state_before": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ toSheafify J\n (((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫\n ((Functor.associator (sheafCompose J (forget D)) (Equivalence.symm (sheafEquivSheafOfTypes J)).inverse\n ((Equivalence.symm (sheafEquivSheafOfTypes J)).functor ⋙ composeAndSheafify J G)).hom.app\n X).val ≫\n ((whiskerLeft (sheafCompose J (forget D))\n (whiskerRight (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n (composeAndSheafify J G))).app\n X).val ≫\n ((adjunction J adj).counit.app X).val =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"tactic": "rw [adjunction_counit_app_val]"
},
{
"state_after": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ ((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).map\n ((sheafToPresheaf J (Type (max u v))).map\n ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit.app\n ((sheafCompose J (forget D)).obj X))) ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) (X.val ⋙ forget D) X.val).symm (𝟙 (X.val ⋙ forget D)) =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"state_before": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ toSheafify J\n (((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj ((sheafEquivSheafOfTypes J).inverse.obj ((sheafForget J).obj X)))) ≫\n ((Functor.associator (sheafCompose J (forget D)) (Equivalence.symm (sheafEquivSheafOfTypes J)).inverse\n ((Equivalence.symm (sheafEquivSheafOfTypes J)).functor ⋙ composeAndSheafify J G)).hom.app\n X).val ≫\n ((whiskerLeft (sheafCompose J (forget D))\n (whiskerRight (Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit\n (composeAndSheafify J G))).app\n X).val ≫\n sheafifyLift J\n (↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) (X.val ⋙ forget D) X.val).symm\n (𝟙 (X.val ⋙ forget D)))\n (_ : Presheaf.IsSheaf J X.val) =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"tactic": "erw [Category.id_comp, J.sheafifyMap_sheafifyLift, J.toSheafify_sheafifyLift]"
},
{
"state_after": "case a.w.h.w\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\nx✝¹ : Cᵒᵖ\nx✝ :\n (forget D).obj\n ((((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj\n (((Equivalence.symm (sheafEquivSheafOfTypes J)).inverse ⋙\n (Equivalence.symm (sheafEquivSheafOfTypes J)).functor).obj\n ((sheafCompose J (forget D)).obj X)))).obj\n x✝¹)\n⊢ (forget D).map\n ((((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).map\n ((sheafToPresheaf J (Type (max u v))).map\n ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit.app\n ((sheafCompose J (forget D)).obj X))) ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) (X.val ⋙ forget D) X.val).symm\n (𝟙 (X.val ⋙ forget D))).app\n x✝¹)\n x✝ =\n (forget D).map\n (((Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1).app x✝¹) x✝",
"state_before": "case a\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\n⊢ ((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).map\n ((sheafToPresheaf J (Type (max u v))).map\n ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit.app\n ((sheafCompose J (forget D)).obj X))) ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) (X.val ⋙ forget D) X.val).symm (𝟙 (X.val ⋙ forget D)) =\n (Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1",
"tactic": "ext"
},
{
"state_after": "case a.w.h.w\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\nx✝¹ : Cᵒᵖ\nx✝ :\n (forget D).obj\n ((((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj\n (((Equivalence.symm (sheafEquivSheafOfTypes J)).inverse ⋙\n (Equivalence.symm (sheafEquivSheafOfTypes J)).functor).obj\n ((sheafCompose J (forget D)).obj X)))).obj\n x✝¹)\n⊢ (forget D).map\n (G.map (𝟙 ((forget D).obj (X.val.obj x✝¹))) ≫\n G.map (𝟙 ((forget D).obj (X.val.obj x✝¹))) ≫\n 𝟙 (G.obj ((forget D).obj (X.val.obj x✝¹))) ≫ adj.counit.app (X.val.obj x✝¹) ≫ 𝟙 (X.val.obj x✝¹))\n x✝ =\n (forget D).map\n (𝟙 (G.obj ((forget D).obj (X.1.obj x✝¹))) ≫\n 𝟙 (G.obj ((forget D).obj (X.1.obj x✝¹))) ≫ adj.counit.app (X.1.obj x✝¹) ≫ 𝟙 (X.val.obj x✝¹))\n x✝",
"state_before": "case a.w.h.w\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\nx✝¹ : Cᵒᵖ\nx✝ :\n (forget D).obj\n ((((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj\n (((Equivalence.symm (sheafEquivSheafOfTypes J)).inverse ⋙\n (Equivalence.symm (sheafEquivSheafOfTypes J)).functor).obj\n ((sheafCompose J (forget D)).obj X)))).obj\n x✝¹)\n⊢ (forget D).map\n ((((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).map\n ((sheafToPresheaf J (Type (max u v))).map\n ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).counit.app\n ((sheafCompose J (forget D)).obj X))) ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) (X.val ⋙ forget D) X.val).symm\n (𝟙 (X.val ⋙ forget D))).app\n x✝¹)\n x✝ =\n (forget D).map\n (((Functor.associator X.1 (forget D) G).hom ≫ (Adjunction.whiskerRight Cᵒᵖ adj).counit.app X.1).app x✝¹) x✝",
"tactic": "dsimp [sheafEquivSheafOfTypes, Equivalence.symm, Equivalence.toAdjunction,\n NatIso.ofComponents, Adjunction.whiskerRight, Adjunction.mkOfUnitCounit]"
},
{
"state_after": "no goals",
"state_before": "case a.w.h.w\nC : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nX : Sheaf J D\nx✝¹ : Cᵒᵖ\nx✝ :\n (forget D).obj\n ((((whiskeringRight Cᵒᵖ (Type (max u v)) D).obj G).obj\n ((sheafToPresheaf J (Type (max u v))).obj\n (((Equivalence.symm (sheafEquivSheafOfTypes J)).inverse ⋙\n (Equivalence.symm (sheafEquivSheafOfTypes J)).functor).obj\n ((sheafCompose J (forget D)).obj X)))).obj\n x✝¹)\n⊢ (forget D).map\n (G.map (𝟙 ((forget D).obj (X.val.obj x✝¹))) ≫\n G.map (𝟙 ((forget D).obj (X.val.obj x✝¹))) ≫\n 𝟙 (G.obj ((forget D).obj (X.val.obj x✝¹))) ≫ adj.counit.app (X.val.obj x✝¹) ≫ 𝟙 (X.val.obj x✝¹))\n x✝ =\n (forget D).map\n (𝟙 (G.obj ((forget D).obj (X.1.obj x✝¹))) ≫\n 𝟙 (G.obj ((forget D).obj (X.1.obj x✝¹))) ≫ adj.counit.app (X.1.obj x✝¹) ≫ 𝟙 (X.val.obj x✝¹))\n x✝",
"tactic": "simp"
}
] |
[
153,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.pullback.lift_fst
|
[] |
[
1173,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1171,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.sInf_adj
|
[] |
[
395,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.div_le_iff'
|
[] |
[
813,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
812,
8
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.restrictScalars_ball
|
[] |
[
1092,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1091,
1
] |
Mathlib/Analysis/Calculus/Taylor.lean
|
taylor_within_apply
|
[
{
"state_after": "case zero\n𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ns : Set ℝ\nx₀ x : ℝ\n⊢ taylorWithinEval f Nat.zero s x₀ x =\n ∑ k in Finset.range (Nat.zero + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀\n\ncase succ\n𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ns : Set ℝ\nx₀ x : ℝ\nk : ℕ\nhk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀\n⊢ taylorWithinEval f (Nat.succ k) s x₀ x =\n ∑ k in Finset.range (Nat.succ k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀",
"state_before": "𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ x : ℝ\n⊢ taylorWithinEval f n s x₀ x = ∑ k in Finset.range (n + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀",
"tactic": "induction' n with k hk"
},
{
"state_after": "case succ\n𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ns : Set ℝ\nx₀ x : ℝ\nk : ℕ\nhk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀\n⊢ ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ +\n (((↑k + 1) * ↑k !)⁻¹ * (x - x₀) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s x₀ =\n ∑ x_1 in Finset.range (k + 1), ((↑x_1 !)⁻¹ * (x - x₀) ^ x_1) • iteratedDerivWithin x_1 f s x₀ +\n ((↑(k + 1)!)⁻¹ * (x - x₀) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s x₀",
"state_before": "case succ\n𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ns : Set ℝ\nx₀ x : ℝ\nk : ℕ\nhk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀\n⊢ taylorWithinEval f (Nat.succ k) s x₀ x =\n ∑ k in Finset.range (Nat.succ k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀",
"tactic": "rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]"
},
{
"state_after": "no goals",
"state_before": "case succ\n𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ns : Set ℝ\nx₀ x : ℝ\nk : ℕ\nhk : taylorWithinEval f k s x₀ x = ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀\n⊢ ∑ k in Finset.range (k + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ +\n (((↑k + 1) * ↑k !)⁻¹ * (x - x₀) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s x₀ =\n ∑ x_1 in Finset.range (k + 1), ((↑x_1 !)⁻¹ * (x - x₀) ^ x_1) • iteratedDerivWithin x_1 f s x₀ +\n ((↑(k + 1)!)⁻¹ * (x - x₀) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s x₀",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case zero\n𝕜 : Type ?u.230356\nE : Type u_1\nF : Type ?u.230362\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\ns : Set ℝ\nx₀ x : ℝ\n⊢ taylorWithinEval f Nat.zero s x₀ x =\n ∑ k in Finset.range (Nat.zero + 1), ((↑k !)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀",
"tactic": "simp"
}
] |
[
123,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Order/GameAdd.lean
|
Sym2.GameAdd.induction
|
[] |
[
250,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_bij
|
[] |
[
553,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.lt_add_of_pos_left
|
[
{
"state_after": "n k : Nat\nh : 0 < k\n⊢ n < n + k",
"state_before": "n k : Nat\nh : 0 < k\n⊢ n < k + n",
"tactic": "rw [Nat.add_comm]"
},
{
"state_after": "no goals",
"state_before": "n k : Nat\nh : 0 < k\n⊢ n < n + k",
"tactic": "exact Nat.lt_add_of_pos_right h"
}
] |
[
93,
53
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
92,
11
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinOn.sup
|
[] |
[
544,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/Algebra/Hom/Units.lean
|
IsUnit.mul_liftRight_inv
|
[
{
"state_after": "F : Type ?u.42331\nG : Type ?u.42334\nα : Type ?u.42337\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (↑f x)\nx x✝ : M\n⊢ ↑(IsUnit.unit (_ : IsUnit (↑f x✝))) = ↑f x✝",
"state_before": "F : Type ?u.42331\nG : Type ?u.42334\nα : Type ?u.42337\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (↑f x)\nx : M\n⊢ ∀ (x : M), ↑(IsUnit.unit (_ : IsUnit (↑f x))) = ↑f x",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.42331\nG : Type ?u.42334\nα : Type ?u.42337\nM : Type u_1\nN : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nf : M →* N\nh : ∀ (x : M), IsUnit (↑f x)\nx x✝ : M\n⊢ ↑(IsUnit.unit (_ : IsUnit (↑f x✝))) = ↑f x✝",
"tactic": "rfl"
}
] |
[
260,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.sign_of_succ
|
[] |
[
199,
62
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
199,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
zpow_one
|
[
{
"state_after": "case h.e'_2\nα : Type ?u.54010\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivInvMonoid G\na : G\n⊢ a ^ 1 = a ^ 1",
"state_before": "α : Type ?u.54010\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivInvMonoid G\na : G\n⊢ a ^ 1 = a",
"tactic": "convert pow_one a using 1"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nα : Type ?u.54010\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : DivInvMonoid G\na : G\n⊢ a ^ 1 = a ^ 1",
"tactic": "exact zpow_ofNat a 1"
}
] |
[
288,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.coe_bind
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9950\nδ : Type ?u.9953\na : α\ns t : Multiset α\nf✝ g : α → Multiset β\nl : List α\nf : α → List β\n⊢ (bind ↑l fun a => ↑(f a)) = join ↑(List.map (ofList ∘ f) l)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9950\nδ : Type ?u.9953\na : α\ns t : Multiset α\nf✝ g : α → Multiset β\nl : List α\nf : α → List β\n⊢ (bind ↑l fun a => ↑(f a)) = ↑(List.bind l f)",
"tactic": "rw [List.bind, ← coe_join, List.map_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.9950\nδ : Type ?u.9953\na : α\ns t : Multiset α\nf✝ g : α → Multiset β\nl : List α\nf : α → List β\n⊢ (bind ↑l fun a => ↑(f a)) = join ↑(List.map (ofList ∘ f) l)",
"tactic": "rfl"
}
] |
[
102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
LinearMap.mulRight_one
|
[
{
"state_after": "case h\nR : Type u_2\nA : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : A\n⊢ ↑(mulRight R 1) x✝ = ↑id x✝",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\n⊢ mulRight R 1 = id",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\nA : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx✝ : A\n⊢ ↑(mulRight R 1) x✝ = ↑id x✝",
"tactic": "simp only [LinearMap.id_coe, mul_one, id.def, mulRight_apply]"
}
] |
[
217,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean
|
HasDerivWithinAt.cexp
|
[] |
[
102,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.coe_trans_to_affineMap
|
[] |
[
346,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
344,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.insert
|
[] |
[
671,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.LpToLpRestrictCLM_coeFn
|
[] |
[
929,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
927,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean
|
WithTop.top_mul'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Zero α\ninst✝ : Mul α\na : WithTop α\n⊢ ⊤ * a = if a = 0 then 0 else ⊤",
"tactic": "induction a using recTopCoe <;> simp [mul_def] <;> rfl"
}
] |
[
57,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/GroupTheory/Commutator.lean
|
commutatorElement_inv
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.10728\nF : Type ?u.10731\ninst✝² : Group G\ninst✝¹ : Group G'\ninst✝ : MonoidHomClass F G G'\nf : F\ng₁ g₂ g₃ g : G\n⊢ ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆",
"tactic": "simp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]"
}
] |
[
63,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/CategoryTheory/Monad/Algebra.lean
|
CategoryTheory.Monad.leftAdjoint_forget
|
[] |
[
243,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.map_extend_nhdsWithin_eq_image
|
[
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.150056\nM' : Type ?u.150059\nH' : Type ?u.150062\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ny : M\nhy : y ∈ f.source\ne : LocalEquiv M E := extend f I\n⊢ map (↑e) (𝓝[s] y) = 𝓝[↑e '' (e.source ∩ s)] ↑e y",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.150056\nM' : Type ?u.150059\nH' : Type ?u.150062\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ny : M\nhy : y ∈ f.source\n⊢ map (↑(extend f I)) (𝓝[s] y) = 𝓝[↑(extend f I) '' ((extend f I).source ∩ s)] ↑(extend f I) y",
"tactic": "set e := f.extend I"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.150056\nM' : Type ?u.150059\nH' : Type ?u.150062\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ny : M\nhy : y ∈ f.source\ne : LocalEquiv M E := extend f I\n⊢ map (↑e) (𝓝[s] y) = 𝓝[↑e '' (e.source ∩ s)] ↑e y",
"tactic": "calc\n map e (𝓝[s] y) = map e (𝓝[e.source ∩ s] y) :=\n congr_arg (map e) (nhdsWithin_inter_of_mem (extend_source_mem_nhdsWithin f I hy)).symm\n _ = 𝓝[e '' (e.source ∩ s)] e y :=\n ((f.extend I).leftInvOn.mono <| inter_subset_left _ _).map_nhdsWithin_eq\n ((f.extend I).left_inv <| by rwa [f.extend_source])\n (continuousAt_extend_symm f I hy).continuousWithinAt\n (continuousAt_extend f I hy).continuousWithinAt"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.150056\nM' : Type ?u.150059\nH' : Type ?u.150062\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ny : M\nhy : y ∈ f.source\ne : LocalEquiv M E := extend f I\n⊢ y ∈ (extend f I).source",
"tactic": "rwa [f.extend_source]"
}
] |
[
896,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
886,
1
] |
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