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/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.of_right
|
[] |
[
537,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Data/Bundle.lean
|
Bundle.coe_snd
|
[] |
[
97,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/MeasureTheory/Covering/Besicovitch.lean
|
Besicovitch.ae_tendsto_rnDeriv
|
[
{
"state_after": "case h\nα : Type ?u.1047956\ninst✝¹¹ : MetricSpace α\nβ : Type u\ninst✝¹⁰ : SecondCountableTopology α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : OpensMeasurableSpace α\ninst✝⁷ : HasBesicovitchCovering α\ninst✝⁶ : MetricSpace β\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : BorelSpace β\ninst✝³ : SecondCountableTopology β\ninst✝² : HasBesicovitchCovering β\nρ μ : MeasureTheory.Measure β\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsLocallyFiniteMeasure ρ\nx : β\nhx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x) (𝓝 (Measure.rnDeriv ρ μ x))\n⊢ Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x))",
"state_before": "α : Type ?u.1047956\ninst✝¹¹ : MetricSpace α\nβ : Type u\ninst✝¹⁰ : SecondCountableTopology α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : OpensMeasurableSpace α\ninst✝⁷ : HasBesicovitchCovering α\ninst✝⁶ : MetricSpace β\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : BorelSpace β\ninst✝³ : SecondCountableTopology β\ninst✝² : HasBesicovitchCovering β\nρ μ : MeasureTheory.Measure β\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsLocallyFiniteMeasure ρ\n⊢ ∀ᵐ (x : β) ∂μ, Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x))",
"tactic": "filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (Besicovitch.vitaliFamily μ) ρ] with x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.1047956\ninst✝¹¹ : MetricSpace α\nβ : Type u\ninst✝¹⁰ : SecondCountableTopology α\ninst✝⁹ : MeasurableSpace α\ninst✝⁸ : OpensMeasurableSpace α\ninst✝⁷ : HasBesicovitchCovering α\ninst✝⁶ : MetricSpace β\ninst✝⁵ : MeasurableSpace β\ninst✝⁴ : BorelSpace β\ninst✝³ : SecondCountableTopology β\ninst✝² : HasBesicovitchCovering β\nρ μ : MeasureTheory.Measure β\ninst✝¹ : IsLocallyFiniteMeasure μ\ninst✝ : IsLocallyFiniteMeasure ρ\nx : β\nhx : Tendsto (fun a => ↑↑ρ a / ↑↑μ a) (VitaliFamily.filterAt (Besicovitch.vitaliFamily μ) x) (𝓝 (Measure.rnDeriv ρ μ x))\n⊢ Tendsto (fun r => ↑↑ρ (closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 (Measure.rnDeriv ρ μ x))",
"tactic": "exact hx.comp (tendsto_filterAt μ x)"
}
] |
[
1165,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1161,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
ZMod.units_pow_card_sub_one_eq_one
|
[
{
"state_after": "no goals",
"state_before": "K : Type ?u.1007117\nR : Type ?u.1007120\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : (ZMod p)ˣ\n⊢ a ^ (p - 1) = 1",
"tactic": "rw [← card_units p, pow_card_eq_one]"
}
] |
[
429,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_bounded_of_finset_linearIndependent_bounded
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\n⊢ (#↑s) ≤ ↑n",
"state_before": "ι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\n⊢ ∀ (s : Set M), LinearIndependent R Subtype.val → (#↑s) ≤ ↑n",
"tactic": "intro s li"
},
{
"state_after": "case H\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\n⊢ ∀ (s_1 : Finset ↑s), Finset.card s_1 ≤ n",
"state_before": "ι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\n⊢ (#↑s) ≤ ↑n",
"tactic": "apply Cardinal.card_le_of"
},
{
"state_after": "case H\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\n⊢ Finset.card t ≤ n",
"state_before": "case H\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\n⊢ ∀ (s_1 : Finset ↑s), Finset.card s_1 ≤ n",
"tactic": "intro t"
},
{
"state_after": "case H\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\n⊢ Finset.card (Finset.map (Embedding.subtype s) t) ≤ n",
"state_before": "case H\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\n⊢ Finset.card t ≤ n",
"tactic": "rw [← Finset.card_map (Embedding.subtype s)]"
},
{
"state_after": "case H.a\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\n⊢ LinearIndependent R fun i => ↑i",
"state_before": "case H\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\n⊢ Finset.card (Finset.map (Embedding.subtype s) t) ≤ n",
"tactic": "apply H"
},
{
"state_after": "no goals",
"state_before": "case H.a\nι : Type u'\nι' : Type ?u.142157\nR : Type u_2\nK : Type ?u.142163\nM : Type u_1\nM' : Type ?u.142169\nM'' : Type ?u.142172\nV : Type u\nV' : Type ?u.142177\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nn : ℕ\nH : ∀ (s : Finset M), (LinearIndependent R fun i => ↑i) → Finset.card s ≤ n\ns : Set M\nli : LinearIndependent R Subtype.val\nt : Finset ↑s\n⊢ LinearIndependent R fun i => ↑i",
"tactic": "apply linearIndependent_finset_map_embedding_subtype _ li"
}
] |
[
342,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.conjTranspose_natCast_smul
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.992438\nm : Type u_2\nn : Type u_3\no : Type ?u.992447\nm' : o → Type ?u.992452\nn' : o → Type ?u.992457\nR : Type u_1\nS : Type ?u.992463\nα : Type v\nβ : Type w\nγ : Type ?u.992470\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid α\ninst✝¹ : StarAddMonoid α\ninst✝ : Module R α\nc : ℕ\nM : Matrix m n α\n⊢ ∀ (i : n) (j : m), (↑c • M)ᴴ i j = (↑c • Mᴴ) i j",
"tactic": "simp"
}
] |
[
2206,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2204,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.surjOn_empty
|
[] |
[
770,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.pure_seq
|
[] |
[
267,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsClosed.hypograph
|
[] |
[
245,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.union_ae_eq_right_of_ae_eq_empty
|
[
{
"state_after": "case h.e'_5\nα : Type u_1\nβ : Type ?u.117614\nγ : Type ?u.117617\nδ : Type ?u.117620\nι : Type ?u.117623\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : s =ᵐ[μ] ∅\n⊢ t = ∅ ∪ t",
"state_before": "α : Type u_1\nβ : Type ?u.117614\nγ : Type ?u.117617\nδ : Type ?u.117620\nι : Type ?u.117623\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : s =ᵐ[μ] ∅\n⊢ s ∪ t =ᵐ[μ] t",
"tactic": "convert ae_eq_set_union h (ae_eq_refl t)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nα : Type u_1\nβ : Type ?u.117614\nγ : Type ?u.117617\nδ : Type ?u.117620\nι : Type ?u.117623\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : s =ᵐ[μ] ∅\n⊢ t = ∅ ∪ t",
"tactic": "rw [empty_union]"
}
] |
[
523,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
521,
1
] |
Mathlib/RingTheory/WittVector/WittPolynomial.lean
|
map_wittPolynomial
|
[
{
"state_after": "p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nn : ℕ\n⊢ ∑ x in range (n + 1), ↑(map f) (↑(monomial (single x (p ^ (n - x)))) (↑p ^ x)) =\n ∑ i in range (n + 1), ↑(monomial (single i (p ^ (n - i)))) (↑p ^ i)",
"state_before": "p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nn : ℕ\n⊢ ↑(map f) (W_ R n) = W_ S n",
"tactic": "rw [wittPolynomial, map_sum, wittPolynomial]"
},
{
"state_after": "p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nn i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ ↑(map f) (↑(monomial (single i (p ^ (n - i)))) (↑p ^ i)) = ↑(monomial (single i (p ^ (n - i)))) (↑p ^ i)",
"state_before": "p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nn : ℕ\n⊢ ∑ x in range (n + 1), ↑(map f) (↑(monomial (single x (p ^ (n - x)))) (↑p ^ x)) =\n ∑ i in range (n + 1), ↑(monomial (single i (p ^ (n - i)))) (↑p ^ i)",
"tactic": "refine sum_congr rfl fun i _ => ?_"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : DecidableEq R\nS : Type u_2\ninst✝ : CommRing S\nf : R →+* S\nn i : ℕ\nx✝ : i ∈ range (n + 1)\n⊢ ↑(map f) (↑(monomial (single i (p ^ (n - i)))) (↑p ^ i)) = ↑(monomial (single i (p ^ (n - i)))) (↑p ^ i)",
"tactic": "rw [map_monomial, RingHom.map_pow, map_natCast]"
}
] |
[
125,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.app_injective_iff_stalkFunctor_map_injective
|
[] |
[
490,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero
|
[] |
[
274,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.fromSubtype_π
|
[
{
"state_after": "case w\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\n⊢ ι (Subtype.restrict p f) i ≫ fromSubtype f p ≫ π f j =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"state_before": "J : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\n⊢ fromSubtype f p ≫ π f j = if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"tactic": "ext i"
},
{
"state_after": "case w\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\n⊢ ι (Subtype.restrict p f) i ≫ fromSubtype f p ≫ π f j =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"state_before": "case w\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\n⊢ ι (Subtype.restrict p f) i ≫ fromSubtype f p ≫ π f j =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"tactic": "dsimp"
},
{
"state_after": "case w\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"state_before": "case w\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\n⊢ ι (Subtype.restrict p f) i ≫ fromSubtype f p ≫ π f j =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"tactic": "rw [biproduct.fromSubtype, biproduct.ι_desc_assoc, biproduct.ι_π]"
},
{
"state_after": "case pos\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0\n\ncase neg\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : ¬p j\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"state_before": "case w\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"tactic": "by_cases h : p j"
},
{
"state_after": "case pos\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n if h_1 : i = { val := j, property := h } then\n eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n else 0",
"state_before": "case pos\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"tactic": "rw [dif_pos h, biproduct.ι_π]"
},
{
"state_after": "case pos.inl.inl\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ↑i = j\nh₂ : i = { val := j, property := h }\n⊢ eqToHom (_ : f ↑i = f j) = eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n\ncase pos.inl.inr\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ↑i = j\nh₂ : ¬i = { val := j, property := h }\n⊢ eqToHom (_ : f ↑i = f j) = 0\n\ncase pos.inr.inl\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ¬↑i = j\nh₂ : i = { val := j, property := h }\n⊢ 0 = eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n\ncase pos.inr.inr\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ¬↑i = j\nh₂ : ¬i = { val := j, property := h }\n⊢ 0 = 0",
"state_before": "case pos\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n if h_1 : i = { val := j, property := h } then\n eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n else 0",
"tactic": "split_ifs with h₁ h₂ h₂"
},
{
"state_after": "no goals",
"state_before": "case pos.inl.inl\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ↑i = j\nh₂ : i = { val := j, property := h }\n⊢ eqToHom (_ : f ↑i = f j) = eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n\ncase pos.inl.inr\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ↑i = j\nh₂ : ¬i = { val := j, property := h }\n⊢ eqToHom (_ : f ↑i = f j) = 0\n\ncase pos.inr.inl\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ¬↑i = j\nh₂ : i = { val := j, property := h }\n⊢ 0 = eqToHom (_ : Subtype.restrict p f i = Subtype.restrict p f { val := j, property := h })\n\ncase pos.inr.inr\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : p j\nh₁ : ¬↑i = j\nh₂ : ¬i = { val := j, property := h }\n⊢ 0 = 0",
"tactic": "exacts [rfl, False.elim (h₂ (Subtype.ext h₁)), False.elim (h₁ (congr_arg Subtype.val h₂)), rfl]"
},
{
"state_after": "no goals",
"state_before": "case neg\nJ : Type w\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroMorphisms C\nf : J → C\ninst✝² : HasBiproduct f\np : J → Prop\ninst✝¹ : HasBiproduct (Subtype.restrict p f)\ninst✝ : DecidablePred p\nj : J\ni : Subtype p\nh : ¬p j\n⊢ (if h : ↑i = j then eqToHom (_ : f ↑i = f j) else 0) =\n ι (Subtype.restrict p f) i ≫ if h : p j then π (Subtype.restrict p f) { val := j, property := h } else 0",
"tactic": "rw [dif_neg h, dif_neg (show (i : J) ≠ j from fun h₂ => h (h₂ ▸ i.2)), comp_zero]"
}
] |
[
592,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Data/Finset/Pi.lean
|
Finset.Pi.cons_ne
|
[] |
[
74,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.dvd_of_mod_eq_zero
|
[
{
"state_after": "m n : Nat\nH : n % m = 0\n⊢ n = m * (n / m)",
"state_before": "m n : Nat\nH : n % m = 0\n⊢ m ∣ n",
"tactic": "exists n / m"
},
{
"state_after": "m n : Nat\nH : n % m = 0\nthis : n = n % m + m * (n / m)\n⊢ n = m * (n / m)",
"state_before": "m n : Nat\nH : n % m = 0\n⊢ n = m * (n / m)",
"tactic": "have := (mod_add_div n m).symm"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nH : n % m = 0\nthis : n = n % m + m * (n / m)\n⊢ n = m * (n / m)",
"tactic": "rwa [H, Nat.zero_add] at this"
}
] |
[
721,
32
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
718,
1
] |
Mathlib/Algebra/Squarefree.lean
|
Squarefree.gcd_left
|
[] |
[
100,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Order/Max.lean
|
not_isMin_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9140\ninst✝ : Preorder α\na b : α\n⊢ ¬IsMin a ↔ ∃ b, b < a",
"tactic": "simp [lt_iff_le_not_le, IsMin, not_forall, exists_prop]"
}
] |
[
341,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Data/Nat/Size.lean
|
Nat.shiftl_eq_mul_pow
|
[
{
"state_after": "m k : ℕ\n⊢ bit0 (shiftl m k) = m * (2 ^ k * 2)",
"state_before": "m k : ℕ\n⊢ shiftl m (k + 1) = m * 2 ^ (k + 1)",
"tactic": "show bit0 (shiftl m k) = m * (2 ^ k * 2)"
},
{
"state_after": "no goals",
"state_before": "m k : ℕ\n⊢ bit0 (shiftl m k) = m * (2 ^ k * 2)",
"tactic": "rw [bit0_val, shiftl_eq_mul_pow m k, mul_comm 2, mul_assoc]"
}
] |
[
27,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
23,
1
] |
Mathlib/CategoryTheory/Sites/Whiskering.lean
|
CategoryTheory.GrothendieckTopology.Cover.multicospanComp_hom_inv_left
|
[] |
[
92,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Std/Data/RBMap/WF.lean
|
Std.RBNode.All.trivial
|
[] |
[
25,
49
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
23,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.iInf_neBot_of_directed'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.110309\nι : Sort x\nf✝ g : Filter α\ns t : Set α\nf : ι → Filter α\ninst✝ : Nonempty ι\nhd : Directed (fun x x_1 => x ≥ x_1) f\n⊢ ¬NeBot (iInf f) → ¬∀ (i : ι), NeBot (f i)",
"tactic": "simpa only [not_forall, not_neBot, ← empty_mem_iff_bot,\nmem_iInf_of_directed hd] using id"
}
] |
[
911,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.sin_sub_two_pi
|
[] |
[
1157,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1156,
1
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.limsup_eq_iInf_iSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.105898\nι : Type ?u.105901\ninst✝ : CompleteLattice α\nf : Filter β\nu : β → α\n⊢ (⨅ (i : Set β) (_ : i ∈ f), sSup (u '' id i)) = ⨅ (s : Set β) (_ : s ∈ f), ⨆ (a : β) (_ : a ∈ s), u a",
"tactic": "simp only [sSup_image, id]"
}
] |
[
696,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
695,
1
] |
Mathlib/Data/Multiset/Nodup.lean
|
Multiset.nodup_zero
|
[] |
[
40,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.measure_eq_right_of_subset_of_measure_add_eq
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.139389\nγ : Type ?u.139392\nδ : Type ?u.139395\nι : Type ?u.139398\nR : Type ?u.139401\nR' : Type ?u.139404\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(ν + μ) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(ν + μ) s = ↑↑(ν + μ) t\n⊢ ↑↑ν s = ↑↑ν t",
"state_before": "α : Type u_1\nβ : Type ?u.139389\nγ : Type ?u.139392\nδ : Type ?u.139395\nι : Type ?u.139398\nR : Type ?u.139401\nR' : Type ?u.139404\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(μ + ν) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(μ + ν) s = ↑↑(μ + ν) t\n⊢ ↑↑ν s = ↑↑ν t",
"tactic": "rw [add_comm] at h'' h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139389\nγ : Type ?u.139392\nδ : Type ?u.139395\nι : Type ?u.139398\nR : Type ?u.139401\nR' : Type ?u.139404\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t✝ s t : Set α\nh : ↑↑(ν + μ) t ≠ ⊤\nh' : s ⊆ t\nh'' : ↑↑(ν + μ) s = ↑↑(ν + μ) t\n⊢ ↑↑ν s = ↑↑ν t",
"tactic": "exact measure_eq_left_of_subset_of_measure_add_eq h h' h''"
}
] |
[
931,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
928,
1
] |
Mathlib/Topology/Algebra/Order/LeftRightLim.lean
|
Monotone.continuousWithinAt_Ioi_iff_rightLim_eq
|
[] |
[
203,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.mem_ball
|
[] |
[
418,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
417,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.tan_zero
|
[
{
"state_after": "no goals",
"state_before": "⊢ tan 0 = 0",
"tactic": "rw [← coe_zero, tan_coe, Real.tan_zero]"
}
] |
[
789,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
789,
1
] |
Std/Data/List/Lemmas.lean
|
List.range'_eq_map_range
|
[
{
"state_after": "s n : Nat\n⊢ range' s n = range' (s + 0) n",
"state_before": "s n : Nat\n⊢ range' s n = map (fun x => s + x) (range n)",
"tactic": "rw [range_eq_range', map_add_range']"
},
{
"state_after": "no goals",
"state_before": "s n : Nat\n⊢ range' s n = range' (s + 0) n",
"tactic": "rfl"
}
] |
[
1898,
44
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1897,
1
] |
Mathlib/NumberTheory/Padics/Hensel.lean
|
newton_seq_dist_tendsto'
|
[] |
[
410,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
408,
9
] |
Mathlib/Logic/Relation.lean
|
Relation.transGen_idem
|
[] |
[
471,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
470,
1
] |
Mathlib/Order/Atoms.lean
|
OrderIso.isCoatomic_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : PartialOrder β\ninst✝¹ : OrderTop α\ninst✝ : OrderTop β\nf : α ≃o β\n⊢ IsCoatomic α ↔ IsCoatomic β",
"tactic": "simp only [← isAtomic_dual_iff_isCoatomic, f.dual.isAtomic_iff]"
}
] |
[
834,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
11
] |
Mathlib/Topology/Algebra/Valuation.lean
|
Valued.hasBasis_uniformity
|
[
{
"state_after": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ Filter.HasBasis (Filter.comap (fun x => x.snd - x.fst) (𝓝 0)) (fun x => True) fun γ => {p | ↑v (p.snd - p.fst) < ↑γ}",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ Filter.HasBasis (uniformity R) (fun x => True) fun γ => {p | ↑v (p.snd - p.fst) < ↑γ}",
"tactic": "rw [uniformity_eq_comap_nhds_zero]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\n⊢ Filter.HasBasis (Filter.comap (fun x => x.snd - x.fst) (𝓝 0)) (fun x => True) fun γ => {p | ↑v (p.snd - p.fst) < ↑γ}",
"tactic": "exact (hasBasis_nhds_zero R Γ₀).comap _"
}
] |
[
129,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Logic/Basic.lean
|
exists_exists_and_eq_and
|
[] |
[
782,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
780,
9
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSpace.comap_mono
|
[] |
[
143,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
summable_congr
|
[] |
[
107,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.ofCauchy_neg
|
[] |
[
128,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean
|
midpoint_add_sub
|
[
{
"state_after": "R : Type u_2\nV : Type u_1\nV' : Type ?u.143234\nP : Type ?u.143237\nP' : Type ?u.143240\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx✝ y✝ z : P\nx y : V\n⊢ midpoint R (x - y) (x + y) = x",
"state_before": "R : Type u_2\nV : Type u_1\nV' : Type ?u.143234\nP : Type ?u.143237\nP' : Type ?u.143240\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx✝ y✝ z : P\nx y : V\n⊢ midpoint R (x + y) (x - y) = x",
"tactic": "rw [midpoint_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nV : Type u_1\nV' : Type ?u.143234\nP : Type ?u.143237\nP' : Type ?u.143240\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\nx✝ y✝ z : P\nx y : V\n⊢ midpoint R (x - y) (x + y) = x",
"tactic": "simp"
}
] |
[
235,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
vadd_left_mem_affineSpan_pair
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type ?u.475169\np₁ p₂ : P\nv : V\n⊢ v +ᵥ p₁ ∈ affineSpan k {p₁, p₂} ↔ ∃ r, r • (p₂ -ᵥ p₁) = v",
"tactic": "rw [vadd_mem_iff_mem_direction _ (left_mem_affineSpan_pair _ _ _), direction_affineSpan,\n mem_vectorSpan_pair_rev]"
}
] |
[
1341,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1338,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.MonoidalClosed.uncurry_natural_left
|
[] |
[
198,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/CategoryTheory/Abelian/LeftDerived.lean
|
CategoryTheory.Abelian.Functor.leftDerivedZeroToSelfAppInv_comp
|
[
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ (inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (leftDerivedObjIso F 0 P).inv) ≫\n (leftDerivedObjIso F 0 P).hom ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ leftDerivedZeroToSelfAppInv F P ≫ leftDerivedZeroToSelfApp F P = 𝟙 (F.obj X)",
"tactic": "dsimp [leftDerivedZeroToSelfApp, leftDerivedZeroToSelfAppInv]"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n ((homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (leftDerivedObjIso F 0 P).inv) ≫\n (leftDerivedObjIso F 0 P).hom ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ (inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (leftDerivedObjIso F 0 P).inv) ≫\n (leftDerivedObjIso F 0 P).hom ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"tactic": "rw [Category.assoc, Category.assoc]"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (leftDerivedObjIso F 0 P).inv ≫\n (leftDerivedObjIso F 0 P).hom ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n ((homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (leftDerivedObjIso F 0 P).inv) ≫\n (leftDerivedObjIso F 0 P).hom ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"tactic": "simp only [Category.assoc]"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (leftDerivedObjIso F 0 P).inv ≫\n (leftDerivedObjIso F 0 P).hom ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"tactic": "rw [← Category.assoc (F.leftDerivedObjIso 0 P).inv, Iso.inv_hom_id]"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"tactic": "simp only [Category.id_comp]"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"tactic": "have : IsIso (cokernel.desc (F.map\n (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0)) (exact_of_map_projectiveResolution F P).w) :=\n isIso_cokernel_desc_of_exact_of_epi _ _ (exact_of_map_projectiveResolution F P)"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) ≫\n 𝟙 (F.obj X)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ inv\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)) ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n 𝟙 (F.obj X)",
"tactic": "rw [IsIso.inv_comp_eq]"
},
{
"state_after": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) ≫\n 𝟙 (F.obj X)",
"tactic": "simp only [Category.comp_id]"
},
{
"state_after": "case h\nC : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ coequalizer.π (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0)) 0 ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n coequalizer.π (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0)) 0 ≫\n cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)",
"state_before": "C : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)",
"tactic": "apply coequalizer.hom_ext"
},
{
"state_after": "case h\nC : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) ≫\n cokernel.π\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (homologyIsoCokernelLift\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).hom ≫\n cokernel.desc\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n F.map (HomologicalComplex.Hom.f P.π 0)",
"state_before": "case h\nC : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ coequalizer.π (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0)) 0 ≫\n cokernel.map (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))))\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) =\n 𝟙 (F.obj (HomologicalComplex.X P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0))) ≫\n kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n homology.desc' (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n coequalizer.π (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0)) 0 ≫\n cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0)",
"tactic": "simp only [cokernel.π_desc_assoc, Category.assoc, cokernel.π_desc, homology.desc']"
},
{
"state_after": "case h\nC : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) ≫\n cokernel.π\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))) ≫\n 𝟙\n (cokernel\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))) ≫\n cokernel.desc\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n F.map (HomologicalComplex.Hom.f P.π 0)",
"state_before": "case h\nC : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) ≫\n cokernel.π\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)) ≫\n (homologyIsoCokernelLift (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).inv ≫\n (homologyIsoCokernelLift\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0)).hom ≫\n cokernel.desc\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι\n (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n F.map (HomologicalComplex.Hom.f P.π 0)",
"tactic": "rw [← Category.assoc, ← Category.assoc (homologyIsoCokernelLift _ _ _).inv, Iso.inv_hom_id]"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u\ninst✝⁶ : Category C\nD : Type u\ninst✝⁵ : Category D\nF : C ⥤ D\nX✝ Y Z : C\nf : X✝ ⟶ Y\ng : Y ⟶ Z\ninst✝⁴ : Abelian C\ninst✝³ : Abelian D\ninst✝² : Functor.Additive F\ninst✝¹ : EnoughProjectives C\ninst✝ : PreservesFiniteColimits F\nX : C\nP : ProjectiveResolution X\nthis :\n IsIso\n (cokernel.desc (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0))\n⊢ (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (𝟙 (F.obj (HomologicalComplex.X P.complex 0)))\n (_ :\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) ≫\n F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)) =\n 0) ≫\n cokernel.π\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))) ≫\n 𝟙\n (cokernel\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))) ≫\n cokernel.desc\n (kernel.lift (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))\n (F.map (HomologicalComplex.d P.complex (ComplexShape.prev (ComplexShape.down ℕ) 0) 0))\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0))\n (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n F.map (HomologicalComplex.Hom.f P.π 0))\n (_ :\n kernel.lift (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (_ :\n HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 ≫\n HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n 0) ≫\n kernel.ι (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0) ≫\n F.map (HomologicalComplex.Hom.f P.π 0) =\n 0) =\n F.map (HomologicalComplex.Hom.f P.π 0)",
"tactic": "simp only [Category.assoc, cokernel.π_desc, kernel.lift_ι_assoc, Category.id_comp]"
}
] |
[
151,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/CategoryTheory/Noetherian.lean
|
CategoryTheory.exists_simple_subobject
|
[
{
"state_after": "C : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\nthis : Nontrivial (Subobject X)\n⊢ ∃ Y, Simple (underlying.obj Y)",
"state_before": "C : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\n⊢ ∃ Y, Simple (underlying.obj Y)",
"tactic": "haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h"
},
{
"state_after": "C : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\nthis✝ : Nontrivial (Subobject X)\nthis : IsAtomic (Subobject X)\n⊢ ∃ Y, Simple (underlying.obj Y)",
"state_before": "C : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\nthis : Nontrivial (Subobject X)\n⊢ ∃ Y, Simple (underlying.obj Y)",
"tactic": "haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)"
},
{
"state_after": "case intro\nC : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\nthis✝ : Nontrivial (Subobject X)\nthis : IsAtomic (Subobject X)\nY : Subobject X\ns : IsAtom Y ∧ Y ≤ ⊤\n⊢ ∃ Y, Simple (underlying.obj Y)",
"state_before": "C : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\nthis✝ : Nontrivial (Subobject X)\nthis : IsAtomic (Subobject X)\n⊢ ∃ Y, Simple (underlying.obj Y)",
"tactic": "obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot"
},
{
"state_after": "no goals",
"state_before": "case intro\nC : Type u_1\ninst✝³ : Category C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroObject C\nX : C\ninst✝ : ArtinianObject X\nh : ¬IsZero X\nthis✝ : Nontrivial (Subobject X)\nthis : IsAtomic (Subobject X)\nY : Subobject X\ns : IsAtom Y ∧ Y ≤ ⊤\n⊢ ∃ Y, Simple (underlying.obj Y)",
"tactic": "exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩"
}
] |
[
90,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Topology/Algebra/Order/ExtrClosure.lean
|
IsLocalMaxOn.closure
|
[
{
"state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\n⊢ IsLocalMaxOn f (closure s) a",
"state_before": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\n⊢ IsLocalMaxOn f (closure s) a",
"tactic": "rcases mem_nhdsWithin.1 h with ⟨U, Uo, aU, hU⟩"
},
{
"state_after": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\n⊢ U ∩ closure s ⊆ {x | (fun x => f x ≤ f a) x}",
"state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\n⊢ IsLocalMaxOn f (closure s) a",
"tactic": "refine' mem_nhdsWithin.2 ⟨U, Uo, aU, _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ x ∈ {x | (fun x => f x ≤ f a) x}",
"state_before": "case intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\n⊢ U ∩ closure s ⊆ {x | (fun x => f x ≤ f a) x}",
"tactic": "rintro x ⟨hxU, hxs⟩"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ x ∈ closure (U ∩ s)\n\ncase intro.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ ContinuousWithinAt f (U ∩ s) x",
"state_before": "case intro.intro.intro.intro\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ x ∈ {x | (fun x => f x ≤ f a) x}",
"tactic": "refine' ContinuousWithinAt.closure_le _ _ continuousWithinAt_const hU"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ U ∈ 𝓝[s] x",
"state_before": "case intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ x ∈ closure (U ∩ s)",
"tactic": "rwa [mem_closure_iff_nhdsWithin_neBot, nhdsWithin_inter_of_mem, ←\n mem_closure_iff_nhdsWithin_neBot]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ U ∈ 𝓝[s] x",
"tactic": "exact nhdsWithin_le_nhds (Uo.mem_nhds hxU)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : Preorder Y\ninst✝ : OrderClosedTopology Y\nf g : X → Y\ns : Set X\na : X\nh : IsLocalMaxOn f s a\nhc : ContinuousOn f (closure s)\nU : Set X\nUo : IsOpen U\naU : a ∈ U\nhU : U ∩ s ⊆ {x | (fun x => f x ≤ f a) x}\nx : X\nhxU : x ∈ U\nhxs : x ∈ closure s\n⊢ ContinuousWithinAt f (U ∩ s) x",
"tactic": "exact (hc _ hxs).mono ((inter_subset_right _ _).trans subset_closure)"
}
] |
[
54,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
11
] |
Mathlib/Topology/Basic.lean
|
isClosed_iUnion
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝¹ : TopologicalSpace α\ninst✝ : Finite ι\ns : ι → Set α\nh : ∀ (i : ι), IsOpen (s iᶜ)\n⊢ IsOpen (⋂ (i : ι), s iᶜ)",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝¹ : TopologicalSpace α\ninst✝ : Finite ι\ns : ι → Set α\nh : ∀ (i : ι), IsClosed (s i)\n⊢ IsClosed (⋃ (i : ι), s i)",
"tactic": "simp only [← isOpen_compl_iff, compl_iUnion] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝¹ : TopologicalSpace α\ninst✝ : Finite ι\ns : ι → Set α\nh : ∀ (i : ι), IsOpen (s iᶜ)\n⊢ IsOpen (⋂ (i : ι), s iᶜ)",
"tactic": "exact isOpen_iInter h"
}
] |
[
262,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/Analysis/Calculus/Taylor.lean
|
taylor_mean_remainder_lagrange
|
[
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "have gcont : ContinuousOn (fun t : ℝ => (x - t) ^ (n + 1)) (Icc x₀ x) := by\n refine' Continuous.continuousOn _\n exact (continuous_const.sub continuous_id').pow _"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "have xy_ne : ∀ y : ℝ, y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0 := by\n intro y hy\n refine' pow_ne_zero _ _\n rw [mem_Ioo] at hy\n rw [sub_ne_zero]\n exact hy.2.ne.symm"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "have hg' : ∀ y : ℝ, y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0 := fun y hy =>\n mul_ne_zero (neg_ne_zero.mpr (Nat.cast_add_one_ne_zero n)) (xy_ne y hy)"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "rcases taylor_mean_remainder hx hf hf' gcont (fun y _ => monomial_has_deriv_aux y x _) hg' with\n ⟨y, hy, h⟩"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ ∃ x' x_1,\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "use y, hy"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n ((x - y) ^ n / ↑n ! * ((x - x) ^ (n + 1) - (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "simp only [sub_self, zero_pow', Ne.def, Nat.succ_ne_zero, not_false_iff, zero_sub, mul_neg] at h"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ ((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1) / ((↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "rw [h, neg_div, ← div_neg, neg_mul, neg_neg]"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / (↑n ! * ((↑n + 1) * (x - y) ^ n)) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ ((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1) / ((↑n + 1) * (x - y) ^ n)) • iteratedDerivWithin (n + 1) f (Icc x₀ x) y =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ↑(n + 1)!",
"tactic": "field_simp"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / ((x - y) ^ n * (↑n ! * (↑n + 1))) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / (↑n ! * ((↑n + 1) * (x - y) ^ n)) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"tactic": "conv_lhs =>\n arg 2\n rw [← mul_assoc, mul_comm]"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / (↑n ! * (↑n + 1)) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - y) ^ n * (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / ((x - y) ^ n * (↑n ! * (↑n + 1))) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"tactic": "rw [mul_assoc, mul_div_mul_left _ _ (xy_ne y hy)]"
},
{
"state_after": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / ((↑n + 1) * ↑n !) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / (↑n ! * (↑n + 1)) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"tactic": "conv_lhs =>\n arg 2\n rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\n𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\nxy_ne : ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0\nhg' : ∀ (y : ℝ), y ∈ Ioo x₀ x → -(↑n + 1) * (x - y) ^ n ≠ 0\ny : ℝ\nhy : y ∈ Ioo x₀ x\nh :\n f x - taylorWithinEval f n (Icc x₀ x) x₀ x =\n (-((x - y) ^ n / ↑n ! * (x - x₀) ^ (n + 1)) / (-(↑n + 1) * (x - y) ^ n)) •\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y\n⊢ (x - x₀) ^ (n + 1) * iteratedDerivWithin (n + 1) f (Icc x₀ x) y / ((↑n + 1) * ↑n !) =\n iteratedDerivWithin (n + 1) f (Icc x₀ x) y * (x - x₀) ^ (n + 1) / ((↑n + 1) * ↑n !)",
"tactic": "nth_rw 1 [mul_comm]"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\n⊢ Continuous fun t => (x - t) ^ (n + 1)",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\n⊢ ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)",
"tactic": "refine' Continuous.continuousOn _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\n⊢ Continuous fun t => (x - t) ^ (n + 1)",
"tactic": "exact (continuous_const.sub continuous_id').pow _"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : y ∈ Ioo x₀ x\n⊢ (x - y) ^ n ≠ 0",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\n⊢ ∀ (y : ℝ), y ∈ Ioo x₀ x → (x - y) ^ n ≠ 0",
"tactic": "intro y hy"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : y ∈ Ioo x₀ x\n⊢ x - y ≠ 0",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : y ∈ Ioo x₀ x\n⊢ (x - y) ^ n ≠ 0",
"tactic": "refine' pow_ne_zero _ _"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : x₀ < y ∧ y < x\n⊢ x - y ≠ 0",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : y ∈ Ioo x₀ x\n⊢ x - y ≠ 0",
"tactic": "rw [mem_Ioo] at hy"
},
{
"state_after": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : x₀ < y ∧ y < x\n⊢ x ≠ y",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : x₀ < y ∧ y < x\n⊢ x - y ≠ 0",
"tactic": "rw [sub_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.315018\nE : Type ?u.315021\nF : Type ?u.315024\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → ℝ\nx x₀ : ℝ\nn : ℕ\nhx : x₀ < x\nhf : ContDiffOn ℝ (↑n) f (Icc x₀ x)\nhf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)\ngcont : ContinuousOn (fun t => (x - t) ^ (n + 1)) (Icc x₀ x)\ny : ℝ\nhy : x₀ < y ∧ y < x\n⊢ x ≠ y",
"tactic": "exact hy.2.ne.symm"
}
] |
[
303,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
nnnorm_le_nnnorm_add_nnnorm_div
|
[] |
[
953,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
952,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.JordanDecomposition.real_smul_posPart_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13833\ninst✝ : MeasurableSpace α\nj : JordanDecomposition α\nr : ℝ\nhr : r < 0\n⊢ (r • j).posPart = Real.toNNReal (-r) • j.negPart",
"tactic": "rw [real_smul_def, ← smul_negPart, if_neg (not_le.2 hr), neg_posPart]"
}
] |
[
163,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
StrictAnti.inv
|
[] |
[
1324,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1323,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.not_nontrivial_singleton
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\nx : α\nH : ∃ y, y ∈ {x} ∧ y ≠ x\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\nx : α\nH : Set.Nontrivial {x}\n⊢ False",
"tactic": "rw [nontrivial_iff_exists_ne (mem_singleton x)] at H"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\nx : α\nH : ∃ y, y ∈ {x} ∧ y ≠ x\ny : α\nhy : y ∈ {x}\nhya : y ≠ x\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\nx : α\nH : ∃ y, y ∈ {x} ∧ y ≠ x\n⊢ False",
"tactic": "let ⟨y, hy, hya⟩ := H"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\nx : α\nH : ∃ y, y ∈ {x} ∧ y ≠ x\ny : α\nhy : y ∈ {x}\nhya : y ≠ x\n⊢ False",
"tactic": "exact hya (mem_singleton_iff.1 hy)"
}
] |
[
2543,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2540,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.le_add_of_sub_left_le
|
[
{
"state_after": "a b c : Int\nh✝ : a - b ≤ c\nh : a - b + b ≤ c + b\n⊢ a ≤ b + c",
"state_before": "a b c : Int\nh : a - b ≤ c\n⊢ a ≤ b + c",
"tactic": "have h := Int.add_le_add_right h b"
},
{
"state_after": "no goals",
"state_before": "a b c : Int\nh✝ : a - b ≤ c\nh : a - b + b ≤ c + b\n⊢ a ≤ b + c",
"tactic": "rwa [Int.sub_add_cancel, Int.add_comm] at h"
}
] |
[
984,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
982,
11
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.sum_eapproxDiff
|
[
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\n⊢ ∑ k in Finset.range (Nat.zero + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f Nat.zero) a\n\ncase succ\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ∑ k in Finset.range (Nat.succ n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f (Nat.succ n)) a",
"state_before": "α : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\nn : ℕ\na : α\n⊢ ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a",
"tactic": "induction' n with n IH"
},
{
"state_after": "case zero\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\n⊢ ↑(↑(eapproxDiff f 0) a) = ↑(eapprox f 0) a",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\n⊢ ∑ k in Finset.range (Nat.zero + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f Nat.zero) a",
"tactic": "simp only [Nat.zero_eq, Nat.zero_add, Finset.sum_singleton, Finset.range_one]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\n⊢ ↑(↑(eapproxDiff f 0) a) = ↑(eapprox f 0) a",
"tactic": "rfl"
},
{
"state_after": "case succ\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ↑(eapprox f (Nat.add n 0 + 1)) a - ↑(eapprox f (Nat.add n 0)) a ≠ ⊤",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ∑ k in Finset.range (Nat.succ n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f (Nat.succ n)) a",
"tactic": "erw [Finset.sum_range_succ, Nat.succ_eq_add_one, IH, eapproxDiff, coe_map, Function.comp_apply,\n coe_sub, Pi.sub_apply, ENNReal.coe_toNNReal,\n add_tsub_cancel_of_le (monotone_eapprox f (Nat.le_succ _) _)]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ↑(eapprox f (Nat.add n 0 + 1)) a - ↑(eapprox f (Nat.add n 0)) a ≤ ↑(eapprox f (n + 1)) a",
"state_before": "case succ\nα : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ↑(eapprox f (Nat.add n 0 + 1)) a - ↑(eapprox f (Nat.add n 0)) a ≠ ⊤",
"tactic": "apply (lt_of_le_of_lt _ (eapprox_lt_top f (n + 1) a)).ne"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ↑(eapprox f (Nat.add n 0 + 1)) a ≤ ↑(eapprox f (n + 1)) a + ↑(eapprox f (Nat.add n 0)) a",
"state_before": "α : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ↑(eapprox f (Nat.add n 0 + 1)) a - ↑(eapprox f (Nat.add n 0)) a ≤ ↑(eapprox f (n + 1)) a",
"tactic": "rw [tsub_le_iff_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.843173\nγ : Type ?u.843176\nδ : Type ?u.843179\ninst✝ : MeasurableSpace α\nK : Type ?u.843185\nf : α → ℝ≥0∞\na : α\nn : ℕ\nIH : ∑ k in Finset.range (n + 1), ↑(↑(eapproxDiff f k) a) = ↑(eapprox f n) a\n⊢ ↑(eapprox f (Nat.add n 0 + 1)) a ≤ ↑(eapprox f (n + 1)) a + ↑(eapprox f (Nat.add n 0)) a",
"tactic": "exact le_self_add"
}
] |
[
949,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
939,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean
|
tendsto_floor_right_pure
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5764\nγ : Type ?u.5767\ninst✝³ : LinearOrderedRing α\ninst✝² : FloorRing α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ Tendsto floor (𝓝[Ici ↑n] ↑n) (pure n)",
"tactic": "simpa only [floor_intCast] using tendsto_floor_right_pure_floor (n : α)"
}
] |
[
77,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Algebra/Order/Ring/WithTop.lean
|
WithBot.mul_def
|
[] |
[
235,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/AlgebraicTopology/DoldKan/Faces.lean
|
AlgebraicTopology.DoldKan.HigherFacesVanish.comp_Hσ_eq_zero
|
[
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\n⊢ φ ≫\n (HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 1) n ≫\n hσ' q n (n + 1) (_ : ComplexShape.Rel c (n + 1) n) +\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1)) =\n 0",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\n⊢ φ ≫ HomologicalComplex.Hom.f (Hσ q) (n + 1) = 0",
"tactic": "simp only [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl)]"
},
{
"state_after": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\n⊢ φ ≫\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1) =\n 0",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\n⊢ φ ≫\n (HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 1) n ≫\n hσ' q n (n + 1) (_ : ComplexShape.Rel c (n + 1) n) +\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1)) =\n 0",
"tactic": "rw [hσ'_eq_zero hqn (c_mk (n + 1) n rfl), comp_zero, zero_add]"
},
{
"state_after": "case pos\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : n + 1 < q\n⊢ φ ≫\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1) =\n 0\n\ncase neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1) =\n 0",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\n⊢ φ ≫\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1) =\n 0",
"tactic": "by_cases hqn' : n + 1 < q"
},
{
"state_after": "no goals",
"state_before": "case pos\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : n + 1 < q\n⊢ φ ≫\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1) =\n 0",
"tactic": "rw [hσ'_eq_zero hqn' (c_mk (n + 2) (n + 1) rfl), zero_comp, comp_zero]"
},
{
"state_after": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ (Finset.sum Finset.univ fun j => φ ≫ σ X 0 ≫ ((-1) ^ ↑j • δ X j)) = 0",
"state_before": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫\n hσ' q (n + 1) (n + 2) (_ : ComplexShape.Rel c (n + 2) (n + 1)) ≫\n HomologicalComplex.d (AlternatingFaceMapComplex.obj X) (n + 2) (n + 1) =\n 0",
"tactic": "simp only [hσ'_eq (show n + 1 = 0 + q by linarith) (c_mk (n + 2) (n + 1) rfl), pow_zero,\n Fin.mk_zero, one_zsmul, eqToHom_refl, comp_id, comp_sum,\n AlternatingFaceMapComplex.obj_d_eq]"
},
{
"state_after": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ (Finset.sum Finset.univ fun i =>\n φ ≫\n σ X 0 ≫\n ((-1) ^ ↑(↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.castLE (_ : 2 ≤ 2 + (n + 1))) i)) •\n δ X (↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.castLE (_ : 2 ≤ 2 + (n + 1))) i)))) =\n 0\n\ncase neg.hf\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ ∀ (j : Fin (n + 1)),\n φ ≫\n σ X 0 ≫\n ((-1) ^ ↑(↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j)) •\n δ X (↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j))) =\n 0",
"state_before": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ (Finset.sum Finset.univ fun j => φ ≫ σ X 0 ≫ ((-1) ^ ↑j • δ X j)) = 0",
"tactic": "rw [← Fin.sum_congr' _ (show 2 + (n + 1) = n + 1 + 2 by linarith), Fin.sum_trunc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ n + 1 = 0 + q",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ 2 + (n + 1) = n + 1 + 2",
"tactic": "linarith"
},
{
"state_after": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫ σ X 0 ≫ ((-1) ^ 0 • δ X { val := 0, isLt := (_ : 0 < n + 1 + 2) }) +\n φ ≫ σ X 0 ≫ ((-1) ^ 1 • δ X { val := 1, isLt := (_ : 1 < n + 1 + 2) }) =\n 0",
"state_before": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ (Finset.sum Finset.univ fun i =>\n φ ≫\n σ X 0 ≫\n ((-1) ^ ↑(↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.castLE (_ : 2 ≤ 2 + (n + 1))) i)) •\n δ X (↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.castLE (_ : 2 ≤ 2 + (n + 1))) i)))) =\n 0",
"tactic": "simp only [Fin.sum_univ_castSucc, Fin.sum_univ_zero, zero_add, Fin.last, Fin.castLE_mk,\n Fin.cast_mk, Fin.castSucc_mk]"
},
{
"state_after": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫ σ X 0 ≫ δ X 0 + -φ ≫ σ X 0 ≫ δ X 1 = 0",
"state_before": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫ σ X 0 ≫ ((-1) ^ 0 • δ X { val := 0, isLt := (_ : 0 < n + 1 + 2) }) +\n φ ≫ σ X 0 ≫ ((-1) ^ 1 • δ X { val := 1, isLt := (_ : 1 < n + 1 + 2) }) =\n 0",
"tactic": "simp only [Fin.mk_zero, Fin.val_zero, pow_zero, one_zsmul, Fin.mk_one, Fin.val_one, pow_one,\n neg_smul, comp_neg]"
},
{
"state_after": "no goals",
"state_before": "case neg\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ φ ≫ σ X 0 ≫ δ X 0 + -φ ≫ σ X 0 ≫ δ X 1 = 0",
"tactic": "erw [δ_comp_σ_self, δ_comp_σ_succ, add_right_neg]"
},
{
"state_after": "case neg.hf\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ φ ≫\n σ X 0 ≫\n ((-1) ^ ↑(↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j)) •\n δ X (↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j))) =\n 0",
"state_before": "case neg.hf\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\n⊢ ∀ (j : Fin (n + 1)),\n φ ≫\n σ X 0 ≫\n ((-1) ^ ↑(↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j)) •\n δ X (↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j))) =\n 0",
"tactic": "intro j"
},
{
"state_after": "case neg.hf\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ φ ≫ σ X 0 ≫ ((-1) ^ (2 + ↑j) • δ X { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }) = 0",
"state_before": "case neg.hf\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ φ ≫\n σ X 0 ≫\n ((-1) ^ ↑(↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j)) •\n δ X (↑(Fin.cast (_ : 2 + (n + 1) = n + 1 + 2)) (↑(Fin.natAdd 2) j))) =\n 0",
"tactic": "dsimp [Fin.cast, Fin.castLE, Fin.castLT]"
},
{
"state_after": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n\ncase neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) ≠\n 0\n\ncase neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q\n\ncase neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n\ncase neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }",
"state_before": "case neg.hf\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ φ ≫ σ X 0 ≫ ((-1) ^ (2 + ↑j) • δ X { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }) = 0",
"tactic": "rw [comp_zsmul, comp_zsmul, δ_comp_σ_of_gt', v.comp_δ_eq_zero_assoc, zero_comp, zsmul_zero]"
},
{
"state_after": "case neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) ≠\n 0\n\ncase neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q",
"state_before": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n\ncase neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) ≠\n 0\n\ncase neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q\n\ncase neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n\ncase neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }",
"tactic": ". simp only [Fin.lt_iff_val_lt_val]\n dsimp [Fin.succ]\n linarith"
},
{
"state_after": "case neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q",
"state_before": "case neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) ≠\n 0\n\ncase neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q",
"tactic": ". intro h\n simp only [Fin.pred, Fin.ext_iff, Nat.pred_eq_sub_one, Nat.succ_add_sub_one, Fin.val_zero,\n add_eq_zero, false_and] at h"
},
{
"state_after": "no goals",
"state_before": "case neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q",
"tactic": ". simp only [Fin.pred, Nat.pred_eq_sub_one, Nat.succ_add_sub_one]\n linarith"
},
{
"state_after": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ ↑(Fin.succ 0) < 2 + ↑j",
"state_before": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.succ 0 < { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }",
"tactic": "simp only [Fin.lt_iff_val_lt_val]"
},
{
"state_after": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ 0 + 1 < 2 + ↑j",
"state_before": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ ↑(Fin.succ 0) < 2 + ↑j",
"tactic": "dsimp [Fin.succ]"
},
{
"state_after": "no goals",
"state_before": "case neg.hf.H\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ 0 + 1 < 2 + ↑j",
"tactic": "linarith"
},
{
"state_after": "case neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\nh :\n Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) =\n 0\n⊢ False",
"state_before": "case neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) ≠\n 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "case neg.hf.hj₁\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\nh :\n Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False) =\n 0\n⊢ False",
"tactic": "simp only [Fin.pred, Fin.ext_iff, Nat.pred_eq_sub_one, Nat.succ_add_sub_one, Fin.val_zero,\n add_eq_zero, false_and] at h"
},
{
"state_after": "case neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤ 1 + ↑j + q",
"state_before": "case neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤\n ↑(Fin.pred { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) }\n (_ : { val := 2 + ↑j, isLt := (_ : ↑(↑(Fin.natAdd 2) j) < n + 1 + 2) } = 0 → False)) +\n q",
"tactic": "simp only [Fin.pred, Nat.pred_eq_sub_one, Nat.succ_add_sub_one]"
},
{
"state_after": "no goals",
"state_before": "case neg.hf.hj₂\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : Preadditive C\nX : SimplicialObject C\nY : C\nn q : ℕ\nφ : Y ⟶ X.obj [n + 1].op\nv : HigherFacesVanish q φ\nhqn : n < q\nhqn' : ¬n + 1 < q\nj : Fin (n + 1)\n⊢ n + 2 ≤ 1 + ↑j + q",
"tactic": "linarith"
}
] |
[
169,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean
|
Polynomial.Monic.geom_sum'
|
[] |
[
249,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
SimpleGraph.coe_cliqueFinset
|
[] |
[
293,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.hasSum_log_one_add_inv
|
[
{
"state_after": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\n⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))",
"state_before": "a : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\n⊢ HasSum (fun k => 2 * (1 / (2 * ↑k + 1)) * (1 / (2 * a + 1)) ^ (2 * k + 1)) (log (1 + a⁻¹))",
"tactic": "convert hasSum_log_sub_log_of_abs_lt_1 h₁ using 1"
},
{
"state_after": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\n⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))",
"state_before": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\n⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))",
"tactic": "have h₂ : (2 : ℝ) * a + 1 ≠ 0 := by linarith"
},
{
"state_after": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))",
"state_before": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\n⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))",
"tactic": "have h₃ := h.ne'"
},
{
"state_after": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ log (1 + a⁻¹) = log ((1 + 1 / (2 * a + 1)) / (1 - 1 / (2 * a + 1)))\n\ncase h.e'_6.hx\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ 1 + 1 / (2 * a + 1) ≠ 0\n\ncase h.e'_6.hy\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ 1 - 1 / (2 * a + 1) ≠ 0",
"state_before": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ log (1 + a⁻¹) = log (1 + 1 / (2 * a + 1)) - log (1 - 1 / (2 * a + 1))",
"tactic": "rw [← log_div]"
},
{
"state_after": "a : ℝ\nh : 0 < a\n⊢ 1 < 2 * a + 1\n\na : ℝ\nh : 0 < a\n⊢ 0 < 2 * a + 1\n\na : ℝ\nh : 0 < a\n⊢ 0 < 1 / (2 * a + 1)",
"state_before": "a : ℝ\nh : 0 < a\n⊢ abs (1 / (2 * a + 1)) < 1",
"tactic": "rw [abs_of_pos, div_lt_one]"
},
{
"state_after": "no goals",
"state_before": "a : ℝ\nh : 0 < a\n⊢ 1 < 2 * a + 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "a : ℝ\nh : 0 < a\n⊢ 0 < 2 * a + 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "a : ℝ\nh : 0 < a\n⊢ 0 < 1 / (2 * a + 1)",
"tactic": "exact div_pos one_pos (by linarith)"
},
{
"state_after": "no goals",
"state_before": "a : ℝ\nh : 0 < a\n⊢ 0 < 2 * a + 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "a : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\n⊢ 2 * a + 1 ≠ 0",
"tactic": "linarith"
},
{
"state_after": "case h.e'_6.e_x\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ 1 + a⁻¹ = (1 + 1 / (2 * a + 1)) / (1 - 1 / (2 * a + 1))",
"state_before": "case h.e'_6\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ log (1 + a⁻¹) = log ((1 + 1 / (2 * a + 1)) / (1 - 1 / (2 * a + 1)))",
"tactic": "congr"
},
{
"state_after": "case h.e'_6.e_x\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ (a + 1) * (2 * a) = (2 * a + 1 + 1) * a",
"state_before": "case h.e'_6.e_x\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ 1 + a⁻¹ = (1 + 1 / (2 * a + 1)) / (1 - 1 / (2 * a + 1))",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6.e_x\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ (a + 1) * (2 * a) = (2 * a + 1 + 1) * a",
"tactic": "linarith"
},
{
"state_after": "case h.e'_6.hx\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ ¬2 * a + 1 + 1 = 0",
"state_before": "case h.e'_6.hx\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ 1 + 1 / (2 * a + 1) ≠ 0",
"tactic": "field_simp"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6.hx\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ ¬2 * a + 1 + 1 = 0",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6.hy\na : ℝ\nh : 0 < a\nh₁ : abs (1 / (2 * a + 1)) < 1\nh₂ : 2 * a + 1 ≠ 0\nh₃ : a ≠ 0\n⊢ 1 - 1 / (2 * a + 1) ≠ 0",
"tactic": "field_simp"
}
] |
[
337,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
AffineEquiv.coe_toHomeomorphOfFiniteDimensional
|
[] |
[
155,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
Submodule.fst_sup_snd
|
[
{
"state_after": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\n⊢ ⊤ ≤ fst R M M₂ ⊔ snd R M M₂",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\n⊢ fst R M M₂ ⊔ snd R M M₂ = ⊤",
"tactic": "rw [eq_top_iff]"
},
{
"state_after": "case mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, n) ∈ fst R M M₂ ⊔ snd R M M₂",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\n⊢ ⊤ ≤ fst R M M₂ ⊔ snd R M M₂",
"tactic": "rintro ⟨m, n⟩ -"
},
{
"state_after": "case mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, 0) + (0, n) ∈ fst R M M₂ ⊔ snd R M M₂",
"state_before": "case mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, n) ∈ fst R M M₂ ⊔ snd R M M₂",
"tactic": "rw [show (m, n) = (m, 0) + (0, n) by simp]"
},
{
"state_after": "case mk.h₁\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, 0) ∈ fst R M M₂ ⊔ snd R M M₂\n\ncase mk.h₂\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (0, n) ∈ fst R M M₂ ⊔ snd R M M₂",
"state_before": "case mk\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, 0) + (0, n) ∈ fst R M M₂ ⊔ snd R M M₂",
"tactic": "apply Submodule.add_mem (Submodule.fst R M M₂ ⊔ Submodule.snd R M M₂)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, n) = (m, 0) + (0, n)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mk.h₁\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (m, 0) ∈ fst R M M₂ ⊔ snd R M M₂",
"tactic": "exact Submodule.mem_sup_left (Submodule.mem_comap.mpr (by simp))"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ ↑(LinearMap.snd R M M₂) (m, 0) ∈ ⊥",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case mk.h₂\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ (0, n) ∈ fst R M M₂ ⊔ snd R M M₂",
"tactic": "exact Submodule.mem_sup_right (Submodule.mem_comap.mpr (by simp))"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.365753\nM₆ : Type ?u.365756\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nm : M\nn : M₂\n⊢ ↑(LinearMap.fst R M M₂) (0, n) ∈ ⊥",
"tactic": "simp"
}
] |
[
682,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
676,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.swap_apply_right
|
[
{
"state_after": "no goals",
"state_before": "α : Sort u_1\ninst✝ : DecidableEq α\na b : α\n⊢ ↑(swap a b) b = a",
"tactic": "by_cases h:b = a <;> simp [swap_apply_def, h]"
}
] |
[
1569,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1568,
1
] |
Mathlib/RingTheory/Localization/Integral.lean
|
IsLocalization.integerNormalization_eval₂_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝⁵ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\nP : Type ?u.47235\ninst✝² : CommRing P\ninst✝¹ : IsLocalization M S\nR' : Type u_2\ninst✝ : CommRing R'\ng : S →+* R'\np : S[X]\nx : R'\nhx : eval₂ g x p = 0\nb : { x // x ∈ M }\nhb : Polynomial.map (algebraMap R S) (integerNormalization M p) = ↑b • p\n⊢ eval₂ g x (Polynomial.map (algebraMap R S) (integerNormalization M p)) = 0",
"tactic": "rw [hb, ← IsScalarTower.algebraMap_smul S (b : R) p, eval₂_smul, hx, MulZeroClass.mul_zero]"
}
] |
[
115,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.coe_map
|
[] |
[
1041,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1040,
1
] |
Mathlib/Algebra/GCDMonoid/Finset.lean
|
Finset.dvd_lcm
|
[] |
[
76,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
AntitoneOn.image_lowerBounds_subset_upperBounds_image
|
[] |
[
1230,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1228,
1
] |
Std/Data/List/Lemmas.lean
|
List.modifyNthTail_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf : List α → List α\nn : Nat\nl₁ l₂ : List α\n⊢ modifyNthTail f (length l₁ + n) (l₁ ++ l₂) = l₁ ++ modifyNthTail f n l₂",
"tactic": "induction l₁ <;> simp [*, Nat.succ_add]"
}
] |
[
745,
42
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
743,
1
] |
Mathlib/Data/Multiset/Sections.lean
|
Multiset.sections_zero
|
[] |
[
35,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.Tape.move_left_right
|
[
{
"state_after": "case mk\nΓ : Type u_1\ninst✝ : Inhabited Γ\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ move Dir.right (move Dir.left { head := head✝, left := left✝, right := right✝ }) =\n { head := head✝, left := left✝, right := right✝ }",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nT : Tape Γ\n⊢ move Dir.right (move Dir.left T) = T",
"tactic": "cases T"
},
{
"state_after": "no goals",
"state_before": "case mk\nΓ : Type u_1\ninst✝ : Inhabited Γ\nhead✝ : Γ\nleft✝ right✝ : ListBlank Γ\n⊢ move Dir.right (move Dir.left { head := head✝, left := left✝, right := right✝ }) =\n { head := head✝, left := left✝, right := right✝ }",
"tactic": "simp [Tape.move]"
}
] |
[
538,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
536,
1
] |
Mathlib/Analysis/Calculus/Deriv/Add.lean
|
HasDerivWithinAt.const_add
|
[] |
[
132,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
8
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.measurable_measure
|
[] |
[
76,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.Bicone.toCone_π_app_mk
|
[] |
[
120,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/NumberTheory/LucasLehmer.lean
|
LucasLehmer.X.zero_snd
|
[] |
[
208,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
9
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.zero_comp
|
[] |
[
339,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean
|
CategoryTheory.GrothendieckTopology.Plus.sep
|
[
{
"state_after": "case intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\ny : (forget D).obj ((plusObj J P).obj X.op)\nSx : Cover J X\nx : Meq P Sx\nh :\n ∀ (I : Cover.Arrow S), (forget D).map ((plusObj J P).map I.f.op) (mk x) = (forget D).map ((plusObj J P).map I.f.op) y\n⊢ mk x = y",
"state_before": "C : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\nx y : (forget D).obj ((plusObj J P).obj X.op)\nh : ∀ (I : Cover.Arrow S), (forget D).map ((plusObj J P).map I.f.op) x = (forget D).map ((plusObj J P).map I.f.op) y\n⊢ x = y",
"tactic": "obtain ⟨Sx, x, rfl⟩ := exists_rep x"
},
{
"state_after": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh :\n ∀ (I : Cover.Arrow S),\n (forget D).map ((plusObj J P).map I.f.op) (mk x) = (forget D).map ((plusObj J P).map I.f.op) (mk y)\n⊢ mk x = mk y",
"state_before": "case intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS : Cover J X\ny : (forget D).obj ((plusObj J P).obj X.op)\nSx : Cover J X\nx : Meq P Sx\nh :\n ∀ (I : Cover.Arrow S), (forget D).map ((plusObj J P).map I.f.op) (mk x) = (forget D).map ((plusObj J P).map I.f.op) y\n⊢ mk x = y",
"tactic": "obtain ⟨Sy, y, rfl⟩ := exists_rep y"
},
{
"state_after": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\n⊢ mk x = mk y",
"state_before": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh :\n ∀ (I : Cover.Arrow S),\n (forget D).map ((plusObj J P).map I.f.op) (mk x) = (forget D).map ((plusObj J P).map I.f.op) (mk y)\n⊢ mk x = mk y",
"tactic": "simp only [res_mk_eq_mk_pullback] at h"
},
{
"state_after": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\n⊢ mk x = mk y",
"state_before": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\n⊢ mk x = mk y",
"tactic": "choose W h1 h2 hh using fun I : S.Arrow => (eq_mk_iff_exists _ _).mp (h I)"
},
{
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"tactic": "rw [eq_mk_iff_exists]"
},
{
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"tactic": "let B : J.Cover X := S.bind W"
},
{
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"tactic": "use B"
},
{
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"state_before": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\n⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2",
"tactic": "let ex : B ⟶ Sx :=\n homOfLE\n (by\n rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩\n rw [← hee]\n apply leOfHom (h1 ⟨_, _, he2⟩)\n exact he1)"
},
{
"state_after": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\n⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2",
"state_before": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\n⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2",
"tactic": "let ey : B ⟶ Sy :=\n homOfLE\n (by\n rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩\n rw [← hee]\n apply leOfHom (h2 ⟨_, _, he2⟩)\n exact he1)"
},
{
"state_after": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\n⊢ Meq.refine x ex = Meq.refine y ey",
"state_before": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\n⊢ ∃ h1 h2, Meq.refine x h1 = Meq.refine y h2",
"tactic": "use ex, ey"
},
{
"state_after": "case intro.intro.intro.intro.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine y ey) I",
"state_before": "case intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\n⊢ Meq.refine x ex = Meq.refine y ey",
"tactic": "ext1 I"
},
{
"state_after": "case intro.intro.intro.intro.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine y ey) I",
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"tactic": "let IS : S.Arrow := I.fromMiddle"
},
{
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"state_before": "case intro.intro.intro.intro.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine y ey) I",
"tactic": "specialize hh IS"
},
{
"state_after": "case intro.intro.intro.intro.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nhh : Meq.refine (Meq.pullback x IS.f) (h1 IS) = Meq.refine (Meq.pullback y IS.f) (h2 IS)\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine y ey) I",
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"tactic": "let IW : (W IS).Arrow := I.toMiddle"
},
{
"state_after": "case intro.intro.intro.intro.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine y ey) I",
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"tactic": "apply_fun fun e => e IW at hh"
},
{
"state_after": "case h.e'_2.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW\n\ncase h.e'_3.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ ↑(Meq.refine y ey) I = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW",
"state_before": "case intro.intro.intro.intro.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine y ey) I",
"tactic": "convert hh using 1"
},
{
"state_after": "case intro.intro.intro.intro.intro\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\nY : C\nf : Y ⟶ X\nZ : C\ne1 : Y ⟶ Z\ne2 : Z ⟶ X\nhe2 : (Cover.sieve S).arrows e2\nhe1 : ((fun Y f hf => Cover.sieve (W { Y := Y, f := f, hf := hf })) Z e2 he2).arrows e1\nhee : e1 ≫ e2 = f\n⊢ ((fun a => ↑a) Sx).arrows f",
"state_before": "C : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nhh : ∀ (I : Cover.Arrow S), Meq.refine (Meq.pullback x I.f) (h1 I) = Meq.refine (Meq.pullback y I.f) (h2 I)\nB : Cover J X := Cover.bind S W\n⊢ B ≤ Sx",
"tactic": "rintro Y f ⟨Z, e1, e2, he2, he1, hee⟩"
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{
"state_after": "case h.e'_2.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRx : Cover.Relation Sx :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sx).arrows I.f),\n h₂ := (_ : (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW",
"state_before": "case h.e'_2.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW",
"tactic": "let Rx : Sx.Relation :=\n ⟨I.Y, I.Y, I.Y, 𝟙 _, 𝟙 _, I.f, I.toMiddleHom ≫ I.fromMiddleHom, leOfHom ex _ I.hf,\n by simpa only [I.middle_spec] using leOfHom ex _ I.hf, by simp [I.middle_spec]⟩"
},
{
"state_after": "case h.e'_2.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRx : Cover.Relation Sx :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sx).arrows I.f),\n h₂ := (_ : (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\nthis :\n (forget D).map (P.map Rx.g₁.op) (↑x (MulticospanIndex.fstTo (Cover.index Sx P) Rx)) =\n (forget D).map (P.map Rx.g₂.op) (↑x (MulticospanIndex.sndTo (Cover.index Sx P) Rx))\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW",
"state_before": "case h.e'_2.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRx : Cover.Relation Sx :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sx).arrows I.f),\n h₂ := (_ : (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW",
"tactic": "have := x.condition Rx"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRx : Cover.Relation Sx :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sx).arrows I.f),\n h₂ := (_ : (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\nthis :\n (forget D).map (P.map Rx.g₁.op) (↑x (MulticospanIndex.fstTo (Cover.index Sx P) Rx)) =\n (forget D).map (P.map Rx.g₂.op) (↑x (MulticospanIndex.sndTo (Cover.index Sx P) Rx))\n⊢ ↑(Meq.refine x ex) I = ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW",
"tactic": "simpa using this"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ (Cover.sieve Sx).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)",
"tactic": "simpa only [I.middle_spec] using leOfHom ex _ I.hf"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I",
"tactic": "simp [I.middle_spec]"
},
{
"state_after": "case h.e'_3.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRy : Cover.Relation Sy :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sy).arrows I.f),\n h₂ := (_ : (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\n⊢ ↑(Meq.refine y ey) I = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW",
"state_before": "case h.e'_3.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ ↑(Meq.refine y ey) I = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW",
"tactic": "let Ry : Sy.Relation :=\n ⟨I.Y, I.Y, I.Y, 𝟙 _, 𝟙 _, I.f, I.toMiddleHom ≫ I.fromMiddleHom, leOfHom ey _ I.hf,\n by simpa only [I.middle_spec] using leOfHom ey _ I.hf, by simp [I.middle_spec]⟩"
},
{
"state_after": "case h.e'_3.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRy : Cover.Relation Sy :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sy).arrows I.f),\n h₂ := (_ : (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\nthis :\n (forget D).map (P.map Ry.g₁.op) (↑y (MulticospanIndex.fstTo (Cover.index Sy P) Ry)) =\n (forget D).map (P.map Ry.g₂.op) (↑y (MulticospanIndex.sndTo (Cover.index Sy P) Ry))\n⊢ ↑(Meq.refine y ey) I = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW",
"state_before": "case h.e'_3.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRy : Cover.Relation Sy :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sy).arrows I.f),\n h₂ := (_ : (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\n⊢ ↑(Meq.refine y ey) I = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW",
"tactic": "have := y.condition Ry"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h\nC : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\nRy : Cover.Relation Sy :=\n { Y₁ := I.Y, Y₂ := I.Y, Z := I.Y, g₁ := 𝟙 I.Y, g₂ := 𝟙 I.Y, f₁ := I.f,\n f₂ := Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I, h₁ := (_ : ((fun a => ↑a) Sy).arrows I.f),\n h₂ := (_ : (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)),\n w := (_ : 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I) }\nthis :\n (forget D).map (P.map Ry.g₁.op) (↑y (MulticospanIndex.fstTo (Cover.index Sy P) Ry)) =\n (forget D).map (P.map Ry.g₂.op) (↑y (MulticospanIndex.sndTo (Cover.index Sy P) Ry))\n⊢ ↑(Meq.refine y ey) I = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW",
"tactic": "simpa using this"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ (Cover.sieve Sy).arrows (Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I)",
"tactic": "simpa only [I.middle_spec] using leOfHom ey _ I.hf"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁶ : Category C\nJ : GrothendieckTopology C\nD : Type w\ninst✝⁵ : Category D\ninst✝⁴ : ConcreteCategory D\ninst✝³ : PreservesLimits (forget D)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\nX : C\nP : Cᵒᵖ ⥤ D\nS Sx : Cover J X\nx : Meq P Sx\nSy : Cover J X\ny : Meq P Sy\nh : ∀ (I : Cover.Arrow S), mk (Meq.pullback x I.f) = mk (Meq.pullback y I.f)\nW : (I : Cover.Arrow S) → Cover J I.Y\nh1 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sx\nh2 : (I : Cover.Arrow S) → W I ⟶ (pullback J I.f).obj Sy\nB : Cover J X := Cover.bind S W\nex : B ⟶ Sx := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sx).arrows f)\ney : B ⟶ Sy := homOfLE (_ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), ((fun a => ↑a) B).arrows f → ((fun a => ↑a) Sy).arrows f)\nI : Cover.Arrow B\nIS : Cover.Arrow S := Cover.Arrow.fromMiddle I\nIW : Cover.Arrow (W IS) := Cover.Arrow.toMiddle I\nhh : ↑(Meq.refine (Meq.pullback x IS.f) (h1 IS)) IW = ↑(Meq.refine (Meq.pullback y IS.f) (h2 IS)) IW\ne_1✝ : (forget D).obj (P.obj I.Y.op) = (forget D).obj (P.obj IW.Y.op)\n⊢ 𝟙 I.Y ≫ I.f = 𝟙 I.Y ≫ Cover.Arrow.toMiddleHom I ≫ Cover.Arrow.fromMiddleHom I",
"tactic": "simp [I.middle_spec]"
}
] |
[
321,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.isAlgebraic_iSup
|
[
{
"state_after": "case mk\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx : x ∈ ⨆ (i : ι), f i\n⊢ IsAlgebraic K { val := x, property := hx }",
"state_before": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\n⊢ Algebra.IsAlgebraic K { x // x ∈ ⨆ (i : ι), f i }",
"tactic": "rintro ⟨x, hx⟩"
},
{
"state_after": "case mk.intro\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx✝ : x ∈ ⨆ (i : ι), f i\ns : Finset ((i : ι) × { x // x ∈ f i })\nhx : x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), K⟮↑i.snd⟯\n⊢ IsAlgebraic K { val := x, property := hx✝ }",
"state_before": "case mk\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx : x ∈ ⨆ (i : ι), f i\n⊢ IsAlgebraic K { val := x, property := hx }",
"tactic": "obtain ⟨s, hx⟩ := exists_finset_of_mem_supr' hx"
},
{
"state_after": "case mk.intro\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx✝ : x ∈ ⨆ (i : ι), f i\ns : Finset ((i : ι) × { x // x ∈ f i })\nhx : x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), K⟮↑i.snd⟯\n⊢ IsAlgebraic K { val := x, property := hx }",
"state_before": "case mk.intro\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx✝ : x ∈ ⨆ (i : ι), f i\ns : Finset ((i : ι) × { x // x ∈ f i })\nhx : x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), K⟮↑i.snd⟯\n⊢ IsAlgebraic K { val := x, property := hx✝ }",
"tactic": "rw [isAlgebraic_iff, Subtype.coe_mk, ← Subtype.coe_mk (p := fun x => x ∈ _) x hx,\n ← isAlgebraic_iff]"
},
{
"state_after": "case mk.intro\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx✝ : x ∈ ⨆ (i : ι), f i\ns : Finset ((i : ι) × { x // x ∈ f i })\nhx : x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), K⟮↑i.snd⟯\nthis : ∀ (i : (i : ι) × { x // x ∈ f i }), FiniteDimensional K { x // x ∈ K⟮↑i.snd⟯ }\n⊢ IsAlgebraic K { val := x, property := hx }",
"state_before": "case mk.intro\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx✝ : x ∈ ⨆ (i : ι), f i\ns : Finset ((i : ι) × { x // x ∈ f i })\nhx : x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), K⟮↑i.snd⟯\n⊢ IsAlgebraic K { val := x, property := hx }",
"tactic": "haveI : ∀ i : Σ i, f i, FiniteDimensional K K⟮(i.2 : L)⟯ := fun ⟨i, x⟩ =>\n adjoin.finiteDimensional (isIntegral_iff.1 (isAlgebraic_iff_isIntegral.1 (h i x)))"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nK : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\nh : ∀ (i : ι), Algebra.IsAlgebraic K { x // x ∈ f i }\nx : L\nhx✝ : x ∈ ⨆ (i : ι), f i\ns : Finset ((i : ι) × { x // x ∈ f i })\nhx : x ∈ ⨆ (i : (i : ι) × { x // x ∈ f i }) (_ : i ∈ s), K⟮↑i.snd⟯\nthis : ∀ (i : (i : ι) × { x // x ∈ f i }), FiniteDimensional K { x // x ∈ K⟮↑i.snd⟯ }\n⊢ IsAlgebraic K { val := x, property := hx }",
"tactic": "apply Algebra.isAlgebraic_of_finite"
}
] |
[
1262,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1253,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.denom_X
|
[] |
[
1454,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1453,
1
] |
Std/Data/List/Lemmas.lean
|
List.suffix_nil
|
[] |
[
1644,
99
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1644,
9
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.SimpleFunc.posPart_sub_negPart
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\n⊢ map (fun b => max b 0) f - map (fun b => max b 0) (-f) = f",
"state_before": "α : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\n⊢ posPart f - negPart f = f",
"tactic": "simp only [posPart, negPart]"
},
{
"state_after": "case H\nα : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\na : α\n⊢ ↑(map (fun b => max b 0) f - map (fun b => max b 0) (-f)) a = ↑f a",
"state_before": "α : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\n⊢ map (fun b => max b 0) f - map (fun b => max b 0) (-f) = f",
"tactic": "ext a"
},
{
"state_after": "case H\nα : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\na : α\n⊢ (↑(map (fun b => max b 0) f) - ↑(map (fun b => max b 0) (-f))) a = ↑f a",
"state_before": "case H\nα : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\na : α\n⊢ ↑(map (fun b => max b 0) f - map (fun b => max b 0) (-f)) a = ↑f a",
"tactic": "rw [coe_sub]"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u_1\nE : Type ?u.94507\nF : Type ?u.94510\n𝕜 : Type ?u.94513\ninst✝² : LinearOrder E\ninst✝¹ : Zero E\ninst✝ : MeasurableSpace α\nf : α →ₛ ℝ\na : α\n⊢ (↑(map (fun b => max b 0) f) - ↑(map (fun b => max b 0) (-f))) a = ↑f a",
"tactic": "exact max_zero_sub_eq_self (f a)"
}
] |
[
289,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_surjective
|
[] |
[
303,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.zero_mem
|
[] |
[
133,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
11
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.openEmbedding_obj_top
|
[
{
"state_after": "case h\nX : TopCat\nU : Opens ↑X\n⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion U)))).obj ⊤) = ↑U",
"state_before": "X : TopCat\nU : Opens ↑X\n⊢ (IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion U)))).obj ⊤ = U",
"tactic": "ext1"
},
{
"state_after": "no goals",
"state_before": "case h\nX : TopCat\nU : Opens ↑X\n⊢ ↑((IsOpenMap.functor (_ : IsOpenMap ((forget TopCat).map (inclusion U)))).obj ⊤) = ↑U",
"tactic": "exact Set.image_univ.trans Subtype.range_coe"
}
] |
[
341,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_eq_zero
|
[
{
"state_after": "ι : Type ?u.869344\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoidWithZero β\nha : a ∈ s\nh : f a = 0\nthis : DecidableEq α\n⊢ ∏ x in s, f x = 0",
"state_before": "ι : Type ?u.869344\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoidWithZero β\nha : a ∈ s\nh : f a = 0\n⊢ ∏ x in s, f x = 0",
"tactic": "haveI := Classical.decEq α"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.869344\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoidWithZero β\nha : a ∈ s\nh : f a = 0\nthis : DecidableEq α\n⊢ ∏ x in s, f x = 0",
"tactic": "rw [← prod_erase_mul _ _ ha, h, mul_zero]"
}
] |
[
1907,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1905,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
Submodule.toLinearPMap_range
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁶ : Ring R\nE : Type u_3\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_2\ninst✝³ : AddCommGroup F\ninst✝² : Module R F\nG : Type ?u.674878\ninst✝¹ : AddCommGroup G\ninst✝ : Module R G\ng : Submodule R (E × F)\nhg : ∀ (x : E × F), x ∈ g → x.fst = 0 → x.snd = 0\n⊢ LinearMap.range (toLinearPMap g hg).toFun = map (LinearMap.snd R E F) g",
"tactic": "rw [← LinearPMap.graph_map_snd_eq_range, toLinearPMap_graph_eq]"
}
] |
[
1005,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
1
] |
Mathlib/Control/Bitraversable/Lemmas.lean
|
Bitraversable.tsnd_tfst
|
[
{
"state_after": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ β₀ β₁ : Type u\nf : α₀ → F α₁\nf' : β₀ → G β₁\nx : t α₀ β₀\n⊢ bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ map f' ∘ pure) x =\n bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ β₀ β₁ : Type u\nf : α₀ → F α₁\nf' : β₀ → G β₁\nx : t α₀ β₀\n⊢ Comp.mk (tsnd f' <$> tfst f x) = bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x",
"tactic": "rw [← comp_bitraverse]"
},
{
"state_after": "no goals",
"state_before": "t : Type u → Type u → Type u\ninst✝⁵ : Bitraversable t\nβ : Type u\nF G : Type u → Type u\ninst✝⁴ : Applicative F\ninst✝³ : Applicative G\ninst✝² : IsLawfulBitraversable t\ninst✝¹ : LawfulApplicative F\ninst✝ : LawfulApplicative G\nα₀ α₁ β₀ β₁ : Type u\nf : α₀ → F α₁\nf' : β₀ → G β₁\nx : t α₀ β₀\n⊢ bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ map f' ∘ pure) x =\n bitraverse (Comp.mk ∘ map pure ∘ f) (Comp.mk ∘ pure ∘ f') x",
"tactic": "simp only [Function.comp, map_pure]"
}
] |
[
97,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.exists_affineBasis
|
[] |
[
331,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.vadd_def
|
[] |
[
420,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
419,
1
] |
Mathlib/Algebra/Category/GroupCat/EpiMono.lean
|
CommGroupCat.range_eq_top_of_epi
|
[] |
[
436,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_inter_union_subset
|
[
{
"state_after": "α : Type u_1\nα' : Type ?u.50587\nβ : Type u_2\nβ' : Type ?u.50593\nγ : Type ?u.50596\nγ' : Type ?u.50599\nδ : Type ?u.50602\nδ' : Type ?u.50605\nε : Type ?u.50608\nε' : Type ?u.50611\nζ : Type ?u.50614\nζ' : Type ?u.50617\nν : Type ?u.50620\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Set α\nt✝ t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → α → β\ns t : Set α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 f (t ∩ s) (s ∪ t) ⊆ image2 f s t",
"state_before": "α : Type u_1\nα' : Type ?u.50587\nβ : Type u_2\nβ' : Type ?u.50593\nγ : Type ?u.50596\nγ' : Type ?u.50599\nδ : Type ?u.50602\nδ' : Type ?u.50605\nε : Type ?u.50608\nε' : Type ?u.50611\nζ : Type ?u.50614\nζ' : Type ?u.50617\nν : Type ?u.50620\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Set α\nt✝ t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → α → β\ns t : Set α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 f (s ∩ t) (s ∪ t) ⊆ image2 f s t",
"tactic": "rw [inter_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.50587\nβ : Type u_2\nβ' : Type ?u.50593\nγ : Type ?u.50596\nγ' : Type ?u.50599\nδ : Type ?u.50602\nδ' : Type ?u.50605\nε : Type ?u.50608\nε' : Type ?u.50611\nζ : Type ?u.50614\nζ' : Type ?u.50617\nν : Type ?u.50620\nf✝ f' : α → β → γ\ng g' : α → β → γ → δ\ns✝ s' : Set α\nt✝ t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\nf : α → α → β\ns t : Set α\nhf : ∀ (a b : α), f a b = f b a\n⊢ image2 f (t ∩ s) (s ∪ t) ⊆ image2 f s t",
"tactic": "exact image2_inter_union_subset_union.trans (union_subset (image2_comm hf).subset Subset.rfl)"
}
] |
[
465,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/RingTheory/Polynomial/Dickson.lean
|
Polynomial.dickson_one_one_charP
|
[
{
"state_after": "case h\nR : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\n⊢ 1 = ↑(ZMod.castHom (_ : p ∣ p) R) 1\n\nR : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nh : 1 = ↑(ZMod.castHom (_ : p ∣ p) R) 1\n⊢ dickson 1 1 p = X ^ p",
"state_before": "R : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\n⊢ dickson 1 1 p = X ^ p",
"tactic": "have h : (1 : R) = ZMod.castHom (dvd_refl p) R 1"
},
{
"state_after": "R : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nh : 1 = ↑(ZMod.castHom (_ : p ∣ p) R) 1\n⊢ dickson 1 1 p = X ^ p",
"state_before": "case h\nR : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\n⊢ 1 = ↑(ZMod.castHom (_ : p ∣ p) R) 1\n\nR : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nh : 1 = ↑(ZMod.castHom (_ : p ∣ p) R) 1\n⊢ dickson 1 1 p = X ^ p",
"tactic": "simp only [ZMod.castHom_apply, ZMod.cast_one']"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.318902\ninst✝³ : CommRing R\ninst✝² : CommRing S\nk : ℕ\na : R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nh : 1 = ↑(ZMod.castHom (_ : p ∣ p) R) 1\n⊢ dickson 1 1 p = X ^ p",
"tactic": "rw [h, ← map_dickson (ZMod.castHom (dvd_refl p) R), dickson_one_one_zmod_p, Polynomial.map_pow,\n map_X]"
}
] |
[
275,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
Antitone.imp
|
[] |
[
368,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/RingTheory/GradedAlgebra/Basic.lean
|
GradedAlgebra.proj_recompose
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type u_3\nA : Type u_2\nσ : Type ?u.286140\ninst✝⁵ : DecidableEq ι\ninst✝⁴ : AddMonoid ι\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\n𝒜 : ι → Submodule R A\ninst✝ : GradedAlgebra 𝒜\na : ⨁ (i : ι), { x // x ∈ 𝒜 i }\ni : ι\n⊢ ↑(proj 𝒜 i) (↑(decompose 𝒜).symm a) = ↑(decompose 𝒜).symm (↑(of (fun i => (fun i => { x // x ∈ 𝒜 i }) i) i) (↑a i))",
"tactic": "rw [GradedAlgebra.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]"
}
] |
[
235,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/CategoryTheory/Iso.lean
|
CategoryTheory.Iso.symm_hom
|
[] |
[
111,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidWithZeroHom.map_one
|
[] |
[
911,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
910,
11
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.coe_one
|
[] |
[
755,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
754,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.le_op_norm_of_le
|
[] |
[
215,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_count
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.1694198\nγ : Type ?u.1694201\nδ : Type ?u.1694204\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (∑' (i : α), ∫⁻ (a : α), f a ∂dirac i) = ∑' (a : α), f a",
"state_before": "α : Type u_1\nβ : Type ?u.1694198\nγ : Type ?u.1694201\nδ : Type ?u.1694204\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (∫⁻ (a : α), f a ∂count) = ∑' (a : α), f a",
"tactic": "rw [count, lintegral_sum_measure]"
},
{
"state_after": "case e_f\nα : Type u_1\nβ : Type ?u.1694198\nγ : Type ?u.1694201\nδ : Type ?u.1694204\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a",
"state_before": "α : Type u_1\nβ : Type ?u.1694198\nγ : Type ?u.1694201\nδ : Type ?u.1694204\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (∑' (i : α), ∫⁻ (a : α), f a ∂dirac i) = ∑' (a : α), f a",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_f\nα : Type u_1\nβ : Type ?u.1694198\nγ : Type ?u.1694201\nδ : Type ?u.1694204\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\nf : α → ℝ≥0∞\n⊢ (fun i => ∫⁻ (a : α), f a ∂dirac i) = fun a => f a",
"tactic": "exact funext fun a => lintegral_dirac a f"
}
] |
[
1405,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1401,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_eq_range_psigma
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.157506\nβ : Type u_1\nγ : Type ?u.157512\nι : Sort u_2\nι' : Sort ?u.157518\nι₂ : Sort ?u.157521\nκ : ι → Sort ?u.157526\nκ₁ : ι → Sort ?u.157531\nκ₂ : ι → Sort ?u.157536\nκ' : ι' → Sort ?u.157541\ns : ι → Set β\n⊢ (⋃ (i : ι), s i) = range fun a => ↑a.snd",
"tactic": "simp [Set.ext_iff]"
}
] |
[
1273,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1272,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.edist_comm
|
[
{
"state_after": "case inl\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp 0 β\n⊢ edist f g = edist g f\n\ncase inr.inl\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp ⊤ β\n⊢ edist f g = edist g f\n\ncase inr.inr\np : ℝ≥0∞\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp p β\nh : 0 < ENNReal.toReal p\n⊢ edist f g = edist g f",
"state_before": "p : ℝ≥0∞\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp p β\n⊢ edist f g = edist g f",
"tactic": "rcases p.trichotomy with (rfl | rfl | h)"
},
{
"state_after": "no goals",
"state_before": "case inl\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp 0 β\n⊢ edist f g = edist g f",
"tactic": "simp only [edist_eq_card, eq_comm, Ne.def]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp ⊤ β\n⊢ edist f g = edist g f",
"tactic": "simp only [edist_eq_iSup, edist_comm]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\np : ℝ≥0∞\n𝕜 : Type ?u.15998\n𝕜' : Type ?u.16001\nι : Type u_1\nα : ι → Type ?u.16009\nβ : ι → Type u_2\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nf g : PiLp p β\nh : 0 < ENNReal.toReal p\n⊢ edist f g = edist g f",
"tactic": "simp only [edist_eq_sum h, edist_comm]"
}
] |
[
185,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
11
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
hasSum_geometric_of_norm_lt_1
|
[
{
"state_after": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹",
"tactic": "have xi_ne_one : ξ ≠ 1 := by\n contrapose! h\n simp [h]"
},
{
"state_after": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹",
"tactic": "have A : Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹)) :=\n ((tendsto_pow_atTop_nhds_0_of_norm_lt_1 h).sub tendsto_const_nhds).mul tendsto_const_nhds"
},
{
"state_after": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Tendsto (fun n => ∑ i in Finset.range n, ξ ^ i) atTop (𝓝 (1 - ξ)⁻¹)\n\nα : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Summable fun i => ‖ξ ^ i‖",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ HasSum (fun n => ξ ^ n) (1 - ξ)⁻¹",
"tactic": "rw [hasSum_iff_tendsto_nat_of_summable_norm]"
},
{
"state_after": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ξ = 1\n⊢ 1 ≤ ‖ξ‖",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\n⊢ ξ ≠ 1",
"tactic": "contrapose! h"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ξ = 1\n⊢ 1 ≤ ‖ξ‖",
"tactic": "simp [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Tendsto (fun n => ∑ i in Finset.range n, ξ ^ i) atTop (𝓝 (1 - ξ)⁻¹)",
"tactic": "simpa [geom_sum_eq, xi_ne_one, neg_inv, div_eq_mul_inv] using A"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.133909\nβ : Type ?u.133912\nι : Type ?u.133915\nK : Type u_1\ninst✝ : NormedField K\nξ : K\nh : ‖ξ‖ < 1\nxi_ne_one : ξ ≠ 1\nA : Tendsto (fun n => (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹))\n⊢ Summable fun i => ‖ξ ^ i‖",
"tactic": "simp [norm_pow, summable_geometric_of_lt_1 (norm_nonneg _) h]"
}
] |
[
294,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Data/List/Duplicate.lean
|
List.Duplicate.duplicate_cons
|
[] |
[
47,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
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