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/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffAt.contDiffBump
|
[] |
[
453,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
18
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingNorm_eq_zero
|
[] |
[
192,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean
|
Set.projIcc_right
|
[] |
[
56,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/NormedSpace/ENorm.lean
|
ENorm.coeFn_injective
|
[
{
"state_after": "case mk\n𝕜 : Type u_2\nV : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne e₂ : ENorm 𝕜 V\ntoFun✝ : V → ℝ≥0∞\neq_zero'✝ : ∀ (x : V), toFun✝ x = 0 → x = 0\nmap_add_le'✝ : ∀ (x y : V), toFun✝ (x + y) ≤ toFun✝ x + toFun✝ y\nmap_smul_le'✝ : ∀ (c : 𝕜) (x : V), toFun✝ (c • x) ≤ ↑‖c‖₊ * toFun✝ x\nh : ↑{ toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ } = ↑e₂\n⊢ { toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ } = e₂",
"state_before": "𝕜 : Type u_2\nV : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne e₁ e₂ : ENorm 𝕜 V\nh : ↑e₁ = ↑e₂\n⊢ e₁ = e₂",
"tactic": "cases e₁"
},
{
"state_after": "case mk.mk\n𝕜 : Type u_2\nV : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne : ENorm 𝕜 V\ntoFun✝¹ : V → ℝ≥0∞\neq_zero'✝¹ : ∀ (x : V), toFun✝¹ x = 0 → x = 0\nmap_add_le'✝¹ : ∀ (x y : V), toFun✝¹ (x + y) ≤ toFun✝¹ x + toFun✝¹ y\nmap_smul_le'✝¹ : ∀ (c : 𝕜) (x : V), toFun✝¹ (c • x) ≤ ↑‖c‖₊ * toFun✝¹ x\ntoFun✝ : V → ℝ≥0∞\neq_zero'✝ : ∀ (x : V), toFun✝ x = 0 → x = 0\nmap_add_le'✝ : ∀ (x y : V), toFun✝ (x + y) ≤ toFun✝ x + toFun✝ y\nmap_smul_le'✝ : ∀ (c : 𝕜) (x : V), toFun✝ (c • x) ≤ ↑‖c‖₊ * toFun✝ x\nh :\n ↑{ toFun := toFun✝¹, eq_zero' := eq_zero'✝¹, map_add_le' := map_add_le'✝¹, map_smul_le' := map_smul_le'✝¹ } =\n ↑{ toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ }\n⊢ { toFun := toFun✝¹, eq_zero' := eq_zero'✝¹, map_add_le' := map_add_le'✝¹, map_smul_le' := map_smul_le'✝¹ } =\n { toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ }",
"state_before": "case mk\n𝕜 : Type u_2\nV : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne e₂ : ENorm 𝕜 V\ntoFun✝ : V → ℝ≥0∞\neq_zero'✝ : ∀ (x : V), toFun✝ x = 0 → x = 0\nmap_add_le'✝ : ∀ (x y : V), toFun✝ (x + y) ≤ toFun✝ x + toFun✝ y\nmap_smul_le'✝ : ∀ (c : 𝕜) (x : V), toFun✝ (c • x) ≤ ↑‖c‖₊ * toFun✝ x\nh : ↑{ toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ } = ↑e₂\n⊢ { toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ } = e₂",
"tactic": "cases e₂"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\n𝕜 : Type u_2\nV : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ne : ENorm 𝕜 V\ntoFun✝¹ : V → ℝ≥0∞\neq_zero'✝¹ : ∀ (x : V), toFun✝¹ x = 0 → x = 0\nmap_add_le'✝¹ : ∀ (x y : V), toFun✝¹ (x + y) ≤ toFun✝¹ x + toFun✝¹ y\nmap_smul_le'✝¹ : ∀ (c : 𝕜) (x : V), toFun✝¹ (c • x) ≤ ↑‖c‖₊ * toFun✝¹ x\ntoFun✝ : V → ℝ≥0∞\neq_zero'✝ : ∀ (x : V), toFun✝ x = 0 → x = 0\nmap_add_le'✝ : ∀ (x y : V), toFun✝ (x + y) ≤ toFun✝ x + toFun✝ y\nmap_smul_le'✝ : ∀ (c : 𝕜) (x : V), toFun✝ (c • x) ≤ ↑‖c‖₊ * toFun✝ x\nh :\n ↑{ toFun := toFun✝¹, eq_zero' := eq_zero'✝¹, map_add_le' := map_add_le'✝¹, map_smul_le' := map_smul_le'✝¹ } =\n ↑{ toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ }\n⊢ { toFun := toFun✝¹, eq_zero' := eq_zero'✝¹, map_add_le' := map_add_le'✝¹, map_smul_le' := map_smul_le'✝¹ } =\n { toFun := toFun✝, eq_zero' := eq_zero'✝, map_add_le' := map_add_le'✝, map_smul_le' := map_smul_le'✝ }",
"tactic": "congr"
}
] |
[
67,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Order/Max.lean
|
NoBotOrder.to_noMinOrder
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.3482\nβ : Type ?u.3485\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : NoBotOrder α\na : α\n⊢ ∃ b, b < a",
"tactic": "simpa [not_le] using exists_not_ge a"
}
] |
[
145,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Equiv.bijective
|
[] |
[
845,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
844,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.normal_le_normalCore
|
[] |
[
2550,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2548,
1
] |
Mathlib/Analysis/Convex/Topology.lean
|
Convex.combo_interior_self_subset_interior
|
[] |
[
136,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Multiset/FinsetOps.lean
|
Multiset.Nodup.ndinsert
|
[] |
[
91,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
upperSemicontinuous_ciInf
|
[] |
[
1054,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Mathlib/Algebra/Order/WithZero.lean
|
mul_lt_right₀
|
[
{
"state_after": "α : Type u_1\na b c✝ d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nc : α\nhc : c ≠ 0\nh : b * c ≤ a * c\n⊢ b ≤ a",
"state_before": "α : Type u_1\na b c✝ d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nc : α\nh : a < b\nhc : c ≠ 0\n⊢ a * c < b * c",
"tactic": "contrapose! h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b c✝ d x y z : α\ninst✝ : LinearOrderedCommGroupWithZero α\nc : α\nhc : c ≠ 0\nh : b * c ≤ a * c\n⊢ b ≤ a",
"tactic": "exact le_of_le_mul_right hc h"
}
] |
[
205,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean
|
Geometry.SimplicialComplex.subset_space
|
[] |
[
96,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
11
] |
Mathlib/Order/WithBot.lean
|
WithBot.strictMono_map_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33858\nδ : Type ?u.33861\na b : α\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α → β\n⊢ ((StrictMono fun a => map f ↑a) ∧ ∀ (x : α), map f ⊥ < map f ↑x) ↔ StrictMono f",
"tactic": "simp [StrictMono, bot_lt_coe]"
}
] |
[
370,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/Topology/MetricSpace/MetrizableUniformity.lean
|
PseudoMetricSpace.dist_ofPreNNDist_le
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nx y : X\n⊢ sum (zipWith d [x] ([] ++ [y])) = d x y",
"tactic": "simp"
}
] |
[
103,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.tensor_comp
|
[
{
"state_after": "case H\nR : Type u\ninst✝ : CommRing R\nX₁ Y₁ Z₁ : ModuleCat R\nX₂ Y₂ Z₂ : ModuleCat R\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : Y₁ ⟶ Z₁\ng₂ : Y₂ ⟶ Z₂\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑X₁ ↑X₂) (tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂)) =\n LinearMap.compr₂ (TensorProduct.mk R ↑X₁ ↑X₂) (tensorHom f₁ f₂ ≫ tensorHom g₁ g₂)",
"state_before": "R : Type u\ninst✝ : CommRing R\nX₁ Y₁ Z₁ : ModuleCat R\nX₂ Y₂ Z₂ : ModuleCat R\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : Y₁ ⟶ Z₁\ng₂ : Y₂ ⟶ Z₂\n⊢ tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom f₁ f₂ ≫ tensorHom g₁ g₂",
"tactic": "apply TensorProduct.ext"
},
{
"state_after": "no goals",
"state_before": "case H\nR : Type u\ninst✝ : CommRing R\nX₁ Y₁ Z₁ : ModuleCat R\nX₂ Y₂ Z₂ : ModuleCat R\nf₁ : X₁ ⟶ Y₁\nf₂ : X₂ ⟶ Y₂\ng₁ : Y₁ ⟶ Z₁\ng₂ : Y₂ ⟶ Z₂\n⊢ LinearMap.compr₂ (TensorProduct.mk R ↑X₁ ↑X₂) (tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂)) =\n LinearMap.compr₂ (TensorProduct.mk R ↑X₁ ↑X₂) (tensorHom f₁ f₂ ≫ tensorHom g₁ g₂)",
"tactic": "rfl"
}
] |
[
77,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/LinearAlgebra/Matrix/IsDiag.lean
|
Matrix.isDiag_smul_one
|
[] |
[
116,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_sum
|
[] |
[
330,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
329,
1
] |
Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean
|
iInter_halfspaces_eq
|
[
{
"state_after": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\n⊢ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s ∧ ↑i x ≤ ↑i y} = s",
"state_before": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\n⊢ (⋂ (l : E →L[ℝ] ℝ), {x | ∃ y, y ∈ s ∧ ↑l x ≤ ↑l y}) = s",
"tactic": "rw [Set.iInter_setOf]"
},
{
"state_after": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s ∧ ↑i x ≤ ↑i y}\n⊢ x ∈ s",
"state_before": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\n⊢ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s ∧ ↑i x ≤ ↑i y} = s",
"tactic": "refine' Set.Subset.antisymm (fun x hx => _) fun x hx l => ⟨x, hx, le_rfl⟩"
},
{
"state_after": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s ∧ ↑i x ≤ ↑i y}\nh : ¬x ∈ s\n⊢ False",
"state_before": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s ∧ ↑i x ≤ ↑i y}\n⊢ x ∈ s",
"tactic": "by_contra h"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns✝ t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s✝ ∧ ↑i x ≤ ↑i y}\nh : ¬x ∈ s✝\nl : E →L[ℝ] ℝ\ns : ℝ\nhlA : ∀ (a : E), a ∈ s✝ → ↑l a < s\nhl : s < ↑l x\n⊢ False",
"state_before": "𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsClosed s\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s ∧ ↑i x ≤ ↑i y}\nh : ¬x ∈ s\n⊢ False",
"tactic": "obtain ⟨l, s, hlA, hl⟩ := geometric_hahn_banach_closed_point hs₁ hs₂ h"
},
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns✝ t : Set E\nx✝ y✝ : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s✝ ∧ ↑i x ≤ ↑i y}\nh : ¬x ∈ s✝\nl : E →L[ℝ] ℝ\ns : ℝ\nhlA : ∀ (a : E), a ∈ s✝ → ↑l a < s\nhl : s < ↑l x\ny : E\nhy : y ∈ s✝\nhxy : ↑l x ≤ ↑l y\n⊢ False",
"state_before": "case intro.intro.intro\n𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns✝ t : Set E\nx✝ y : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s✝ ∧ ↑i x ≤ ↑i y}\nh : ¬x ∈ s✝\nl : E →L[ℝ] ℝ\ns : ℝ\nhlA : ∀ (a : E), a ∈ s✝ → ↑l a < s\nhl : s < ↑l x\n⊢ False",
"tactic": "obtain ⟨y, hy, hxy⟩ := hx l"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type ?u.130888\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : TopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ns✝ t : Set E\nx✝ y✝ : E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s✝\nhs₂ : IsClosed s✝\nx : E\nhx : x ∈ {x | ∀ (i : E →L[ℝ] ℝ), ∃ y, y ∈ s✝ ∧ ↑i x ≤ ↑i y}\nh : ¬x ∈ s✝\nl : E →L[ℝ] ℝ\ns : ℝ\nhlA : ∀ (a : E), a ∈ s✝ → ↑l a < s\nhl : s < ↑l x\ny : E\nhy : y ∈ s✝\nhxy : ↑l x ≤ ↑l y\n⊢ False",
"tactic": "exact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl"
}
] |
[
215,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk'_def
|
[] |
[
150,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
toDual_hnot
|
[] |
[
1147,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1146,
1
] |
Mathlib/Data/Polynomial/Div.lean
|
Polynomial.rootMultiplicity_pos'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ q p : R[X]\nx : R\n⊢ 0 < rootMultiplicity x p ↔ p ≠ 0 ∧ IsRoot p x",
"tactic": "rw [pos_iff_ne_zero, Ne.def, rootMultiplicity_eq_zero_iff, not_imp, and_comm]"
}
] |
[
574,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.norm_p
|
[] |
[
312,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
HomogeneousIdeal.toIdeal_sSup
|
[] |
[
385,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Data/List/Forall2.lean
|
List.forall₂_nil_right_iff
|
[
{
"state_after": "case nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.15064\nδ : Type ?u.15067\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\n⊢ [] = []",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15064\nδ : Type ?u.15067\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl : List α\nH : Forall₂ R l []\n⊢ l = []",
"tactic": "cases H"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.15064\nδ : Type ?u.15067\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\n⊢ [] = []",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15064\nδ : Type ?u.15067\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\n⊢ Forall₂ R [] []",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15064\nδ : Type ?u.15067\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl : List α\n⊢ l = [] → Forall₂ R l []",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.15064\nδ : Type ?u.15067\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\n⊢ Forall₂ R [] []",
"tactic": "exact Forall₂.nil"
}
] |
[
87,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snorm_smul_measure_of_ne_zero
|
[
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : p = 0\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ\n\ncase neg\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : ¬p = 0\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"state_before": "α : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"tactic": "by_cases hp0 : p = 0"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : ¬p = 0\nhp_top : p = ⊤\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ\n\ncase neg\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : ¬p = 0\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"tactic": "by_cases hp_top : p = ∞"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : ¬p = 0\nhp_top : ¬p = ⊤\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"tactic": "exact snorm_smul_measure_of_ne_zero_of_ne_top hp0 hp_top c"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : p = 0\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"tactic": "simp [hp0]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.2146317\nF : Type u_2\nG : Type ?u.2146323\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\nhp0 : ¬p = 0\nhp_top : p = ⊤\n⊢ snorm f p (c • μ) = c ^ ENNReal.toReal (1 / p) • snorm f p μ",
"tactic": "simp [hp_top, snormEssSup_smul_measure hc]"
}
] |
[
631,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
625,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.valMinAbs_natCast_eq_self
|
[
{
"state_after": "n a : ℕ\ninst✝ : NeZero n\nha : valMinAbs ↑a = ↑a\n⊢ a ≤ n / 2",
"state_before": "n a : ℕ\ninst✝ : NeZero n\n⊢ valMinAbs ↑a = ↑a ↔ a ≤ n / 2",
"tactic": "refine' ⟨fun ha => _, valMinAbs_natCast_of_le_half⟩"
},
{
"state_after": "n a : ℕ\ninst✝ : NeZero n\nha : valMinAbs ↑a = ↑a\n⊢ Int.natAbs (valMinAbs ↑a) ≤ n / 2",
"state_before": "n a : ℕ\ninst✝ : NeZero n\nha : valMinAbs ↑a = ↑a\n⊢ a ≤ n / 2",
"tactic": "rw [← Int.natAbs_ofNat a, ← ha]"
},
{
"state_after": "no goals",
"state_before": "n a : ℕ\ninst✝ : NeZero n\nha : valMinAbs ↑a = ↑a\n⊢ Int.natAbs (valMinAbs ↑a) ≤ n / 2",
"tactic": "exact natAbs_valMinAbs_le a"
}
] |
[
1087,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1084,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
Real.deriv_rpow_const'
|
[] |
[
369,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Data/List/Infix.lean
|
List.inits_eq_tails
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.42454\nl l₁ l₂ l₃ : List α\na b : α\nm n : ℕ\n⊢ inits [] = reverse (map reverse (tails (reverse [])))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.42454\nl✝ l₁ l₂ l₃ : List α\na✝ b : α\nm n : ℕ\na : α\nl : List α\n⊢ inits (a :: l) = reverse (map reverse (tails (reverse (a :: l))))",
"tactic": "simp [inits_eq_tails l, map_eq_map_iff, reverse_map]"
}
] |
[
397,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
EMetric.continuous_infEdist
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.10627\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ 1 ≠ ⊤",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.10627\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : α\ns t : Set α\nΦ : α → β\n⊢ ∀ (x y : α), infEdist x s ≤ infEdist y s + 1 * edist x y",
"tactic": "simp only [one_mul, infEdist_le_infEdist_add_edist, forall₂_true_iff]"
}
] |
[
131,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
AEMeasurable.ennreal_toNNReal
|
[] |
[
1899,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1897,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean
|
LieSubalgebra.inf_coe
|
[] |
[
489,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/MeasureTheory/Measure/GiryMonad.lean
|
MeasureTheory.Measure.join_eq_bind
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.41117\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure (Measure α)\n⊢ join μ = bind μ id",
"tactic": "rw [bind, map_id]"
}
] |
[
211,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Analysis/Calculus/Deriv/Prod.lean
|
HasDerivAt.prod
|
[] |
[
71,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
8
] |
Mathlib/LinearAlgebra/FreeModule/PID.lean
|
Ideal.selfBasis_def
|
[] |
[
614,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
612,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioo_insert_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.48593\ninst✝ : PartialOrder α\na b c : α\nh : a < b\n⊢ insert b (Ioo a b) = Ioc a b",
"tactic": "rw [insert_eq, union_comm, Ioo_union_right h]"
}
] |
[
892,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
891,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.range_succ
|
[
{
"state_after": "n✝ m n : ℕ\n⊢ Set.range ↑(succAbove 0) = {0}ᶜ",
"state_before": "n✝ m n : ℕ\n⊢ Set.range succ = {0}ᶜ",
"tactic": "rw [← succAbove_zero]"
},
{
"state_after": "no goals",
"state_before": "n✝ m n : ℕ\n⊢ Set.range ↑(succAbove 0) = {0}ᶜ",
"tactic": "exact range_succAbove (0 : Fin (n + 1))"
}
] |
[
2176,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2174,
1
] |
Mathlib/Algebra/CovariantAndContravariant.lean
|
act_rel_act_of_rel
|
[] |
[
154,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean
|
fixingSubmonoid_antitone
|
[] |
[
76,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.sub_limZero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝² : LinearOrderedField α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : LimZero f\nhg : LimZero g\n⊢ LimZero (f - g)",
"tactic": "simpa only [sub_eq_add_neg] using add_limZero hf (neg_limZero hg)"
}
] |
[
447,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.isAdjointPair_equiv'
|
[
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "have h' : IsUnit P.det := P.isUnit_iff_isUnit_det.mp h"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "let u := P.nonsingInvUnit h'"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "have coe_u : (u : Matrix n n R₃) = P := rfl"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "have coe_u_inv : (↑u⁻¹ : Matrix n n R₃) = P⁻¹ := rfl"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "let v := Pᵀ.nonsingInvUnit (P.isUnit_det_transpose h')"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "have coe_v : (v : Matrix n n R₃) = Pᵀ := rfl"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "have coe_v_inv : (↑v⁻¹ : Matrix n n R₃) = P⁻¹ᵀ := P.transpose_nonsing_inv.symm"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "set x := Aᵀ * Pᵀ * J with x_def"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "set y := J * P * A' with y_def"
},
{
"state_after": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ Aᵀ * (Pᵀ * J * P) = Pᵀ * J * P * A' ↔ (P * A * P⁻¹)ᵀ * J = J * (P * A' * P⁻¹)",
"state_before": "R : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ IsAdjointPair (Pᵀ ⬝ J ⬝ P) (Pᵀ ⬝ J ⬝ P) A A' ↔ IsAdjointPair J J (P ⬝ A ⬝ P⁻¹) (P ⬝ A' ⬝ P⁻¹)",
"tactic": "simp only [Matrix.IsAdjointPair, ← Matrix.mul_eq_mul]"
},
{
"state_after": "case calc_1\nR : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ Aᵀ * (Pᵀ * J * P) = Pᵀ * J * P * A' ↔ x * ↑u = ↑v * y\n\ncase calc_2\nR : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ x * ↑u = ↑v * y ↔ ↑v⁻¹ * x = y * ↑u⁻¹\n\ncase calc_3\nR : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ ↑v⁻¹ * x = y * ↑u⁻¹ ↔ (P * A * P⁻¹)ᵀ * J = J * (P * A' * P⁻¹)",
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"tactic": "calc (Aᵀ * (Pᵀ * J * P) = Pᵀ * J * P * A')\n ↔ (x * ↑u = ↑v * y) := ?_\n _ ↔ (↑v⁻¹ * x = y * ↑u⁻¹) := ?_\n _ ↔ ((P * A * P⁻¹)ᵀ * J = J * (P * A' * P⁻¹)) := ?_"
},
{
"state_after": "no goals",
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"tactic": "simp only [mul_assoc, x_def, y_def, coe_u, coe_v]"
},
{
"state_after": "no goals",
"state_before": "case calc_2\nR : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ x * ↑u = ↑v * y ↔ ↑v⁻¹ * x = y * ↑u⁻¹",
"tactic": "rw [Units.eq_mul_inv_iff_mul_eq, mul_assoc ↑v⁻¹ x, Units.inv_mul_eq_iff_eq_mul]"
},
{
"state_after": "case calc_3\nR : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ P⁻¹ᵀ * (Aᵀ * Pᵀ * J) = J * P * A' * P⁻¹ ↔ (P * A * P⁻¹)ᵀ * J = J * (P * A' * P⁻¹)",
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"tactic": "rw [x_def, y_def, coe_u_inv, coe_v_inv]"
},
{
"state_after": "no goals",
"state_before": "case calc_3\nR : Type ?u.2171788\nM : Type ?u.2171791\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.2171827\nM₁ : Type ?u.2171830\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type ?u.2172439\nM₂ : Type ?u.2172442\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type u_2\nM₃ : Type ?u.2172632\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.2173220\nK : Type ?u.2173223\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\ninst✝¹ : Fintype n\nb : Basis n R₃ M₃\nJ J₃ A A' : Matrix n n R₃\ninst✝ : DecidableEq n\nP : Matrix n n R₃\nh : IsUnit P\nh' : IsUnit (det P)\nu : (Matrix n n R₃)ˣ := nonsingInvUnit P h'\ncoe_u : ↑u = P\ncoe_u_inv : ↑u⁻¹ = P⁻¹\nv : (Matrix n n R₃)ˣ := nonsingInvUnit Pᵀ (_ : IsUnit (det Pᵀ))\ncoe_v : ↑v = Pᵀ\ncoe_v_inv : ↑v⁻¹ = P⁻¹ᵀ\nx : Matrix n n R₃ := Aᵀ * Pᵀ * J\nx_def : x = Aᵀ * Pᵀ * J\ny : Matrix n n R₃ := J * P * A'\ny_def : y = J * P * A'\n⊢ P⁻¹ᵀ * (Aᵀ * Pᵀ * J) = J * P * A' * P⁻¹ ↔ (P * A * P⁻¹)ᵀ * J = J * (P * A' * P⁻¹)",
"tactic": "simp only [Matrix.mul_eq_mul, Matrix.mul_assoc, Matrix.transpose_mul]"
}
] |
[
494,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
|
MvQPF.recF_eq_of_wEquiv
|
[
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α),\n WEquiv (MvPFunctor.wMk (P F) a f' f) y → recF u (MvPFunctor.wMk (P F) a f' f) = recF u y",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\n⊢ WEquiv x y → recF u x = recF u y",
"tactic": "apply q.P.w_cases _ x"
},
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\n⊢ WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) y → recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u y",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α),\n WEquiv (MvPFunctor.wMk (P F) a f' f) y → recF u (MvPFunctor.wMk (P F) a f' f) = recF u y",
"tactic": "intro a₀ f'₀ f₀"
},
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α),\n WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a f' f) →\n recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a f' f)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\n⊢ WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) y → recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u y",
"tactic": "apply q.P.w_cases _ y"
},
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\n⊢ WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁) →\n recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a₁ f'₁ f₁)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α),\n WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a f' f) →\n recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a f' f)",
"tactic": "intro a₁ f'₁ f₁"
},
{
"state_after": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a₁ f'₁ f₁)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\n⊢ WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁) →\n recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a₁ f'₁ f₁)",
"tactic": "intro h"
},
{
"state_after": "case refine'_1\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α)\n (a_1 : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)),\n (∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))) →\n (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a f' f₀) (MvPFunctor.wMk (P F) a f' f₁)\n (_ : WEquiv (MvPFunctor.wMk (P F) a f' f₀) (MvPFunctor.wMk (P F) a f' f₁))\n\ncase refine'_2\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ ∀ (a₀ : (P F).A) (f'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α)\n (f₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α) (a₁ : (P F).A)\n (f'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α)\n (f₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α)\n (a :\n abs { fst := a₀, snd := MvPFunctor.appendContents (P F) f'₀ f₀ } =\n abs { fst := a₁, snd := MvPFunctor.appendContents (P F) f'₁ f₁ }),\n (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n (_ : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁))\n\ncase refine'_3\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ ∀ (u_1 v w : MvPFunctor.W (P F) α) (a : WEquiv u_1 v) (a_1 : WEquiv v w),\n (fun a a' x => recF u a = recF u a') u_1 v a →\n (fun a a' x => recF u a = recF u a') v w a_1 → (fun a a' x => recF u a = recF u a') u_1 w (_ : WEquiv u_1 w)",
"state_before": "n : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a₁ f'₁ f₁)",
"tactic": "refine' @WEquiv.recOn _ _ _ _ _ (λ a a' _ => recF u a = recF u a') _ _ h _ _ _"
},
{
"state_after": "case refine'_1\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\n⊢ recF u (MvPFunctor.wMk (P F) a f' f₀) = recF u (MvPFunctor.wMk (P F) a f' f₁)",
"state_before": "case refine'_1\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ ∀ (a : (P F).A) (f' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α)\n (f₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α)\n (a_1 : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)),\n (∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))) →\n (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a f' f₀) (MvPFunctor.wMk (P F) a f' f₁)\n (_ : WEquiv (MvPFunctor.wMk (P F) a f' f₀) (MvPFunctor.wMk (P F) a f' f₁))",
"tactic": "intros a f' f₀ f₁ _h ih"
},
{
"state_after": "case refine'_1\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\n⊢ u (abs { fst := a, snd := splitFun f' fun x => recF u (f₀ x) }) =\n u (abs { fst := a, snd := splitFun f' fun x => recF u (f₁ x) })",
"state_before": "case refine'_1\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\n⊢ recF u (MvPFunctor.wMk (P F) a f' f₀) = recF u (MvPFunctor.wMk (P F) a f' f₁)",
"tactic": "simp only [recF_eq, Function.comp]"
},
{
"state_after": "case refine'_1.e_a.e_a.e_snd\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\n⊢ (splitFun f' fun x => recF u (f₀ x)) = splitFun f' fun x => recF u (f₁ x)",
"state_before": "case refine'_1\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\n⊢ u (abs { fst := a, snd := splitFun f' fun x => recF u (f₀ x) }) =\n u (abs { fst := a, snd := splitFun f' fun x => recF u (f₁ x) })",
"tactic": "congr"
},
{
"state_after": "case refine'_1.e_a.e_a.e_snd.h.h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\nx✝¹ : Fin2 (n + 1)\nx✝ : MvPFunctor.B (P F) a x✝¹\n⊢ splitFun f' (fun x => recF u (f₀ x)) x✝¹ x✝ = splitFun f' (fun x => recF u (f₁ x)) x✝¹ x✝",
"state_before": "case refine'_1.e_a.e_a.e_snd\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\n⊢ (splitFun f' fun x => recF u (f₀ x)) = splitFun f' fun x => recF u (f₁ x)",
"tactic": "funext"
},
{
"state_after": "case refine'_1.e_a.e_a.e_snd.h.h.e_g\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\nx✝¹ : Fin2 (n + 1)\nx✝ : MvPFunctor.B (P F) a x✝¹\n⊢ (fun x => recF u (f₀ x)) = fun x => recF u (f₁ x)",
"state_before": "case refine'_1.e_a.e_a.e_snd.h.h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\nx✝¹ : Fin2 (n + 1)\nx✝ : MvPFunctor.B (P F) a x✝¹\n⊢ splitFun f' (fun x => recF u (f₀ x)) x✝¹ x✝ = splitFun f' (fun x => recF u (f₁ x)) x✝¹ x✝",
"tactic": "congr"
},
{
"state_after": "case refine'_1.e_a.e_a.e_snd.h.h.e_g.h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\nx✝² : Fin2 (n + 1)\nx✝¹ : MvPFunctor.B (P F) a x✝²\nx✝ : last (MvPFunctor.B (P F) a)\n⊢ recF u (f₀ x✝) = recF u (f₁ x✝)",
"state_before": "case refine'_1.e_a.e_a.e_snd.h.h.e_g\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\nx✝¹ : Fin2 (n + 1)\nx✝ : MvPFunctor.B (P F) a x✝¹\n⊢ (fun x => recF u (f₀ x)) = fun x => recF u (f₁ x)",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.e_a.e_a.e_snd.h.h.e_g.h\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀✝) (MvPFunctor.wMk (P F) a₁ f'₁ f₁✝)\na : (P F).A\nf' : MvPFunctor.B (MvPFunctor.drop (P F)) a ⟹ α\nf₀ f₁ : PFunctor.B (MvPFunctor.last (P F)) a → MvPFunctor.W (P F) α\n_h : ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a), WEquiv (f₀ x) (f₁ x)\nih :\n ∀ (x : PFunctor.B (MvPFunctor.last (P F)) a),\n (fun a a' x => recF u a = recF u a') (f₀ x) (f₁ x) (_ : WEquiv (f₀ x) (f₁ x))\nx✝² : Fin2 (n + 1)\nx✝¹ : MvPFunctor.B (P F) a x✝²\nx✝ : last (MvPFunctor.B (P F) a)\n⊢ recF u (f₀ x✝) = recF u (f₁ x✝)",
"tactic": "apply ih"
},
{
"state_after": "case refine'_2\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀✝ : (P F).A\nf'₀✝ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀✝ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀✝ → MvPFunctor.W (P F) α\na₁✝ : (P F).A\nf'₁✝ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁✝ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁✝ → MvPFunctor.W (P F) α\nh✝ : WEquiv (MvPFunctor.wMk (P F) a₀✝ f'₀✝ f₀✝) (MvPFunctor.wMk (P F) a₁✝ f'₁✝ f₁✝)\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh :\n abs { fst := a₀, snd := MvPFunctor.appendContents (P F) f'₀ f₀ } =\n abs { fst := a₁, snd := MvPFunctor.appendContents (P F) f'₁ f₁ }\n⊢ recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a₁ f'₁ f₁)",
"state_before": "case refine'_2\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ ∀ (a₀ : (P F).A) (f'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α)\n (f₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α) (a₁ : (P F).A)\n (f'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α)\n (f₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α)\n (a :\n abs { fst := a₀, snd := MvPFunctor.appendContents (P F) f'₀ f₀ } =\n abs { fst := a₁, snd := MvPFunctor.appendContents (P F) f'₁ f₁ }),\n (fun a a' x => recF u a = recF u a') (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n (_ : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁))",
"tactic": "intros a₀ f'₀ f₀ a₁ f'₁ f₁ h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀✝ : (P F).A\nf'₀✝ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀✝ ⟹ α\nf₀✝ : PFunctor.B (MvPFunctor.last (P F)) a₀✝ → MvPFunctor.W (P F) α\na₁✝ : (P F).A\nf'₁✝ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁✝ ⟹ α\nf₁✝ : PFunctor.B (MvPFunctor.last (P F)) a₁✝ → MvPFunctor.W (P F) α\nh✝ : WEquiv (MvPFunctor.wMk (P F) a₀✝ f'₀✝ f₀✝) (MvPFunctor.wMk (P F) a₁✝ f'₁✝ f₁✝)\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh :\n abs { fst := a₀, snd := MvPFunctor.appendContents (P F) f'₀ f₀ } =\n abs { fst := a₁, snd := MvPFunctor.appendContents (P F) f'₁ f₁ }\n⊢ recF u (MvPFunctor.wMk (P F) a₀ f'₀ f₀) = recF u (MvPFunctor.wMk (P F) a₁ f'₁ f₁)",
"tactic": "simp only [recF_eq', abs_map, MvPFunctor.wDest'_wMk, h]"
},
{
"state_after": "case refine'_3\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx✝ y✝ : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\nx y z : MvPFunctor.W (P F) α\n_e₁ : WEquiv x y\n_e₂ : WEquiv y z\nih₁ : recF u x = recF u y\nih₂ : recF u y = recF u z\n⊢ recF u x = recF u z",
"state_before": "case refine'_3\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx y : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\n⊢ ∀ (u_1 v w : MvPFunctor.W (P F) α) (a : WEquiv u_1 v) (a_1 : WEquiv v w),\n (fun a a' x => recF u a = recF u a') u_1 v a →\n (fun a a' x => recF u a = recF u a') v w a_1 → (fun a a' x => recF u a = recF u a') u_1 w (_ : WEquiv u_1 w)",
"tactic": "intros x y z _e₁ _e₂ ih₁ ih₂"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nn : ℕ\nF : TypeVec (n + 1) → Type u\ninst✝ : MvFunctor F\nq : MvQPF F\nα : TypeVec n\nβ : Type u\nu : F (α ::: β) → β\nx✝ y✝ : MvPFunctor.W (P F) α\na₀ : (P F).A\nf'₀ : MvPFunctor.B (MvPFunctor.drop (P F)) a₀ ⟹ α\nf₀ : PFunctor.B (MvPFunctor.last (P F)) a₀ → MvPFunctor.W (P F) α\na₁ : (P F).A\nf'₁ : MvPFunctor.B (MvPFunctor.drop (P F)) a₁ ⟹ α\nf₁ : PFunctor.B (MvPFunctor.last (P F)) a₁ → MvPFunctor.W (P F) α\nh : WEquiv (MvPFunctor.wMk (P F) a₀ f'₀ f₀) (MvPFunctor.wMk (P F) a₁ f'₁ f₁)\nx y z : MvPFunctor.W (P F) α\n_e₁ : WEquiv x y\n_e₂ : WEquiv y z\nih₁ : recF u x = recF u y\nih₂ : recF u y = recF u z\n⊢ recF u x = recF u z",
"tactic": "exact Eq.trans ih₁ ih₂"
}
] |
[
107,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.fst_surjective
|
[] |
[
87,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Algebra.comap_top
|
[] |
[
938,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
937,
1
] |
Mathlib/RingTheory/FiniteType.lean
|
AddMonoidAlgebra.mvPolynomial_aeval_of_surjective_of_closure
|
[
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf : AddMonoidAlgebra R M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\n⊢ Surjective ↑(MvPolynomial.aeval fun s => of' R M ↑s)",
"tactic": "intro f"
},
{
"state_after": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd m)\n\ncase hadd\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf g : AddMonoidAlgebra R M\nihf : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = g\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f + g\n\ncase hsmul\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nf : AddMonoidAlgebra R M\nih : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = r • f",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf : AddMonoidAlgebra R M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f",
"tactic": "induction' f using induction_on with m f g ihf ihg r f ih"
},
{
"state_after": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd m)",
"state_before": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd m)",
"tactic": "have : m ∈ closure S := hS.symm ▸ mem_top _"
},
{
"state_after": "case hM.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm✝ : M\nthis : m✝ ∈ closure S\nm : M\nhm : m ∈ S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd m)\n\ncase hM.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd 0)\n\ncase hM.refine'_3\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∀ (x y : M),\n (∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd x)) →\n (∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd y)) →\n ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd (x + y))",
"state_before": "case hM\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd m)",
"tactic": "refine' closure_induction this (fun m hm => _) _ _"
},
{
"state_after": "no goals",
"state_before": "case hM.refine'_1\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm✝ : M\nthis : m✝ ∈ closure S\nm : M\nhm : m ∈ S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd m)",
"tactic": "exact ⟨MvPolynomial.X ⟨m, hm⟩, MvPolynomial.aeval_X _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case hM.refine'_2\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd 0)",
"tactic": "exact ⟨1, AlgHom.map_one _⟩"
},
{
"state_after": "case hM.refine'_3.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₁ = ↑(of R M) (↑Multiplicative.ofAdd m₁)\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₂ = ↑(of R M) (↑Multiplicative.ofAdd m₂)\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd (m₁ + m₂))",
"state_before": "case hM.refine'_3\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\n⊢ ∀ (x y : M),\n (∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd x)) →\n (∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd y)) →\n ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd (x + y))",
"tactic": "rintro m₁ m₂ ⟨P₁, hP₁⟩ ⟨P₂, hP₂⟩"
},
{
"state_after": "no goals",
"state_before": "case hM.refine'_3.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₁ = ↑(of R M) (↑Multiplicative.ofAdd m₁)\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₂ = ↑(of R M) (↑Multiplicative.ofAdd m₂)\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(of R M) (↑Multiplicative.ofAdd (m₁ + m₂))",
"tactic": "exact\n ⟨P₁ * P₂, by\n rw [AlgHom.map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single,\n one_mul]; rfl⟩"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₁ = ↑(of R M) (↑Multiplicative.ofAdd m₁)\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₂ = ↑(of R M) (↑Multiplicative.ofAdd m₂)\n⊢ single (↑Multiplicative.toAdd (↑Multiplicative.ofAdd m₁) + ↑Multiplicative.toAdd (↑Multiplicative.ofAdd m₂)) 1 =\n single (↑Multiplicative.toAdd (↑Multiplicative.ofAdd (m₁ + m₂))) 1",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₁ = ↑(of R M) (↑Multiplicative.ofAdd m₁)\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₂ = ↑(of R M) (↑Multiplicative.ofAdd m₂)\n⊢ ↑(MvPolynomial.aeval fun s => of' R M ↑s) (P₁ * P₂) = ↑(of R M) (↑Multiplicative.ofAdd (m₁ + m₂))",
"tactic": "rw [AlgHom.map_mul, hP₁, hP₂, of_apply, of_apply, of_apply, single_mul_single,\n one_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nm : M\nthis : m ∈ closure S\nm₁ m₂ : M\nP₁ : MvPolynomial (↑S) R\nhP₁ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₁ = ↑(of R M) (↑Multiplicative.ofAdd m₁)\nP₂ : MvPolynomial (↑S) R\nhP₂ : ↑(MvPolynomial.aeval fun s => of' R M ↑s) P₂ = ↑(of R M) (↑Multiplicative.ofAdd m₂)\n⊢ single (↑Multiplicative.toAdd (↑Multiplicative.ofAdd m₁) + ↑Multiplicative.toAdd (↑Multiplicative.ofAdd m₂)) 1 =\n single (↑Multiplicative.toAdd (↑Multiplicative.ofAdd (m₁ + m₂))) 1",
"tactic": "rfl"
},
{
"state_after": "case hadd.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\ng : AddMonoidAlgebra R M\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = g\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(MvPolynomial.aeval fun s => of' R M ↑s) P + g",
"state_before": "case hadd\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nf g : AddMonoidAlgebra R M\nihf : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = g\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f + g",
"tactic": "rcases ihf with ⟨P, rfl⟩"
},
{
"state_after": "case hadd.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nP Q : MvPolynomial (↑S) R\n⊢ ∃ a,\n ↑(MvPolynomial.aeval fun s => of' R M ↑s) a =\n ↑(MvPolynomial.aeval fun s => of' R M ↑s) P + ↑(MvPolynomial.aeval fun s => of' R M ↑s) Q",
"state_before": "case hadd.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\ng : AddMonoidAlgebra R M\nihg : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = g\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = ↑(MvPolynomial.aeval fun s => of' R M ↑s) P + g",
"tactic": "rcases ihg with ⟨Q, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case hadd.intro.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nP Q : MvPolynomial (↑S) R\n⊢ ∃ a,\n ↑(MvPolynomial.aeval fun s => of' R M ↑s) a =\n ↑(MvPolynomial.aeval fun s => of' R M ↑s) P + ↑(MvPolynomial.aeval fun s => of' R M ↑s) Q",
"tactic": "exact ⟨P + Q, AlgHom.map_add _ _ _⟩"
},
{
"state_after": "case hsmul.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = r • ↑(MvPolynomial.aeval fun s => of' R M ↑s) P",
"state_before": "case hsmul\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nf : AddMonoidAlgebra R M\nih : ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = f\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = r • f",
"tactic": "rcases ih with ⟨P, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case hsmul.intro\nR : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : CommSemiring R\nS : Set M\nhS : closure S = ⊤\nr : R\nP : MvPolynomial (↑S) R\n⊢ ∃ a, ↑(MvPolynomial.aeval fun s => of' R M ↑s) a = r • ↑(MvPolynomial.aeval fun s => of' R M ↑s) P",
"tactic": "exact ⟨r • P, AlgHom.map_smul _ _ _⟩"
}
] |
[
443,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_map_le_prod_map
|
[] |
[
407,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
405,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.abs_cos_eq_abs_sin_of_two_nsmul_add_two_nsmul_eq_pi
|
[
{
"state_after": "θ ψ : Angle\nh : θ = ↑(π / 2) - ψ ∨ θ = ↑(π / 2) - ψ + ↑π\n⊢ abs (cos θ) = abs (sin ψ)",
"state_before": "θ ψ : Angle\nh : 2 • θ + 2 • ψ = ↑π\n⊢ abs (cos θ) = abs (sin ψ)",
"tactic": "rw [← eq_sub_iff_add_eq, ← two_nsmul_coe_div_two, ← nsmul_sub, two_nsmul_eq_iff] at h"
},
{
"state_after": "no goals",
"state_before": "θ ψ : Angle\nh : θ = ↑(π / 2) - ψ ∨ θ = ↑(π / 2) - ψ + ↑π\n⊢ abs (cos θ) = abs (sin ψ)",
"tactic": "rcases h with (rfl | rfl) <;> simp [cos_pi_div_two_sub]"
}
] |
[
765,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
762,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidHom.congr_fun
|
[] |
[
684,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Std/Data/List/Lemmas.lean
|
List.nil_sublist
|
[] |
[
320,
37
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
318,
9
] |
Mathlib/InformationTheory/Hamming.lean
|
Hamming.dist_eq_hammingDist
|
[] |
[
416,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.sUnion
|
[
{
"state_after": "ι : Type ?u.3562\nα : Type u_1\nβ : Type ?u.3568\nγ : Type ?u.3571\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ns : Set (Set α)\nhs : Set.Countable s\nh : ∀ (t : Set α), t ∈ s → NullMeasurableSet t\n⊢ NullMeasurableSet (⋃ (i : Set α) (_ : i ∈ s), i)",
"state_before": "ι : Type ?u.3562\nα : Type u_1\nβ : Type ?u.3568\nγ : Type ?u.3571\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ns : Set (Set α)\nhs : Set.Countable s\nh : ∀ (t : Set α), t ∈ s → NullMeasurableSet t\n⊢ NullMeasurableSet (⋃₀ s)",
"tactic": "rw [sUnion_eq_biUnion]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.3562\nα : Type u_1\nβ : Type ?u.3568\nγ : Type ?u.3571\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ns : Set (Set α)\nhs : Set.Countable s\nh : ∀ (t : Set α), t ∈ s → NullMeasurableSet t\n⊢ NullMeasurableSet (⋃ (i : Set α) (_ : i ∈ s), i)",
"tactic": "exact MeasurableSet.biUnion hs h"
}
] |
[
168,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
11
] |
Mathlib/Analysis/BoxIntegral/Partition/Measure.lean
|
BoxIntegral.Box.volume_apply'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\nI : Box ι\n⊢ ENNReal.toReal (↑↑volume ↑I) = ∏ i : ι, (upper I i - lower I i)",
"tactic": "rw [coe_eq_pi, Real.volume_pi_Ioc_toReal I.lower_le_upper]"
}
] |
[
132,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/CategoryTheory/Sites/DenseSubsite.lean
|
CategoryTheory.CoverDense.Types.pushforwardFamily_def
|
[] |
[
175,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
9
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.map_map
|
[
{
"state_after": "case h₁\nC : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ inl ≫ map f g ≫ map h k = inl ≫ map (f ≫ h) (g ≫ k)\n\ncase h₂\nC : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ inr ≫ map f g ≫ map h k = inr ≫ map (f ≫ h) (g ≫ k)",
"state_before": "C : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ map f g ≫ map h k = map (f ≫ h) (g ≫ k)",
"tactic": "apply coprod.hom_ext"
},
{
"state_after": "case h₂\nC : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ inr ≫ map f g ≫ map h k = inr ≫ map (f ≫ h) (g ≫ k)",
"state_before": "case h₁\nC : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ inl ≫ map f g ≫ map h k = inl ≫ map (f ≫ h) (g ≫ k)\n\ncase h₂\nC : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ inr ≫ map f g ≫ map h k = inr ≫ map (f ≫ h) (g ≫ k)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h₂\nC : Type u\ninst✝³ : Category C\nX Y A₁ A₂ A₃ B₁ B₂ B₃ : C\ninst✝² : HasBinaryCoproduct A₁ B₁\ninst✝¹ : HasBinaryCoproduct A₂ B₂\ninst✝ : HasBinaryCoproduct A₃ B₃\nf : A₁ ⟶ A₂\ng : B₁ ⟶ B₂\nh : A₂ ⟶ A₃\nk : B₂ ⟶ B₃\n⊢ inr ≫ map f g ≫ map h k = inr ≫ map (f ≫ h) (g ≫ k)",
"tactic": "simp"
}
] |
[
885,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/Data/Part.lean
|
Part.mem_assert_iff
|
[] |
[
471,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
LinearMap.map_comp_rTensor
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹⁴ : CommSemiring R\nR' : Type ?u.1281534\ninst✝¹³ : Monoid R'\nR'' : Type ?u.1281540\ninst✝¹² : Semiring R''\nM : Type u_2\nN : Type u_4\nP : Type u_3\nQ : Type u_5\nS : Type u_6\ninst✝¹¹ : AddCommMonoid M\ninst✝¹⁰ : AddCommMonoid N\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : AddCommMonoid Q\ninst✝⁷ : AddCommMonoid S\ninst✝⁶ : Module R M\ninst✝⁵ : Module R N\ninst✝⁴ : Module R P\ninst✝³ : Module R Q\ninst✝² : Module R S\ninst✝¹ : DistribMulAction R' M\ninst✝ : Module R'' M\ng✝ : P →ₗ[R] Q\nf✝ : N →ₗ[R] P\nf : M →ₗ[R] P\ng : N →ₗ[R] Q\nf' : S →ₗ[R] M\n⊢ comp (map f g) (rTensor N f') = map (comp f f') g",
"tactic": "simp only [lTensor, rTensor, ← map_comp, id_comp, comp_id]"
}
] |
[
1131,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1129,
1
] |
Mathlib/Topology/Algebra/FilterBasis.lean
|
ModuleFilterBasis.smul_left
|
[] |
[
362,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
LinearMap.injective_iff_surjective_of_finrank_eq_finrank
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\n⊢ Injective ↑f ↔ Surjective ↑f",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\n⊢ Injective ↑f ↔ Surjective ↑f",
"tactic": "have := finrank_range_add_finrank_ker f"
},
{
"state_after": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\n⊢ ker f = ⊥ ↔ range f = ⊤",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\n⊢ Injective ↑f ↔ Surjective ↑f",
"tactic": "rw [← ker_eq_bot, ← range_eq_top]"
},
{
"state_after": "case refine'_1\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\nh : ker f = ⊥\n⊢ range f = ⊤\n\ncase refine'_2\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\nh : range f = ⊤\n⊢ ker f = ⊥",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\n⊢ ker f = ⊥ ↔ range f = ⊤",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } = finrank K V₂\nh : ker f = ⊥\n⊢ range f = ⊤",
"state_before": "case refine'_1\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\nh : ker f = ⊥\n⊢ range f = ⊤",
"tactic": "rw [h, finrank_bot, add_zero, H] at this"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } = finrank K V₂\nh : ker f = ⊥\n⊢ range f = ⊤",
"tactic": "exact eq_top_of_finrank_eq this"
},
{
"state_after": "case refine'_2\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K V₂ + finrank K { x // x ∈ ker f } = finrank K V₂\nh : range f = ⊤\n⊢ ker f = ⊥",
"state_before": "case refine'_2\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V\nh : range f = ⊤\n⊢ ker f = ⊥",
"tactic": "rw [h, finrank_top, H] at this"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nK : Type u\nV : Type v\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : FiniteDimensional K V\ninst✝ : FiniteDimensional K V₂\nH : finrank K V = finrank K V₂\nf : V →ₗ[K] V₂\nthis : finrank K V₂ + finrank K { x // x ∈ ker f } = finrank K V₂\nh : range f = ⊤\n⊢ ker f = ⊥",
"tactic": "exact finrank_eq_zero.1 (add_right_injective _ this)"
}
] |
[
1053,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1045,
1
] |
Mathlib/Algebra/Category/Ring/Colimits.lean
|
CommRingCat.Colimits.quot_mul
|
[] |
[
239,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
StrictMono.const_mul_of_neg
|
[] |
[
736,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
734,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.rmap_compose
|
[] |
[
88,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurable_inclusion
|
[] |
[
580,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Topology/Category/TopCat/Opens.lean
|
TopologicalSpace.Opens.map_obj
|
[] |
[
161,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_lt_div_of_lt
|
[
{
"state_after": "ι : Type ?u.63758\nα : Type u_1\nβ : Type ?u.63764\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhc : 0 < c\nh : a < b\n⊢ a * (1 / c) < b * (1 / c)",
"state_before": "ι : Type ?u.63758\nα : Type u_1\nβ : Type ?u.63764\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhc : 0 < c\nh : a < b\n⊢ a / c < b / c",
"tactic": "rw [div_eq_mul_one_div a c, div_eq_mul_one_div b c]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.63758\nα : Type u_1\nβ : Type ?u.63764\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nhc : 0 < c\nh : a < b\n⊢ a * (1 / c) < b * (1 / c)",
"tactic": "exact mul_lt_mul_of_pos_right h (one_div_pos.2 hc)"
}
] |
[
362,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
360,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.coe_mul
|
[] |
[
744,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
PerfectClosure.int_cast
|
[
{
"state_after": "case negSucc\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\na✝ : ℕ\n⊢ -mk K p (0, ↑(a✝ + 1)) = mk K p (0, -↑(a✝ + 1))",
"state_before": "K : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\nx : ℤ\n⊢ ↑x = mk K p (0, ↑x)",
"tactic": "induction x <;> simp only [Int.ofNat_eq_coe, Int.cast_ofNat, Int.cast_negSucc, nat_cast K p 0]"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nK : Type u\ninst✝² : CommRing K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP K p\na✝ : ℕ\n⊢ -mk K p (0, ↑(a✝ + 1)) = mk K p (0, -↑(a✝ + 1))",
"tactic": "rfl"
}
] |
[
434,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.liftAux_apply_inr
|
[
{
"state_after": "no goals",
"state_before": "S : Type ?u.861406\nR R' : Type u\nM : Type v\ninst✝¹⁷ : CommSemiring S\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : CommSemiring R'\ninst✝¹⁴ : AddCommMonoid M\ninst✝¹³ : Algebra S R\ninst✝¹² : Algebra S R'\ninst✝¹¹ : Module S M\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module Rᵐᵒᵖ M\ninst✝⁸ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁷ : IsScalarTower S R M\ninst✝⁶ : IsScalarTower S Rᵐᵒᵖ M\ninst✝⁵ : Module R' M\ninst✝⁴ : Module R'ᵐᵒᵖ M\ninst✝³ : IsCentralScalar R' M\ninst✝² : IsScalarTower S R' M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R' A\nf : M →ₗ[R'] A\nhf : ∀ (x y : M), ↑f x * ↑f y = 0\nm : M\n⊢ ↑(algebraMap R' A) 0 + ↑f m = ↑f m",
"tactic": "rw [RingHom.map_zero, zero_add]"
}
] |
[
821,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.closure_eq_of_le
|
[] |
[
346,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/LinearAlgebra/Dimension.lean
|
rank_range_of_surjective
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\nV V₁ V₂ V₃ : Type v\nV' V'₁ : Type v'\nV'' : Type v''\nι : Type w\nι' : Type w'\nη : Type u₁'\nφ : η → Type ?u.83941\nR : Type u\ninst✝⁶ : Ring R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nM' : Type v'\ninst✝³ : AddCommGroup M'\ninst✝² : Module R M'\nM₁ : Type v\ninst✝¹ : AddCommGroup M₁\ninst✝ : Module R M₁\nf : M →ₗ[R] M'\nh : Surjective ↑f\n⊢ Module.rank R { x // x ∈ LinearMap.range f } = Module.rank R M'",
"tactic": "rw [LinearMap.range_eq_top.2 h, rank_top]"
}
] |
[
227,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
csInf_insert
|
[] |
[
714,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.log_pow_eq_self
|
[] |
[
487,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Data/Fin/Interval.lean
|
Fin.card_fintypeIoo
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\na b : Fin n\n⊢ Fintype.card ↑(Set.Ioo a b) = ↑b - ↑a - 1",
"tactic": "rw [← card_Ioo, Fintype.card_ofFinset]"
}
] |
[
119,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_iInf_of_finite
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.76872\nι✝ : Sort x\nf✝ g : Filter α✝\ns✝ t : Set α✝\nι : Type u_1\ninst✝ : Finite ι\nα : Type u_2\nf : ι → Filter α\ns : Set α\n⊢ (∃ t, (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ (i : ι), t i) → s ∈ ⨅ (i : ι), f i",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.76872\nι✝ : Sort x\nf✝ g : Filter α✝\ns✝ t : Set α✝\nι : Type u_1\ninst✝ : Finite ι\nα : Type u_2\nf : ι → Filter α\ns : Set α\n⊢ (s ∈ ⨅ (i : ι), f i) ↔ ∃ t, (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ (i : ι), t i",
"tactic": "refine' ⟨exists_iInter_of_mem_iInf, _⟩"
},
{
"state_after": "case intro.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.76872\nι✝ : Sort x\nf✝ g : Filter α✝\ns t✝ : Set α✝\nι : Type u_1\ninst✝ : Finite ι\nα : Type u_2\nf : ι → Filter α\nt : ι → Set α\nht : ∀ (i : ι), t i ∈ f i\n⊢ (⋂ (i : ι), t i) ∈ ⨅ (i : ι), f i",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.76872\nι✝ : Sort x\nf✝ g : Filter α✝\ns✝ t : Set α✝\nι : Type u_1\ninst✝ : Finite ι\nα : Type u_2\nf : ι → Filter α\ns : Set α\n⊢ (∃ t, (∀ (i : ι), t i ∈ f i) ∧ s = ⋂ (i : ι), t i) → s ∈ ⨅ (i : ι), f i",
"tactic": "rintro ⟨t, ht, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.76872\nι✝ : Sort x\nf✝ g : Filter α✝\ns t✝ : Set α✝\nι : Type u_1\ninst✝ : Finite ι\nα : Type u_2\nf : ι → Filter α\nt : ι → Set α\nht : ∀ (i : ι), t i ∈ f i\n⊢ (⋂ (i : ι), t i) ∈ ⨅ (i : ι), f i",
"tactic": "exact iInter_mem.2 fun i => mem_iInf_of_mem i (ht i)"
}
] |
[
641,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
637,
1
] |
Mathlib/Order/Atoms/Finite.lean
|
Fintype.IsSimpleOrder.card
|
[] |
[
56,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.dvd_of_mem_factors
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : a = 0\n⊢ p ∣ a\n\ncase neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : ¬a = 0\n⊢ p ∣ a",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\n⊢ p ∣ a",
"tactic": "by_cases ha0 : a = 0"
},
{
"state_after": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : ¬a = 0\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ a",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : ¬a = 0\n⊢ p ∣ a",
"tactic": "obtain ⟨a0, nza, ha'⟩ := exists_non_zero_rep ha0"
},
{
"state_after": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : ¬a = 0\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ FactorSet.prod (factors a)",
"state_before": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : ¬a = 0\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ a",
"tactic": "rw [← Associates.factors_prod a]"
},
{
"state_after": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ FactorSet.prod ↑(factors' a0)",
"state_before": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : ¬a = 0\na0 : α\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ FactorSet.prod (factors a)",
"tactic": "rw [← ha', factors_mk a0 nza] at hm⊢"
},
{
"state_after": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ prod (map Subtype.val (factors' a0))",
"state_before": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ FactorSet.prod ↑(factors' a0)",
"tactic": "rw [prod_coe]"
},
{
"state_after": "case neg.intro.intro.a\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∈ map Subtype.val (factors' a0)",
"state_before": "case neg.intro.intro\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∣ prod (map Subtype.val (factors' a0))",
"tactic": "apply Multiset.dvd_prod"
},
{
"state_after": "case neg.intro.intro.a\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ ∃ a, a ∈ factors' a0 ∧ ↑a = p",
"state_before": "case neg.intro.intro.a\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ p ∈ map Subtype.val (factors' a0)",
"tactic": "apply Multiset.mem_map.mpr"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.a\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nha0 : ¬a = 0\na0 : α\nhm : p ∈ ↑(factors' a0)\nnza : a0 ≠ 0\nha' : Associates.mk a0 = a\n⊢ ∃ a, a ∈ factors' a0 ∧ ↑a = p",
"tactic": "exact ⟨⟨p, hp⟩, mem_factorSet_some.mp hm, rfl⟩"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : a = 0\n⊢ p ∣ 0",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : a = 0\n⊢ p ∣ a",
"tactic": "rw [ha0]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na p : Associates α\nhp : Irreducible p\nhm : p ∈ factors a\nha0 : a = 0\n⊢ p ∣ 0",
"tactic": "exact dvd_zero p"
}
] |
[
1593,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1583,
1
] |
Mathlib/CategoryTheory/Abelian/LeftDerived.lean
|
CategoryTheory.Abelian.Functor.leftDerivedZeroToSelfApp_comp_inv
|
[
{
"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⊢ ((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 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 𝟙 ((leftDerived F 0).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⊢ leftDerivedZeroToSelfApp F P ≫ leftDerivedZeroToSelfAppInv F P = 𝟙 ((leftDerived F 0).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⊢ ((((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 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 ((leftDerived F 0).obj X) ≫ (leftDerivedObjIso F 0 P).hom",
"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⊢ ((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 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 𝟙 ((leftDerived F 0).obj X)",
"tactic": "rw [← Category.assoc, ← Category.assoc, ← Category.assoc, Iso.comp_inv_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\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 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).hom",
"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⊢ ((((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 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 ((leftDerived F 0).obj X) ≫ (leftDerivedObjIso F 0 P).hom",
"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\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 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).hom",
"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⊢ ((((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 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).hom",
"tactic": "rw [Category.assoc, Category.assoc, Category.assoc]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((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 (HomologicalComplex.dFrom ((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\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 𝟙 ((homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))",
"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⊢ (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 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).hom",
"tactic": "convert Category.comp_id (leftDerivedObjIso F 0 P).hom"
},
{
"state_after": "case h.e'_2.h.h.e'_7.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((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 (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\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)).hom",
"state_before": "case h.e'_2.h.h.e'_7.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((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 (HomologicalComplex.dFrom ((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\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 𝟙 ((homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))",
"tactic": "rw [← Category.assoc, ← Category.assoc, Iso.comp_inv_eq]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((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 (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\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)).hom",
"state_before": "case h.e'_2.h.h.e'_7.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((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 (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\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)).hom",
"tactic": "simp only [Category.id_comp]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 (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 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 homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\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)).hom",
"state_before": "case h.e'_2.h.h.e'_7.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((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 (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\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)).hom",
"tactic": "apply homology.hom_from_ext"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ ((homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 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 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 (homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))) ≫\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)).hom",
"state_before": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 (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 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 homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)) ≫\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)).hom",
"tactic": "simp only [← Category.assoc]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 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 cokernel.π\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))",
"state_before": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ ((homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 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 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 (homology.π' (HomologicalComplex.dTo ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)\n (HomologicalComplex.dFrom ((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 𝟙\n ((homologyFunctor D (ComplexShape.down ℕ) 0).obj\n ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))) ≫\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)).hom",
"tactic": "erw [homology.π'_desc', Category.assoc, Category.assoc, ←\n Category.assoc (F.map _), Abelian.cokernel.desc.inv _ _ (exact_of_map_projectiveResolution F P),\n cokernel.π_desc, homology.π', Category.comp_id, Category.assoc (cokernel.π _), Iso.inv_hom_id,\n Category.comp_id, ← Category.assoc]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 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 (kernel (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)) ≫\n cokernel.π\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))",
"state_before": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 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 cokernel.π\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))",
"tactic": "conv_rhs => rw [← Category.id_comp (cokernel.π _)]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.h.e_a\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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 𝟙 (kernel (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0))",
"state_before": "case h.e'_2.h.h.e'_7.h.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 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 (kernel (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)) ≫\n cokernel.π\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))",
"tactic": "congr"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.h.e_a.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0 =\n 𝟙 (kernel (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)) ≫\n equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0",
"state_before": "case h.e'_2.h.h.e'_7.h.h.e_a\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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 𝟙 (kernel (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0))",
"tactic": "apply equalizer.hom_ext"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.h.e_a.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0 =\n equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0",
"state_before": "case h.e'_2.h.h.e'_7.h.h.e_a.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0 =\n 𝟙 (kernel (HomologicalComplex.dFrom ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0)) ≫\n equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0",
"tactic": "simp only [Category.id_comp]"
},
{
"state_after": "case h.e'_2.h.h.e'_7.h.h.e_a.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) =\n kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))",
"state_before": "case h.e'_2.h.h.e'_7.h.h.e_a.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ (kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (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 equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0 =\n equalizer.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) 0",
"tactic": "rw [Category.assoc, equalizer_as_kernel, kernel.lift_ι]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.h.e'_7.h.h.e_a.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\ne_1✝ :\n ((leftDerived F 0).obj X ⟶\n homology (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 ((leftDerived F 0).obj X ⟶\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex))\ne_4✝ :\n HomologicalComplex.homology ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex) 0 =\n (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\ne_5✝ :\n homology (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 (homologyFunctor D (ComplexShape.down ℕ) 0).obj ((mapHomologicalComplex F (ComplexShape.down ℕ)).obj P.complex)\n⊢ kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0))) ≫\n 𝟙 (F.obj (HomologicalComplex.X P.complex 0)) =\n kernel.ι (F.map (HomologicalComplex.d P.complex 0 (ComplexShape.next (ComplexShape.down ℕ) 0)))",
"tactic": "simp only [Category.comp_id]"
}
] |
[
124,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.diam_le_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.283736\ninst✝ : PseudoEMetricSpace α\nx y z : α\nε ε₁ ε₂ : ℝ≥0∞\ns t : Set α\nd : ℝ≥0∞\n⊢ diam s ≤ d ↔ ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → edist x y ≤ d",
"tactic": "simp only [diam, iSup_le_iff]"
}
] |
[
879,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
878,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
RingEquiv.ofLeftInverseS_symm_apply
|
[] |
[
1301,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1299,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Set.Finite.summable
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.35832\nδ : Type ?u.35835\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf✝ g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhs : Set.Finite s\nf : β → α\nthis : Summable (f ∘ Subtype.val)\n⊢ Summable (f ∘ Subtype.val)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.35832\nδ : Type ?u.35835\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf✝ g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhs : Set.Finite s\nf : β → α\n⊢ Summable (f ∘ Subtype.val)",
"tactic": "have := hs.toFinset.summable f"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.35832\nδ : Type ?u.35835\ninst✝¹ : AddCommMonoid α\ninst✝ : TopologicalSpace α\nf✝ g : β → α\na b : α\ns✝ : Finset β\ns : Set β\nhs : Set.Finite s\nf : β → α\nthis : Summable (f ∘ Subtype.val)\n⊢ Summable (f ∘ Subtype.val)",
"tactic": "rwa [hs.coe_toFinset] at this"
}
] |
[
191,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
11
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.intDegree_zero
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\n⊢ intDegree 0 = 0",
"tactic": "rw [intDegree, num_zero, natDegree_zero, denom_zero, natDegree_one, sub_self]"
}
] |
[
1572,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1571,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
|
Real.tan_add
|
[
{
"state_after": "no goals",
"state_before": "x y : ℝ\nh :\n ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * π / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * π / 2) ∨\n (∃ k, x = (2 * ↑k + 1) * π / 2) ∧ ∃ l, y = (2 * ↑l + 1) * π / 2\n⊢ tan (x + y) = (tan x + tan y) / (1 - tan x * tan y)",
"tactic": "simpa only [← Complex.ofReal_inj, Complex.ofReal_sub, Complex.ofReal_add, Complex.ofReal_div,\n Complex.ofReal_mul, Complex.ofReal_tan] using\n @Complex.tan_add (x : ℂ) (y : ℂ) (by convert h <;> norm_cast)"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nh :\n ((∀ (k : ℤ), x ≠ (2 * ↑k + 1) * π / 2) ∧ ∀ (l : ℤ), y ≠ (2 * ↑l + 1) * π / 2) ∨\n (∃ k, x = (2 * ↑k + 1) * π / 2) ∧ ∃ l, y = (2 * ↑l + 1) * π / 2\n⊢ ((∀ (k : ℤ), ↑x ≠ (2 * ↑k + 1) * ↑π / 2) ∧ ∀ (l : ℤ), ↑y ≠ (2 * ↑l + 1) * ↑π / 2) ∨\n (∃ k, ↑x = (2 * ↑k + 1) * ↑π / 2) ∧ ∃ l, ↑y = (2 * ↑l + 1) * ↑π / 2",
"tactic": "convert h <;> norm_cast"
}
] |
[
37,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
31,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioi_def
|
[] |
[
116,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.univ_filter_mem_range
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.29339\ninst✝³ : Fintype α\ns t : Finset α\nf : α → β\ninst✝² : Fintype β\ninst✝¹ : DecidablePred fun y => y ∈ Set.range f\ninst✝ : DecidableEq β\nthis : DecidablePred fun y => ∃ x, f x = y := inst✝¹\n⊢ filter (fun y => y ∈ Set.range f) univ = image f univ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.29339\ninst✝³ : Fintype α\ns t : Finset α\nf : α → β\ninst✝² : Fintype β\ninst✝¹ : DecidablePred fun y => y ∈ Set.range f\ninst✝ : DecidableEq β\n⊢ filter (fun y => y ∈ Set.range f) univ = image f univ",
"tactic": "letI : DecidablePred (fun y => ∃ x, f x = y) := by simpa using ‹_›"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.29339\ninst✝³ : Fintype α\ns t : Finset α\nf : α → β\ninst✝² : Fintype β\ninst✝¹ : DecidablePred fun y => y ∈ Set.range f\ninst✝ : DecidableEq β\nthis : DecidablePred fun y => ∃ x, f x = y := inst✝¹\n⊢ filter (fun y => y ∈ Set.range f) univ = image f univ",
"tactic": "exact univ_filter_exists f"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.29339\ninst✝³ : Fintype α\ns t : Finset α\nf : α → β\ninst✝² : Fintype β\ninst✝¹ : DecidablePred fun y => y ∈ Set.range f\ninst✝ : DecidableEq β\n⊢ DecidablePred fun y => ∃ x, f x = y",
"tactic": "simpa using ‹_›"
}
] |
[
318,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.lt_left_of_right_lt
|
[
{
"state_after": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.610093\nα : Type ?u.610096\nβ : Type u_3\nι : Type ?u.610102\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhyz : f y < f (a • x + b • y)\n⊢ f (a • x + b • y) < f x",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.610093\nα : Type ?u.610096\nβ : Type u_3\nι : Type ?u.610102\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y z : E\nhx : x ∈ s\nhy : y ∈ s\nhz : z ∈ openSegment 𝕜 x y\nhyz : f y < f z\n⊢ f z < f x",
"tactic": "obtain ⟨a, b, ha, hb, hab, rfl⟩ := hz"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.610093\nα : Type ?u.610096\nβ : Type u_3\nι : Type ?u.610102\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhyz : f y < f (a • x + b • y)\n⊢ f (a • x + b • y) < f x",
"tactic": "exact hf.lt_left_of_right_lt' hx hy ha hb hab hyz"
}
] |
[
806,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
803,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.nonzero_image_of_nonzero
|
[] |
[
155,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformContinuousOn_iff_restrict
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.149530\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\n⊢ Tendsto (fun x => (f x.fst, f x.snd)) (𝓤 α ⊓ 𝓟 (s ×ˢ s)) (𝓤 β) ↔\n Tendsto (fun x => (restrict s f x.fst, restrict s f x.snd)) (𝓤 ↑s) (𝓤 β)",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.149530\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\n⊢ UniformContinuousOn f s ↔ UniformContinuous (restrict s f)",
"tactic": "delta UniformContinuousOn UniformContinuous"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.149530\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\n⊢ Tendsto ((fun x => (f x.fst, f x.snd)) ∘ Prod.map Subtype.val Subtype.val) (𝓤 ↑s) (𝓤 β) ↔\n Tendsto (fun x => (restrict s f x.fst, restrict s f x.snd)) (𝓤 ↑s) (𝓤 β)",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.149530\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\n⊢ Tendsto (fun x => (f x.fst, f x.snd)) (𝓤 α ⊓ 𝓟 (s ×ˢ s)) (𝓤 β) ↔\n Tendsto (fun x => (restrict s f x.fst, restrict s f x.snd)) (𝓤 ↑s) (𝓤 β)",
"tactic": "rw [← map_uniformity_set_coe, tendsto_map'_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.149530\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\n⊢ Tendsto ((fun x => (f x.fst, f x.snd)) ∘ Prod.map Subtype.val Subtype.val) (𝓤 ↑s) (𝓤 β) ↔\n Tendsto (fun x => (restrict s f x.fst, restrict s f x.snd)) (𝓤 ↑s) (𝓤 β)",
"tactic": "rfl"
}
] |
[
1504,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1501,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.embedding_coeFn
|
[] |
[
291,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Topology/MetricSpace/IsometricSMul.lean
|
Metric.smul_closedBall
|
[] |
[
444,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/RingTheory/Ideal/LocalRing.lean
|
LocalRing.ResidueField.map_comp
|
[] |
[
426,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Combinatorics/Configuration.lean
|
Configuration.HasLines.card_le
|
[
{
"state_after": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\n⊢ False",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\n⊢ Fintype.card P ≤ Fintype.card L",
"tactic": "by_contra hc₂"
},
{
"state_after": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\n⊢ False",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\n⊢ False",
"tactic": "obtain ⟨f, hf₁, hf₂⟩ := Nondegenerate.exists_injective_of_card_le (le_of_not_le hc₂)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\nthis : ∑ p : P, lineCount L p < ∑ p : P, lineCount L p\n⊢ False",
"tactic": "exact lt_irrefl _ this"
},
{
"state_after": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\n⊢ (∃ a, a ∈ Finset.univ ∧ f a = p) → ∃ a ha, p = (fun l x => f l) a ha",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\n⊢ p ∈ Finset.image f Finset.univ → ∃ a ha, p = (fun l x => f l) a ha",
"tactic": "rw [Finset.mem_image]"
},
{
"state_after": "no goals",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\n⊢ (∃ a, a ∈ Finset.univ ∧ f a = p) → ∃ a ha, p = (fun l x => f l) a ha",
"tactic": "exact fun ⟨a, ⟨h, h'⟩⟩ => ⟨a, ⟨h, h'.symm⟩⟩"
},
{
"state_after": "case intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ ∑ p in Finset.image f Finset.univ, lineCount L p < ∑ p : P, lineCount L p",
"state_before": "P : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\n⊢ ∑ p in Finset.image f Finset.univ, lineCount L p < ∑ p : P, lineCount L p",
"tactic": "obtain ⟨p, hp⟩ := not_forall.mp (mt (Fintype.card_le_of_surjective f) hc₂)"
},
{
"state_after": "case intro.refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ ¬p ∈ Finset.image f Finset.univ\n\ncase intro.refine'_2\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ 0 < lineCount L p",
"state_before": "case intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ ∑ p in Finset.image f Finset.univ, lineCount L p < ∑ p : P, lineCount L p",
"tactic": "refine'\n Finset.sum_lt_sum_of_subset (Finset.univ.image f).subset_univ (Finset.mem_univ p) _ _\n fun p _ _ => zero_le (lineCount L p)"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ ¬p ∈ Finset.image f Finset.univ",
"tactic": "simpa only [Finset.mem_image, exists_prop, Finset.mem_univ, true_and_iff]"
},
{
"state_after": "case intro.refine'_2\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ Nonempty { l // p ∈ l }",
"state_before": "case intro.refine'_2\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ 0 < lineCount L p",
"tactic": "rw [lineCount, Nat.card_eq_fintype_card, Fintype.card_pos_iff]"
},
{
"state_after": "case intro.refine'_2.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\nl : L\nh✝ : ¬p ∈ l\n⊢ Nonempty { l // p ∈ l }",
"state_before": "case intro.refine'_2\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\n⊢ Nonempty { l // p ∈ l }",
"tactic": "obtain ⟨l, _⟩ := @exists_line P L _ _ p"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2.intro\nP : Type u_1\nL : Type u_2\ninst✝³ : Membership P L\ninst✝² : HasLines P L\ninst✝¹ : Fintype P\ninst✝ : Fintype L\nhc₂ : ¬Fintype.card P ≤ Fintype.card L\nf : L → P\nhf₁ : Function.Injective f\nhf₂ : ∀ (l : L), ¬f l ∈ l\np : P\nhp : ¬∃ a, f a = p\nl : L\nh✝ : ¬p ∈ l\n⊢ Nonempty { l // p ∈ l }",
"tactic": "exact\n let this := not_exists.mp hp l\n ⟨⟨mkLine this, (mkLine_ax this).2⟩⟩"
}
] |
[
256,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
230,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
ciSup_unique
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.59227\nγ : Type ?u.59230\nι : Sort u_1\ninst✝¹ : ConditionallyCompleteLattice α\ns✝ t : Set α\na b : α\ninst✝ : Unique ι\ns : ι → α\nthis : ∀ (i : ι), s i = s default\n⊢ (⨆ (i : ι), s i) = s default",
"state_before": "α : Type u_2\nβ : Type ?u.59227\nγ : Type ?u.59230\nι : Sort u_1\ninst✝¹ : ConditionallyCompleteLattice α\ns✝ t : Set α\na b : α\ninst✝ : Unique ι\ns : ι → α\n⊢ (⨆ (i : ι), s i) = s default",
"tactic": "have : ∀ i, s i = s default := fun i => congr_arg s (Unique.eq_default i)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.59227\nγ : Type ?u.59230\nι : Sort u_1\ninst✝¹ : ConditionallyCompleteLattice α\ns✝ t : Set α\na b : α\ninst✝ : Unique ι\ns : ι → α\nthis : ∀ (i : ι), s i = s default\n⊢ (⨆ (i : ι), s i) = s default",
"tactic": "simp only [this, ciSup_const]"
}
] |
[
843,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
841,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean
|
Real.lt_logb_iff_rpow_lt_of_base_lt_one
|
[
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nb_pos : 0 < b\nb_lt_one : b < 1\nhy : 0 < y\n⊢ x < logb b y ↔ y < b ^ x",
"tactic": "rw [← rpow_lt_rpow_left_iff_of_base_lt_one b_pos b_lt_one, rpow_logb b_pos (b_ne_one b_lt_one) hy]"
}
] |
[
285,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
differentiableAt_euclidean
|
[
{
"state_after": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\n⊢ (∀ (i : ι), DifferentiableAt 𝕜 (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔\n ∀ (i : ι), DifferentiableAt 𝕜 (fun x => f x i) y",
"state_before": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\n⊢ DifferentiableAt 𝕜 f y ↔ ∀ (i : ι), DifferentiableAt 𝕜 (fun x => f x i) y",
"tactic": "rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiableAt_iff, differentiableAt_pi]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nι : Type u_3\nH : Type u_2\ninst✝³ : IsROrC 𝕜\ninst✝² : NormedAddCommGroup H\ninst✝¹ : NormedSpace 𝕜 H\ninst✝ : Fintype ι\nf : H → EuclideanSpace 𝕜 ι\nf' : H →L[𝕜] EuclideanSpace 𝕜 ι\nt : Set H\ny : H\n⊢ (∀ (i : ι), DifferentiableAt 𝕜 (fun x => (↑(EuclideanSpace.equiv ι 𝕜) ∘ f) x i) y) ↔\n ∀ (i : ι), DifferentiableAt 𝕜 (fun x => f x i) y",
"tactic": "rfl"
}
] |
[
308,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/CategoryTheory/Preadditive/Opposite.lean
|
CategoryTheory.op_add
|
[] |
[
65,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Connected.map
|
[] |
[
1959,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1956,
1
] |
Mathlib/Topology/PathConnected.lean
|
pathComponent.nonempty
|
[] |
[
909,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
AlgebraicIndependent.aeval_comp_repr
|
[] |
[
411,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean
|
Set.isUnit_singleton
|
[] |
[
1188,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1187,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.bit0_div_bit0
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ bit0 n / bit0 m = n / m",
"tactic": "rw [bit0_eq_two_mul m, ← Nat.div_div_eq_div_mul, bit0_div_two]"
}
] |
[
255,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Std/Data/List/Lemmas.lean
|
List.infix_append
|
[] |
[
1563,
83
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1563,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.image_symm_image
|
[] |
[
2058,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2057,
1
] |
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