module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 15
} | {
"line": 477,
"column": 16
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nn : ℕ∞\nk : ℕ\nih : ∀ ⦃x : M⦄, ↑k ≤ n → ((f - μ • 1) ^ k) x = 0 → ((f - μ • 1) ^ k) (f x) = 0\nx : M\nhk : ↑(k + 1) ≤ n\nhx : ((f - μ • 1) ^ k) ((f - μ • 1) x) = 0\n⊢ ((f - μ • 1) ^ k) ((f - μ ... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nn : ℕ∞\nk : ℕ\nih : ∀ ⦃x : M⦄, ↑k ≤ n → ((f - μ • 1) ^ k) x = 0 → ((f - μ • 1) ^ k) (f x) = 0\nx : M\nhk : ↑(k + 1) ≤ n\nhx : ((f - μ • 1) ^ k) ((f - μ • 1) x) = 0\n⊢ ((f - μ • 1) ^ k) (f (f x)) - μ • ((f ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 572,
"column": 14
} | {
"line": 572,
"column": 25
} | {
"line": 572,
"column": 26
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : f.HasGenEigenvalue μ k\n⊢ ↑k ≤ ↑m",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"ins... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk m : ℕ\nhm : k ≤ m\nhk : f.HasGenEigenvalue μ k\n⊢ k ≤ m"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 577,
"column": 37
} | {
"line": 577,
"column": 48
} | {
"line": 577,
"column": 49
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\n⊢ 1 ≤ ↑k",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"instCharZeroENat",
"instAddMo... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\n⊢ 1 ≤ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 586,
"column": 14
} | {
"line": 586,
"column": 25
} | {
"line": 586,
"column": 26
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : f.HasEigenvalue μ\n⊢ 0 < ↑k",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinearOrderENat",
"ChainComple... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\nμ : R\nk : ℕ\nhk : 0 < k\nhμ : f.HasEigenvalue μ\n⊢ 0 < k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 663,
"column": 2
} | {
"line": 663,
"column": 40
} | {
"line": 663,
"column": 41
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nf : End R M\nk : ℕ∞\nμ₁ : R\na✝ : μ₁ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}\nμ₂ : R\nhμ₂ : μ₂ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}\nhμ₁₂ : (fun x ↦ (f.genEigenspace x) k) μ... | [
"R : Type v\nM : Type w\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R M\nf : End R M\nk : ℕ∞\nμ₁ : R\na✝ : μ₁ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}\nμ₂ : R\nhμ₂ : μ₂ ∈ {μ | (f.genEigenspace μ) k ≠ ⊥}\nhμ₁₂ : (fun x ↦ (f.genEigenspace x) k) μ₁ = (fun x ↦... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.DirectSum.LinearMap | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 39
} | {
"line": 136,
"column": 40
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_4\nN : ι → Submodule R M\ns : Set ι\nf : End R M\nhf : ∀ (i : ι), Submodule.map f (N i) ≤ N i\n⊢ Submodule.map f (⨆ i ∈ s, N i) ≤ ⨆ i ∈ s, N i",
"ppTerm": "?m.122",
"assigned": true,
"us... | [
"R : Type u_2\nM : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nι : Type u_4\nN : ι → Submodule R M\ns : Set ι\nf : End R M\nhf : ∀ (i : ι), Submodule.map f (N i) ≤ N i\n⊢ ⨆ i ∈ s, Submodule.map f (N i) ≤ ⨆ i ∈ s, N i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 13
} | {
"line": 806,
"column": 14
} | [
{
"pp": "R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhfp : ∀ x ∈ p, f x ∈ p\nμ : R\nhμp : Disjoint (f.eigenspace μ) p\nx : ↥p\nhx : x ∈ eigenspace (LinearMap.restrict f hfp) μ\n⊢ x ∈ ⊥",
"ppTerm": "?m.56",
"assigned": true,
... | [
"R : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhfp : ∀ x ∈ p, f x ∈ p\nμ : R\nhμp : Disjoint (f.eigenspace μ) p\nx : ↥p\nhx : x ∈ eigenspace (LinearMap.restrict f hfp) μ\n⊢ x = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 816,
"column": 63
} | {
"line": 816,
"column": 74
} | {
"line": 816,
"column": 75
} | [
{
"pp": "K : Type v\nV : Type w\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : End K V\nk : ℕ\nμ : K\nhx : f.HasEigenvalue μ\nhk : 0 < k\n⊢ 1 ≤ ↑k",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instCompleteLinear... | [
"K : Type v\nV : Type w\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : End K V\nk : ℕ\nμ : K\nhx : f.HasEigenvalue μ\nhk : 0 < k\n⊢ 1 ≤ k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Trace | {
"line": 270,
"column": 2
} | {
"line": 271,
"column": 21
} | {
"line": 272,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] M\ng : N →ₗ[R] N\nh :\n (((mapBilinear (Ri... | [
"R : Type u_1\ninst✝⁸ : CommRing R\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u_3\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : Free R M\ninst✝² : Module.Finite R M\ninst✝¹ : Free R N\ninst✝ : Module.Finite R N\nf : M →ₗ[R] M\ng : N →ₗ[R] N\nh : (trace R (M ⊗[R] N)) (map f g) = ... | simp only [compr₂_apply, mapBilinear_apply, compl₁₂_apply, lsmul_apply,
smul_eq_mul] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Trace | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 17
} | {
"line": 320,
"column": 18
} | [
{
"pp": "K : Type u_6\nV : Type u_7\nW : Type u_8\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\nF : Type u_9\ninst✝¹ : EquivLike F (End K V) (End K W)\ninst✝ : AlgEquivClass F K (End K V) (End K W)\nf : F\nx : End K V\nw✝ : V ≃ₗ[K] W\nh : ↑f = Lin... | [
"K : Type u_6\nV : Type u_7\nW : Type u_8\ninst✝⁶ : Field K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\ninst✝³ : AddCommGroup W\ninst✝² : Module K W\nF : Type u_9\ninst✝¹ : EquivLike F (End K V) (End K W)\ninst✝ : AlgEquivClass F K (End K V) (End K W)\nf : F\nx : End K V\nw✝ : V ≃ₗ[K] W\nh : ↑f = LinearEquiv.con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Trace | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 56
} | {
"line": 326,
"column": 4
} | [
{
"pp": "K : Type u_6\nm : Type u_7\nn : Type u_8\ninst✝⁶ : Field K\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nF : Type u_9\ninst✝¹ : EquivLike F (Matrix m m K) (Matrix n n K)\ninst✝ : AlgEquivClass F K (Matrix m m K) (Matrix n n K)\nf : F\nx : Matrix m m K\n⊢ (f x)... | [
"K : Type u_6\nm : Type u_7\nn : Type u_8\ninst✝⁶ : Field K\ninst✝⁵ : Fintype m\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\nF : Type u_9\ninst✝¹ : EquivLike F (Matrix m m K) (Matrix n n K)\ninst✝ : AlgEquivClass F K (Matrix m m K) (Matrix n n K)\nf : F\nx : Matrix m m K\n⊢ (f x).trace = x.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Basic | {
"line": 852,
"column": 4
} | {
"line": 852,
"column": 35
} | {
"line": 852,
"column": 36
} | [
{
"pp": "case h.left\nR : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf₁ f₂ : End R M\nμ₁ μ₂ : R\nk₁ k₂ : ℕ∞\nm : M\nl₁ : ℕ\nhlk₁ : ↑l₁ ≤ k₁\nhl₁ : ((f₁ - μ₁ • 1) ^ l₁) m = 0\nl₂ : ℕ\nhlk₂ : ↑l₂ ≤ k₂\nhl₂ : ((f₂ - μ₂ • 1) ^ l₂) m = 0\nthis : f₁ + f₂ - (μ₁ + μ₂) • 1 = f₁... | [
"case h.left\nR : Type v\nM : Type w\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf₁ f₂ : End R M\nμ₁ μ₂ : R\nk₁ k₂ : ℕ∞\nm : M\nl₁ : ℕ\nhlk₁ : ↑l₁ ≤ k₁\nhl₁ : ((f₁ - μ₁ • 1) ^ l₁) m = 0\nl₂ : ℕ\nhlk₂ : ↑l₂ ≤ k₂\nhl₂ : ((f₂ - μ₂ • 1) ^ l₂) m = 0\nthis : f₁ + f₂ - (μ₁ + μ₂) • 1 = f₁ - μ₁ • 1 + ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.GeomSum | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 35
} | {
"line": 58,
"column": 36
} | [
{
"pp": "K : Type u_2\ninst✝ : DivisionRing K\nx : K\nhx : x ≠ 1\nm n : ℕ\nhmn : m ≤ n\n⊢ ∑ i ∈ Ico m n, x ^ i = (x ^ m - x ^ n) / (1 - x)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddGroup... | [
"K : Type u_2\ninst✝ : DivisionRing K\nx : K\nhx : x ≠ 1\nm n : ℕ\nhmn : m ≤ n\n⊢ (x ^ n - x ^ m) / (x - 1) = (x ^ m - x ^ n) / (1 - x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.MinimalAxioms | {
"line": 75,
"column": 51
} | {
"line": 76,
"column": 25
} | {
"line": 77,
"column": 6
} | [
{
"pp": "G : Type u\ninst✝² : Mul G\ninst✝¹ : Inv G\ninst✝ : One G\nassoc : ∀ (a b c : G), a * b * c = a * (b * c)\nmul_one : ∀ (a : G), a * 1 = a\nmul_inv_cancel : ∀ (a : G), a * a⁻¹ = 1\na : G\n⊢ a⁻¹ * a * (a⁻¹ * a * (a⁻¹ * a)⁻¹) = a⁻¹ * (a * a⁻¹) * a * (a⁻¹ * a)⁻¹",
"ppTerm": "?m.169",
"assigned": tr... | [] | by
simp only [assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Field.Periodic | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 50
} | {
"line": 49,
"column": 51
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝² : AddCommMonoid α\ninst✝¹ : DivisionSemiring γ\ninst✝ : Module γ α\nh : Periodic f c\na : γ\nx : α\nha : ¬a = 0\n⊢ (fun x ↦ f (a • x)) (x + a⁻¹ • c) = (fun x ↦ f (a • x)) x",
"ppTerm": "?neg✝",
"assigned": true,
"u... | [
"case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝² : AddCommMonoid α\ninst✝¹ : DivisionSemiring γ\ninst✝ : Module γ α\nh : Periodic f c\na : γ\nx : α\nha : ¬a = 0\n⊢ f (a • x + c) = f (a • x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 28
} | {
"line": 57,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝² : AddCommMonoid α\ninst✝¹ : DivisionSemiring γ\ninst✝ : Module γ α\nh : Periodic f c\na : γ\n⊢ Periodic (fun x ↦ f (a⁻¹ • x)) (a • c)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝² : AddCommMonoid α\ninst✝¹ : DivisionSemiring γ\ninst✝ : Module γ α\nh : Periodic f c\na : γ\n⊢ Periodic (fun x ↦ f (a⁻¹ • x)) (a • c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 68,
"column": 48
} | {
"line": 68,
"column": 81
} | {
"line": 68,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝ : DivisionSemiring α\nh : Periodic f c\na : α\n⊢ Periodic (fun x ↦ f (x * a)) (c / a)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"GroupWi... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝ : DivisionSemiring α\nh : Periodic f c\na : α\n⊢ Periodic (fun x ↦ f (x * a)) (c * a⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 75,
"column": 48
} | {
"line": 75,
"column": 81
} | {
"line": 75,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝ : DivisionSemiring α\nh : Periodic f c\na : α\n⊢ Periodic (fun x ↦ f (x / a)) (c * a)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"GroupWi... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝ : DivisionSemiring α\nh : Periodic f c\na : α\n⊢ Periodic (fun x ↦ f (x * a⁻¹)) (c * a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 23
} | {
"line": 127,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : NonAssocSemiring α\ninst✝ : Ring β\nh : Antiperiodic f c\nn : ℕ\n⊢ f (x + ↑n * c) = (-1) ^ n * f x",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : NonAssocSemiring α\ninst✝ : Ring β\nh : Antiperiodic f c\nn : ℕ\n⊢ f (x + ↑n * c) = (-1) ^ n * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 23
} | {
"line": 132,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : NonAssocRing α\ninst✝ : Ring β\nh : Antiperiodic f c\nn : ℕ\n⊢ f (x - ↑n * c) = (-1) ^ n * f x",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : NonAssocRing α\ninst✝ : Ring β\nh : Antiperiodic f c\nn : ℕ\n⊢ f (x - ↑n * c) = (-1) ^ n * f x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 136,
"column": 2
} | {
"line": 137,
"column": 23
} | {
"line": 137,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : NonAssocRing α\ninst✝ : Ring β\nh : Antiperiodic f c\nn : ℕ\n⊢ f (↑n * c - x) = (-1) ^ n * f (-x)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc x : α\ninst✝¹ : NonAssocRing α\ninst✝ : Ring β\nh : Antiperiodic f c\nn : ℕ\n⊢ f (↑n * c - x) = (-1) ^ n * f (-x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 141,
"column": 14
} | {
"line": 141,
"column": 60
} | {
"line": 141,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : AddMonoid α\ninst✝² : Neg β\ninst✝¹ : GroupWithZero γ\ninst✝ : DistribMulAction γ α\nh : Antiperiodic f c\na : γ\nha : a ≠ 0\nx : α\n⊢ (fun x ↦ f (a • x)) (x + a⁻¹ • c) = -(fun x ↦ f (a • x)) x",
"ppTerm": "?m.26",
"assigned":... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : AddMonoid α\ninst✝² : Neg β\ninst✝¹ : GroupWithZero γ\ninst✝ : DistribMulAction γ α\nh : Antiperiodic f c\na : γ\nha : a ≠ 0\nx : α\n⊢ f (a • x + c) = -f (a • x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 28
} | {
"line": 149,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : AddMonoid α\ninst✝² : Neg β\ninst✝¹ : GroupWithZero γ\ninst✝ : DistribMulAction γ α\nh : Antiperiodic f c\na : γ\nha : a ≠ 0\n⊢ Antiperiodic (fun x ↦ f (a⁻¹ • x)) (a • c)",
"ppTerm": "?m.25",
"assigned": false,
"usedConsta... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : α → β\nc : α\ninst✝³ : AddMonoid α\ninst✝² : Neg β\ninst✝¹ : GroupWithZero γ\ninst✝ : DistribMulAction γ α\nh : Antiperiodic f c\na : γ\nha : a ≠ 0\n⊢ Antiperiodic (fun x ↦ f (a⁻¹ • x)) (a • c)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 35
} | {
"line": 161,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : DivisionSemiring α\ninst✝ : Neg β\nh : Antiperiodic f c\na : α\nha : a ≠ 0\n⊢ Antiperiodic (fun x ↦ f (x * a)) (c / a)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : DivisionSemiring α\ninst✝ : Neg β\nh : Antiperiodic f c\na : α\nha : a ≠ 0\n⊢ Antiperiodic (fun x ↦ f (x * a)) (c * a⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Field.Periodic | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 35
} | {
"line": 169,
"column": 36
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : DivisionSemiring α\ninst✝ : Neg β\nh : Antiperiodic f c\na : α\nha : a ≠ 0\n⊢ Antiperiodic (fun x ↦ f (x / a)) (c * a)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nc : α\ninst✝¹ : DivisionSemiring α\ninst✝ : Neg β\nh : Antiperiodic f c\na : α\nha : a ≠ 0\n⊢ Antiperiodic (fun x ↦ f (x * a⁻¹)) (c * a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 67
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 72,
"column": 46
} | {
"line": 72,
"column": 57
} | {
"line": 72,
"column": 58
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FreeAlgebra.Cardinality | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 13
} | {
"line": 65,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nX : Type u\n⊢ #(FreeAlgebra R X) ≤ max (max #R #X) ℵ₀",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"R : Type u\ninst✝ : CommSemiring R\nX : Type u\n⊢ (#(FreeAlgebra R X) ≤ #R ∨ #(FreeAlgebra R X) ≤ #X) ∨ #(FreeAlgebra R X) ≤ ℵ₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 75,
"column": 40
} | {
"line": 75,
"column": 51
} | {
"line": 75,
"column": 52
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FreeAlgebra.Cardinality | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 13
} | {
"line": 80,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type u\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\n⊢ #↥(adjoin R s) ≤ max (max #R #↑s) ℵ₀",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Car... | [
"R : Type u\ninst✝² : CommSemiring R\nA : Type u\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\ns : Set A\n⊢ (#↥(adjoin R s) ≤ #R ∨ #↥(adjoin R s) ≤ #↑s) ∨ (↑(adjoin R s)).Countable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FreeMonoid.Count | {
"line": 61,
"column": 98
} | {
"line": 62,
"column": 45
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nx y : α\n⊢ (count x) (of y) = Pi.mulSingle x (Multiplicative.ofAdd 1) y",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"CancelMonoid.toRightCancelMonoid",
"FreeMonoid",
"MonoidHom.instFunLike",
"Equiv.instEquivLike",
... | [] | by
simp [count, countP_of, Pi.mulSingle_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.FreeMonoid.FreeSemigroup | {
"line": 54,
"column": 4
} | {
"line": 54,
"column": 72
} | {
"line": 54,
"column": 73
} | [
{
"pp": "α : Type u_1\nx : α\nxs : List α\nthis : ∀ (x : FreeMonoid α), List.foldl (fun x1 x2 ↦ x1 * x2) x (List.map FreeMonoid.of xs) = x * ofList xs\n⊢ toFreeMonoid { head := x, tail := xs } = ofList (x :: xs)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"FreeSemigroup.lift._pro... | [
"α : Type u_1\nx : α\nxs : List α\nthis : ∀ (x : FreeMonoid α), List.foldl (fun x1 x2 ↦ x1 * x2) x (List.map FreeMonoid.of xs) = x * ofList xs\n⊢ List.foldl HMul.hMul (FreeMonoid.of x) (List.map FreeMonoid.of xs) = FreeMonoid.of x * ofList xs"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FreeMonoid.FreeSemigroup | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 13
} | {
"line": 61,
"column": 14
} | [
{
"pp": "α : Type u_1\nx : α\nxs : List α\ny : α\nys : List α\nh : x :: xs = y :: ys\n⊢ { head := x, tail := xs } = { head := y, tail := ys }",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FreeSemigroup.mk.injEq",
"FreeSemigroup.mk",
"id",
"List",
... | [
"α : Type u_1\nx : α\nxs : List α\ny : α\nys : List α\nh : x :: xs = y :: ys\n⊢ x = y ∧ xs = ys"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 103,
"column": 42
} | {
"line": 103,
"column": 53
} | {
"line": 103,
"column": 54
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 105,
"column": 39
} | {
"line": 105,
"column": 50
} | {
"line": 105,
"column": 51
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 107,
"column": 36
} | {
"line": 107,
"column": 47
} | {
"line": 107,
"column": 48
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nM₄ : Type u_4\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\nN₄ : Type u_9\ninst✝⁷ : Group M₁\ninst✝⁶ : Group M₂\ninst✝⁵ : Group M₃\ninst✝⁴ : Group M₄\ninst✝³ : Group N₁\ninst✝² : Group N₂\ninst✝¹ : Group N₃\ninst✝ : Group N₄\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\nf₃ : M₃ →* M₄\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 123,
"column": 49
} | {
"line": 123,
"column": 60
} | {
"line": 123,
"column": 61
} | [
{
"pp": "M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝⁵ : Group M₁\ninst✝⁴ : Group M₂\ninst✝³ : Group M₃\ninst✝² : Group N₁\ninst✝¹ : Group N₂\ninst✝ : Group N₃\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\ng₁ : N₁ →* N₂\ng₂ : N₂ →* N₃\ni₁ : M₁ →* N₁\ni₂ : M₂ →* N₂\ni₃ : M₃ →*... | [
"M₁ : Type u_1\nM₂ : Type u_2\nM₃ : Type u_3\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝⁵ : Group M₁\ninst✝⁴ : Group M₂\ninst✝³ : Group M₃\ninst✝² : Group N₁\ninst✝¹ : Group N₂\ninst✝ : Group N₃\nf₁ : M₁ →* M₂\nf₂ : M₂ →* M₃\ng₁ : N₁ →* N₂\ng₂ : N₂ →* N₃\ni₁ : M₁ →* N₁\ni₂ : M₂ →* N₂\ni₃ : M₃ →* N₃\nhc₁ : g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.FiveLemma | {
"line": 227,
"column": 69
} | {
"line": 227,
"column": 80
} | {
"line": 227,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝¹² : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type u_7\nN₂ : Type u_8\nN₃ : Type u_9\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : AddCommGroup M₂\ninst✝⁹ : AddCommGroup M₃\ninst✝⁸ : Module R M₁\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\ninst✝⁵ : AddCommGroup N₁\ninst✝... | [
"R : Type u_1\ninst✝¹² : CommRing R\nM₁ : Type u_2\nM₂ : Type u_3\nM₃ : Type u_4\nN₁ : Type u_7\nN₂ : Type u_8\nN₃ : Type u_9\ninst✝¹¹ : AddCommGroup M₁\ninst✝¹⁰ : AddCommGroup M₂\ninst✝⁹ : AddCommGroup M₃\ninst✝⁸ : Module R M₁\ninst✝⁷ : Module R M₂\ninst✝⁶ : Module R M₃\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : AddCommG... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Polynomial.Eisenstein.Criterion | {
"line": 94,
"column": 4
} | {
"line": 96,
"column": 37
} | {
"line": 97,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f g : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_lC : (algebraMap R K) f.leadingCoeff ≠ 0\nhf_prim : f.IsPrimitive\nhfmodP : map (algebraMap R K) f = C ... | [
"R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\nK : Type u_2\ninst✝¹ : Field K\ninst✝ : Algebra R K\nq f g : R[X]\np : ℕ\nhq_irr : Irreducible (map (algebraMap R K) q)\nhq_monic : q.Monic\nhf_lC : (algebraMap R K) f.leadingCoeff ≠ 0\nhf_prim : f.IsPrimitive\nhfmodP : map (algebraMap R K) f = C ((algebraMap... | suffices C (algebraMap R K g.leadingCoeff) = u by
simp [r, ← this, Polynomial.map_sub, ← hu, Polynomial.map_mul, map_C,
Polynomial.map_pow, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.Polynomial.Eisenstein.Basic | {
"line": 108,
"column": 35
} | {
"line": 108,
"column": 46
} | {
"line": 108,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : f.Monic\nhf : f.IsWeaklyEisensteinAt (R ∙ p)\nhx : x ^ (Polynomial.map (algebraMap R S) f).natDegree = -∑ i, (Polynomial.map (algebraMap R S) f).coeff ↑i * x ^ ↑i\nn : ℕ\nhn : n < f.natD... | [
"R : Type u\ninst✝² : CommRing R\nf : R[X]\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R\nx : S\nhmo : f.Monic\nhf : f.IsWeaklyEisensteinAt (R ∙ p)\nhx : x ^ (Polynomial.map (algebraMap R S) f).natDegree = -∑ i, (Polynomial.map (algebraMap R S) f).coeff ↑i * x ^ ↑i\nn : ℕ\nhn : n < f.natDegree\n⊢ f.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Quaternion | {
"line": 329,
"column": 32
} | {
"line": 329,
"column": 53
} | {
"line": 331,
"column": 0
} | [
{
"pp": "case re\nS : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x✝ y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝² : SMul S R\ninst✝¹ : SMul T R\ns✝ : S\ninst✝ : SMulCommClass S T R\ns : S\nt : T\nx : ℍ[R,c₁,c₂,c₃]\n⊢ (s • t • x).re = (t • s • x).re",
"ppTerm": "?re",
"assigned": true,
"usedConstants": [
... | [] | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Quaternion | {
"line": 329,
"column": 32
} | {
"line": 329,
"column": 53
} | {
"line": 331,
"column": 0
} | [
{
"pp": "case imI\nS : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x✝ y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝² : SMul S R\ninst✝¹ : SMul T R\ns✝ : S\ninst✝ : SMulCommClass S T R\ns : S\nt : T\nx : ℍ[R,c₁,c₂,c₃]\n⊢ (s • t • x).imI = (t • s • x).imI",
"ppTerm": "?imI",
"assigned": true,
"usedConstants"... | [] | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Quaternion | {
"line": 329,
"column": 32
} | {
"line": 329,
"column": 53
} | {
"line": 331,
"column": 0
} | [
{
"pp": "case imJ\nS : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x✝ y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝² : SMul S R\ninst✝¹ : SMul T R\ns✝ : S\ninst✝ : SMulCommClass S T R\ns : S\nt : T\nx : ℍ[R,c₁,c₂,c₃]\n⊢ (s • t • x).imJ = (t • s • x).imJ",
"ppTerm": "?imJ",
"assigned": true,
"usedConstants"... | [] | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Quaternion | {
"line": 329,
"column": 32
} | {
"line": 329,
"column": 53
} | {
"line": 331,
"column": 0
} | [
{
"pp": "case imK\nS : Type u_1\nT : Type u_2\nR : Type u_3\nc₁ c₂ c₃ r x✝ y : R\na b : ℍ[R,c₁,c₂,c₃]\ninst✝² : SMul S R\ninst✝¹ : SMul T R\ns✝ : S\ninst✝ : SMulCommClass S T R\ns : S\nt : T\nx : ℍ[R,c₁,c₂,c₃]\n⊢ (s • t • x).imK = (t • s • x).imK",
"ppTerm": "?imK",
"assigned": true,
"usedConstants"... | [] | exact smul_comm _ _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.IntegralClosure.IntegrallyClosed | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 24
} | {
"line": 326,
"column": 25
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsIntegrallyClosed R\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nha : a = 0\n⊢ a ∣ b",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
... | [
"case pos\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsIntegrallyClosed R\nn : ℕ\nhn : n ≠ 0\na b : R\nx✝ : a ^ n ∣ b ^ n\nx : R\nhx : b ^ n = a ^ n * x\nha : a = 0\n⊢ b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Pointwise.Finset | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 52
} | {
"line": 283,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ns : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ s\na n : ℕ\n⊢ (fun x ↦ x ∈ a +ᵥ s) n ↔ a ≤ n ∧ n - a ∈ s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
"Eq.mpr",
"_private.Mathlib.A... | [] | simp only [Set.mem_vadd_set, vadd_eq_add]; aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Pointwise.Finset | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 52
} | {
"line": 283,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ns : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ s\na n : ℕ\n⊢ (fun x ↦ x ∈ a +ᵥ s) n ↔ a ≤ n ∧ n - a ∈ s",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Nat.le_add_right._simp_1",
"Eq.mpr",
"_private.Mathlib.A... | [] | simp only [Set.mem_vadd_set, vadd_eq_add]; aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Ideal | {
"line": 83,
"column": 10
} | {
"line": 83,
"column": 35
} | {
"line": 83,
"column": 36
} | [
{
"pp": "case inr\nM : Type u_1\ninst✝ : Semigroup M\ns : Set M\nx y z : M\nhz : z ∈ s\n⊢ x • (fun x1 x2 ↦ x1 * x2) y z ∈ s ∪ univ * s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Algebra.Group.Ideal.0.SemigroupIdeal.coe_closure._simp_1_1",
"... | [
"case inr\nM : Type u_1\ninst✝ : Semigroup M\ns : Set M\nx y z : M\nhz : z ∈ s\n⊢ x * y * z ∈ s ∨ x * y * z ∈ univ * s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 37
} | {
"line": 127,
"column": 38
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommGroup G\nh : M\nf g : M → G\nn : ℕ\n⊢ Δ_[h]^[n] (f + g) = Δ_[h]^[n] f + Δ_[h]^[n] g",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommGroup G\nh : M\nf g : M → G\nn : ℕ\n⊢ Δ_[h]^[n] (f + g) = Δ_[h]^[n] f + Δ_[h]^[n] g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 37
} | {
"line": 137,
"column": 38
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommGroup G\nh : M\nα : Type u_3\ns : Finset α\nf : α → M → G\nn : ℕ\n⊢ Δ_[h]^[n] (∑ k ∈ s, f k) = ∑ k ∈ s, Δ_[h]^[n] (f k)",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"M : Type u_1\nG : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : AddCommGroup G\nh : M\nα : Type u_3\ns : Finset α\nf : α → M → G\nn : ℕ\n⊢ Δ_[h]^[n] (∑ k ∈ s, f k) = ∑ k ∈ s, Δ_[h]^[n] (f k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 46
} | {
"line": 90,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁵ : Mul M\ninst✝⁴ : Pow M ℕ+\ninst✝³ : PNatPowAssoc M\ninst✝² : Mul N\ninst✝¹ : Pow N ℕ+\ninst✝ : PNatPowAssoc N\nx✝² x✝¹ : ℕ+\nx✝ : M × N\n⊢ x✝ ^ (x✝² + x✝¹) = x✝ ^ x✝² * x✝ ^ x✝¹",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [] | ext <;> simp [ppow_add] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 46
} | {
"line": 90,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁵ : Mul M\ninst✝⁴ : Pow M ℕ+\ninst✝³ : PNatPowAssoc M\ninst✝² : Mul N\ninst✝¹ : Pow N ℕ+\ninst✝ : PNatPowAssoc N\nx✝² x✝¹ : ℕ+\nx✝ : M × N\n⊢ x✝ ^ (x✝² + x✝¹) = x✝ ^ x✝² * x✝ ^ x✝¹",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [] | ext <;> simp [ppow_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.PNatPowAssoc | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 46
} | {
"line": 90,
"column": 2
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁵ : Mul M\ninst✝⁴ : Pow M ℕ+\ninst✝³ : PNatPowAssoc M\ninst✝² : Mul N\ninst✝¹ : Pow N ℕ+\ninst✝ : PNatPowAssoc N\nx✝² x✝¹ : ℕ+\nx✝ : M × N\n⊢ x✝ ^ (x✝² + x✝¹) = x✝ ^ x✝² * x✝ ^ x✝¹",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"HMul.hMul",
... | [] | ext <;> simp [ppow_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 200,
"column": 4
} | {
"line": 203,
"column": 59
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case inl\nm n : ℕ\nhmn : m < n\n⊢ Δ_[1]^[n] (fun x ↦ ↑(x.choose m)) 0 = if n = m then 1 else 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"fwdDiff_const",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
... | [] | rcases Nat.exists_eq_add_of_lt hmn with ⟨k, rfl⟩
simp_rw [hmn.ne', if_false, (by ring : m + k + 1 = k + 1 + m), iterate_add_apply,
add_zero m ▸ fwdDiff_iter_choose 0 m, choose_zero_right, iterate_one, cast_one, fwdDiff_const,
fwdDiff_iter_eq_sum_shift, smul_zero, sum_const_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 200,
"column": 4
} | {
"line": 203,
"column": 59
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case inl\nm n : ℕ\nhmn : m < n\n⊢ Δ_[1]^[n] (fun x ↦ ↑(x.choose m)) 0 = if n = m then 1 else 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"fwdDiff_const",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
... | [] | rcases Nat.exists_eq_add_of_lt hmn with ⟨k, rfl⟩
simp_rw [hmn.ne', if_false, (by ring : m + k + 1 = k + 1 + m), iterate_add_apply,
add_zero m ▸ fwdDiff_iter_choose 0 m, choose_zero_right, iterate_one, cast_one, fwdDiff_const,
fwdDiff_iter_eq_sum_shift, smul_zero, sum_const_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.ForwardDiff | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 36
} | {
"line": 263,
"column": 37
} | [
{
"pp": "case succ\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\nIH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !\nthis : (Δ_[1] fun r ↦ r ^ (n + 1)) = ∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i\n⊢ ((∑ x ∈ range n, (n + 1).choose x • Δ_[1]^[n] fun r ↦ r ^ x) + (n + 1).choose n • Δ_[1]^[n] fun r ↦ r ^ n) = ↑(n + 1)!... | [
"case succ\nR : Type u_3\ninst✝ : CommRing R\nn : ℕ\nIH : (Δ_[1]^[n] fun r ↦ r ^ n) = ↑n !\nthis : (Δ_[1] fun r ↦ r ^ (n + 1)) = ∑ i ∈ range (n + 1), (n + 1).choose i • fun r ↦ r ^ i\n⊢ (∑ x ∈ range n, ↑((n + 1).choose x) * Δ_[1]^[n] fun r ↦ r ^ x) = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 43,
"column": 29
} | {
"line": 43,
"column": 40
} | {
"line": 43,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : Preorder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftMono α\ninst✝¹ : MulRightMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Icc a b * Icc c d) ⊆ ↑(Icc (a * c) (b * d))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : Preorder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftMono α\ninst✝¹ : MulRightMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ Set.Icc a b * Set.Icc c d ⊆ Set.Icc (a * c) (b * d)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 47,
"column": 29
} | {
"line": 47,
"column": 40
} | {
"line": 47,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : Preorder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftMono α\ninst✝¹ : MulRightMono α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ ↑(Iic a * Iic b) ⊆ ↑(Iic (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMu... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : Preorder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftMono α\ninst✝¹ : MulRightMono α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ Set.Iic a * Set.Iic b ⊆ Set.Iic (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 51,
"column": 29
} | {
"line": 51,
"column": 40
} | {
"line": 51,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : Preorder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftMono α\ninst✝¹ : MulRightMono α\ninst✝ : LocallyFiniteOrderTop α\na b : α\n⊢ ↑(Ici a * Ici b) ⊆ ↑(Ici (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMu... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : Preorder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftMono α\ninst✝¹ : MulRightMono α\ninst✝ : LocallyFiniteOrderTop α\na b : α\n⊢ Set.Ici a * Set.Ici b ⊆ Set.Ici (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 63,
"column": 29
} | {
"line": 63,
"column": 40
} | {
"line": 63,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Icc a b * Ico c d) ⊆ ↑(Ico (a * c) (b * d))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ Set.Icc a b * Set.Ico c d ⊆ Set.Ico (a * c) (b * d)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 68,
"column": 29
} | {
"line": 68,
"column": 40
} | {
"line": 68,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Ico a b * Icc c d) ⊆ ↑(Ico (a * c) (b * d))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ Set.Ico a b * Set.Icc c d ⊆ Set.Ico (a * c) (b * d)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 73,
"column": 29
} | {
"line": 73,
"column": 40
} | {
"line": 73,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Ioc a b * Ico c d) ⊆ ↑(Ioo (a * c) (b * d))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ Set.Ioc a b * Set.Ico c d ⊆ Set.Ioo (a * c) (b * d)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 78,
"column": 29
} | {
"line": 78,
"column": 40
} | {
"line": 78,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ ↑(Ico a b * Ioc c d) ⊆ ↑(Ioo (a * c) (b * d))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrder α\na b c d : α\n⊢ Set.Ico a b * Set.Ioc c d ⊆ Set.Ioo (a * c) (b * d)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 82,
"column": 29
} | {
"line": 82,
"column": 40
} | {
"line": 82,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ ↑(Iic a * Iio b) ⊆ ↑(Iio (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ Set.Iic a * Set.Iio b ⊆ Set.Iio (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 86,
"column": 29
} | {
"line": 86,
"column": 40
} | {
"line": 86,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ ↑(Iio a * Iic b) ⊆ ↑(Iio (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderBot α\na b : α\n⊢ Set.Iio a * Set.Iic b ⊆ Set.Iio (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 90,
"column": 29
} | {
"line": 90,
"column": 40
} | {
"line": 90,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderTop α\na b : α\n⊢ ↑(Ioi a * Ici b) ⊆ ↑(Ioi (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderTop α\na b : α\n⊢ Set.Ioi a * Set.Ici b ⊆ Set.Ioi (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Interval | {
"line": 94,
"column": 29
} | {
"line": 94,
"column": 40
} | {
"line": 94,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderTop α\na b : α\n⊢ ↑(Ici a * Ioi b) ⊆ ↑(Ioi (a * b))",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"α : Type u_1\ninst✝⁵ : Mul α\ninst✝⁴ : PartialOrder α\ninst✝³ : DecidableEq α\ninst✝² : MulLeftStrictMono α\ninst✝¹ : MulRightStrictMono α\ninst✝ : LocallyFiniteOrderTop α\na b : α\n⊢ Set.Ici a * Set.Ioi b ⊆ Set.Ioi (a * b)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.ZPowers.Lemmas | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 35
} | {
"line": 49,
"column": 36
} | [
{
"pp": "R : Type u_4\ninst✝ : Ring R\nr : R\nk : ℤ\n⊢ ↑k * r ∈ zmultiples r",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"instHSMul",
"HMul.hMul",
"Ring.toNonAssocRing",
"AddGroupWithOne.toAddGroup",
"congrArg",
"_pr... | [
"R : Type u_4\ninst✝ : Ring R\nr : R\nk : ℤ\n⊢ k • r ∈ zmultiples r"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Action.Pointwise.Finset | {
"line": 89,
"column": 22
} | {
"line": 89,
"column": 33
} | {
"line": 89,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : SMulWithZero α β\ns : Finset β\nh : s.Nonempty\n⊢ ↑(0 • s) = ↑0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"SMulWithZero.to... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : DecidableEq β\ninst✝² : Zero α\ninst✝¹ : Zero β\ninst✝ : SMulWithZero α β\ns : Finset β\nh : s.Nonempty\n⊢ 0 • ↑s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Submonoid.Instances | {
"line": 51,
"column": 4
} | {
"line": 51,
"column": 15
} | {
"line": 51,
"column": 16
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : GroupWithZero G\ninst✝ : GroupWithZero H\nf : G →*₀ H\na : H\nha : a ∈ MonoidHom.mrange f\nh : ¬a = ↑0\n⊢ ⟨a, ha⟩ * ⟨(↑⟨a, ha⟩)⁻¹, ⋯⟩ = 1",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | [
"G : Type u_1\nH : Type u_2\ninst✝¹ : GroupWithZero G\ninst✝ : GroupWithZero H\nf : G →*₀ H\na : H\nha : a ∈ MonoidHom.mrange f\nh : ¬a = ↑0\n⊢ a * a⁻¹ = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 41,
"column": 25
} | {
"line": 41,
"column": 36
} | {
"line": 41,
"column": 37
} | [
{
"pp": "M : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Subsingleton Mˣ\nx : M\nhx : x ≠ 1\nh : IsMulIndecomposable id univ x\n⊢ ∀ ⦃a b : M⦄, x = a * b → IsUnit a ∨ IsUnit b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toM... | [
"M : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Subsingleton Mˣ\nx : M\nhx : x ≠ 1\nh : IsMulIndecomposable id univ x\n⊢ ∀ ⦃a b : M⦄, x = a * b → a = 1 ∨ b = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 41,
"column": 52
} | {
"line": 41,
"column": 63
} | {
"line": 41,
"column": 64
} | [
{
"pp": "M : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Subsingleton Mˣ\nx : M\nhx : x ≠ 1\nh : Irreducible x\n⊢ IsMulIndecomposable id univ x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"M : Type u_2\ninst✝¹ : Monoid M\ninst✝ : Subsingleton Mˣ\nx : M\nhx : x ≠ 1\nh : Irreducible x\n⊢ ∀ (j k : M), x = j * k → j = 1 ∨ k = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 15
} | {
"line": 69,
"column": 16
} | [
{
"pp": "ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\nthis : ⇑invMonoidHom ∘ v '' baseOf v f ⊆ ⇑(MonoidHom.id G) '' v '' baseOf v (inv... | [
"ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\nthis : ⇑invMonoidHom ∘ v '' baseOf v f ⊆ ⇑(MonoidHom.id G) '' v '' baseOf v (invMonoidHom.co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 72,
"column": 19
} | {
"line": 72,
"column": 39
} | {
"line": 72,
"column": 40
} | [
{
"pp": "ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\ni : ι\nhi : i ∈ {i | 1 < (invMonoidHom.comp f) (v i)}\nhi' :\n ∀ j ∈ {i | 1 < (... | [
"ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\ni : ι\nhi : i ∈ {i | 1 < (invMonoidHom.comp f) (v i)}\nhi' :\n ∀ j ∈ {i | 1 < (invMonoidHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 203,
"column": 4
} | {
"line": 203,
"column": 42
} | {
"line": 203,
"column": 43
} | [
{
"pp": "case mpr\nA : Type u_1\nB : Type u_2\ninst✝¹ : GroupWithZero A\ninst✝ : GroupWithZero B\nf : A →*₀ B\nx : B\nhx₀ : ¬x = 0\ny : A\nhy : f y = x\n⊢ ∃ y, f y = ↑(Units.mk0 x hx₀)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.toMonoidWithZero",... | [
"case mpr\nA : Type u_1\nB : Type u_2\ninst✝¹ : GroupWithZero A\ninst✝ : GroupWithZero B\nf : A →*₀ B\nx : B\nhx₀ : ¬x = 0\ny : A\nhy : f y = x\n⊢ ∃ y, f y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 22
} | {
"line": 171,
"column": 23
} | [
{
"pp": "ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\nhf : ∀ (i : ι), f (v i) ≠ 1\ns : Set ι\nhst : s ⊆ {j | IsMulIndecomposable v {i ... | [
"ι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nhv_inv : ∀ (i : ι), v i⁻¹ = (v i)⁻¹\nf : G →* S\nhf : ∀ (i : ι), f (v i) ≠ 1\ns : Set ι\nhst : s ⊆ {j | IsMulIndecomposable v {i | 1 < f (v i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GroupWithZero.Range | {
"line": 242,
"column": 6
} | {
"line": 242,
"column": 54
} | {
"line": 242,
"column": 55
} | [
{
"pp": "case refine_1.mul\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\ny c d : Bˣ\nhc : c ∈ Subgroup.closure (Units.val ⁻¹' range ⇑f)\nhd : d ∈ Subgroup.closure (Units.val ⁻¹' range ⇑f)\nu : A\nhu : f u ≠ 0\na : A\nha : f u * ↑c = f a\nv : A\nhv : f v ≠ 0\nb... | [
"case refine_1.mul\nA : Type u_1\nB : Type u_2\ninst✝¹ : MonoidWithZero A\ninst✝ : CommGroupWithZero B\nf : A →*₀ B\ny c d : Bˣ\nhc : c ∈ Subgroup.closure (Units.val ⁻¹' range ⇑f)\nhd : d ∈ Subgroup.closure (Units.val ⁻¹' range ⇑f)\nu : A\nhu : f u ≠ 0\na : A\nha : f u * ↑c = f a\nv : A\nhv : f v ≠ 0\nb : A\nhb : f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 28
} | {
"line": 206,
"column": 29
} | [
{
"pp": "case inr.inr\nι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝³ : Monoid M\ninst✝² : LinearOrder S\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\ni : ι\nhv_one : v i ≠ 1\nhi : v i ∈ closure (v '' {i | 1 < f (v i)})\nx y : M\nx✝¹ : x ∈ closure (v '' {i | 1 < f (v i)})\nx✝... | [
"case inr.inr\nι : Type u_1\nM : Type u_2\nS : Type u_4\ninst✝³ : Monoid M\ninst✝² : LinearOrder S\ninst✝¹ : CommMonoid S\ninst✝ : IsOrderedCancelMonoid S\nv : ι → M\nf : M →* S\ni : ι\nhv_one : v i ≠ 1\nhi : v i ∈ closure (v '' {i | 1 < f (v i)})\nx y : M\nx✝¹ : x ∈ closure (v '' {i | 1 < f (v i)})\nx✝ : y ∈ closu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 24
} | {
"line": 220,
"column": 25
} | [
{
"pp": "case inr.inr\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\nhf : ∀ i ∈ s, 1 < f (v i)\ni : ι\nhv_one : v i ≠ 1\nhv_inv : v i⁻¹ = (v i)⁻¹\nthis :\n ∀ {ι... | [
"case inr.inr\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\nhf : ∀ i ∈ s, 1 < f (v i)\ni : ι\nhv_one : v i ≠ 1\nhv_inv : v i⁻¹ = (v i)⁻¹\nthis :\n ∀ {ι : Type u_1}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.FintypeCat | {
"line": 216,
"column": 4
} | {
"line": 216,
"column": 15
} | {
"line": 216,
"column": 16
} | [
{
"pp": "X✝ Y✝ : Skeleton\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : incl.map a₁✝ = incl.map a₂✝\n⊢ a₁✝ = a₂✝",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"X✝ Y✝ : Skeleton\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : incl.map a₁✝ = incl.map a₂✝\n⊢ a₁✝ = a₂✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Irreducible.Indecomposable | {
"line": 226,
"column": 2
} | {
"line": 226,
"column": 13
} | {
"line": 226,
"column": 14
} | [
{
"pp": "case inr.inr\nι✝ : Type u_1\nG✝ : Type u_3\nS✝ : Type u_4\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : LinearOrder S✝\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\... | [
"case inr.inr\nι✝ : Type u_1\nG✝ : Type u_3\nS✝ : Type u_4\ninst✝⁶ : CommGroup G✝\ninst✝⁵ : LinearOrder S✝\nι : Type u_1\nG : Type u_3\nS : Type u_4\ninst✝⁴ : CommGroup G\ninst✝³ : LinearOrder S\ninst✝² : InvolutiveInv ι\ninst✝¹ : CommGroup S\ninst✝ : IsOrderedMonoid S\nv : ι → G\nf : G →* S\ns : Set ι\nhf : ∀ i ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Category.NonemptyFinLinOrd | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 27
} | {
"line": 206,
"column": 28
} | [
{
"pp": "case refine_1\nX Y : NonemptyFinLinOrd\nf : X ⟶ Y\nhf : Epi f\nH✝ : ∀ (y : ↑Y.toLinOrd), Nonempty ↑(⇑(ConcreteCategory.hom f) ⁻¹' {y})\nφ : ↑Y.toLinOrd → ↑X.toLinOrd := fun y ↦ ↑⋯.some\nhφ : ∀ (y : ↑Y.toLinOrd), (ConcreteCategory.hom f) (φ y) = y\na b : ↑Y.1\nh : φ b < φ a\nH : (ConcreteCategory.hom f)... | [
"case refine_1\nX Y : NonemptyFinLinOrd\nf : X ⟶ Y\nhf : Epi f\nH✝ : ∀ (y : ↑Y.toLinOrd), Nonempty ↑(⇑(ConcreteCategory.hom f) ⁻¹' {y})\nφ : ↑Y.toLinOrd → ↑X.toLinOrd := fun y ↦ ↑⋯.some\nhφ : ∀ (y : ↑Y.toLinOrd), (ConcreteCategory.hom f) (φ y) = y\na b : ↑Y.1\nh : φ b < φ a\nH : (ConcreteCategory.hom f) (φ b) ≤ (Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.Basic | {
"line": 99,
"column": 39
} | {
"line": 100,
"column": 74
} | {
"line": 102,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nX : C\np : X ⟶ X\nhp : p ≫ p = p\n⊢ (𝟙 X - p) ≫ (𝟙 X - p) = 𝟙 X - p",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"sub_self",
"... | [] | by
simp only [comp_sub, sub_comp, id_comp, comp_id, hp, sub_self, sub_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 82
} | {
"line": 65,
"column": 83
} | [
{
"pp": "case mk.mk\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX✝ : C\np✝¹ : X✝ ⟶ X✝\nidem✝¹ : p✝¹ ≫ p✝¹ = p✝¹\np✝ : X✝ ⟶ X✝\nidem✝ : p✝ ≫ p✝ = p✝\nh_p : p✝¹ ≫ eqToHom ⋯ = eqToHom ⋯ ≫ p✝\n⊢ { X := X✝, p := p✝¹, idem := idem✝¹ } = { X := X✝, p := p✝, idem := idem✝ }",
"ppTerm": "?mk.mk",
"assigned": t... | [
"case mk.mk\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX✝ : C\np✝¹ : X✝ ⟶ X✝\nidem✝¹ : p✝¹ ≫ p✝¹ = p✝¹\np✝ : X✝ ⟶ X✝\nidem✝ : p✝ ≫ p✝ = p✝\nh_p : p✝¹ ≫ eqToHom ⋯ = eqToHom ⋯ ≫ p✝\n⊢ p✝¹ = p✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 27
} | {
"line": 110,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP Q : Karoubi C\nf g : P ⟶ Q\nh : f.f = g.f\n⊢ f = g",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"CategoryTheory.Idempotents.Karoubi",
"CategoryTheory.Cat... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nP Q : Karoubi C\nf g : P ⟶ Q\nh : f.f = g.f\n⊢ f.f = g.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 159,
"column": 21
} | {
"line": 159,
"column": 67
} | {
"line": 159,
"column": 68
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ P.p ≫ (-f.f) ≫ Q.p = -f.f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"CategoryTheory.Idempotents.Karoubi.Hom.f",
"Subtrac... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Preadditive C\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ P.p ≫ f.f ≫ Q.p = f.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 300,
"column": 4
} | {
"line": 300,
"column": 15
} | {
"line": 300,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Epi f\nZ✝ : Karoubi C\ng h : (toKaroubi C).obj Y✝ ⟶ Z✝\neq : (toKaroubi C).map f ≫ g = (toKaroubi C).map f ≫ h\n⊢ f ≫ g.f = f ≫ h.f",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Epi f\nZ✝ : Karoubi C\ng h : (toKaroubi C).obj Y✝ ⟶ Z✝\neq : (toKaroubi C).map f ≫ g = (toKaroubi C).map f ≫ h\n⊢ f ≫ g.f = f ≫ h.f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Idempotents.Karoubi | {
"line": 307,
"column": 4
} | {
"line": 307,
"column": 15
} | {
"line": 307,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Mono f\nZ✝ : Karoubi C\ng h : Z✝ ⟶ (toKaroubi C).obj X✝\neq : g ≫ (toKaroubi C).map f = h ≫ (toKaroubi C).map f\n⊢ g.f ≫ f = h.f ≫ f",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Mono f\nZ✝ : Karoubi C\ng h : Z✝ ⟶ (toKaroubi C).obj X✝\neq : g ≫ (toKaroubi C).map f = h ≫ (toKaroubi C).map f\n⊢ g.f ≫ f = h.f ≫ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 67,
"column": 64
} | {
"line": 68,
"column": 43
} | {
"line": 70,
"column": 0
} | [
{
"pp": "n : SimplexCategory\nf g : ⦋0⦌ ⟶ n\nh0 : (toOrderHom f) 0 = (toOrderHom g) 0\n⊢ f = g",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"PartialOrder.toPreorder",
"Eq.rec",
"Fin.instOfNat",
"SimplexCategory.Hom.ext",
"instOf... | [] | by
ext i; match i with | 0 => exact h0 ▸ rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 299,
"column": 40
} | {
"line": 306,
"column": 15
} | {
"line": 308,
"column": 0
} | [
{
"pp": "n : ℕ\ni : Fin (n + 1)\n⊢ δ i.castSucc ≫ σ i = 𝟙 ⦋n⦌",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Fin.predAbove._proof_2",
"Fin.dite_val",
"False",
"Nat.instMulZeroClass",
"Nat.instOrderedSub",
"... | [] | by
rcases i with ⟨i, hi⟩
ext ⟨j, hj⟩
dsimp [σ, δ, Fin.predAbove, Fin.succAbove]
simp only [Fin.lt_def, Fin.dite_val, Fin.ite_val, Fin.val_pred]
split_ifs
any_goals simp
all_goals lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicTopology.SimplicialObject.Basic | {
"line": 510,
"column": 8
} | {
"line": 510,
"column": 86
} | {
"line": 510,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX✝ X : SimplicialObject C\nX₀ : C\nf : X _⦋0⦌ ⟶ X₀\nw : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), X.map g₁.op ≫ f = X.map g₂.op ≫ f\ni j : SimplexCategoryᵒᵖ\ng : i ⟶ j\n⊢ X.map g.unop.op ≫ X.map (⦋0⦌.const (unop j) 0).op ≫ f = (X.map (⦋0⦌.const (unop i) 0).op ≫ f... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX✝ X : SimplicialObject C\nX₀ : C\nf : X _⦋0⦌ ⟶ X₀\nw : ∀ (i : SimplexCategory) (g₁ g₂ : ⦋0⦌ ⟶ i), X.map g₁.op ≫ f = X.map g₂.op ≫ f\ni j : SimplexCategoryᵒᵖ\ng : i ⟶ j\n⊢ X.map (⦋0⦌.const (unop j) 0 ≫ g.unop).op ≫ f = X.map (⦋0⦌.const (unop i) 0).op ≫ f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 13
} | {
"line": 639,
"column": 14
} | [
{
"pp": "x y : SimplexCategory\nf : x ⟶ y\ninst✝ : Mono f\n⊢ x.len ≤ y.len",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : SimplexCategory\nf : x ⟶ y\ninst✝ : Mono f\n⊢ x.len ≤ y.len"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplexCategory.Basic | {
"line": 647,
"column": 2
} | {
"line": 647,
"column": 13
} | {
"line": 647,
"column": 14
} | [
{
"pp": "x y : SimplexCategory\nf : x ⟶ y\ninst✝ : Epi f\n⊢ y.len ≤ x.len",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"x y : SimplexCategory\nf : x ⟶ y\ninst✝ : Epi f\n⊢ y.len ≤ x.len"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 37
} | {
"line": 134,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn : ℕ\n⊢ (H.h 0 ≫ p.app (op ⦋n + 1⦌)) ≫ Y'.δ 0 = (g ≫ p).app (op ⦋n⦌)",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Categor... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn : ℕ\n⊢ H.h 0 ≫ p.app (op ⦋n + 1⦌) ≫ Y'.δ 0 = g.app (op ⦋n⦌) ≫ p.app (op ⦋n⦌)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplexCategory.DeltaZeroIter | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 33
} | {
"line": 197,
"column": 34
} | [
{
"pp": "case pos\nn : ℕ\ni : Fin (n + 2)\nj m : ℕ\nh : m + (j + 1) = n + 1\nhi' : ↑i ≤ j + 1\nk : Fin (⦋n⦌.len + 1)\nhk : j ≤ ↑k\n⊢ ↑((ConcreteCategory.hom (σ₀Iter (j + 1) ⋯)) ((ConcreteCategory.hom (δ i)) k)) =\n ↑((ConcreteCategory.hom (σ₀Iter j ⋯)) k)",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case pos\nn : ℕ\ni : Fin (n + 2)\nj m : ℕ\nh : m + (j + 1) = n + 1\nhi' : ↑i ≤ j + 1\nk : Fin (⦋n⦌.len + 1)\nhk : j ≤ ↑k\n⊢ ↑((ConcreteCategory.hom (σ₀Iter (j + 1) ⋯)) ((ConcreteCategory.hom (δ i)) k)) = ↑k - j"
] | σ₀Iter_coe_eq_of_ge j .., | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 37
} | {
"line": 136,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn : ℕ\n⊢ (H.h (Fin.last n) ≫ p.app (op ⦋n + 1⦌)) ≫ Y'.δ (Fin.last (n + 1)) = (f ≫ p).app (op ⦋n⦌)",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn : ℕ\n⊢ H.h (Fin.last n) ≫ p.app (op ⦋n + 1⦌) ≫ Y'.δ (Fin.last (n + 1)) = f.app (op ⦋n⦌) ≫ p.app (op ⦋n⦌)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 15
} | {
"line": 138,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\ni : Fin (n✝ + 2)\nj : Fin (n✝ + 1)\nhij : i ≤ j.castSucc\n⊢ (H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌)) ≫ Y'.δ i.castSucc = X.δ i ≫ H.h j ≫ p.app (op ⦋n✝ + 1⦌)",
"... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\ni : Fin (n✝ + 2)\nj : Fin (n✝ + 1)\nhij : i ≤ j.castSucc\n⊢ H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌) ≫ Y'.δ i.castSucc = X.δ i ≫ H.h j ≫ p.app (op ⦋n✝ + 1⦌)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 15
} | {
"line": 140,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\nj : Fin (n✝ + 1)\n⊢ (H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌)) ≫ Y'.δ j.castSucc.succ =\n (H.h j.castSucc ≫ p.app (op ⦋n✝ + 1 + 1⦌)) ≫ Y'.δ j.castSucc.succ",
"... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\nj : Fin (n✝ + 1)\n⊢ H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌) ≫ Y'.δ j.castSucc.succ =\n H.h j.castSucc ≫ p.app (op ⦋n✝ + 1 + 1⦌) ≫ Y'.δ j.castSucc.succ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 15
} | {
"line": 142,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\ni : Fin (n✝ + 2)\nj : Fin (n✝ + 1)\nhji : j.castSucc < i\n⊢ (H.h j.castSucc ≫ p.app (op ⦋n✝ + 1 + 1⦌)) ≫ Y'.δ i.succ = X.δ i ≫ H.h j ≫ p.app (op ⦋n✝ + 1⦌)",
"... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\ni : Fin (n✝ + 2)\nj : Fin (n✝ + 1)\nhji : j.castSucc < i\n⊢ H.h j.castSucc ≫ p.app (op ⦋n✝ + 1 + 1⦌) ≫ Y'.δ i.succ = X.δ i ≫ H.h j ≫ p.app (op ⦋n✝ + 1⦌)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicTopology.SimplicialObject.Homotopy | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 15
} | {
"line": 144,
"column": 16
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\ni j : Fin (n✝ + 1)\nhij : i ≤ j\n⊢ (H.h j ≫ p.app (op ⦋n✝ + 1⦌)) ≫ Y'.σ i.castSucc = X.σ i ≫ H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌)",
"ppTerm": "?m.246",
"a... | [
"C : Type u\ninst✝ : Category.{v, u} C\nX Y : SimplicialObject C\nf g : X ⟶ Y\nH : Homotopy f g\nY' : SimplicialObject C\np : Y ⟶ Y'\nn✝ : ℕ\ni j : Fin (n✝ + 1)\nhij : i ≤ j\n⊢ H.h j ≫ p.app (op ⦋n✝ + 1⦌) ≫ Y'.σ i.castSucc = X.σ i ≫ H.h j.succ ≫ p.app (op ⦋n✝ + 1 + 1⦌)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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