module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.DFinsupp.Defs | {
"line": 935,
"column": 19
} | {
"line": 935,
"column": 30
} | {
"line": 935,
"column": 31
} | [
{
"pp": "ι : Type u\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\nh : Function.Surjective (mapRange f hf)\ni : ι\nu : β₂ i\nx : Π₀ (i : ι), β₁ i\nhx : mapRange f hf x = single i u\n⊢ f i (x i) = u",
"... | [
"ι : Type u\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\ninst✝¹ : (i : ι) → Zero (β₁ i)\ninst✝ : (i : ι) → Zero (β₂ i)\nf : (i : ι) → β₁ i → β₂ i\nhf : ∀ (i : ι), f i 0 = 0\nh : Function.Surjective (mapRange f hf)\ni : ι\nu : β₂ i\nx : Π₀ (i : ι), β₁ i\nhx : mapRange f hf x = single i u\n⊢ f i (x i) = u"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 296,
"column": 34
} | {
"line": 296,
"column": 45
} | {
"line": 296,
"column": 46
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : AddCommMonoid γ\ni : ι\nφ : ZeroHom (β i) γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\nsf : Multiset ι\nhf : ∀ (i : ι), i ∈ sf ∨ f i = 0\nhi : i ∉ sf.toFinset\n⊢ i ∉ sf... | [
"ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : AddCommMonoid γ\ni : ι\nφ : ZeroHom (β i) γ\nf : (i : ι) → β i\nsupport'✝ : Trunc { s // ∀ (i : ι), i ∈ s ∨ f i = 0 }\nsf : Multiset ι\nhf : ∀ (i : ι), i ∈ sf ∨ f i = 0\nhi : i ∉ sf.toFinset\n⊢ i ∉ sf"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 370,
"column": 2
} | {
"line": 372,
"column": 9
} | {
"line": 372,
"column": 10
} | [
{
"pp": "case mk.mk\nι₁ : Type u_4\nι₂ : Type u_5\nβ₁ : ι₁ → Type u_1\nβ₂ : ι₂ → Type u_2\nγ : Type u_3\ninst✝⁴ : DecidableEq ι₁\ninst✝³ : DecidableEq ι₂\ninst✝² : (i : ι₁) → AddZeroClass (β₁ i)\ninst✝¹ : (i : ι₂) → AddZeroClass (β₂ i)\ninst✝ : AddCommMonoid γ\nh : (i : ι₁) → (j : ι₂) → β₁ i →+ β₂ j →+ γ\nf₁ : ... | [
"case mk.mk\nι₁ : Type u_4\nι₂ : Type u_5\nβ₁ : ι₁ → Type u_1\nβ₂ : ι₂ → Type u_2\nγ : Type u_3\ninst✝⁴ : DecidableEq ι₁\ninst✝³ : DecidableEq ι₂\ninst✝² : (i : ι₁) → AddZeroClass (β₁ i)\ninst✝¹ : (i : ι₂) → AddZeroClass (β₂ i)\ninst✝ : AddCommMonoid γ\nh : (i : ι₁) → (j : ι₂) → β₁ i →+ β₂ j →+ γ\nf₁ : (i : ι₁) → β... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 311,
"column": 53
} | {
"line": 311,
"column": 78
} | {
"line": 311,
"column": 79
} | [
{
"pp": "K : Type u_1\nV : Type u_2\ninst✝⁸ : CommSemiring K\ninst✝⁷ : AddCommMonoid V\ninst✝⁶ : Module K V\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : IsReflexive R M\nr : R\nhr : IsReg... | [
"K : Type u_1\nV : Type u_2\ninst✝⁸ : CommSemiring K\ninst✝⁷ : AddCommMonoid V\ninst✝⁶ : Module K V\nR : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : IsReflexive R M\nr : R\nhr : IsRegular r\nm₁ m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 18
} | {
"line": 254,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\ninst✝² : Semiring R\nβ₂ : ι → Type u_9\ninst✝¹ : (i : ι) → AddCommMonoid (β₂ i)\ninst✝ : (i : ι) → Module R (β₂ i)\nx✝ : Π₀ (i : ι), β₂ i\ni✝ : ι\n⊢ ((linearMap fun i ↦ LinearMap.id) x✝) i✝ = (LinearMap.id x✝) i✝",
"ppTerm": "?m.73",
"assigned": true,
"usedConsta... | [] | simp [linearMap] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 97,
"column": 25
} | {
"line": 97,
"column": 36
} | {
"line": 97,
"column": 37
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\nh : LinearIndependent R ![x, y]\ns t : R\nh' : s • x = t • y\n⊢ s • x + 0 • y = 0 • x + t • y",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\nh : LinearIndependent R ![x, y]\ns t : R\nh' : s • x = t • y\n⊢ s • x = t • y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 102,
"column": 14
} | {
"line": 102,
"column": 25
} | {
"line": 102,
"column": 26
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\ns t : R\nh' : s • x + t • y = 0\nh : Finsupp.single 0 s + Finsupp.single 1 t = 0\n⊢ s = 0",
"ppTerm": "?m.64",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\ns t : R\nh' : s • x + t • y = 0\nh : Finsupp.single 0 s + Finsupp.single 1 t = 0\n⊢ s = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 102,
"column": 42
} | {
"line": 102,
"column": 53
} | {
"line": 102,
"column": 54
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\ns t : R\nh' : s • x + t • y = 0\nh : Finsupp.single 0 s + Finsupp.single 1 t = 0\n⊢ t = 0",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals"... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nx y : M\ns t : R\nh' : s • x + t • y = 0\nh : Finsupp.single 0 s + Finsupp.single 1 t = 0\n⊢ t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 121,
"column": 61
} | {
"line": 121,
"column": 72
} | {
"line": 121,
"column": 73
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Set ι)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, LinearIndepOn R v a\n⊢ ∀ (i : ↑s), LinearIndepOn R v ↑i",
"ppTerm": "?m.44",
"assigned": true,
"usedC... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Set ι)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, LinearIndepOn R v a\n⊢ ∀ a ∈ s, LinearIndepOn R v a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 127,
"column": 82
} | {
"line": 127,
"column": 93
} | {
"line": 127,
"column": 94
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nη : Type u_6\ns : Set η\nt : η → Set ι\nhs : DirectedOn (t ⁻¹'o fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, LinearIndepOn R v (t a)\n⊢ ∀ (i : ↑s), LinearIndepOn R v (t ↑i)",
"ppTerm": "?m... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\nv : ι → M\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nη : Type u_6\ns : Set η\nt : η → Set ι\nhs : DirectedOn (t ⁻¹'o fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, LinearIndepOn R v (t a)\n⊢ ∀ a ∈ s, LinearIndepOn R v (t a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 29
} | {
"line": 504,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : ι → Submodule R N\nh : Function.Injective ⇑((lsum ℕ) fun i ↦ (p i).subtype)\ni : ι\nx : ↥(p i)\nv : Π₀ (i : ι), ↥(p i)\nhv : ((lsum ℕ) fun i ↦ (p i).subtype) (erase i... | [] | simpa [eq_comm] using! this | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 37
} | {
"line": 174,
"column": 38
} | [
{
"pp": "ι : Type u'\nι' : Type u_1\nR : Type u_2\nM : Type u_4\nM' : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nv : ι → M\nv' : ι' → M'\nhv : LinearIndependent R v\nhv' : LinearIndependent R v'\nthis :\n linearCombination R (Su... | [
"ι : Type u'\nι' : Type u_1\nR : Type u_2\nM : Type u_4\nM' : Type u_5\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M'\ninst✝¹ : Module R M\ninst✝ : Module R M'\nv : ι → M\nv' : ι' → M'\nhv : LinearIndependent R v\nhv' : LinearIndependent R v'\nthis :\n linearCombination R (Sum.elim (⇑(in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 27
} | {
"line": 206,
"column": 4
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\nindep : Set ι → Prop := fun s ↦ LinearIndepOn R v s\nX : Type (max 0 u') := { I // indep I }\nr : X → X → Prop := fun I J ↦ ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ LinearIndepOn R v (⋃... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\nindep : Set ι → Prop := fun s ↦ LinearIndepOn R v s\nX : Type (max 0 u') := { I // indep I }\nr : X → X → Prop := fun I J ↦ ↑I ⊆ ↑J\nc : Set X\nhc : IsChain r c\n⊢ ∀ f ∈ Finsupp.supported R R (⋃ I... | rw [linearIndepOn_iffₛ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 597,
"column": 8
} | {
"line": 597,
"column": 23
} | {
"line": 597,
"column": 24
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\ni : ι\nf : ι →₀ N\nhf : ∀ (i_1 : ι), f i_1 ∈ ⨆ (_ : i_1 ≠ i), p i_1\nhx : (f.sum fun _i xi ↦ xi) ∈ p i\nhsup : (f.sum fun _i xi ↦ xi) ∈ ⨆ j, ⨆ (_ : j ≠ i), p j\nh✝ : ... | [
"case neg\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\ni : ι\nf : ι →₀ N\nhf : ∀ (i_1 : ι), f i_1 ∈ ⨆ (_ : i_1 ≠ i), p i_1\nhx : (f.sum fun _i xi ↦ xi) ∈ p i\nhsup : (f.sum fun _i xi ↦ xi) ∈ ⨆ j, ⨆ (_ : j ≠ i), p j\nh✝ : (f.sum fun _... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 258,
"column": 4
} | {
"line": 258,
"column": 82
} | {
"line": 258,
"column": 83
} | [
{
"pp": "case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\nc d : R\nh :\n ↑{i, j} ⊆ {i, j} →\n (∀ i_1 ∉ {i, j}, (Pi.single i c + Pi.single j d) i_1 = 0) →\n ∑ i_1 ∈ {i, j}, (Pi.single i c + Pi.single j ... | [
"case refine_1\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\nc d : R\nh :\n ↑{i, j} ⊆ {i, j} →\n (∀ i_1 ∉ {i, j}, (Pi.single i c + Pi.single j d) i_1 = 0) →\n ∑ i_1 ∈ {i, j}, (Pi.single i c + Pi.single j d) i_1 • f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 28
} | {
"line": 260,
"column": 29
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\nh : ∀ (c d : R), c • f i + d • f j = 0 → c = 0 ∧ d = 0\nt : Finset ι\ng : ι → R\nht : ↑t ⊆ {i, j}\nhg0 : ∀ i ∉ t, g i = 0\nh0 : ∑ i ∈ t, g i • f i = 0\n... | [
"case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\nh : ∀ (c d : R), c • f i + d • f j = 0 → c = 0 ∧ d = 0\nt : Finset ι\ng : ι → R\nht : ↑t ⊆ {i, j}\nhg0 : ∀ i ∉ t, g i = 0\nh0 : ∑ x ∈ {i, j}, g x • f x = 0\nht' : t... | Finset.sum_subset ht', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 612,
"column": 4
} | {
"line": 612,
"column": 29
} | {
"line": 612,
"column": 30
} | [
{
"pp": "case mp\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (s : Finset ι) (v : ι → N), (∀ i ∈ s, v i ∈ p i) → ∑ i ∈ s, v i = 0 → ∀ i ∈ s, v i = 0\ns : Finset ι\nv w : ι → N\nhvw : ∀ i ∈ s, v i ∈ p i ∧ w i ∈ p i\n⊢ ∑ i ∈ ... | [
"case mp\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (s : Finset ι) (v : ι → N), (∀ i ∈ s, v i ∈ p i) → ∑ i ∈ s, v i = 0 → ∀ i ∈ s, v i = 0\ns : Finset ι\nv w : ι → N\nhvw : ∀ i ∈ s, v i ∈ p i ∧ w i ∈ p i\n⊢ ∑ i ∈ s, v i = ∑ i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 287,
"column": 54
} | {
"line": 287,
"column": 65
} | {
"line": 287,
"column": 66
} | [
{
"pp": "case refine_1\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • -x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • x + t • y = 0\n⊢ s = 0 ∧ t = 0",
"ppTerm": "?refine_1",
"assigned": false,
"usedConstants": [],
"used... | [
"case refine_1\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • -x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • x + t • y = 0\n⊢ s = 0 ∧ t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 287,
"column": 79
} | {
"line": 287,
"column": 90
} | {
"line": 287,
"column": 91
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • -x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • x + t • y = 0\n⊢ -s • -x + t • y = 0",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtrac... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • -x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • x + t • y = 0\n⊢ s • x + t • y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 287,
"column": 54
} | {
"line": 287,
"column": 65
} | {
"line": 287,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • -x + t • y = 0\n⊢ s = 0 ∧ t = 0",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"used... | [
"case refine_2\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • -x + t • y = 0\n⊢ s = 0 ∧ t = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 287,
"column": 79
} | {
"line": 287,
"column": 90
} | {
"line": 287,
"column": 91
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • -x + t • y = 0\n⊢ -s • x + t • y = 0",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Neg... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nh : ∀ (s t : R), s • x + t • y = 0 → s = 0 ∧ t = 0\ns t : R\nhst : s • -x + t • y = 0\n⊢ -(s • x) + t • y = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Algebra.Opposite | {
"line": 49,
"column": 43
} | {
"line": 50,
"column": 89
} | {
"line": 52,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nr : R\nx... | [] | by
simp only [RingHom.toOpposite_apply, Function.comp_apply, ← op_mul, Algebra.commutes] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 335,
"column": 4
} | {
"line": 335,
"column": 17
} | {
"line": 336,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_2\nM : Type u_4\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nx y : M\nS : Type u_6\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module S R\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass S R M\ninst✝² : IsScalarTower S R M\ninst✝¹ : IsTorsionFree S R\na b c ... | [
"case inl\nR : Type u_2\nM : Type u_4\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nx y : M\nS : Type u_6\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module S R\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass S R M\ninst✝² : IsScalarTower S R M\ninst✝¹ : IsTorsionFree S R\na b c d : S\ninst✝... | rw [pair_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 340,
"column": 6
} | {
"line": 340,
"column": 42
} | {
"line": 341,
"column": 6
} | [
{
"pp": "case neg\nR : Type u_2\nM : Type u_4\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nx y : M\nS : Type u_6\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module S R\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass S R M\ninst✝² : IsScalarTower S R M\ninst✝¹ : IsTorsionFree S R\na b c ... | [
"case neg\nR : Type u_2\nM : Type u_4\ninst✝¹⁰ : Ring R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\nx y : M\nS : Type u_6\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : Module S R\ninst✝⁴ : Module S M\ninst✝³ : SMulCommClass S R M\ninst✝² : IsScalarTower S R M\ninst✝¹ : IsTorsionFree S R\na b c d : S\ninst✝... | refine ⟨c • 1, -a • 1, ?_, by aesop⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 13
} | {
"line": 356,
"column": 14
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nc : S\nhS : Nontri... | [
"R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nc : S\nhS : Nontrivial S\na✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 13
} | {
"line": 362,
"column": 14
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nb : S\nhS : Nontri... | [
"R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\nb : S\nhS : Nontrivial S\na✝ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 367,
"column": 26
} | {
"line": 367,
"column": 37
} | {
"line": 367,
"column": 38
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nthis : ∀ (x y : M), LinearIndependent R ![x, x + y] → LinearIndependent R ![x, y]\nh : LinearIndependent R ![x, y]\n⊢ LinearIndependent R ![x, x + y]",
"ppTerm": "?m.71",
"assigned": true,
"us... | [
"R : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx y : M\nthis : ∀ (x y : M), LinearIndependent R ![x, x + y] → LinearIndependent R ![x, y]\nh : LinearIndependent R ![x, y]\n⊢ LinearIndependent R ![x, x + y]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.AddChar | {
"line": 425,
"column": 26
} | {
"line": 425,
"column": 74
} | {
"line": 425,
"column": 75
} | [
{
"pp": "A : Type u_1\nM₀ : Type u_2\ninst✝² : AddGroup A\ninst✝¹ : MonoidWithZero M₀\ninst✝ : Nontrivial M₀\nψ : AddChar A M₀\nh : ψ 0 = 0 0\n⊢ False",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Type u_1\nM₀ : Type u_2\ninst✝² : AddGroup A\ninst✝¹ : MonoidWithZero M₀\ninst✝ : Nontrivial M₀\nψ : AddChar A M₀\nh : ψ 0 = 0 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 52
} | {
"line": 522,
"column": 53
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nf : ι → Dual R M\nh1 : Pairwise fun i j ↦ (f i) (v j) = 0\nh2 : ∀ (i : ι), (f i) (v i) = 1\ns : Finset ι\ng : ι → R\nhrel : ∑ i ∈ s, g i • v i = 0\ni : ι\nhi : i ∈ s\naux : ∀ j ∈ s, j ≠ i →... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\nf : ι → Dual R M\nh1 : Pairwise fun i j ↦ (f i) (v j) = 0\nh2 : ∀ (i : ι), (f i) (v i) = 1\ns : Finset ι\ng : ι → R\nhrel : ∑ i ∈ s, g i • v i = 0\ni : ι\nhi : i ∈ s\naux : ∀ j ∈ s, j ≠ i → g j * (f i)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 575,
"column": 69
} | {
"line": 579,
"column": 52
} | {
"line": 581,
"column": 0
} | [
{
"pp": "ι : Type u'\nK : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : ι → V\ns : Set ι\nx : ι\nhs : LinearIndepOn K v s\nhx : v x ∉ span K (v '' s)\n⊢ LinearIndepOn K v (insert x s)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"S... | [] | by
rw [← union_singleton]
have x0 : v x ≠ 0 := fun h => hx (h ▸ zero_mem _)
apply hs.union (LinearIndepOn.singleton x0)
rwa [image_singleton, disjoint_span_singleton' x0] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Module | {
"line": 268,
"column": 38
} | {
"line": 268,
"column": 49
} | {
"line": 268,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ns : Set M\nm : M\ninst✝ : Nontrivial R\nh : (of R M) m ∈ Submodule.span R ↑(Submonoid.map (of R M) (Submonoid.closure s))\n⊢ m ∈ Submonoid.closure s",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"use... | [
"R : Type u_1\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ns : Set M\nm : M\ninst✝ : Nontrivial R\nh : (of R M) m ∈ Submodule.span R ↑(Submonoid.map (of R M) (Submonoid.closure s))\n⊢ m ∈ Submonoid.closure s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 13
} | {
"line": 627,
"column": 14
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\na : V\ns : Set V\n⊢ LinearIndepOn K id (insert a s) ↔ LinearIndepOn K id s ∧ (a ∈ span K s → a ∈ s)",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\na : V\ns : Set V\n⊢ LinearIndepOn K id (insert a s) ↔ LinearIndepOn K id s ∧ (a ∈ span K s → a ∈ s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 642,
"column": 2
} | {
"line": 642,
"column": 13
} | {
"line": 642,
"column": 14
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\na : V\nh : LinearIndepOn K id s\n⊢ a ∈ span K s ↔ LinearIndepOn K id (insert a s) → a ∈ s",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoa... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\na : V\nh : LinearIndepOn K id s\n⊢ a ∈ span K s ↔ LinearIndepOn K id (insert a s) → a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 720,
"column": 6
} | {
"line": 720,
"column": 66
} | {
"line": 720,
"column": 67
} | [
{
"pp": "R : Type u_6\nK : Type u_7\nM : Type u_8\ninst✝⁷ : CommRing R\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R K\ninst✝³ : Module K M\ninst✝² : Module R M\ninst✝¹ : IsScalarTower R K M\ninst✝ : FaithfulSMul R K\nn : ℕ\nv : Fin n → M\nhv : LinearIndependent R v\nx : M\nhx : x ∉ span... | [
"R : Type u_6\nK : Type u_7\nM : Type u_8\ninst✝⁷ : CommRing R\ninst✝⁶ : DivisionRing K\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Algebra R K\ninst✝³ : Module K M\ninst✝² : Module R M\ninst✝¹ : IsScalarTower R K M\ninst✝ : FaithfulSMul R K\nn : ℕ\nv : Fin n → M\nhv : LinearIndependent R v\nx : M\nhx : x ∉ span K (range v)... | (eq_inv_smul_iff₀ hc').mpr (eq_neg_of_add_eq_zero_left heq), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 20
} | {
"line": 111,
"column": 21
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+ S\nhe : Injective ⇑f\nthis : map f = ⇑coeffEquiv.symm ∘ mapRange ⇑f ⋯ ∘ ⇑coeffEquiv\n⊢ Injective (map f)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemir... | [
"R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+ S\nhe : Injective ⇑f\nthis : map f = ⇑coeffEquiv.symm ∘ mapRange ⇑f ⋯ ∘ ⇑coeffEquiv\n⊢ Injective (mapRange ⇑f ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 20
} | {
"line": 118,
"column": 21
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+ S\nhe : Surjective ⇑f\nthis : map f = ⇑coeffEquiv.symm ∘ mapRange ⇑f ⋯ ∘ ⇑coeffEquiv\n⊢ Surjective (map f)",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSem... | [
"R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+ S\nhe : Surjective ⇑f\nthis : map f = ⇑coeffEquiv.symm ∘ mapRange ⇑f ⋯ ∘ ⇑coeffEquiv\n⊢ Surjective (mapRange ⇑f ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 56
} | {
"line": 162,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_6\nN : Type u_7\ninst✝ : Semiring R\nf : M → N\nx : R[N]\nhx : ↑x.coeff.support ⊆ Set.range f\nhf : Injective f\n⊢ mapDomain f (comapDomain f hf x) = x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.coeff",
"MonoidAlgebra.ext",
... | [] | ext : 1; exact Finsupp.mapDomain_comapDomain _ hf _ hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 56
} | {
"line": 162,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_6\nN : Type u_7\ninst✝ : Semiring R\nf : M → N\nx : R[N]\nhx : ↑x.coeff.support ⊆ Set.range f\nhf : Injective f\n⊢ mapDomain f (comapDomain f hf x) = x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.coeff",
"MonoidAlgebra.ext",
... | [] | ext : 1; exact Finsupp.mapDomain_comapDomain _ hf _ hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 854,
"column": 78
} | {
"line": 854,
"column": 89
} | {
"line": 854,
"column": 90
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt✝ : Finset V\nhs : LinearIndepOn K id s\nhst : s ⊆ ↑(span K ↑t✝)\nt s' : Finset V\nhs' : ↑s' ⊆ s\nx✝ : s ∩ ↑∅ = ∅\nhss' : s ⊆ ↑(span K ↑(s' ∪ ∅))\n⊢ s ⊆ ↑(span K ↑s')",
"ppTerm": "?m.95",
... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt✝ : Finset V\nhs : LinearIndepOn K id s\nhst : s ⊆ ↑(span K ↑t✝)\nt s' : Finset V\nhs' : ↑s' ⊆ s\nx✝ : s ∩ ↑∅ = ∅\nhss' : s ⊆ ↑(span K ↑(s' ∪ ∅))\n⊢ s ⊆ ↑(span K ↑s')"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 883,
"column": 12
} | {
"line": 883,
"column": 71
} | {
"line": 883,
"column": 72
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt✝¹ : Finset V\nhs : LinearIndepOn K id s\nhst✝¹ : s ⊆ ↑(span K ↑t✝¹)\nt✝ : Finset V\nb₁ : V\nt : Finset V\nhb₁t : b₁ ∉ t\nih :\n ∀ (s' : Finset V),\n ↑s' ⊆ s → s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt✝¹ : Finset V\nhs : LinearIndepOn K id s\nhst✝¹ : s ⊆ ↑(span K ↑t✝¹)\nt✝ : Finset V\nb₁ : V\nt : Finset V\nhb₁t : b₁ ∉ t\nih :\n ∀ (s' : Finset V),\n ↑s' ⊆ s → s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ∪ t)) → ∃ t'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 374,
"column": 63
} | {
"line": 374,
"column": 88
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_6\nN : Type u_7\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\nr : R\nm : M\n⊢ (mapDomainRingEquiv R e) (single m r) = single (e m) r",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"MulEquiv.in... | [] | simp [mapDomainRingEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 374,
"column": 63
} | {
"line": 374,
"column": 88
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_6\nN : Type u_7\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\nr : R\nm : M\n⊢ (mapDomainRingEquiv R e) (single m r) = single (e m) r",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"MulEquiv.in... | [] | simp [mapDomainRingEquiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 374,
"column": 63
} | {
"line": 374,
"column": 88
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_3\nM : Type u_6\nN : Type u_7\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\ne : M ≃* N\nr : R\nm : M\n⊢ (mapDomainRingEquiv R e) (single m r) = single (e m) r",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"MulEquiv.in... | [] | simp [mapDomainRingEquiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 886,
"column": 47
} | {
"line": 886,
"column": 58
} | {
"line": 886,
"column": 59
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt✝¹ : Finset V\nhs : LinearIndepOn K id s\nhst✝¹ : s ⊆ ↑(span K ↑t✝¹)\nt✝ : Finset V\nb₁ : V\nt : Finset V\nhb₁t : b₁ ∉ t\nih :\n ∀ (s' : Finset V),\n ↑s' ⊆ s → s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt✝¹ : Finset V\nhs : LinearIndepOn K id s\nhst✝¹ : s ⊆ ↑(span K ↑t✝¹)\nt✝ : Finset V\nb₁ : V\nt : Finset V\nhb₁t : b₁ ∉ t\nih :\n ∀ (s' : Finset V),\n ↑s' ⊆ s → s ∩ ↑t = ∅ → s ⊆ ↑(span K ↑(s' ∪ t)) → ∃ t'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Opposite | {
"line": 38,
"column": 4
} | {
"line": 38,
"column": 15
} | {
"line": 38,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Mul M\n⊢ ∀ (a a_1 : R[M]) (a_2 : M),\n (a_1.coeff.sum fun m₁ r₁ ↦ a.coeff.sum fun m₂ r₂ ↦ if m₁ * m₂ = a_2 then r₁ * r₂ else 0) =\n a.coeff.sum fun i n ↦ a_1.coeff.sum fun i_1 n_1 ↦ if i_1 * i = a_2 then n_1 * n else 0",
"ppTerm": "?m... | [
"R : Type u_1\nM : Type u_2\ninst✝¹ : Semiring R\ninst✝ : Mul M\n⊢ ∀ (a a_1 : R[M]) (a_2 : M),\n (a_1.coeff.sum fun m₁ r₁ ↦ a.coeff.sum fun m₂ r₂ ↦ if m₁ * m₂ = a_2 then r₁ * r₂ else 0) =\n a.coeff.sum fun i n ↦ a_1.coeff.sum fun i_1 n_1 ↦ if i_1 * i = a_2 then n_1 * n else 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 410,
"column": 25
} | {
"line": 410,
"column": 55
} | {
"line": 410,
"column": 56
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nN : Type u_8\ninst✝ : AddZeroClass N\nf g : R[M] →+ N\nhfg : ∀ (m : M), f.comp (singleAddHom m) = g.comp (singleAddHom m)\n⊢ ∀ (m : M) (r : R), f (single m r) = g (single m r)",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"... | [
"R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nN : Type u_8\ninst✝ : AddZeroClass N\nf g : R[M] →+ N\nhfg : ∀ (m : M), f.comp (singleAddHom m) = g.comp (singleAddHom m)\n⊢ ∀ (m : M) (r : R), f (single m r) = g (single m r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 441,
"column": 8
} | {
"line": 441,
"column": 19
} | {
"line": 441,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝ : Semiring R\nmotive : R[M] → Prop\nx : R[M]\nzero : motive 0\nsingle_add : ∀ (m : M) (r : R) (x : R[M]), m ∉ x.coeff.support → r ≠ 0 → motive x → motive (single m r + x)\n⊢ motive (ofCoeff 0)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nM : Type u_4\ninst✝ : Semiring R\nmotive : R[M] → Prop\nx : R[M]\nzero : motive 0\nsingle_add : ∀ (m : M) (r : R) (x : R[M]), m ∉ x.coeff.support → r ≠ 0 → motive x → motive (single m r + x)\n⊢ motive 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Sort | {
"line": 65,
"column": 55
} | {
"line": 65,
"column": 66
} | {
"line": 65,
"column": 67
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nr : α → α → Prop\ninst✝⁷ : DecidableRel r\ninst✝⁶ : IsTrans α r\ninst✝⁵ : Std.Antisymm r\ninst✝⁴ : Std.Total r\nr' : β → β → Prop\ninst✝³ : DecidableRel r'\ninst✝² : IsTrans β r'\ninst✝¹ : Std.Antisymm r'\ninst✝ : Std.Total r'\nl : List α\nh : ∀ (a : α), a ∈ Quot.... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nr : α → α → Prop\ninst✝⁷ : DecidableRel r\ninst✝⁶ : IsTrans α r\ninst✝⁵ : Std.Antisymm r\ninst✝⁴ : Std.Total r\nr' : β → β → Prop\ninst✝³ : DecidableRel r'\ninst✝² : IsTrans β r'\ninst✝¹ : Std.Antisymm r'\ninst✝ : Std.Total r'\nl : List α\nh : ∀ (a : α), a ∈ Quot.mk (⇑(isSeto... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Sort | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 42
} | {
"line": 69,
"column": 43
} | [
{
"pp": "α : Type u_1\na : α\ns : Multiset α\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : Std.Antisymm r\ninst✝ : Std.Total r\nl : List α\n⊢ (∀ (b : α), b ∈ Quot.mk (⇑(isSetoid α)) l → r a b) →\n (a ::ₘ Quot.mk (⇑(isSetoid α)) l).sort r = a :: sort (Quot.mk (⇑(isSetoid α)) l) r"... | [
"α : Type u_1\na : α\ns : Multiset α\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : Std.Antisymm r\ninst✝ : Std.Total r\nl : List α\n⊢ (∀ (b : α), b ∈ l → r a b) → orderedInsert r a (insertionSort r l) = a :: insertionSort r l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 24
} | {
"line": 190,
"column": 25
} | [
{
"pp": "G : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\nh : UniqueMul (map { toFun := op, inj' := ⋯ } B) (map { toFun := op, inj' := ⋯ } A) (op b0) (op a0)\na b : G\naA : a ∈ A\nbB : b ∈ B\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
... | [
"G : Type u_1\ninst✝ : Mul G\nA B : Finset G\na0 b0 : G\nh : UniqueMul (map { toFun := op, inj' := ⋯ } B) (map { toFun := op, inj' := ⋯ } A) (op b0) (op a0)\na b : G\naA : a ∈ A\nbB : b ∈ B\nab : a * b = a0 * b0\n⊢ a = a0 ∧ b = b0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 593,
"column": 46
} | {
"line": 593,
"column": 67
} | {
"line": 593,
"column": 68
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Mul M\nr : R\ng g' : M\nx : R[M]\nh : ¬∃ d, g' = d * g\n⊢ ∀ (d : M), d * g ≠ g'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Ne",
"_private.Mat... | [
"R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Mul M\nr : R\ng g' : M\nx : R[M]\nh : ¬∃ d, g' = d * g\n⊢ ∀ (d : M), ¬g' = d * g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 599,
"column": 46
} | {
"line": 599,
"column": 67
} | {
"line": 599,
"column": 68
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Mul M\nr : R\ng g' : M\nx : R[M]\nh : ¬∃ d, g' = g * d\n⊢ ∀ (d : M), g * d ≠ g'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"_private.Mathlib.Algebra.MonoidAlgeb... | [
"R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Mul M\nr : R\ng g' : M\nx : R[M]\nh : ¬∃ d, g' = g * d\n⊢ ∀ (d : M), ¬g' = g * d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 664,
"column": 2
} | {
"line": 664,
"column": 57
} | {
"line": 664,
"column": 58
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Nontrivial R\na b : M\nh : (of R M) a = (of R M) b\n⊢ a = b",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Nontrivial R\na b : M\nh : (of R M) a = (of R M) b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 707,
"column": 18
} | {
"line": 707,
"column": 48
} | {
"line": 707,
"column": 49
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Semiring S\nf g : R[M] →+* S\nh₁ : f.comp singleOneRingHom = g.comp singleOneRingHom\nh_of : (↑f).comp (of R M) = (↑g).comp (of R M)\n⊢ ∀ (r : R), f (single 1 r) = g (single 1 r)",
"ppTerm": "?m.62",
... | [
"R : Type u_1\nS : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Semiring S\nf g : R[M] →+* S\nh₁ : f.comp singleOneRingHom = g.comp singleOneRingHom\nh_of : (↑f).comp (of R M) = (↑g).comp (of R M)\n⊢ ∀ (r : R), f (single 1 r) = g (single 1 r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 707,
"column": 57
} | {
"line": 707,
"column": 87
} | {
"line": 707,
"column": 88
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Semiring S\nf g : R[M] →+* S\nh₁ : f.comp singleOneRingHom = g.comp singleOneRingHom\nh_of : (↑f).comp (of R M) = (↑g).comp (of R M)\n⊢ ∀ (m : M), f (single m 1) = g (single m 1)",
"ppTerm": "?m.63",
... | [
"R : Type u_1\nS : Type u_2\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Semiring S\nf g : R[M] →+* S\nh₁ : f.comp singleOneRingHom = g.comp singleOneRingHom\nh_of : (↑f).comp (of R M) = (↑g).comp (of R M)\n⊢ ∀ (m : M), f (single m 1) = g (single m 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 726,
"column": 8
} | {
"line": 726,
"column": 19
} | {
"line": 726,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Monoid M\np : R[M] → Prop\nx : R[M]\nhM : ∀ (m : M), p ((of R M) m)\nhadd : ∀ (x y : R[M]), p x → p y → p (x + y)\nhsmul : ∀ (r : R) (x : R[M]), p x → p (r • x)\n⊢ p (ofCoeff 0)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": ... | [
"R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Monoid M\np : R[M] → Prop\nx : R[M]\nhM : ∀ (m : M), p ((of R M) m)\nhadd : ∀ (x y : R[M]), p x → p y → p (x + y)\nhsmul : ∀ (r : R) (x : R[M]), p x → p (r • x)\n⊢ p 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 728,
"column": 17
} | {
"line": 728,
"column": 28
} | {
"line": 728,
"column": 29
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Monoid M\np : R[M] → Prop\nx : R[M]\nhM : ∀ (m : M), p ((of R M) m)\nhadd : ∀ (x y : R[M]), p x → p y → p (x + y)\nhsmul : ∀ (r : R) (x : R[M]), p x → p (r • x)\nm : M\nr : R\n⊢ p (ofCoeff (Finsupp.single m r))",
"ppTerm": "?m.43",
"assig... | [
"R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Monoid M\np : R[M] → Prop\nx : R[M]\nhM : ∀ (m : M), p ((of R M) m)\nhadd : ∀ (x y : R[M]), p x → p y → p (x + y)\nhsmul : ∀ (r : R) (x : R[M]), p x → p (r • x)\nm : M\nr : R\n⊢ p (single m r)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 736,
"column": 6
} | {
"line": 736,
"column": 40
} | {
"line": 736,
"column": 41
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝² : Semiring R\nx✝ y : R[M]\nr r₁ r₂ : R\nm m' m₁ m₂ m₁' m₂' : M\ninst✝¹ : Monoid M\ninst✝ : Monoid N\na : R\nx : R[M]\nhax : singleOneRingHom a * x = 1\nhxa : x * singleOneRingHom a = ... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝² : Semiring R\nx✝ y : R[M]\nr r₁ r₂ : R\nm m' m₁ m₂ m₁' m₂' : M\ninst✝¹ : Monoid M\ninst✝ : Monoid N\na : R\nx : R[M]\nhax : singleOneRingHom a * x = 1\nhxa : x * singleOneRingHom a = 1\n⊢ a * x.c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 737,
"column": 6
} | {
"line": 737,
"column": 40
} | {
"line": 737,
"column": 41
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝² : Semiring R\nx✝ y : R[M]\nr r₁ r₂ : R\nm m' m₁ m₂ m₁' m₂' : M\ninst✝¹ : Monoid M\ninst✝ : Monoid N\na : R\nx : R[M]\nhax : singleOneRingHom a * x = 1\nhxa : x * singleOneRingHom a = ... | [
"case refine_2\nR : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝² : Semiring R\nx✝ y : R[M]\nr r₁ r₂ : R\nm m' m₁ m₂ m₁' m₂' : M\ninst✝¹ : Monoid M\ninst✝ : Monoid N\na : R\nx : R[M]\nhax : singleOneRingHom a * x = 1\nhxa : x * singleOneRingHom a = 1\n⊢ x.coeff... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 971,
"column": 2
} | {
"line": 971,
"column": 57
} | {
"line": 971,
"column": 58
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : AddZeroClass M\na b : Multiplicative M\nh : (of R M) a = (of R M) b\n⊢ a = b",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : AddZeroClass M\na b : Multiplicative M\nh : (of R M) a = (of R M) b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 992,
"column": 8
} | {
"line": 992,
"column": 19
} | {
"line": 992,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : AddMonoid M\np : R[M] → Prop\nx : R[M]\nhM : ∀ (m : M), p ((of R M) (Multiplicative.ofAdd m))\nhadd : ∀ (x y : R[M]), p x → p y → p (x + y)\nhsmul : ∀ (r : R) (x : R[M]), p x → p (r • x)\n⊢ p (ofCoeff 0)",
"ppTerm": "?m.30",
"assigned": t... | [
"R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : AddMonoid M\np : R[M] → Prop\nx : R[M]\nhM : ∀ (m : M), p ((of R M) (Multiplicative.ofAdd m))\nhadd : ∀ (x y : R[M]), p x → p y → p (x + y)\nhsmul : ∀ (r : R) (x : R[M]), p x → p (r • x)\n⊢ p 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 838,
"column": 2
} | {
"line": 838,
"column": 13
} | {
"line": 838,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nhs : 1 ∈ s\nhn : n ≠ 0\n⊢ s ⊆ s ^ n",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nhs : 1 ∈ s\nhn : n ≠ 0\n⊢ s ⊆ s ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 867,
"column": 2
} | {
"line": 867,
"column": 13
} | {
"line": 867,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\na : α\nn : ℕ\nha : a ∈ s\n⊢ a ^ n ∈ s ^ n",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\na : α\nn : ℕ\nha : a ∈ s\n⊢ a ^ n ∈ s ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 869,
"column": 64
} | {
"line": 869,
"column": 75
} | {
"line": 869,
"column": 76
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nhs : 1 ∈ s\n⊢ 1 ∈ s ^ n",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Monoid α\ns : Finset α\nn : ℕ\nhs : 1 ∈ s\n⊢ 1 ∈ s ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 999,
"column": 21
} | {
"line": 999,
"column": 32
} | {
"line": 999,
"column": 33
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nhs : s.Nonempty\nn : ℕ\n⊢ (s ^ Int.negSucc n).Nonempty",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvO... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nhs : s.Nonempty\nn : ℕ\n⊢ (s ^ (n + 1)).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 419,
"column": 8
} | {
"line": 419,
"column": 95
} | {
"line": 419,
"column": 96
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhcard : 1 < #C ∨ 1 < #D\nhC : 1 ∈ C\nhD : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nd1 : G\nh... | [
"case pos\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhcard : 1 < #C ∨ 1 < #D\nhC : 1 ∈ C\nhD : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nd1 : G\nhd1 : d1 ∈ D\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 424,
"column": 8
} | {
"line": 424,
"column": 19
} | {
"line": 424,
"column": 20
} | [
{
"pp": "case neg.inl\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc✝ : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhC✝ : 1 ∈ C\nhD : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nhc1✝ : 1 ∈ C\nhd2 : 1 ∈ D\nh... | [
"case neg.inl\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc✝ : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhC✝ : 1 ∈ C\nhD : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nhc1✝ : 1 ∈ C\nhd2 : 1 ∈ D\nhc2 : 1 ∈ C\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 427,
"column": 8
} | {
"line": 427,
"column": 19
} | {
"line": 427,
"column": 20
} | [
{
"pp": "case neg.inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhC : 1 ∈ C\nhD✝ : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nhc1 : 1 ∈ C\nhd2 : 1 ∈ D\nhc2... | [
"case neg.inr\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : UniqueProds G\nA B : Finset G\nhc : A.Nonempty ∧ B.Nonempty ∧ (1 < #A ∨ 1 < #B)\na : G\nha : a ∈ A\nb : G\nhb : b ∈ B\nhu✝ : UniqueMul A B a b\nC D : Finset G\nhC : 1 ∈ C\nhD✝ : 1 ∈ D\nx✝ : Mul (Finset G) := Finset.mul\nhc1 : 1 ∈ C\nhd2 : 1 ∈ D\nhc2 : 1 ∈ C\nhd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1216,
"column": 19
} | {
"line": 1216,
"column": 41
} | {
"line": 1216,
"column": 42
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelMonoid α\ns : Finset α\nhs : s.Nontrivial\nn : ℕ\nx✝ : n + 2 ≠ 0\n⊢ (s ^ (n + 2)).Nontrivial",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CancelMonoid.toRightCancelMonoid",
"HMul.hMul",
"Mono... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelMonoid α\ns : Finset α\nhs : s.Nontrivial\nn : ℕ\nx✝ : n + 2 ≠ 0\n⊢ (s ^ n * s * s).Nontrivial"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1233,
"column": 2
} | {
"line": 1233,
"column": 13
} | {
"line": 1233,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelMonoid α\ns : Finset α\nn : ℕ\nhn : n ≠ 0\n⊢ #s ≤ #(s ^ n)",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : CancelMonoid α\ns : Finset α\nn : ℕ\nhn : n ≠ 0\n⊢ #s ≤ #(s ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Basic | {
"line": 202,
"column": 48
} | {
"line": 202,
"column": 59
} | {
"line": 202,
"column": 60
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nf : End R M\nh : ∀ (g : End R M) (x : M), g (f x) = f (g x)\nh✝ : Nontrivial M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\ni : Free.ChooseBasisIndex R M := ⋯.some\nx... | [
"R : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nf : End R M\nh : ∀ (g : End R M) (x : M), g (f x) = f (g x)\nh✝ : Nontrivial M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\ni : Free.ChooseBasisIndex R M := ⋯.some\nx : M\n⊢ f x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.FreeModule.Basic | {
"line": 203,
"column": 43
} | {
"line": 203,
"column": 54
} | {
"line": 203,
"column": 55
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nf : End R M\nh : ∀ (g : End R M) (x : M), g (f x) = f (g x)\nh✝ : Nontrivial M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\ni : Free.ChooseBasisIndex R M := ⋯.some\nH... | [
"R : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nf : End R M\nh : ∀ (g : End R M) (x : M), g (f x) = f (g x)\nh✝ : Nontrivial M\nb : Basis (Free.ChooseBasisIndex R M) R M := Free.chooseBasis R M\ni : Free.ChooseBasisIndex R M := ⋯.some\nH : ∀ (x : M)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sort | {
"line": 183,
"column": 12
} | {
"line": 183,
"column": 23
} | {
"line": 183,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\ns : Finset α\nh : s.Nonempty\n⊢ (s.sort fun a b ↦ a ≤ b).length - 1 < (s.sort fun a b ↦ a ≤ b).length",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"HSub.h... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\ns : Finset α\nh : s.Nonempty\n⊢ #s - 1 < #s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sort | {
"line": 184,
"column": 33
} | {
"line": 184,
"column": 44
} | {
"line": 184,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\nh : s.Nonempty\n⊢ (s.sort fun a b ↦ a ≤ b).length - 1 < (s.sort fun a b ↦ a ≤ b).length",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"HSub.hSub",
"P... | [
"α : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\nh : s.Nonempty\n⊢ #s - 1 < #s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 64,
"column": 2
} | {
"line": 69,
"column": 62
} | {
"line": 71,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns t : Set α\nhs : Disjoint s t\ni : α\n⊢ (⨅ a ∈ sᶜ, (lapply a).ker) ⊓ ⨅ a ∈ tᶜ, (lapply a).ker ≤ (lapply i).ker",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"Disj... | [] | classical
by_cases his : i ∈ s
· by_cases hit : i ∈ t
· exact (hs.le_bot ⟨his, hit⟩).elim
exact inf_le_of_right_le (iInf_le_of_le i <| iInf_le _ hit)
exact inf_le_of_left_le (iInf_le_of_le i <| iInf_le _ his) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.Polynomial.Basic | {
"line": 317,
"column": 55
} | {
"line": 317,
"column": 66
} | {
"line": 317,
"column": 67
} | [
{
"pp": "R : Type u\na b : R\nm n : ℕ\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝² : Semiring S\ninst✝¹ : Module S R\ninst✝ : IsTorsionFree S R\ns : S\nhs : IsRegular s\nf g : R[ℕ]\nhfg : (fun x ↦ s • x) { toFinsupp := f } = (fun x ↦ s • x) { toFinsupp := g }\n⊢ (fun x ↦ s • x) f = (fun x ↦ s • x) g",... | [
"R : Type u\na b : R\nm n : ℕ\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_1\ninst✝² : Semiring S\ninst✝¹ : Module S R\ninst✝ : IsTorsionFree S R\ns : S\nhs : IsRegular s\nf g : R[ℕ]\nhfg : (fun x ↦ s • x) { toFinsupp := f } = (fun x ↦ s • x) { toFinsupp := g }\n⊢ s • f = s • g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 13
} | {
"line": 497,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\na : R\nhe : X = C a\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\na : R\nhe : X = C a\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sort | {
"line": 393,
"column": 4
} | {
"line": 393,
"column": 35
} | {
"line": 393,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Finite α\nf : α →o α\nhf : Function.Injective ⇑f\n⊢ Set.range ⇑f = Set.range ⇑id",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"OrderHom.id",
"Eq.mpr",
"OrderHom.id_coe",
"congrArg",
"Set.univ",
"_... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : Finite α\nf : α →o α\nhf : Function.Injective ⇑f\n⊢ Function.Surjective ⇑f"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 30
} | {
"line": 151,
"column": 31
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set σ\nf : σ → R\n⊢ f '' s • ⊤ = ((lsum R) fun i ↦ f ↑i • LinearMap.id).range",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"R : Type u_1\nM : Type u_2\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set σ\nf : σ → R\n⊢ f '' s • ⊤ = ((lsum R) fun i ↦ f ↑i • LinearMap.id).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 13
} | {
"line": 156,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\n⊢ s • ⊤ = ((lsum R) fun i ↦ ↑i • LinearMap.id).range",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set R\n⊢ s • ⊤ = ((lsum R) fun i ↦ ↑i • LinearMap.id).range"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 39
} | {
"line": 698,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\n⊢ (∀ (f g : R[X]), f = g) ↔ ∀ (a b : R), a = b",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"_private.Mathlib.Algebra.Polynomial.Basic.0.Polynomial.forall_eq_iff_forall_eq._simp_1_1",
"Poly... | [
"R : Type u\ninst✝ : Semiring R\n⊢ Subsingleton R[X] ↔ Subsingleton R"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 765,
"column": 2
} | {
"line": 765,
"column": 40
} | {
"line": 765,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nc : R\n⊢ (C c * X).support ⊆ {1}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.C_mul_X_eq_monomial",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"Finset",
"Par... | [
"R : Type u\ninst✝ : Semiring R\nc : R\n⊢ ((monomial 1) c).support ⊆ {1}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 776,
"column": 2
} | {
"line": 776,
"column": 44
} | {
"line": 776,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\nc : R\n⊢ (C c * X ^ n).support ⊆ {n}",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
... | [
"R : Type u\ninst✝ : Semiring R\nn : ℕ\nc : R\n⊢ ((monomial n) c).support ⊆ {n}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1020,
"column": 2
} | {
"line": 1020,
"column": 38
} | {
"line": 1022,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn n✝ : ℕ\n⊢ (p.update n 0).coeff n✝ = (erase n p).coeff n✝",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Polynomial.coeff_erase",
"Polynomial.update",
"Polynomial.coeff_update_apply",
... | [] | rw [coeff_update_apply, coeff_erase] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.Submodule.Invariant | {
"line": 180,
"column": 2
} | {
"line": 180,
"column": 18
} | {
"line": 180,
"column": 19
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhp : p ∈ f.invtSubmodule\nq : Submodule R ↥p\nx : M\nhx : x ∈ p\nhq : ⟨f ↑⟨x, hx⟩, ⋯⟩ ∈ q\nhx' : ⟨x, hx⟩ ∈ ↑q\n⊢ p.subtype ⟨x, hx⟩ ∈ Submodule.comap f (Submodule.map p.subtype ... | [
"R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf : End R M\np : Submodule R M\nhp : p ∈ f.invtSubmodule\nq : Submodule R ↥p\nx : M\nhx : x ∈ p\nhq : ⟨f ↑⟨x, hx⟩, ⋯⟩ ∈ q\nhx' : ⟨x, hx⟩ ∈ ↑q\n⊢ f x ∈ p"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projection | {
"line": 461,
"column": 6
} | {
"line": 461,
"column": 38
} | {
"line": 462,
"column": 6
} | [
{
"pp": "case fst\nR : Type u_1\ninst✝¹² : Ring R\nE : Type u_2\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module R E\nF : Type u_3\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module R F\nG : Type u_4\ninst✝⁷ : AddCommGroup G\ninst✝⁶ : Module R G\np✝ q✝ : Submodule R E\nS : Type u_5\ninst✝⁵ : Semiring S\nM : Type u_6\ninst✝⁴ ... | [
"case snd\nR : Type u_1\ninst✝¹² : Ring R\nE : Type u_2\ninst✝¹¹ : AddCommGroup E\ninst✝¹⁰ : Module R E\nF : Type u_3\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module R F\nG : Type u_4\ninst✝⁷ : AddCommGroup G\ninst✝⁶ : Module R G\np✝ q✝ : Submodule R E\nS : Type u_5\ninst✝⁵ : Semiring S\nM : Type u_6\ninst✝⁴ : AddCommMon... | · exact ofIsCompl_apply_left h x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.SuccPred.WithBot | {
"line": 24,
"column": 57
} | {
"line": 24,
"column": 68
} | {
"line": 24,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderBot α\ninst✝¹ : AddMonoidWithOne α\ninst✝ : SuccAddOrder α\n⊢ succ 0 = 1",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderBot α\ninst✝¹ : AddMonoidWithOne α\ninst✝ : SuccAddOrder α\n⊢ succ 0 = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.SuccPred.WithBot | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 34
} | {
"line": 28,
"column": 35
} | [
{
"pp": "α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderBot α\ninst✝¹ : AddMonoidWithOne α\ninst✝ : SuccAddOrder α\n⊢ succ 1 = 2",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderBot α\ninst✝¹ : AddMonoidWithOne α\ninst✝ : SuccAddOrder α\n⊢ succ 1 = 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENat.Lattice | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 33
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℕ∞\n⊢ ⨅ i, f i = 0 ↔ ∃ i, f i = 0",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Sort u_1\nf : ι → ℕ∞\n⊢ ⨅ i, f i = 0 ↔ ∃ i, f i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projection | {
"line": 638,
"column": 4
} | {
"line": 638,
"column": 15
} | {
"line": 638,
"column": 16
} | [
{
"pp": "case mp\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nf : M →ₗ[S] M\nhf : IsProj ⊥ f\nx✝ : M\n⊢ f x✝ = 0 x✝",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"LinearMap.instFunLike",
"id",
"LinearMap",
"Zero.toO... | [
"case mp\nS : Type u_5\ninst✝² : Semiring S\nM : Type u_6\ninst✝¹ : AddCommMonoid M\ninst✝ : Module S M\nf : M →ₗ[S] M\nhf : IsProj ⊥ f\nx✝ : M\n⊢ f x✝ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 166,
"column": 43
} | {
"line": 166,
"column": 77
} | {
"line": 166,
"column": 78
} | [
{
"pp": "M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ ... | [
"M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ →* N₁\nτ₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projection | {
"line": 695,
"column": 22
} | {
"line": 695,
"column": 65
} | {
"line": 695,
"column": 66
} | [
{
"pp": "S : Type u_5\ninst✝² : Semiring S\nE : Type u_7\ninst✝¹ : AddCommGroup E\ninst✝ : Module S E\np : E →ₗ[S] E\nhp : IsIdempotentElem p\nx : E\nhx : x ∈ (id - p).ker\n⊢ p x = x",
"ppTerm": "?m.101",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"S : Type u_5\ninst✝² : Semiring S\nE : Type u_7\ninst✝¹ : AddCommGroup E\ninst✝ : Module S E\np : E →ₗ[S] E\nhp : IsIdempotentElem p\nx : E\nhx : x ∈ (id - p).ker\n⊢ p x = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Exact.Basic | {
"line": 171,
"column": 26
} | {
"line": 171,
"column": 51
} | {
"line": 171,
"column": 52
} | [
{
"pp": "M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ ... | [
"M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ →* N₁\nτ₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Finiteness.Finsupp | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 43
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh_exact : Function.Exact ⇑f ⇑g\nh_surj : Surjec... | [
"case refine_1\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nh_exact : Function.Exact ⇑f ⇑g\nh_surj : g.range = ⊤\ninst✝... | rw [← LinearMap.range_eq_top] at h_surj | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Exact.Basic | {
"line": 177,
"column": 26
} | {
"line": 177,
"column": 51
} | {
"line": 177,
"column": 52
} | [
{
"pp": "M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ ... | [
"M₁ : Type u_8\nM₂ : Type u_9\nM₃ : Type u_10\nN₁ : Type u_11\nN₂ : Type u_12\nN₃ : Type u_13\ninst✝⁵ : CommMonoid M₁\ninst✝⁴ : CommMonoid M₂\ninst✝³ : CommMonoid M₃\ninst✝² : CommMonoid N₁\ninst✝¹ : CommMonoid N₂\ninst✝ : CommMonoid N₃\nf : M₁ →* M₂\ng : M₂ →* M₃\nf' : N₁ →* N₂\ng' : N₂ →* N₃\nτ₁ : M₁ →* N₁\nτ₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENat.Lattice | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 13
} | {
"line": 243,
"column": 14
} | [
{
"pp": "ι : Sort u_2\nR : Type u_4\ninst✝¹ : SMul R ℕ∞\ninst✝ : IsScalarTower R ℕ∞ ℕ∞\nf : ι → ℕ∞\nc : R\n⊢ c • ⨆ i, f i = ⨆ i, c • f i",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Sort u_2\nR : Type u_4\ninst✝¹ : SMul R ℕ∞\ninst✝ : IsScalarTower R ℕ∞ ℕ∞\nf : ι → ℕ∞\nc : R\n⊢ c • ⨆ i, f i = ⨆ i, c • f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.RelSeries | {
"line": 111,
"column": 25
} | {
"line": 111,
"column": 45
} | {
"line": 111,
"column": 46
} | [
{
"pp": "α : Type u_1\nr : SetRel α α\nx : RelSeries r\ni : ℕ\nh : i < x.length\n⊢ i < x.length",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nr : SetRel α α\nx : RelSeries r\ni : ℕ\nh : i < x.length\n⊢ i < x.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Rel | {
"line": 360,
"column": 43
} | {
"line": 360,
"column": 60
} | {
"line": 360,
"column": 60
} | [
{
"pp": "α : Type u_1\nR : SetRel α α\ninst✝ : R.IsRefl\n⊢ SetRel.id ⊆ R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"SetRel.id",
"Membership.mem",
"Prod.mk",
"Eq.ndrec",
"Prod",
"Prod.casesOn",
"Set.instMembership",
"Set"
],
"used... | [
"α : Type u_1\nR : SetRel α α\ninst✝ : R.IsRefl\nfst✝ : α\n⊢ (fst✝, fst✝) ∈ R"
] | rintro ⟨_, _⟩ rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.Rel | {
"line": 388,
"column": 2
} | {
"line": 388,
"column": 13
} | {
"line": 388,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nS : SetRel β β\nR : SetRel α β\ninst✝ : S.IsRefl\n⊢ R ⊆ R ○ S",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nS : SetRel β β\nR : SetRel α β\ninst✝ : S.IsRefl\n⊢ R ⊆ R ○ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Rel | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 13
} | {
"line": 391,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nR : SetRel α α\ninst✝ : R.IsRefl\nS : SetRel α β\n⊢ S ⊆ R ○ S",
"ppTerm": "?m.7",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nR : SetRel α α\ninst✝ : R.IsRefl\nS : SetRel α β\n⊢ S ⊆ R ○ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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