module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.Congruence.Defs | {
"line": 408,
"column": 8
} | {
"line": 408,
"column": 65
} | {
"line": 409,
"column": 8
} | [
{
"pp": "M : Type u_1\ninst✝ : Mul M\nr : M → M → Prop\ns : Con M\nhs : s ∈ {s | ∀ (x y : M), r x y → s x y}\nx y : M\nhxy : (conGen r) x y\n⊢ s x y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"ConGen.Rel.recOn",
"Con",
"ConGen.Rel",
"Con.instFunLikeForallProp",... | [
"case of\nM : Type u_1\ninst✝ : Mul M\nr : M → M → Prop\ns : Con M\nhs : s ∈ {s | ∀ (x y : M), r x y → s x y}\nx y : M\nhxy : (conGen r) x y\n⊢ ∀ (x y : M), r x y → s x y",
"case refl\nM : Type u_1\ninst✝ : Mul M\nr : M → M → Prop\ns : Con M\nhs : s ∈ {s | ∀ (x y : M), r x y → s x y}\nx y : M\nhxy : (conGen r) x ... | apply ConGen.Rel.recOn (motive := fun x y _ => s x y) hxy | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 234,
"column": 48
} | {
"line": 234,
"column": 59
} | {
"line": 234,
"column": 60
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nt : Finset κ\nf : ι → M\ng : κ → M\ne : ι → κ\nhe : Set.InjOn e ↑s\nhest : Set.MapsTo e ↑s ↑t\nh' : ∀ i ∈ t, i ∉ e '' ↑s → g i = 1\nh : ∀ i ∈ s, f i = g (e i)\n⊢ ∀ x ∈ t, x ∉ image e s → g x = 1",
"ppTerm": "?m.69",
"... | [
"ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nt : Finset κ\nf : ι → M\ng : κ → M\ne : ι → κ\nhe : Set.InjOn e ↑s\nhest : Set.MapsTo e ↑s ↑t\nh' : ∀ i ∈ t, i ∉ e '' ↑s → g i = 1\nh : ∀ i ∈ s, f i = g (e i)\n⊢ ∀ x ∈ t, (∀ x_1 ∈ s, ¬e x_1 = x) → g x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 588,
"column": 21
} | {
"line": 588,
"column": 32
} | {
"line": 588,
"column": 33
} | [
{
"pp": "M : Type u_4\ninst✝ : Monoid M\nc : Con M\nw x : M\nx✝ : c w x\n⊢ c (w ^ 0) (x ^ 0)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"id",
"MulOne.toMul",
"instOfNatNat",
"... | [
"M : Type u_4\ninst✝ : Monoid M\nc : Con M\nw x : M\nx✝ : c w x\n⊢ c 1 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 589,
"column": 30
} | {
"line": 589,
"column": 52
} | {
"line": 589,
"column": 53
} | [
{
"pp": "M : Type u_4\ninst✝ : Monoid M\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ n.succ) (x ^ n.succ)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"pow_succ",
"id",
"MulOne.toMul"... | [
"M : Type u_4\ninst✝ : Monoid M\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ n * w) (x ^ n * x)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 662,
"column": 2
} | {
"line": 662,
"column": 35
} | {
"line": 662,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝ : Group M\nc : Con M\nw x y z : M\nh1 : c w x\nh2 : c y z\n⊢ c (w / y) (x / z)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"M : Type u_1\ninst✝ : Group M\nc : Con M\nw x y z : M\nh1 : c w x\nh2 : c y z\n⊢ c (w * y⁻¹) (x * z⁻¹)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 667,
"column": 31
} | {
"line": 667,
"column": 84
} | {
"line": 667,
"column": 85
} | [
{
"pp": "M : Type u_1\ninst✝ : Group M\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ Int.ofNat n) (x ^ Int.ofNat n)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"DivInvMonoid.toZPow",
"id... | [
"M : Type u_1\ninst✝ : Group M\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ n) (x ^ n)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Congruence.Defs | {
"line": 668,
"column": 33
} | {
"line": 668,
"column": 64
} | {
"line": 668,
"column": 65
} | [
{
"pp": "M : Type u_1\ninst✝ : Group M\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ Int.negSucc n) (x ^ Int.negSucc n)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Monoid.toMulOneClass",
"congrArg",
"zpow_negSucc",
... | [
"M : Type u_1\ninst✝ : Group M\nc : Con M\nn : ℕ\nw x : M\nh : c w x\n⊢ c (w ^ (n + 1))⁻¹ (x ^ (n + 1))⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 315,
"column": 33
} | {
"line": 315,
"column": 44
} | {
"line": 315,
"column": 45
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\np : ι → Prop\ninst✝ : DecidablePred p\nhp : ∀ x ∈ s, f x ≠ 1 → p x\nx : ι\nh₁ : x ∈ s\nh₂ : ¬f x = 1\n⊢ p x",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\np : ι → Prop\ninst✝ : DecidablePred p\nhp : ∀ x ∈ s, f x ≠ 1 → p x\nx : ι\nh₁ : x ∈ s\nh₂ : ¬f x = 1\n⊢ p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 329,
"column": 48
} | {
"line": 329,
"column": 59
} | {
"line": 329,
"column": 60
} | [
{
"pp": "case hc\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M\na : ι\nh : a ∈ {a ∈ s | p a}\n⊢ p a",
"ppTerm": "?hc",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case hc\nι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M\na : ι\nh : a ∈ {a ∈ s | p a}\n⊢ p a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 333,
"column": 25
} | {
"line": 333,
"column": 36
} | {
"line": 333,
"column": 37
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M\nx : ι\nhs : x ∈ s\nh : x ∈ s → ¬p x\n⊢ ¬p x",
"ppTerm": "?m.126",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M\nx : ι\nhs : x ∈ s\nh : x ∈ s → ¬p x\n⊢ ¬p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 174,
"column": 35
} | {
"line": 174,
"column": 46
} | {
"line": 174,
"column": 47
} | [
{
"pp": "α : Type u\ntl x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\n⊢ ∃ L₅, Step ((x✝¹, x✝) :: (x✝¹, !x✝) :: tl ++ x✝⁴) L₅ ∧ Step ([] ++ tl.append ((x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁴)) L₅",
"ppTerm": "?m.863",
"assigned": true,
"usedConstants": [
"FreeGroup.Red.Step",
"... | [
"α : Type u\ntl x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\n⊢ ∃ L₅, Step ((x✝¹, x✝) :: (x✝¹, !x✝) :: (tl ++ x✝⁴)) L₅ ∧ Step (tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁴) L₅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 178,
"column": 29
} | {
"line": 178,
"column": 45
} | {
"line": 179,
"column": 4
} | [
{
"pp": "α : Type u\nx3 : α\nb3 : Bool\ntl x✝⁵ : List (α × Bool)\nx4 : α\nb4 : Bool\ntl2 x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\nH : (x3, b3) :: tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁵ = (x4, b4) :: tl2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: x✝⁴\nH1 : (x3, b3) = (x4, b4)\nH2 : tl.append ((x✝³, x✝²... | [] | by simp [H1, H3] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coset.Defs | {
"line": 229,
"column": 45
} | {
"line": 229,
"column": 56
} | {
"line": 229,
"column": 57
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\nx : α\nx✝ : ∃ x_1 ∈ s, x_1⁻¹ * x ∈ N\ny : α\nhs : y ∈ s\nhN : y⁻¹ * x ∈ N\n⊢ x * (y⁻¹ * x)⁻¹ ∈ s",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOn... | [
"α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\nx : α\nx✝ : ∃ x_1 ∈ s, x_1⁻¹ * x ∈ N\ny : α\nhs : y ∈ s\nhN : y⁻¹ * x ∈ N\n⊢ y ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 179,
"column": 57
} | {
"line": 179,
"column": 73
} | {
"line": 179,
"column": 74
} | [
{
"pp": "α : Type u\nx3 : α\nb3 : Bool\ntl x✝⁵ : List (α × Bool)\nx4 : α\nb4 : Bool\ntl2 x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\nH : (x3, b3) :: tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁵ = (x4, b4) :: tl2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: x✝⁴\nH1 : (x3, b3) = (x4, b4)\nH2 : tl.append ((x✝³, x✝²... | [
"α : Type u\nx3 : α\nb3 : Bool\ntl x✝⁵ : List (α × Bool)\nx4 : α\nb4 : Bool\ntl2 x✝⁴ : List (α × Bool)\nx✝³ : α\nx✝² : Bool\nx✝¹ : α\nx✝ : Bool\nH : (x3, b3) :: tl ++ (x✝³, x✝²) :: (x✝³, !x✝²) :: x✝⁵ = (x4, b4) :: tl2 ++ (x✝¹, x✝) :: (x✝¹, !x✝) :: x✝⁴\nH1 : (x3, b3) = (x4, b4)\nH2 : tl.append ((x✝³, x✝²) :: (x✝³, !... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Coset.Defs | {
"line": 230,
"column": 22
} | {
"line": 230,
"column": 33
} | {
"line": 230,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\nx : α\nx✝ : ∃ x_1 ∈ N, x * x_1 ∈ s\nz : α\nhz : z ∈ N\nhxz : x * z ∈ s\n⊢ (x * z)⁻¹ * x ∈ N",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Subgroup.instSubgroupClass",
... | [
"α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\nx : α\nx✝ : ∃ x_1 ∈ N, x * x_1 ∈ s\nz : α\nhz : z ∈ N\nhxz : x * z ∈ s\n⊢ z ∈ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 13
} | {
"line": 436,
"column": 14
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ x ∈ s, p x\n⊢ ∀ x ∈ s, p x",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\np : ι → Prop\ninst✝ : DecidablePred p\nh : ∀ x ∈ s, p x\n⊢ ∀ x ∈ s, p x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 506,
"column": 59
} | {
"line": 506,
"column": 70
} | {
"line": 506,
"column": 71
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nt : Finset κ\nf : ι → M\ng : κ → M\ni : (a : ι) → a ∈ s → f a ≠ 1 → κ\nhi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),... | [
"ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝ : CommMonoid M\ns : Finset ι\nt : Finset κ\nf : ι → M\ng : κ → M\ni : (a : ι) → a ∈ s → f a ≠ 1 → κ\nhi : ∀ (a : ι) (h₁ : a ∈ s) (h₂ : f a ≠ 1), i a h₁ h₂ ∈ t\ni_inj :\n ∀ (a₁ : ι) (h₁₁ : a₁ ∈ s) (h₁₂ : f a₁ ≠ 1) (a₂ : ι) (h₂₁ : a₂ ∈ s) (h₂₂ : f a₂ ≠ 1),\n i a₁ h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Finiteness | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 39
} | {
"line": 99,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nι : Type u_3\ns : Finset ι\nP : ι → Submonoid M\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : Monoid M\nι : Type u_3\ns : Finset ι\nP : ι → Submonoid M\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Finiteness | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 13
} | {
"line": 104,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nι : Type u_3\ns : Set ι\nhs : s.Finite\nP : ι → Submonoid M\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : Monoid M\nι : Type u_3\ns : Set ι\nhs : s.Finite\nP : ι → Submonoid M\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Finiteness | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 31
} | {
"line": 109,
"column": 32
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\nι : Sort u_3\ninst✝ : Finite ι\nP : ι → Submonoid M\nhP : ∀ (i : ι), (P i).FG\n⊢ (_root_.iSup P).FG",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : Monoid M\nι : Sort u_3\ninst✝ : Finite ι\nP : ι → Submonoid M\nhP : ∀ (i : ι), (P i).FG\n⊢ (_root_.iSup P).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 61
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case tail\nα : Type u\nL₁ L₂ b✝ c✝ : List (α × Bool)\n_h₁₂ : ReflTransGen Step L₁ b✝\nh₂₃ : Step b✝ c✝\nn : ℕ\neq : L₁.length = b✝.length + 2 * n\n⊢ L₁.length = c✝.length + 2 * (1 + n)",
"ppTerm": "?tail",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"congrArg",
"add_... | [] | simp [Nat.mul_add, eq, (Step.length h₂₃).symm, add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 578,
"column": 51
} | {
"line": 578,
"column": 62
} | {
"line": 578,
"column": 63
} | [
{
"pp": "M : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq M\ns : List M\n⊢ s.prod = ∏ m ∈ s.toFinset, m ^ count m s",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq M\ns : List M\n⊢ s.prod = ∏ m ∈ s.toFinset, m ^ count m s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Finiteness | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 39
} | {
"line": 361,
"column": 40
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\nι : Type u_5\ns : Finset ι\nP : ι → Subgroup G\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_3\ninst✝ : Group G\nι : Type u_5\ns : Finset ι\nP : ι → Subgroup G\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Finiteness | {
"line": 366,
"column": 2
} | {
"line": 366,
"column": 13
} | {
"line": 366,
"column": 14
} | [
{
"pp": "G : Type u_3\ninst✝ : Group G\nι : Type u_5\ns : Set ι\nhs : s.Finite\nP : ι → Subgroup G\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_3\ninst✝ : Group G\nι : Type u_5\ns : Set ι\nhs : s.Finite\nP : ι → Subgroup G\nhP : ∀ i ∈ s, (P i).FG\n⊢ (⨆ i ∈ s, P i).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Finiteness | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 31
} | {
"line": 371,
"column": 32
} | [
{
"pp": "G : Type u_3\ninst✝¹ : Group G\nι : Sort u_5\ninst✝ : Finite ι\nP : ι → Subgroup G\nhP : ∀ (i : ι), (P i).FG\n⊢ (_root_.iSup P).FG",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_3\ninst✝¹ : Group G\nι : Sort u_5\ninst✝ : Finite ι\nP : ι → Subgroup G\nhP : ∀ (i : ι), (P i).FG\n⊢ (_root_.iSup P).FG"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 595,
"column": 15
} | {
"line": 595,
"column": 47
} | {
"line": 595,
"column": 48
} | [
{
"pp": "α : Type u\nL₁ L₂ : List (α × Bool)\nh : Step (FreeGroup.invRev L₁) (FreeGroup.invRev L₂)\n⊢ Step L₁ L₂",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nL₁ L₂ : List (α × Bool)\nh : Step (FreeGroup.invRev L₁) (FreeGroup.invRev L₂)\n⊢ Step L₁ L₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 599,
"column": 15
} | {
"line": 599,
"column": 47
} | {
"line": 599,
"column": 48
} | [
{
"pp": "α : Type u\nL₁ L₂ : List (α × Bool)\nh : Red (invRev L₁) (invRev L₂)\n⊢ Red L₁ L₂",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nL₁ L₂ : List (α × Bool)\nh : Red (invRev L₁) (invRev L₂)\n⊢ Red L₁ L₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 42,
"column": 2
} | {
"line": 44,
"column": 9
} | {
"line": 44,
"column": 10
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\n⊢ leftRel N = ⊤ ↔ N = ⊤",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Setoid.completeLattice",
... | [
"G : Type u_1\ninst✝ : Group G\nN : Subgroup G\n⊢ (∀ (x x_1 : G), x_1⁻¹ * x ∈ N) ↔ N = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 44,
"column": 27
} | {
"line": 44,
"column": 38
} | {
"line": 44,
"column": 39
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nh : ∀ (x x_1 : G), x_1⁻¹ * x ∈ N\nx✝ : G\n⊢ x✝ ∈ N ↔ x✝ ∈ ⊤",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"iff_true",
"Membership.mem",
"id",
"Subgroup",
"Iff",
"... | [
"G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nh : ∀ (x x_1 : G), x_1⁻¹ * x ∈ N\nx✝ : G\n⊢ x✝ ∈ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 47,
"column": 2
} | {
"line": 49,
"column": 9
} | {
"line": 49,
"column": 10
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\n⊢ rightRel N = ⊤ ↔ N = ⊤",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"_private.Mathlib.GroupTheory.QuotientGroup.Defs.0.QuotientGroup.rightRel_eq_top._simp_1_2",
"Eq.mpr",
"instHSMul",
"Lattice.toSemilattic... | [
"G : Type u_1\ninst✝ : Group G\nN : Subgroup G\n⊢ (∀ (x x_1 : G), x * x_1⁻¹ ∈ N) ↔ N = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 49,
"column": 27
} | {
"line": 49,
"column": 38
} | {
"line": 49,
"column": 39
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nh : ∀ (x x_1 : G), x * x_1⁻¹ ∈ N\nx✝ : G\n⊢ x✝ ∈ N ↔ x✝ ∈ ⊤",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"iff_true",
"Membership.mem",
"id",
"Subgroup",
"Iff",
"... | [
"G : Type u_1\ninst✝ : Group G\nN : Subgroup G\nh : ∀ (x x_1 : G), x * x_1⁻¹ ∈ N\nx✝ : G\n⊢ x✝ ∈ N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 192,
"column": 23
} | {
"line": 192,
"column": 34
} | {
"line": 192,
"column": 35
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\na✝ b✝ : G\nhx : a✝ ∈ {x | c x 1}\nhy : b✝ ∈ {x | c x 1}\n⊢ a✝ * b✝ ∈ {x | c x 1}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\na✝ b✝ : G\nhx : a✝ ∈ {x | c x 1}\nhy : b✝ ∈ {x | c x 1}\n⊢ c (a✝ * b✝) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 193,
"column": 19
} | {
"line": 193,
"column": 30
} | {
"line": 193,
"column": 31
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\nx✝ : G\nh : x✝ ∈ {x | c x 1}\n⊢ x✝⁻¹ ∈ {x | c x 1}",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
... | [
"G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\nx✝ : G\nh : x✝ ∈ {x | c x 1}\n⊢ c x✝⁻¹ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 201,
"column": 19
} | {
"line": 201,
"column": 30
} | {
"line": 201,
"column": 31
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\nx : G\nhx : x ∈ c.subgroup\ng : G\n⊢ g * x * g⁻¹ ∈ c.subgroup",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\nx : G\nhx : x ∈ c.subgroup\ng : G\n⊢ c (g * x * g⁻¹) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 201,
"column": 16
} | {
"line": 201,
"column": 73
} | {
"line": 201,
"column": 73
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nI : Type u_3\nM : Type u_4\ninst✝² : Group G\ninst✝¹ : Group H\ninst✝ : Monoid M\nN : Subgroup G\nnN : N.Normal\nc : Con G\nx : G\nhx : x ∈ c.subgroup\ng : G\n⊢ g * x * g⁻¹ ∈ c.subgroup",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by simpa using (c.mul (c.mul (c.refl g) hx) (c.refl g⁻¹)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 214,
"column": 20
} | {
"line": 214,
"column": 31
} | {
"line": 214,
"column": 32
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nc : Con G\nx y : G\nh : c (x⁻¹ * y) 1\n⊢ c x y",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nc : Con G\nx y : G\nh : c (x⁻¹ * y) 1\n⊢ c x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.QuotientGroup.Defs | {
"line": 215,
"column": 15
} | {
"line": 215,
"column": 26
} | {
"line": 215,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nc : Con G\nx y : G\nh : c x y\n⊢ c (x⁻¹ * y) 1",
"ppTerm": "?m.50",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nc : Con G\nx y : G\nh : c x y\n⊢ c (x⁻¹ * y) 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 711,
"column": 21
} | {
"line": 711,
"column": 32
} | {
"line": 711,
"column": 33
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : Group β\nf : α → β\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : (lift f) (Quot.mk Red.Step tl) ∈ s\nx : α\nb : Bool\n⊢ (lift f) (Quot.mk Red.Step ((x, false) :: tl)) ∈ s",
"ppTerm": "?m.61",
... | [
"α : Type u\nβ : Type v\ninst✝ : Group β\nf : α → β\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : (lift f) (Quot.mk Red.Step tl) ∈ s\nx : α\nb : Bool\n⊢ (f x)⁻¹ * (List.map (fun x ↦ bif x.2 then f x.1 else (f x.1)⁻¹) tl).prod ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 712,
"column": 10
} | {
"line": 712,
"column": 21
} | {
"line": 712,
"column": 22
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : Group β\nf : α → β\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : (lift f) (Quot.mk Red.Step tl) ∈ s\nx : α\nb : Bool\n⊢ (lift f) (Quot.mk Red.Step ((x, true) :: tl)) ∈ s",
"ppTerm": "?m.62",
... | [
"α : Type u\nβ : Type v\ninst✝ : Group β\nf : α → β\ns : Subgroup β\nH : Set.range f ⊆ ↑s\nw✝ : FreeGroup α\nL : List (α × Bool)\nx✝ : α × Bool\ntl : List (α × Bool)\nih : (lift f) (Quot.mk Red.Step tl) ∈ s\nx : α\nb : Bool\n⊢ f x * (List.map (fun x ↦ bif x.2 then f x.1 else (f x.1)⁻¹) tl).prod ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 677,
"column": 17
} | {
"line": 677,
"column": 34
} | {
"line": 677,
"column": 35
} | [
{
"pp": "case inl\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝ : CommMonoid M\nf : ι → M\ns : Finset ι\nih :\n ∀ t ⊂ s,\n ∀ (g : (a : ι) → a ∈ t → ι),\n (∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1) →\n (∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) →\n ∀ (g_mem : ∀ (a : ι) (ha : ... | [
"case inl\nι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝ : CommMonoid M\nf : ι → M\ns : Finset ι\nih :\n ∀ t ⊂ s,\n ∀ (g : (a : ι) → a ∈ t → ι),\n (∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1) →\n (∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) →\n ∀ (g_mem : ∀ (a : ι) (ha : a ∈ t), g a ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 760,
"column": 73
} | {
"line": 760,
"column": 92
} | {
"line": 762,
"column": 0
} | [
{
"pp": "case mk\nα : Type u\nx : FreeGroup α\nL : List (α × Bool)\n⊢ (map _root_.id) (Quot.mk Red.Step L) = Quot.mk Red.Step L",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"List.map_id'",
"FreeGroup.Red.Step",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.to... | [] | simp [List.map_id'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 680,
"column": 44
} | {
"line": 680,
"column": 75
} | {
"line": 680,
"column": 75
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns✝ : Finset ι\ninst✝ : CommMonoid M\nf : ι → M\ns : Finset ι\nih :\n ∀ t ⊂ s,\n ∀ (g : (a : ι) → a ∈ t → ι),\n (∀ (a : ι) (ha : a ∈ t), f a * f (g a ha) = 1) →\n (∀ (a : ι) (ha : a ∈ t), f a ≠ 1 → g a ha ≠ a) →\n ∀ (g_mem : ∀ (a : ι) (ha : a ∈ t), g ... | [] | by simp [insert_subset_iff, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 964,
"column": 19
} | {
"line": 964,
"column": 45
} | {
"line": 964,
"column": 46
} | [
{
"pp": "case succ\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : ℕ\nhx : (fun x ↦ ((map 1) x).sum) ((fun x ↦ of default ^ x) ↑x) = ↑x\n⊢ (fun x ↦ ((map 1) x).sum) ((fun x ↦ of default ^ x) (↑x + 1)) = ↑x + 1",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"zpow_... | [
"case succ\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : ℕ\nhx : (fun x ↦ ((map 1) x).sum) ((fun x ↦ of default ^ x) ↑x) = ↑x\n⊢ (of 1 ^ x).sum = ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 965,
"column": 19
} | {
"line": 965,
"column": 63
} | {
"line": 965,
"column": 64
} | [
{
"pp": "case pred\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : ℕ\nhx : (fun x ↦ ((map 1) x).sum) ((fun x ↦ of default ^ x) (-↑x)) = -↑x\n⊢ (fun x ↦ ((map 1) x).sum) ((fun x ↦ of default ^ x) (-↑x - 1)) = -↑x - 1",
"ppTerm": "?pred",
"assigned": true,
"usedConstants": [
... | [
"case pred\nα : Type u\nL L₁ L₂ L₃ L₄ : List (α × Bool)\ninst✝ : Unique α\nx : ℕ\nhx : (fun x ↦ ((map 1) x).sum) ((fun x ↦ of default ^ x) (-↑x)) = -↑x\n⊢ (of 1 ^ x).sum = ↑x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 797,
"column": 2
} | {
"line": 807,
"column": 67
} | {
"line": 809,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\ns : Finset κ\nt : κ → Finset ι\nhs : (↑s).Pairwise fun i j ↦ ∀ k ∈ t i ∩ t j, f k = 1\n⊢ ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | classical
let t' k := (t k).filter (fun i ↦ f i ≠ 1)
have : s.biUnion t' = (s.biUnion t).filter (fun i ↦ f i ≠ 1) := by grind
rw [← prod_filter_ne_one, ← this, prod_biUnion]
swap
· intro i hi j hj hij a hai haj k hk
have hki : k ∈ t' i := hai hk
have hkj : k ∈ t' j := haj hk
simp only [ne_eq, mem_... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 797,
"column": 2
} | {
"line": 807,
"column": 67
} | {
"line": 809,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\ns : Finset κ\nt : κ → Finset ι\nhs : (↑s).Pairwise fun i j ↦ ∀ k ∈ t i ∩ t j, f k = 1\n⊢ ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | classical
let t' k := (t k).filter (fun i ↦ f i ≠ 1)
have : s.biUnion t' = (s.biUnion t).filter (fun i ↦ f i ≠ 1) := by grind
rw [← prod_filter_ne_one, ← this, prod_biUnion]
swap
· intro i hi j hj hij a hai haj k hk
have hki : k ∈ t' i := hai hk
have hkj : k ∈ t' j := haj hk
simp only [ne_eq, mem_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 797,
"column": 2
} | {
"line": 807,
"column": 67
} | {
"line": 809,
"column": 0
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\ns : Finset κ\nt : κ → Finset ι\nhs : (↑s).Pairwise fun i j ↦ ∀ k ∈ t i ∩ t j, f k = 1\n⊢ ∏ x ∈ s.biUnion t, f x = ∏ x ∈ s, ∏ i ∈ t x, f i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | classical
let t' k := (t k).filter (fun i ↦ f i ≠ 1)
have : s.biUnion t' = (s.biUnion t).filter (fun i ↦ f i ≠ 1) := by grind
rw [← prod_filter_ne_one, ← this, prod_biUnion]
swap
· intro i hi j hj hij a hai haj k hk
have hki : k ∈ t' i := hai hk
have hkj : k ∈ t' j := haj hk
simp only [ne_eq, mem_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 830,
"column": 6
} | {
"line": 830,
"column": 17
} | {
"line": 830,
"column": 17
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq ι\nf : κ → ι\ng : ι → M\nI : Finset κ\nhf : (↑I).Pairwise fun i j ↦ f i = f j → g (f i) = 1\n⊢ ∏ s ∈ image f I, g s = ∏ i ∈ I, g (f i)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq ι\nf : κ → ι\ng : ι → M\nI : Finset κ\nhf : (↑I).Pairwise fun i j ↦ f i = f j → g (f i) = 1\n⊢ ∏ i ∈ I, ?h i = ∏ i ∈ I, g (f i)",
"case h\nι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\ninst✝ : DecidableEq ι\nf... | prod_image' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 324,
"column": 85
} | {
"line": 324,
"column": 96
} | {
"line": 324,
"column": 96
} | [
{
"pp": "α : Type u\nG : Type u_1\nβ α✝ β✝ γ✝ : Type u\nx✝ : FreeAbelianGroup α✝\nf : α✝ → FreeAbelianGroup β✝\ng : β✝ → FreeAbelianGroup γ✝\nx : α✝\nih : pure x >>= f >>= g = pure x >>= fun x ↦ f x >>= g\n⊢ -(pure x >>= f) >>= g = -pure x >>= fun x ↦ f x >>= g",
"ppTerm": "?m.125",
"assigned": false,
... | [
"α : Type u\nG : Type u_1\nβ α✝ β✝ γ✝ : Type u\nx✝ : FreeAbelianGroup α✝\nf : α✝ → FreeAbelianGroup β✝\ng : β✝ → FreeAbelianGroup γ✝\nx : α✝\nih : pure x >>= f >>= g = pure x >>= fun x ↦ f x >>= g\n⊢ -(pure x >>= f) >>= g = -pure x >>= fun x ↦ f x >>= g"
] | try rw [ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 324,
"column": 85
} | {
"line": 324,
"column": 96
} | {
"line": 324,
"column": 96
} | [
{
"pp": "α : Type u\nG : Type u_1\nβ α✝ β✝ γ✝ : Type u\nx✝ : FreeAbelianGroup α✝\nf : α✝ → FreeAbelianGroup β✝\ng : β✝ → FreeAbelianGroup γ✝\nx : α✝\nih : pure x >>= f >>= g = pure x >>= fun x ↦ f x >>= g\n⊢ -(pure x >>= f >>= g) = -pure x >>= fun x ↦ f x >>= g",
"ppTerm": "?m.130",
"assigned": true,
... | [
"α : Type u\nG : Type u_1\nβ α✝ β✝ γ✝ : Type u\nx✝ : FreeAbelianGroup α✝\nf : α✝ → FreeAbelianGroup β✝\ng : β✝ → FreeAbelianGroup γ✝\nx : α✝\nih : pure x >>= f >>= g = pure x >>= fun x ↦ f x >>= g\n⊢ -(pure x >>= fun x ↦ f x >>= g) = -pure x >>= fun x ↦ f x >>= g"
] | try rw [ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 960,
"column": 2
} | {
"line": 960,
"column": 37
} | {
"line": 960,
"column": 38
} | [
{
"pp": "ι : Type u_1\nκ : Type u_5\ninst✝ : DecidableEq κ\ns : Finset ι\nt : Finset κ\ng : ι → κ\n⊢ ∑ j ∈ t, #({i ∈ s | g i = j}) = #({i ∈ s | g i ∈ t})",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.card_eq_sum_ones",
"congrArg",
"Finset",
... | [
"ι : Type u_1\nκ : Type u_5\ninst✝ : DecidableEq κ\ns : Finset ι\nt : Finset κ\ng : ι → κ\n⊢ ∑ x ∈ t, ∑ x ∈ s with g x = x, 1 = ∑ x ∈ s with g x ∈ t, 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 968,
"column": 43
} | {
"line": 968,
"column": 54
} | {
"line": 968,
"column": 55
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝ : DecidableEq M\nt : ι → Finset M\nh : (↑s).PairwiseDisjoint t\n⊢ #(s.biUnion t) = ∑ u ∈ s, #(t u)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝ : DecidableEq M\nt : ι → Finset M\nh : (↑s).PairwiseDisjoint t\n⊢ #(s.biUnion t) = ∑ u ∈ s, #(t u)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 335,
"column": 18
} | {
"line": 335,
"column": 61
} | {
"line": 336,
"column": 6
} | [
{
"pp": "case pure.pure\nα : Type u\nG : Type u_1\nβ α✝ β✝ : Type u\np : α✝\nq : β✝\n⊢ Prod.mk p <$> pure q = (fun b a ↦ (a, b)) <$> pure q <*> pure p",
"ppTerm": "?pure.pure",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"FreeAbelianGroup.pure_seq",
"congrArg"... | [] | rw [map_pure, map_pure, pure_seq, map_pure] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 335,
"column": 18
} | {
"line": 335,
"column": 61
} | {
"line": 336,
"column": 6
} | [
{
"pp": "case pure.pure\nα : Type u\nG : Type u_1\nβ α✝ β✝ : Type u\np : α✝\nq : β✝\n⊢ Prod.mk p <$> pure q = (fun b a ↦ (a, b)) <$> pure q <*> pure p",
"ppTerm": "?pure.pure",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"FreeAbelianGroup.pure_seq",
"congrArg"... | [] | rw [map_pure, map_pure, pure_seq, map_pure] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 335,
"column": 18
} | {
"line": 335,
"column": 61
} | {
"line": 336,
"column": 6
} | [
{
"pp": "case pure.pure\nα : Type u\nG : Type u_1\nβ α✝ β✝ : Type u\np : α✝\nq : β✝\n⊢ Prod.mk p <$> pure q = (fun b a ↦ (a, b)) <$> pure q <*> pure p",
"ppTerm": "?pure.pure",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"FreeAbelianGroup.pure_seq",
"congrArg"... | [] | rw [map_pure, map_pure, pure_seq, map_pure] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 1001,
"column": 41
} | {
"line": 1001,
"column": 52
} | {
"line": 1001,
"column": 53
} | [
{
"pp": "M : Type u_4\nκ : Type u_6\nι : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : Fintype κ\ninst✝ : CommMonoid M\ne : ι → κ\nhe : Injective e\nf : ι → M\ng : κ → M\nh' : ∀ i ∉ Set.range e, g i = 1\nh : ∀ (i : ι), f i = g (e i)\n⊢ ∀ i ∈ univ, i ∉ e '' ↑univ → g i = 1",
"ppTerm": "?m.40",
"assigned": true,... | [
"M : Type u_4\nκ : Type u_6\nι : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : Fintype κ\ninst✝ : CommMonoid M\ne : ι → κ\nhe : Injective e\nf : ι → M\ng : κ → M\nh' : ∀ i ∉ Set.range e, g i = 1\nh : ∀ (i : ι), f i = g (e i)\n⊢ ∀ (i : κ), (∀ (x : ι), ¬e x = i) → g i = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 1058,
"column": 2
} | {
"line": 1058,
"column": 31
} | {
"line": 1058,
"column": 32
} | [
{
"pp": "ι : Type u_1\ninst✝ : DecidableEq ι\nl : List ι\n⊢ ∑ a ∈ l.toFinset, count a l = l.length",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\ninst✝ : DecidableEq ι\nl : List ι\n⊢ ∑ a ∈ l.toFinset, count a l = l.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 1080,
"column": 2
} | {
"line": 1080,
"column": 13
} | {
"line": 1080,
"column": 14
} | [
{
"pp": "ι : Type u_1\ninst✝ : DecidableEq ι\ns : Multiset ι\n⊢ ∑ a ∈ s.toFinset, count a s = s.card",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\ninst✝ : DecidableEq ι\ns : Multiset ι\n⊢ ∑ a ∈ s.toFinset, count a s = s.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 1086,
"column": 2
} | {
"line": 1086,
"column": 13
} | {
"line": 1086,
"column": 14
} | [
{
"pp": "case e_s\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nm : Multiset ι\nhms : ∀ a ∈ m, a ∈ s\na : ι\n⊢ a ∈ {x ∈ s | count x m ≠ 0} ↔ a ∈ m.toFinset",
"ppTerm": "?e_s",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"instDecidableNot",
"Finse... | [
"case e_s\nι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nm : Multiset ι\nhms : ∀ a ∈ m, a ∈ s\na : ι\n⊢ a ∈ m → a ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 1116,
"column": 19
} | {
"line": 1116,
"column": 30
} | {
"line": 1116,
"column": 31
} | [
{
"pp": "case cons\nM : Type u_4\ninst✝ : CommMonoid M\na : M\ns : Multiset M\nih : IsUnit s.prod ↔ ∀ a ∈ s, IsUnit a\n⊢ IsUnit (a ::ₘ s).prod ↔ ∀ a_1 ∈ a ::ₘ s, IsUnit a_1",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Multiset.mem_cons._simp_1",
"Eq.mpr",
"HMul.hMul",... | [
"case cons\nM : Type u_4\ninst✝ : CommMonoid M\na : M\ns : Multiset M\nih : IsUnit s.prod ↔ ∀ a ∈ s, IsUnit a\n⊢ IsUnit a → (IsUnit s.prod ↔ ∀ a ∈ s, IsUnit a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 15
} | {
"line": 411,
"column": 16
} | [
{
"pp": "α : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\ninst✝ : Mul α\na : FreeAbelianGroup α\nh : 0 * a + 0 * a = 0 * a\n⊢ 0 * a = 0",
"ppTerm": "?m.33",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\ninst✝ : Mul α\na : FreeAbelianGroup α\nh : 0 * a + 0 * a = 0 * a\n⊢ 0 * a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 446,
"column": 27
} | {
"line": 446,
"column": 67
} | {
"line": 448,
"column": 0
} | [
{
"pp": "case add\nα : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\ninst✝ : Semigroup α\nx y z₁ z₂ : FreeAbelianGroup α\nih₁ : x * y * z₁ = x * (y * z₁)\nih₂ : x * y * z₂ = x * (y * z₂)\n⊢ x * y * (z₁ + z₂) = x * (y * (z₁ + z₂))",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Distri... | [] | rw [mul_add, mul_add, mul_add, ih₁, ih₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 446,
"column": 27
} | {
"line": 446,
"column": 67
} | {
"line": 448,
"column": 0
} | [
{
"pp": "case add\nα : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\ninst✝ : Semigroup α\nx y z₁ z₂ : FreeAbelianGroup α\nih₁ : x * y * z₁ = x * (y * z₁)\nih₂ : x * y * z₂ = x * (y * z₂)\n⊢ x * y * (z₁ + z₂) = x * (y * (z₁ + z₂))",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Distri... | [] | rw [mul_add, mul_add, mul_add, ih₁, ih₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 446,
"column": 27
} | {
"line": 446,
"column": 67
} | {
"line": 448,
"column": 0
} | [
{
"pp": "case add\nα : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\ninst✝ : Semigroup α\nx y z₁ z₂ : FreeAbelianGroup α\nih₁ : x * y * z₁ = x * (y * z₁)\nih₂ : x * y * z₂ = x * (y * z₂)\n⊢ x * y * (z₁ + z₂) = x * (y * (z₁ + z₂))",
"ppTerm": "?add",
"assigned": true,
"usedConstants": [
"Distri... | [] | rw [mul_add, mul_add, mul_add, ih₁, ih₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup | {
"line": 56,
"column": 29
} | {
"line": 56,
"column": 88
} | {
"line": 56,
"column": 89
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommGroup G\nm₁ m₂ : M\ns₁ s₂ : ↥⊤\nh : (r ⊤) (m₁, s₁) (m₂, s₂)\n⊢ ⋯ ▸ mk ↑s₁ ⟨m₁, ⋯⟩ = mk ↑s₂ ⟨m₂, ⋯⟩",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Localization.mk",
"Submon... | [
"M : Type u_1\nG : Type u_2\ninst✝¹ : CommMonoid M\ninst✝ : CommGroup G\nm₁ m₂ : M\ns₁ s₂ : ↥⊤\nh : (r ⊤) (m₁, s₁) (m₂, s₂)\n⊢ ∃ a, a * (m₁ * ↑s₂) = a * (m₂ * ↑s₁)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Hom.Monoid | {
"line": 187,
"column": 17
} | {
"line": 187,
"column": 28
} | {
"line": 187,
"column": 29
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : Group α\ninst✝⁵ : Monoid β\nF : Type u_6\ninst✝⁴ : FunLike F α β\ninst✝³ : MonoidHomClass F α β\ninst✝² : LE β\ninst✝¹ : MulRightMono β\ninst✝ : MulLeftMono β\nf g : F\nx : α\nh : ∀ (f g : F) (x : α), f x⁻¹ ≤ g x⁻¹ → g x ≤ f x\n⊢ g x ≤ f x → f x⁻¹ ≤ g x⁻¹",
"ppT... | [
"α : Type u_2\nβ : Type u_3\ninst✝⁶ : Group α\ninst✝⁵ : Monoid β\nF : Type u_6\ninst✝⁴ : FunLike F α β\ninst✝³ : MonoidHomClass F α β\ninst✝² : LE β\ninst✝¹ : MulRightMono β\ninst✝ : MulLeftMono β\nf g : F\nx : α\nh : ∀ (f g : F) (x : α), f x⁻¹ ≤ g x⁻¹ → g x ≤ f x\n⊢ g x ≤ f x → f x⁻¹ ≤ g x⁻¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.MonoidLocalization.Maps | {
"line": 84,
"column": 4
} | {
"line": 89,
"column": 10
} | {
"line": 91,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nx y : N\n⊢ g (f.sec (x * y)).1 * ↑((IsUnit.liftRight (g.restrict S) hg) (f.sec (x * y)).2)⁻¹ =\n g (f.se... | [] | rw [mul_inv_left hg, ← mul_assoc, ← mul_assoc, mul_inv_right hg, mul_comm _ (g (f.sec y).1), ←
mul_assoc, ← mul_assoc, mul_inv_right hg]
repeat rw [← g.map_mul]
refine f.eq_of_eq hg ?_
simp_rw [map_mul, sec_spec', ← toMonoidHom_apply]
ac_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.MonoidLocalization.Maps | {
"line": 84,
"column": 4
} | {
"line": 89,
"column": 10
} | {
"line": 91,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\nf : S.LocalizationMap N\ng : M →* P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\nx y : N\n⊢ g (f.sec (x * y)).1 * ↑((IsUnit.liftRight (g.restrict S) hg) (f.sec (x * y)).2)⁻¹ =\n g (f.se... | [] | rw [mul_inv_left hg, ← mul_assoc, ← mul_assoc, mul_inv_right hg, mul_comm _ (g (f.sec y).1), ←
mul_assoc, ← mul_assoc, mul_inv_right hg]
repeat rw [← g.map_mul]
refine f.eq_of_eq hg ?_
simp_rw [map_mul, sec_spec', ← toMonoidHom_apply]
ac_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.MonoidLocalization.Maps | {
"line": 208,
"column": 26
} | {
"line": 208,
"column": 36
} | {
"line": 208,
"column": 37
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝² : CommMonoid N\nf : S.LocalizationMap N\nT : Submonoid M\nhST : S ≤ T\nQ : Type u_4\ninst✝¹ : CommMonoid Q\nk : T.LocalizationMap Q\nA : Type u_5\ninst✝ : CommMonoid A\nl : M →* A\nhl : ∀ (w : ↥T), IsUnit (l ↑w)\nx : M\n⊢ ((k.li... | [
"M : Type u_1\ninst✝³ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝² : CommMonoid N\nf : S.LocalizationMap N\nT : Submonoid M\nhST : S ≤ T\nQ : Type u_4\ninst✝¹ : CommMonoid Q\nk : T.LocalizationMap Q\nA : Type u_5\ninst✝ : CommMonoid A\nl : M →* A\nhl : ∀ (w : ↥T), IsUnit (l ↑w)\nx : M\n⊢ ((k.lift hl).comp ... | lift_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 282,
"column": 21
} | {
"line": 282,
"column": 55
} | {
"line": 282,
"column": 56
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\np : Sort u\nx : Localization S\nf : M → ↥S → p\nH : ∀ {a c : M} {b d : ↥S}, (r S) (a, b) (c, d) → f a b = f c d\na✝ c✝ : M\nb✝ d✝ : ↥S\nh : (r S) (a✝, b✝) (c✝, d✝)\n⊢ ⋯ ▸ f a✝ ... | [
"M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type u_3\ninst✝ : CommMonoid P\np : Sort u\nx : Localization S\nf : M → ↥S → p\nH : ∀ {a c : M} {b d : ↥S}, (r S) (a, b) (c, d) → f a b = f c d\na✝ c✝ : M\nb✝ d✝ : ↥S\nh : (r S) (a✝, b✝) (c✝, d✝)\n⊢ f a✝ b✝ = f c✝ d✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 25
} | {
"line": 452,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\nz w : N\n⊢ ∃ z' w' d, z * f ↑d = f z' ∧ w * f ↑d = f w'",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"Member... | [
"M : Type u_1\ninst✝¹ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\nz w : N\na : M × ↥S\nha : z * f ↑a.2 = f a.1\n⊢ ∃ z' w' d, z * f ↑d = f z' ∧ w * f ↑d = f w'"
] | let ⟨a, ha⟩ := surj f z | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 387,
"column": 60
} | {
"line": 387,
"column": 68
} | {
"line": 387,
"column": 68
} | [
{
"pp": "case c.c.c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ * (r₂ /ₒ s₂) = ra * r₂ /ₒ (sa * s₁)\n⊢ (r₁ /ₒ s₁ * (r₂ /ₒ s₂)) • (r₃ /ₒ s₃) ... | [
"case c.c.c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ * (r₂ /ₒ s₂) = ra * r₂ /ₒ (sa * s₁)\n⊢ (ra * r₂ /ₒ (sa * s₁)) • (r₃ /ₒ s₃) = (r₁ /ₒ s₁) ... | rw [ha'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 569,
"column": 2
} | {
"line": 569,
"column": 19
} | {
"line": 570,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\nx : M\n⊢ f.mk' x 1 = f x",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"IsUnit.liftRight",
"Units.val",
"Eq.mpr",
"MonoidHom.instMonoidHomCl... | [
"M : Type u_1\ninst✝¹ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\nx : M\n⊢ f x * ↑1⁻¹ = f x"
] | rw [mk', map_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 439,
"column": 88
} | {
"line": 440,
"column": 49
} | {
"line": 442,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nX : Type u_2\ninst✝ : MulAction R X\nr : X\ns t : ↥S\n⊢ (↑s /ₒ t) • (r /ₒ s) = r /ₒ t",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"OreLocalization.instSMul",
"MulOne.toOne",
"instHSMul",
... | [] | by
simp [oreDiv_smul_char s.1 r t s 1 1 (by simp)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 146,
"column": 43
} | {
"line": 146,
"column": 80
} | {
"line": 146,
"column": 81
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : DecidableEq α\nR : Type u_2\nS : Type u_3\ninst✝⁵ : MulZeroOneClass R\ninst✝⁴ : DecidableEq R\ninst✝³ : Nontrivial R\ninst✝² : MulZeroOneClass S\ninst✝¹ : DecidableEq S\ninst✝ : Nontrivial S\nf : R →*₀ S\nhf : Function.Injective ⇑f\nthis : ∀ (z : WithTop R), map (⇑f) z = 0 ↔ z = ... | [
"α : Type u_1\ninst✝⁶ : DecidableEq α\nR : Type u_2\nS : Type u_3\ninst✝⁵ : MulZeroOneClass R\ninst✝⁴ : DecidableEq R\ninst✝³ : Nontrivial R\ninst✝² : MulZeroOneClass S\ninst✝¹ : DecidableEq S\ninst✝ : Nontrivial S\nf : R →*₀ S\nhf : Function.Injective ⇑f\nthis : ∀ (z : WithTop R), map (⇑f) z = 0 ↔ z = 0\nx : R\nhx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 651,
"column": 23
} | {
"line": 651,
"column": 50
} | {
"line": 651,
"column": 51
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\na₁ b₁ : M\na₂ b₂ : ↥S\nH : b₁ * ↑a₂ = a₁ * ↑b₂\n⊢ ↑a₂ * b₁ = ↑b₂ * a₁",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommMo... | [
"M : Type u_1\ninst✝¹ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoid N\nf : S.LocalizationMap N\na₁ b₁ : M\na₂ b₂ : ↥S\nH : b₁ * ↑a₂ = a₁ * ↑b₂\n⊢ b₁ * ↑a₂ = a₁ * ↑b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 513,
"column": 62
} | {
"line": 513,
"column": 70
} | {
"line": 513,
"column": 70
} | [
{
"pp": "case c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type ?u.10\ninst✝¹ : MulAction R X\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ ... | [
"case c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type ?u.10\ninst✝¹ : MulAction R X\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\nha' : r₁ /ₒ s₁ * (r₂ /ₒ s₂)... | rw [ha'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 514,
"column": 4
} | {
"line": 517,
"column": 56
} | {
"line": 519,
"column": 0
} | [
{
"pp": "case c.c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type ?u.10\ninst✝¹ : MulAction R X\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * r₁ = ra * ↑s₂\n⊢ liftExpand (f... | [] | rw [liftExpand_of, liftExpand_of, liftExpand_of, Units.inv_mul_eq_iff_eq_mul, map_mul, map_mul,
Units.val_mul, mul_assoc, ← mul_assoc (fS s₁ : T), ← mul_assoc (fS s₁ : T), Units.mul_inv,
one_mul, ← hf, ← mul_assoc, ← map_mul _ _ r₁, ha, map_mul, hf s₂, mul_assoc,
← mul_assoc (fS s₂ : T), (fS s₂).mul_i... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.MonoidLocalization.Basic | {
"line": 778,
"column": 2
} | {
"line": 778,
"column": 42
} | {
"line": 778,
"column": 43
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Submonoid α\nb : ↥s\nc d : α\nh : (fun a ↦ mk a b) c = (fun a ↦ mk a b) d\n⊢ c = d",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : CommMonoid α\ninst✝ : IsCancelMul α\ns : Submonoid α\nb : ↥s\nc d : α\nh : (fun a ↦ mk a b) c = (fun a ↦ mk a b) d\n⊢ c = d"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 555,
"column": 4
} | {
"line": 556,
"column": 53
} | {
"line": 556,
"column": 53
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' ... | [] | rw [← mul_one (oreDenom (c • 1) s), ← oreDiv_smul_oreDiv, ← mul_one (oreDenom (c • 1) _),
← oreDiv_smul_oreDiv, ← OreLocalization.expand] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 555,
"column": 4
} | {
"line": 556,
"column": 53
} | {
"line": 556,
"column": 53
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' ... | [] | rw [← mul_one (oreDenom (c • 1) s), ← oreDiv_smul_oreDiv, ← mul_one (oreDenom (c • 1) _),
← oreDiv_smul_oreDiv, ← OreLocalization.expand] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 555,
"column": 4
} | {
"line": 556,
"column": 53
} | {
"line": 556,
"column": 53
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nM : Type u_3\nX : Type u_4\ninst✝¹² : Monoid M\nS : Submonoid M\ninst✝¹¹ : OreSet S\ninst✝¹⁰ : MulAction M X\ninst✝⁹ : SMul R X\ninst✝⁸ : SMul R M\ninst✝⁷ : IsScalarTower R M M\ninst✝⁶ : IsScalarTower R M X\ninst✝⁵ : SMul R' X\ninst✝⁴ : SMul R' M\ninst✝³ : IsScalarTower R' ... | [] | rw [← mul_one (oreDenom (c • 1) s), ← oreDiv_smul_oreDiv, ← mul_one (oreDenom (c • 1) _),
← oreDiv_smul_oreDiv, ← OreLocalization.expand] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 269,
"column": 20
} | {
"line": 269,
"column": 45
} | {
"line": 269,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\nb₂ : WithTop α\nthis : MulPosStrictMono α\na₁ b₁ : α\nhb : ↑b₁ < b₂\nha : ↑a₁ < ⊤\n⊢ b₂ ≠ 0",
"ppTerm": "?m.124",
"assigned"... | [
"α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\nb₂ : WithTop α\nthis : MulPosStrictMono α\na₁ b₁ : α\nhb : ↑b₁ < b₂\nha : ↑a₁ < ⊤\n⊢ ¬b₂ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Iterate | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 31
} | {
"line": 126,
"column": 32
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\nf : α → α\nh : id ≤ f\nn : ℕ\n⊢ id ≤ f^[n]",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : Preorder α\nf : α → α\nh : id ≤ f\nn : ℕ\n⊢ id ≤ f^[n]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Ring.WithTop | {
"line": 272,
"column": 20
} | {
"line": 272,
"column": 45
} | {
"line": 272,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\na₂ : WithTop α\nthis : MulPosStrictMono α\na₁ : α\nha : ↑a₁ < a₂\nb₁ : α\nha₂ : a₂ ≠ ⊤\nhb : ↑b₁ < ⊤\n⊢ a₂ ≠ 0",
"ppTerm": "?m.1... | [
"α : Type u_1\ninst✝⁵ : DecidableEq α\ninst✝⁴ : CommSemiring α\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : CanonicallyOrderedAdd α\ninst✝ : PosMulStrictMono α\na₂ : WithTop α\nthis : MulPosStrictMono α\na₁ : α\nha : ↑a₁ < a₂\nb₁ : α\nha₂ : a₂ ≠ ⊤\nhb : ↑b₁ < ⊤\n⊢ ¬a₂ = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Iterate | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 27
} | {
"line": 188,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nf g : α → α\nh : Commute f g\nhf : Monotone f\nhg : StrictMono g\nx : α\nn : ℕ\nhn : 0 < n\n⊢ f^[n] x ≤ g^[n] x ↔ f x ≤ g x",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : LinearOrder α\nf g : α → α\nh : Commute f g\nhf : Monotone f\nhg : StrictMono g\nx : α\nn : ℕ\nhn : 0 < n\n⊢ f^[n] x ≤ g^[n] x ↔ f x ≤ g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Iterate | {
"line": 192,
"column": 2
} | {
"line": 192,
"column": 27
} | {
"line": 192,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\nf g : α → α\nh : Commute f g\nhf : StrictMono f\nhg : Monotone g\nx : α\nn : ℕ\nhn : 0 < n\n⊢ f^[n] x ≤ g^[n] x ↔ f x ≤ g x",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : LinearOrder α\nf g : α → α\nh : Commute f g\nhf : StrictMono f\nhg : Monotone g\nx : α\nn : ℕ\nhn : 0 < n\n⊢ f^[n] x ≤ g^[n] x ↔ f x ≤ g x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.WithBot | {
"line": 40,
"column": 52
} | {
"line": 40,
"column": 63
} | {
"line": 40,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : OrderBot α\ninst✝ : SuccOrder α\na b : α\nhab : ↑a ≤ ↑b\n⊢ a ≤ b",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : OrderBot α\ninst✝ : SuccOrder α\na b : α\nhab : ↑a ≤ ↑b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.WithBot | {
"line": 44,
"column": 52
} | {
"line": 44,
"column": 63
} | {
"line": 44,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\na b : α\nhab : ↑a < ↑b\n⊢ a < b",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderBot α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\na b : α\nhab : ↑a < ↑b\n⊢ a < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.WithBot | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 65
} | {
"line": 57,
"column": 66
} | [
{
"pp": "case coe\nα : Type u_2\ninst✝³ : Nontrivial α\ninst✝² : LinearOrder α\ninst✝¹ : OrderBot α\ninst✝ : SuccOrder α\na✝ : α\n⊢ (↑a✝).succ = ⊥ ↔ ↑a✝ = ⊥",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"WithBot.some",
"WithBot",
"Order.suc... | [
"case coe\nα : Type u_2\ninst✝³ : Nontrivial α\ninst✝² : LinearOrder α\ninst✝¹ : OrderBot α\ninst✝ : SuccOrder α\na✝ : α\n⊢ ¬Order.succ a✝ = ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.WithBot | {
"line": 81,
"column": 52
} | {
"line": 81,
"column": 63
} | {
"line": 81,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : OrderTop α\ninst✝ : PredOrder α\na b : α\nhab : ↑a ≤ ↑b\n⊢ a ≤ b",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : OrderTop α\ninst✝ : PredOrder α\na b : α\nhab : ↑a ≤ ↑b\n⊢ a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.WithBot | {
"line": 85,
"column": 52
} | {
"line": 85,
"column": 63
} | {
"line": 85,
"column": 64
} | [
{
"pp": "α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderTop α\ninst✝¹ : PredOrder α\ninst✝ : NoMinOrder α\na b : α\nhab : ↑a < ↑b\n⊢ a < b",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝³ : Preorder α\ninst✝² : OrderTop α\ninst✝¹ : PredOrder α\ninst✝ : NoMinOrder α\na b : α\nhab : ↑a < ↑b\n⊢ a < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Basic | {
"line": 106,
"column": 74
} | {
"line": 106,
"column": 85
} | {
"line": 106,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\nsucc : α → α\nhn : ∀ {a : α}, ¬IsMax a → ∀ (b : α), a < b ↔ succ a ≤ b\nhm : ∀ (a : α), IsMax a → succ a = a\na : α\nh : ¬IsMax a\n⊢ a < succ a",
"ppTerm": "?m.66",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\nsucc : α → α\nhn : ∀ {a : α}, ¬IsMax a → ∀ (b : α), a < b ↔ succ a ≤ b\nhm : ∀ (a : α), IsMax a → succ a = a\na : α\nh : ¬IsMax a\n⊢ a < succ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Basic | {
"line": 107,
"column": 50
} | {
"line": 107,
"column": 61
} | {
"line": 107,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\nsucc : α → α\nhn : ∀ {a : α}, ¬IsMax a → ∀ (b : α), a < b ↔ succ a ≤ b\nhm : ∀ (a : α), IsMax a → succ a = a\na : α\nh : ¬IsMax a\n⊢ ¬succ a ≤ a",
"ppTerm": "?m.79",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT"... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\nsucc : α → α\nhn : ∀ {a : α}, ¬IsMax a → ∀ (b : α), a < b ↔ succ a ≤ b\nhm : ∀ (a : α), IsMax a → succ a = a\na : α\nh : ¬IsMax a\n⊢ a < succ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Basic | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 23
} | {
"line": 200,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nh : a ≤ b\n⊢ succ a ≤ succ b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Order.succ",
"Classical.propDecidable",
"Preorder.toLE",
"LE.le",
"dite",
"Not",
"IsMax"
],
... | [
"case pos\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nh : a ≤ b\nhb : IsMax b\n⊢ succ a ≤ succ b",
"case neg\nα : Type u_1\ninst✝¹ : Preorder α\ninst✝ : SuccOrder α\na b : α\nh : a ≤ b\nhb : ¬IsMax b\n⊢ succ a ≤ succ b"
] | by_cases hb : IsMax b | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Order.SuccPred.Basic | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 28
} | {
"line": 334,
"column": 29
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : SuccOrder α\na b : α\nha : IsMax a\n⊢ a ≤ b ↔ a = b ∨ succ a ≤ b",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Order.succ",
"congrArg",
"iff_or_self._simp_1",
"PartialOrder.toPreord... | [
"case pos\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : SuccOrder α\na b : α\nha : IsMax a\n⊢ a = b → a ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Basic | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 23
} | {
"line": 469,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\n⊢ a ≤ succ b ↔ a = succ b ∨ a ≤ b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Order.succ",
"PartialOrder.toPreorder",
"Classical.propDecidable",
"Preorder.toLE",
"SemilatticeInf.... | [
"case pos\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nhb : IsMax b\n⊢ a ≤ succ b ↔ a = succ b ∨ a ≤ b",
"case neg\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nhb : ¬IsMax b\n⊢ a ≤ succ b ↔ a = succ b ∨ a ≤ b"
] | by_cases hb : IsMax b | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Order.SuccPred.Basic | {
"line": 698,
"column": 8
} | {
"line": 698,
"column": 19
} | {
"line": 698,
"column": 20
} | [
{
"pp": "case zero\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nhn : ¬IsMax (succ^[0 - 1] i) → pred^[0] (succ^[0] i) = i\nhin : ¬IsMax (succ^[0] i)\n⊢ ¬IsMax (succ^[0 - 1] i)",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case zero\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nhn : ¬IsMax (succ^[0 - 1] i) → pred^[0] (succ^[0] i) = i\nhin : ¬IsMax (succ^[0] i)\n⊢ ∃ b, i < b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.SuccPred.Basic | {
"line": 753,
"column": 6
} | {
"line": 753,
"column": 38
} | {
"line": 754,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : (a : α) → Decidable (succ a = a)\na✝ : α\nha' : succ a✝ = a✝\nha : ⊤ ≤ ↑a✝\n⊢ IsMax ↑a✝",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"WithTop.instPreorder",
"False.elim",
... | [] | exact (not_top_le_coe _ ha).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.SuccPred.Basic | {
"line": 753,
"column": 6
} | {
"line": 753,
"column": 38
} | {
"line": 754,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : (a : α) → Decidable (succ a = a)\na✝ : α\nha' : succ a✝ = a✝\nha : ⊤ ≤ ↑a✝\n⊢ IsMax ↑a✝",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"WithTop.instPreorder",
"False.elim",
... | [] | exact (not_top_le_coe _ ha).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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