module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.List.Pairwise | {
"line": 71,
"column": 16
} | {
"line": 71,
"column": 44
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case a\nα : Type u_1\nR : α → α → Prop\nl : List α\ninst✝ : Std.Refl R\nh : ∀ (a : α), a ∈ l → ∀ (b : α), b ∈ l → a ≠ b → R a b\na b : α\nhab : [a, b] <+ l\nheq : ¬a = b\n⊢ a ≠ b",
"ppTerm": "?a✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case a\nα : Type u_1\nR : α → α → Prop\nl : List α\ninst✝ : Std.Refl R\nh : ∀ (a : α), a ∈ l → ∀ (b : α), b ∈ l → a ≠ b → R a b\na b : α\nhab : [a, b] <+ l\nheq : ¬a = b\n⊢ a ≠ b"
] | try (apply hab.subset; simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticTry__1 | Lean.Parser.Tactic.tacticTry_ |
Mathlib.Data.List.Pairwise | {
"line": 139,
"column": 33
} | {
"line": 139,
"column": 44
} | {
"line": 139,
"column": 45
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\ninst✝ : DecidableRel R\nl : List α\nh✝ : Pairwise R l\na b : α\nh : R a b\n⊢ Decidable.decide (R a b) = true",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"id",
"Bool.true",
"Bool",
"decide_eq_true_eq",
... | [
"α : Type u_1\nR : α → α → Prop\ninst✝ : DecidableRel R\nl : List α\nh✝ : Pairwise R l\na b : α\nh : R a b\n⊢ R a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Multiset.Defs | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 33
} | {
"line": 62,
"column": 2
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\ns : Multiset M\n⊢ s.toList.prod = s.prod",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Multiset.coe_toList",
"Monoid.toMulOneClass",
"congrArg",
"Multiset.prod",
"Eq.rec",
... | [
"M : Type u_3\ninst✝ : CommMonoid M\ns : Multiset M\n⊢ s.toList.prod = (↑s.toList).prod"
] | conv_rhs => rw [← coe_toList s] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Algebra.BigOperators.Group.List.Lemmas | {
"line": 107,
"column": 2
} | {
"line": 107,
"column": 13
} | {
"line": 107,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\nl : List α\n⊢ (map (fun x ↦ count x l) l.dedup).sum = l.length",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : DecidableEq α\nl : List α\n⊢ (map (fun x ↦ count x l) l.dedup).sum = l.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Chain | {
"line": 296,
"column": 4
} | {
"line": 296,
"column": 53
} | {
"line": 296,
"column": 54
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\nl₁ l₂ l₃ : List α\nh₁ : IsChain R (l₁ ++ l₂)\nh₂ : IsChain R (l₂ ++ l₃)\nhn : l₂ ≠ []\n⊢ ∀ (x : α), x ∈ (l₁ ++ l₂).getLast? → ∀ (y : α), y ∈ l₃.head? → R x y",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"List.head?",
"Eq.mpr",
"... | [
"α : Type u_1\nR : α → α → Prop\nl₁ l₂ l₃ : List α\nh₁ : IsChain R (l₁ ++ l₂)\nh₂ : IsChain R (l₂ ++ l₃)\nhn : l₂ ≠ []\n⊢ ∀ (x : α), x ∈ l₂.getLast? → ∀ (y : α), y ∈ l₃.head? → R x y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 13
} | {
"line": 256,
"column": 14
} | [
{
"pp": "α : Type u\nl : List α\nn : ℕ\nk : Fin l.length\n⊢ l.get k = (l.rotate n).get ⟨(l.length - n % l.length + ↑k) % l.length, ⋯⟩",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"congrArg",
"List.get",
"GetElem.getElem.congr_simp",... | [
"α : Type u\nl : List α\nn : ℕ\nk : Fin l.length\n⊢ l[↑k] = l[(l.length - n % l.length + ↑k + n) % l.length]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.BigOperators.Group.Multiset.Basic | {
"line": 148,
"column": 41
} | {
"line": 148,
"column": 52
} | {
"line": 148,
"column": 53
} | [
{
"pp": "M : Type u_5\ninst✝ : CommMonoid M\ns : Multiset M\na✝ : M\nl : List M\na : M\nh : a ∈ ⟦l⟧\n⊢ a ∣ prod ⟦l⟧",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Dvd.dvd",
"Multiset.prod",
"semigroupDvd",
"id",
"Quotient.mk",
"List",
"CommMonoid... | [
"M : Type u_5\ninst✝ : CommMonoid M\ns : Multiset M\na✝ : M\nl : List M\na : M\nh : a ∈ ⟦l⟧\n⊢ a ∣ l.prod"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Bind | {
"line": 89,
"column": 23
} | {
"line": 89,
"column": 34
} | {
"line": 89,
"column": 35
} | [
{
"pp": "case cons\nα : Type u_1\nβ : Type v\nr : α → β → Prop\ns : Multiset (Multiset α)\nt : Multiset (Multiset β)\na✝ : Multiset α\nb✝ : Multiset β\nas✝ : Multiset (Multiset α)\nbs✝ : Multiset (Multiset β)\nhab : Rel r a✝ b✝\nhst : Rel (Rel r) as✝ bs✝\nih : Rel r as✝.join bs✝.join\n⊢ Rel r (a✝ ::ₘ as✝).join ... | [
"case cons\nα : Type u_1\nβ : Type v\nr : α → β → Prop\ns : Multiset (Multiset α)\nt : Multiset (Multiset β)\na✝ : Multiset α\nb✝ : Multiset β\nas✝ : Multiset (Multiset α)\nbs✝ : Multiset (Multiset β)\nhab : Rel r a✝ b✝\nhst : Rel (Rel r) as✝ bs✝\nih : Rel r as✝.join bs✝.join\n⊢ Rel r (a✝ + as✝.join) (b✝ + bs✝.join... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 341,
"column": 2
} | {
"line": 342,
"column": 42
} | {
"line": 342,
"column": 43
} | [
{
"pp": "α : Type u\nl : List α\nhl : l.Nodup\nhn : l ≠ []\ni j : ℕ\nh : l.rotate (i % l.length) = l.rotate (j % l.length)\n⊢ i % l.length = j % l.length",
"ppTerm": "?m.28",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nl : List α\nhl : l.Nodup\nhn : l ≠ []\ni j : ℕ\nh : l.rotate (i % l.length) = l.rotate (j % l.length)\n⊢ i % l.length = j % l.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 415,
"column": 21
} | {
"line": 415,
"column": 32
} | {
"line": 415,
"column": 33
} | [
{
"pp": "α : Type u\nl : List α\nx✝ : l ~r []\nn : ℕ\nhn : l.rotate n = []\n⊢ l = []",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nl : List α\nx✝ : l ~r []\nn : ℕ\nhn : l.rotate n = []\n⊢ l = []"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 423,
"column": 21
} | {
"line": 423,
"column": 32
} | {
"line": 423,
"column": 33
} | [
{
"pp": "α : Type u\nl : List α\nx : α\nx✝ : l ~r [x]\nn : ℕ\nhn : l.rotate n = [x]\n⊢ l = [x]",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nl : List α\nx : α\nx✝ : l ~r [x]\nn : ℕ\nhn : l.rotate n = [x]\n⊢ l = [x]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 17
} | {
"line": 442,
"column": 18
} | [
{
"pp": "case mp\nα : Type u\nl l' : List α\nh : l.reverse ~r l'\n⊢ l ~r l'.reverse",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mp\nα : Type u\nl l' : List α\nh : l.reverse ~r l'\n⊢ l ~r l'.reverse"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 17
} | {
"line": 442,
"column": 18
} | [
{
"pp": "case mpr\nα : Type u\nl l' : List α\nh : l ~r l'.reverse\n⊢ l.reverse ~r l'",
"ppTerm": "?mpr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case mpr\nα : Type u\nl l' : List α\nh : l ~r l'.reverse\n⊢ l.reverse ~r l'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 13
} | {
"line": 470,
"column": 14
} | [
{
"pp": "α : Type u\nl : List α\na : α\n⊢ a :: l ~r l ++ [a]",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nl : List α\na : α\n⊢ a :: l ~r l ++ [a]"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 47
} | {
"line": 478,
"column": 4
} | [
{
"pp": "case append_singleton\nα : Type u\na : List α\nL : α\na✝ : ∀ (hL : a ≠ []), a.getLast hL :: a.dropLast ~r a\nhL : a ++ [L] ≠ []\n⊢ (a ++ [L]).getLast hL :: (a ++ [L]).dropLast ~r a ++ [L]",
"ppTerm": "?append_singleton",
"assigned": true,
"usedConstants": [
"List.getLast",
"Eq.m... | [
"case append_singleton\nα : Type u\na : List α\nL : α\na✝ : ∀ (hL : a ≠ []), a.getLast hL :: a.dropLast ~r a\nhL : a ++ [L] ≠ []\n⊢ (if h' : [L].isEmpty = true then a.getLast ⋯ else [L].getLast ⋯) :: a ~r a ++ [L]"
] | simp only [getLast_append, dropLast_concat] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.Rotate | {
"line": 527,
"column": 18
} | {
"line": 527,
"column": 29
} | {
"line": 527,
"column": 30
} | [
{
"pp": "α : Type u\na : α\nl : List α\nh : (a :: l).cyclicPermutations = []\n⊢ False",
"ppTerm": "?m.10",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\na : α\nl : List α\nh : (a :: l).cyclicPermutations = []\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 537,
"column": 39
} | {
"line": 537,
"column": 50
} | {
"line": 537,
"column": 51
} | [
{
"pp": "α : Type u\nn : ℕ\na : α\nl : List α\nh : n < (a :: l).cyclicPermutations.length\n⊢ n ≤ (a :: l).length",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"id",
"LE.le",
"instLENat",
"List.cons",
"Nat",
"List.length"
],
"usedFVars": [
... | [
"α : Type u\nn : ℕ\na : α\nl : List α\nh : n < (a :: l).cyclicPermutations.length\n⊢ n ≤ l.length + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 554,
"column": 2
} | {
"line": 554,
"column": 13
} | {
"line": 554,
"column": 14
} | [
{
"pp": "α : Type u\nl l' : List α\nh : l.cyclicPermutations = l'.cyclicPermutations\n⊢ l = l'",
"ppTerm": "?m.6",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nl l' : List α\nh : l.cyclicPermutations = l'.cyclicPermutations\n⊢ l = l'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 567,
"column": 2
} | {
"line": 567,
"column": 13
} | {
"line": 567,
"column": 14
} | [
{
"pp": "α : Type u\nl : List α\n⊢ l ∈ l.cyclicPermutations",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nl : List α\n⊢ l ∈ l.cyclicPermutations"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 54
} | {
"line": 607,
"column": 55
} | [
{
"pp": "case inr\nα : Type u\nl : List α\nhn : l.Nodup\nhl : l ≠ []\ni : ℕ\nhi✝ : i < l.cyclicPermutations.length\nj : ℕ\nhj✝ : j < l.cyclicPermutations.length\nh : l.cyclicPermutations.get ⟨i, hi✝⟩ = l.cyclicPermutations.get ⟨j, hj✝⟩\nhi : i < l.length\nhj : j < l.length\n⊢ ⟨i, hi✝⟩ = ⟨j, hj✝⟩",
"ppTerm":... | [
"case inr\nα : Type u\nl : List α\nhn : l.Nodup\nhl : l ≠ []\ni : ℕ\nhi✝ : i < l.cyclicPermutations.length\nj : ℕ\nhj✝ : j < l.cyclicPermutations.length\nh : l.cyclicPermutations.get ⟨i, hi✝⟩ = l.cyclicPermutations.get ⟨j, hj✝⟩\nhi : i < l.length\nhj : j < l.length\n⊢ i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.List.Rotate | {
"line": 612,
"column": 2
} | {
"line": 612,
"column": 20
} | {
"line": 614,
"column": 0
} | [
{
"pp": "α : Type u\nl : List α\nk : ℕ\n⊢ l.cyclicPermutations ~r (l.rotate k).cyclicPermutations",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"congrArg",
"List.cyclicPermutations",
"List",
"Nat",
"True",
"eq_self",
"Exists.intro",
"List.c... | [] | exact ⟨k, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.Union | {
"line": 109,
"column": 18
} | {
"line": 109,
"column": 29
} | {
"line": 109,
"column": 30
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nb : α\n⊢ b ∈ t.disjiUnion (fun a ↦ {x ∈ s | f x = a}) ⋯ ↔ b ∈ {c ∈ s | f c ∈ t}",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_filter._simp_1",
"congr... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nb : α\n⊢ f b ∈ t ∧ b ∈ s ↔ b ∈ s ∧ f b ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Multiset.Bind | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 29
} | {
"line": 411,
"column": 30
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nσ : α → Type u_5\nt : (a : α) → Multiset (σ a)\nl₁ : List α\nf : (a : α) → List (σ a)\nhf : ∀ (a : α), ⟦f a⟧ = t a\n⊢ Nodup (Quot.mk (⇑(List.isSetoid α)) l₁) →\n (∀ (a : α), (t a).Nodup) → (Multiset.sigma (Quot.mk (⇑(List.isSetoid α)) l₁) t).Nodup",
"ppTerm": "?m.21... | [
"α : Type u_1\ns : Multiset α\nσ : α → Type u_5\nt : (a : α) → Multiset (σ a)\nl₁ : List α\nf : (a : α) → List (σ a)\nhf : ∀ (a : α), ⟦f a⟧ = t a\n⊢ l₁.Nodup → (∀ (a : α), (f a).Nodup) → (l₁.sigma f).Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Union | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 41
} | {
"line": 162,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Finset β\nt : Finset α\nf : α → Finset β\nhf : (↑t).PairwiseDisjoint f\n⊢ Disjoint s (t.disjiUnion f hf) ↔ ∀ i ∈ t, Disjoint s (f i)",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Finset β\nt : Finset α\nf : α → Finset β\nhf : (↑t).PairwiseDisjoint f\n⊢ Disjoint s (t.disjiUnion f hf) ↔ ∀ i ∈ t, Disjoint s (f i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Union | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 41
} | {
"line": 281,
"column": 42
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\ns : Finset β\nt : Finset α\nf : α → Finset β\n⊢ Disjoint s (t.biUnion f) ↔ ∀ i ∈ t, Disjoint s (f i)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\ns : Finset β\nt : Finset α\nf : α → Finset β\n⊢ Disjoint s (t.biUnion f) ↔ ∀ i ∈ t, Disjoint s (f i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Prod | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 25
} | {
"line": 113,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf : α ↪ β\ng : γ ↪ δ\ns : Finset α\nt : Finset γ\n⊢ map (f.prodMap g) (s ×ˢ t) = map f s ×ˢ map g t",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
"congrArg",... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf : α ↪ β\ng : γ ↪ δ\ns : Finset α\nt : Finset γ\n⊢ (fun a ↦ Prod.map (⇑f) (⇑g) a) '' ↑s ×ˢ ↑t = (⇑f '' ↑s) ×ˢ (⇑g '' ↑t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Prod | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 13
} | {
"line": 154,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Finset α\nt : Finset β\np : α → Prop\ninst✝ : DecidablePred p\n⊢ {x ∈ s ×ˢ t | p x.1} = filter p s ×ˢ t",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Finset α\nt : Finset β\np : α → Prop\ninst✝ : DecidablePred p\n⊢ {x ∈ s ×ˢ t | p x.1} = filter p s ×ˢ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Prod | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 13
} | {
"line": 158,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns : Finset α\nt : Finset β\nq : β → Prop\ninst✝ : DecidablePred q\n⊢ {x ∈ s ×ˢ t | q x.2} = s ×ˢ filter q t",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\ns : Finset α\nt : Finset β\nq : β → Prop\ninst✝ : DecidablePred q\n⊢ {x ∈ s ×ˢ t | q x.2} = s ×ˢ filter q t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Prod | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 13
} | {
"line": 367,
"column": 14
} | [
{
"pp": "ι : Type u_4\ninst✝² : PartialOrder ι\ninst✝¹ : DecidableLE ι\ninst✝ : DecidableLT ι\ns : Finset ι\na✝ : ι × ι\n⊢ a✝ ∈ {i ∈ s.offDiag | i.1 < i.2} ↔ a✝ ∈ {i ∈ s.offDiag | i.1 ≤ i.2}",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_filter._simp_1",
... | [
"ι : Type u_4\ninst✝² : PartialOrder ι\ninst✝¹ : DecidableLE ι\ninst✝ : DecidableLT ι\ns : Finset ι\na✝ : ι × ι\n⊢ a✝.1 ∈ s → a✝.2 ∈ s → ¬a✝.1 = a✝.2 → (a✝.1 < a✝.2 ↔ a✝.1 ≤ a✝.2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fintype.Pi | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 22
} | {
"line": 210,
"column": 23
} | [
{
"pp": "ι : Type u_3\ninst✝ : Finite ι\nκ : ι → Type u_4\nt : (i : ι) → Set (κ i)\nht : ∀ (i : ι), (t i).Finite\n⊢ {f | ∀ (i : ι), f i ∈ t i}.Finite",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_3\ninst✝ : Finite ι\nκ : ι → Type u_4\nt : (i : ι) → Set (κ i)\nht : ∀ (i : ι), (t i).Finite\n⊢ {f | ∀ (i : ι), f i ∈ t i}.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Hom | {
"line": 51,
"column": 23
} | {
"line": 51,
"column": 44
} | {
"line": 51,
"column": 45
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nα : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulAction M α\ninst✝ : Monoid N\ng : N →* M\nx✝² x✝¹ : N\nx✝ : α\n⊢ (x✝² * x✝¹) • x✝ = x✝² • x✝¹ • x✝",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"... | [
"M : Type u_1\nN : Type u_2\nα : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulAction M α\ninst✝ : Monoid N\ng : N →* M\nx✝² x✝¹ : N\nx✝ : α\n⊢ SMul.smul (g x✝² * g x✝¹) x✝ = SMul.smul (g x✝²) (SMul.smul (g x✝¹) x✝)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Hom | {
"line": 50,
"column": 19
} | {
"line": 50,
"column": 40
} | {
"line": 50,
"column": 41
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nα : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulAction M α\ninst✝ : Monoid N\ng : N →* M\nx✝ : α\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"instHSMul",
... | [
"M : Type u_1\nN : Type u_2\nα : Type u_3\ninst✝² : Monoid M\ninst✝¹ : MulAction M α\ninst✝ : Monoid N\ng : N →* M\nx✝ : α\n⊢ SMul.smul 1 x✝ = x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Lattice.Image | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 25
} | {
"line": 103,
"column": 4
} | [
{
"pp": "ι : Type u_9\nα : ι → Type u_10\nv : (i : ι) → Set (α i)\nhv : (univ.pi v).Nonempty\ni : ι\n⊢ (fun x ↦ x i) '' ⋂ k, (fun x ↦ x k) ⁻¹' v k = v i",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.Subset.antisymm",
"Set.iInter",
"Set.preimage",
"Set.image"
... | [
"case h₁\nι : Type u_9\nα : ι → Type u_10\nv : (i : ι) → Set (α i)\nhv : (univ.pi v).Nonempty\ni : ι\n⊢ (fun x ↦ x i) '' ⋂ k, (fun x ↦ x k) ⁻¹' v k ⊆ v i",
"case h₂\nι : Type u_9\nα : ι → Type u_10\nv : (i : ι) → Set (α i)\nhv : (univ.pi v).Nonempty\ni : ι\n⊢ v i ⊆ (fun x ↦ x i) '' ⋂ k, (fun x ↦ x k) ⁻¹' v k"
] | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Set.Lattice.Image | {
"line": 109,
"column": 33
} | {
"line": 109,
"column": 44
} | {
"line": 109,
"column": 45
} | [
{
"pp": "ι : Type u_9\nα : ι → Type u_10\nv : (i : ι) → Set (α i)\ni : ι\ny : α i\ny_in : y ∈ v i\nz : (i : ι) → α i\nhz : z ∈ univ.pi v\nj : ι\nx✝ : j ≠ i\n⊢ z j ∈ v j",
"ppTerm": "?m.55",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_9\nα : ι → Type u_10\nv : (i : ι) → Set (α i)\ni : ι\ny : α i\ny_in : y ∈ v i\nz : (i : ι) → α i\nhz : z ∈ univ.pi v\nj : ι\nx✝ : j ≠ i\n⊢ z j ∈ v j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Lattice.Image | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 72
} | {
"line": 200,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nf : α → β\nU : ι → Set β\nhU : iUnion U = univ\n⊢ Surjective f ↔ ∀ (i : ι), Surjective ((U i).restrictPreimage f)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.restrictPreimage",
"Set.restrictPreimage_surjective",
... | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_5\nf : α → β\nU : ι → Set β\nhU : iUnion U = univ\nH : ∀ (i : ι), Surjective ((U i).restrictPreimage f)\nx : β\n⊢ ∃ a, f a = x"
] | refine ⟨fun H i => (U i).restrictPreimage_surjective H, fun H x => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Set.Lattice.Image | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 42
} | {
"line": 236,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ i, ⋃ (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n⊢ InjOn f (⋃ i, s ↑i ⋯)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"id",
"Subtype",
... | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_5\np : ι → Prop\ns : (i : ι) → p i → Set α\nhp : ∃ i, p i\nf : α → β\nh : InjOn f (⋃ i, ⋃ (hi : p i), s i hi)\nthis : Nonempty { i // p i }\n⊢ InjOn f (⨆ i, s ↑i ⋯)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Lattice.Image | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 37
} | {
"line": 252,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : ι → Set α\nhs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ i, s i\ny : α\nhy : y ∈ ⋃ i, s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\n⊢ x = y",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants"... | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : ι → Set α\nhs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\nf : α → β\nhf : ∀ (i : ι), InjOn f (s i)\nx : α\nhx✝ : x ∈ ⋃ i, s i\ny : α\nhy✝ : y ∈ ⋃ i, s i\nhxy : f x = f y\ni : ι\nhx : x ∈ s i\nj : ι\nhy : y ∈ s j\n⊢ x = y"
] | rcases mem_iUnion.1 hy with ⟨j, hy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Set.Lattice.Image | {
"line": 320,
"column": 55
} | {
"line": 320,
"column": 80
} | {
"line": 322,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nκ : ι → Sort u_8\nf : α → β\ns : (i : ι) → κ i → Set α\n⊢ f '' ⋃ i, ⋃ j, s i j = ⋃ i, ⋃ j, f '' s i j",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.image_iUnion",
"funext",
"True",
"eq_self"... | [] | by simp_rw [image_iUnion] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Lattice.Image | {
"line": 336,
"column": 54
} | {
"line": 336,
"column": 65
} | {
"line": 336,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nf : ι → α\ng : α → Set β\n⊢ ⋃ x, ⋃ y, ⋃ (_ : f y = x), g x = ⋃ y, g (f y)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_5\nf : ι → α\ng : α → Set β\n⊢ ⋃ x, ⋃ y, ⋃ (_ : f y = x), g x = ⋃ y, g (f y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Set.Lattice.Image | {
"line": 343,
"column": 54
} | {
"line": 343,
"column": 65
} | {
"line": 343,
"column": 66
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\nf : ι → α\ng : α → Set β\n⊢ ⋂ x, ⋂ y, ⋂ (_ : f y = x), g x = ⋂ y, g (f y)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_5\nf : ι → α\ng : α → Set β\n⊢ ⋂ x, ⋂ y, ⋂ (_ : f y = x), g x = ⋂ y, g (f y)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pi.Lemmas | {
"line": 452,
"column": 21
} | {
"line": 452,
"column": 32
} | {
"line": 452,
"column": 33
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nI : Type u\nf✝ : I → Type v\nM : ι → Type u_3\nN : ι → Type u_4\ni : I\nη : Type v\nR : Type w\ns : ι → η\ninst✝ : MulOneClass R\nf g : ι → R\n⊢ extend s (f * g) 1 = extend s f 1 * extend s g 1",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"use... | [
"ι : Type u_1\nα : Type u_2\nI : Type u\nf✝ : I → Type v\nM : ι → Type u_3\nN : ι → Type u_4\ni : I\nη : Type v\nR : Type w\ns : ι → η\ninst✝ : MulOneClass R\nf g : ι → R\n⊢ extend s (f * g) 1 = extend s f 1 * extend s g 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Basic | {
"line": 232,
"column": 27
} | {
"line": 232,
"column": 38
} | {
"line": 232,
"column": 39
} | [
{
"pp": "M : Type u_1\ninst✝ : MulOneClass M\ns : Set M\nx : M\nhx : x ∈ {1}\n⊢ x = 1",
"ppTerm": "?m.152",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝ : MulOneClass M\ns : Set M\nx : M\nhx : x ∈ {1}\n⊢ x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 264,
"column": 6
} | {
"line": 264,
"column": 40
} | {
"line": 264,
"column": 40
} | [
{
"pp": "α : Type u_2\ninst✝ : InvolutiveInv α\ns : Set α\np : α → Prop\n⊢ (∀ (x : α), x⁻¹ ∈ s → p x) ↔ ∀ (x : α), x ∈ s → p x⁻¹",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.inv",
"congrArg",
"InvolutiveInv.toInv",
... | [
"α : Type u_2\ninst✝ : InvolutiveInv α\ns : Set α\np : α → Prop\n⊢ (∀ (a : α), ((Equiv.inv α) a)⁻¹ ∈ s → p ((Equiv.inv α) a)) ↔ ∀ (x : α), x ∈ s → p x⁻¹"
] | ← (Equiv.inv _).forall_congr_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 662,
"column": 2
} | {
"line": 662,
"column": 13
} | {
"line": 662,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\ns : Set 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✝ : Monoid α\ns : Set 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.Set.Basic | {
"line": 691,
"column": 2
} | {
"line": 691,
"column": 13
} | {
"line": 691,
"column": 14
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\ns : Set 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✝ : Monoid α\ns : Set 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.Set.Basic | {
"line": 693,
"column": 64
} | {
"line": 693,
"column": 75
} | {
"line": 693,
"column": 76
} | [
{
"pp": "α : Type u_2\ninst✝ : Monoid α\ns : Set α\nn : ℕ\nhs : 1 ∈ s\n⊢ 1 ∈ s ^ n",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\ninst✝ : Monoid α\ns : Set α\nn : ℕ\nhs : 1 ∈ s\n⊢ 1 ∈ s ^ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 757,
"column": 19
} | {
"line": 757,
"column": 41
} | {
"line": 757,
"column": 42
} | [
{
"pp": "α : Type u_2\ninst✝ : CancelMonoid α\ns : Set 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",
"Monoid.toMulOneClass",
"c... | [
"α : Type u_2\ninst✝ : CancelMonoid α\ns : Set 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.Set.Basic | {
"line": 840,
"column": 21
} | {
"line": 840,
"column": 32
} | {
"line": 840,
"column": 33
} | [
{
"pp": "α : Type u_2\ninst✝ : DivisionMonoid α\ns : Set α\nhs : s.Nonempty\nn : ℕ\n⊢ (s ^ Int.negSucc n).Nonempty",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Set.ZPow",... | [
"α : Type u_2\ninst✝ : DivisionMonoid α\ns : Set α\nhs : s.Nonempty\nn : ℕ\n⊢ (s ^ (n + 1)).Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 1004,
"column": 16
} | {
"line": 1004,
"column": 38
} | {
"line": 1004,
"column": 39
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nf : F\ns : Set β\nn : ℕ\n⊢ (⇑f ⁻¹' s) ^ (n + 1) ⊆ ⇑f ⁻¹' s ^ (n + 1)",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Monoid α\ninst✝² : Monoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nf : F\ns : Set β\nn : ℕ\n⊢ (⇑f ⁻¹' s) ^ n * ⇑f ⁻¹' s ⊆ ⇑f ⁻¹' (s ^ n * s)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 217,
"column": 77
} | {
"line": 219,
"column": 59
} | {
"line": 221,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ H ≠ ⊥ ↔ ∃ a, a ≠ 1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Iff.of_eq",
"congrArg",
"_private.Mathlib.Al... | [] | by
rw [← nontrivial_iff_ne_bot, nontrivial_iff_exists_ne_one]
simp only [ne_eq, Subtype.exists, mk_eq_one, exists_prop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 487,
"column": 2
} | {
"line": 487,
"column": 31
} | {
"line": 487,
"column": 32
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx : G\n⊢ x ∈ closure {x}",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nx : G\n⊢ x ∈ closure {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 540,
"column": 6
} | {
"line": 540,
"column": 57
} | {
"line": 540,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Type u_2\np : ι → Prop\nK : ι → Subgroup G\ni : ι\nhp : p i\nhK : DirectedOn ((fun x1 x2 ↦ x1 ≤ x2) on K) {i | p i}\nx : G\nthis : x ∈ closure (⋃ i, ⋃ (_ : p i), ↑(K i)) → ∃ i, p i ∧ x ∈ K i\n⊢ x ∈ ⨆ i, ⨆ (_ : p i), K i → ∃ i, p i ∧ x ∈ K i",
"ppTerm": "?m.81",
... | [
"G : Type u_1\ninst✝ : Group G\nι : Type u_2\np : ι → Prop\nK : ι → Subgroup G\ni : ι\nhp : p i\nhK : DirectedOn ((fun x1 x2 ↦ x1 ≤ x2) on K) {i | p i}\nx : G\nthis : x ∈ closure (⋃ i, ⋃ (_ : p i), ↑(K i)) → ∃ i, p i ∧ x ∈ K i\n⊢ x ∈ ⨆ i, ⨆ (_ : p i), K i → ∃ i, p i ∧ x ∈ K i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 154,
"column": 17
} | {
"line": 154,
"column": 28
} | {
"line": 154,
"column": 29
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nx y : G\nh : x ∈ H\nhba : y * x ∈ H\n⊢ y ∈ H",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nx y : G\nh : x ∈ H\nhba : y * x ∈ H\n⊢ y ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 158,
"column": 17
} | {
"line": 158,
"column": 28
} | {
"line": 158,
"column": 29
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nx y : G\nh : x ∈ H\nhab : x * y ∈ H\n⊢ y ∈ H",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝² : Group G\nS : Type u_4\nH : S\ninst✝¹ : SetLike S G\ninst✝ : SubgroupClass S G\nx y : G\nh : x ∈ H\nhab : x * y ∈ H\n⊢ y ∈ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 420,
"column": 20
} | {
"line": 420,
"column": 36
} | {
"line": 420,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K✝ K : Subgroup G\ns : Set G\nhs : s = ↑K\nx✝ : G\nhx : x✝ ∈ s\n⊢ x✝⁻¹ ∈ s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Subgroup.instSubgroupClass",
... | [
"G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K✝ K : Subgroup G\ns : Set G\nhs : s = ↑K\nx✝ : G\nhx : x✝ ∈ s\n⊢ x✝ ∈ K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 15
} | {
"line": 482,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K : Subgroup G\ns : Set G\nhsn : s.Nonempty\nhs : ∀ (x : G), x ∈ s → ∀ (y : G), y ∈ s → x * y⁻¹ ∈ s\nx : G\nhx : x ∈ s\n⊢ 1 ∈ s",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoal... | [
"G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K : Subgroup G\ns : Set G\nhsn : s.Nonempty\nhs : ∀ (x : G), x ∈ s → ∀ (y : G), y ∈ s → x * y⁻¹ ∈ s\nx : G\nhx : x ∈ s\n⊢ 1 ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 483,
"column": 56
} | {
"line": 483,
"column": 67
} | {
"line": 483,
"column": 68
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K : Subgroup G\ns : Set G\nhsn : s.Nonempty\nhs : ∀ (x : G), x ∈ s → ∀ (y : G), y ∈ s → x * y⁻¹ ∈ s\none_mem : 1 ∈ s\nx : G\nhx : x ∈ s\n⊢ x⁻¹ ∈ s",
"ppTerm": "?m.77",
"assigned": false,
"usedConstants": [],
"usedFVars"... | [
"G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K : Subgroup G\ns : Set G\nhsn : s.Nonempty\nhs : ∀ (x : G), x ∈ s → ∀ (y : G), y ∈ s → x * y⁻¹ ∈ s\none_mem : 1 ∈ s\nx : G\nhx : x ∈ s\n⊢ x⁻¹ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 487,
"column": 32
} | {
"line": 487,
"column": 43
} | {
"line": 487,
"column": 44
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K : Subgroup G\ns : Set G\nhsn : s.Nonempty\nhs : ∀ (x : G), x ∈ s → ∀ (y : G), y ∈ s → x * y⁻¹ ∈ s\none_mem : 1 ∈ s\ninv_mem : ∀ (x : G), x ∈ s → x⁻¹ ∈ s\na✝ b✝ : G\nhx : a✝ ∈ s\nhy : b✝ ∈ s\n⊢ a✝ * b✝ ∈ s",
"ppTerm": "?m.89",
... | [
"G : Type u_1\ninst✝¹ : Group G\nA : Type u_2\ninst✝ : AddGroup A\nH K : Subgroup G\ns : Set G\nhsn : s.Nonempty\nhs : ∀ (x : G), x ∈ s → ∀ (y : G), y ∈ s → x * y⁻¹ ∈ s\none_mem : 1 ∈ s\ninv_mem : ∀ (x : G), x ∈ s → x⁻¹ ∈ s\na✝ b✝ : G\nhx : a✝ ∈ s\nhy : b✝ ∈ s\n⊢ a✝ * b✝ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 13
} | {
"line": 221,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nN : Type u_5\ninst✝¹ : Group N\nι : Sort u_7\ninst✝ : Nonempty ι\nf : G →* N\nhf : Injective ⇑f\ns : ι → Subgroup G\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"Monoi... | [
"G : Type u_1\ninst✝² : Group G\nN : Type u_5\ninst✝¹ : Group N\nι : Sort u_7\ninst✝ : Nonempty ι\nf : G →* N\nhf : Injective ⇑f\ns : ι → Subgroup G\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 351,
"column": 2
} | {
"line": 351,
"column": 81
} | {
"line": 351,
"column": 82
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH₁ H₂ K : Subgroup G\n⊢ H₁.subgroupOf K = H₂.subgroupOf K ↔ H₁ ⊓ K = H₂ ⊓ K",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.subgroupOf",
"congrArg",
"Membership.mem",
"id",
"Subtype",
"Sub... | [
"G : Type u_1\ninst✝ : Group G\nH₁ H₂ K : Subgroup G\n⊢ (∀ (x : ↥K), ↑x ∈ H₁ ↔ ↑x ∈ H₂) ↔ ∀ x ∈ K, x ∈ H₁ ↔ x ∈ H₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 379,
"column": 2
} | {
"line": 379,
"column": 23
} | {
"line": 379,
"column": 24
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\ninst✝⁴ : Group G''\nA : Type u_4\ninst✝³ : AddGroup A\nH✝ K : Subgroup G\nk : Set G\nN : Type u_5\ninst✝² : Group N\nP : Type u_6\ninst✝¹ : Group P\nH : Subgroup G\nf : G →* G'\ninst✝ : IsMulCommutative ↥H\na : G\nha : a ... | [
"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁶ : Group G\ninst✝⁵ : Group G'\ninst✝⁴ : Group G''\nA : Type u_4\ninst✝³ : AddGroup A\nH✝ K : Subgroup G\nk : Set G\nN : Type u_5\ninst✝² : Group N\nP : Type u_6\ninst✝¹ : Group P\nH : Subgroup G\nf : G →* G'\ninst✝ : IsMulCommutative ↥H\na : G\nha : a ∈ ↑H\nb : G\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 15
} | {
"line": 385,
"column": 16
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\ninst✝² : Group G\ninst✝¹ : Group G'\nH : Subgroup G\nf : G' →* G\nhf : Injective ⇑f\ninst✝ : IsMulCommutative ↥H\na : G'\nha : f a ∈ H\nb : G'\nhb : f b ∈ H\n⊢ f (a * b) = f (b * a)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"G : Type u_1\nG' : Type u_2\ninst✝² : Group G\ninst✝¹ : Group G'\nH : Subgroup G\nf : G' →* G\nhf : Injective ⇑f\ninst✝ : IsMulCommutative ↥H\na : G'\nha : f a ∈ H\nb : G'\nhb : f b ∈ H\n⊢ f a * f b = f b * f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 413,
"column": 17
} | {
"line": 413,
"column": 51
} | {
"line": 414,
"column": 6
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nH : Type u_5\ninst✝ : Group H\nf : G ≃* H\nsg1 sg2 : Subgroup H\nh :\n { toFun := Subgroup.comap ↑f, invFun := Subgroup.comap ↑f.symm, left_inv := ⋯, right_inv := ⋯ ... | [
"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁴ : Group G\ninst✝³ : Group G'\ninst✝² : Group G''\nA : Type u_4\ninst✝¹ : AddGroup A\nH : Type u_5\ninst✝ : Group H\nf : G ≃* H\nsg1 sg2 : Subgroup H\nh :\n { toFun := Subgroup.comap ↑f, invFun := Subgroup.comap ↑f.symm, left_inv := ⋯, right_inv := ⋯ } sg1 ≤\n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 606,
"column": 12
} | {
"line": 616,
"column": 81
} | {
"line": 616,
"column": 81
} | [
{
"pp": "C : Type u_2\ninst✝ : CommGroup C\ns t : Subgroup C\nx : C\nh : x ∈ s ⊔ t\n⊢ ∃ y ∈ s, ∃ z ∈ t, y * z = x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"DivInvMonoid.toInv",
"Subgroup.instSubgroupClass",
"Lattice.toSemilatticeSup",
... | [] | by
rw [sup_eq_closure] at h
refine Subgroup.closure_induction ?_ ?_ ?_ ?_ h
· rintro y (h | h)
· exact ⟨y, h, 1, t.one_mem, by simp⟩
· exact ⟨1, s.one_mem, y, h, by simp⟩
· exact ⟨1, s.one_mem, 1, ⟨t.one_mem, mul_one 1⟩⟩
· rintro _ _ _ _ ⟨y₁, hy₁, z₁, hz₁, rfl⟩ ⟨y₂, hy₂, z₂, hz₂, rfl⟩
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Map | {
"line": 439,
"column": 17
} | {
"line": 439,
"column": 47
} | {
"line": 440,
"column": 6
} | [
{
"pp": "G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH✝ : Type u_5\ninst✝¹ : Group H✝\nH : Type u_6\ninst✝ : Group H\nf : G ≃* H\nsg1 sg2 : Subgroup G\nh :\n { toFun := Subgroup.map ↑f, invFun := Subgroup.map ↑f.symm, ... | [
"G : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nA : Type u_4\ninst✝² : AddGroup A\nH✝ : Type u_5\ninst✝¹ : Group H✝\nH : Type u_6\ninst✝ : Group H\nf : G ≃* H\nsg1 sg2 : Subgroup G\nh :\n { toFun := Subgroup.map ↑f, invFun := Subgroup.map ↑f.symm, left_inv := ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 682,
"column": 15
} | {
"line": 682,
"column": 32
} | {
"line": 682,
"column": 33
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH₁ H₂ : Subgroup G\nh : ∀ {x y : G}, x ∈ H₁ → y ∈ H₂ → x = y → x = 1\nx y : G\nhx : x ∈ H₁\nhy : y ∈ H₂\nhxy : x * y = 1\nhx1 : x = 1 := h hx (Subgroup.inv_mem H₂ hy) (eq_inv_iff_mul_eq_one.mpr hxy)\n⊢ y = 1",
"ppTerm": "?m.60",
"assigned": false,
"usedConstan... | [
"G : Type u_1\ninst✝ : Group G\nH₁ H₂ : Subgroup G\nh : ∀ {x y : G}, x ∈ H₁ → y ∈ H₂ → x = y → x = 1\nx y : G\nhx : x ∈ H₁\nhy : y ∈ H₂\nhxy : x * y = 1\nhx1 : x = 1 := h hx (Subgroup.inv_mem H₂ hy) (eq_inv_iff_mul_eq_one.mpr hxy)\n⊢ y = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 13
} | {
"line": 299,
"column": 14
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\nF : Type u_4\ninst✝¹ : FunLike F M N\nmc : MonoidHomClass F M N\nι : Sort u_5\ninst✝ : Nonempty ι\nf : F\nhf : Injective ⇑f\ns : ι → Submonoid M\n⊢ ↑(map f (iInf s)) = ↑(⨅ i, map f (s i))",
"ppTerm": "?m.39",
"assigned"... | [
"M : Type u_1\nN : Type u_2\ninst✝³ : MulOneClass M\ninst✝² : MulOneClass N\nF : Type u_4\ninst✝¹ : FunLike F M N\nmc : MonoidHomClass F M N\nι : Sort u_5\ninst✝ : Nonempty ι\nf : F\nhf : Injective ⇑f\ns : ι → Submonoid M\n⊢ ⇑f '' ⋂ i, ↑(s i) = ⋂ i, ⇑f '' ↑(s i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Ker | {
"line": 151,
"column": 26
} | {
"line": 151,
"column": 37
} | {
"line": 151,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nx : N\n⊢ x ∈ range 1 ↔ x ∈ ⊥",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MonoidHom.range",
"MonoidHom.instFunLike",
"InvOneClass.toOne",
"DivInvOneMono... | [
"G : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\nx : N\n⊢ 1 = x ↔ x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 17
} | {
"line": 579,
"column": 18
} | [
{
"pp": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\n⊢ (inl M N) x1 ∈ map (inl M N) s",
"ppTerm": "?m.147",
... | [
"N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\n⊢ x1 ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 582,
"column": 6
} | {
"line": 582,
"column": 17
} | {
"line": 582,
"column": 18
} | [
{
"pp": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\nh1' : (inl M N) x1 ∈ u\n⊢ (inr M N) x2 ∈ map (inr M N) t",
... | [
"N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\nh1' : (inl M N) x1 ∈ u\n⊢ x2 ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 582,
"column": 6
} | {
"line": 582,
"column": 20
} | {
"line": 583,
"column": 4
} | [
{
"pp": "N : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\nh1' : (inl M N) x1 ∈ u\n⊢ (inr M N) x2 ∈ map (inr M N) t",
... | [] | simpa using h2 | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 15
} | {
"line": 583,
"column": 16
} | [
{
"pp": "case mpr\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\nh1' : (inl M N) x1 ∈ u\nh2' : (inr M N) x2 ∈ u\n⊢... | [
"case mpr\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\nhH : map (inl M N) s ≤ u\nhK : map (inr M N) t ≤ u\nx1 : M\nx2 : N\nh1 : (x1, x2).1 ∈ ↑s\nh2 : (x1, x2).2 ∈ ↑t\nh1' : (inl M N) x1 ∈ u\nh2' : (inr M N) x2 ∈ u\n⊢ (x1, x2) ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 576,
"column": 4
} | {
"line": 583,
"column": 43
} | {
"line": 585,
"column": 0
} | [
{
"pp": "case mpr\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u → s.prod t ≤ u",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.... | [] | rintro ⟨hH, hK⟩ ⟨x1, x2⟩ ⟨h1, h2⟩
have h1' : inl M N x1 ∈ u := by
apply hH
simpa using h1
have h2' : inr M N x2 ∈ u := by
apply hK
simpa using h2
simpa using Submonoid.mul_mem _ h1' h2' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 576,
"column": 4
} | {
"line": 583,
"column": 43
} | {
"line": 585,
"column": 0
} | [
{
"pp": "case mpr\nN : Type u_2\ninst✝¹ : MulOneClass N\nM : Type u_5\ninst✝ : MulOneClass M\ns : Submonoid M\nt : Submonoid N\nu : Submonoid (M × N)\n⊢ map (inl M N) s ≤ u ∧ map (inr M N) t ≤ u → s.prod t ≤ u",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.... | [] | rintro ⟨hH, hK⟩ ⟨x1, x2⟩ ⟨h1, h2⟩
have h1' : inl M N x1 ∈ u := by
apply hH
simpa using h1
have h2' : inr M N x2 ∈ u := by
apply hK
simpa using h2
simpa using Submonoid.mul_mem _ h1' h2' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 674,
"column": 2
} | {
"line": 674,
"column": 34
} | {
"line": 674,
"column": 35
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\ng : N →* P\nf : M →* N\n⊢ map g (mrange f) = mrange (g.comp f)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
... | [
"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\ng : N →* P\nf : M →* N\n⊢ map g (map f ⊤) = map (g.comp f) ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 725,
"column": 2
} | {
"line": 725,
"column": 26
} | {
"line": 727,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : MulOneClass M\nN : Type u_5\nS : Type u_6\ninst✝² : MulOneClass N\nf : M →* N\ninst✝¹ : SetLike S M\ninst✝ : SubmonoidClass S M\ns : S\n⊢ f.restrict s = 1 ↔ ∀ x ∈ s, f x = 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"_privat... | [] | simp [MonoidHom.ext_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 725,
"column": 2
} | {
"line": 725,
"column": 26
} | {
"line": 727,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : MulOneClass M\nN : Type u_5\nS : Type u_6\ninst✝² : MulOneClass N\nf : M →* N\ninst✝¹ : SetLike S M\ninst✝ : SubmonoidClass S M\ns : S\n⊢ f.restrict s = 1 ↔ ∀ x ∈ s, f x = 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"_privat... | [] | simp [MonoidHom.ext_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 725,
"column": 2
} | {
"line": 725,
"column": 26
} | {
"line": 727,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : MulOneClass M\nN : Type u_5\nS : Type u_6\ninst✝² : MulOneClass N\nf : M →* N\ninst✝¹ : SetLike S M\ninst✝ : SubmonoidClass S M\ns : S\n⊢ f.restrict s = 1 ↔ ∀ x ∈ s, f x = 1",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"_privat... | [] | simp [MonoidHom.ext_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 889,
"column": 55
} | {
"line": 889,
"column": 87
} | {
"line": 889,
"column": 88
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\n⊢ mrange (inl M N) = ⊤.prod ⊥",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike",
"MonoidHom",
"congrArg",
... | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\n⊢ map (inl M N) ⊤ = ⊤.prod ⊥"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 892,
"column": 55
} | {
"line": 892,
"column": 87
} | {
"line": 892,
"column": 88
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\n⊢ mrange (inr M N) = ⊥.prod ⊤",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MonoidHom.instFunLike",
"MonoidHom.inr",
"MonoidHom",
... | [
"M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\n⊢ map (inr M N) ⊤ = ⊥.prod ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 962,
"column": 2
} | {
"line": 962,
"column": 13
} | {
"line": 962,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝¹ : MulOneClass M\nS : Submonoid M\ninst✝ : Subsingleton ↥S\ny : M\nhy : y ∈ S\n⊢ y = 1",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝¹ : MulOneClass M\nS : Submonoid M\ninst✝ : Subsingleton ↥S\ny : M\nhy : y ∈ S\n⊢ y = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 1152,
"column": 2
} | {
"line": 1152,
"column": 37
} | {
"line": 1152,
"column": 38
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\nF : Type u_4\ninst✝ : FunLike F M N\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\nh : S ≤ MonoidHom.mrange f\n⊢ map f (comap f S) = S",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"M : Type u_1\nN : Type u_2\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\nF : Type u_4\ninst✝ : FunLike F M N\nmc : MonoidHomClass F M N\nf : F\nS : Submonoid N\nh : S ≤ MonoidHom.mrange f\n⊢ map f (comap f S) = S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Countable.Defs | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 36
} | {
"line": 131,
"column": 37
} | [
{
"pp": "α : Sort u\n⊢ ¬Uncountable α ↔ Countable α",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"uncountable_iff_not_countable",
"Uncountable",
"Iff",
"propext",
"Countable",
"Eq",
"Not"
],
"... | [
"α : Sort u\n⊢ ¬¬Countable α ↔ Countable α"
] | uncountable_iff_not_countable, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Bits | {
"line": 106,
"column": 15
} | {
"line": 106,
"column": 26
} | {
"line": 106,
"column": 27
} | [
{
"pp": "case bit\nb : Bool\nn : ℕ\n⊢ (bit b n).bodd.toNat + 2 * (bit b n).div2 = bit b n",
"ppTerm": "?bit",
"assigned": true,
"usedConstants": [
"Nat.bit",
"Eq.mpr",
"Nat.bodd_bit",
"HMul.hMul",
"Nat.div2_bit",
"congrArg",
"Bool.toNat",
"id",
"... | [
"case bit\nb : Bool\nn : ℕ\n⊢ b.toNat + n * 2 = bit b n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Bits | {
"line": 169,
"column": 16
} | {
"line": 169,
"column": 27
} | {
"line": 169,
"column": 28
} | [
{
"pp": "n : ℕ\n⊢ (n + 2).bodd = (1 &&& n + 2 != 0)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"bne",
"Bool.not",
"Nat.instAndOp",
"congrArg",
"Bool.not_not",
"id",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"... | [
"n : ℕ\n⊢ n.bodd = (n % 2 == 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Bits | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 35
} | {
"line": 224,
"column": 36
} | [
{
"pp": "motive : ℕ → Sort u\nH₁ H₂ : (b : Bool) → (n : ℕ) → motive (bit b n)\nh : (fun H n ↦ bitCasesOn n H) H₁ = (fun H n ↦ bitCasesOn n H) H₂\nb : Bool\nn : ℕ\n⊢ H₁ b n = H₂ b n",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"motive : ℕ → Sort u\nH₁ H₂ : (b : Bool) → (n : ℕ) → motive (bit b n)\nh : (fun H n ↦ bitCasesOn n H) H₁ = (fun H n ↦ bitCasesOn n H) H₂\nb : Bool\nn : ℕ\n⊢ H₁ b n = H₂ b n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Pairing | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 17
} | {
"line": 58,
"column": 18
} | [
{
"pp": "n a b : ℕ\nH : unpair n = (a, b)\n⊢ pair a b = n",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n a b : ℕ\nH : unpair n = (a, b)\n⊢ pair a b = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Pairing | {
"line": 102,
"column": 52
} | {
"line": 102,
"column": 63
} | {
"line": 102,
"column": 64
} | [
{
"pp": "a b : ℕ\n⊢ a ≤ pair a b",
"ppTerm": "?m.3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a b : ℕ\n⊢ a ≤ pair a b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Pairing | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 25
} | {
"line": 106,
"column": 26
} | [
{
"pp": "case pos\na b : ℕ\nh : a < b\n⊢ b ≤ pair a b",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"instMulNat",
"LE.le",
"ite_cond_eq_true",
"instLENat",
"Nat.pair",
"instHAdd",
"... | [
"case pos\na b : ℕ\nh : a < b\n⊢ b ≤ b * b + a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Pairing | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "n : ℕ\n⊢ (unpair n).2 ≤ n",
"ppTerm": "?m.5",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : ℕ\n⊢ (unpair n).2 ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 25
} | {
"line": 407,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ns : Set G\nhH : ∀ h ∈ H, ∀ g ∈ s, h * g * h⁻¹ ∈ s\nh : G\nhh : h ∈ H\nk : G\nhk : h * k * h⁻¹ ∈ s\n⊢ k ∈ s",
"ppTerm": "?m.47",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ns : Set G\nhH : ∀ h ∈ H, ∀ g ∈ s, h * g * h⁻¹ ∈ s\nh : G\nhh : h ∈ H\nk : G\nhk : h * k * h⁻¹ ∈ s\n⊢ k ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 412,
"column": 2
} | {
"line": 412,
"column": 25
} | {
"line": 412,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhH : ∀ h ∈ H, ∀ k ∈ K, h * k * h⁻¹ ∈ K\nh : G\nhh : h ∈ H\nk : G\nhk : h * k * h⁻¹ ∈ ↑K\n⊢ k ∈ ↑K",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Membership.mem",
"id",
... | [
"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhH : ∀ h ∈ H, ∀ k ∈ K, h * k * h⁻¹ ∈ K\nh : G\nhh : h ∈ H\nk : G\nhk : h * k * h⁻¹ ∈ ↑K\n⊢ k ∈ K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.Pairing | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 59
} | {
"line": 128,
"column": 60
} | [
{
"pp": "case pos\na b₁ b₂ : ℕ\nh : b₁ < b₂\nh₁ : a < b₁\n⊢ pair a b₁ < pair a b₂",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
"lt_trans",
"id",
"instMulNat",
"Nat.add_lt_add_iff_right._s... | [
"case pos\na b₁ b₂ : ℕ\nh : b₁ < b₂\nh₁ : a < b₁\n⊢ b₁ * b₁ < b₂ * b₂"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 561,
"column": 28
} | {
"line": 561,
"column": 45
} | {
"line": 561,
"column": 45
} | [
{
"pp": "G : Type u_5\ninst✝ : Group G\ns t : Set G\n⊢ ⋃ a ∈ s ∪ t, conjugatesOf a = (⋃ a ∈ s, conjugatesOf a) ∪ ⋃ a ∈ t, conjugatesOf a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"conjugatesOf",
"Set.instUnion",
"DivInvMonoi... | [] | Set.biUnion_union | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 673,
"column": 6
} | {
"line": 673,
"column": 18
} | {
"line": 673,
"column": 19
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na : G\nh : a ∈ H.normalCore\n⊢ a ∈ H",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Membership.mem",
"id",
"Mu... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\na : G\nh : a ∈ H.normalCore\n⊢ a * 1 ∈ H"
] | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.ZPowers.Basic | {
"line": 143,
"column": 72
} | {
"line": 144,
"column": 80
} | {
"line": 146,
"column": 0
} | [
{
"pp": "a : ℤ\n⊢ AddSubgroup.zmultiples ↑a.natAbs = AddSubgroup.zmultiples a",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddSubgroup.zmultiples_le._simp_1",
"Dvd.dvd",
"congrArg",
"and_self",
"_private.Mathlib.Algebra.Group.Subgroup.ZPowers.Basic.0.Int.zm... | [] | by
simp [le_antisymm_iff, Int.mem_zmultiples_iff, Int.dvd_natAbs, Int.natAbs_dvd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 13
} | {
"line": 328,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns t : Set β\na : α\n⊢ Disjoint (a • s) t ↔ Disjoint s (a⁻¹ • t)",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns t : Set β\na : α\n⊢ Disjoint (a • s) t ↔ Disjoint s (a⁻¹ • t)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 13
} | {
"line": 332,
"column": 14
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns t : Set β\na : α\n⊢ Disjoint s (a • t) ↔ Disjoint (a⁻¹ • s) t",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns t : Set β\na : α\n⊢ Disjoint s (a • t) ↔ Disjoint (a⁻¹ • s) t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 338,
"column": 22
} | {
"line": 338,
"column": 33
} | {
"line": 338,
"column": 34
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns : Set β\nh : ∀ ⦃i j : α⦄, ((j⁻¹ * i) • s ∩ s).Nonempty → i = j\na : α\n⊢ (a • s ∩ s).Nonempty → a = 1",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns : Set β\nh : ∀ ⦃i j : α⦄, ((j⁻¹ * i) • s ∩ s).Nonempty → i = j\na : α\n⊢ (a • s ∩ s).Nonempty → a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 339,
"column": 22
} | {
"line": 339,
"column": 59
} | {
"line": 339,
"column": 60
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns : Set β\nh : ∀ (a : α), (a • s ∩ s).Nonempty → a = 1\ni j : α\nne : ((j⁻¹ * i) • s ∩ s).Nonempty\n⊢ i = j",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns : Set β\nh : ∀ (a : α), (a • s ∩ s).Nonempty → a = 1\ni j : α\nne : ((j⁻¹ * i) • s ∩ s).Nonempty\n⊢ i = j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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