module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Fin.SuccPred | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 21
} | {
"line": 255,
"column": 2
} | [
{
"pp": "n : ℕ\ni : Fin n.succ\nhi : i ∈ Set.range castSucc\n⊢ ↑((Equiv.ofInjective castSucc ⋯).symm ⟨i, hi⟩) = ↑i",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"congrArg",
"Membership.mem",
"Set.Elem",
"id",
"Equ... | [
"n : ℕ\ni : Fin n.succ\nhi : i ∈ Set.range castSucc\n⊢ ↑((Equiv.ofInjective castSucc ⋯).symm ⟨i, hi⟩).castSucc = ↑i"
] | rw [← val_castSucc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.Duplicate | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 23
} | {
"line": 67,
"column": 24
} | [
{
"pp": "case cons_duplicate\nα : Type u_1\nl : List α\nx y y✝ : α\nl✝ : List α\nh : x ∈+ l✝\na_ih✝ : l✝ ≠ [y]\n⊢ y✝ :: l✝ ≠ [y]",
"ppTerm": "?cons_duplicate",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
"congrArg",
"List.ne_nil_of_mem",
"List.Duplicate.me... | [] | | cons_duplicate h => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.List.Duplicate | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 23
} | {
"line": 98,
"column": 4
} | [
{
"pp": "case cons_cons\nα : Type u_1\nl : List α\nx : α\nl' l₁✝ l₂✝ : List α\ny : α\nh : l₁✝ <+ l₂✝\nIH : x ∈+ l₁✝ → x ∈+ l₂✝\nhx : x ∈+ y :: l₁✝\n⊢ x ∈+ y :: l₂✝",
"ppTerm": "?cons_cons",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"and_self",
"true_or",
... | [] | | cons_cons y h IH => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.List.NodupEquivFin | {
"line": 122,
"column": 10
} | {
"line": 122,
"column": 35
} | {
"line": 122,
"column": 36
} | [
{
"pp": "α : Type u_1\nhd : α\ntl : List α\nIH : ∀ {l' : List α} (f : ℕ ↪o ℕ), (∀ (ix : ℕ), tl[ix]? = l'[f ix]?) → tl <+ l'\nl' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), (hd :: tl)[ix]? = l'[f ix]?\nw : f 0 < l'.length\nh : l'[f 0] = hd\na b : ℕ\n⊢ f (a + 1) - (f 0 + 1) ≤ f (b + 1) - (f 0 + 1) ↔ a ≤ b",
"ppTer... | [
"α : Type u_1\nhd : α\ntl : List α\nIH : ∀ {l' : List α} (f : ℕ ↪o ℕ), (∀ (ix : ℕ), tl[ix]? = l'[f ix]?) → tl <+ l'\nl' : List α\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), (hd :: tl)[ix]? = l'[f ix]?\nw : f 0 < l'.length\nh : l'[f 0] = hd\na b : ℕ\n⊢ f (a + 1) ≤ f (b + 1) ↔ a ≤ b",
"α : Type u_1\nhd : α\ntl : List α\nIH : ∀ {... | Nat.sub_le_sub_iff_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.NodupEquivFin | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 20
} | {
"line": 150,
"column": 4
} | [
{
"pp": "case mp.cons\nα : Type u_1\nl l' l₁✝ l₂✝ : List α\na✝¹ : α\na✝ : l₁✝ <+ l₂✝\nf : ℕ ↪o ℕ\nhf : ∀ (ix : ℕ), l₁✝[ix]? = l₂✝[f ix]?\n⊢ ∀ (ix : ℕ), l₁✝[ix]? = (a✝¹ :: l₂✝)[(RelEmbedding.trans f (OrderEmbedding.ofStrictMono (fun x ↦ x + 1) ⋯)) ix]?",
"ppTerm": "?mp.cons",
"assigned": true,
"usedC... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.List.NodupEquivFin | {
"line": 218,
"column": 10
} | {
"line": 218,
"column": 100
} | {
"line": 219,
"column": 10
} | [
{
"pp": "case mpr.refine_1.succ.succ\nα : Type u_1\nl : List α\nx : α\nn m : Fin l.length\nhnm : n < m\nh : x = l.get n\nh' : x = l.get m\nn✝¹ : ℕ\nhi : n✝¹ + 1 < (replicate 2 x).length\nn✝ : ℕ\nhj : n✝ + 1 < (replicate 2 x).length\n⊢ ⟨n✝¹ + 1, hi⟩ < ⟨n✝ + 1, hj⟩ →\n (fun i ↦ if ↑i = 0 then n else m) ⟨n✝¹ + ... | [
"case mpr.refine_1.succ.succ\nα : Type u_1\nl : List α\nx : α\nn m : Fin l.length\nhnm : n < m\nh : x = l.get n\nh' : x = l.get m\nn✝¹ : ℕ\nhi✝ : n✝¹ + 1 < (replicate 2 x).length\nn✝ : ℕ\nhj✝ : n✝ + 1 < (replicate 2 x).length\nhi : n✝¹ = 0\nhj : n✝ = 0\n⊢ ⟨n✝¹ + 1, hi✝⟩ < ⟨n✝ + 1, hj✝⟩ →\n (fun i ↦ if ↑i = 0 the... | simp only [Nat.lt_succ_iff, Nat.succ_le_succ_iff, replicate, length, Nat.le_zero] at hi hj | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Multiset.Fold | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 23
} | {
"line": 64,
"column": 24
} | [
{
"pp": "α : Type u_1\nop : α → α → α\nhc : Std.Commutative op\nha : Std.Associative op\nb a : α\ns : Multiset α\n⊢ fold op b (a ::ₘ s) = fold op (op a b) s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Multiset.cons",
"id",
"Multiset.... | [
"α : Type u_1\nop : α → α → α\nhc : Std.Commutative op\nha : Std.Associative op\nb a : α\ns : Multiset α\n⊢ fold op (op b a) s = fold op (op a b) s"
] | fold_cons'_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Fold | {
"line": 138,
"column": 8
} | {
"line": 138,
"column": 88
} | {
"line": 140,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nop : β → β → β\nhc : Std.Commutative op\nha : Std.Associative op\nf : α → β\nb : β\ns✝ : Finset α\ng : α → β\nhb : op b b = b\np : α → Prop\ninst✝ : DecidablePred p\nx : α\ns : Finset α\nhx : x ∉ s\nIH : fold op b (fun i ↦ if p i then f i else g i) s = op (fold op ... | [] | simp [Finset.filter_insert, h, Finset.fold_insert this, IH, ← ha.assoc, hc.comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 916,
"column": 80
} | {
"line": 917,
"column": 69
} | {
"line": 919,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Fin (n + 1) → Sort u_1\ni j : Fin (n + 1)\nh : j < i\nx : α i\np : (k : Fin n) → α (i.succAbove k)\n⊢ i.insertNth x p j = ⋯ ▸ p (j.castPred ⋯)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"Fin.succAboveCases",
"Fin.s... | [] | by
rw [insertNth, succAboveCases, dif_neg (Fin.ne_of_lt h), dif_pos h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Finite.Basic | {
"line": 952,
"column": 2
} | {
"line": 952,
"column": 42
} | {
"line": 953,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : LinearOrder α\nh : ∀ ⦃x y z : α⦄, x < y → y < z → False\na✝ : Nontrivial α\n⊢ Finite α",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Finite",
"Exists",
"Ne",
"Exists.casesOn",
"exists_pair_ne"
],
"usedFVars": [
"α"... | [
"α : Type u\ninst✝ : LinearOrder α\nh : ∀ ⦃x y z : α⦄, x < y → y < z → False\na✝ : Nontrivial α\nx y : α\nhne : x ≠ y\n⊢ Finite α"
] | rcases exists_pair_ne α with ⟨x, y, hne⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 1000,
"column": 69
} | {
"line": 1001,
"column": 42
} | {
"line": 1003,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Fin (n + 1) → Type u_3\ninst✝ : (i : Fin (n + 1)) → Preorder (α i)\ni : Fin (n + 1)\nx : α i\np : (j : Fin n) → α (i.succAbove j)\nq : (j : Fin (n + 1)) → α j\n⊢ i.insertNth x p ≤ q ↔ x ≤ q i ∧ p ≤ fun j ↦ q (i.succAbove j)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": ... | [] | by
simp [Pi.le_def, forall_iff_succAbove i] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 1004,
"column": 71
} | {
"line": 1005,
"column": 42
} | {
"line": 1007,
"column": 0
} | [
{
"pp": "n : ℕ\nα : Fin (n + 1) → Type u_3\ninst✝ : (i : Fin (n + 1)) → Preorder (α i)\ni : Fin (n + 1)\nx : α i\np : (j : Fin n) → α (i.succAbove j)\nq : (j : Fin (n + 1)) → α j\n⊢ q ≤ i.insertNth x p ↔ q i ≤ x ∧ (fun j ↦ q (i.succAbove j)) ≤ p",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants"... | [] | by
simp [Pi.le_def, forall_iff_succAbove i] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 1271,
"column": 6
} | {
"line": 1271,
"column": 22
} | {
"line": 1272,
"column": 2
} | [
{
"pp": "case inl.h\nn : ℕ\nα : Sort u_1\nj : Fin (n + 1)\nop : α → α → α\ng : Fin (n + 1) → α\nk : Fin n\nhjk : ↑j ≠ ↑k\nh : ↑k < ↑j\n⊢ k.castSucc < j",
"ppTerm": "?inl.h✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Fin.val",
... | [] | rwa [Fin.lt_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 1271,
"column": 6
} | {
"line": 1271,
"column": 22
} | {
"line": 1272,
"column": 2
} | [
{
"pp": "case inl.h\nn : ℕ\nα : Sort u_1\nj : Fin (n + 1)\nop : α → α → α\ng : Fin (n + 1) → α\nk : Fin n\nhjk : ↑j ≠ ↑k\nh : ↑k < ↑j\n⊢ k.castSucc < j",
"ppTerm": "?inl.h✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Fin.val",
... | [] | rwa [Fin.lt_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 1271,
"column": 6
} | {
"line": 1271,
"column": 22
} | {
"line": 1272,
"column": 2
} | [
{
"pp": "case inl.h\nn : ℕ\nα : Sort u_1\nj : Fin (n + 1)\nop : α → α → α\ng : Fin (n + 1) → α\nk : Fin n\nhjk : ↑j ≠ ↑k\nh : ↑k < ↑j\n⊢ k.castSucc < j",
"ppTerm": "?inl.h✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"Fin.val",
... | [] | rwa [Fin.lt_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Multiset.Defs | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 29
} | {
"line": 117,
"column": 4
} | [
{
"pp": "case cons\nM : Type u_3\ninst✝ : CommMonoid M\ns✝ : Multiset M\np : M → Prop\np_mul : ∀ (a b : M), p a → p b → p (a * b)\na : M\ns : Multiset M\nhsa : s ≠ ∅ → (∀ (a : M), a ∈ s → p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ (a_1 : M), a_1 ∈ a ::ₘ s → p a_1\n⊢ p (a * s.prod)",
"ppTerm": "?cons",
"... | [
"case pos\nM : Type u_3\ninst✝ : CommMonoid M\ns✝ : Multiset M\np : M → Prop\np_mul : ∀ (a b : M), p a → p b → p (a * b)\na : M\ns : Multiset M\nhsa : s ≠ ∅ → (∀ (a : M), a ∈ s → p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ (a_1 : M), a_1 ∈ a ::ₘ s → p a_1\nhs_empty : s = ∅\n⊢ p (a * s.prod)",
"case neg\nM : Type u... | by_cases hs_empty : s = ∅ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Data.List.Rotate | {
"line": 59,
"column": 42
} | {
"line": 59,
"column": 69
} | {
"line": 59,
"column": 69
} | [
{
"pp": "α : Type u\na : α\nl : List α\nn : ℕ\n⊢ ((l ++ [a]).rotate' n).length = (a :: l).length",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"List.cons",
"instHAppendOfAppend",
"List",
"List.rotate'",
"Nat",
... | [
"α : Type u\na : α\nl : List α\nn : ℕ\n⊢ (l ++ [a]).length = (a :: l).length"
] | length_rotate' (l ++ [a]) n | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Rotate | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 52
} | {
"line": 135,
"column": 53
} | [
{
"pp": "case inr\nα : Type u\nl : List α\nn : ℕ\nhl : 0 < l.length\n⊢ l.rotate n = drop (n % l.length) l ++ take (n % l.length) l",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"LT.lt.le",
"Nat.instMod",
"instHMod",
"Na... | [
"case inr\nα : Type u\nl : List α\nn : ℕ\nhl : 0 < l.length\n⊢ l.rotate n = l.rotate (n % l.length)"
] | ← rotate_eq_drop_append_take (n.mod_lt hl).le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Enum | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 67
} | {
"line": 33,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : List α\nn : ℕ\np : α × ℕ → Prop\n⊢ (∃ x, x ∈ l.zipIdx n ∧ p x) ↔ ∃ i x, p (l[i], n + i)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"List.getElem_zipIdx._proof_1",
"Iff.of_eq",
"congrArg",
"Membership.mem",
"Exis... | [] | simp only [exists_mem_iff_getElem, getElem_zipIdx, length_zipIdx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.Enum | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 67
} | {
"line": 33,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : List α\nn : ℕ\np : α × ℕ → Prop\n⊢ (∃ x, x ∈ l.zipIdx n ∧ p x) ↔ ∃ i x, p (l[i], n + i)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"List.getElem_zipIdx._proof_1",
"Iff.of_eq",
"congrArg",
"Membership.mem",
"Exis... | [] | simp only [exists_mem_iff_getElem, getElem_zipIdx, length_zipIdx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Enum | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 67
} | {
"line": 33,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : List α\nn : ℕ\np : α × ℕ → Prop\n⊢ (∃ x, x ∈ l.zipIdx n ∧ p x) ↔ ∃ i x, p (l[i], n + i)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"List.getElem_zipIdx._proof_1",
"Iff.of_eq",
"congrArg",
"Membership.mem",
"Exis... | [] | simp only [exists_mem_iff_getElem, getElem_zipIdx, length_zipIdx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Pi | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 23
} | {
"line": 73,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\nδ : α → Sort u_2\nm : Multiset α\na : α\nf : (a' : α) → a' ∈ a ::ₘ m → δ a'\na' : α\nh' : a' ∈ a ::ₘ m\nh : ¬a' = a\n⊢ cons m a (f a ⋯) (fun a' ha' ↦ f a' ⋯) a' h' = f a' h'",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | · rw [Pi.cons_ne _ h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Action.Pretransitive | {
"line": 101,
"column": 21
} | {
"line": 101,
"column": 36
} | {
"line": 101,
"column": 36
} | [
{
"pp": "G : Type u_2\nX : Type u_5\ninst✝¹ : Group G\ninst✝ : MulAction G X\nx₀ : X\nha : ∀ (x : X), ∃ g, g • x₀ = x\ng h : G\n⊢ (h * g⁻¹) • g • x₀ = h • x₀",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"DivInvOneMonoi... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Action.Pretransitive | {
"line": 101,
"column": 21
} | {
"line": 101,
"column": 36
} | {
"line": 101,
"column": 36
} | [
{
"pp": "G : Type u_2\nX : Type u_5\ninst✝¹ : Group G\ninst✝ : MulAction G X\nx₀ : X\nha : ∀ (x : X), ∃ g, g • x₀ = x\ng h : G\n⊢ (h * g⁻¹) • g • x₀ = h • x₀",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"DivInvOneMonoi... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Pretransitive | {
"line": 101,
"column": 21
} | {
"line": 101,
"column": 36
} | {
"line": 101,
"column": 36
} | [
{
"pp": "G : Type u_2\nX : Type u_5\ninst✝¹ : Group G\ninst✝ : MulAction G X\nx₀ : X\nha : ∀ (x : X), ∃ g, g • x₀ = x\ng h : G\n⊢ (h * g⁻¹) • g • x₀ = h • x₀",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
"DivInvOneMonoi... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subsemigroup.Basic | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 17
} | {
"line": 232,
"column": 18
} | [
{
"pp": "M : Type u_1\ninst✝ : Mul M\nm : M\np : Subsemigroup M\n⊢ closure {m} ≤ p ↔ m ∈ p",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Subsemigroup.instPartialOrder",
"Preorder.toLE",
"Membership.mem"... | [
"M : Type u_1\ninst✝ : Mul M\nm : M\np : Subsemigroup M\n⊢ {m} ⊆ ↑p ↔ m ∈ p"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Lattice.Image | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 41
} | {
"line": 221,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ninst✝ : Nonempty ι\ns : ι → Set α\nf : α → β\nh : InjOn f (⋃ i, s i)\ninhabited_h : Inhabited ι\ny : β\nhy : y ∈ ⋂ i, f '' s i\n⊢ y ∈ f '' ⋂ i, s i",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Set.Lattice.Im... | [
"α : Type u_1\nβ : Type u_2\nι : Sort u_5\ninst✝ : Nonempty ι\ns : ι → Set α\nf : α → β\nh : InjOn f (⋃ i, s i)\ninhabited_h : Inhabited ι\ny : β\nhy : ∀ (i : ι), ∃ x ∈ s i, f x = y\n⊢ y ∈ f '' ⋂ i, s i"
] | simp only [mem_iInter, mem_image] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Pi.Lemmas | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 46
} | {
"line": 531,
"column": 0
} | [
{
"pp": "α : Type u_5\nβ : α → Type u_6\nγ : (a : α) → β a → Type u_7\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (β a)\ninst✝ : (a : α) → (b : β a) → One (γ a b)\na : α\nb : β a\nx : γ a b\n⊢ uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x",
"ppTerm": "?m.24",
"assigned"... | [] | rw [← curry_mulSingle ⟨a, b⟩, uncurry_curry] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Pi.Lemmas | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 46
} | {
"line": 531,
"column": 0
} | [
{
"pp": "α : Type u_5\nβ : α → Type u_6\nγ : (a : α) → β a → Type u_7\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (β a)\ninst✝ : (a : α) → (b : β a) → One (γ a b)\na : α\nb : β a\nx : γ a b\n⊢ uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x",
"ppTerm": "?m.24",
"assigned"... | [] | rw [← curry_mulSingle ⟨a, b⟩, uncurry_curry] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Pi.Lemmas | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 46
} | {
"line": 531,
"column": 0
} | [
{
"pp": "α : Type u_5\nβ : α → Type u_6\nγ : (a : α) → β a → Type u_7\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (β a)\ninst✝ : (a : α) → (b : β a) → One (γ a b)\na : α\nb : β a\nx : γ a b\n⊢ uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle ⟨a, b⟩ x",
"ppTerm": "?m.24",
"assigned"... | [] | rw [← curry_mulSingle ⟨a, b⟩, uncurry_curry] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 869,
"column": 61
} | {
"line": 870,
"column": 75
} | {
"line": 872,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\ns t : Set α\n⊢ 1 ∈ s / t ↔ ¬Disjoint s t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"_private.Mathlib.Algebra.Group.Pointwise.Set.Basic.0.Set.one_mem_div_i... | [] | by
simp [not_disjoint_iff_nonempty_inter, mem_div, div_eq_one, Set.Nonempty] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Basic | {
"line": 266,
"column": 26
} | {
"line": 266,
"column": 36
} | {
"line": 266,
"column": 36
} | [
{
"pp": "M : Type u_1\ninst✝ : MulOneClass M\nN N' : Submonoid M\n⊢ N ⊔ N' = closure ↑N ⊔ closure ↑N'",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",
"SemilatticeSup.toMax",
"Submonoid.closur... | [] | closure_eq | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.Submonoid.Basic | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 17
} | {
"line": 274,
"column": 18
} | [
{
"pp": "M : Type u_1\ninst✝ : MulOneClass M\nm : M\np : Submonoid M\n⊢ closure {m} ≤ p ↔ m ∈ p",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Set.instSingletonSet",
... | [
"M : Type u_1\ninst✝ : MulOneClass M\nm : M\np : Submonoid M\n⊢ {m} ⊆ ↑p ↔ m ∈ p"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 1015,
"column": 2
} | {
"line": 1018,
"column": 27
} | {
"line": 1020,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Group α\ninst✝² : DivisionMonoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nm : F\ns t : Set β\nhs : s ⊆ range ⇑m\nht : t ⊆ range ⇑m\n⊢ s / t ⊆ range ⇑m",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv... | [] | rintro _ ⟨a, ha, b, hb, rfl⟩
obtain ⟨a, rfl⟩ := hs ha
obtain ⟨b, rfl⟩ := ht hb
exact ⟨a / b, map_div ..⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Pointwise.Set.Basic | {
"line": 1015,
"column": 2
} | {
"line": 1018,
"column": 27
} | {
"line": 1020,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Group α\ninst✝² : DivisionMonoid β\ninst✝¹ : FunLike F α β\ninst✝ : MonoidHomClass F α β\nm : F\ns t : Set β\nhs : s ⊆ range ⇑m\nht : t ⊆ range ⇑m\n⊢ s / t ⊆ range ⇑m",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"instHDiv... | [] | rintro _ ⟨a, ha, b, hb, rfl⟩
obtain ⟨a, rfl⟩ := hs ha
obtain ⟨b, rfl⟩ := ht hb
exact ⟨a / b, map_div ..⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 450,
"column": 26
} | {
"line": 450,
"column": 36
} | {
"line": 450,
"column": 36
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH H' : Subgroup G\n⊢ H ⊔ H' = closure ↑H ⊔ closure ↑H'",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Subgroup.closure_eq",
"Lattice.toSemilatticeSup",
"Subgroup.closure",
"CompleteLattice.toLattice",
"congrArg",
... | [] | closure_eq | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 461,
"column": 75
} | {
"line": 461,
"column": 85
} | {
"line": 461,
"column": 85
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Sort u_2\np : ι → Subgroup G\n⊢ ⨆ i, p i = ⨆ i, closure ↑(p i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Subgroup.closure_eq",
"Subgroup.closure",
"congrArg",
"iSup",
"Subgroup",
"funext",
"SetLike... | [] | closure_eq | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Group.Subgroup.Lattice | {
"line": 468,
"column": 92
} | {
"line": 479,
"column": 26
} | {
"line": 481,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nx y : G\n⊢ y ∈ closure {x} ↔ ∃ n, x ^ n = y",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.mem_singleton",
"Eq.mpr",
"Subgroup.instSubgroupClass",
"zpow_add",
"InvOneClass.toOne",
"Set.eq_of_mem_singleton",
... | [] | by
refine
⟨fun hy => closure_induction ?_ ?_ ?_ ?_ hy, fun ⟨n, hn⟩ =>
hn ▸ zpow_mem (subset_closure <| mem_singleton x) n⟩
· intro y hy
rw [eq_of_mem_singleton hy]
exact ⟨1, zpow_one x⟩
· exact ⟨0, zpow_zero x⟩
· rintro _ _ _ _ ⟨n, rfl⟩ ⟨m, rfl⟩
exact ⟨n + m, zpow_add x n m⟩
rintro _ _ ⟨... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Defs | {
"line": 696,
"column": 14
} | {
"line": 696,
"column": 58
} | {
"line": 696,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Set G\ng : G\nh : ∀ (h : G), h * g ∈ S ↔ g * h ∈ S\nn : G\n⊢ n ∈ S ↔ g * n * g⁻¹ ∈ S",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.to... | [] | by rw [mul_assoc, ← h, inv_mul_cancel_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Subgroup.Ker | {
"line": 135,
"column": 39
} | {
"line": 135,
"column": 65
} | {
"line": 135,
"column": 65
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nN : Type u_7\ninst✝ : Group N\nf : G →* N\n⊢ ↑f.range = ↑⊤ ↔ Set.range ⇑f = Set.univ",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneCla... | [] | by rw [coe_range, coe_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Pairing | {
"line": 136,
"column": 75
} | {
"line": 138,
"column": 56
} | {
"line": 140,
"column": 0
} | [
{
"pp": "m n : ℕ\n⊢ pair m n < (max m n + 1) ^ 2",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Nat.Pairing.0.Nat.pair_lt_max_add_one_sq._simp_1_2",
"instPowNat",
"Eq.mpr",
"False",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
... | [] | by
simp only [pair, Nat.pow_two, Nat.mul_add, Nat.add_mul, Nat.mul_one, Nat.one_mul, Nat.add_assoc]
split_ifs <;> simp [Nat.le_of_lt, not_lt.1, *] <;> lia | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 808,
"column": 2
} | {
"line": 808,
"column": 17
} | {
"line": 810,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nx✝ : M\n⊢ x✝ ∈ mker 1 ↔ x✝ ∈ ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.instFunLike",
"MonoidHom.mker",
"Monoi... | [] | simp [mem_mker] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 824,
"column": 2
} | {
"line": 824,
"column": 17
} | {
"line": 826,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nx : M\n⊢ x ∈ mker (inl M N) ↔ x ∈ ⊥",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.instFunLike",
"MonoidHom.mker",
"... | [] | simp [mem_mker] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Submonoid.Operations | {
"line": 829,
"column": 2
} | {
"line": 829,
"column": 17
} | {
"line": 831,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : MulOneClass M\ninst✝ : MulOneClass N\nx : N\n⊢ x ∈ mker (inr M N) ↔ x ∈ ⊥",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MonoidHom.instMonoidHomClass",
"MulOne.toOne",
"MonoidHom.instFunLike",
"MonoidHom.mker",
"... | [] | simp [mem_mker] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 395,
"column": 6
} | {
"line": 395,
"column": 17
} | {
"line": 395,
"column": 18
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ns : Set G\nhH : ∀ h ∈ H, ∀ g ∈ s, h * g * h⁻¹ ∈ closure s\nh : G\nhh : h ∈ H\n⊢ closure s ≤ closure (⇑↑(MulAut.conj h) '' s)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"Monoid... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\ns : Set G\nhH : ∀ h ∈ H, ∀ g ∈ s, h * g * h⁻¹ ∈ closure s\nh : G\nhh : h ∈ H\n⊢ s ⊆ ↑(closure (⇑↑(MulAut.conj h) '' s))"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.ZPowers.Basic | {
"line": 120,
"column": 26
} | {
"line": 120,
"column": 37
} | {
"line": 120,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ng : G\nH : Subgroup G\n⊢ closure {g} ≤ H ↔ g ∈ H",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subgroup.closure",
"congrArg",
"PartialOrder.toPreorder",
"Subgroup.closure_le",
"Preorder.toLE",
"M... | [
"G : Type u_1\ninst✝ : Group G\ng : G\nH : Subgroup G\n⊢ {g} ⊆ ↑H ↔ g ∈ H"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 670,
"column": 53
} | {
"line": 680,
"column": 25
} | {
"line": 682,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ H.normalCore = ⨅ g, map (↑(MulAut.conj g)) H",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulEquiv.refl",
"le_refl",
"MulEquiv.instEquivLik... | [] | by
have : (⨅ g : G, H.map (MulAut.conj g) : Subgroup G).Normal := by
refine normal_iff_map_conj_eq.mpr fun g ↦ ?_
conv_rhs => rw [← Equiv.iInf_comp (Equiv.mulLeft g)]
rw [map_iInf _ (MulAut.conj g).injective]
simp [map_map, MulAut.mul_def]
refine le_antisymm (le_iInf fun g ↦ ?_) ?_
· grw [← Normal... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Idempotent | {
"line": 98,
"column": 4
} | {
"line": 99,
"column": 59
} | {
"line": 100,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\na : M\nh : IsUnit a\nidem : IsIdempotentElem a\n⊢ a = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"mul_asso... | [] | have ⟨q, eq⟩ := h.exists_left_inv
rw [← eq, ← idem.eq, ← mul_assoc, eq, one_mul, idem.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Idempotent | {
"line": 98,
"column": 4
} | {
"line": 99,
"column": 59
} | {
"line": 100,
"column": 2
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\na : M\nh : IsUnit a\nidem : IsIdempotentElem a\n⊢ a = 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"mul_asso... | [] | have ⟨q, eq⟩ := h.exists_left_inv
rw [← eq, ← idem.eq, ← mul_assoc, eq, one_mul, idem.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 979,
"column": 2
} | {
"line": 979,
"column": 43
} | {
"line": 981,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nι : Sort u_6\na : ι → Subgroup G\nnorm : ∀ (i : ι), (a i).Normal\ng : G\ng_in_iInf : ∀ (i : ι), g ∈ a i\nh : G\ni : ι\n⊢ h * g * h⁻¹ ∈ a i",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Subgroup.Normal.conj_mem"
],
"usedFVars": [
... | [] | exact (norm i).conj_mem g (g_in_iInf i) h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 337,
"column": 44
} | {
"line": 337,
"column": 75
} | {
"line": 337,
"column": 75
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns : Set β\n⊢ (∀ ⦃i j : α⦄, ¬Disjoint ((j⁻¹ * i) • s) s → i = j) ↔ ∀ (a : α), (a • s ∩ s).Nonempty → a = 1",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instHSMul"... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns : Set β\n⊢ (∀ ⦃i j : α⦄, ((j⁻¹ * i) • s ∩ s).Nonempty → i = j) ↔ ∀ (a : α), (a • s ∩ s).Nonempty → a = 1"
] | not_disjoint_iff_nonempty_inter | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Sigma.Lex | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 23
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case left\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b : (i : ι) × α i\ninst✝¹ : Std.Irrefl r\ninst✝ : ∀ (i : ι), Std.Irrefl (s i)\ni✝ : ι\na : α i✝\nhi : r i✝ i✝\n⊢ False",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
... | [] | exact irrefl _ hi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Sigma.Lex | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 23
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case left\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b : (i : ι) × α i\ninst✝¹ : Std.Irrefl r\ninst✝ : ∀ (i : ι), Std.Irrefl (s i)\ni✝ : ι\na : α i✝\nhi : r i✝ i✝\n⊢ False",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
... | [] | exact irrefl _ hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sigma.Lex | {
"line": 89,
"column": 6
} | {
"line": 89,
"column": 23
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case left\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b : (i : ι) × α i\ninst✝¹ : Std.Irrefl r\ninst✝ : ∀ (i : ι), Std.Irrefl (s i)\ni✝ : ι\na : α i✝\nhi : r i✝ i✝\n⊢ False",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
... | [] | exact irrefl _ hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sigma.Lex | {
"line": 131,
"column": 4
} | {
"line": 134,
"column": 50
} | {
"line": 135,
"column": 4
} | [
{
"pp": "case inr.inl\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b✝ : (i : ι) × α i\ninst✝¹ : Std.Trichotomous r\ninst✝ : ∀ (i : ι), Std.Trichotomous (s i)\ni : ι\na b : α i\n⊢ Lex r s ⟨i, a⟩ ⟨i, b⟩ ∨ ⟨i, a⟩ = ⟨i, b⟩ ∨ Lex r s ⟨i, b⟩ ⟨i, a⟩",
"ppTerm": ... | [
"case inr.inr\nι : Type u_1\nα : ι → Type u_2\nr r₁ r₂ : ι → ι → Prop\ns s₁ s₂ : (i : ι) → α i → α i → Prop\na✝ b✝ : (i : ι) × α i\ninst✝¹ : Std.Trichotomous r\ninst✝ : ∀ (i : ι), Std.Trichotomous (s i)\ni : ι\na : α i\nj : ι\nb : α j\nhji : r j i\n⊢ Lex r s ⟨i, a⟩ ⟨j, b⟩ ∨ ⟨i, a⟩ = ⟨j, b⟩ ∨ Lex r s ⟨j, b⟩ ⟨i, a⟩"
... | · obtain hab | rfl | hba := trichotomous_of (s i) a b
· exact Or.inl (Lex.right _ _ hab)
· exact Or.inr (Or.inl rfl)
· exact Or.inr (Or.inr <| Lex.right _ _ hba) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Choose.Basic | {
"line": 145,
"column": 16
} | {
"line": 145,
"column": 51
} | {
"line": 146,
"column": 2
} | [
{
"pp": "x✝ : ℕ\nhk : x✝ ≤ 0\n⊢ choose 0 x✝ * x✝! * (0 - x✝)! = 0!",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Nat.choose",
"HMul.hMul",
"Nat.mul_one",
"congrArg",
"HSub.hSub",
"instSubNat",
"instMulNat",
"instOfNatNat",
"Nat.sub_s... | [] | by simp [Nat.eq_zero_of_le_zero hk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 80,
"column": 2
} | {
"line": 84,
"column": 34
} | {
"line": 86,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns : Finset β\nf g : β → α\n⊢ s.sup (f ⊔ g) = s.sup f ⊔ s.sup g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"sup_sup_sup_comm",
"Finset.cons_induction",
"Finset.cons",
... | [] | induction s using Finset.cons_induction with
| empty => rw [sup_empty, sup_empty, sup_empty, bot_sup_eq]
| cons _ _ _ ih =>
rw [sup_cons, sup_cons, sup_cons, ih]
exact sup_sup_sup_comm _ _ _ _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 80,
"column": 2
} | {
"line": 84,
"column": 34
} | {
"line": 86,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns : Finset β\nf g : β → α\n⊢ s.sup (f ⊔ g) = s.sup f ⊔ s.sup g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"sup_sup_sup_comm",
"Finset.cons_induction",
"Finset.cons",
... | [] | induction s using Finset.cons_induction with
| empty => rw [sup_empty, sup_empty, sup_empty, bot_sup_eq]
| cons _ _ _ ih =>
rw [sup_cons, sup_cons, sup_cons, ih]
exact sup_sup_sup_comm _ _ _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 80,
"column": 2
} | {
"line": 84,
"column": 34
} | {
"line": 86,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns : Finset β\nf g : β → α\n⊢ s.sup (f ⊔ g) = s.sup f ⊔ s.sup g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"sup_sup_sup_comm",
"Finset.cons_induction",
"Finset.cons",
... | [] | induction s using Finset.cons_induction with
| empty => rw [sup_empty, sup_empty, sup_empty, bot_sup_eq]
| cons _ _ _ ih =>
rw [sup_cons, sup_cons, sup_cons, ih]
exact sup_sup_sup_comm _ _ _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.Powerset | {
"line": 325,
"column": 6
} | {
"line": 325,
"column": 58
} | {
"line": 326,
"column": 6
} | [
{
"pp": "α : Type u_1\ns : Multiset α\nl : List α\nh : Nodup ⟦l⟧\n⊢ (powerset ⟦l⟧).Nodup",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"List.sublists'",
"Eq.mpr",
"Multiset.powerset_coe'",
"Multiset.Nodup",
"congrArg",
"List.map",
"Multiset.power... | [
"α : Type u_1\ns : Multiset α\nl : List α\nh : Nodup ⟦l⟧\n⊢ (List.map ofList l.sublists').Nodup"
] | simp only [quot_mk_to_coe, powerset_coe', coe_nodup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 620,
"column": 2
} | {
"line": 620,
"column": 43
} | {
"line": 620,
"column": 44
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_5\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : F\ns : Finset ι\nhs : s.Nonempty\ng : ι → α\n⊢ f (s.sup' hs g) = s.sup' hs (⇑f ∘ g)",
"ppTerm": "?m.18",
"assigned": true,
"us... | [
"case refine_1\nF : Type u_1\nα : Type u_2\nβ : Type u_3\nι : Type u_5\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : F\ns : Finset ι\nhs : s.Nonempty\ng : ι → α\na✝ : ι\n⊢ f ({a✝}.sup' ⋯ g) = {a✝}.sup' ⋯ (⇑f ∘ g)",
"case refine_2\nF : Type u_1\nα : T... | refine hs.cons_induction ?_ ?_ <;> intros | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 725,
"column": 30
} | {
"line": 725,
"column": 52
} | {
"line": 725,
"column": 52
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nι : Type u_5\ninst✝³ : LinearOrder α\ns : Finset ι\nf : ι → α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\ng : α → β\nmono_g : Monotone g\nH : s.Nonempty\n⊢ g (s.sup' H f) = s.sup (g ∘ f)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstan... | [
"α : Type u_2\nβ : Type u_3\nι : Type u_5\ninst✝³ : LinearOrder α\ns : Finset ι\nf : ι → α\ninst✝² : OrderBot α\ninst✝¹ : SemilatticeSup β\ninst✝ : OrderBot β\ng : α → β\nmono_g : Monotone g\nH : s.Nonempty\n⊢ g (s.sup' H f) = s.sup' H (g ∘ f)"
] | ← Finset.sup'_eq_sup H | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Finite.Lattice | {
"line": 166,
"column": 38
} | {
"line": 166,
"column": 71
} | {
"line": 167,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type u_1\ns : ι → Set α\nt : Set ι\nht : t.Finite\nhs : ∀ i ∈ t, (s i).Finite\nhe : ∀ i ∉ t, s i = ∅\nthis : ⋃ i, s i ⊆ ⋃ i ∈ t, s i\n⊢ (⋃ i, s i).Finite",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.Finite.subset",
"Set.F... | [] | exact (ht.biUnion hs).subset this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Set.Finite.Lattice | {
"line": 166,
"column": 38
} | {
"line": 166,
"column": 71
} | {
"line": 167,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type u_1\ns : ι → Set α\nt : Set ι\nht : t.Finite\nhs : ∀ i ∈ t, (s i).Finite\nhe : ∀ i ∉ t, s i = ∅\nthis : ⋃ i, s i ⊆ ⋃ i ∈ t, s i\n⊢ (⋃ i, s i).Finite",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.Finite.subset",
"Set.F... | [] | exact (ht.biUnion hs).subset this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Finite.Lattice | {
"line": 166,
"column": 38
} | {
"line": 166,
"column": 71
} | {
"line": 167,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type u_1\ns : ι → Set α\nt : Set ι\nht : t.Finite\nhs : ∀ i ∈ t, (s i).Finite\nhe : ∀ i ∉ t, s i = ∅\nthis : ⋃ i, s i ⊆ ⋃ i ∈ t, s i\n⊢ (⋃ i, s i).Finite",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.Finite.subset",
"Set.F... | [] | exact (ht.biUnion hs).subset this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Finite.Lattice | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 20
} | {
"line": 245,
"column": 21
} | [
{
"pp": "α : Type u\ns : Set α\nhs : s.Finite\nι : Type u_1\nt : ι → Set α\nh : s ⊆ ⋃ i, t i\nthis : Finite ↑s\nf : ↑s → ι\nhf : ∀ (x : ↑s), ↑x ∈ t (f x)\nx : α\nhx : x ∈ s\n⊢ x ∈ ⋃ i ∈ range f, t i",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Se... | [
"α : Type u\ns : Set α\nhs : s.Finite\nι : Type u_1\nt : ι → Set α\nh : s ⊆ ⋃ i, t i\nthis : Finite ↑s\nf : ↑s → ι\nhf : ∀ (x : ↑s), ↑x ∈ t (f x)\nx : α\nhx : x ∈ s\n⊢ x ∈ ⋃ y, t (f y)"
] | biUnion_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Finite.Lattice | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 9
} | {
"line": 380,
"column": 10
} | [
{
"pp": "α : Type u\nι : Sort v\nκ : ι → Sort w\ninst✝¹ : Order.Frame α\ninst✝ : Finite ι\nf : (a : ι) → κ a → α\n⊢ ∀ {ι : Type v} {κ : ι → Type w} [Finite ι] (f : (a : ι) → κ a → α), ⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [],
"usedFVars": [... | [
"α : Type u\nι✝ : Sort v\nκ : ι✝ → Sort w\ninst✝¹ : Order.Frame α\ninst✝ : Finite ι✝\nf : (a : ι✝) → κ a → α\nι : Type v\n⊢ ∀ {κ : ι → Type w} [Finite ι] (f : (a : ι) → κ a → α), ⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)"
] | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 455,
"column": 4
} | {
"line": 458,
"column": 64
} | {
"line": 459,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nι : Sort u_4\ninst✝ : ConditionallyCompleteLinearOrder α\np : ι → Prop\nf : Subtype p → α\nhp : ¬∀ (i : ι), p i\nle : sSup ∅ ≤ iSup f\n⊢ ⨆ i, ⨆ (h : p i), f ⟨i, h⟩ = max (iSup f) (sSup ∅)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [max_eq_left le]
by_cases bdd : BddAbove (range f)
· rw [← ciSup_subtype bdd le]
· rw [ciSup_of_not_bddAbove bdd, cbiSup_of_not_bddAbove bdd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 455,
"column": 4
} | {
"line": 458,
"column": 64
} | {
"line": 459,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nι : Sort u_4\ninst✝ : ConditionallyCompleteLinearOrder α\np : ι → Prop\nf : Subtype p → α\nhp : ¬∀ (i : ι), p i\nle : sSup ∅ ≤ iSup f\n⊢ ⨆ i, ⨆ (h : p i), f ⟨i, h⟩ = max (iSup f) (sSup ∅)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [max_eq_left le]
by_cases bdd : BddAbove (range f)
· rw [← ciSup_subtype bdd le]
· rw [ciSup_of_not_bddAbove bdd, cbiSup_of_not_bddAbove bdd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Sigma | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 15
} | {
"line": 179,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\nα : ι → Type u_2\nβ : ι → Type u_3\nγ : ι → Type u_4\ninst✝ : DecidableEq ι\nf : ⦃i : ι⦄ → α i → β i → Finset (γ i)\nx : Sigma γ\na : α x.1\nb : β x.1\nh : x.1 ≠ x.1\nh✝ : x.snd ∈ f (⋯ ▸ ⟨x.1, a⟩.snd) (⋯ ▸ ⟨x.1, b⟩.snd)\n⊢ False",
"ppTerm": "?inr",
"assigned": true,
... | [] | exact h rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Cover | {
"line": 618,
"column": 2
} | {
"line": 618,
"column": 33
} | {
"line": 619,
"column": 2
} | [
{
"pp": "ι : Type u_3\nα : ι → Type u_4\ninst✝ : (i : ι) → Preorder (α i)\na b : (i : ι) → α i\nh : (a ≤ b ∧ ∃ i, a i < b i) ∧ ∀ ⦃c : (i : ι) → α i⦄, a ≤ c → ∀ (x : ι), a x < c x → c ≤ b → ∀ (x : ι), ¬c x < b x\n⊢ ∃ i, ∀ (j : ι), j ≠ i → AntisymmRel (fun x1 x2 ↦ x1 ≤ x2) (a j) (b j)",
"ppTerm": "?m.24",
... | [
"ι : Type u_3\nα : ι → Type u_4\ninst✝ : (i : ι) → Preorder (α i)\na b : (i : ι) → α i\nh : ∀ ⦃c : (i : ι) → α i⦄, a ≤ c → ∀ (x : ι), a x < c x → c ≤ b → ∀ (x : ι), ¬c x < b x\nhab : a ≤ b\ni : ι\nhi : a i < b i\n⊢ ∃ i, ∀ (j : ι), j ≠ i → AntisymmRel (fun x1 x2 ↦ x1 ≤ x2) (a j) (b j)"
] | obtain ⟨⟨hab, ⟨i, hi⟩⟩, h⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Cover | {
"line": 621,
"column": 2
} | {
"line": 621,
"column": 71
} | {
"line": 622,
"column": 2
} | [
{
"pp": "ι : Type u_3\nα : ι → Type u_4\ninst✝ : (i : ι) → Preorder (α i)\na b : (i : ι) → α i\nh : ∀ ⦃c : (i : ι) → α i⦄, a ≤ c → ∀ (x : ι), a x < c x → c ≤ b → ∀ (x : ι), ¬c x < b x\nhab : a ≤ b\ni : ι\nhi : a i < b i\nj : ι\nhj : j ≠ i\nc : (i : ι) → α i := Function.update a i (b i)\n⊢ AntisymmRel (fun x1 x2... | [
"ι : Type u_3\nα : ι → Type u_4\ninst✝ : (i : ι) → Preorder (α i)\na b : (i : ι) → α i\nh : ∀ ⦃c : (i : ι) → α i⦄, a ≤ c → ∀ (x : ι), a x < c x → c ≤ b → ∀ (x : ι), ¬c x < b x\nhab : a ≤ b\ni : ι\nhi : a i < b i\nj : ι\nhj : j ≠ i\nc : (i : ι) → α i := Function.update a i (b i)\nh₁ : c ≤ b\n⊢ AntisymmRel (fun x1 x2... | have h₁ : c ≤ b := by simpa [update_le_iff, c] using fun k hk ↦ hab k | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.Interval.Multiset | {
"line": 230,
"column": 56
} | {
"line": 230,
"column": 72
} | {
"line": 230,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na : α\n⊢ (Finset.Icc a a).val = {a}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Multiset",
"id",
"Finset.I... | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na : α\n⊢ {a}.val = {a}"
] | Finset.Icc_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Finset.Nat | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 69
} | {
"line": 158,
"column": 4
} | [
{
"pp": "case succ\na n : ℕ\nih : Set.InjOn (fun x ↦ x % a) ↑(Ico n (n + a))\nk l : ℕ\nhkl : k % a = l % a\nha : 0 < a\nhk : k ≠ n ∧ (k = n + a ∨ k ∈ Ico n (n + a))\nhl : l ≠ n ∧ (l = n + a ∨ l ∈ Ico n (n + a))\n⊢ k = l",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Finset",
... | [
"case succ.inl.inl\na n : ℕ\nih : Set.InjOn (fun x ↦ x % a) ↑(Ico n (n + a))\nha : 0 < a\nhkn : n + a ≠ n\nhkl : (n + a) % a = (n + a) % a\nhln : n + a ≠ n\n⊢ n + a = n + a",
"case succ.inl.inr\na n : ℕ\nih : Set.InjOn (fun x ↦ x % a) ↑(Ico n (n + a))\nl : ℕ\nha : 0 < a\nhkl : (n + a) % a = l % a\nhkn : n + a ≠ n... | rcases hk with ⟨hkn, rfl | hk⟩ <;> rcases hl with ⟨hln, rfl | hl⟩ | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Order.Interval.Finset.Defs | {
"line": 578,
"column": 26
} | {
"line": 578,
"column": 70
} | {
"line": 580,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Preorder α\ninst✝³ : Preorder β\ninst✝² : Fintype α\ninst✝¹ : DecidableLT α\ninst✝ : DecidableLE α\na b x : α\n⊢ x ∈ (Set.Ioo a b).toFinset ↔ a < x ∧ x < b",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Set.decidableMemIoo",
"Preo... | [] | by simp only [Set.mem_toFinset, Set.mem_Ioo] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.OrderIsoNat | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 73
} | {
"line": 252,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : WellFoundedGT α\na : ℕ →o α\nm : ℕ\nhm : monotonicSequenceLimitIndex a < m\nh : ∃ n, ∀ (m : ℕ), n ≤ m → a n = a m\n⊢ a m ≤ monotonicSequenceLimit a",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.ge",
"Partial... | [] | exact (Nat.sInf_mem (s := {n | ∀ m, n ≤ m → a n = a m}) h m hm.le).ge | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Finset.Basic | {
"line": 129,
"column": 46
} | {
"line": 129,
"column": 86
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_2\na b : α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\n⊢ a ∈ Icc a b ↔ a ≤ b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Preorder.toLE",
"Membership.mem",
"_private.Mathlib.Order.Interval.Finset.Basic.0.Fi... | [] | by simp only [mem_Icc, true_and, le_rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Basic | {
"line": 135,
"column": 47
} | {
"line": 135,
"column": 87
} | {
"line": 137,
"column": 0
} | [
{
"pp": "α : Type u_2\na b : α\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrder α\n⊢ b ∈ Ioc a b ↔ a < b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.right_mem_Ioc._simp_1_1",
"and_true",
"co... | [] | by simp only [mem_Ioc, and_true, le_rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Defs | {
"line": 595,
"column": 4
} | {
"line": 597,
"column": 47
} | {
"line": 598,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nh₀_finset_Icc h₀_finset_Ico h₀_finset_Ioc h₀_finset_Ioo : α → α → Finset α\nh₀_finset_mem_Icc : ∀ (a b x : α), x ∈ h₀_finset_Icc a b ↔ a ≤ x ∧ x ≤ b\nh₀_finset_mem_Ico : ∀ (a b x : α), x ∈ h₀_finset_Ico a b ↔ a ≤ x ∧ x < b\nh₀_finset_... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nh₀_finset_Icc h₀_finset_Ico h₀_finset_Ioc h₀_finset_Ioo : α → α → Finset α\nh₀_finset_mem_Icc : ∀ (a b x : α), x ∈ h₀_finset_Icc a b ↔ a ≤ x ∧ x ≤ b\nh₀_finset_mem_Ico : ∀ (a b x : α), x ∈ h₀_finset_Ico a b ↔ a ≤ x ∧ x < b\nh₀_finset_mem_Ioc : ∀ ... | have hIoo : h₀_finset_Ioo = h₁_finset_Ioo := by
ext a b x
rw [h₀_finset_mem_Ioo, h₁_finset_mem_Ioo] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.WellQuasiOrder | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 60
} | {
"line": 82,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nr : α → α → Prop\ns : β → β → Prop\ninst✝ : IsPreorder α r\nhr : WellQuasiOrdered r\nhs : WellQuasiOrdered s\nf : ℕ → α × β\n⊢ ∃ m n, m < n ∧ (fun a b ↦ r a.1 b.1 ∧ s a.2 b.2) (f m) (f n)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Function.c... | [
"α : Type u_1\nβ : Type u_2\nr : α → α → Prop\ns : β → β → Prop\ninst✝ : IsPreorder α r\nhr : WellQuasiOrdered r\nhs : WellQuasiOrdered s\nf : ℕ → α × β\ng : ℕ ↪o ℕ\nh₁ : ∀ (m n : ℕ), m ≤ n → r ((Prod.fst ∘ f) (g m)) ((Prod.fst ∘ f) (g n))\n⊢ ∃ m n, m < n ∧ (fun a b ↦ r a.1 b.1 ∧ s a.2 b.2) (f m) (f n)"
] | obtain ⟨g, h₁⟩ := hr.exists_monotone_subseq (Prod.fst ∘ f) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Interval.Finset.Basic | {
"line": 586,
"column": 70
} | {
"line": 587,
"column": 99
} | {
"line": 589,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a ≤ b\n⊢ insert a (Ioc a b) = Icc a b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Set.insert_eq",
"congrArg",
"Finset",... | [] | by
rw [← coe_inj, coe_insert, coe_Ioc, coe_Icc, Set.insert_eq, Set.union_comm, Set.Ioc_union_left h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Basic | {
"line": 636,
"column": 32
} | {
"line": 636,
"column": 49
} | {
"line": 636,
"column": 49
} | [
{
"pp": "α : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b : α\nh : a ≤ b\n⊢ Icc a b = insert a (Ioc a b)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Classical.propDecidable... | [
"α : Type u_2\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b : α\nh : a ≤ b\n⊢ Icc a b = Icc a b"
] | Ioc_insert_left h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 107,
"column": 54
} | {
"line": 107,
"column": 64
} | {
"line": 107,
"column": 64
} | [
{
"pp": "M : Type u_3\ninst✝ : Monoid M\nH K : Submonoid M\n⊢ H ⊔ closure ↑K ≤ H ⊔ K",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Monoid.toMulOneClass",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"M : Type u_3\ninst✝ : Monoid M\nH K : Submonoid M\n⊢ H ⊔ K ≤ H ⊔ K"
] | closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 173,
"column": 8
} | {
"line": 173,
"column": 19
} | {
"line": 173,
"column": 20
} | [
{
"pp": "case a\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ closure s⁻¹ ≤ (closure s)⁻¹",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submonoid.inv",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"PartialOrder.toPreorde... | [
"case a\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ s⁻¹ ⊆ ↑(closure s)⁻¹"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 175,
"column": 16
} | {
"line": 175,
"column": 27
} | {
"line": 175,
"column": 28
} | [
{
"pp": "case a\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ closure s ≤ (closure s⁻¹)⁻¹",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submonoid.inv",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"PartialOrder.toPreorde... | [
"case a\nG : Type u_2\ninst✝ : Group G\ns : Set G\n⊢ s ⊆ ↑(closure s⁻¹)⁻¹"
] | closure_le, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.WellFoundedSet | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 19
} | {
"line": 108,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\nβ : Type u_3\nr : α → α → Prop\nf : β → α\nf' : β → ↑(range f) := fun c ↦ ⟨f c, ⋯⟩\nh : WellFounded (r on f)\nc : β\n⊢ ∀ {a : β} {b : ↑(range f)}, Subrel r (fun x ↦ x ∈ range f) b (f' a) → ∃ c, f' c = b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants":... | [] | rintro _ ⟨_, c', rfl⟩ -
exact ⟨c', rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.WellFoundedSet | {
"line": 106,
"column": 4
} | {
"line": 107,
"column": 19
} | {
"line": 108,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\nβ : Type u_3\nr : α → α → Prop\nf : β → α\nf' : β → ↑(range f) := fun c ↦ ⟨f c, ⋯⟩\nh : WellFounded (r on f)\nc : β\n⊢ ∀ {a : β} {b : ↑(range f)}, Subrel r (fun x ↦ x ∈ range f) b (f' a) → ∃ c, f' c = b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants":... | [] | rintro _ ⟨_, c', rfl⟩ -
exact ⟨c', rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellFoundedSet | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 47
} | {
"line": 306,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns : Set α\nhs : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ s) → ∃ m n, m < n ∧ r (f m) (f n)\nhf : ∀ a₁ ∈ s, ∀ a₂ ∈ s, r a₁ a₂ → r' (f a₁) (f a₂)\ng : ℕ → α\nhgs : ∀ (n : ℕ), g n ∈ s\nheq : ∀ (n : ℕ), f (g n) = (f ∘ g) n\nm n : ℕ\nhlt : ... | [] | exact ⟨m, n, hlt, hf _ (hgs m) _ (hgs n) hmn⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Subgroup.Center | {
"line": 95,
"column": 6
} | {
"line": 98,
"column": 15
} | {
"line": 99,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nh : center G = ⊤\n⊢ ∀ (a b : G), a * b = b * a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Group",
"Subgroup.mem_center_iff",
"Membership.mem",
"Eq.mp",
... | [] | rw [eq_top_iff'] at h
intro x y
apply Subgroup.mem_center_iff.mp _ x
exact h y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Subgroup.Center | {
"line": 95,
"column": 6
} | {
"line": 98,
"column": 15
} | {
"line": 99,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nh : center G = ⊤\n⊢ ∀ (a b : G), a * b = b * a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Group",
"Subgroup.mem_center_iff",
"Membership.mem",
"Eq.mp",
... | [] | rw [eq_top_iff'] at h
intro x y
apply Subgroup.mem_center_iff.mp _ x
exact h y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.WellFoundedSet | {
"line": 885,
"column": 4
} | {
"line": 886,
"column": 14
} | {
"line": 887,
"column": 4
} | [
{
"pp": "case h.left\nα : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\ns : Set (Lex (α × β))\nhα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x ↦ (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)\nhβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO\nf : ℕ → Lex (α × β)\nhf : ∀ (n : ℕ), f n ∈ s\ng : ℕ ↪o ℕ... | [
"case h.right\nα : Type u_2\nβ : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\ns : Set (Lex (α × β))\nhα : ∀ (f : ℕ → α), (∀ (n : ℕ), f n ∈ (fun x ↦ (ofLex x).1) '' s) → ∃ g, Monotone (f ∘ ⇑g)\nhβ : ∀ (a : α), {y | toLex (a, y) ∈ s}.IsPWO\nf : ℕ → Lex (α × β)\nhf : ∀ (n : ℕ), f n ∈ s\ng : ℕ ↪o ℕ\nhg : Mono... | · by_contra hx
simp_all | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 249,
"column": 54
} | {
"line": 249,
"column": 64
} | {
"line": 249,
"column": 64
} | [
{
"pp": "G : Type u_2\ninst✝ : Group G\nH K : Subgroup G\n⊢ H ⊔ closure ↑K ≤ H ⊔ K",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Subgroup.closure_eq",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Subgroup.closure",
"congrArg",
"PartialOrder.toPreorder",
... | [
"G : Type u_2\ninst✝ : Group G\nH K : Subgroup G\n⊢ H ⊔ K ≤ H ⊔ K"
] | closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Subgroup.Pointwise | {
"line": 314,
"column": 2
} | {
"line": 314,
"column": 35
} | {
"line": 315,
"column": 2
} | [
{
"pp": "case mpr\nG : Type u_2\ninst✝ : Group G\nA B C : Subgroup G\nh : A ≤ C\nx✝ : G\n⊢ (∃ x ∈ ↑A, ∃ y ∈ ↑B, x * y = x✝) ∧ x✝ ∈ ↑C → ∃ x ∈ ↑A, ∃ y, (y ∈ ↑B ∧ y ∈ ↑C) ∧ x * y = x✝",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"Membe... | [
"case mpr\nG : Type u_2\ninst✝ : Group G\nA B C : Subgroup G\nh : A ≤ C\ny : G\nhy : y ∈ ↑A\nz : G\nhz : z ∈ ↑B\nhyz : y * z ∈ ↑C\n⊢ ∃ x ∈ ↑A, ∃ y_1, (y_1 ∈ ↑B ∧ y_1 ∈ ↑C) ∧ x * y_1 = y * z"
] | rintro ⟨⟨y, hy, z, hz, rfl⟩, hyz⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 59
} | {
"line": 141,
"column": 59
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\nh : Disjoint s₁ s₂\n⊢ ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Finset.instUn... | [
"ι : Type u_1\nM : Type u_4\ns₁ s₂ : Finset ι\ninst✝¹ : CommMonoid M\nf : ι → M\ninst✝ : DecidableEq ι\nh : Disjoint s₁ s₂\n⊢ ∏ x ∈ s₁ ∪ s₂, f x = (∏ x ∈ s₁ ∪ s₂, f x) * ∏ x ∈ ∅, f x"
] | rw [← prod_union_inter, disjoint_iff_inter_eq_empty.mp h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Coset.Defs | {
"line": 226,
"column": 2
} | {
"line": 227,
"column": 18
} | {
"line": 228,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\nx : α\n⊢ x ∈ mk ⁻¹' mk '' s ↔ x ∈ ⋃ x, (fun x_1 ↦ x_1 * ↑x) ⁻¹' s",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Iff.of_eq",
"Monoid.to... | [
"α : Type u_1\ninst✝ : Group α\nN : Subgroup α\ns : Set α\nx : α\n⊢ (∃ x_1 ∈ s, x_1⁻¹ * x ∈ N) ↔ ∃ x_1 ∈ N, x * x_1 ∈ s"
] | simp only [QuotientGroup.eq, SetLike.exists, exists_prop, Set.mem_preimage, Set.mem_iUnion,
Set.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 107,
"column": 42
} | {
"line": 107,
"column": 85
} | {
"line": 107,
"column": 85
} | [
{
"pp": "α : Type u\nL1 L2 : List (α × Bool)\nx : α\nb : Bool\n⊢ (L1 ++ L2).length + 2 = (L1 ++ (x, b) :: (x, !b) :: L2).length",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Bool.not",
"congrArg",
"List.length_append",
"id",
"Prod.mk",
... | [
"α : Type u\nL1 L2 : List (α × Bool)\nx : α\nb : Bool\n⊢ L1.length + L2.length + 2 = L1.length + ((x, b) :: (x, !b) :: L2).length"
] | rw [List.length_append, List.length_append] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 504,
"column": 4
} | {
"line": 508,
"column": 44
} | {
"line": 509,
"column": 2
} | [] | [
"case calc_1\nι : 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)... | ∏ x ∈ s, f x = ∏ x ∈ s with f x ≠ 1, f x := by rw [prod_filter_ne_one]
_ = ∏ x ∈ t with g x ≠ 1, g x :=
prod_bij (fun a ha => i a (mem_filter.mp ha).1 <| by simpa using (mem_filter.mp ha).2)
?_ ?_ ?_ ?_
_ = ∏ x ∈ t, g x := prod_filter_ne_one _ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 57
} | {
"line": 659,
"column": 4
} | [
{
"pp": "case succ\nM : Type u_4\ninst✝ : CommMonoid M\nf : ℕ → M\nn : ℕ\nih : ∏ r ∈ range (n + 1), f (n - r) = ∏ k ∈ range (n + 1), f k\n⊢ ∏ r ∈ range (n + 1 + 1), f (n + 1 - r) = ∏ k ∈ range (n + 1 + 1), f k",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.pr... | [
"case succ\nM : Type u_4\ninst✝ : CommMonoid M\nf : ℕ → M\nn : ℕ\nih : ∏ r ∈ range (n + 1), f (n - r) = ∏ k ∈ range (n + 1), f k\n⊢ (∏ k ∈ range (n + 1), f (n + 1 - (k + 1))) * f (n + 1 - 0) = (∏ x ∈ range n.succ, f x) * f n.succ"
] | rw [prod_range_succ', prod_range_succ _ (Nat.succ n)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 675,
"column": 8
} | {
"line": 675,
"column": 26
} | {
"line": 675,
"column": 27
} | [
{
"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 ... | [
"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 a ha ∈ t), (... | ← prod_sdiff this, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Finiteness | {
"line": 478,
"column": 4
} | {
"line": 478,
"column": 35
} | {
"line": 480,
"column": 0
} | [
{
"pp": "case refine_2\nG : Type u_3\ninst✝ : Group G\nS : Set G\nhfin : Finite ↑S\nφ : FreeGroup ↑S →* G\nhφ : Function.Surjective ⇑φ\n⊢ FG G",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Group.fg_of_surjective",
"Set.Elem",
"instFGFreeGroupOfFinite",
"FreeG... | [] | exact Group.fg_of_surjective hφ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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