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.Algebra.BigOperators.Group.Finset.Basic | {
"line": 267,
"column": 62
} | {
"line": 267,
"column": 86
} | {
"line": 268,
"column": 4
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\ns : Finset ι\nt : Finset κ\ninst✝ : DecidableEq κ\ng : ι → κ\nh : ∀ i ∈ s, g i ∈ t\nf : κ → M\ny : κ\nx✝ : y ∈ t\nx : ι\nhx : x ∈ {i ∈ s | g i = y}\n⊢ f y = f (g x)",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
... | [] | rw [(mem_filter.1 hx).2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 267,
"column": 62
} | {
"line": 267,
"column": 86
} | {
"line": 268,
"column": 4
} | [
{
"pp": "ι : Type u_1\nκ : Type u_2\nM : Type u_4\ninst✝¹ : CommMonoid M\ns : Finset ι\nt : Finset κ\ninst✝ : DecidableEq κ\ng : ι → κ\nh : ∀ i ∈ s, g i ∈ t\nf : κ → M\ny : κ\nx✝ : y ∈ t\nx : ι\nhx : x ∈ {i ∈ s | g i = y}\n⊢ f y = f (g x)",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
... | [] | rw [(mem_filter.1 hx).2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 331,
"column": 6
} | {
"line": 333,
"column": 44
} | {
"line": 335,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M\n⊢ (∏ a ∈ s with p a, if p a then f a else 1) = ∏ a ∈ s, if p a then f a else 1",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toM... | [] | { refine prod_subset (filter_subset _ s) fun x hs h => ?_
rw [mem_filter, not_and] at h
exact if_neg (by simpa using h hs) } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 331,
"column": 6
} | {
"line": 333,
"column": 44
} | {
"line": 335,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ns : Finset ι\ninst✝¹ : CommMonoid M\np : ι → Prop\ninst✝ : DecidablePred p\nf : ι → M\n⊢ (∏ a ∈ s with p a, if p a then f a else 1) = ∏ a ∈ s, if p a then f a else 1",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toM... | [] | { refine prod_subset (filter_subset _ s) fun x hs h => ?_
rw [mem_filter, not_and] at h
exact if_neg (by simpa using h hs) } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 48
} | {
"line": 312,
"column": 4
} | [
{
"pp": "α : Type u\nL₂ : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nH1 : (x1, b1) ≠ (x2, b2)\nL₃ L₄ : List (α × Bool)\nh₂✝ : Red L₄ L₂\nh₁✝ : Red ((x1, b1) :: L₃) [(x2, b2)]\nH2 : Red ((x1, b1) :: L₃.append L₄) ((x2, b2) :: L₂)\nthis : Red ((x1, b1) :: L₃.append L₄) ([(x2, b2)] ++ L₂)\nh₁ : Red ((x... | [
"α : Type u\nL₂ : List (α × Bool)\nx1 : α\nb1 : Bool\nx2 : α\nb2 : Bool\nH1 : (x1, b1) ≠ (x2, b2)\nL₃ L₄ : List (α × Bool)\nh₂✝¹ : Red L₄ L₂\nh₁✝¹ : Red ((x1, b1) :: L₃) [(x2, b2)]\nH2 : Red ((x1, b1) :: L₃.append L₄) ((x2, b2) :: L₂)\nthis : Red ((x1, b1) :: L₃.append L₄) ([(x2, b2)] ++ L₂)\nh₁✝ : Red ((x1, !b1) :... | rcases church_rosser h₁ h₂ with ⟨L', h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 625,
"column": 4
} | {
"line": 625,
"column": 64
} | {
"line": 626,
"column": 4
} | [
{
"pp": "case succ\nM : Type u_4\ninst✝ : CommMonoid M\nf s : ℕ → M\nbase : s 0 = 1\nk : ℕ\nhk : (∀ k_1 < k, s (k_1 + 1) = s k_1 * f k_1) → ∏ k ∈ range k, f k = s k\nstep : ∀ k_1 < k + 1, s (k_1 + 1) = s k_1 * f k_1\n⊢ ∏ k ∈ range (k + 1), f k = s (k + 1)",
"ppTerm": "?succ",
"assigned": true,
"used... | [
"case succ\nM : Type u_4\ninst✝ : CommMonoid M\nf s : ℕ → M\nbase : s 0 = 1\nk : ℕ\nhk : (∀ k_1 < k, s (k_1 + 1) = s k_1 * f k_1) → ∏ k ∈ range k, f k = s k\nstep : ∀ k_1 < k + 1, s (k_1 + 1) = s k_1 * f k_1\n⊢ ∀ k_1 < k, s (k_1 + 1) = s k_1 * f k_1"
] | rw [Finset.prod_range_succ, step _ (Nat.lt_succ_self _), hk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeGroup.Basic | {
"line": 350,
"column": 71
} | {
"line": 356,
"column": 61
} | {
"line": 358,
"column": 0
} | [
{
"pp": "α : Type u\nL₁ L₂ : List (α × Bool)\nh : Red L₁ L₂\n⊢ ∃ n, L₁.length = L₂.length + 2 * n",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"FreeGroup.Red.Step",
"HMul.hMul",
"congrArg",
"add_assoc",
"Nat.instMulOneClass",
"Exists",
"instMulN... | [] | by
induction h with
| refl => exact ⟨0, rfl⟩
| tail _h₁₂ h₂₃ ih =>
rcases ih with ⟨n, eq⟩
exists 1 + n
simp [Nat.mul_add, eq, (Step.length h₂₃).symm, add_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Finiteness | {
"line": 329,
"column": 59
} | {
"line": 329,
"column": 63
} | {
"line": 329,
"column": 63
} | [
{
"pp": "case mpr.refine_1\nG : Type u_3\ninst✝ : Group G\nP : Subgroup G\nS : Finset G\nhS : Submonoid.closure ↑S = P.toSubmonoid\n⊢ ↑S ⊆ ↑P.toSubmonoid",
"ppTerm": "?mpr.refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | [
"case mpr.refine_1\nG : Type u_3\ninst✝ : Group G\nP : Subgroup G\nS : Finset G\nhS : Submonoid.closure ↑S = P.toSubmonoid\n⊢ ↑S ⊆ ↑(Submonoid.closure ↑S)"
] | ← hS | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Finiteness | {
"line": 558,
"column": 8
} | {
"line": 558,
"column": 34
} | {
"line": 558,
"column": 34
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝⁵ : Monoid M\nG✝ : Type u_3\nH : Type u_4\ninst✝⁴ : Group G✝\ninst✝³ : AddGroup H\nι : Type u_5\ninst✝² : Finite ι\nG : ι → Type u_6\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : ∀ (i : ι), Group.FG (G i)\n⊢ ⊤.FG",
"ppTerm": "?m.8",
"assigned": true,
"usedConstan... | [
"M : Type u_1\nN : Type u_2\ninst✝⁵ : Monoid M\nG✝ : Type u_3\nH : Type u_4\ninst✝⁴ : Group G✝\ninst✝³ : AddGroup H\nι : Type u_5\ninst✝² : Finite ι\nG : ι → Type u_6\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : ∀ (i : ι), Group.FG (G i)\n⊢ (Subgroup.pi Set.univ fun i ↦ ⊤).FG"
] | ← Subgroup.pi_top Set.univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 54
} | {
"line": 185,
"column": 2
} | [
{
"pp": "α : Type u\nG : Type u_1\ninst✝ : AddCommGroup G\nf g : α → G\na : FreeAbelianGroup α\n⊢ (lift (f + g)) a = (lift f) a + (lift g) a",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"AddMonoidHom.map_zero",
"AddMonoidHom.instAddMonoidHomClass"... | [] | induction a using FreeAbelianGroup.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.BigOperators.Group.Finset.Basic | {
"line": 1024,
"column": 21
} | {
"line": 1024,
"column": 48
} | {
"line": 1024,
"column": 48
} | [
{
"pp": "M : Type u_4\nι : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : CommMonoid M\ninst✝ : Subsingleton ι\nf : ι → M\na : ι\nthis : Unique ι\n⊢ f default = f a",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"congrArg",
"id",
"Uniqu... | [
"M : Type u_4\nι : Type u_7\ninst✝² : Fintype ι\ninst✝¹ : CommMonoid M\ninst✝ : Subsingleton ι\nf : ι → M\na : ι\nthis : Unique ι\n⊢ f a = f a"
] | Subsingleton.elim default a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 546,
"column": 8
} | {
"line": 546,
"column": 39
} | {
"line": 546,
"column": 39
} | [
{
"pp": "α : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\nT : Type u_2\ninst✝ : Unique T\nz : FreeAbelianGroup T\n⊢ (fun n ↦ n • of default) ((lift fun x ↦ 1) 0) = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Inhabited.default",
"instHSMul",... | [] | simp only [zero_smul, map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 546,
"column": 8
} | {
"line": 546,
"column": 39
} | {
"line": 546,
"column": 39
} | [
{
"pp": "α : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\nT : Type u_2\ninst✝ : Unique T\nz : FreeAbelianGroup T\n⊢ (fun n ↦ n • of default) ((lift fun x ↦ 1) 0) = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Inhabited.default",
"instHSMul",... | [] | simp only [zero_smul, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeAbelianGroup | {
"line": 546,
"column": 8
} | {
"line": 546,
"column": 39
} | {
"line": 546,
"column": 39
} | [
{
"pp": "α : Type u\nG : Type u_1\nβ : Type v\nγ : Type w\nT : Type u_2\ninst✝ : Unique T\nz : FreeAbelianGroup T\n⊢ (fun n ↦ n • of default) ((lift fun x ↦ 1) 0) = 0",
"ppTerm": "?m.93",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Inhabited.default",
"instHSMul",... | [] | simp only [zero_smul, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.OreLocalization.Basic | {
"line": 538,
"column": 24
} | {
"line": 538,
"column": 50
} | {
"line": 538,
"column": 50
} | [
{
"pp": "case c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nφ : OreLocalization S R →* T\nhuniv : ∀ (r : R), φ (numeratorHom r) = f r\nr : R\ns : ↥S\n⊢ ↑(fS s)⁻¹ * φ (↑s /ₒ 1 * (r /ₒ s)) = ↑(fS ... | [
"case c\nR : Type u_1\ninst✝² : Monoid R\nS : Submonoid R\ninst✝¹ : OreSet S\nT : Type u_2\ninst✝ : Monoid T\nf : R →* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nφ : OreLocalization S R →* T\nhuniv : ∀ (r : R), φ (numeratorHom r) = f r\nr : R\ns : ↥S\n⊢ ↑(fS s)⁻¹ * φ (r /ₒ 1) = ↑(fS s)⁻¹ * φ (r /ₒ 1)"
] | OreLocalization.mul_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.MonoidLocalization.Maps | {
"line": 207,
"column": 26
} | {
"line": 208,
"column": 47
} | {
"line": 210,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝² : CommMonoid N\nf : S.LocalizationMap N\nT : Submonoid M\nhST : S ≤ T\nQ : Type u_4\ninst✝¹ : CommMonoid Q\nk : T.LocalizationMap Q\nA : Type u_5\ninst✝ : CommMonoid A\nl : M →* A\nhl : ∀ (w : ↥T), IsUnit (l ↑w)\nx : M\n⊢ ((k.li... | [] | by rw [← toMonoidHom_apply, ← MonoidHom.comp_apply,
MonoidHom.comp_assoc, lift_comp, lift_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SuccPred.Basic | {
"line": 420,
"column": 64
} | {
"line": 422,
"column": 45
} | {
"line": 424,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : SuccOrder α\na b : α\nh : a < succ b\n⊢ a ≤ b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Order.succ",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
... | [] | by
by_contra! nh
exact (h.trans_le (succ_le_of_lt nh)).false | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SuccPred.Basic | {
"line": 612,
"column": 41
} | {
"line": 612,
"column": 67
} | {
"line": 612,
"column": 68
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : SuccOrder α\na : α\n⊢ sInf (Ioi a) = sInf (range fun x ↦ ↑x)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"setOf",
... | [
"α : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : SuccOrder α\na : α\n⊢ sInf (Ioi a) = sInf {x | x > a}"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.SuccPred | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 24
} | {
"line": 276,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : LinearOrder α\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : SuccAddOrder α\ninst✝² : IsBotZeroClass α\ninst✝¹ : NeZero 1\ninst✝ : NoMaxOrder α\n⊢ Set.Iic 2 = {0, 1, 2}",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"AddMono... | [] | ext; simp [le_two_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.SuccPred | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 24
} | {
"line": 276,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : LinearOrder α\ninst✝⁴ : AddMonoidWithOne α\ninst✝³ : SuccAddOrder α\ninst✝² : IsBotZeroClass α\ninst✝¹ : NeZero 1\ninst✝ : NoMaxOrder α\n⊢ Set.Iic 2 = {0, 1, 2}",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"AddMono... | [] | ext; simp [le_two_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.Basic | {
"line": 685,
"column": 4
} | {
"line": 699,
"column": 22
} | {
"line": 701,
"column": 0
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nn : ℕ\nhn : ¬IsMax (succ^[n - 1] i) → pred^[n] (succ^[n] i) = i\nhin : ¬IsMax (succ^[n + 1 - 1] i)\n⊢ pred^[n + 1] (succ^[n + 1] i) = i",
"ppTerm": "?succ",
"assigned": true,
"usedConstants":... | [] | rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hin
have h_not_max : ¬IsMax (succ^[n - 1] i) := by
rcases n with - | n
· simpa using hin
rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hn ⊢
have h_sub_le : succ^[n] i ≤ succ^[n.succ] i := by
rw [Function.iterate_succ']
exact le_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.Basic | {
"line": 685,
"column": 4
} | {
"line": 699,
"column": 22
} | {
"line": 701,
"column": 0
} | [
{
"pp": "case succ\nα : Type u_1\ninst✝² : PartialOrder α\ninst✝¹ : SuccOrder α\ninst✝ : PredOrder α\ni : α\nn : ℕ\nhn : ¬IsMax (succ^[n - 1] i) → pred^[n] (succ^[n] i) = i\nhin : ¬IsMax (succ^[n + 1 - 1] i)\n⊢ pred^[n + 1] (succ^[n + 1] i) = i",
"ppTerm": "?succ",
"assigned": true,
"usedConstants":... | [] | rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hin
have h_not_max : ¬IsMax (succ^[n - 1] i) := by
rcases n with - | n
· simpa using hin
rw [Nat.succ_sub_succ_eq_sub, Nat.sub_zero] at hn ⊢
have h_sub_le : succ^[n] i ≤ succ^[n.succ] i := by
rw [Function.iterate_succ']
exact le_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Sum | {
"line": 50,
"column": 63
} | {
"line": 50,
"column": 84
} | {
"line": 52,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : Fintype β\nu : Finset (α ⊕ β)\n⊢ u.toLeft = ∅ ↔ u ⊆ map Function.Embedding.inr univ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.toLeft",
"Finset.univ",
"and_true",
"congrArg",
"Finset.subset_univ._simp_1... | [] | simp [subset_map_inr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Fintype.Sum | {
"line": 50,
"column": 63
} | {
"line": 50,
"column": 84
} | {
"line": 52,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : Fintype β\nu : Finset (α ⊕ β)\n⊢ u.toLeft = ∅ ↔ u ⊆ map Function.Embedding.inr univ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.toLeft",
"Finset.univ",
"and_true",
"congrArg",
"Finset.subset_univ._simp_1... | [] | simp [subset_map_inr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Sum | {
"line": 50,
"column": 63
} | {
"line": 50,
"column": 84
} | {
"line": 52,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝ : Fintype β\nu : Finset (α ⊕ β)\n⊢ u.toLeft = ∅ ↔ u ⊆ map Function.Embedding.inr univ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.toLeft",
"Finset.univ",
"and_true",
"congrArg",
"Finset.subset_univ._simp_1... | [] | simp [subset_map_inr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Control.Applicative | {
"line": 57,
"column": 6
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 4
} | [
{
"pp": "F : Type u → Type u_1\nF1 : Functor F\np1 : {α : Type u} → α → F α\ns1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β\nsl1 : {α β : Type u} → F α → (Unit → F β) → F α\nsr1 : {α β : Type u} → F α → (Unit → F β) → F β\nF2 : Functor F\ns2 : {α β : Type u} → F (α → β) → (Unit → F α) → F β\nsl2 : {α β : ... | [] | funext α β f x
exact H2 f (x Unit.unit) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Control.Applicative | {
"line": 57,
"column": 6
} | {
"line": 58,
"column": 30
} | {
"line": 59,
"column": 4
} | [
{
"pp": "F : Type u → Type u_1\nF1 : Functor F\np1 : {α : Type u} → α → F α\ns1 : {α β : Type u} → F (α → β) → (Unit → F α) → F β\nsl1 : {α β : Type u} → F α → (Unit → F β) → F α\nsr1 : {α β : Type u} → F α → (Unit → F β) → F β\nF2 : Functor F\ns2 : {α β : Type u} → F (α → β) → (Unit → F α) → F β\nsl2 : {α β : ... | [] | funext α β f x
exact H2 f (x Unit.unit) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sym.Basic | {
"line": 552,
"column": 53
} | {
"line": 554,
"column": 35
} | {
"line": 556,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\na x : α\ni : Fin (n + 1)\ns : Sym α (n - ↑i)\nhx : x ≠ a\n⊢ count x ↑(fill a i s) = count x ↑s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"instNeZeroNatHAdd_1",
"False",
"eq_false",
"congrArg",
"AddMono... | [] | by
suffices x ∉ Multiset.replicate i a by simp [coe_fill, coe_replicate, this]
simp [Multiset.mem_replicate, hx] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Hom.Lex | {
"line": 171,
"column": 38
} | {
"line": 171,
"column": 69
} | {
"line": 172,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : Unique α\ninst✝ : LE β\nx : Lex (α × β)\nx✝ : α × β\na : α\nb : β\n⊢ (fun x ↦ toLex (default, x)) ((fun x ↦ (ofLex x).2) (toLex (a, b))) = toLex (a, b)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"E... | [] | simpa using Unique.default_eq a | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Hom.Lex | {
"line": 171,
"column": 38
} | {
"line": 171,
"column": 69
} | {
"line": 172,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : Unique α\ninst✝ : LE β\nx : Lex (α × β)\nx✝ : α × β\na : α\nb : β\n⊢ (fun x ↦ toLex (default, x)) ((fun x ↦ (ofLex x).2) (toLex (a, b))) = toLex (a, b)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"E... | [] | simpa using Unique.default_eq a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Hom.Lex | {
"line": 171,
"column": 38
} | {
"line": 171,
"column": 69
} | {
"line": 172,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝² : Preorder α\ninst✝¹ : Unique α\ninst✝ : LE β\nx : Lex (α × β)\nx✝ : α × β\na : α\nb : β\n⊢ (fun x ↦ toLex (default, x)) ((fun x ↦ (ofLex x).2) (toLex (a, b))) = toLex (a, b)",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"E... | [] | simpa using Unique.default_eq a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Defs | {
"line": 473,
"column": 2
} | {
"line": 473,
"column": 84
} | {
"line": 475,
"column": 0
} | [
{
"pp": "ι : Type u\nc : ι → Type v\n⊢ #(ULift ((i : ι) → c i)) = #((i : ι) → ULift (c i))",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Equiv.trans",
"ULift",
"Equiv.ulift",
"Equiv.piCongrRight",
"Equiv.symm",
"Cardinal.mk_congr"
],
"usedFVar... | [] | exact mk_congr (Equiv.ulift.trans <| Equiv.piCongrRight fun i => Equiv.ulift.symm) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Sum.Order | {
"line": 554,
"column": 38
} | {
"line": 554,
"column": 65
} | {
"line": 554,
"column": 66
} | [
{
"pp": "case inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\n... | [
"case inl.inl.inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\nb : ... | rcases b with ((_ | _) | _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Sum.Order | {
"line": 554,
"column": 38
} | {
"line": 554,
"column": 65
} | {
"line": 554,
"column": 66
} | [
{
"pp": "case inl.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\n... | [
"case inl.inr.inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\nb : ... | rcases b with ((_ | _) | _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Sum.Order | {
"line": 554,
"column": 38
} | {
"line": 554,
"column": 65
} | {
"line": 554,
"column": 66
} | [
{
"pp": "case inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\nb✝ :... | [
"case inr.inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\nb : β✝\n... | rcases b with ((_ | _) | _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inl.inl.inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\n... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inl.inl.inl.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\n... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inl.inl.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : ... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inl.inr.inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\n... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inl.inr.inl.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\n... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inl.inr.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : ... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inr.inl.inl\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : ... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inr.inl.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : ... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sum.Order | {
"line": 555,
"column": 6
} | {
"line": 555,
"column": 27
} | {
"line": 555,
"column": 28
} | [
{
"pp": "case inr.inr\nα✝ : Type u_1\nβ✝ : Type u_2\nγ✝ : Type u_3\nα₁ : Type u_4\nα₂ : Type u_5\nβ₁ : Type u_6\nβ₂ : Type u_7\nγ₁ : Type u_8\nγ₂ : Type u_9\ninst✝¹¹ : LE α✝\ninst✝¹⁰ : LE β✝\ninst✝⁹ : LE γ✝\ninst✝⁸ : LE α₁\ninst✝⁷ : LE α₂\ninst✝⁶ : LE β₁\ninst✝⁵ : LE β₂\ninst✝⁴ : LE γ₁\ninst✝³ : LE γ₂\na : α✝\n... | [] | simp [Equiv.sumAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.UpperLower.Basic | {
"line": 161,
"column": 6
} | {
"line": 161,
"column": 23
} | {
"line": 161,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ↪o β\nhe : IsUpperSet (range ⇑e)\na : α\n⊢ ⇑e '' Ioi a = Ioi (e a)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"congrArg",
"Preorder.toLE",
"id",
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ↪o β\nhe : IsUpperSet (range ⇑e)\na : α\n⊢ ⇑e '' ⇑e ⁻¹' Ioi (e a) = Ioi (e a)"
] | ← e.preimage_Ioi, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Hom.Order | {
"line": 109,
"column": 58
} | {
"line": 110,
"column": 29
} | {
"line": 112,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\nι : Sort u_3\ninst✝ : CompleteLattice β\nf : ι → α →o β\n⊢ ⇑(⨆ i, f i) = ⨆ i, ⇑(f i)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"congrArg",
"iSup",
"OrderHom.instSupSet",
"PartialOrder.toPreorder",
... | [] | by
funext x; simp [iSup_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.BourbakiWitt | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 22
} | {
"line": 133,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\ny : α\nle_map : ∀ (x : α), x ≤ f x\nhy : IsExtremePt x f y\n⊢ {z | z ∈ bot x f ∧ (z ≤ y ∨ f y ≤ z)} = bot x f",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.subse... | [
"α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\ny : α\nle_map : ∀ (x : α), x ≤ f x\nhy : IsExtremePt x f y\n⊢ {z | z ∈ bot x f ∧ (z ≤ y ∨ f y ≤ z)} ⊆ bot x f",
"α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\ny : α\nle_map : ∀ (x : α), x ≤ f x\nhy : IsExtremePt x f y\n⊢ ... | ← subset_bot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Zorn | {
"line": 130,
"column": 4
} | {
"line": 131,
"column": 55
} | {
"line": 133,
"column": 0
} | [
{
"pp": "case refine_1.inr\nα : Type u_1\ninst✝ : Preorder α\ns : Set α\nih : ∀ c ⊆ s, IsChain (fun x1 x2 ↦ x1 ≤ x2) c → ∀ y ∈ c, ∃ ub ∈ s, ∀ z ∈ c, z ≤ ub\nx : α\nhxs : x ∈ s\nc : Set α\nhcs : c ⊆ {y | y ∈ s ∧ x ≤ y}\nhc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\ny : α\nhy : y ∈ c\n⊢ ∃ ub ∈ {y | y ∈ s ∧ x ≤ y}, ∀ z ∈ ... | [] | · rcases ih c (fun z hz => (hcs hz).1) hc y hy with ⟨z, hzs, hz⟩
exact ⟨z, ⟨hzs, (hcs hy).2.trans <| hz _ hy⟩, hz⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.BourbakiWitt | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 37
} | {
"line": 188,
"column": 4
} | [
{
"pp": "case cSup_mem\nα : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\nc : NonemptyChain α\nhc : ↑c ⊆ {y | IsExtremePt x f y}\ny : α\nhy : y ∈ bot x f\nhy' : y < cSup c\nz : α\nhz : z ∈ c\nhzy : ¬f z ≤ y\n⊢ f y ≤ cSup c",
"ppTerm": "?cSup_mem",
"assigne... | [
"case cSup_mem\nα : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\nc : NonemptyChain α\nhc : ↑c ⊆ {y | IsExtremePt x f y}\ny : α\nhy : y ∈ bot x f\nhy' : y < cSup c\nz : α\nhz : z ∈ c\nhzy : ¬f z ≤ y\nh : y ≤ z\n⊢ f y ≤ cSup c"
] | have h : y ≤ z := by
rw [← bot_eq_of_le_or_map_le le_map (hc hz)] at hy
exact Or.resolve_right hy.2 hzy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Order.BourbakiWitt | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 22
} | {
"line": 197,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\n⊢ {y | IsExtremePt x f y} = bot x f",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.subset_bot_iff",
"congrArg",
"setOf",
... | [
"α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\n⊢ {y | IsExtremePt x f y} ⊆ bot x f",
"α : Type u_1\ninst✝ : ChainCompletePartialOrder α\nx : α\nf : α → α\nle_map : ∀ (x : α), x ≤ f x\n⊢ IsAdmissible x f {y | IsExtremePt x f y}"
] | ← subset_bot_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 70,
"column": 58
} | {
"line": 70,
"column": 61
} | {
"line": 70,
"column": 61
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : Nonempty β\nF : Set α →o Set α := { toFun := fun s ↦ (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }\ns : Set α := OrderHom.lfp F\nhs : (g '' (f '' s)ᶜ)ᶜ ... | [
"α : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : Nonempty β\nF : Set α →o Set α := { toFun := fun s ↦ (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }\ns : Set α := OrderHom.lfp F\nhs : (g '' (f '' s)ᶜ)ᶜ = s\nhns : g... | hns | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 71,
"column": 58
} | {
"line": 71,
"column": 61
} | {
"line": 71,
"column": 61
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : Nonempty β\nF : Set α →o Set α := { toFun := fun s ↦ (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }\ns : Set α := OrderHom.lfp F\nhs : (g '' (f '' s)ᶜ)ᶜ ... | [
"α : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : Nonempty β\nF : Set α →o Set α := { toFun := fun s ↦ (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }\ns : Set α := OrderHom.lfp F\nhs : (g '' (f '' s)ᶜ)ᶜ = s\nhns : g... | hns | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.SchroederBernstein | {
"line": 75,
"column": 60
} | {
"line": 75,
"column": 63
} | {
"line": 75,
"column": 63
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : Nonempty β\nF : Set α →o Set α := { toFun := fun s ↦ (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }\ns : Set α := OrderHom.lfp F\nhs : (g '' (f '' s)ᶜ)ᶜ ... | [
"α : Type u\nβ : Type v\nf : α → β\ng : β → α\nhf : Injective f\nhg : Injective g\nR : α → β → Prop\nhp₁ : ∀ (a : α), R a (f a)\nhp₂ : ∀ (b : β), R (g b) b\nhβ : Nonempty β\nF : Set α →o Set α := { toFun := fun s ↦ (g '' (f '' s)ᶜ)ᶜ, monotone' := ⋯ }\ns : Set α := OrderHom.lfp F\nhs : (g '' (f '' s)ᶜ)ᶜ = s\nhns : g... | hns | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.OmegaCompletePartialOrder | {
"line": 691,
"column": 6
} | {
"line": 691,
"column": 31
} | {
"line": 692,
"column": 6
} | [
{
"pp": "case a\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : OmegaCompletePartialOrder α\ninst✝¹ : OmegaCompletePartialOrder β\ninst✝ : OmegaCompletePartialOrder γ\nf : α → β →𝒄 γ\nhf : ωScottContinuous f\ng : α → β\nhg : ωScottContinuous g\nc : Chain α\ni : ℕ\n⊢ (f (c i)) (ωSup (c.map { toFun := g, mon... | [
"case a\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : OmegaCompletePartialOrder α\ninst✝¹ : OmegaCompletePartialOrder β\ninst✝ : OmegaCompletePartialOrder γ\nf : α → β →𝒄 γ\nhf : ωScottContinuous f\ng : α → β\nhg : ωScottContinuous g\nc : Chain α\ni : ℕ\n⊢ ωSup ((c.map { toFun := g, monotone' := ⋯ }).map ↑(f... | rw [(f (c i)).continuous] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Cardinal.ToNat | {
"line": 44,
"column": 42
} | {
"line": 44,
"column": 57
} | {
"line": 44,
"column": 58
} | [
{
"pp": "c : Cardinal.{u}\n⊢ toENat c = 0 ∨ toENat c = ⊤ ↔ c = 0 ∨ ℵ₀ ≤ c",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"Cardinal.toENat_eq_zero",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring"... | [
"c : Cardinal.{u}\n⊢ c = 0 ∨ toENat c = ⊤ ↔ c = 0 ∨ ℵ₀ ≤ c"
] | toENat_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Order | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 51
} | {
"line": 338,
"column": 51
} | [
{
"pp": "a : Cardinal.{u}\n⊢ a < 2 ^ a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Cardinal.instPowCardinal",
"Cardinal",
"PartialOrder.toPreorder",
"Nat.instAtLeastTwoHAddOfNat",
"instOfNatNat",
"Cardinal.partialOrder",
... | [
"case mk\nα : Type u\n⊢ #α < 2 ^ #α"
] | induction a using Cardinal.inductionOn with | _ α
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 30
} | {
"line": 324,
"column": 31
} | [
{
"pp": "c : Cardinal.{u_1}\n⊢ 1 ≤ c ↔ c ≠ 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal.instOne",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"id",
"Ne",
"LE.le",
"Cardinal.instLE",
... | [
"c : Cardinal.{u_1}\n⊢ 0 < c ↔ c ≠ 0"
] | Cardinal.one_le_iff_pos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 780,
"column": 84
} | {
"line": 782,
"column": 20
} | {
"line": 784,
"column": 0
} | [
{
"pp": "α : Type u\nA : Set (Set α)\n⊢ #↑(⋃₀ A) ≤ #↑A * ⨆ s, #↑↑s",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Cardinal",
"congrArg",
"iSup",
"Set.sUnion_eq_iUnion",
"Set.sUnion",
"Cardinal.mk",
"Membership.m... | [] | by
rw [sUnion_eq_iUnion]
apply mk_iUnion_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 955,
"column": 6
} | {
"line": 955,
"column": 12
} | {
"line": 955,
"column": 12
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nt : Set β\nh : t ⊆ range fun x ↦ f ↑x\n⊢ lift.{v, u} #↑{x | x ∈ s ∧ f x ∈ t} = lift.{v, u} #↑((fun x ↦ f ↑x) ⁻¹' t)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"setOf",
... | [
"α : Type u\nβ : Type v\nf : α → β\ns : Set α\nt : Set β\nh : t ⊆ range fun x ↦ f ↑x\n⊢ lift.{v, u} #↑{x | f ↑x ∈ t} = lift.{v, u} #↑((fun x ↦ f ↑x) ⁻¹' t)"
] | mk_sep | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Basic | {
"line": 962,
"column": 6
} | {
"line": 962,
"column": 12
} | {
"line": 962,
"column": 12
} | [
{
"pp": "α β : Type u\nf : α → β\ns : Set α\nt : Set β\nh : t ⊆ range fun x ↦ f ↑x\n⊢ #↑{x | x ∈ s ∧ f x ∈ t} = #↑((fun x ↦ f ↑x) ⁻¹' t)",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"setOf",
"Cardinal.mk",
"Membership... | [
"α β : Type u\nf : α → β\ns : Set α\nt : Set β\nh : t ⊆ range fun x ↦ f ↑x\n⊢ #↑{x | f ↑x ∈ t} = #↑((fun x ↦ f ↑x) ⁻¹' t)"
] | mk_sep | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction | {
"line": 454,
"column": 75
} | {
"line": 457,
"column": 28
} | {
"line": 459,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝¹ : Group R\ninst✝ : MulAction R M\np : SubMulAction R M\n⊢ MulAction.orbitRel R ↥p = Setoid.comap Subtype.val (MulAction.orbitRel R M)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Setoid.comap",
"SubMulAction.mulAction",
"Iff.mpr... | [] | by
refine Setoid.ext_iff.2 (fun x y ↦ ?_)
rw [Setoid.comap_rel]
exact mem_orbit_subMul_iff | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.Map | {
"line": 401,
"column": 2
} | {
"line": 401,
"column": 65
} | {
"line": 402,
"column": 2
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nσ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nhf : Injective ⇑f\np q : Submodule R M\... | [
"R : Type u_1\nR₂ : Type u_3\nM : Type u_5\nM₂ : Type u_7\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M\ninst✝¹ : Module R₂ M₂\nσ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective σ₁₂\nf : M →ₛₗ[σ₁₂] M₂\nhf : Injective ⇑f\np q : Submodule R M\nh : p ⋖ q\n... | refine h.2 ?_ (Submodule.comap_lt_of_lt_map_of_injective hf h₂) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 173,
"column": 42
} | {
"line": 173,
"column": 54
} | {
"line": 173,
"column": 54
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nσ : Type u_4\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS✝ : Subsemigroup M\nf : M →ₙ* N\nS : Subsemigroup N\na✝ b✝ : M\nha : a✝ ∈ ⇑f ⁻¹' ↑S\nhb : b✝ ∈ ⇑f ⁻¹' ↑S\n⊢ f (a✝ * b✝) ∈ S",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
... | [
"M : Type u_1\nN : Type u_2\nP : Type u_3\nσ : Type u_4\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS✝ : Subsemigroup M\nf : M →ₙ* N\nS : Subsemigroup N\na✝ b✝ : M\nha : a✝ ∈ ⇑f ⁻¹' ↑S\nhb : b✝ ∈ ⇑f ⁻¹' ↑S\n⊢ f a✝ * f b✝ ∈ S"
] | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 200,
"column": 68
} | {
"line": 200,
"column": 80
} | {
"line": 200,
"column": 80
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nσ : Type u_4\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS✝ : Subsemigroup M\nf : M →ₙ* N\nS : Subsemigroup M\nx : M\nhx : x ∈ ↑S\ny : M\nhy : y ∈ ↑S\n⊢ f (x * y) = f x * f y",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"MulHo... | [] | rw [map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 200,
"column": 68
} | {
"line": 200,
"column": 80
} | {
"line": 200,
"column": 80
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nσ : Type u_4\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS✝ : Subsemigroup M\nf : M →ₙ* N\nS : Subsemigroup M\nx : M\nhx : x ∈ ↑S\ny : M\nhy : y ∈ ↑S\n⊢ f (x * y) = f x * f y",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"MulHo... | [] | rw [map_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subsemigroup.Operations | {
"line": 200,
"column": 68
} | {
"line": 200,
"column": 80
} | {
"line": 200,
"column": 80
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\nσ : Type u_4\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Mul P\nS✝ : Subsemigroup M\nf : M →ₙ* N\nS : Subsemigroup M\nx : M\nhx : x ∈ ↑S\ny : M\nhy : y ∈ ↑S\n⊢ f (x * y) = f x * f y",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"MulHo... | [] | rw [map_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Basic | {
"line": 405,
"column": 44
} | {
"line": 406,
"column": 89
} | {
"line": 408,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : FaithfulSMul R A\ninst✝ : CharZero R\n⊢ FaithfulSMul ℕ R",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | [] | by
simpa only [faithfulSMul_iff_algebraMap_injective] using (algebraMap ℕ R).injective_nat | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 219,
"column": 92
} | {
"line": 221,
"column": 46
} | {
"line": 223,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : MonoidWithZero M\na b : M\nh : a ~ᵤ b\n⊢ a = 0 ↔ b = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOneClass",
"congrArg",
"MulZeroClass.zero_mu... | [] | by
obtain ⟨u, rfl⟩ := h
rw [← Units.eq_mul_inv_iff_mul_eq, zero_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 317,
"column": 8
} | {
"line": 318,
"column": 47
} | {
"line": 318,
"column": 47
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\na b c d : M\nh : a * b ~ᵤ c * d\nh₁ : a ~ᵤ c\nha : a ≠ 0\nu : Mˣ\nhu : a * b * ↑u = c * d\nv : Mˣ\nhv : c * ↑v = a\n⊢ a * (b * ↑(u * v)) = a * d",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"CommMonoi... | [] | rw [← hv, mul_assoc c (v : M) d, mul_left_comm c, ← hu]
simp [hv.symm, mul_comm, mul_left_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Associated | {
"line": 317,
"column": 8
} | {
"line": 318,
"column": 47
} | {
"line": 318,
"column": 47
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : IsCancelMulZero M\na b c d : M\nh : a * b ~ᵤ c * d\nh₁ : a ~ᵤ c\nha : a ≠ 0\nu : Mˣ\nhu : a * b * ↑u = c * d\nv : Mˣ\nhv : c * ↑v = a\n⊢ a * (b * ↑(u * v)) = a * d",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"CommMonoi... | [] | rw [← hv, mul_assoc c (v : M) d, mul_left_comm c, ← hu]
simp [hv.symm, mul_comm, mul_left_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.NonZeroDivisors | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 32
} | {
"line": 272,
"column": 33
} | [
{
"pp": "M₀ : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : MonoidWithZero M₀\ninst✝² : MonoidWithZero S\ninst✝¹ : EquivLike F M₀ S\ninst✝ : MulEquivClass F M₀ S\nh✝ : F\nh : M₀ ≃* S := ↑h✝\nx✝ : S\n⊢ ((∀ (x : M₀), h.symm x✝ * x = 0 → x = 0) ∧ ∀ (x : M₀), x * h.symm x✝ = 0 → x = 0) ↔\n (∀ (x : S), x✝ * x = ... | [
"M₀ : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : MonoidWithZero M₀\ninst✝² : MonoidWithZero S\ninst✝¹ : EquivLike F M₀ S\ninst✝ : MulEquivClass F M₀ S\nh✝ : F\nh : M₀ ≃* S := ↑h✝\nx✝ : S\n⊢ ((∀ (a : S), h.symm x✝ * h.symm.toEquiv a = 0 → h.symm.toEquiv a = 0) ∧\n ∀ (a : S), h.symm.toEquiv a * h.symm x✝ = 0... | ← h.symm.forall_congr_right, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 14
} | {
"line": 418,
"column": 14
} | [
{
"pp": "case le\nR : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass S R M\nr : S\nm : M\nhm : m ∈ ↑N\n⊢ ∀ p ∈ {p | ∀ ⦃r_1 : S⦄ {n : M}, r_1 ∈ {r} → n ∈ N → r_1 •... | [
"case le\nR : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass S R M\nr : S\nm : M\nhm : m ∈ ↑N\np✝ : Submodule R M\nhp : p✝ ∈ {p | ∀ ⦃r_1 : S⦄ {n : M}, r_1 ∈ {r} → n ∈... | intro _ hp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.SupClosed | {
"line": 50,
"column": 78
} | {
"line": 50,
"column": 94
} | {
"line": 51,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\n⊢ SupClosed ∅",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"Membership.mem",
"SemilatticeSup.toMax",
"imp_self._simp_1",
"Max.max",
"implies_congr",
... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.SupClosed | {
"line": 50,
"column": 78
} | {
"line": 50,
"column": 94
} | {
"line": 51,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\n⊢ SupClosed ∅",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"Membership.mem",
"SemilatticeSup.toMax",
"imp_self._simp_1",
"Max.max",
"implies_congr",
... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupClosed | {
"line": 50,
"column": 78
} | {
"line": 50,
"column": 94
} | {
"line": 51,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\n⊢ SupClosed ∅",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"Membership.mem",
"SemilatticeSup.toMax",
"imp_self._simp_1",
"Max.max",
"implies_congr",
... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupClosed | {
"line": 51,
"column": 84
} | {
"line": 51,
"column": 100
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\na : α\n⊢ SupClosed {a}",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"instReflLe",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Std.le_refl._simp_1",
"Membership.mem",
"SemilatticeSup.toMax",
"Set.i... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.SupClosed | {
"line": 51,
"column": 84
} | {
"line": 51,
"column": 100
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\na : α\n⊢ SupClosed {a}",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"instReflLe",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Std.le_refl._simp_1",
"Membership.mem",
"SemilatticeSup.toMax",
"Set.i... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupClosed | {
"line": 51,
"column": 84
} | {
"line": 51,
"column": 100
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\na : α\n⊢ SupClosed {a}",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"instReflLe",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Std.le_refl._simp_1",
"Membership.mem",
"SemilatticeSup.toMax",
"Set.i... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupClosed | {
"line": 53,
"column": 80
} | {
"line": 53,
"column": 96
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\n⊢ SupClosed univ",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"SemilatticeSup.toMax",
"imp_self._simp_1",
"Max.max",
"implies_congr",
"Tr... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.SupClosed | {
"line": 53,
"column": 80
} | {
"line": 53,
"column": 96
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\n⊢ SupClosed univ",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"SemilatticeSup.toMax",
"imp_self._simp_1",
"Max.max",
"implies_congr",
"Tr... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupClosed | {
"line": 53,
"column": 80
} | {
"line": 53,
"column": 96
} | {
"line": 55,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\n⊢ SupClosed univ",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"SemilatticeSup.toMax",
"imp_self._simp_1",
"Max.max",
"implies_congr",
"Tr... | [] | simp [SupClosed] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupClosed | {
"line": 102,
"column": 2
} | {
"line": 103,
"column": 7
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\ns : Set α\na : α\nhs : SupClosed s\nha : a ∈ upperBounds s\n⊢ SupClosed (insert a s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instReflLe",
"congrArg",
"true_or",
"PartialOrder.toPreorder",
... | [] | rw [SupClosed]
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupClosed | {
"line": 102,
"column": 2
} | {
"line": 103,
"column": 7
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : SemilatticeSup α\ns : Set α\na : α\nhs : SupClosed s\nha : a ∈ upperBounds s\n⊢ SupClosed (insert a s)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instReflLe",
"congrArg",
"true_or",
"PartialOrder.toPreorder",
... | [] | rw [SupClosed]
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SupClosed | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 65
} | {
"line": 230,
"column": 4
} | [
{
"pp": "ι : Sort u_1\nF : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Finset α\nht : t.Nonempty\nhts : ↑t ⊆ s\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\n⊢ t.sup' ht id ⊔ u.sup' hu id ∈ {a | ∃ t, ∃ (ht : t.Nonempty), ↑t ⊆ s... | [
"ι : Sort u_1\nF : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ns✝ t✝ : Set α\na b : α\ns : Set α\nt : Finset α\nht : t.Nonempty\nhts : ↑t ⊆ s\nu : Finset α\nhu : u.Nonempty\nhus : ↑u ⊆ s\n⊢ ↑(t ∪ u) ⊆ s"
] | refine ⟨_, ht.mono subset_union_left, ?_, sup'_union ht hu _⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 19
} | {
"line": 255,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns t : Set M\nhst : Disjoint (span R s) (span R t)\nh0s : 0 ∉ s\n⊢ Disjoint (s \\ {0}) t",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Disjoint.of_span"
],
"usedFVars": [... | [] | exact hst.of_span | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.SupClosed | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 46
} | {
"line": 363,
"column": 2
} | [
{
"pp": "case right.mem\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ ∀ a ∈ f '' s, a ∈ f '' latticeClosure s",
"ppTerm": "?right.mem",
"assigned": true,
"usedCon... | [] | exact Set.image_mono subset_latticeClosure | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.SupClosed | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 46
} | {
"line": 363,
"column": 2
} | [
{
"pp": "case right.mem\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ ∀ a ∈ f '' s, a ∈ f '' latticeClosure s",
"ppTerm": "?right.mem",
"assigned": true,
"usedCon... | [] | exact Set.image_mono subset_latticeClosure | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SupClosed | {
"line": 362,
"column": 4
} | {
"line": 362,
"column": 46
} | {
"line": 363,
"column": 2
} | [
{
"pp": "case right.mem\nα : Type u_3\nβ : Type u_4\ninst✝¹ : Lattice α\ninst✝ : Lattice β\ns : Set α\nf : α → β\nmap_sup : ∀ (a b : α), f (a ⊔ b) = f a ⊔ f b\nmap_inf : ∀ (a b : α), f (a ⊓ b) = f a ⊓ f b\n⊢ ∀ a ∈ f '' s, a ∈ f '' latticeClosure s",
"ppTerm": "?right.mem",
"assigned": true,
"usedCon... | [] | exact Set.image_mono subset_latticeClosure | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Atoms | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 38
} | {
"line": 453,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα✝¹ : Type u_2\nβ : Type u_3\ninst✝⁴ : PartialOrder α✝¹\nα✝ : Type u_4\na✝ b : α✝\ninst✝³ : Preorder α✝\nα : Type u_5\ninst✝² : PartialOrder α\ninst✝¹ : OrderBot α\ninst✝ : IsStronglyAtomic α\na : α\nhlt : ⊥ < a\nx : α\nhx : ⊥ ⋖ x\nhxa : x ≤ a\n⊢ ∃ a_1, IsAtom a_1 ∧ a_1 ≤ a",
"ppTerm"... | [] | exact ⟨x, bot_covBy_iff.1 hx, hxa⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Set.OrderIso | {
"line": 33,
"column": 2
} | {
"line": 33,
"column": 22
} | {
"line": 35,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ne : α ≃o β\nb : β\nx : α\n⊢ x ∈ ⇑e ⁻¹' Iio b ↔ x ∈ Iio (e.symm b)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"OrderIso.lt_iff_lt",
"Preorder.toLT",
"OrderIso.apply_symm_apply",
"congr... | [] | simp [← e.lt_iff_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Atoms | {
"line": 471,
"column": 39
} | {
"line": 471,
"column": 52
} | {
"line": 473,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα✝ : Type u_2\nβ : Type u_3\ninst✝³ : PartialOrder α✝\nα : Type u_4\na b : α\ninst✝² : Preorder α\ninst✝¹ : IsStronglyAtomic α\ns : Set α\ninst✝ : s.OrdConnected\n⊢ s.OrdConnected",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"inst✝"
]... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Atoms | {
"line": 474,
"column": 41
} | {
"line": 474,
"column": 54
} | {
"line": 476,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα✝ : Type u_2\nβ : Type u_3\ninst✝² : PartialOrder α✝\nα : Type u_4\na b : α\ninst✝¹ : Preorder α\ninst✝ : IsStronglyCoatomic α\ns : Set α\nh : s.OrdConnected\n⊢ s.OrdConnected",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h"
],
"... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Atoms | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 36
} | {
"line": 610,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : IsAtomistic α\ninst✝ : OrderTop α\n⊢ IsLUB {a | IsAtom a} ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"and_true",
"congrArg",
"le_top._simp_2",
"PartialOrder.toPreorder",
"setO... | [] | simpa using isLUB_atoms_le (⊤ : α) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Atoms | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 36
} | {
"line": 610,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : IsAtomistic α\ninst✝ : OrderTop α\n⊢ IsLUB {a | IsAtom a} ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"and_true",
"congrArg",
"le_top._simp_2",
"PartialOrder.toPreorder",
"setO... | [] | simpa using isLUB_atoms_le (⊤ : α) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Atoms | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 36
} | {
"line": 610,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : PartialOrder α\ninst✝² : OrderBot α\ninst✝¹ : IsAtomistic α\ninst✝ : OrderTop α\n⊢ IsLUB {a | IsAtom a} ⊤",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"and_true",
"congrArg",
"le_top._simp_2",
"PartialOrder.toPreorder",
"setO... | [] | simpa using isLUB_atoms_le (⊤ : α) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 556,
"column": 11
} | {
"line": 556,
"column": 33
} | {
"line": 556,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Sort u_9\np : ι → Submodule R M\nh : ι → Prop\n⊢ ⨆ i, ⨆ (_ : h i), p i = span R (⋃ i, ⋃ (_ : h i), ↑(p i))",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Subm... | [
"R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Sort u_9\np : ι → Submodule R M\nh : ι → Prop\n⊢ ⨆ i, ⨆ (_ : h i), p i = ⨆ i, ⨆ (_ : h i), span R ↑(p i)"
] | ← Submodule.iSup_span, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Span.Defs | {
"line": 604,
"column": 4
} | {
"line": 604,
"column": 34
} | {
"line": 606,
"column": 0
} | [
{
"pp": "case refine_4\nR : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Set M\nx✝ : M\nhx : x✝ ∈ span R S\na : R\nx : M\nT : Finset M\nhT : ↑T ⊆ S\nh2 : x ∈ span R ↑T\n⊢ ∃ T, ↑T ⊆ S ∧ a • x ∈ span R ↑T",
"ppTerm": "?refine_4",
"assigned": true,
"use... | [] | exact ⟨T, hT, smul_mem _ _ h2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Atoms | {
"line": 765,
"column": 20
} | {
"line": 770,
"column": 12
} | {
"line": 772,
"column": 0
} | [
{
"pp": "ι : Sort u_1\nα✝ : Type u_2\nβ : Type u_3\ninst✝⁵ : PartialOrder α✝\ninst✝⁴ : BoundedOrder α✝\ninst✝³ : IsSimpleOrder α✝\nα : Type ?u.12\ninst✝² : LE α\ninst✝¹ : BoundedOrder α\ninst✝ : IsSimpleOrder α\na b c : α\n⊢ a ≤ b → b ≤ c → a ≤ c",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants... | [] | by
rcases eq_bot_or_eq_top a with (rfl | rfl)
· simp
· rcases eq_bot_or_eq_top b with (rfl | rfl)
· rcases eq_bot_or_eq_top c with (rfl | rfl) <;> simp
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompactlyGenerated.Basic | {
"line": 508,
"column": 4
} | {
"line": 508,
"column": 68
} | {
"line": 509,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nη : Type u_3\ns : η → Set α\nhs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ (i : η), sSupIndep (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ i, s i\n⊢ sSupIndep ↑t",
"ppTerm": "?pos✝",
"assigned": true,
"u... | [
"case pos\nα : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\nη : Type u_3\ns : η → Set α\nhs : Directed (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ (i : η), sSupIndep (s i)\nhη : Nonempty η\nt : Finset α\nht : ↑t ⊆ ⋃ i, s i\nI : Set η\nfi : I.Finite\nhI : ↑t ⊆ ⋃ i ∈ I, s i\n⊢ sSupIndep ↑t"
] | obtain ⟨I, fi, hI⟩ := Set.finite_subset_iUnion t.finite_toSet ht | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.BigOperators.Pi | {
"line": 89,
"column": 4
} | {
"line": 90,
"column": 64
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\nj : κ\nhj : j ∉ ⋂ x ∈ s, f x\n⊢ ∏ i ∈ s, (f i).indicator (g i) j = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemi... | [] | obtain ⟨i, hi, hj⟩ := by simpa using hj
exact Finset.prod_eq_zero hi <| Set.indicator_of_notMem hj _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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