module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Order.WellFounded | {
"line": 172,
"column": 2
} | {
"line": 175,
"column": 31
} | {
"line": 177,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\n⊢ IsWellOrder α r ↔ ∀ (s : Set α), s.Nonempty → ∃ m, m ∈ s ∧ ∀ (x : α), x ∈ s → ¬r x m ∧ (m = x ∨ r m x)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"IsWellFounded.mk",
"_private.Mathlib.Order.WellFounded.0.WellFounded.isWellOrder_if... | [] | refine ⟨fun h s hs ↦ ?_, fun h ↦ { wf := ?_, trichotomous a b := ?_ }⟩
· grind [h.wf.has_min, trichotomous_of r]
· grind [wellFounded_iff_has_min]
· grind [h {a, b} <| by simp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Basic | {
"line": 602,
"column": 29
} | {
"line": 602,
"column": 56
} | {
"line": 602,
"column": 57
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Ico a b \\ {a} ⊆ s\nhc : s ⊆ Icc a b\nha : a ∈ s\nhb : b ∈ s\n⊢ Icc a b ⊆ s",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
"Se... | [
"case pos\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Ico a b ⊆ insert a s\nhc : s ⊆ Icc a b\nha : a ∈ s\nhb : b ∈ s\n⊢ Icc a b ⊆ s"
] | sdiff_singleton_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Basic | {
"line": 603,
"column": 8
} | {
"line": 603,
"column": 35
} | {
"line": 603,
"column": 36
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Icc a b \\ {b} ⊆ s\nhc : s ⊆ Icc a b\nha : a ∈ s\nhb : b ∈ s\n⊢ Icc a b ⊆ s",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
"Se... | [
"case pos\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Icc a b ⊆ insert b s\nhc : s ⊆ Icc a b\nha : a ∈ s\nhb : b ∈ s\n⊢ Icc a b ⊆ s"
] | sdiff_singleton_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Logic.Equiv.Set | {
"line": 352,
"column": 52
} | {
"line": 352,
"column": 73
} | {
"line": 353,
"column": 4
} | [
{
"pp": "α✝¹ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα✝ : Type u_1\nβ : Type u_2\nα : Type u\ns t : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\n⊢ ↑(s ∪ t) ⊕ ↑(s ∩ t) ≃ ↑(s ∪ t \\ s) ⊕ ↑(s ∩ t)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion... | [] | rw [union_sdiff_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Logic.Equiv.Set | {
"line": 352,
"column": 52
} | {
"line": 352,
"column": 73
} | {
"line": 353,
"column": 4
} | [
{
"pp": "α✝¹ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα✝ : Type u_1\nβ : Type u_2\nα : Type u\ns t : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\n⊢ ↑(s ∪ t) ⊕ ↑(s ∩ t) ≃ ↑(s ∪ t \\ s) ⊕ ↑(s ∩ t)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion... | [] | rw [union_sdiff_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Set | {
"line": 352,
"column": 52
} | {
"line": 352,
"column": 73
} | {
"line": 353,
"column": 4
} | [
{
"pp": "α✝¹ : Sort u\nβ✝ : Sort v\nγ : Sort w\nα✝ : Type u_1\nβ : Type u_2\nα : Type u\ns t : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\n⊢ ↑(s ∪ t) ⊕ ↑(s ∩ t) ≃ ↑(s ∪ t \\ s) ⊕ ↑(s ∩ t)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion... | [] | rw [union_sdiff_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.Basic | {
"line": 607,
"column": 31
} | {
"line": 607,
"column": 58
} | {
"line": 607,
"column": 59
} | [
{
"pp": "case neg.refine_2\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Ico a b \\ {a} ⊆ s\nhc : s ⊆ Icc a b\nha : a ∈ s\nhb : ¬b ∈ s\n⊢ Ico a b ⊆ s",
"ppTerm": "?neg.refine_2✝",
"assigned": true,
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Members... | [
"case neg.refine_2\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Ico a b ⊆ insert a s\nhc : s ⊆ Icc a b\nha : a ∈ s\nhb : ¬b ∈ s\n⊢ Ico a b ⊆ s"
] | sdiff_singleton_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.Basic | {
"line": 611,
"column": 32
} | {
"line": 611,
"column": 59
} | {
"line": 611,
"column": 60
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Ioc a b \\ {b} ⊆ s\nhc : s ⊆ Icc a b\nha : ¬a ∈ s\nhb : b ∈ s\n⊢ Ioc a b ⊆ s",
"ppTerm": "?pos.refine_2✝",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"congrArg",
"PartialOrder.toPreorder... | [
"case pos.refine_2\nα : Type u_1\ninst✝ : PartialOrder α\na b : α\ns : Set α\nho : Ioc a b ⊆ insert b s\nhc : s ⊆ Icc a b\nha : ¬a ∈ s\nhb : b ∈ s\n⊢ Ioc a b ⊆ s"
] | sdiff_singleton_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.NormNum.Basic | {
"line": 448,
"column": 29
} | {
"line": 462,
"column": 72
} | {
"line": 464,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : Semiring α\nf : α → α → α\na b : α\nna nb nc da db dc k : ℕ\n⊢ f = HMul.hMul →\n IsNNRat a na da → IsNNRat b nb db → na.mul nb = k.mul nc → da.mul db = k.mul dc → IsNNRat (f a b) nc dc",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | by
rintro rfl ⟨_, rfl⟩ ⟨_, rfl⟩ (h₁ : na * nb = k * nc) (h₂ : da * db = k * dc)
have : Invertible (↑(da * db) : α) := by simpa using invertibleMul (da:α) db
have := invertibleOfMul' (α := α) h₂
refine ⟨this, ?_⟩
have H := (Nat.cast_commute (α := α) da db).invOf_left.invOf_right.right_comm
have h₁ := congr_a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Field.Basic | {
"line": 254,
"column": 2
} | {
"line": 258,
"column": 25
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : Semifield α\ninst✝² : PartialOrder α\ninst✝¹ : PosMulReflectLT α\na b : α\ninst✝ : IsStrictOrderedRing α\ns : Set α\nha : 0 ≤ a\nhs : IsGLB s b\n⊢ IsGLB ((fun b ↦ a * b) '' s) (a * b)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq... | [] | rcases lt_or_eq_of_le ha with (ha | rfl)
· exact (OrderIso.mulLeft₀ _ ha).isGLB_image'.2 hs
· simp_rw [zero_mul]
rw [hs.nonempty.image_const]
exact isGLB_singleton | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Field.Basic | {
"line": 254,
"column": 2
} | {
"line": 258,
"column": 25
} | {
"line": 260,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : Semifield α\ninst✝² : PartialOrder α\ninst✝¹ : PosMulReflectLT α\na b : α\ninst✝ : IsStrictOrderedRing α\ns : Set α\nha : 0 ≤ a\nhs : IsGLB s b\n⊢ IsGLB ((fun b ↦ a * b) '' s) (a * b)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq... | [] | rcases lt_or_eq_of_le ha with (ha | rfl)
· exact (OrderIso.mulLeft₀ _ ha).isGLB_image'.2 hs
· simp_rw [zero_mul]
rw [hs.nonempty.image_const]
exact isGLB_singleton | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Field.Basic | {
"line": 443,
"column": 2
} | {
"line": 444,
"column": 25
} | {
"line": 446,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : Field α\ninst✝² : PartialOrder α\ninst✝¹ : PosMulReflectLT α\ninst✝ : IsStrictOrderedRing α\na b : α\nhb : b < 0\n⊢ AntitoneOn (fun x ↦ x⁻¹) (Set.Icc a b)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inv",
"DivisionCommMonoid.... | [] | convert! sub_inv_antitoneOn_Icc_left hb
exact (sub_zero _).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Field.Basic | {
"line": 443,
"column": 2
} | {
"line": 444,
"column": 25
} | {
"line": 446,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : Field α\ninst✝² : PartialOrder α\ninst✝¹ : PosMulReflectLT α\ninst✝ : IsStrictOrderedRing α\na b : α\nhb : b < 0\n⊢ AntitoneOn (fun x ↦ x⁻¹) (Set.Icc a b)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inv",
"DivisionCommMonoid.... | [] | convert! sub_inv_antitoneOn_Icc_left hb
exact (sub_zero _).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Field.Basic | {
"line": 685,
"column": 6
} | {
"line": 685,
"column": 17
} | {
"line": 685,
"column": 18
} | [
{
"pp": "α : Type u_2\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nε : α\nhε : 0 < ε\nn : ℕ\nB : α\nB_pos : 0 < B\npos : 0 < 1 + ↑n * (B + 1) ^ (n - 1)\nq r : α\nhr : |r| ≤ B\nhqr : |q - r| ≤ min 1 (ε / (1 + ↑n * (B + 1) ^ (n - 1)))\n⊢ |q - r| * ↑n * max |q| |r| ^ (n - 1) < ε",
... | [
"α : Type u_2\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nε : α\nhε : 0 < ε\nn : ℕ\nB : α\nB_pos : 0 < B\npos : 0 < 1 + ↑n * (B + 1) ^ (n - 1)\nq r : α\nhr : |r| ≤ B\nhqr : |q - r| ≤ 1 ∧ |q - r| ≤ ε / (1 + ↑n * (B + 1) ^ (n - 1))\n⊢ |q - r| * ↑n * max |q| |r| ^ (n - 1) < ε"
] | le_inf_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Perm.Basic | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 22
} | {
"line": 182,
"column": 23
} | [
{
"pp": "case trans\nα : Type u_1\nβ : Type u_2\nf : α → β → β\nl₁ l₂ : List α\nlcomm : LeftCommutative f\nl₁✝ l₂✝ l₃✝ : List α\nh₁✝ : l₁✝ ~ l₂✝\nh₂✝ : l₂✝ ~ l₃✝\nr₁ : ∀ (b : β), foldr f b l₁✝ = foldr f b l₂✝\nr₂ : ∀ (b : β), foldr f b l₂✝ = foldr f b l₃✝\nb : β\n⊢ foldr f b l₁✝ = foldr f b l₃✝",
"ppTerm": ... | [] | | trans _ _ r₁ r₂ => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.BigOperators.Group.List.Basic | {
"line": 181,
"column": 6
} | {
"line": 181,
"column": 91
} | {
"line": 181,
"column": 91
} | [
{
"pp": "case cons.inr\nM : Type u_4\ninst✝¹ : Monoid M\nl✝ : List M\na : M\ninst✝ : DecidableEq M\nb : M\nl : List M\nih : a ∈ l → (∀ (x : M), x ∈ l → ∀ (y : M), y ∈ l → x * y = y * x) → a * (l.erase a).prod = l.prod\nha : a ∈ b :: l\ncomm : ∀ (x : M), x ∈ b :: l → ∀ (y : M), y ∈ b :: l → x * y = y * x\nne : a... | [
"case cons.inr\nM : Type u_4\ninst✝¹ : Monoid M\nl✝ : List M\na : M\ninst✝ : DecidableEq M\nb : M\nl : List M\nih : a ∈ l → (∀ (x : M), x ∈ l → ∀ (y : M), y ∈ l → x * y = y * x) → a * (l.erase a).prod = l.prod\nha : a ∈ b :: l\ncomm : ∀ (x : M), x ∈ b :: l → ∀ (y : M), y ∈ b :: l → x * y = y * x\nne : a ≠ b\nh : a ... | ih h fun x hx y hy ↦ comm _ (List.mem_cons_of_mem b hx) _ (List.mem_cons_of_mem b hy) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Basic | {
"line": 547,
"column": 44
} | {
"line": 550,
"column": 9
} | {
"line": 552,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\nl : List α\na : α\n⊢ idxOf a l = 0 ↔ l = [] ∨ l.head? = some a",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"List.head?",
"False",
"Option.ctorIdx",
"congrArg",
"False.elim",
"List.idxOf_of_not... | [] | by
cases l
· simp
· grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Rat.Defs | {
"line": 144,
"column": 69
} | {
"line": 144,
"column": 75
} | {
"line": 144,
"column": 75
} | [
{
"pp": "q a b c : ℚ\n⊢ 1 ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"instDecidableNot",
"of_decide_eq_true",
"Rat",
"id",
"Ne",
"Bool.true",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"instDecidableEqRat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Rat.Defs | {
"line": 144,
"column": 69
} | {
"line": 144,
"column": 75
} | {
"line": 144,
"column": 75
} | [
{
"pp": "q a b c : ℚ\n⊢ 1 ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"instDecidableNot",
"of_decide_eq_true",
"Rat",
"id",
"Ne",
"Bool.true",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"instDecidableEqRat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rat.Defs | {
"line": 144,
"column": 69
} | {
"line": 144,
"column": 75
} | {
"line": 144,
"column": 75
} | [
{
"pp": "q a b c : ℚ\n⊢ 1 ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Rat.instOfNat",
"instDecidableNot",
"of_decide_eq_true",
"Rat",
"id",
"Ne",
"Bool.true",
"Bool",
"Eq.refl",
"OfNat.ofNat",
"instDecidableEqRat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.List.Basic | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 91
} | {
"line": 442,
"column": 0
} | [
{
"pp": "G : Type u_7\ninst✝ : CommGroup G\na b : G\nl : List G\n⊢ (a :: b :: l).alternatingProd = a * (b :: l).alternatingProd⁻¹",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid... | [] | rw [alternatingProd_cons_cons', alternatingProd_cons' b l, mul_inv, inv_inv, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Group.List.Basic | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 91
} | {
"line": 442,
"column": 0
} | [
{
"pp": "G : Type u_7\ninst✝ : CommGroup G\na b : G\nl : List G\n⊢ (a :: b :: l).alternatingProd = a * (b :: l).alternatingProd⁻¹",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid... | [] | rw [alternatingProd_cons_cons', alternatingProd_cons' b l, mul_inv, inv_inv, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Group.List.Basic | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 91
} | {
"line": 442,
"column": 0
} | [
{
"pp": "G : Type u_7\ninst✝ : CommGroup G\na b : G\nl : List G\n⊢ (a :: b :: l).alternatingProd = a * (b :: l).alternatingProd⁻¹",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"InvOneClass.toOne",
"HMul.hMul",
"DivisionCommMonoid... | [] | rw [alternatingProd_cons_cons', alternatingProd_cons' b l, mul_inv, inv_inv, mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNRat.Defs | {
"line": 75,
"column": 75
} | {
"line": 75,
"column": 81
} | {
"line": 75,
"column": 81
} | [
{
"pp": "p q : ℚ≥0\n⊢ 1 ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"NonAssocSemiring.toAddCommMonoidWithOne",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"CommSemiring.toSemiring",
"id",
"NNRat",
"Ne",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.NNRat.Defs | {
"line": 75,
"column": 75
} | {
"line": 75,
"column": 81
} | {
"line": 75,
"column": 81
} | [
{
"pp": "p q : ℚ≥0\n⊢ 1 ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"NonAssocSemiring.toAddCommMonoidWithOne",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"CommSemiring.toSemiring",
"id",
"NNRat",
"Ne",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.NNRat.Defs | {
"line": 75,
"column": 75
} | {
"line": 75,
"column": 81
} | {
"line": 75,
"column": 81
} | [
{
"pp": "p q : ℚ≥0\n⊢ 1 ≠ 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"NonAssocSemiring.toAddCommMonoidWithOne",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"CommSemiring.toSemiring",
"id",
"NNRat",
"Ne",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Unbundled.Rat | {
"line": 41,
"column": 19
} | {
"line": 41,
"column": 25
} | {
"line": 41,
"column": 25
} | [
{
"pp": "m e : ℕ\n⊢ ¬false = true",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instDecidableEqBool",
"Bool.true",
"Bool",
"Eq.refl",
"Bool.false",
"Decidable.decide",
"Eq",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Algebra.Order.Ring.Unbundled.Rat | {
"line": 41,
"column": 19
} | {
"line": 41,
"column": 25
} | {
"line": 41,
"column": 25
} | [
{
"pp": "m e : ℕ\n⊢ ¬false = true",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instDecidableEqBool",
"Bool.true",
"Bool",
"Eq.refl",
"Bool.false",
"Decidable.decide",
"Eq",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Unbundled.Rat | {
"line": 41,
"column": 19
} | {
"line": 41,
"column": 25
} | {
"line": 41,
"column": 25
} | [
{
"pp": "m e : ℕ\n⊢ ¬false = true",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instDecidableEqBool",
"Bool.true",
"Bool",
"Eq.refl",
"Bool.false",
"Decidable.decide",
"Eq",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.NNRat.Defs | {
"line": 373,
"column": 54
} | {
"line": 373,
"column": 70
} | {
"line": 375,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ divNat n 0 = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"NNRat.divNat",
"Rat.instOfNat",
"Subtype.mk.congr_simp",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"Rat",
"AddGroupWithOne.toAddMonoidWi... | [] | by simp [divNat] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Divisibility.Units | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 76
} | {
"line": 218,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CommMonoid α\nx y z : α\nhu : IsUnit x\n⊢ IsRelPrime (y * x) (z * x) ↔ IsRelPrime y z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Iff.rfl",
"IsRelPrime",
... | [] | rw [isRelPrime_mul_unit_right_left hu, isRelPrime_mul_unit_right_right hu] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Divisibility.Units | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 76
} | {
"line": 218,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CommMonoid α\nx y z : α\nhu : IsUnit x\n⊢ IsRelPrime (y * x) (z * x) ↔ IsRelPrime y z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Iff.rfl",
"IsRelPrime",
... | [] | rw [isRelPrime_mul_unit_right_left hu, isRelPrime_mul_unit_right_right hu] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Divisibility.Units | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 76
} | {
"line": 218,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CommMonoid α\nx y z : α\nhu : IsUnit x\n⊢ IsRelPrime (y * x) (z * x) ↔ IsRelPrime y z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Iff.rfl",
"IsRelPrime",
... | [] | rw [isRelPrime_mul_unit_right_left hu, isRelPrime_mul_unit_right_right hu] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Divisibility.Units | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 92
} | {
"line": 255,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoid α\nx y z : α\ninst✝ : DecompositionMonoid α\nH1 : IsRelPrime x z\nH2 : IsRelPrime y z\na b : α\nha : a ∣ x\nhb : b ∣ y\nh : a * b ∣ x * y\nhz : a * b ∣ z\n⊢ IsUnit (a * b)",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
... | [] | exact (H1 ha <| (dvd_mul_right a b).trans hz).mul (H2 hb <| (dvd_mul_left b a).trans hz) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Rat.Lemmas | {
"line": 73,
"column": 2
} | {
"line": 75,
"column": 65
} | {
"line": 77,
"column": 0
} | [
{
"pp": "q₁ q₂ : ℚ\n⊢ ↑(q₁.den.gcd q₂.den) ∣ q₁.num * ↑q₂.den + q₂.num * ↑q₁.den",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Nat.gcd_dvd_left",
"Eq.mpr",
"Rat.num",
"Dvd.dvd",
"HMul.hMul",
"dvd_mul_of_dvd_right",
"congrArg",
... | [] | apply Int.dvd_add
<;> apply dvd_mul_of_dvd_right <;> rw [Int.natCast_dvd_natCast]
<;> [exact Nat.gcd_dvd_right _ _; exact Nat.gcd_dvd_left _ _] | Batteries.Tactic._aux_Batteries_Tactic_SeqFocus___macroRules_Batteries_Tactic_seq_focus_1 | Batteries.Tactic.seq_focus |
Mathlib.Tactic.NormNum.Pow | {
"line": 51,
"column": 16
} | {
"line": 51,
"column": 45
} | {
"line": 51,
"column": 45
} | [
{
"pp": "a b c : ℕ\nh1 : a.pow b = c\n⊢ a.pow (2 * b) = c.mul c",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instPowNat",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"two_mul",
"pow_add",
"MulOne.toMul",
"Dist... | [] | simp [two_mul, pow_add, ← h1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.NormNum.Pow | {
"line": 51,
"column": 16
} | {
"line": 51,
"column": 45
} | {
"line": 51,
"column": 45
} | [
{
"pp": "a b c : ℕ\nh1 : a.pow b = c\n⊢ a.pow (2 * b) = c.mul c",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instPowNat",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"two_mul",
"pow_add",
"MulOne.toMul",
"Dist... | [] | simp [two_mul, pow_add, ← h1] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.NormNum.Pow | {
"line": 51,
"column": 16
} | {
"line": 51,
"column": 45
} | {
"line": 51,
"column": 45
} | [
{
"pp": "a b c : ℕ\nh1 : a.pow b = c\n⊢ a.pow (2 * b) = c.mul c",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"instPowNat",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instMonoid",
"two_mul",
"pow_add",
"MulOne.toMul",
"Dist... | [] | simp [two_mul, pow_add, ← h1] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteLattice.Defs | {
"line": 327,
"column": 6
} | {
"line": 327,
"column": 17
} | {
"line": 327,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLinearOrder α\ns : Set α\n⊢ sSup s = ⊤ ↔ ∀ b < ⊤, ∃ a ∈ s, b < a",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"eq_top_iff",
"congrArg",
"PartialOrder.toPreor... | [
"α : Type u_1\ninst✝ : CompleteLinearOrder α\ns : Set α\n⊢ ⊤ ≤ sSup s ↔ ∀ b < ⊤, ∃ a ∈ s, b < a"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompleteLattice.Defs | {
"line": 331,
"column": 6
} | {
"line": 331,
"column": 17
} | {
"line": 331,
"column": 18
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\ninst✝ : CompleteLinearOrder α\nf : ι → α\n⊢ iSup f = ⊤ ↔ ∀ b < ⊤, ∃ i, b < f i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"eq_top_iff",
"congrArg",
"iSup",
... | [
"α : Type u_1\nι : Sort u_4\ninst✝ : CompleteLinearOrder α\nf : ι → α\n⊢ ⊤ ≤ iSup f ↔ ∀ b < ⊤, ∃ i, b < f i"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.GaloisConnection.Basic | {
"line": 148,
"column": 29
} | {
"line": 148,
"column": 44
} | {
"line": 148,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ns : Set α\nt : Set β\nl : α → β → γ\nu₁ : β → γ → α\nu₂ : α → γ → β\na₀ : α\nb₀ : β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nha₀ : ∀ (b : α), a₀ ≤... | [
"α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ns : Set α\nt : Set β\nl : α → β → γ\nu₁ : β → γ → α\nu₂ : α → γ → β\na₀ : α\nb₀ : β\nh₁ : ∀ (b : β), GaloisConnection (swap l b) (u₁ b)\nh₂ : ∀ (a : α), GaloisConnection (l a) (u₂ a)\nha₀ : ∀ (b : α), a₀ ≤ b ↔ ∀ x ∈ s... | mem_upperBounds | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.CompleteLattice.Basic | {
"line": 474,
"column": 24
} | {
"line": 474,
"column": 34
} | {
"line": 474,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\na : α\ns : Set β\nhs : s.Nonempty\nthis : Nonempty ↑s\n⊢ ⨆ i, a = a",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"iSup",
"Set.Elem",
"id",
"CompleteSemilatticeSup.to... | [
"α : Type u_1\nβ : Type u_2\ninst✝ : CompleteLattice α\na : α\ns : Set β\nhs : s.Nonempty\nthis : Nonempty ↑s\n⊢ a = a"
] | iSup_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompleteLattice.Basic | {
"line": 496,
"column": 19
} | {
"line": 496,
"column": 29
} | {
"line": 496,
"column": 29
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\ninst✝¹ : CompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ (⨆ x, f x) ⊔ a = (⨆ x, f x) ⊔ ⨆ x, a",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg... | [
"α : Type u_1\nι : Sort u_4\ninst✝¹ : CompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ (⨆ x, f x) ⊔ a = (⨆ x, f x) ⊔ a"
] | iSup_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.CompleteLattice.Basic | {
"line": 500,
"column": 19
} | {
"line": 500,
"column": 29
} | {
"line": 500,
"column": 29
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\ninst✝¹ : CompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ a ⊔ ⨆ x, f x = (⨆ x, a) ⊔ ⨆ x, f x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"congrArg",... | [
"α : Type u_1\nι : Sort u_4\ninst✝¹ : CompleteLattice α\ninst✝ : Nonempty ι\nf : ι → α\na : α\n⊢ a ⊔ ⨆ x, f x = a ⊔ ⨆ x, f x"
] | iSup_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Lattice | {
"line": 1052,
"column": 64
} | {
"line": 1052,
"column": 89
} | {
"line": 1052,
"column": 89
} | [
{
"pp": "β : Type u_2\nι : Type u_12\nx : ι → β\nt : ι → Set β\n⊢ (⋃ i, t i) ∪ ⋃ i, {x i} = (⋃ i, t i) ∪ range x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.iUnion_singleton_eq_range",
"Set.instUnion",
"Set.instSingletonSet",
"True",
... | [] | iUnion_singleton_eq_range | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Lattice | {
"line": 1093,
"column": 50
} | {
"line": 1093,
"column": 76
} | {
"line": 1095,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Sort u_5\ns : ι → Set α\nt : α → Set β\n⊢ ⋃ x ∈ ⋃ i, s i, t x = ⋃ i, ⋃ x ∈ s i, t x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.of_eq",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membership.mem",
"Exists",
"f... | [] | by simp [@iUnion_comm _ ι] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompleteBooleanAlgebra | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 55
} | {
"line": 678,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort w\ninst✝¹ : CompleteBooleanAlgebra α\nf : ι → α\ninst✝ : Nonempty ι\na : α\n⊢ a ∆ ⨆ i, f i ≤ ⨆ i, a ∆ f i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Complete... | [] | simpa [symmDiff_comm] using iSup_symmDiff_le (a := a) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.CompleteBooleanAlgebra | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 55
} | {
"line": 678,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort w\ninst✝¹ : CompleteBooleanAlgebra α\nf : ι → α\ninst✝ : Nonempty ι\na : α\n⊢ a ∆ ⨆ i, f i ≤ ⨆ i, a ∆ f i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Complete... | [] | simpa [symmDiff_comm] using iSup_symmDiff_le (a := a) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.CompleteBooleanAlgebra | {
"line": 676,
"column": 2
} | {
"line": 676,
"column": 55
} | {
"line": 678,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort w\ninst✝¹ : CompleteBooleanAlgebra α\nf : ι → α\ninst✝ : Nonempty ι\na : α\n⊢ a ∆ ⨆ i, f i ≤ ⨆ i, a ∆ f i",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Complete... | [] | simpa [symmDiff_comm] using iSup_symmDiff_le (a := a) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Lattice | {
"line": 1353,
"column": 2
} | {
"line": 1353,
"column": 23
} | {
"line": 1354,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nt : α → Set β\ns₁ s₂ : Set α\n⊢ s₁ ⊆ s₂ ∪ s₁ \\ s₂",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"LE.le",
"Set.union_sdiff_self",
"SDiff.sdiff",
"Set.instLE",... | [
"α : Type u_1\nβ : Type u_2\nt : α → Set β\ns₁ s₂ : Set α\n⊢ s₁ ⊆ s₂ ∪ s₁"
] | rw [union_sdiff_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Interval.Set.LinearOrder | {
"line": 54,
"column": 54
} | {
"line": 54,
"column": 93
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : α\n⊢ Ioi a \\ Ioi b = Ioc a b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Set.Ioi_inter_Iic",
"Eq.mpr",
"Set.Ioc",
"Set.Ioi",
"Lattice.toSemilatticeSup",
"congrArg",
"Compl.compl",
"Par... | [] | rw [sdiff_eq, compl_Ioi, Ioi_inter_Iic] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Interval.Set.LinearOrder | {
"line": 54,
"column": 54
} | {
"line": 54,
"column": 93
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : α\n⊢ Ioi a \\ Ioi b = Ioc a b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Set.Ioi_inter_Iic",
"Eq.mpr",
"Set.Ioc",
"Set.Ioi",
"Lattice.toSemilatticeSup",
"congrArg",
"Compl.compl",
"Par... | [] | rw [sdiff_eq, compl_Ioi, Ioi_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.LinearOrder | {
"line": 54,
"column": 54
} | {
"line": 54,
"column": 93
} | {
"line": 56,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : α\n⊢ Ioi a \\ Ioi b = Ioc a b",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Set.Ioi_inter_Iic",
"Eq.mpr",
"Set.Ioc",
"Set.Ioi",
"Lattice.toSemilatticeSup",
"congrArg",
"Compl.compl",
"Par... | [] | rw [sdiff_eq, compl_Ioi, Ioi_inter_Iic] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompleteLattice.Basic | {
"line": 793,
"column": 2
} | {
"line": 793,
"column": 60
} | {
"line": 794,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : ConditionallyCompleteLattice β\nf : α → β\ns t : Set α\nht : BddBelow (f '' t)\nhf : MonotoneOn f t\nhst : s ⊆ t\nh : ∀ y ∈ t, ∃ x ∈ s, x ≤ y\nhs : s.Nonempty\n⊢ sInf (f '' s) = sInf (f '' t)",
"ppTerm": "?inr",
"assigned": true... | [
"case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder α\ninst✝ : ConditionallyCompleteLattice β\nf : α → β\ns t : Set α\nht : BddBelow (f '' t)\nhf : MonotoneOn f t\nhst : s ⊆ t\nh : ∀ y ∈ t, ∃ x ∈ s, x ≤ y\nhs : s.Nonempty\n⊢ sInf (f '' s) ≤ sInf (f '' t)"
] | refine le_antisymm ?_ (by gcongr; exacts [ht, hs.image f]) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Interval.Set.UnorderedInterval | {
"line": 258,
"column": 41
} | {
"line": 258,
"column": 48
} | {
"line": 258,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b c : α\n⊢ b < a ∧ a ≤ c ∨ a ∈ Ioc c b ↔ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b c : α\n⊢ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b ↔ b < a ∧ a ≤ c ∨ c < a ∧ a ≤ b"
] | mem_Ioc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Antichain | {
"line": 199,
"column": 87
} | {
"line": 200,
"column": 60
} | {
"line": 202,
"column": 0
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ns t : Set α\n⊢ IsAntichain r (s ∪ t) ↔ IsAntichain r s ∧ IsAntichain r t ∧ ∀ a ∈ s, ∀ b ∈ t, a ≠ b → rᶜ a b ∧ rᶜ b a",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Iff.rfl",
"Prop.... | [] | by
rw [IsAntichain, IsAntichain, IsAntichain, pairwise_union] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Preorder.Chain | {
"line": 305,
"column": 50
} | {
"line": 305,
"column": 76
} | {
"line": 306,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ns : Set α\nhs : IsChain r s\nh : ∃ t, IsChain r s ∧ SuperChain r s t\n⊢ IsChain r (SuccChain r s)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"succChain_spec",
"SemilatticeInf.t... | [] | exact (succChain_spec h).1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Preorder.Chain | {
"line": 305,
"column": 50
} | {
"line": 305,
"column": 76
} | {
"line": 306,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ns : Set α\nhs : IsChain r s\nh : ∃ t, IsChain r s ∧ SuperChain r s t\n⊢ IsChain r (SuccChain r s)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"succChain_spec",
"SemilatticeInf.t... | [] | exact (succChain_spec h).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Preorder.Chain | {
"line": 305,
"column": 50
} | {
"line": 305,
"column": 76
} | {
"line": 306,
"column": 2
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ns : Set α\nhs : IsChain r s\nh : ∃ t, IsChain r s ∧ SuperChain r s t\n⊢ IsChain r (SuccChain r s)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"succChain_spec",
"SemilatticeInf.t... | [] | exact (succChain_spec h).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteLatticeIntervals | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 83
} | {
"line": 189,
"column": 83
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\ns : Set α\ninst✝¹ : ConditionallyCompleteLattice α\na b : α\ninst✝ : Fact (a ≤ b)\nS : Set ↑(Icc a b)\nc : ↑(Icc a b)\nhc : c ∈ S\n⊢ sInf (Subtype.val '' S) ≤ b",
"ppTerm": "?m.320",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"lo... | [] | exact le_trans (csInf_le ⟨a, fun _ ⟨d, _, hd⟩ ↦ hd ▸ d.2.1⟩ ⟨c, hc, rfl⟩) c.2.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Tactic.NormNum.Eq | {
"line": 42,
"column": 81
} | {
"line": 44,
"column": 53
} | {
"line": 46,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Ring α\nn₁ n₂ : ℤ\na₁ a₂ : α\ninst✝¹ : Invertible a₁\ninst✝ : Invertible a₂\n⊢ ↑n₁ * ⅟a₁ = ↑n₂ * ⅟a₂ ↔ ↑n₁ * a₂ = ↑n₂ * a₁",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
... | [] | by
rw [mul_invOf_eq_iff_eq_mul_right, ← Int.commute_cast, mul_assoc,
← mul_left_eq_iff_eq_invOf_mul, Int.commute_cast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Notation.Support | {
"line": 98,
"column": 11
} | {
"line": 98,
"column": 45
} | {
"line": 98,
"column": 46
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝ : One M\nf : ι → M\ns : Set ι\n⊢ Disjoint (mulSupport f) s ↔ EqOn f 1 s",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Disjoint",
"SemilatticeInf.toPartialOrder",
"id",
... | [
"ι : Type u_1\nM : Type u_3\ninst✝ : One M\nf : ι → M\ns : Set ι\n⊢ mulSupport f ⊆ sᶜ ↔ EqOn f 1 s"
] | ← subset_compl_iff_disjoint_right, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.PNat.Basic | {
"line": 165,
"column": 31
} | {
"line": 165,
"column": 37
} | {
"line": 165,
"column": 37
} | [
{
"pp": "p : ℕ+ → Sort u_1\none : p 1\nsucc : (n : ℕ+) → p n → p (n + 1)\nh : 0 < 0\n⊢ ¬0 < 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.PNat.Basic | {
"line": 165,
"column": 31
} | {
"line": 165,
"column": 37
} | {
"line": 165,
"column": 37
} | [
{
"pp": "p : ℕ+ → Sort u_1\none : p 1\nsucc : (n : ℕ+) → p n → p (n + 1)\nh : 0 < 0\n⊢ ¬0 < 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.PNat.Basic | {
"line": 165,
"column": 31
} | {
"line": 165,
"column": 37
} | {
"line": 165,
"column": 37
} | [
{
"pp": "p : ℕ+ → Sort u_1\none : p 1\nsucc : (n : ℕ+) → p n → p (n + 1)\nh : 0 < 0\n⊢ ¬0 < 0",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.PNat.Basic | {
"line": 179,
"column": 34
} | {
"line": 179,
"column": 40
} | {
"line": 179,
"column": 40
} | [
{
"pp": "p : ℕ+ → Sort u_1\none : p 1\nsucc : (n : ℕ+) → p n → p (n + 1)\nh : 0 < 0\n⊢ ¬0 < 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.PNat.Basic | {
"line": 179,
"column": 34
} | {
"line": 179,
"column": 40
} | {
"line": 179,
"column": 40
} | [
{
"pp": "p : ℕ+ → Sort u_1\none : p 1\nsucc : (n : ℕ+) → p n → p (n + 1)\nh : 0 < 0\n⊢ ¬0 < 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.PNat.Basic | {
"line": 179,
"column": 34
} | {
"line": 179,
"column": 40
} | {
"line": 179,
"column": 40
} | [
{
"pp": "p : ℕ+ → Sort u_1\none : p 1\nsucc : (n : ℕ+) → p n → p (n + 1)\nh : 0 < 0\n⊢ ¬0 < 0",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT.lt",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Round | {
"line": 163,
"column": 4
} | {
"line": 168,
"column": 34
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_2\ninst✝³ : Ring α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : FloorRing α\nx : α\nz : ℤ\nhx : x < ↑z\n⊢ 1 - fract x ≤ |x - ↑z|",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"sub_neg",
"IsRightCancelAdd.addRightStrictMono_of_a... | [] | rw [abs_eq_neg_self.mpr (sub_neg.mpr hx).le]
conv_rhs => rw [← fract_add_floor x]
rw [add_sub_assoc, add_comm, neg_add, neg_sub, le_add_neg_iff_add_le, sub_add_cancel,
le_sub_comm]
norm_cast
rwa [le_sub_one_iff, floor_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Round | {
"line": 163,
"column": 4
} | {
"line": 168,
"column": 34
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_2\ninst✝³ : Ring α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : FloorRing α\nx : α\nz : ℤ\nhx : x < ↑z\n⊢ 1 - fract x ≤ |x - ↑z|",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"sub_neg",
"IsRightCancelAdd.addRightStrictMono_of_a... | [] | rw [abs_eq_neg_self.mpr (sub_neg.mpr hx).le]
conv_rhs => rw [← fract_add_floor x]
rw [add_sub_assoc, add_comm, neg_add, neg_sub, le_add_neg_iff_add_le, sub_add_cancel,
le_sub_comm]
norm_cast
rwa [le_sub_one_iff, floor_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.Ring.PNat | {
"line": 48,
"column": 35
} | {
"line": 48,
"column": 62
} | {
"line": 48,
"column": 62
} | [
{
"pp": "n : ℕ+\nn' k : ℕ\nh1 : CSLiftVal n n'\n⊢ n' ^ k = CSLift.lift (n ^ k)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"PNat.val",
"outParam",
"congrArg",
"Nat.instMonoid",
"Mathlib.Tactic.Ring.instCSLiftPNatNat",
"NPow.toPow",
"HPow.hPow",... | [] | by simp [h1.1, CSLift.lift] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Group.Basic | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 43
} | {
"line": 47,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na : α\nha : 1 ≤ a\n⊢ Monotone fun n ↦ a ^ n",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"DivInvMonoid.toZPow",
"Int",
"monotone_int_of_le_... | [
"α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na : α\nha : 1 ≤ a\nn : ℤ\n⊢ a ^ n ≤ a ^ (n + 1)"
] | refine monotone_int_of_le_succ fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Group.Action.Basic | {
"line": 169,
"column": 26
} | {
"line": 169,
"column": 67
} | {
"line": 170,
"column": 2
} | [
{
"pp": "G : Type u_1\nM : Type u_2\nA : Type u_3\nB : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁴ : Monoid M\ninst✝³ : Monoid A\ninst✝² : MulDistribMulAction M A\ninst✝¹ : Monoid B\ninst✝ : SMul M B\nf : B →* A\nhf : Injective ⇑f\nsmul : ∀ (c : M) (x : B), f (c • x) = c • f x\nc : M\nx y : B\n⊢ f (c • (x * y)... | [] | by simp only [smul, f.map_mul, smul_mul'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 599,
"column": 39
} | {
"line": 599,
"column": 47
} | {
"line": 599,
"column": 47
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : FloorRing R\na : R\nha : -1 < a\n⊢ 0 - 1 < a",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Preorder.toLT",
"AddGroupWithOne.toAddGroup",
"congr... | [
"R : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : FloorRing R\na : R\nha : -1 < a\n⊢ -1 < a"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 689,
"column": 2
} | {
"line": 689,
"column": 18
} | {
"line": 691,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorRing R\ninst✝ : IsOrderedRing R\na : R\n⊢ ⌈a⌉ < ⌈a⌉ + 1",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Int.instNeZeroOfNatOfNat",
"instIsLeftCancelAddOfAddLeftReflectLE",
"lt_add_one",
"... | [] | apply lt_add_one | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 738,
"column": 4
} | {
"line": 738,
"column": 31
} | {
"line": 739,
"column": 4
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorRing R\ninst✝ : IsOrderedRing R\na : R\nha : fract a ≠ 0\nthis : ↑⌈a⌉ = ↑⌊a⌋ + 1\n⊢ fract a = a + 1 - ↑⌈a⌉",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.floor",
... | [
"R : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorRing R\ninst✝ : IsOrderedRing R\na : R\nha : fract a ≠ 0\nthis : ↑⌈a⌉ = ↑⌊a⌋ + 1\n⊢ fract a = a + 1 - (a - fract a + 1)"
] | rw [this, ← self_sub_fract] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Archimedean.Basic | {
"line": 363,
"column": 2
} | {
"line": 363,
"column": 40
} | {
"line": 364,
"column": 2
} | [
{
"pp": "K : Type u_4\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : Archimedean K\nx y : K\nn : ℕ\nh : x < y\nnh : (y - x)⁻¹ < ↑n\n⊢ ∃ z, x < ↑z / ↑n ∧ ↑z / ↑n < y",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Int.cast",
"exists_floor",
... | [
"K : Type u_4\ninst✝³ : Field K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : Archimedean K\nx y : K\nn : ℕ\nh : x < y\nnh : (y - x)⁻¹ < ↑n\nz : ℤ\nzh : ∀ (z_1 : ℤ), z_1 ≤ z ↔ ↑z_1 ≤ x * ↑n\n⊢ ∃ z, x < ↑z / ↑n ∧ ↑z / ↑n < y"
] | obtain ⟨z, zh⟩ := exists_floor (x * n) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Multiset.Defs | {
"line": 261,
"column": 8
} | {
"line": 261,
"column": 20
} | {
"line": 261,
"column": 20
} | [
{
"pp": "α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\nf : (a : α) → p a → β\ns : Multiset α\nl₁ l₂ : List α\npp : l₁ ~ l₂\nH₂ : ∀ (a : α), a ∈ l₂ → p a\nH₁ : ∀ (a : α), a ∈ l₁ → p a\n⊢ ∀ {s₂ : Multiset α} {e : ↑l₁ = s₂} {H : ∀ (a : α), a ∈ s₂ → p a},\n Eq.ndrec (motive := fun s ↦ (∀ (a : α), a ∈ s →... | [
"α : Type u_1\nβ : Type v\nγ : Type u_2\np : α → Prop\nf : (a : α) → p a → β\ns : Multiset α\nl₁ l₂ : List α\npp : l₁ ~ l₂\nH₂ : ∀ (a : α), a ∈ l₂ → p a\nH₁ : ∀ (a : α), a ∈ l₁ → p a\ns₂ : Multiset α\ne : ↑l₁ = s₂\nH✝ : ∀ (a : α), a ∈ s₂ → p a\n⊢ Eq.ndrec (motive := fun s ↦ (∀ (a : α), a ∈ s → p a) → Multiset β) (f... | intro s₂ e _ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.Multiset.Replicate | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 23
} | {
"line": 106,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nm : Multiset α\nx : α\nn : ℕ\nh : m ≤ replicate n x\n⊢ ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Multiset.replicate_succ",
"Preorder.toLE... | [
"case mpr\nα : Type u_1\nm : Multiset α\nx : α\nn : ℕ\nh : m ≤ replicate n x\n⊢ ∃ a, a ::ₘ m ≤ x ::ₘ replicate n x"
] | rw [replicate_succ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Multiset.Replicate | {
"line": 104,
"column": 4
} | {
"line": 106,
"column": 31
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nm : Multiset α\nx : α\nn : ℕ\n⊢ m ≤ replicate n x → ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Multiset.replicate_succ",
"Preorder.toLE",
... | [] | intro h
rw [replicate_succ]
exact ⟨x, cons_le_cons _ h⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Multiset.Replicate | {
"line": 104,
"column": 4
} | {
"line": 106,
"column": 31
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nm : Multiset α\nx : α\nn : ℕ\n⊢ m ≤ replicate n x → ∃ a, a ::ₘ m ≤ replicate (n + 1) x",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Multiset.replicate_succ",
"Preorder.toLE",
... | [] | intro h
rw [replicate_succ]
exact ⟨x, cons_le_cons _ h⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.List | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 37
} | {
"line": 47,
"column": 2
} | [
{
"pp": "α : Type u_1\nl : List α\n⊢ (range fun x ↦ l[x]?) = insert none (some '' {x | x ∈ l})",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Set.range_comp",
"Eq.mpr",
"congrArg",
"List.get",
"setOf",
"List.instGetElem?NatLtLength",
"Option.some... | [
"α : Type u_1\nl : List α\n⊢ (range fun x ↦ l[x]?) = insert none (range (some ∘ l.get))"
] | rw [← range_list_get, ← range_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.List.Dedup | {
"line": 145,
"column": 25
} | {
"line": 145,
"column": 86
} | {
"line": 145,
"column": 86
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nxs ys : List α\nh : xs ⊆ ys\n⊢ xs ∪ ys.dedup = ys.dedup",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.Subset.union_eq_right",
"congrArg",
"List.dedup",
"id",
"instBEqOfDecidableEq",
"L... | [
"α : Type u_1\ninst✝ : DecidableEq α\nxs ys : List α\nh : xs ⊆ ys\n⊢ ys.dedup = ys.dedup"
] | Subset.union_eq_right (List.Subset.trans h <| subset_dedup _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.MapFold | {
"line": 375,
"column": 4
} | {
"line": 375,
"column": 8
} | {
"line": 376,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ ↑(List.foldl List.erase l₁ l₂) = foldl erase ↑l₁ ↑l₂",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Multiset",
"Multiset.instRightCommutativeErase",
"List.foldl",
"instBEqOfDecidable... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns t : Multiset α\nl₁ l₂ : List α\n⊢ foldl erase ↑l₁ ↑l₂ = ↑(List.foldl List.erase l₁ l₂)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Multiset.FinsetOps | {
"line": 184,
"column": 84
} | {
"line": 186,
"column": 77
} | {
"line": 188,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Multiset α\nh : s ⊆ t\n⊢ s.ndunion t = t",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"List.Subset.union_eq_right",
"Multiset.instHasSubset",
"Multiset",
"HasSubset.Subset",
"instBEqOfDecidableEq",
"Q... | [] | by
induction s, t using Quot.induction_on₂
exact congr_arg ((↑) : List α → Multiset α) <| List.Subset.union_eq_right h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Lattice.Lemmas | {
"line": 130,
"column": 48
} | {
"line": 130,
"column": 98
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns₁ s₂ : Finset α\na : α\nh : a ∈ s₁\n⊢ s₁ ∩ insert a s₂ = insert a (s₁ ∩ s₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.inter_comm",
"Finset.insert_inter_of_mem",
... | [] | rw [inter_comm, insert_inter_of_mem h, inter_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Finset.Lattice.Lemmas | {
"line": 130,
"column": 48
} | {
"line": 130,
"column": 98
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns₁ s₂ : Finset α\na : α\nh : a ∈ s₁\n⊢ s₁ ∩ insert a s₂ = insert a (s₁ ∩ s₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.inter_comm",
"Finset.insert_inter_of_mem",
... | [] | rw [inter_comm, insert_inter_of_mem h, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Lattice.Lemmas | {
"line": 130,
"column": 48
} | {
"line": 130,
"column": 98
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns₁ s₂ : Finset α\na : α\nh : a ∈ s₁\n⊢ s₁ ∩ insert a s₂ = insert a (s₁ ∩ s₂)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Finset.inter_comm",
"Finset.insert_inter_of_mem",
... | [] | rw [inter_comm, insert_inter_of_mem h, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Basic | {
"line": 293,
"column": 69
} | {
"line": 294,
"column": 35
} | {
"line": 296,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Finset α\n⊢ s.attach = ∅ ↔ s = ∅",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.Finset.Basic.0.Finset.attach_eq_empty_iff._simp_1_1",
"False",
"congrArg",
"Finset",
"Subtype.forall._simp_1",
"Membership.m... | [] | by
simp [eq_empty_iff_forall_notMem] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Basic | {
"line": 483,
"column": 2
} | {
"line": 483,
"column": 65
} | {
"line": 485,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\n⊢ s.toFinset.Nonempty ↔ s ≠ 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"congrArg",
"Finset",
"_private.Mathlib.Data.Finset.Basic.0.Multiset.toFinset_nonempty._simp_1_1",
"... | [] | simp only [toFinset_eq_empty, Ne, Finset.nonempty_iff_ne_empty] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.Basic | {
"line": 483,
"column": 2
} | {
"line": 483,
"column": 65
} | {
"line": 485,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\n⊢ s.toFinset.Nonempty ↔ s ≠ 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"congrArg",
"Finset",
"_private.Mathlib.Data.Finset.Basic.0.Multiset.toFinset_nonempty._simp_1_1",
"... | [] | simp only [toFinset_eq_empty, Ne, Finset.nonempty_iff_ne_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Basic | {
"line": 483,
"column": 2
} | {
"line": 483,
"column": 65
} | {
"line": 485,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\n⊢ s.toFinset.Nonempty ↔ s ≠ 0",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"congrArg",
"Finset",
"_private.Mathlib.Data.Finset.Basic.0.Multiset.toFinset_nonempty._simp_1_1",
"... | [] | simp only [toFinset_eq_empty, Ne, Finset.nonempty_iff_ne_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Image | {
"line": 395,
"column": 89
} | {
"line": 396,
"column": 92
} | {
"line": 398,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : DecidableEq β\nf : α → β\ns₁ s₂ : Finset α\nh : s₁ ⊆ s₂\n⊢ image f s₁ ⊆ image f s₂",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Multiset.map",
"Finset",
"Multiset.instHasSubset",
"PartialOrder.toPreorder",
"Mult... | [] | by
simp only [subset_def, image_val, subset_dedup', dedup_subset', Multiset.map_subset_map h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.Basic | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 46
} | {
"line": 344,
"column": 2
} | [
{
"pp": "n : ℕ\ni j : Fin n\nhij : i < j\n⊢ ↑0 ≠ ↑((j - i).castLT ⋯)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Fin.instSub",
"congrArg",
"Zero.ofOfNat0",
"HSub.hSub",
"id",
"Fin.instOfNat",
"instSubNat",
... | [
"n : ℕ\ni j : Fin n\nhij : i < j\n⊢ ¬0 = ↑j - ↑i"
] | simp [coe_sub_iff_le.mpr (Fin.le_of_lt hij)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Fin.Basic | {
"line": 274,
"column": 2
} | {
"line": 283,
"column": 40
} | {
"line": 285,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ Injective predAbove",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Fin.succ",
"congrArg",
"Fin.ne_of_lt",
"False.elim",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Exists",... | [] | intro i j hij
obtain ⟨n, rfl⟩ := Nat.exists_add_one_eq.2 i.size_positive
wlog! h : i < j generalizing i j
· obtain h | rfl := h.lt_or_eq
· exact (this hij.symm h).symm
· rfl
replace hij := congr_fun hij i.succ
rw [predAbove_succ_self, Fin.predAbove_of_le_castSucc _ _ (by simpa),
← Fin.castSucc_inj... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Fin.Basic | {
"line": 274,
"column": 2
} | {
"line": 283,
"column": 40
} | {
"line": 285,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ Injective predAbove",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Fin.succ",
"congrArg",
"Fin.ne_of_lt",
"False.elim",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Exists",... | [] | intro i j hij
obtain ⟨n, rfl⟩ := Nat.exists_add_one_eq.2 i.size_positive
wlog! h : i < j generalizing i j
· obtain h | rfl := h.lt_or_eq
· exact (this hij.symm h).symm
· rfl
replace hij := congr_fun hij i.succ
rw [predAbove_succ_self, Fin.predAbove_of_le_castSucc _ _ (by simpa),
← Fin.castSucc_inj... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.FinRange | {
"line": 52,
"column": 82
} | {
"line": 60,
"column": 24
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\nf : Fin n → α\n⊢ (ofFn f).Nodup ↔ Function.Injective f",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Fin.cons_succ",
"List.nodup_ofFn_ofInjective",
"Fin.succ",
"congrArg",
"_private.Mat... | [] | by
refine ⟨?_, nodup_ofFn_ofInjective⟩
refine Fin.consInduction ?_ (fun x₀ xs ih => ?_) f
· intro _
exact Function.injective_of_subsingleton _
· intro h
rw [Fin.cons_injective_iff]
simp_rw [ofFn_succ, Fin.cons_succ, nodup_cons, Fin.cons_zero, mem_ofFn] at h
exact h.imp_right ih | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fintype.Card | {
"line": 433,
"column": 2
} | {
"line": 433,
"column": 6
} | {
"line": 434,
"column": 2
} | [
{
"pp": "α : Type u_4\ninst✝ : Fintype α\nx : α\nh : Fintype.card α = 1\n⊢ univ = {x}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.univ",
"Finset",
"Finset.instSingleton",
"Singleton.singleton",
"Eq.symm"
],
"usedFVars": [
"α",
"... | [
"α : Type u_4\ninst✝ : Fintype α\nx : α\nh : Fintype.card α = 1\n⊢ {x} = univ"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Fintype.Card | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 52
} | {
"line": 532,
"column": 0
} | [
{
"pp": "n : ℕ\ns : Finset (Fin n)\n⊢ #s ≤ n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"congrArg",
"Eq.mp",
"Fintype.card",
"LE.le",
"instLENat",
"Fin.fintype",
"Finset.card_le_univ",
"Nat",
"Finset.card",... | [] | simpa only [Fintype.card_fin] using s.card_le_univ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Fintype.Card | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 52
} | {
"line": 532,
"column": 0
} | [
{
"pp": "n : ℕ\ns : Finset (Fin n)\n⊢ #s ≤ n",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Fintype.card_fin",
"congrArg",
"Eq.mp",
"Fintype.card",
"LE.le",
"instLENat",
"Fin.fintype",
"Finset.card_le_univ",
"Nat",
"Finset.card",... | [] | simpa only [Fintype.card_fin] using s.card_le_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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