module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.List.Basic | {
"line": 812,
"column": 21
} | {
"line": 812,
"column": 32
} | {
"line": 812,
"column": 33
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β → β\nb : β\nhf : ∀ (a : α), f a b = b\na : α\nl : List α\n⊢ foldr f b (a :: l) = b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.foldr_cons",
"congrArg",
"id",
"List.cons",
"List.foldr",
... | [
"α : Type u\nβ : Type v\nf : α → β → β\nb : β\nhf : ∀ (a : α), f a b = b\na : α\nl : List α\n⊢ f a (foldr f b l) = b"
] | foldr_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Basic | {
"line": 912,
"column": 4
} | {
"line": 912,
"column": 31
} | {
"line": 913,
"column": 2
} | [
{
"pp": "case cons.some\nα : Type u\nβ : Type v\nf : α → Option β\ng : α → β\na : α\nl : List α\nih : filterMap f l = map g l → ∀ (x : α), x ∈ l → f x = some (g x)\nb : β\nha : f a = some b\n⊢ filterMap f (a :: l) = map g (a :: l) → ∀ (x : α), x ∈ a :: l → f x = some (g x)",
"ppTerm": "?cons.some",
"ass... | [] | · simp +contextual [ha, ih] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Ring.Unbundled.Rat | {
"line": 39,
"column": 6
} | {
"line": 39,
"column": 22
} | {
"line": 39,
"column": 22
} | [
{
"pp": "m : ℕ\ns : Bool\ne : ℕ\n⊢ 0 ≤ Rat.ofScientific m s e",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"instPowNat",
"Rat.instOfNat",
"Eq.mpr",
"HMul.hMul",
"Rat.ofScientific._proof_2",
"Rat.ofScientific.eq_1",
"congrArg",
"Rat",
... | [
"m : ℕ\ns : Bool\ne : ℕ\n⊢ 0 ≤ if s = true then normalize (↑m) (10 ^ e) ⋯ else ↑(m * 10 ^ e)"
] | Rat.ofScientific | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Semiconj.Basic | {
"line": 52,
"column": 4
} | {
"line": 52,
"column": 21
} | {
"line": 54,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\na x y : G\nh : SemiconjBy a x y\nn : ℕ\n⊢ SemiconjBy a (x ^ (n + 1)) (y ^ (n + 1))",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"DivInvMonoid.toMonoid",
"instOfNatNat",
"Group.toDivInvMonoid",
"instHAdd",
"SemiconjBy.... | [] | apply pow_right h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.GroupWithZero.Divisibility | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 21
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\na : α\nm n : ℕ\nha₀ : a ≠ 0\nha : ¬IsUnit a\n⊢ n ≤ m → a ^ n ∣ a ^ m",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"pow_dvd_pow",
"CommMonoidWithZero.toMonoidWithZero",
"MonoidWith... | [] | apply pow_dvd_pow | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.GroupWithZero.Divisibility | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 21
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\na : α\nm n : ℕ\nha₀ : a ≠ 0\nha : ¬IsUnit a\n⊢ n ≤ m → a ^ n ∣ a ^ m",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"pow_dvd_pow",
"CommMonoidWithZero.toMonoidWithZero",
"MonoidWith... | [] | apply pow_dvd_pow | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Divisibility | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 21
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\na : α\nm n : ℕ\nha₀ : a ≠ 0\nha : ¬IsUnit a\n⊢ n ≤ m → a ^ n ∣ a ^ m",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"pow_dvd_pow",
"CommMonoidWithZero.toMonoidWithZero",
"MonoidWith... | [] | apply pow_dvd_pow | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rat.Lemmas | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 54
} | {
"line": 114,
"column": 0
} | [
{
"pp": "q : ℚ\nn : ℤ\n⊢ (q - ↑n).den = q.den",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Rat.instSub",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [] | rw [sub_eq_add_neg, ← Int.cast_neg, add_intCast_den] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Rat.Lemmas | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 54
} | {
"line": 114,
"column": 0
} | [
{
"pp": "q : ℚ\nn : ℤ\n⊢ (q - ↑n).den = q.den",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Rat.instSub",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [] | rw [sub_eq_add_neg, ← Int.cast_neg, add_intCast_den] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rat.Lemmas | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 54
} | {
"line": 114,
"column": 0
} | [
{
"pp": "q : ℚ\nn : ℤ\n⊢ (q - ↑n).den = q.den",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Rat.instSub",
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
... | [] | rw [sub_eq_add_neg, ← Int.cast_neg, add_intCast_den] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rat.Cast.Defs | {
"line": 68,
"column": 4
} | {
"line": 69,
"column": 10
} | {
"line": 70,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝ : DivisionSemiring α\na b : ℕ\nhb : ↑b ≠ 0\nd : ℕ\nh : d ≠ 0\nn : ℕ\nc : (↑n).natAbs.Coprime d\nhn : 0 ≤ { num := ↑n, den := d, den_nz := h, reduced := c }\nhd : ↑d ≠ 0\nhb' : b ≠ 0\ne : a * d = n * b\n⊢ ↑n * ↑b = ↑a * ↑d",
"ppTerm": "?m.150",
"assigned": true,
"usedCons... | [] | norm_cast
rw [e] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Rat.Cast.Defs | {
"line": 68,
"column": 4
} | {
"line": 69,
"column": 10
} | {
"line": 70,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝ : DivisionSemiring α\na b : ℕ\nhb : ↑b ≠ 0\nd : ℕ\nh : d ≠ 0\nn : ℕ\nc : (↑n).natAbs.Coprime d\nhn : 0 ≤ { num := ↑n, den := d, den_nz := h, reduced := c }\nhd : ↑d ≠ 0\nhb' : b ≠ 0\ne : a * d = n * b\n⊢ ↑n * ↑b = ↑a * ↑d",
"ppTerm": "?m.150",
"assigned": true,
"usedCons... | [] | norm_cast
rw [e] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rat.Cast.Defs | {
"line": 218,
"column": 55
} | {
"line": 218,
"column": 67
} | {
"line": 218,
"column": 68
} | [
{
"pp": "F : Type u_1\nα : Type u_3\nβ : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : DivisionRing α\ninst✝¹ : DivisionRing β\ninst✝ : RingHomClass F α β\nf : F\nq : ℚ\n⊢ ↑q.num / f ↑q.den = ↑q",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocS... | [
"F : Type u_1\nα : Type u_3\nβ : Type u_4\ninst✝³ : FunLike F α β\ninst✝² : DivisionRing α\ninst✝¹ : DivisionRing β\ninst✝ : RingHomClass F α β\nf : F\nq : ℚ\n⊢ ↑q.num / ↑q.den = ↑q"
] | map_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Rat.Lemmas | {
"line": 263,
"column": 4
} | {
"line": 263,
"column": 12
} | {
"line": 264,
"column": 4
} | [
{
"pp": "case pos\nn : ℤ\nhn : n = 0\n⊢ ↑(n / n) = ↑n / ↑n",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.cast",
"Int.instDiv",
"instHDiv",
"Rat",
"Rat.instIntCast",
"HDiv.hDiv",
"Int",
"instOfNat",
"Eq.ndrec",
"OfNat.ofNat"... | [
"case pos\n⊢ ↑(0 / 0) = ↑0 / ↑0"
] | subst hn | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Data.Tree.Basic | {
"line": 89,
"column": 52
} | {
"line": 92,
"column": 47
} | {
"line": 94,
"column": 0
} | [
{
"pp": "α : Type u\nt : BinaryTree α\n⊢ map id t = t",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"BinaryTree.map.eq_1",
"BinaryTree",
"BinaryTree.rec",
"BinaryTree.nil",
"id",
"BinaryTree.map.eq_2",
"BinaryTree... | [] | by
induction t with
| nil => rw [map]
| node v l r hl hr => rw [map, hl, hr, id_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.CompleteBooleanAlgebra | {
"line": 600,
"column": 20
} | {
"line": 600,
"column": 90
} | {
"line": 602,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort w'\ninst✝¹ : CompletelyDistribLattice α\ninst✝ : CompletelyDistribLattice β\nι✝ : Type (max u v)\nκ✝ : ι✝ → Type (max u v)\nf : (a : ι✝) → κ✝ a → α × β\n⊢ ⨅ a, ⨆ b, f a b = ⨆ g, ⨅ a, f a (g a)",
"ppTerm": "?m.16",
"assigned": true,
"usedConst... | [] | by ext <;> simp [fst_iSup, fst_iInf, snd_iSup, snd_iInf, iInf_iSup_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Set.Image | {
"line": 249,
"column": 74
} | {
"line": 250,
"column": 48
} | {
"line": 252,
"column": 0
} | [
{
"pp": "α : Type u_1\np q r : α → α → Prop\na b : α\nc : { x // p a x ∧ q x b }\nh : ∀ {x : α}, r (↑c) x → p a x\n⊢ {x | p a x ∧ q x b} ∩ {y | r (↑c) y} = {y | r (↑c) y ∧ q y b}",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.ext",
"congrArg",
"setOf",
"Member... | [] | by
ext; simp +contextual [@and_comm (r _ _), h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.ConditionallyCompleteLattice.Basic | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 43
} | {
"line": 105,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LE α\ninst✝ : SupSet α\ns : Set α\nhs : BddAbove s\n⊢ ↑(sSup s) =\n if ⊤ ∈ (fun a ↦ ↑a) '' s then ⊤\n else if BddAbove ((fun a ↦ ↑a) ⁻¹' (fun a ↦ ↑a) '' s) then ↑(sSup ((fun a ↦ ↑a) ⁻¹' (fun a ↦ ↑a) '' s)) else ⊤",
"ppTerm": "?m.22",
"assigned": true,
"usedConst... | [
"case h\nα : Type u_1\ninst✝¹ : LE α\ninst✝ : SupSet α\ns : Set α\nhs : BddAbove s\n⊢ Injective fun a ↦ ↑a",
"case hnc\nα : Type u_1\ninst✝¹ : LE α\ninst✝ : SupSet α\ns : Set α\nhs : BddAbove s\n⊢ ⊤ ∉ (fun a ↦ ↑a) '' s"
] | rw [if_neg, preimage_image_eq, if_pos hs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.ConditionallyCompleteLattice.Basic | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 15
} | {
"line": 429,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : ¬BddAbove s\n⊢ ¬BddAbove univ",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Set.univ",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Mathlib.Tactic.Contrapose.contrapose₄",
"C... | [
"α : Type u_1\ninst✝ : ConditionallyCompleteLinearOrder α\ns : Set α\nhs : BddAbove univ\n⊢ BddAbove s"
] | contrapose hs | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose_1 | Mathlib.Tactic.Contrapose.contrapose |
Mathlib.Order.ConditionallyCompleteLattice.Basic | {
"line": 473,
"column": 6
} | {
"line": 473,
"column": 38
} | {
"line": 474,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLinearOrder α\ns t : Set α\nhs : ∀ x ∈ s, ∃ y ∈ t, x ≤ y\nht : ∀ y ∈ t, ∃ x ∈ s, y ≤ x\ns_ne : s.Nonempty\nt_ne : t.Nonempty\nB : BddAbove s ∨ BddAbove t\nBs : BddAbove s\nBt : BddAbove t\nx : α\nhx : x ∈ s\ny : α\nyt : y ∈ t\nhxy : x ≤ y\n⊢ x ≤ sSup t",
"... | [] | exact hxy.trans (le_csSup Bt yt) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.ConditionallyCompleteLattice.Basic | {
"line": 739,
"column": 24
} | {
"line": 739,
"column": 34
} | {
"line": 739,
"column": 34
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLinearOrderBot α\ns : Set α\nhs : s.Nonempty\nh's : BddBelow s\n⊢ sInf ((fun a ↦ ↑a) '' s) = ⨅ a ∈ s, ↑a",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WithTop.instInfSet",
"Eq.mpr",
"WithTop.instCompleteLinearOrder",
... | [
"α : Type u_1\ninst✝ : ConditionallyCompleteLinearOrderBot α\ns : Set α\nhs : s.Nonempty\nh's : BddBelow s\n⊢ ⨅ a ∈ s, ↑a = ⨅ a ∈ s, ↑a"
] | sInf_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Set.LinearOrder | {
"line": 112,
"column": 54
} | {
"line": 115,
"column": 38
} | {
"line": 117,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : α\n⊢ Ioi b ⊆ Ioi a ↔ a ≤ b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"not_le",
"Set.Ioi",
"Preorder.toLT",
"lt_irrefl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOr... | [] | by
refine ⟨fun h => ?_, Ioi_subset_Ioi⟩
by_contra ba
exact lt_irrefl _ (h (not_le.mp ba)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Pairwise.Basic | {
"line": 131,
"column": 2
} | {
"line": 131,
"column": 40
} | {
"line": 133,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nι : Type u_4\ninst✝¹ : Nonempty ι\ns : Set α\nf : α → ι\nr : ι → ι → Prop\ninst✝ : IsEquiv ι r\nhne : s.Nonempty\n⊢ s.Pairwise (r on f) ↔ ∃ z, ∀ (x : α), x ∈ s → r (f x) z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.Nonempty.pairwise_iff_exist... | [] | · exact hne.pairwise_iff_exists_forall | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.SetLike.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 41
} | {
"line": 258,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : SetLike A B\ninst✝¹ : LE A\ninst✝ : IsConcreteLE A B\nS T : A\n⊢ S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Membership.mem",
"LE.le",
"i... | [] | simp [← coe_subset_coe, Set.subset_def] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.SetLike.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 41
} | {
"line": 258,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : SetLike A B\ninst✝¹ : LE A\ninst✝ : IsConcreteLE A B\nS T : A\n⊢ S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Membership.mem",
"LE.le",
"i... | [] | simp [← coe_subset_coe, Set.subset_def] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.SetLike.Basic | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 41
} | {
"line": 258,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝² : SetLike A B\ninst✝¹ : LE A\ninst✝ : IsConcreteLE A B\nS T : A\n⊢ S ≤ T ↔ ∀ ⦃x : B⦄, x ∈ S → x ∈ T",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"congrArg",
"Membership.mem",
"LE.le",
"i... | [] | simp [← coe_subset_coe, Set.subset_def] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.OrdConnected | {
"line": 140,
"column": 12
} | {
"line": 140,
"column": 58
} | {
"line": 140,
"column": 58
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\ns : Set α\nh : (⇑ofDual ⁻¹' s).OrdConnected\n⊢ s.OrdConnected",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"OrderDual.ofDual",
"Membership.mem",
"Eq.mp",
"id",
"Equiv",
... | [] | by simpa only [ordConnected_def] using! h.dual | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Nonneg.Ring | {
"line": 66,
"column": 24
} | {
"line": 66,
"column": 50
} | {
"line": 66,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝³ : Semiring α\ninst✝² : PartialOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : ExistsAddOfLE α\na b : { x // 0 ≤ x }\nh : ↑a ≤ ↑b\nc : α\nhc : ↑b = ↑a + c\n⊢ b = a + ⟨c, ?m.49⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMo... | [] | simp [Subtype.ext_iff, hc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Nonneg.Ring | {
"line": 66,
"column": 24
} | {
"line": 66,
"column": 50
} | {
"line": 66,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝³ : Semiring α\ninst✝² : PartialOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : ExistsAddOfLE α\na b : { x // 0 ≤ x }\nh : ↑a ≤ ↑b\nc : α\nhc : ↑b = ↑a + c\n⊢ b = a + ⟨c, ?m.49⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMo... | [] | simp [Subtype.ext_iff, hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Nonneg.Ring | {
"line": 66,
"column": 24
} | {
"line": 66,
"column": 50
} | {
"line": 66,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝³ : Semiring α\ninst✝² : PartialOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : ExistsAddOfLE α\na b : { x // 0 ≤ x }\nh : ↑a ≤ ↑b\nc : α\nhc : ↑b = ↑a + c\n⊢ b = a + ⟨c, ?m.49⟩",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMo... | [] | simp [Subtype.ext_iff, hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rat.Cast.Order | {
"line": 149,
"column": 69
} | {
"line": 153,
"column": 8
} | {
"line": 155,
"column": 0
} | [
{
"pp": "K : Type u_5\ninst✝² : Semifield K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\np q : ℚ≥0\nh : p < q\n⊢ ↑p < ↑q",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [] | by
rwa [NNRat.cast_def, NNRat.cast_def, div_lt_div_iff₀, ← Nat.cast_mul, ← Nat.cast_mul,
Nat.cast_lt (α := K), ← NNRat.lt_def]
· simp
· simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.NormNum.Ineq | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 31
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Ring α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nnum✝¹ num✝ : ℤ\nda db : ℕ\ninv✝¹ : Invertible ↑da\ninv✝ : Invertible ↑db\nh✝¹ : decide (num✝¹.mul (Int.ofNat db) ≤ num✝.mul (Int.ofNat da)) = true\nh✝ : ↑(num✝¹.mul (Int.ofNat db)) ≤ ↑(num✝.mul (Int.ofNat da))\nha : 0... | [] | rwa [Int.commute_cast] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Order.Group.PosPart | {
"line": 141,
"column": 87
} | {
"line": 143,
"column": 28
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : Group α\ninst✝ : MulLeftMono α\na : α\n⊢ a⁺ᵐ / a⁻ᵐ = a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"Lattice.toSemilatticeSup",
"instHDiv",
"HMul.hMul"... | [] | by
rw [div_eq_mul_inv, mul_inv_eq_iff_eq_mul, leOnePart_def, mul_sup, mul_one, mul_inv_cancel,
sup_comm, oneLePart_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.NormNum.Ineq | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 31
} | {
"line": 140,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Ring α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nnum✝¹ num✝ : ℤ\nda db : ℕ\ninv✝¹ : Invertible ↑da\ninv✝ : Invertible ↑db\nh✝¹ : decide (num✝¹ * ↑db < num✝ * ↑da) = true\nh✝ : (fun x ↦ ↑x) (num✝¹ * ↑db) < (fun x ↦ ↑x) (num✝ * ↑da)\nha : 0 < ⅟↑da\nhb : 0 < ⅟↑db\nh : ... | [] | rwa [Int.commute_cast] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Notation.Support | {
"line": 177,
"column": 18
} | {
"line": 178,
"column": 83
} | {
"line": 180,
"column": 0
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\nN : Type u_4\ninst✝¹ : One M\ninst✝ : One N\nf : ι → M\ng : ι → N\nx : ι\n⊢ (x ∈ mulSupport fun x ↦ (f x, g x)) ↔ x ∈ mulSupport f ∪ mulSupport g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"congrArg",
"setOf",
"Membership.mem",
... | [] | by
simp only [mulSupport, not_and_or, mem_union, mem_setOf_eq, Prod.mk_eq_one, Ne] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorial.Basic | {
"line": 296,
"column": 4
} | {
"line": 298,
"column": 86
} | {
"line": 300,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ (n + 1).ascFactorial (k + 1) ≤ (n + (k + 1)) ^ (k + 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.mul_le_mul_right",
"Nat.ascFactorial",
"Nat.ascFactorial_succ",
"... | [] | rw [ascFactorial_succ, Nat.pow_succ, Nat.mul_comm, ← Nat.add_assoc, Nat.add_right_comm n 1 k]
exact Nat.mul_le_mul_right _
(Nat.le_trans (ascFactorial_le_pow_add _ k) (Nat.pow_le_pow_left (le_succ _) _)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Factorial.Basic | {
"line": 296,
"column": 4
} | {
"line": 298,
"column": 86
} | {
"line": 300,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ (n + 1).ascFactorial (k + 1) ≤ (n + (k + 1)) ^ (k + 1)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.mul_le_mul_right",
"Nat.ascFactorial",
"Nat.ascFactorial_succ",
"... | [] | rw [ascFactorial_succ, Nat.pow_succ, Nat.mul_comm, ← Nat.add_assoc, Nat.add_right_comm n 1 k]
exact Nat.mul_le_mul_right _
(Nat.le_trans (ascFactorial_le_pow_add _ k) (Nat.pow_le_pow_left (le_succ _) _)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Factorial.Basic | {
"line": 362,
"column": 14
} | {
"line": 362,
"column": 83
} | {
"line": 364,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ n.succ.descFactorial n.succ = n.succ !",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.factorial_succ",
"id",
"instMulNat",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat.fac... | [] | by rw [succ_descFactorial_succ, descFactorial_self n, factorial_succ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Ring.Pow | {
"line": 53,
"column": 8
} | {
"line": 54,
"column": 72
} | {
"line": 55,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\na b : R\nHcomm : Commute a b\nha : 0 ≤ a\nHsq : 0 ≤ b ^ 2\nHsq' : 0 ≤ (a + b) ^ 2\nH : 0 ≤ 2 * a + b\nn : ℕ\n⊢ (a + b) ^ 2 * (a ^ (n + 1) + ↑(n + 1) * a ^ n * b) ≤ (a + b) ^ 2 * (a + b) ^ (n + 1)",
"ppTerm": "?m.45... | [] | gcongr
apply Commute.pow_add_mul_le_add_pow_of_sq_nonneg <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Pow | {
"line": 53,
"column": 8
} | {
"line": 54,
"column": 72
} | {
"line": 55,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\na b : R\nHcomm : Commute a b\nha : 0 ≤ a\nHsq : 0 ≤ b ^ 2\nHsq' : 0 ≤ (a + b) ^ 2\nH : 0 ≤ 2 * a + b\nn : ℕ\n⊢ (a + b) ^ 2 * (a ^ (n + 1) + ↑(n + 1) * a ^ n * b) ≤ (a + b) ^ 2 * (a + b) ^ (n + 1)",
"ppTerm": "?m.45... | [] | gcongr
apply Commute.pow_add_mul_le_add_pow_of_sq_nonneg <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rat.Floor | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 82
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : FloorRing α\nn d : ℕ\ninv : Invertible ↑d\n⊢ fract (↑n * ⅟↑d) = ↑(n % d) * ⅟↑d",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"in... | [] | simp only [invOf_eq_inv, ← div_eq_mul_inv, fract_div_natCast_eq_div_natCast_mod] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 473,
"column": 20
} | {
"line": 473,
"column": 39
} | {
"line": 473,
"column": 39
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorRing R\ninst✝ : IsOrderedRing R\nx : R\nhx : fract x ≠ 0\n| -x",
"ppTerm": "?m.99",
"assigned": true,
"usedConstants": [
"Int.cast",
"NegZeroClass.toNeg",
"Int.floor",
"congrArg",
"Int.fract",... | [
"R : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorRing R\ninst✝ : IsOrderedRing R\nx : R\nhx : fract x ≠ 0\n| -(↑⌊x⌋ + fract x)"
] | ← floor_add_fract x | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.AddConstMap.Basic | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 47
} | {
"line": 260,
"column": 4
} | [
{
"pp": "F : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁷ : FunLike F G H\na : G\nb : H\ninst✝⁶ : AddCommGroup G\ninst✝⁵ : LinearOrder G\ninst✝⁴ : IsOrderedAddMonoid G\ninst✝³ : Archimedean G\ninst✝² : AddGroup H\ninst✝¹ : AddConstMapClass F G H a b\nf : F\nR : H → H → Prop\ninst✝ : IsTrans H R\nha : 0 < a\nl :... | [
"case inl\nF : Type u_1\nG : Type u_2\nH : Type u_3\ninst✝⁷ : FunLike F G H\na : G\nb : H\ninst✝⁶ : AddCommGroup G\ninst✝⁵ : LinearOrder G\ninst✝⁴ : IsOrderedAddMonoid G\ninst✝³ : Archimedean G\ninst✝² : AddGroup H\ninst✝¹ : AddConstMapClass F G H a b\nf : F\nR : H → H → Prop\ninst✝ : IsTrans H R\nha : 0 < a\nl : G... | rcases lt_trichotomy n 0 with hn | rfl | hn | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 11
} | {
"line": 501,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝³ : Ring R\ninst✝² : LinearOrder R\ninst✝¹ : FloorRing R\ninst✝ : IsOrderedRing R\ns : Set R\nx : R\nhms : x - ↑⌊x⌋ ∈ s\nhm0 : 0 ≤ x - ↑⌊x⌋\nhm1 : x - ↑⌊x⌋ < 1\n⊢ fract x ∈ s",
"ppTerm": "?m.127",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hms"
]... | [] | exact hms | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 539,
"column": 4
} | {
"line": 542,
"column": 44
} | {
"line": 544,
"column": 0
} | [
{
"pp": "case inr.refine_3\nk : Type u_4\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsOrderedRing k\ninst✝ : FloorRing k\nm n : ℕ\nhn : n > 0\nhn' : 0 < ↑n\n⊢ ↑m / ↑n - ↑(m % n) / ↑n = ↑(↑m / ↑n)",
"ppTerm": "?inr.refine_3",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",... | [] | rw [sub_eq_iff_eq_add', ← mul_right_inj' hn'.ne', mul_div_cancel₀ _ hn'.ne', mul_add,
mul_div_cancel₀ _ hn'.ne']
norm_cast
rw [← Nat.cast_add, Nat.mod_add_div m n] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 539,
"column": 4
} | {
"line": 542,
"column": 44
} | {
"line": 544,
"column": 0
} | [
{
"pp": "case inr.refine_3\nk : Type u_4\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsOrderedRing k\ninst✝ : FloorRing k\nm n : ℕ\nhn : n > 0\nhn' : 0 < ↑n\n⊢ ↑m / ↑n - ↑(m % n) / ↑n = ↑(↑m / ↑n)",
"ppTerm": "?inr.refine_3",
"assigned": true,
"usedConstants": [
"Nat.cast_mul._simp_1",... | [] | rw [sub_eq_iff_eq_add', ← mul_right_inj' hn'.ne', mul_div_cancel₀ _ hn'.ne', mul_add,
mul_div_cancel₀ _ hn'.ne']
norm_cast
rw [← Nat.cast_add, Nat.mod_add_div m n] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Archimedean.Basic | {
"line": 258,
"column": 15
} | {
"line": 258,
"column": 68
} | {
"line": 258,
"column": 68
} | [
{
"pp": "K : Type u_4\ninst✝⁴ : Semifield K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : Archimedean K\nx y : K\ninst✝ : ExistsAddOfLE K\nhx : 0 < x\nhy : y < 1\ny_pos : 0 < y\nq : ℕ\nhq : x⁻¹ < y⁻¹ ^ q\n⊢ y ^ q < x",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
... | [] | rwa [inv_pow, inv_lt_inv₀ hx (pow_pos y_pos _)] at hq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Order.Archimedean.Basic | {
"line": 258,
"column": 15
} | {
"line": 258,
"column": 68
} | {
"line": 258,
"column": 68
} | [
{
"pp": "K : Type u_4\ninst✝⁴ : Semifield K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : Archimedean K\nx y : K\ninst✝ : ExistsAddOfLE K\nhx : 0 < x\nhy : y < 1\ny_pos : 0 < y\nq : ℕ\nhq : x⁻¹ < y⁻¹ ^ q\n⊢ y ^ q < x",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
... | [] | rwa [inv_pow, inv_lt_inv₀ hx (pow_pos y_pos _)] at hq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Basic | {
"line": 258,
"column": 15
} | {
"line": 258,
"column": 68
} | {
"line": 258,
"column": 68
} | [
{
"pp": "K : Type u_4\ninst✝⁴ : Semifield K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : Archimedean K\nx y : K\ninst✝ : ExistsAddOfLE K\nhx : 0 < x\nhy : y < 1\ny_pos : 0 < y\nq : ℕ\nhq : x⁻¹ < y⁻¹ ^ q\n⊢ y ^ q < x",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
... | [] | rwa [inv_pow, inv_lt_inv₀ hx (pow_pos y_pos _)] at hq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Floor.Ring | {
"line": 598,
"column": 61
} | {
"line": 599,
"column": 48
} | {
"line": 601,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : FloorRing R\na : R\nha : -1 < a\n⊢ 0 ≤ ⌈a⌉",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"Preorder.toLT",
"AddGroupWithOne.toAddGrou... | [] | by
rwa [Int.le_ceil_iff, Int.cast_zero, zero_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Archimedean.Basic | {
"line": 296,
"column": 6
} | {
"line": 305,
"column": 83
} | {
"line": 307,
"column": 0
} | [
{
"pp": "case inr.inr\nK : Type u_4\ninst✝⁴ : Semifield K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : Archimedean K\ninst✝ : ExistsAddOfLE K\na b c : K\nh : a < b * c\nhb₀ : 0 < b\nhc₀ : 0 < c\nhc₁ : c < 1\nha : 0 < a\nhb₁ : 1 < b\n⊢ ∃ n, a < c ^ n ∧ c ^ n < b",
"ppTerm": "?inr.inr",
... | [] | rcases lt_or_ge a 1 with ha₁ | ha₁
· refine ⟨0, ?_⟩
rw [zpow_zero]
exact ⟨ha₁, hb₁⟩
· have : b⁻¹ < a⁻¹ * c := by rwa [lt_inv_mul_iff₀' ha, inv_mul_lt_iff₀ hb₀]
obtain ⟨n, hn₁, hn₂⟩ :=
exists_pow_btwn_of_lt_mul this (inv_pos_of_pos ha) (inv_le_one_of_one_le₀ ha₁) hc₀ hc₁
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Basic | {
"line": 296,
"column": 6
} | {
"line": 305,
"column": 83
} | {
"line": 307,
"column": 0
} | [
{
"pp": "case inr.inr\nK : Type u_4\ninst✝⁴ : Semifield K\ninst✝³ : LinearOrder K\ninst✝² : IsStrictOrderedRing K\ninst✝¹ : Archimedean K\ninst✝ : ExistsAddOfLE K\na b c : K\nh : a < b * c\nhb₀ : 0 < b\nhc₀ : 0 < c\nhc₁ : c < 1\nha : 0 < a\nhb₁ : 1 < b\n⊢ ∃ n, a < c ^ n ∧ c ^ n < b",
"ppTerm": "?inr.inr",
... | [] | rcases lt_or_ge a 1 with ha₁ | ha₁
· refine ⟨0, ?_⟩
rw [zpow_zero]
exact ⟨ha₁, hb₁⟩
· have : b⁻¹ < a⁻¹ * c := by rwa [lt_inv_mul_iff₀' ha, inv_mul_lt_iff₀ hb₀]
obtain ⟨n, hn₁, hn₂⟩ :=
exists_pow_btwn_of_lt_mul this (inv_pos_of_pos ha) (inv_le_one_of_one_le₀ ha₁) hc₀ hc₁
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Torsor.Basic | {
"line": 251,
"column": 28
} | {
"line": 251,
"column": 40
} | {
"line": 251,
"column": 41
} | [
{
"pp": "G : Type u_3\nP : Type u_4\ninst✝ : AddGroup G\nT : AddTorsor G P\nx y : P\nh : Injective fun x ↦ 2 • x\n⊢ x -ᵥ y + (x -ᵥ y) = 0 ↔ y = x",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"instHSMul",
"congrArg",
... | [
"G : Type u_3\nP : Type u_4\ninst✝ : AddGroup G\nT : AddTorsor G P\nx y : P\nh : Injective fun x ↦ 2 • x\n⊢ 2 • (x -ᵥ y) = 0 ↔ y = x"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Torsor.Basic | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 16
} | {
"line": 259,
"column": 17
} | [
{
"pp": "G : Type u_5\nP : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : AddTorsor G P\nh : Injective fun x ↦ 2 • x\ny x₁ x₂ : P\nhy : x₁ -ᵥ x₂ + (x₁ -ᵥ x₂) = 0\n⊢ x₁ = x₂",
"ppTerm": "?m.100",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"instHSMul",
"congrArg... | [
"G : Type u_5\nP : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : AddTorsor G P\nh : Injective fun x ↦ 2 • x\ny x₁ x₂ : P\nhy : 2 • (x₁ -ᵥ x₂) = 0\n⊢ x₁ = x₂"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Nodup | {
"line": 313,
"column": 2
} | {
"line": 315,
"column": 39
} | {
"line": 317,
"column": 0
} | [
{
"pp": "α : Type u\nl₂ : List α\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\nl₁ : List α\nh : l₂.Nodup\n⊢ (l₁ ∪ l₂).Nodup",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"List.Nodup.insert",
"List.rec",
"List.Nodup",
"List",
"List.instUnionOfBEq_batteries",
"Un... | [] | induction l₁ generalizing l₂ with
| nil => exact h
| cons a l₁ ih => exact (ih h).insert | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.List.Nodup | {
"line": 313,
"column": 2
} | {
"line": 315,
"column": 39
} | {
"line": 317,
"column": 0
} | [
{
"pp": "α : Type u\nl₂ : List α\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\nl₁ : List α\nh : l₂.Nodup\n⊢ (l₁ ∪ l₂).Nodup",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"List.Nodup.insert",
"List.rec",
"List.Nodup",
"List",
"List.instUnionOfBEq_batteries",
"Un... | [] | induction l₁ generalizing l₂ with
| nil => exact h
| cons a l₁ ih => exact (ih h).insert | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Nodup | {
"line": 313,
"column": 2
} | {
"line": 315,
"column": 39
} | {
"line": 317,
"column": 0
} | [
{
"pp": "α : Type u\nl₂ : List α\ninst✝¹ : BEq α\ninst✝ : LawfulBEq α\nl₁ : List α\nh : l₂.Nodup\n⊢ (l₁ ∪ l₂).Nodup",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"List.Nodup.insert",
"List.rec",
"List.Nodup",
"List",
"List.instUnionOfBEq_batteries",
"Un... | [] | induction l₁ generalizing l₂ with
| nil => exact h
| cons a l₁ ih => exact (ih h).insert | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Dedup | {
"line": 103,
"column": 16
} | {
"line": 103,
"column": 25
} | {
"line": 103,
"column": 26
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\na : α\nl' : List α\nh : a ∈ l ∧ ¬a ∈ l' ∧ l.dedup.tail = l'\nthis : l.dedup.head! :: l.dedup.tail = l.dedup\nhal : a ∈ l.dedup.head! :: l.dedup.tail\n⊢ l.dedup = a :: l'",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants... | [
"case refine_2\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\na : α\nl' : List α\nh : a ∈ l ∧ ¬a ∈ l' ∧ l.dedup.tail = l'\nthis : l.dedup.head! :: l.dedup.tail = l.dedup\nhal : a = l.dedup.head! ∨ a ∈ l.dedup.tail\n⊢ l.dedup = a :: l'"
] | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Dedup | {
"line": 101,
"column": 4
} | {
"line": 104,
"column": 83
} | {
"line": 106,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\na : α\nl' : List α\nh : a ∈ l ∧ ¬a ∈ l' ∧ l.dedup.tail = l'\n⊢ l.dedup = a :: l'",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Classical.or_iff_not_imp_right",
"congrArg",
"Li... | [] | have := @List.cons_head!_tail α ⟨a⟩ _ (ne_nil_of_mem (mem_dedup.2 h.1))
have hal : a ∈ l.dedup := mem_dedup.2 h.1
rw [← this, mem_cons, or_iff_not_imp_right] at hal
exact this ▸ h.2.2.symm ▸ cons_eq_cons.2 ⟨(hal (h.2.2.symm ▸ h.2.1)).symm, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Dedup | {
"line": 101,
"column": 4
} | {
"line": 104,
"column": 83
} | {
"line": 106,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\na : α\nl' : List α\nh : a ∈ l ∧ ¬a ∈ l' ∧ l.dedup.tail = l'\n⊢ l.dedup = a :: l'",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Classical.or_iff_not_imp_right",
"congrArg",
"Li... | [] | have := @List.cons_head!_tail α ⟨a⟩ _ (ne_nil_of_mem (mem_dedup.2 h.1))
have hal : a ∈ l.dedup := mem_dedup.2 h.1
rw [← this, mem_cons, or_iff_not_imp_right] at hal
exact this ▸ h.2.2.symm ▸ cons_eq_cons.2 ⟨(hal (h.2.2.symm ▸ h.2.1)).symm, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.MapFold | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 33
} | {
"line": 143,
"column": 34
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type v\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f u = t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ map f ((a ::ₘ u).erase a) = t",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multi... | [
"case mpr\nα : Type u_1\nβ : Type v\ninst✝ : DecidableEq α\nf : α → β\nt : Multiset β\na : α\nu : Multiset α\nh1 : a ∈ a ::ₘ u\nh : map f u = t\nthis : f a ∈ map f (a ::ₘ u)\n⊢ map f u = t"
] | Multiset.erase_cons_head, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.UnionInter | {
"line": 169,
"column": 80
} | {
"line": 176,
"column": 63
} | {
"line": 178,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t u : Multiset α\n⊢ s ∩ t + u = (s + u) ∩ (t + u)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"Multiset.lt_iff_cons_le",
"Multiset.instInter",
"congrArg",
"PartialOrder.toPreor... | [] | by
by_contra! h
obtain ⟨a, ha⟩ := lt_iff_cons_le.1 <| h.lt_of_le <| le_inter
(Multiset.add_le_add_right inter_le_left) (Multiset.add_le_add_right inter_le_right)
rw [← cons_add] at ha
exact (lt_cons_self (s ∩ t) a).not_ge <| le_inter
(Multiset.le_of_add_le_add_right (ha.trans inter_le_left))
(Multis... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Dedup | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 27
} | {
"line": 149,
"column": 27
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nl l' : List α\nhl : l.Nodup\nhl' : l'.Nodup\nh : l.toFinset = l'.toFinset\n⊢ l ~ l'",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Multiset",
"id",
"List.Perm",
"Multiset.coe_eq_coe",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\nl l' : List α\nhl : l.Nodup\nhl' : l'.Nodup\nh : l.toFinset = l'.toFinset\n⊢ ↑l = ↑l'"
] | ← Multiset.coe_eq_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Infix | {
"line": 221,
"column": 27
} | {
"line": 221,
"column": 39
} | {
"line": 221,
"column": 39
} | [
{
"pp": "α : Type u_1\ns✝ : List α\na : α\nt : List α\nmi : s✝ <+: a :: t\nb : α\ns r : List α\nhr : b :: s ++ r = a :: t\nba : b ≍ a\nst : s ++ r ≍ t\n⊢ ∃ l, l ∈ t.inits ∧ a :: l = b :: s",
"ppTerm": "?m.168",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Membership.... | [
"α : Type u_1\ns✝ : List α\na : α\nt : List α\nmi : s✝ <+: a :: t\nb : α\ns r : List α\nhr : b :: s ++ r = a :: t\nba : b ≍ a\nst : s ++ r ≍ t\n⊢ ∃ l, l ∈ t.inits ∧ a :: l = a :: s"
] | eq_of_heq ba | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.BooleanAlgebra | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 23
} | {
"line": 213,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Set.InjOn (fun a ↦ insert a s) (↑s)ᶜ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Finset.insert_inj_on"
],
"usedFVars": [
"α",
"inst✝",
"s"
],
"usedGoals": []
}
] | [] | exact s.insert_inj_on | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fintype.Sets | {
"line": 84,
"column": 94
} | {
"line": 85,
"column": 42
} | {
"line": 87,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\ninst✝¹ : Fintype ↑s\ninst✝ : Fintype ↑t\n⊢ s.toFinset ⊆ t.toFinset ↔ s ⊆ t",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"LE.le",
... | [] | by
simp [Finset.subset_iff, Set.subset_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Image | {
"line": 224,
"column": 22
} | {
"line": 224,
"column": 77
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α ↪ β\na : α\n⊢ ↑(map f {a}) = ↑{f a}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.coe_singleton",
"Set.image_singleton",
"congrArg",
"Finset",
"Finset.map",
"Set.instSingletonSet",
"Function.Embe... | [] | simp only [coe_map, coe_singleton, Set.image_singleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.Image | {
"line": 224,
"column": 22
} | {
"line": 224,
"column": 77
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α ↪ β\na : α\n⊢ ↑(map f {a}) = ↑{f a}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.coe_singleton",
"Set.image_singleton",
"congrArg",
"Finset",
"Finset.map",
"Set.instSingletonSet",
"Function.Embe... | [] | simp only [coe_map, coe_singleton, Set.image_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Image | {
"line": 224,
"column": 22
} | {
"line": 224,
"column": 77
} | {
"line": 226,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α ↪ β\na : α\n⊢ ↑(map f {a}) = ↑{f a}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finset.coe_singleton",
"Set.image_singleton",
"congrArg",
"Finset",
"Finset.map",
"Set.instSingletonSet",
"Function.Embe... | [] | simp only [coe_map, coe_singleton, Set.image_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fintype.Sets | {
"line": 258,
"column": 22
} | {
"line": 258,
"column": 34
} | {
"line": 260,
"column": 0
} | [
{
"pp": "⊢ univ.val = {True, False}.val",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Multiset.ndinsert_of_notMem",
"False",
"Finset.univ",
"LinearOrder.toDecidableEq",
"iff_false",
"congrArg",
"Finset",
"Prop.fintype",
... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Image | {
"line": 633,
"column": 15
} | {
"line": 633,
"column": 43
} | {
"line": 635,
"column": 0
} | [
{
"pp": "α : Type u_1\np : α → Prop\ninst✝ : DecidablePred p\ns : Finset α\na : α\nha : p a\n⊢ ⟨a, ha⟩ ∈ Finset.subtype p s ↔ ↑⟨a, ha⟩ ∈ s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Finset.mem_filter._simp_1",
"and_true",
"Iff.of_eq",
"congrArg",
"Finset... | [] | by simp [Finset.subtype, ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.Basic | {
"line": 99,
"column": 29
} | {
"line": 99,
"column": 35
} | {
"line": 99,
"column": 35
} | [
{
"pp": "n : ℕ\na b : Fin (n + 1)\nha : a ≠ last n\nhab : b ≤ a\n⊢ a - b < last n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.instSub",
"congrArg",
"HSub.hSub",
"id",
"instOfNatNat",
"Fin.val",
"instHAdd",
"Fin.lt_def... | [
"n : ℕ\na b : Fin (n + 1)\nha : a ≠ last n\nhab : b ≤ a\n⊢ ↑(a - b) < ↑(last n)"
] | lt_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Fin.Basic | {
"line": 312,
"column": 92
} | {
"line": 313,
"column": 14
} | {
"line": 315,
"column": 0
} | [
{
"pp": "m n : ℕ\nh : n = m\n⊢ (castOrderIso h).toEquiv = Equiv.cast ⋯",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Fin.castOrderIso",
"Equiv.cast",
"Eq.rec",
"Equiv",
"LE.le",
"instLEFin",
"Nat",
"Eq.refl",
"RelIso.toEquiv",
... | [] | by
subst h; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.OfFn | {
"line": 56,
"column": 4
} | {
"line": 57,
"column": 50
} | {
"line": 59,
"column": 0
} | [
{
"pp": "case succ\nα : Type u\nn : ℕ\nIH : ∀ (f : Fin n.succ → α), ofFn f = (ofFn fun i ↦ f i.castSucc).concat (f (Fin.last n))\nf : Fin (n + 1).succ → α\n⊢ ofFn f = (ofFn fun i ↦ f i.castSucc).concat (f (Fin.last (n + 1)))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [ofFn_succ, IH, ofFn_succ, concat_cons, Fin.castSucc_zero, Fin.succ_last]
simp only [succ_eq_add_one, Fin.castSucc_succ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.OfFn | {
"line": 56,
"column": 4
} | {
"line": 57,
"column": 50
} | {
"line": 59,
"column": 0
} | [
{
"pp": "case succ\nα : Type u\nn : ℕ\nIH : ∀ (f : Fin n.succ → α), ofFn f = (ofFn fun i ↦ f i.castSucc).concat (f (Fin.last n))\nf : Fin (n + 1).succ → α\n⊢ ofFn f = (ofFn fun i ↦ f i.castSucc).concat (f (Fin.last (n + 1)))",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [ofFn_succ, IH, ofFn_succ, concat_cons, Fin.castSucc_zero, Fin.succ_last]
simp only [succ_eq_add_one, Fin.castSucc_succ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.OfFn | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 57
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u\nR : α → α → Prop\nn : ℕ\nf : Fin n → α\n⊢ Pairwise R (ofFn f) ↔ ∀ ⦃i j : Fin n⦄, i < j → R (f i) (f j)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.List.OfFn.0.List.pairwise_ofFn._simp_1_3",
"List.Pairwise",
"congrArg",
... | [] | simp only [pairwise_iff_getElem, length_ofFn, List.getElem_ofFn,
Fin.forall_iff,
Fin.mk_lt_mk, forall_comm (α := (_ : Prop)) (β := ℕ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.List.OfFn | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 57
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u\nR : α → α → Prop\nn : ℕ\nf : Fin n → α\n⊢ Pairwise R (ofFn f) ↔ ∀ ⦃i j : Fin n⦄, i < j → R (f i) (f j)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.List.OfFn.0.List.pairwise_ofFn._simp_1_3",
"List.Pairwise",
"congrArg",
... | [] | simp only [pairwise_iff_getElem, length_ofFn, List.getElem_ofFn,
Fin.forall_iff,
Fin.mk_lt_mk, forall_comm (α := (_ : Prop)) (β := ℕ)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.OfFn | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 57
} | {
"line": 125,
"column": 0
} | [
{
"pp": "α : Type u\nR : α → α → Prop\nn : ℕ\nf : Fin n → α\n⊢ Pairwise R (ofFn f) ↔ ∀ ⦃i j : Fin n⦄, i < j → R (f i) (f j)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Data.List.OfFn.0.List.pairwise_ofFn._simp_1_3",
"List.Pairwise",
"congrArg",
... | [] | simp only [pairwise_iff_getElem, length_ofFn, List.getElem_ofFn,
Fin.forall_iff,
Fin.mk_lt_mk, forall_comm (α := (_ : Prop)) (β := ℕ)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.SuccPred | {
"line": 434,
"column": 40
} | {
"line": 435,
"column": 44
} | {
"line": 437,
"column": 0
} | [
{
"pp": "n : ℕ\na b : Fin (n + 1)\nha : a.succ ≠ last (n + 1)\n⊢ a.succ.castPred ha ≤ b ↔ a < b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.succ",
"congrArg",
"Iff.rfl",
"id",
"Fin.castPred",
"instOfNatNat",
"Fin.castPred_l... | [] | by
rw [castPred_le_iff, succ_le_castSucc_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.SuccPred | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 64
} | {
"line": 661,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ\ninst✝ : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : i.castSucc < p\n⊢ 0 < p.succAbove i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"congrArg",
"Fin.castSucc_pos_iff._simp... | [] | simpa [succAbove_of_castSucc_lt _ _ H] using castSucc_pos' h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Fin.SuccPred | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 64
} | {
"line": 661,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ\ninst✝ : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : i.castSucc < p\n⊢ 0 < p.succAbove i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"congrArg",
"Fin.castSucc_pos_iff._simp... | [] | simpa [succAbove_of_castSucc_lt _ _ H] using castSucc_pos' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.SuccPred | {
"line": 660,
"column": 4
} | {
"line": 660,
"column": 64
} | {
"line": 661,
"column": 2
} | [
{
"pp": "case pos\nn : ℕ\ninst✝ : NeZero n\np : Fin (n + 1)\ni : Fin n\nh : 0 < i\nH : i.castSucc < p\n⊢ 0 < p.succAbove i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"congrArg",
"Fin.castSucc_pos_iff._simp... | [] | simpa [succAbove_of_castSucc_lt _ _ H] using castSucc_pos' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.SuccPred | {
"line": 737,
"column": 2
} | {
"line": 737,
"column": 33
} | {
"line": 737,
"column": 33
} | [
{
"pp": "n : ℕ\nj : Fin n\n⊢ succAbove 1 j.succ = j.succ.succ",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"instNeZeroNatHAdd_1",
"Fin.succ_succAbove_succ",
"Fin.succ",
"Fin.instOfNat",
"instOfNatNat",
"instHAdd",
"HAdd.hA... | [
"n : ℕ\nj : Fin n\nthis : (succ 0).succAbove j.succ = (succAbove 0 j).succ\n⊢ succAbove 1 j.succ = j.succ.succ"
] | have := succ_succAbove_succ 0 j | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Fin.SuccPred | {
"line": 877,
"column": 8
} | {
"line": 877,
"column": 81
} | {
"line": 878,
"column": 6
} | [
{
"pp": "case inr\nn : ℕ\np : Fin n\ni : Fin (n + 1)\nh✝ : i ≠ p.succ\nh : p.succ < i\n⊢ p.succ.succAbove (p.predAbove i) = i",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Eq.mpr",
"Fin.castSucc_le_succ",
"Fin.ne_zero_of_lt",
"Fin.succ",
... | [
"case inr\nn : ℕ\np : Fin n\ni : Fin (n + 1)\nh✝ : i ≠ p.succ\nh : p.succ < i\n⊢ p.succ.succAbove (i.pred ⋯) = i"
] | predAbove_of_castSucc_lt _ _ (Fin.lt_of_le_of_lt (p.castSucc_le_succ) h), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 813,
"column": 16
} | {
"line": 813,
"column": 34
} | {
"line": 814,
"column": 2
} | [
{
"pp": "case cast\nn : ℕ\nP : Fin (n + 1) → Prop\ni✝ : Fin n\nhi : P i✝.castSucc\n⊢ P (last n) ∨ ∃ i, P i.castSucc",
"ppTerm": "?cast",
"assigned": true,
"usedConstants": [
"Exists",
"Fin.last",
"Exists.intro",
"Fin.castSucc",
"Fin",
"Or.inr"
],
"usedFVar... | [] | exact .inr ⟨_, hi⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 813,
"column": 16
} | {
"line": 813,
"column": 34
} | {
"line": 814,
"column": 2
} | [
{
"pp": "case cast\nn : ℕ\nP : Fin (n + 1) → Prop\ni✝ : Fin n\nhi : P i✝.castSucc\n⊢ P (last n) ∨ ∃ i, P i.castSucc",
"ppTerm": "?cast",
"assigned": true,
"usedConstants": [
"Exists",
"Fin.last",
"Exists.intro",
"Fin.castSucc",
"Fin",
"Or.inr"
],
"usedFVar... | [] | exact .inr ⟨_, hi⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 813,
"column": 16
} | {
"line": 813,
"column": 34
} | {
"line": 814,
"column": 2
} | [
{
"pp": "case cast\nn : ℕ\nP : Fin (n + 1) → Prop\ni✝ : Fin n\nhi : P i✝.castSucc\n⊢ P (last n) ∨ ∃ i, P i.castSucc",
"ppTerm": "?cast",
"assigned": true,
"usedConstants": [
"Exists",
"Fin.last",
"Exists.intro",
"Fin.castSucc",
"Fin",
"Or.inr"
],
"usedFVar... | [] | exact .inr ⟨_, hi⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 24
} | {
"line": 827,
"column": 2
} | [
{
"pp": "case p\nn : ℕ\nP : Fin (n + 1) → Prop\np : Fin (n + 1)\nj✝ : Fin n\nhi : P (p.succAbove j✝)\n⊢ P p ∨ ∃ i, P (p.succAbove i)",
"ppTerm": "?p",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Exists",
"Exists.intro",
"Fin",
"Or.inr"
],
"usedFVars":... | [] | exact .inr ⟨_, hi⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 24
} | {
"line": 827,
"column": 2
} | [
{
"pp": "case p\nn : ℕ\nP : Fin (n + 1) → Prop\np : Fin (n + 1)\nj✝ : Fin n\nhi : P (p.succAbove j✝)\n⊢ P p ∨ ∃ i, P (p.succAbove i)",
"ppTerm": "?p",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Exists",
"Exists.intro",
"Fin",
"Or.inr"
],
"usedFVars":... | [] | exact .inr ⟨_, hi⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fin.Tuple.Basic | {
"line": 826,
"column": 6
} | {
"line": 826,
"column": 24
} | {
"line": 827,
"column": 2
} | [
{
"pp": "case p\nn : ℕ\nP : Fin (n + 1) → Prop\np : Fin (n + 1)\nj✝ : Fin n\nhi : P (p.succAbove j✝)\n⊢ P p ∨ ∃ i, P (p.succAbove i)",
"ppTerm": "?p",
"assigned": true,
"usedConstants": [
"Fin.succAbove",
"Exists",
"Exists.intro",
"Fin",
"Or.inr"
],
"usedFVars":... | [] | exact .inr ⟨_, hi⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Lex | {
"line": 70,
"column": 6
} | {
"line": 70,
"column": 42
} | {
"line": 71,
"column": 6
} | [
{
"pp": "case inr.inl\nα : Type u\nr : α → α → Prop\ninst✝¹ : IsOrderConnected α r\ninst✝ : Std.Trichotomous r\na : α\nl₁ l₂ l₃ : List α\nh : Lex r l₁ l₃\n⊢ Lex r (a :: l₁) (a :: l₂) ∨ Lex r (a :: l₂) (a :: l₃)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"List.cons",
"Or... | [
"case inr.inr\nα : Type u\nr : α → α → Prop\ninst✝¹ : IsOrderConnected α r\ninst✝ : Std.Trichotomous r\na : α\nl₁ : List α\nb : α\nl₂ l₃ : List α\nh : Lex r l₁ l₃\nab : r b a\n⊢ Lex r (a :: l₁) (b :: l₂) ∨ Lex r (b :: l₂) (a :: l₃)"
] | · exact (aux _ l₂ _ h).imp cons cons | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.BigOperators.Group.Multiset.Defs | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 39
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\na : M\n⊢ (replicate n a).prod = a ^ n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"congrArg",
"Multiset.prod",
"NPow.toPow",
"HPow.hPow",
"CommMonoid.toMonoid",
"Nat",
"True",
"eq_self",
... | [] | simp [replicate, List.prod_replicate] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Group.Multiset.Defs | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 39
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\na : M\n⊢ (replicate n a).prod = a ^ n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"congrArg",
"Multiset.prod",
"NPow.toPow",
"HPow.hPow",
"CommMonoid.toMonoid",
"Nat",
"True",
"eq_self",
... | [] | simp [replicate, List.prod_replicate] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Group.Multiset.Defs | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 39
} | {
"line": 88,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\nn : ℕ\na : M\n⊢ (replicate n a).prod = a ^ n",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"congrArg",
"Multiset.prod",
"NPow.toPow",
"HPow.hPow",
"CommMonoid.toMonoid",
"Nat",
"True",
"eq_self",
... | [] | simp [replicate, List.prod_replicate] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Multiset.Defs | {
"line": 117,
"column": 4
} | {
"line": 117,
"column": 28
} | {
"line": 118,
"column": 4
} | [
{
"pp": "case pos\nM : Type u_3\ninst✝ : CommMonoid M\ns✝ : Multiset M\np : M → Prop\np_mul : ∀ (a b : M), p a → p b → p (a * b)\na : M\ns : Multiset M\nhsa : s ≠ ∅ → (∀ (a : M), a ∈ s → p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ (a_1 : M), a_1 ∈ a ::ₘ s → p a_1\nhs_empty : s = ∅\n⊢ p (a * s.prod)",
"ppTerm... | [
"case neg\nM : Type u_3\ninst✝ : CommMonoid M\ns✝ : Multiset M\np : M → Prop\np_mul : ∀ (a b : M), p a → p b → p (a * b)\na : M\ns : Multiset M\nhsa : s ≠ ∅ → (∀ (a : M), a ∈ s → p a) → p s.prod\nhs : a ::ₘ s ≠ ∅\np_s : ∀ (a_1 : M), a_1 ∈ a ::ₘ s → p a_1\nhs_empty : ¬s = ∅\n⊢ p (a * s.prod)"
] | · simp [hs_empty, p_s a] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.List.Chain | {
"line": 304,
"column": 8
} | {
"line": 304,
"column": 17
} | {
"line": 304,
"column": 18
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\nl₁ l₂ : List α\nL : List (List α)\nhL : ¬[] ∈ l₁ :: l₂ :: L\n⊢ IsChain R (l₁ :: l₂ :: L).flatten ↔\n (∀ (l : List α), l ∈ l₁ :: l₂ :: L → IsChain R l) ∧\n IsChain (fun l₁ l₂ ↦ ∀ (x : α), x ∈ l₁.getLast? → ∀ (y : α), y ∈ l₂.head? → R x y) (l₁ :: l₂ :: L)",
"pp... | [
"α : Type u_1\nR : α → α → Prop\nl₁ l₂ : List α\nL : List (List α)\nhL : ¬([] = l₁ ∨ [] ∈ l₂ :: L)\n⊢ IsChain R (l₁ :: l₂ :: L).flatten ↔\n (∀ (l : List α), l ∈ l₁ :: l₂ :: L → IsChain R l) ∧\n IsChain (fun l₁ l₂ ↦ ∀ (x : α), x ∈ l₁.getLast? → ∀ (y : α), y ∈ l₂.head? → R x y) (l₁ :: l₂ :: L)"
] | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.List.Chain | {
"line": 306,
"column": 8
} | {
"line": 306,
"column": 17
} | {
"line": 306,
"column": 18
} | [
{
"pp": "α : Type u_1\nR : α → α → Prop\nl₁ l₂ : List α\nL : List (List α)\nhL : [] ≠ l₁ ∧ ¬[] ∈ l₂ :: L\n⊢ (IsChain R l₁ ∧\n ((IsChain R l₂ ∧ ∀ (x : List α), x ∈ L → IsChain R x) ∧\n IsChain (fun l₁ l₂ ↦ ∀ (x : α), x ∈ l₁.getLast? → ∀ (y : α), y ∈ l₂.head? → R x y) (l₂ :: L)) ∧\n ∀ (x : α)... | [
"α : Type u_1\nR : α → α → Prop\nl₁ l₂ : List α\nL : List (List α)\nhL : [] ≠ l₁ ∧ ¬([] = l₂ ∨ [] ∈ L)\n⊢ (IsChain R l₁ ∧\n ((IsChain R l₂ ∧ ∀ (x : List α), x ∈ L → IsChain R x) ∧\n IsChain (fun l₁ l₂ ↦ ∀ (x : α), x ∈ l₁.getLast? → ∀ (y : α), y ∈ l₂.head? → R x y) (l₂ :: L)) ∧\n ∀ (x : α), x ∈ ... | mem_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Group.Multiset.Basic | {
"line": 210,
"column": 43
} | {
"line": 210,
"column": 64
} | {
"line": 212,
"column": 0
} | [
{
"pp": "case empty\nn : ℕ\n⊢ sum 0 % n = (map (fun x ↦ x % n) 0).sum % n",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"Nat",
"eq_self",
"of_eq_true",
"OfNat.ofNat",
"Eq"
],
"usedFVars": [],
"usedGoals": []
}
] | [] | simp [Nat.add_mod, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Group.Multiset.Basic | {
"line": 210,
"column": 43
} | {
"line": 210,
"column": 64
} | {
"line": 212,
"column": 0
} | [
{
"pp": "case cons\nn a✝¹ : ℕ\ns✝ : Multiset ℕ\na✝ : s✝.sum % n = (map (fun x ↦ x % n) s✝).sum % n\n⊢ (a✝¹ ::ₘ s✝).sum % n = (map (fun x ↦ x % n) (a✝¹ ::ₘ s✝)).sum % n",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Dvd.dvd",
"Multiset.map_cons",
"M... | [] | simp [Nat.add_mod, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Lattice.Image | {
"line": 490,
"column": 28
} | {
"line": 490,
"column": 73
} | {
"line": 490,
"column": 74
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nS : Set (Set α)\nhS : S.Nonempty\nt : Set β\n⊢ ⋂₀ S ×ˢ ⋂₀ {t} = ⋂ s ∈ S, s ×ˢ ⋂₀ {t}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.singleton_nonempty",
"Eq.mpr",
"SProd.sprod",
"congrArg",
"Set... | [
"α : Type u_1\nβ : Type u_2\nS : Set (Set α)\nhS : S.Nonempty\nt : Set β\n⊢ ⋂ r ∈ S ×ˢ {t}, r.1 ×ˢ r.2 = ⋂ s ∈ S, s ×ˢ ⋂₀ {t}"
] | sInter_prod_sInter hS (singleton_nonempty t), | Lean.Elab.Tactic.evalRewriteSeq | null |
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