module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Nat.Pairing | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 67
} | {
"line": 144,
"column": 2
} | [
{
"pp": "case inl\nm n : ℕ\nh : m < n\n⊢ max m n ^ 2 + min m n ≤ if m < n then n * n + m else m * m + m + n",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"le_refl",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
"PartialOrder.to... | [
"case inr\nm n : ℕ\nh : n ≤ m\n⊢ max m n ^ 2 + min m n ≤ if m < n then n * n + m else m * m + m + n"
] | · rw [if_pos h, max_eq_right h.le, min_eq_left h.le, Nat.pow_two] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Subgroup.ZPowers.Basic | {
"line": 126,
"column": 61
} | {
"line": 126,
"column": 97
} | {
"line": 128,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ng : G\n⊢ zpowers g = ⊥ ↔ g = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"OrderBot.toBot",
"Iff.rfl",
"PartialOrder.toPreorder"... | [] | rw [eq_bot_iff, zpowers_le, mem_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Subgroup.ZPowers.Basic | {
"line": 126,
"column": 61
} | {
"line": 126,
"column": 97
} | {
"line": 128,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ng : G\n⊢ zpowers g = ⊥ ↔ g = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"OrderBot.toBot",
"Iff.rfl",
"PartialOrder.toPreorder"... | [] | rw [eq_bot_iff, zpowers_le, mem_bot] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.ZPowers.Basic | {
"line": 126,
"column": 61
} | {
"line": 126,
"column": 97
} | {
"line": 128,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\ng : G\n⊢ zpowers g = ⊥ ↔ g = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"OrderBot.toBot",
"Iff.rfl",
"PartialOrder.toPreorder"... | [] | rw [eq_bot_iff, zpowers_le, mem_bot] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 714,
"column": 2
} | {
"line": 714,
"column": 44
} | {
"line": 715,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\nthis : (⨅ g, map (↑(MulAut.conj g)) H).Normal\n⊢ H.normalCore = ⨅ g, map (↑(MulAut.conj g)) H",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"MulEquiv.instEquivLike",
"iInf",
"MonoidHom.instFunLike",
"MonoidHo... | [
"case refine_1\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nthis : (⨅ g, map (↑(MulAut.conj g)) H).Normal\ng : G\n⊢ H.normalCore ≤ map (↑(MulAut.conj g)) H",
"case refine_2\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nthis : (⨅ g, map (↑(MulAut.conj g)) H).Normal\n⊢ ⨅ g, map (↑(MulAut.conj g)) H ≤ H.normalC... | refine le_antisymm (le_iInf fun g ↦ ?_) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 716,
"column": 2
} | {
"line": 718,
"column": 25
} | {
"line": 720,
"column": 0
} | [
{
"pp": "case refine_2\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nthis : (⨅ g, map (↑(MulAut.conj g)) H).Normal\n⊢ ⨅ g, map (↑(MulAut.conj g)) H ≤ H.normalCore",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClass",
"MulEquiv.ref... | [] | · rw [normal_le_normalCore]
apply iInf_le_of_le 1
simp [MulAut.one_def] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 183,
"column": 19
} | {
"line": 183,
"column": 64
} | {
"line": 185,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : Monoid α\ninst✝ : MulAction α β\nx✝ : Set β\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
"congrArg",
"Set.image_id'... | [] | simp only [← image_smul, one_smul, image_id'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 183,
"column": 19
} | {
"line": 183,
"column": 64
} | {
"line": 185,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : Monoid α\ninst✝ : MulAction α β\nx✝ : Set β\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
"congrArg",
"Set.image_id'... | [] | simp only [← image_smul, one_smul, image_id'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 183,
"column": 19
} | {
"line": 183,
"column": 64
} | {
"line": 185,
"column": 0
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝¹ : Monoid α\ninst✝ : MulAction α β\nx✝ : Set β\n⊢ 1 • x✝ = x✝",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
"congrArg",
"Set.image_id'... | [] | simp only [← image_smul, one_smul, image_id'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 955,
"column": 4
} | {
"line": 955,
"column": 34
} | {
"line": 956,
"column": 2
} | [
{
"pp": "case a\nG : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ map f (normalClosure s) ≤ normalClosure (⇑f '' s)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Subgroup.map_normalCl... | [] | exact map_normalClosure_le s f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 955,
"column": 4
} | {
"line": 955,
"column": 34
} | {
"line": 956,
"column": 2
} | [
{
"pp": "case a\nG : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ map f (normalClosure s) ≤ normalClosure (⇑f '' s)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Subgroup.map_normalCl... | [] | exact map_normalClosure_le s f | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Basic | {
"line": 955,
"column": 4
} | {
"line": 955,
"column": 34
} | {
"line": 956,
"column": 2
} | [
{
"pp": "case a\nG : Type u_1\ninst✝¹ : Group G\nN : Type u_5\ninst✝ : Group N\ns : Set G\nf : G →* N\nhf : Surjective ⇑f\nthis : (map f (normalClosure s)).Normal\n⊢ map f (normalClosure s) ≤ normalClosure (⇑f '' s)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Subgroup.map_normalCl... | [] | exact map_normalClosure_le s f | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 282,
"column": 65
} | {
"line": 283,
"column": 51
} | {
"line": 285,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : Group α\ns t : Set α\nx : α\n⊢ (x • s ∩ t).Nonempty ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a / b = x",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"instHDiv",
"instSMulOfMul",
"HMul.hMul",... | [] | by
simp_rw [smul_inter_nonempty_iff, div_eq_mul_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Action.Pointwise.Set.Basic | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 30
} | {
"line": 298,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝ : Group α\ns t : Set α\nx : αᵐᵒᵖ\n⊢ (∃ a b, (a ∈ s ∧ b ∈ t) ∧ a⁻¹ * b = unop x) → (x • s ∩ t).Nonempty",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"instHSMul",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneCl... | [
"case mpr\nα : Type u_2\ninst✝ : Group α\ns t : Set α\nx : αᵐᵒᵖ\na b : α\nH : a⁻¹ * b = unop x\nha : a ∈ s\nhb : b ∈ t\n⊢ (x • s ∩ t).Nonempty"
] | rintro ⟨a, b, ⟨ha, hb⟩, H⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.FreeMonoid.Basic | {
"line": 430,
"column": 6
} | {
"line": 435,
"column": 38
} | {
"line": 436,
"column": 2
} | [
{
"pp": "case mp.mul_of\nα : Type u_1\nβ : Type u_2\nf : α → β\nfs : Function.Surjective ⇑(map f)\nd : β\nhead : α\na✝¹ : FreeMonoid α\na✝ : (map f) a✝¹ = of d → ∃ a, f a = d\nhb : (map f) (of head * a✝¹) = of d\n⊢ ∃ a, f a = d",
"ppTerm": "?mp.mul_of",
"assigned": true,
"usedConstants": [
"Mo... | [] | simp only [map_mul, map_of] at hb
use head
have H := congr_arg length hb
simp only [length_mul, length_of, add_eq_left, length_eq_zero] at H
rw [H, mul_one] at hb
exact FreeMonoid.of_injective hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.FreeMonoid.Basic | {
"line": 430,
"column": 6
} | {
"line": 435,
"column": 38
} | {
"line": 436,
"column": 2
} | [
{
"pp": "case mp.mul_of\nα : Type u_1\nβ : Type u_2\nf : α → β\nfs : Function.Surjective ⇑(map f)\nd : β\nhead : α\na✝¹ : FreeMonoid α\na✝ : (map f) a✝¹ = of d → ∃ a, f a = d\nhb : (map f) (of head * a✝¹) = of d\n⊢ ∃ a, f a = d",
"ppTerm": "?mp.mul_of",
"assigned": true,
"usedConstants": [
"Mo... | [] | simp only [map_mul, map_of] at hb
use head
have H := congr_arg length hb
simp only [length_mul, length_of, add_eq_left, length_eq_zero] at H
rw [H, mul_one] at hb
exact FreeMonoid.of_injective hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Group.List | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 19
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case cons\nι : Type u_1\nM : Type u_3\ninst✝⁵ : Monoid M\ninst✝⁴ : Preorder M\ninst✝³ : MulLeftStrictMono M\ninst✝² : MulLeftMono M\ninst✝¹ : MulRightStrictMono M\ninst✝ : MulRightMono M\nf g : ι → M\ni : ι\nl : List ι\nihl : (∀ (i : ι), i ∈ l → f i ≤ g i) → (∃ i, i ∈ l ∧ f i < g i) → (map f l).prod < ... | [] | | cons i l ihl => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Logic.Embedding.Set | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 14
} | {
"line": 105,
"column": 6
} | [
{
"pp": "case inr\nα : Type u_1\np q : α → Prop\ninst✝ : DecidablePred p\nh : Disjoint p q\nx : { x // q x }\n⊢ (subtypeOrLeftEmbedding p q)\n (Sum.elim (⇑(Subtype.impEmbedding p (fun x ↦ p x ∨ q x) ⋯)) (⇑(Subtype.impEmbedding q (fun x ↦ p x ∨ q x) ⋯))\n (Sum.inr x)) =\n Sum.inr x",
"ppTerm":... | [] | | inr x => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Logic.Denumerable | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 36
} | {
"line": 119,
"column": 6
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\n⊢ ∃ a ∈ decode 0, encode a = 0",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Option.instMembership",
"Membership.mem",
"Option.encodable",
"instOfNatNat",
"Option... | [
"case zero\nα : Type u_1\nβ : Type u_2\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\n⊢ none ∈ decode 0"
] | refine ⟨none, ?_, encode_none⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Group.Submonoid.Membership | {
"line": 482,
"column": 2
} | {
"line": 482,
"column": 7
} | {
"line": 484,
"column": 0
} | [
{
"pp": "N : Type u_4\ninst✝ : CommMonoid N\nP : N → Prop\ns t : Submonoid N\n⊢ (∀ (x x_1 : N), x_1 ∈ s → ∀ x_2 ∈ t, x_1 * x_2 = x → P x) ↔ ∀ x₁ ∈ s, ∀ x₂ ∈ t, P (x₁ * x₂)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"Membership.mem... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finset.Sum | {
"line": 117,
"column": 57
} | {
"line": 117,
"column": 62
} | {
"line": 117,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Finset α\nt : Finset β\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\na : α\nb : β\nu : Finset (α ⊕ β)\n⊢ ∀ (a a' : α ⊕ β), ∀ b ∈ Sum.elim some (fun x ↦ none) a, b ∈ Sum.elim some (fun x ↦ none) a' → a = a'",
"ppTerm": "?m.11",
"assigned": true,
"usedCons... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finset.Sum | {
"line": 126,
"column": 59
} | {
"line": 126,
"column": 64
} | {
"line": 126,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Finset α\nt : Finset β\ns₁ s₂ : Finset α\nt₁ t₂ : Finset β\na : α\nb : β\nu : Finset (α ⊕ β)\n⊢ ∀ (a a' : α ⊕ β), ∀ b ∈ Sum.elim (fun x ↦ none) some a, b ∈ Sum.elim (fun x ↦ none) some a' → a = a'",
"ppTerm": "?m.11",
"assigned": true,
"usedCons... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.List.Sublists | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 17
} | {
"line": 63,
"column": 0
} | [
{
"pp": "α : Type u\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n (Array.foldl (fun r l ↦ r.push (x :: l)) y.toList.toArray y.toList.toArray).toList =\n (Array.foldl (fun r l ↦ r.push (x :: l)) y y).toList",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Array.push",
... | [] | intros; congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Sublists | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 17
} | {
"line": 63,
"column": 0
} | [
{
"pp": "α : Type u\nl : List α\n⊢ ∀ (x : α) (y : Array (List α)),\n (Array.foldl (fun r l ↦ r.push (x :: l)) y.toList.toArray y.toList.toArray).toList =\n (Array.foldl (fun r l ↦ r.push (x :: l)) y y).toList",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Array.push",
... | [] | intros; congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Choose.Basic | {
"line": 225,
"column": 2
} | {
"line": 232,
"column": 21
} | {
"line": 234,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ n.choose k * (n + 1) = (n + 1).choose k * (n + 1 - k)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.mul_sub_left_distrib",
"Preorder.toLT",
"Nat.choose",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
... | [] | cases k with
| zero => simp
| succ k =>
obtain hk | hk := le_or_gt (k + 1) (n + 1)
· rw [choose_succ_succ, Nat.add_mul, succ_sub_succ, ← choose_succ_right_eq, ← succ_sub_succ,
Nat.mul_sub_left_distrib, Nat.add_sub_cancel' (Nat.mul_le_mul_left _ hk)]
· rw [choose_eq_zero_of_lt hk, choose_eq_zero_... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Data.Nat.Choose.Basic | {
"line": 225,
"column": 2
} | {
"line": 232,
"column": 21
} | {
"line": 234,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ n.choose k * (n + 1) = (n + 1).choose k * (n + 1 - k)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.mul_sub_left_distrib",
"Preorder.toLT",
"Nat.choose",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
... | [] | cases k with
| zero => simp
| succ k =>
obtain hk | hk := le_or_gt (k + 1) (n + 1)
· rw [choose_succ_succ, Nat.add_mul, succ_sub_succ, ← choose_succ_right_eq, ← succ_sub_succ,
Nat.mul_sub_left_distrib, Nat.add_sub_cancel' (Nat.mul_le_mul_left _ hk)]
· rw [choose_eq_zero_of_lt hk, choose_eq_zero_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Choose.Basic | {
"line": 225,
"column": 2
} | {
"line": 232,
"column": 21
} | {
"line": 234,
"column": 0
} | [
{
"pp": "n k : ℕ\n⊢ n.choose k * (n + 1) = (n + 1).choose k * (n + 1 - k)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.mul_sub_left_distrib",
"Preorder.toLT",
"Nat.choose",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
... | [] | cases k with
| zero => simp
| succ k =>
obtain hk | hk := le_or_gt (k + 1) (n + 1)
· rw [choose_succ_succ, Nat.add_mul, succ_sub_succ, ← choose_succ_right_eq, ← succ_sub_succ,
Nat.mul_sub_left_distrib, Nat.add_sub_cancel' (Nat.mul_le_mul_left _ hk)]
· rw [choose_eq_zero_of_lt hk, choose_eq_zero_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.List.Zip | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 27
} | {
"line": 57,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nf : α → β → γ\np : γ → Prop\na : α\nl₁ : List α\nb : β\nl₂ : List β\nh : (a :: l₁).length = (b :: l₂).length\n⊢ Forall p (zipWith f (a :: l₁) (b :: l₂)) ↔ Forall₂ (fun x y ↦ p (f x y)) (a :: l₁) (b :: l₂)",
"ppTerm": "?m.27",
"assigned": true,
"usedCo... | [] | simp only [length_cons, succ_inj] at h
simp [forall_zipWith h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.List.Zip | {
"line": 54,
"column": 4
} | {
"line": 55,
"column": 27
} | {
"line": 57,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type u_1\nγ : Type u_2\nf : α → β → γ\np : γ → Prop\na : α\nl₁ : List α\nb : β\nl₂ : List β\nh : (a :: l₁).length = (b :: l₂).length\n⊢ Forall p (zipWith f (a :: l₁) (b :: l₂)) ↔ Forall₂ (fun x y ↦ p (f x y)) (a :: l₁) (b :: l₂)",
"ppTerm": "?m.27",
"assigned": true,
"usedCo... | [] | simp only [length_cons, succ_inj] at h
simp [forall_zipWith h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Pairwise.Lattice | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 39
} | {
"line": 38,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nκ : Sort u_4\nr : α → α → Prop\nf : κ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) f\nH : ∀ (n : κ), (f n).Pairwise r\ni : α\nhi : i ∈ ⋃ n, f n\nj : α\nhj : j ∈ ⋃ n, f n\nhij : i ≠ j\nm : κ\nhm : i ∈ f m\n⊢ r i j",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants":... | [
"case mpr\nα : Type u_1\nκ : Sort u_4\nr : α → α → Prop\nf : κ → Set α\nhd : Directed (fun x1 x2 ↦ x1 ⊆ x2) f\nH : ∀ (n : κ), (f n).Pairwise r\ni : α\nhi : i ∈ ⋃ n, f n\nj : α\nhj : j ∈ ⋃ n, f n\nhij : i ≠ j\nm : κ\nhm : i ∈ f m\nn : κ\nhn : j ∈ f n\n⊢ r i j"
] | rcases mem_iUnion.1 hj with ⟨n, hn⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Data.List.Sublists | {
"line": 246,
"column": 21
} | {
"line": 248,
"column": 96
} | {
"line": 250,
"column": 0
} | [
{
"pp": "α : Type u\nn : ℕ\na : α\nl : List α\n⊢ sublistsLen (n + 1) (a :: l) <+ (a :: l).sublists'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"List.sublists'",
"Eq.mpr",
"congrArg",
"List.map",
"List.Sublist.map",
"id",
"List.sublistsLen",
... | [] | by
rw [sublistsLen_succ_cons, sublists'_cons]
exact (sublistsLen_sublist_sublists' _ _).append ((sublistsLen_sublist_sublists' _ _).map _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 59
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ns₁ s₂ : Finset β\nf : β → α\ninst✝ : DecidableEq β\nc : α\n⊢ (s₁ ∪ s₂).sup f ≤ c ↔ s₁.sup f ⊔ s₂.sup f ≤ c",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finset.instUnion",
"congrArg",
... | [] | simp [or_imp, forall_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 59
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ns₁ s₂ : Finset β\nf : β → α\ninst✝ : DecidableEq β\nc : α\n⊢ (s₁ ∪ s₂).sup f ≤ c ↔ s₁.sup f ⊔ s₂.sup f ≤ c",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finset.instUnion",
"congrArg",
... | [] | simp [or_imp, forall_and] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 129,
"column": 34
} | {
"line": 129,
"column": 59
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ns₁ s₂ : Finset β\nf : β → α\ninst✝ : DecidableEq β\nc : α\n⊢ (s₁ ∪ s₂).sup f ≤ c ↔ s₁.sup f ⊔ s₂.sup f ≤ c",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finset.instUnion",
"congrArg",
... | [] | simp [or_imp, forall_and] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Powerset | {
"line": 161,
"column": 27
} | {
"line": 161,
"column": 40
} | {
"line": 161,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ t ≠ s ∧ t ∈ s.powerset ↔ t ⊂ s",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.... | [
"α : Type u_1\ninst✝ : DecidableEq α\ns t : Finset α\n⊢ t ≠ s ∧ t ⊆ s ↔ t ⊂ s"
] | mem_powerset, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Pairwise.Lattice | {
"line": 172,
"column": 75
} | {
"line": 176,
"column": 49
} | {
"line": 178,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\nha : a ∉ s\n⊢ (𝒫 s).PairwiseDisjoint fun t ↦ {t, insert a t}",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.LeftInvOn",
"False",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"tru... | [] | by
rw [pairwiseDisjoint_iff]
rintro i hi j hj
have := insert_erase_invOn.2.injOn (notMem_subset hi ha) (notMem_subset hj ha)
aesop (add simp [Set.Nonempty, Set.subset_def]) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Finite.Range | {
"line": 105,
"column": 33
} | {
"line": 105,
"column": 38
} | {
"line": 105,
"column": 38
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nu : Set β\nhu : u.Finite\nthis : Finite ↑u\ng : ⦃a : β⦄ → a ∈ u → α\nhg : ∀ ⦃a : β⦄ (a_1 : a ∈ u), g a_1 ∈ s\nhg' : ∀ ⦃a : β⦄ (a_1 : a ∈ u), f (g a_1) = a\ng' : ↑u → α := fun x ↦ g ⋯\na : α\nha : a ∈ range g'\n⊢ a ∈ s",
"ppTerm": "?m.46",
"assigned"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Finite.Range | {
"line": 105,
"column": 33
} | {
"line": 105,
"column": 38
} | {
"line": 105,
"column": 38
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nu : Set β\nhu : u.Finite\nthis : Finite ↑u\ng : ⦃a : β⦄ → a ∈ u → α\nhg : ∀ ⦃a : β⦄ (a_1 : a ∈ u), g a_1 ∈ s\nhg' : ∀ ⦃a : β⦄ (a_1 : a ∈ u), f (g a_1) = a\ng' : ↑u → α := fun x ↦ g ⋯\na : α\nha : a ∈ range g'\n⊢ a ∈ s",
"ppTerm": "?m.46",
"assigned"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Finite.Range | {
"line": 105,
"column": 33
} | {
"line": 105,
"column": 38
} | {
"line": 105,
"column": 38
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nu : Set β\nhu : u.Finite\nthis : Finite ↑u\ng : ⦃a : β⦄ → a ∈ u → α\nhg : ∀ ⦃a : β⦄ (a_1 : a ∈ u), g a_1 ∈ s\nhg' : ∀ ⦃a : β⦄ (a_1 : a ∈ u), f (g a_1) = a\ng' : ↑u → α := fun x ↦ g ⋯\na : α\nha : a ∈ range g'\n⊢ a ∈ s",
"ppTerm": "?m.46",
"assigned"... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Finite.Range | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 64
} | {
"line": 105,
"column": 64
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nu : Set β\nhu : u.Finite\nthis : Finite ↑u\ng : ⦃a : β⦄ → a ∈ u → α\nhg : ∀ ⦃a : β⦄ (a_1 : a ∈ u), g a_1 ∈ s\nhg' : ∀ ⦃a : β⦄ (a_1 : a ∈ u), f (g a_1) = a\ng' : ↑u → α := fun x ↦ g ⋯\n⊢ f '' range g' = u",
"ppTerm": "?m.54",
"assigned": true,
"u... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Finite.Range | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 64
} | {
"line": 105,
"column": 64
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nu : Set β\nhu : u.Finite\nthis : Finite ↑u\ng : ⦃a : β⦄ → a ∈ u → α\nhg : ∀ ⦃a : β⦄ (a_1 : a ∈ u), g a_1 ∈ s\nhg' : ∀ ⦃a : β⦄ (a_1 : a ∈ u), f (g a_1) = a\ng' : ↑u → α := fun x ↦ g ⋯\n⊢ f '' range g' = u",
"ppTerm": "?m.54",
"assigned": true,
"u... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Finite.Range | {
"line": 105,
"column": 59
} | {
"line": 105,
"column": 64
} | {
"line": 105,
"column": 64
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nu : Set β\nhu : u.Finite\nthis : Finite ↑u\ng : ⦃a : β⦄ → a ∈ u → α\nhg : ∀ ⦃a : β⦄ (a_1 : a ∈ u), g a_1 ∈ s\nhg' : ∀ ⦃a : β⦄ (a_1 : a ∈ u), f (g a_1) = a\ng' : ↑u → α := fun x ↦ g ⋯\n⊢ f '' range g' = u",
"ppTerm": "?m.54",
"assigned": true,
"u... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Powerset | {
"line": 265,
"column": 66
} | {
"line": 265,
"column": 92
} | {
"line": 265,
"column": 92
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : Finset α\nh : #s < n\n⊢ (#s).choose n = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"congrArg",
"id",
"instOfNatNat",
"Nat",
"Nat.choose_eq_zero_of_lt",
"Finset.card",
"OfN... | [
"α : Type u_1\nn : ℕ\ns : Finset α\nh : #s < n\n⊢ 0 = 0"
] | Nat.choose_eq_zero_of_lt h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Powerset | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 53
} | {
"line": 316,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ns a : Finset α\nha : a ∈ (range (#s + 1)).disjiUnion (fun i ↦ powersetCard i s) ⋯\n⊢ a ∈ s.powerset",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disj... | [] | rcases mem_disjiUnion.mp ha with ⟨i, _hi, ha⟩
exact mem_powerset.mpr (mem_powersetCard.mp ha).1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Powerset | {
"line": 313,
"column": 4
} | {
"line": 314,
"column": 53
} | {
"line": 316,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ns a : Finset α\nha : a ∈ (range (#s + 1)).disjiUnion (fun i ↦ powersetCard i s) ⋯\n⊢ a ∈ s.powerset",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disj... | [] | rcases mem_disjiUnion.mp ha with ⟨i, _hi, ha⟩
exact mem_powerset.mpr (mem_powersetCard.mp ha).1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Powerset | {
"line": 313,
"column": 2
} | {
"line": 314,
"column": 53
} | {
"line": 316,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ns a : Finset α\nha : a ∈ (range (#s + 1)).disjiUnion (fun i ↦ powersetCard i s) ⋯\n⊢ a ∈ s.powerset",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disj... | [] | · rcases mem_disjiUnion.mp ha with ⟨i, _hi, ha⟩
exact mem_powerset.mpr (mem_powersetCard.mp ha).1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 361,
"column": 2
} | {
"line": 361,
"column": 46
} | {
"line": 362,
"column": 2
} | [
{
"pp": "α : Type u_2\nι : Type u_5\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\n⊢ s.sup f ⊓ a = s.sup fun i ↦ f i ⊓ a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Finset.sup_in... | [
"α : Type u_2\nι : Type u_5\ninst✝¹ : DistribLattice α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\na : α\n⊢ (s.sup fun i ↦ a ⊓ f i) = s.sup fun i ↦ f i ⊓ a"
] | rw [_root_.inf_comm, s.sup_inf_distrib_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 571,
"column": 34
} | {
"line": 571,
"column": 59
} | {
"line": 573,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : DecidableEq β\ns₁ s₂ : Finset β\nh₁ : s₁.Nonempty\nh₂ : s₂.Nonempty\nf : β → α\na : α\n⊢ (s₁ ∪ s₂).sup' ⋯ f ≤ a ↔ s₁.sup' h₁ f ⊔ s₂.sup' h₂ f ≤ a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Finset.instUnion"... | [] | simp [or_imp, forall_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 571,
"column": 34
} | {
"line": 571,
"column": 59
} | {
"line": 573,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : DecidableEq β\ns₁ s₂ : Finset β\nh₁ : s₁.Nonempty\nh₂ : s₂.Nonempty\nf : β → α\na : α\n⊢ (s₁ ∪ s₂).sup' ⋯ f ≤ a ↔ s₁.sup' h₁ f ⊔ s₂.sup' h₂ f ≤ a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Finset.instUnion"... | [] | simp [or_imp, forall_and] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Lattice.Fold | {
"line": 571,
"column": 34
} | {
"line": 571,
"column": 59
} | {
"line": 573,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SemilatticeSup α\ninst✝ : DecidableEq β\ns₁ s₂ : Finset β\nh₁ : s₁.Nonempty\nh₂ : s₂.Nonempty\nf : β → α\na : α\n⊢ (s₁ ∪ s₂).sup' ⋯ f ≤ a ↔ s₁.sup' h₁ f ⊔ s₂.sup' h₂ f ≤ a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Finset.instUnion"... | [] | simp [or_imp, forall_and] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompletePartialOrder.Indexed | {
"line": 177,
"column": 68
} | {
"line": 177,
"column": 73
} | {
"line": 177,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : ConditionallyCompletePartialOrderSup α\ninst✝ : Preorder β\nf : β → α\na : β\nhf : Monotone f\nhd : Directed (fun x1 x2 ↦ x1 ≤ x2) fun x ↦ f ↑x\nx✝ : α\n⊢ (x✝ ∈ range fun x ↦ f ↑x) → x✝ ≤ f a",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.ConditionallyCompletePartialOrder.Indexed | {
"line": 177,
"column": 68
} | {
"line": 177,
"column": 73
} | {
"line": 177,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : ConditionallyCompletePartialOrderSup α\ninst✝ : Preorder β\nf : β → α\na : β\nhf : Monotone f\nhd : Directed (fun x1 x2 ↦ x1 ≤ x2) fun x ↦ f ↑x\nx✝ : α\n⊢ (x✝ ∈ range fun x ↦ f ↑x) → x✝ ≤ f a",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.ConditionallyCompletePartialOrder.Indexed | {
"line": 177,
"column": 68
} | {
"line": 177,
"column": 73
} | {
"line": 177,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : ConditionallyCompletePartialOrderSup α\ninst✝ : Preorder β\nf : β → α\na : β\nhf : Monotone f\nhd : Directed (fun x1 x2 ↦ x1 ≤ x2) fun x ↦ f ↑x\nx✝ : α\n⊢ (x✝ ∈ range fun x ↦ f ↑x) → x✝ ≤ f a",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompleteLattice.Finset | {
"line": 79,
"column": 4
} | {
"line": 80,
"column": 77
} | {
"line": 81,
"column": 4
} | [
{
"pp": "case refine_1\nι : Type u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder α\nf : ι → α\ns : Finset ι\nh : ∃ x ∈ s, sSup ∅ ≤ f x\nh' : (image f s).Nonempty\n⊢ ∀ x ∈ Set.range fun i ↦ ⨆ (_ : i ∈ s), f i, ∃ y ∈ f '' ↑s, x ≤ y",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants"... | [
"case refine_1\nι : Type u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder α\nf : ι → α\ns : Finset ι\nh : ∃ x ∈ s, sSup ∅ ≤ f x\nh' : (image f s).Nonempty\n⊢ ∀ (a : ι), ∃ a_1 ∈ s, (if a ∈ s then f a else sSup ∅) ≤ f a_1"
] | simp only [ciSup_eq_ite, dite_eq_ite, Set.mem_range, Set.mem_image, mem_coe,
exists_exists_and_eq_and, forall_exists_index, forall_apply_eq_imp_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.ConditionallyCompleteLattice.Finset | {
"line": 93,
"column": 42
} | {
"line": 99,
"column": 7
} | {
"line": 101,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder α\nf : ι → α\ns : Finset ι\nh : ∃ x ∈ s, f x ≤ sInf ∅\nh' : (image f s).Nonempty\n⊢ ⨅ i ∈ s, f i = (image f s).min' h'",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Finset.min'",
"Iff.mpr",
"... | [] | by
classical
rw [← OrderDual.toDual_inj, toDual_min', toDual_iInf]
simp only [toDual_iInf]
rw [ciSup_eq_max'_image _ h]
simp only [image_image]
congr | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.ConditionallyCompleteLattice.Finset | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 82
} | {
"line": 131,
"column": 6
} | [
{
"pp": "case mp.refine_1\nι : Type u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\ns : Set ι\nf : ι → α\nhs : s.Finite\nh✝ : ∃ x ∈ s, sSup ∅ ≤ f x\nh : ⨆ i ∈ s, f i < a\nx : ι\nhx : x ∈ s\n⊢ BddAbove (range fun i ↦ ⨆ (_ : i ∈ s), f i)",
"ppTerm": "?mp.refine_1",
"assigned": true,
... | [
"case mp.refine_1\nι : Type u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder α\na : α\ns : Set ι\nf : ι → α\nhs : s.Finite\nh✝ : ∃ x ∈ s, sSup ∅ ≤ f x\nh : ⨆ i ∈ s, f i < a\nx : ι\nhx : x ∈ s\n⊢ (range fun i ↦ ⨆ (_ : i ∈ s), f i) ⊆ f '' s ∪ {sSup ∅}"
] | refine (((hs.image f).union (finite_singleton (sSup ∅))).subset ?_).bddAbove | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Set.Finite.Lattice | {
"line": 236,
"column": 45
} | {
"line": 236,
"column": 50
} | {
"line": 238,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nhimage : (f '' s).Finite\nhfibers : ∀ x ∈ f '' s, (s ∩ f ⁻¹' {x}).Finite\nx : α\n⊢ x ∈ s → x ∈ ⋃ i ∈ f '' s, s ∩ f ⁻¹' {i}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.iUnion_iUnion_eq_right",
"Eq.mpr",
"Iff.o... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Finite.Lattice | {
"line": 236,
"column": 45
} | {
"line": 236,
"column": 50
} | {
"line": 238,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nhimage : (f '' s).Finite\nhfibers : ∀ x ∈ f '' s, (s ∩ f ⁻¹' {x}).Finite\nx : α\n⊢ x ∈ s → x ∈ ⋃ i ∈ f '' s, s ∩ f ⁻¹' {i}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.iUnion_iUnion_eq_right",
"Eq.mpr",
"Iff.o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Finite.Lattice | {
"line": 236,
"column": 45
} | {
"line": 236,
"column": 50
} | {
"line": 238,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\ns : Set α\nhimage : (f '' s).Finite\nhfibers : ∀ x ∈ f '' s, (s ∩ f ⁻¹' {x}).Finite\nx : α\n⊢ x ∈ s → x ∈ ⋃ i ∈ f '' s, s ∩ f ⁻¹' {i}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Set.iUnion_iUnion_eq_right",
"Eq.mpr",
"Iff.o... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Finite.Lattice | {
"line": 362,
"column": 66
} | {
"line": 366,
"column": 35
} | {
"line": 368,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\ninst✝¹ : LinearOrder ι'\ninst✝ : Nonempty ι'\nα : ι → Type u_3\nI : Set ι\ns : (i : ι) → ι' → Set (α i)\nhI : I.Finite\nhs : ∀ i ∈ I, Monotone (s i)\n⊢ (⋃ j, I.pi fun i ↦ s i j) = I.pi fun i ↦ ⋃ j, s i j",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
... | [] | by
simp only [pi_def, biInter_eq_iInter, preimage_iUnion]
haveI := hI.fintype.finite
refine iUnion_iInter_of_monotone (ι' := ι') (fun (i : I) j₁ j₂ h => ?_)
exact preimage_mono <| hs i i.2 h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Finite.Lattice | {
"line": 399,
"column": 4
} | {
"line": 400,
"column": 63
} | {
"line": 401,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type v\nκ : ι → Sort w\ninst✝¹ : Nonempty ((a : ι) → κ a)\ninst✝ : Order.Frame α\ns : Set ι\nhs : s.Finite\nf : (a : ι) → κ a → α\nh : ∀ {κ : ι → Type w} [Nonempty ((a : ι) → κ a)] (f : (a : ι) → κ a → α), ⨅ a ∈ s, ⨆ b, f a b = ⨆ g, ⨅ a ∈ s, f a (g a)\nthis : Nonempty ((a : ι) → PLift (... | [] | simpa [← Equiv.plift.symm.iSup_comp, ← (Equiv.piCongrRight fun _ => Equiv.plift).symm.iSup_comp]
using h (κ := fun a => PLift (κ a)) fun a b => f a b.down | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.Set.Finite.Lattice | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 63
} | {
"line": 401,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type v\nκ : ι → Sort w\ninst✝¹ : Nonempty ((a : ι) → κ a)\ninst✝ : Order.Frame α\ns : Set ι\nhs : s.Finite\nf : (a : ι) → κ a → α\nh : ∀ {κ : ι → Type w} [Nonempty ((a : ι) → κ a)] (f : (a : ι) → κ a → α), ⨅ a ∈ s, ⨆ b, f a b = ⨆ g, ⨅ a ∈ s, f a (g a)\n⊢ ⨅ a ∈ s, ⨆ b, f a b = ⨆ g, ⨅ a ∈... | [] | haveI : Nonempty (Π a, PLift (κ a)) := (Equiv.piCongrRight fun _ => Equiv.plift).nonempty
simpa [← Equiv.plift.symm.iSup_comp, ← (Equiv.piCongrRight fun _ => Equiv.plift).symm.iSup_comp]
using h (κ := fun a => PLift (κ a)) fun a b => f a b.down | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Finite.Lattice | {
"line": 398,
"column": 4
} | {
"line": 400,
"column": 63
} | {
"line": 401,
"column": 2
} | [
{
"pp": "α : Type u\nι : Type v\nκ : ι → Sort w\ninst✝¹ : Nonempty ((a : ι) → κ a)\ninst✝ : Order.Frame α\ns : Set ι\nhs : s.Finite\nf : (a : ι) → κ a → α\nh : ∀ {κ : ι → Type w} [Nonempty ((a : ι) → κ a)] (f : (a : ι) → κ a → α), ⨅ a ∈ s, ⨆ b, f a b = ⨆ g, ⨅ a ∈ s, f a (g a)\n⊢ ⨅ a ∈ s, ⨆ b, f a b = ⨆ g, ⨅ a ∈... | [] | haveI : Nonempty (Π a, PLift (κ a)) := (Equiv.piCongrRight fun _ => Equiv.plift).nonempty
simpa [← Equiv.plift.symm.iSup_comp, ← (Equiv.piCongrRight fun _ => Equiv.plift).symm.iSup_comp]
using h (κ := fun a => PLift (κ a)) fun a b => f a b.down | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 24
} | {
"line": 387,
"column": 2
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\ninst✝ : ConditionallyCompleteLattice α\np : ι → Prop\nf : Exists p → α\n⊢ ⨆ (ih : Exists p), f ih ≤ ⨆ i, ⨆ (h : p i), f ⋯",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"iSup",
"PartialOrder.t... | [
"case pos\nα : Type u_1\nι : Sort u_4\ninst✝ : ConditionallyCompleteLattice α\np : ι → Prop\nf : Exists p → α\nh : Exists p\n⊢ ⨆ (ih : Exists p), f ih ≤ ⨆ i, ⨆ (h : p i), f ⋯",
"case neg\nα : Type u_1\nι : Sort u_4\ninst✝ : ConditionallyCompleteLattice α\np : ι → Prop\nf : Exists p → α\nh : ∀ (x : ι), ¬p x\n⊢ ⨆ (... | by_cases! h : Exists p | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.Data.Set.Sigma | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 7
} | {
"line": 206,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ns : Set ι\nt : (i : ι) → Set (α i)\nγ : Type u_7\nf : (i : ι) × α i → γ\n⊢ f '' s.sigma t = ⋃ i ∈ s, f ∘ Sigma.mk i '' t i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Sigma.exists._simp_1",
"Iff.of_e... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Sigma | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 7
} | {
"line": 206,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ns : Set ι\nt : (i : ι) → Set (α i)\nγ : Type u_7\nf : (i : ι) × α i → γ\n⊢ f '' s.sigma t = ⋃ i ∈ s, f ∘ Sigma.mk i '' t i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Sigma.exists._simp_1",
"Iff.of_e... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Sigma | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 7
} | {
"line": 206,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ns : Set ι\nt : (i : ι) → Set (α i)\nγ : Type u_7\nf : (i : ι) × α i → γ\n⊢ f '' s.sigma t = ⋃ i ∈ s, f ∘ Sigma.mk i '' t i",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Sigma.exists._simp_1",
"Iff.of_e... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Sigma | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 7
} | {
"line": 220,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ns : Set ι\nt : (i : ι) → Set (α i)\n⊢ s.sigma t = ⋃ i ∈ s, Sigma.mk i '' t i",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.mem_image._simp_1",
"Set.mem_iUnion._... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Set.Sigma | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 7
} | {
"line": 220,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ns : Set ι\nt : (i : ι) → Set (α i)\n⊢ s.sigma t = ⋃ i ∈ s, Sigma.mk i '' t i",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.mem_image._simp_1",
"Set.mem_iUnion._... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Sigma | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 7
} | {
"line": 220,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ns : Set ι\nt : (i : ι) → Set (α i)\n⊢ s.sigma t = ⋃ i ∈ s, Sigma.mk i '' t i",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Iff.of_eq",
"congrArg",
"Set.mem_image._simp_1",
"Set.mem_iUnion._... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.ConditionallyCompleteLattice.Indexed | {
"line": 398,
"column": 2
} | {
"line": 398,
"column": 37
} | {
"line": 398,
"column": 38
} | [
{
"pp": "α : Type u_1\ninst✝ : ConditionallyCompleteLattice α\np q : Prop\nf : p ∧ q → α\n⊢ ⨆ (ih : p ∧ q), f ih = ⨆ (h₁ : p), ⨆ (h₂ : q), f ⋯",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"iSup",
"Classical.propDecidable",
"dite",
"And",
"And.intro",
... | [
"case pos\nα : Type u_1\ninst✝ : ConditionallyCompleteLattice α\np q : Prop\nf : p ∧ q → α\nhp : p\nhq : q\n⊢ ⨆ (ih : p ∧ q), f ih = ⨆ (h₁ : p), ⨆ (h₂ : q), f ⋯",
"case neg\nα : Type u_1\ninst✝ : ConditionallyCompleteLattice α\np q : Prop\nf : p ∧ q → α\nhp : p\nhq : ¬q\n⊢ ⨆ (ih : p ∧ q), f ih = ⨆ (h₁ : p), ⨆ (h₂... | by_cases hp : p <;> by_cases hq : q | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.Finset.Preimage | {
"line": 103,
"column": 63
} | {
"line": 103,
"column": 68
} | {
"line": 103,
"column": 68
} | [
{
"pp": "α : Type u\nβ : Type v\ns : Finset β\nf : α → β\nhf : InjOn f (f ⁻¹' ↑s)\ninst✝ : DecidablePred fun x ↦ x ∈ Set.range f\nb : β\nhb : b ∈ ↑({x ∈ s | x ∈ Set.range f})\n⊢ b ∈ f '' ↑(s.preimage f hf)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_c... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.Finset.Preimage | {
"line": 103,
"column": 63
} | {
"line": 103,
"column": 68
} | {
"line": 103,
"column": 68
} | [
{
"pp": "α : Type u\nβ : Type v\ns : Finset β\nf : α → β\nhf : InjOn f (f ⁻¹' ↑s)\ninst✝ : DecidablePred fun x ↦ x ∈ Set.range f\nb : β\nhb : b ∈ ↑({x ∈ s | x ∈ Set.range f})\n⊢ b ∈ f '' ↑(s.preimage f hf)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_c... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Preimage | {
"line": 103,
"column": 63
} | {
"line": 103,
"column": 68
} | {
"line": 103,
"column": 68
} | [
{
"pp": "α : Type u\nβ : Type v\ns : Finset β\nf : α → β\nhf : InjOn f (f ⁻¹' ↑s)\ninst✝ : DecidablePred fun x ↦ x ∈ Set.range f\nb : β\nhb : b ∈ ↑({x ∈ s | x ∈ Set.range f})\n⊢ b ∈ f '' ↑(s.preimage f hf)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_c... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Cover | {
"line": 381,
"column": 29
} | {
"line": 381,
"column": 38
} | {
"line": 381,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝ : PartialOrder α\na b : α\nh : a ⋖ b\n⊢ Ioo a b ∪ {a} = {a}",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Set.instUnion",
"Set.instSingletonSet",
"id",
"Set.instEmptyC... | [
"α : Type u_1\ninst✝ : PartialOrder α\na b : α\nh : a ⋖ b\n⊢ ∅ ∪ {a} = {a}"
] | h.Ioo_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Cover | {
"line": 415,
"column": 35
} | {
"line": 415,
"column": 44
} | {
"line": 415,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : α\nh : a ⋖ b\n⊢ Ioo a b ∪ Ici b = Ici b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ici",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"Set.instUnion",
... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : α\nh : a ⋖ b\n⊢ ∅ ∪ Ici b = Ici b"
] | h.Ioo_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Cover | {
"line": 419,
"column": 35
} | {
"line": 419,
"column": 44
} | {
"line": 419,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\na b : α\nh : a ⋖ b\n⊢ Iic a ∪ Ioo a b = Iic a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"SemilatticeInf.toPartialOrder",
"Set.instUnion",
"DistribLattice... | [
"α : Type u_1\ninst✝ : LinearOrder α\na b : α\nh : a ⋖ b\n⊢ Iic a ∪ ∅ = Iic a"
] | h.Ioo_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Minimal | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 51
} | {
"line": 276,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_2\nP Q : α → Prop\nx : α\ninst✝ : PartialOrder α\nhPQ : ∀ ⦃x : α⦄, Q x → P x\nh : ∀ ⦃x : α⦄, P x → ∃ y, y ≤ x ∧ Q y\nh' : Minimal P x\ny : α\nhyx : y ≤ x\nhy : Q y\n⊢ Q x",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg"... | [] | rwa [((h'.le_of_le (hPQ hy)) hyx).antisymm hyx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Order.Minimal | {
"line": 433,
"column": 21
} | {
"line": 433,
"column": 34
} | {
"line": 433,
"column": 34
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nx : α\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set α\nf : α → β\nhf : ∀ ⦃x y : α⦄, x ∈ s → y ∈ s → (f x ≤ f y ↔ x ≤ y)\nhx : Minimal (fun x ↦ x ∈ s) x\ny : α\nhy : y ∈ s\n⊢ f y ≤ f x → x ≤ y",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq... | [
"α : Type u_2\nβ : Type u_3\nx : α\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns : Set α\nf : α → β\nhf : ∀ ⦃x y : α⦄, x ∈ s → y ∈ s → (f x ≤ f y ↔ x ≤ y)\nhx : Minimal (fun x ↦ x ∈ s) x\ny : α\nhy : y ∈ s\n⊢ y ≤ x → x ≤ y"
] | hf hy hx.prop | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Minimal | {
"line": 528,
"column": 2
} | {
"line": 528,
"column": 7
} | {
"line": 530,
"column": 0
} | [
{
"pp": "case e'_3\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Preorder α\ninst✝ : Preorder β\nf : α ≃o β\nP : α → Prop\nx✝¹ x✝ : β\n⊢ P (f.symm x✝) ↔ ∃ x₀, P x₀ ∧ f x₀ = x✝",
"ppTerm": "?e'_3",
"assigned": true,
"usedConstants": [
"OrderIso.apply_symm_apply",
"congrArg",
"Preorder.toLE"... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.Interval.Multiset | {
"line": 244,
"column": 8
} | {
"line": 244,
"column": 15
} | {
"line": 244,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b c d : α\nh : b ≤ c\na✝ : α\nhab : a✝ ∈ Ico a b\nhbc : a✝ ∈ Ico c d\n⊢ False",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"Pre... | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\na b c d : α\nh : b ≤ c\na✝ : α\nhab : a ≤ a✝ ∧ a✝ < b\nhbc : c ≤ a✝ ∧ a✝ < d\n⊢ False"
] | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Lattice.Nat | {
"line": 88,
"column": 40
} | {
"line": 88,
"column": 63
} | {
"line": 90,
"column": 0
} | [
{
"pp": "case inr\ns : Set ℕ\nm : ℕ\nh : s.Nonempty\nhm : m < Nat.find h\n⊢ m ∉ s",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Membership.mem",
"Nat.find_min",
"Nat",
"Set.instMembership",
"Set"
],
"usedFVars": [
... | [] | exact Nat.find_min h hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Finset.Nat | {
"line": 121,
"column": 22
} | {
"line": 121,
"column": 29
} | {
"line": 121,
"column": 29
} | [
{
"pp": "a b c : ℕ\nh : c ≤ a\nx : ℕ\n⊢ (∃ a_1 ∈ Ico a b, a_1 - c = x) ↔ x ∈ Ico (a - c) (b - c)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"HSub.hSub",
"Preorder.toLE",
"Nat.instLocallyFiniteOr... | [
"a b c : ℕ\nh : c ≤ a\nx : ℕ\n⊢ (∃ a_1, (a ≤ a_1 ∧ a_1 < b) ∧ a_1 - c = x) ↔ a - c ≤ x ∧ x < b - c"
] | mem_Ico | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Interval.Finset.Nat | {
"line": 129,
"column": 22
} | {
"line": 129,
"column": 29
} | {
"line": 129,
"column": 29
} | [
{
"pp": "a b c : ℕ\nhac : a ≤ c\nx : ℕ\n⊢ (∃ a_1 ∈ Ico a b, c - a_1 = x) ↔ x ∈ Ico (c + 1 - b) (c + 1 - a)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"HSub.hSub",
"Preorder.toLE",
"Nat.instLocal... | [
"a b c : ℕ\nhac : a ≤ c\nx : ℕ\n⊢ (∃ a_1, (a ≤ a_1 ∧ a_1 < b) ∧ c - a_1 = x) ↔ c + 1 - b ≤ x ∧ x < c + 1 - a"
] | mem_Ico | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Lattice.Nat | {
"line": 147,
"column": 43
} | {
"line": 147,
"column": 53
} | {
"line": 147,
"column": 54
} | [
{
"pp": "s : Set ℕ\nhs : ¬BddBelow s\n⊢ sInf s = sInf ∅",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"OrderBot.bddBelow._simp_1",
"False",
"Lattice.toSemilatticeSup",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"Preorder.toLE",
"E... | [] | simp at hs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Lattice.Nat | {
"line": 147,
"column": 43
} | {
"line": 147,
"column": 53
} | {
"line": 147,
"column": 54
} | [
{
"pp": "s : Set ℕ\nhs : ¬BddBelow s\n⊢ sInf s = sInf ∅",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"OrderBot.bddBelow._simp_1",
"False",
"Lattice.toSemilatticeSup",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"Preorder.toLE",
"E... | [] | simp at hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Lattice.Nat | {
"line": 147,
"column": 43
} | {
"line": 147,
"column": 53
} | {
"line": 147,
"column": 54
} | [
{
"pp": "s : Set ℕ\nhs : ¬BddBelow s\n⊢ sInf s = sInf ∅",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"OrderBot.bddBelow._simp_1",
"False",
"Lattice.toSemilatticeSup",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"Preorder.toLE",
"E... | [] | simp at hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Finset.Nat | {
"line": 136,
"column": 22
} | {
"line": 136,
"column": 29
} | {
"line": 136,
"column": 29
} | [
{
"pp": "a b x : ℕ\n⊢ x ∈ Ico a.succ b ↔ x ≠ a ∧ x ∈ Ico a b",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"Preorder.toLE",
"Nat.instLocallyFiniteOrder",
"Membership.mem",
"id",
"Finset... | [
"a b x : ℕ\n⊢ a.succ ≤ x ∧ x < b ↔ x ≠ a ∧ a ≤ x ∧ x < b"
] | mem_Ico | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Interval.Finset.Nat | {
"line": 141,
"column": 23
} | {
"line": 141,
"column": 30
} | {
"line": 141,
"column": 30
} | [
{
"pp": "a b : ℕ\nh : a ≤ b\nx : ℕ\n⊢ x ∈ Ico a b.succ ↔ x = b ∨ x ∈ Ico a b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"Finset",
"Preorder.toLE",
"Nat.instLocallyFiniteOrder",
"Membership.mem",
"id... | [
"a b : ℕ\nh : a ≤ b\nx : ℕ\n⊢ a ≤ x ∧ x < b.succ ↔ x = b ∨ a ≤ x ∧ x < b"
] | mem_Ico | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.WellQuasiOrder | {
"line": 182,
"column": 6
} | {
"line": 183,
"column": 67
} | {
"line": 185,
"column": 0
} | [
{
"pp": "case inr.refine_2\nα : Type u_1\ninst✝ : Preorder α\nx✝ : WellFoundedLT α ∧ ∀ (s : Set α), IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s → s.Finite\nhwf : WellFoundedLT α\nf : ℕ → α\ng : ℕ ↪o ℕ\nh2 : ∀ (m n : ℕ), m < n → ¬f (g m) > f (g n)\nhc : ∀ (m n : ℕ), m < n → ¬f m ≤ f n\nm n : ℕ\nhf : f (g m) = f (g n)\n⊢... | [] | obtain h | rfl | h := lt_trichotomy m n <;>
(first | rfl | cases (hf ▸ hc _ _ (g.strictMono h)) le_rfl) | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Order.WellQuasiOrder | {
"line": 205,
"column": 30
} | {
"line": 205,
"column": 50
} | {
"line": 205,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\n⊢ (WellFoundedLT α ∧ ∀ (s : Set α), IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s → s.Finite) ↔ WellFoundedLT α",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"WellFoundedLT",
"PartialO... | [
"α : Type u_1\ninst✝ : LinearOrder α\n⊢ WellFoundedLT α → ∀ (s : Set α), IsAntichain (fun x1 x2 ↦ x1 ≤ x2) s → s.Finite"
] | and_iff_left_iff_imp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Interval.Finset.Basic | {
"line": 344,
"column": 65
} | {
"line": 345,
"column": 46
} | {
"line": 347,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝³ : Preorder α\ninst✝² : LocallyFiniteOrderTop α\ninst✝¹ : OrderBot α\ninst✝ : Fintype α\n⊢ Ici ⊥ = univ",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.Ici_bot._simp_1_3",
"Finset.univ",
"... | [] | by
ext a; simp only [mem_Ici, bot_le, mem_univ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Defs | {
"line": 978,
"column": 6
} | {
"line": 978,
"column": 13
} | {
"line": 978,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝² : Preorder α\np : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : LocallyFiniteOrder α\na b : Subtype p\nhp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x\nx : α\nhx : x ∈ Ico ↑a ↑b\n⊢ p x",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
... | [
"α : Type u_1\ninst✝² : Preorder α\np : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : LocallyFiniteOrder α\na b : Subtype p\nhp : ∀ ⦃a b x : α⦄, a ≤ x → x ≤ b → p a → p b → p x\nx : α\nhx : ↑a ≤ x ∧ x < ↑b\n⊢ p x"
] | mem_Ico | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 60,
"column": 19
} | {
"line": 60,
"column": 76
} | {
"line": 62,
"column": 0
} | [
{
"pp": "M : Type u_3\nS : Type u_6\ninst✝² : Monoid M\ninst✝¹ : SetLike S M\ninst✝ : SubmonoidClass S M\nn : ℕ\nx✝ : n + 2 ≠ 0\nH : S\n⊢ ↑H ^ (n + 2) = ↑H",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"coe_mul_coe",
"Monoid.toMulOneClass",
... | [] | by rw [pow_succ, coe_set_pow n.succ_ne_zero, coe_mul_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Basic | {
"line": 598,
"column": 68
} | {
"line": 598,
"column": 90
} | {
"line": 600,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a ≤ b\n⊢ Icc a b \\ Ico a b = {b}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.coe_Ico",
"Finset.coe_singleton",
"congrArg",
"Finset",
... | [] | by simp [← coe_inj, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Basic | {
"line": 603,
"column": 68
} | {
"line": 603,
"column": 90
} | {
"line": 605,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a ≤ b\n⊢ Icc a b \\ Ioc a b = {a}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"Finset.coe_singleton",
"congrArg",
"Finset",
"_priv... | [] | by simp [← coe_inj, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 75,
"column": 18
} | {
"line": 75,
"column": 36
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case zero\nM : Type u_3\ninst✝ : Monoid M\ns : Set M\nS : Submonoid M\nhs : s ⊆ ↑S\n⊢ s ^ 0 ⊆ ↑S",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Membership.mem",
"Su... | [] | simp [pow_succ, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Submonoid.Pointwise | {
"line": 75,
"column": 18
} | {
"line": 75,
"column": 36
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case succ\nM : Type u_3\ninst✝ : Monoid M\ns : Set M\nS : Submonoid M\nhs : s ⊆ ↑S\nn✝ : ℕ\na✝ : s ^ n✝ ⊆ ↑S\n⊢ s ^ (n✝ + 1) ⊆ ↑S",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"pow_suc... | [] | simp [pow_succ, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Interval.Finset.Basic | {
"line": 608,
"column": 71
} | {
"line": 608,
"column": 93
} | {
"line": 610,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a ≤ b\n⊢ Icc a b \\ Ioo a b = {a, b}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Order.Interval.Finset.Basic.0.Finset.Icc_sdiff_Ioo_self._simp_1_1"... | [] | by simp [← coe_inj, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Interval.Finset.Basic | {
"line": 613,
"column": 68
} | {
"line": 613,
"column": 90
} | {
"line": 615,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝² : PartialOrder α\ninst✝¹ : LocallyFiniteOrder α\na b : α\ninst✝ : DecidableEq α\nh : a < b\n⊢ Ico a b \\ Ioo a b = {a}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Finset.coe_Ico",
"Finset.coe_singleton",
"congrArg",
"Finset",
... | [] | by simp [← coe_inj, h] | [anonymous] | Lean.Parser.Term.byTactic |
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