module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 92,
"column": 41
} | {
"line": 92,
"column": 59
} | {
"line": 92,
"column": 60
} | [
{
"pp": "α : Type u_2\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : IsOrderedCancelAddMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image (fun x ↦ x + c) (Ioo a b) = image (⇑(addRightEmbedding c)) (Ioo a b)",
"ppTerm": "?m.45",
"ass... | [
"α : Type u_2\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : IsOrderedCancelAddMonoid α\ninst✝² : ExistsAddOfLE α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b c : α\n⊢ image (fun x ↦ x + c) (Ioo a b) = image (⇑{ toFun := fun h ↦ h + c, inj' := ⋯ }) (Ioo a b)"
] | addRightEmbedding, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 188,
"column": 6
} | {
"line": 188,
"column": 31
} | {
"line": 188,
"column": 32
} | [
{
"pp": "n : ℕ\n⊢ ∑ i ∈ range n, i = n * (n - 1) / 2",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"id",
"HDiv.hDiv",
"instSubNat",
"instMulNat",
"instOfNatNat",
... | [
"n : ℕ\n⊢ ∑ i ∈ range n, i = (∑ i ∈ range n, i) * 2 / 2"
] | ← sum_range_id_mul_two n, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Defs | {
"line": 97,
"column": 15
} | {
"line": 97,
"column": 53
} | {
"line": 98,
"column": 2
} | [
{
"pp": "α : Type u\nβ : Type v\nF : Type w\nR : Type ?u.7\nM : Type ?u.9\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nm m' : M\n⊢ 0 ∈ {r | r • m = r • m'}",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"Set... | [] | by simp_rw [Set.mem_ofPred, zero_smul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Prime | {
"line": 85,
"column": 11
} | {
"line": 85,
"column": 29
} | {
"line": 85,
"column": 30
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\n⊢ ¬I.IsPrime ↔ I = ⊤ ∨ ∃ x, ∃ (_ : x ∉ I), ∃ y, ∃ (_ : y ∉ I), x * y ∈ I",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"HMul.hMul",
"congrArg",
"Membership.mem",
"Exis... | [
"α : Type u\ninst✝ : Semiring α\nI : Ideal α\n⊢ ¬(I ≠ ⊤ ∧ ∀ {x y : α}, x * y ∈ I → x ∈ I ∨ y ∈ I) ↔ I = ⊤ ∨ ∃ x, ∃ (_ : x ∉ I), ∃ y, ∃ (_ : y ∉ I), x * y ∈ I"
] | Ideal.isPrime_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 13
} | {
"line": 93,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\n⊢ (∃ M, M.IsMaximal ∧ I ≤ M) → I ≠ ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.not_and_eq",
"Semiring.toModule",
"Mathlib.Tacti... | [
"α : Type u\ninst✝ : Semiring α\nI : Ideal α\n⊢ I = ⊤ → ∀ (M : Ideal α), M.IsMaximal → ¬I ≤ M"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.RingTheory.Ideal.Basic | {
"line": 68,
"column": 2
} | {
"line": 71,
"column": 13
} | {
"line": 73,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_5\ninst✝¹ : (i : ι) → Semiring (R i)\nI : (i : ι) → Ideal (R i)\ninst✝ : DecidableEq ι\ni : ι\nr : R i\nhr : r ∈ I i\n⊢ Pi.single i r ∈ pi I",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Submodule.addSubmonoidCla... | [] | intro j
obtain rfl | ne := eq_or_ne i j
· simpa
· simp [ne] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Basic | {
"line": 68,
"column": 2
} | {
"line": 71,
"column": 13
} | {
"line": 73,
"column": 0
} | [
{
"pp": "ι : Type u_1\nR : ι → Type u_5\ninst✝¹ : (i : ι) → Semiring (R i)\nI : (i : ι) → Ideal (R i)\ninst✝ : DecidableEq ι\ni : ι\nr : R i\nhr : r ∈ I i\n⊢ Pi.single i r ∈ pi I",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Submodule.addSubmonoidCla... | [] | intro j
obtain rfl | ne := eq_or_ne i j
· simpa
· simp [ne] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Basic | {
"line": 137,
"column": 10
} | {
"line": 137,
"column": 13
} | {
"line": 138,
"column": 2
} | [
{
"pp": "case cons\nα : Type u_2\ninst✝¹ : CommSemiring α\ninst✝ : DecidableEq α\nn : ℕ\na : α\ns : Multiset α\nhs : s.sum ^ (s.card * n + 1) ∈ span ↑(Multiset.map (fun x ↦ x ^ (n + 1)) s).toFinset\nc : ℕ\n⊢ c ∈ Finset.range ((s.card + 1) * n + 1 + 1) →\n a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * ↑(((s.ca... | [
"case cons\nα : Type u_2\ninst✝¹ : CommSemiring α\ninst✝ : DecidableEq α\nn : ℕ\na : α\ns : Multiset α\nhs : s.sum ^ (s.card * n + 1) ∈ span ↑(Multiset.map (fun x ↦ x ^ (n + 1)) s).toFinset\nc : ℕ\n_hc : c ∈ Finset.range ((s.card + 1) * n + 1 + 1)\n⊢ a ^ c * s.sum ^ ((s.card + 1) * n + 1 - c) * ↑(((s.card + 1) * n ... | _hc | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Ideal.Basic | {
"line": 130,
"column": 65
} | {
"line": 153,
"column": 29
} | {
"line": 155,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝¹ : CommSemiring α\ninst✝ : DecidableEq α\ns : Multiset α\nn : ℕ\n⊢ s.sum ^ (s.card * n + 1) ∈ span ↑(Multiset.map (fun x ↦ x ^ (n + 1)) s).toFinset",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Multiset.toFinset",
"add_mul",... | [] | by
induction s using Multiset.induction_on with
| empty => simp
| cons a s hs => ?_
simp only [Finset.coe_insert, Multiset.map_cons, Multiset.toFinset_cons, Multiset.sum_cons,
Multiset.card_cons, add_pow]
refine Submodule.sum_mem _ ?_
intro c _hc
rw [mem_span_insert]
by_cases! h : n + 1 ≤ c
· refi... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Basic | {
"line": 211,
"column": 10
} | {
"line": 211,
"column": 91
} | {
"line": 211,
"column": 91
} | [
{
"pp": "case single\nι : Type u_5\ninst✝² : DecidableEq ι\ninst✝¹ : Finite ι\nR : ι → Type u_6\ninst✝ : (i : ι) → Semiring (R i)\ni : ι\nr : R i\n⊢ Pi.single i r ∈ span (range fun i ↦ Pi.single i 1)",
"ppTerm": "?single",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.... | [
"case single\nι : Type u_5\ninst✝² : DecidableEq ι\ninst✝¹ : Finite ι\nR : ι → Type u_6\ninst✝ : (i : ι) → Semiring (R i)\ni : ι\nr : R i\n⊢ Pi.single i r * Pi.single i 1 ∈ span (range fun i ↦ Pi.single i 1)"
] | show Pi.single i r = Pi.single i r * Pi.single i 1 by simp [← Pi.single_mul_left] | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Basic | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 34
} | {
"line": 256,
"column": 2
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nhf : ¬IsField R\nthis : ∃ a, a ≠ 0 ∧ ∀ (b : R), a * b ≠ 1\n⊢ ∃ x, ∃ (_ : x ≠ 0), ∀ (b : R), x * b ≠ 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",... | [
"R : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nhf : ¬IsField R\nx : R\nhx : x ≠ 0\nnot_unit : ∀ (b : R), x * b ≠ 1\n⊢ ∃ x, ∃ (_ : x ≠ 0), ∀ (b : R), x * b ≠ 1"
] | obtain ⟨x, hx, not_unit⟩ := this | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Filter.AtTopBot.Defs | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 23
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : LinearOrder α\ns : Set α\nhs : ¬BddAbove s\n⊢ atTop = generate (Ici '' s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exists",
"Semilattice... | [] | refine atTop_eq_generate_of_forall_exists_le fun x ↦ ?_
obtain ⟨y, hy, hy'⟩ := not_bddAbove_iff.mp hs x
exact ⟨y, hy, hy'.le⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Defs | {
"line": 101,
"column": 2
} | {
"line": 103,
"column": 23
} | {
"line": 105,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : LinearOrder α\ns : Set α\nhs : ¬BddAbove s\n⊢ atTop = generate (Ici '' s)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exists",
"Semilattice... | [] | refine atTop_eq_generate_of_forall_exists_le fun x ↦ ?_
obtain ⟨y, hy, hy'⟩ := not_bddAbove_iff.mp hs x
exact ⟨y, hy, hy'.le⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 543,
"column": 2
} | {
"line": 543,
"column": 86
} | {
"line": 544,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Filter β\nm : α → β\n⊢ (comap m f).NeBot ↔ ∀ t ∈ f, ∃ a, m a ∈ t",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Filter.NeBot",
"_private.Mathlib.Order.Filter.Map.0.Filter... | [
"α : Type u_1\nβ : Type u_2\nf : Filter β\nm : α → β\n⊢ (∀ (s : Set α), ∀ x ∈ f, m ⁻¹' x ⊆ s → s.Nonempty) ↔ ∀ t ∈ f, ∃ a, m a ∈ t"
] | simp only [← forall_mem_nonempty_iff_neBot, mem_comap, forall_exists_index, and_imp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.AtTopBot.Tendsto | {
"line": 149,
"column": 6
} | {
"line": 149,
"column": 36
} | {
"line": 149,
"column": 37
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β\ne : β → γ\nl : Filter α\nhm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂\nhu : ∀ (c : γ), ∃ b, c ≤ e b\n⊢ Tendsto (e ∘ f) l atTop ↔ Tendsto f l atTop",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants... | [
"α : Type u_3\nβ : Type u_4\nγ : Type u_5\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\nf : α → β\ne : β → γ\nl : Filter α\nhm : ∀ (b₁ b₂ : β), e b₁ ≤ e b₂ ↔ b₁ ≤ b₂\nhu : ∀ (c : γ), ∃ b, c ≤ e b\n⊢ Tendsto (e ∘ f) l atTop ↔ Tendsto f l (comap e atTop)"
] | ← comap_embedding_atTop hm hu, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 124,
"column": 50
} | {
"line": 133,
"column": 26
} | {
"line": 135,
"column": 0
} | [
{
"pp": "P : ℕ → ℕ → Prop\nh : ∀ (n : ℕ), ∃ᶠ (k : ℕ) in atTop, P n k\n⊢ ∃ φ, StrictMono φ ∧ ∀ (n : ℕ), P n (φ n)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"instDistribLatticeNat",
"StrictMono",
"_private.Mathl... | [] | by
simp only [frequently_atTop'] at h
choose u hu hu' using h
use (fun n => Nat.recOn n (u 0 0) fun n v => u (n + 1) v : ℕ → ℕ)
constructor
· apply strictMono_nat_of_lt_succ
intro n
apply hu
· intro n
cases n <;> simp [hu'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.NoZeroSMulDivisors.Defs | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 22
} | {
"line": 60,
"column": 23
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : r • m₁ = r • m₂\n⊢ m₁ = m₂",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\nM : Type u_2\nG : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : IsDomain R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : NoZeroSMulDivisors R M\nr : R\nhr : IsRegular r\nm₁ m₂ : M\nhm : r • m₁ - r • m₂ = 0\n⊢ m₁ = m₂"
] | ← sub_eq_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 967,
"column": 38
} | {
"line": 967,
"column": 48
} | {
"line": 967,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosMono α β\n_b : βᵒᵈ\nhb : 0 ≤ _b\na₁ a₂ : α\nha : a₁ ≤ a₂\n⊢ a₂ • -_b ≤ -(a₁ • _b... | [
"α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosMono α β\n_b : βᵒᵈ\nhb : 0 ≤ _b\na₁ a₂ : α\nha : a₁ ≤ a₂\n⊢ a₂ • -_b ≤ a₁ • -_b"
] | ← smul_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 972,
"column": 38
} | {
"line": 972,
"column": 48
} | {
"line": 972,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosStrictMono α β\n_b : βᵒᵈ\nhb : 0 < _b\na₁ a₂ : α\nha : a₁ < a₂\n⊢ a₂ • -_b < -(a... | [
"α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosStrictMono α β\n_b : βᵒᵈ\nhb : 0 < _b\na₁ a₂ : α\nha : a₁ < a₂\n⊢ a₂ • -_b < a₁ • -_b"
] | ← smul_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 977,
"column": 38
} | {
"line": 977,
"column": 48
} | {
"line": 977,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosReflectLT α β\n_b : βᵒᵈ\nhb : 0 ≤ _b\na₁ a₂ : α\nh : a₂ • -_b < -(a₁ • _b)\n⊢ a₁... | [
"α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosReflectLT α β\n_b : βᵒᵈ\nhb : 0 ≤ _b\na₁ a₂ : α\nh : a₂ • -_b < a₁ • -_b\n⊢ a₁ < a₂"
] | ← smul_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 982,
"column": 38
} | {
"line": 982,
"column": 48
} | {
"line": 982,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosReflectLE α β\n_b : βᵒᵈ\nhb : 0 < _b\na₁ a₂ : α\nh : a₂ • -_b ≤ -(a₁ • _b)\n⊢ a₁... | [
"α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosReflectLE α β\n_b : βᵒᵈ\nhb : 0 < _b\na₁ a₂ : α\nh : a₂ • -_b ≤ a₁ • -_b\n⊢ a₁ ≤ a₂"
] | ← smul_neg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1046,
"column": 6
} | {
"line": 1046,
"column": 18
} | {
"line": 1046,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ a • b₂ ≤ a • b₁",
"ppTerm": "?m... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulMono α β\nh : b₁ ≤ b₂\nha : a ≤ 0\n⊢ - -a • b₂ ≤ - -a • b₁"
] | ← neg_neg a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1062,
"column": 6
} | {
"line": 1062,
"column": 18
} | {
"line": 1062,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\nhb : b₁ < b₂\nha : a < 0\n⊢ a • b₂ < a • b₁",
"ppTer... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulStrictMono α β\nhb : b₁ < b₂\nha : a < 0\n⊢ - -a • b₂ < - -a • b₁"
] | ← neg_neg a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1076,
"column": 6
} | {
"line": 1076,
"column": 18
} | {
"line": 1076,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulReflectLE α β\nh : a • b₁ ≤ a • b₂\nha : a < 0\n⊢ b₂ ≤ b₁",
"ppTerm"... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulReflectLE α β\nh : - -a • b₁ ≤ - -a • b₂\nha : a < 0\n⊢ b₂ ≤ b₁"
] | ← neg_neg a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1081,
"column": 6
} | {
"line": 1081,
"column": 18
} | {
"line": 1081,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulReflectLT α β\nh : a • b₁ < a • b₂\nha : a ≤ 0\n⊢ b₂ < b₁",
"ppTerm"... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁷ : Ring α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : PosSMulReflectLT α β\nh : - -a • b₁ < - -a • b₂\nha : a ≤ 0\n⊢ b₂ < b₁"
] | ← neg_neg a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1090,
"column": 6
} | {
"line": 1090,
"column": 18
} | {
"line": 1090,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : a < 0\n⊢ a • b₁ ≤ a • b₂ ↔... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulMono α β\ninst✝ : PosSMulReflectLE α β\nha : a < 0\n⊢ - -a • b₁ ≤ - -a • b₂ ↔ b₂ ≤ ... | ← neg_neg a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1097,
"column": 6
} | {
"line": 1097,
"column": 18
} | {
"line": 1097,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : a < 0\n⊢ a • b₁ < a ... | [
"α : Type u_1\nβ : Type u_2\na : α\nb₁ b₂ : β\ninst✝⁸ : Ring α\ninst✝⁷ : PartialOrder α\ninst✝⁶ : IsOrderedRing α\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module α β\ninst✝¹ : PosSMulStrictMono α β\ninst✝ : PosSMulReflectLT α β\nha : a < 0\n⊢ - -a • b₁ < - -a • b₂ ↔... | ← neg_neg a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.IsNormal | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 23
} | {
"line": 59,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\nhf : IsNormal f\na : α\nha : IsSuccLimit a\n⊢ f a ∈ upperBounds (f '' Iio a)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"Members... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\nhf : IsNormal f\na : α\nha : IsSuccLimit a\nb : α\nhb : b ∈ Iio a\n⊢ f b ≤ f a"
] | rintro - ⟨b, hb, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Order.IsNormal | {
"line": 62,
"column": 36
} | {
"line": 63,
"column": 83
} | {
"line": 65,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nf : α → β\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nhf : IsNormal f\nha : IsSuccLimit a\nb : β\n⊢ f a ≤ b ↔ ∀ a' < a, f a' ≤ b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"congrArg",
"Order.IsNormal.is... | [] | by
simpa [mem_upperBounds] using isLUB_le_iff (hf.isLUB_image_Iio_of_isSuccLimit ha) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Cofinality.Enum | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 30
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : IsRegularCardinalOrder α\ns : Set α\nhs : IsCofinal s\ninst✝ : SuccOrder α\na o : α\nha : o ∈ s\nH : ↑((enum s hs) a) < o\nb : α\nhb : b < succ a\n⊢ ↑((enum s hs) b) ≤ ↑((enum s hs) a)",
"ppTerm": "?m.43",
"assigned": true... | [] | simpa using le_of_lt_succ hb | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 63
} | {
"line": 489,
"column": 2
} | [
{
"pp": "case refine_1\na : Ordinal.{u_4}\nh : 0 < a\nb : Ordinal.{u_4}\n⊢ a * b < a * succ b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Order.suc... | [] | simpa [mul_add_one] using (add_lt_add_iff_left (a * b)).2 h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 63
} | {
"line": 489,
"column": 2
} | [
{
"pp": "case refine_1\na : Ordinal.{u_4}\nh : 0 < a\nb : Ordinal.{u_4}\n⊢ a * b < a * succ b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Order.suc... | [] | simpa [mul_add_one] using (add_lt_add_iff_left (a * b)).2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 488,
"column": 4
} | {
"line": 488,
"column": 63
} | {
"line": 489,
"column": 2
} | [
{
"pp": "case refine_1\na : Ordinal.{u_4}\nh : 0 < a\nb : Ordinal.{u_4}\n⊢ a * b < a * succ b",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Order.suc... | [] | simpa [mul_add_one] using (add_lt_add_iff_left (a * b)).2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1118,
"column": 2
} | {
"line": 1118,
"column": 19
} | {
"line": 1118,
"column": 19
} | [
{
"pp": "o : Ordinal.{u_1}\nc : Cardinal.{u_1}\nho : o ≤ c.ord\n⊢ o.card ≤ c",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"id",
"LE.le",
"Cardinal.ord",
"Cardinal.instLE",
"Ordinal.card",
"Cardinal.ca... | [
"o : Ordinal.{u_1}\nc : Cardinal.{u_1}\nho : o ≤ c.ord\n⊢ o.card ≤ c.ord.card"
] | rw [← card_ord c] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 546,
"column": 13
} | {
"line": 546,
"column": 26
} | {
"line": 546,
"column": 26
} | [
{
"pp": "a b c : Ordinal.{u_4}\nba : b + a = a\nl : IsSuccLimit c\nIH : ∀ c' < c, (a + b) * succ c' = a * succ c' + b\n⊢ a * c ≤ (a + b) * c",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"le_refl",
"Ordinal.mulRightMono",
"HMul.hMul",
"Ordinal.partialOrder",
... | [
"a b c : Ordinal.{u_4}\nba : b + a = a\nl : IsSuccLimit c\nIH : ∀ c' < c, (a + b) * succ c' = a * succ c' + b\n⊢ a * c ≤ a * c"
] | ← le_self_add | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 45
} | {
"line": 566,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_4}\n⊢ o * 2 = o + o",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"MulZeroClass.toMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"MulOn... | [] | rw [← one_add_one_eq_two, mul_add, mul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 45
} | {
"line": 566,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_4}\n⊢ o * 2 = o + o",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"MulZeroClass.toMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"MulOn... | [] | rw [← one_add_one_eq_two, mul_add, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 564,
"column": 2
} | {
"line": 564,
"column": 45
} | {
"line": 566,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_4}\n⊢ o * 2 = o + o",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"MulZeroClass.toMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"MulOn... | [] | rw [← one_add_one_eq_two, mul_add, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 657,
"column": 61
} | {
"line": 657,
"column": 74
} | {
"line": 657,
"column": 74
} | [
{
"pp": "case a\na c : Ordinal.{u_4}\nhc : c < a\nb d : Ordinal.{u_4}\nhd : d ≠ 0\nH : a * d ≠ 0\n⊢ a * b ≤ a * b + c",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"le_refl",
"Semigroup.toMul",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"... | [
"case a\na c : Ordinal.{u_4}\nhc : c < a\nb d : Ordinal.{u_4}\nhd : d ≠ 0\nH : a * d ≠ 0\n⊢ a * b ≤ a * b"
] | ← le_self_add | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Data.Nat.Log | {
"line": 216,
"column": 2
} | {
"line": 216,
"column": 45
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case inr\nb m n : ℕ\nh₁ : b ^ m ≤ n\nh₂ : n < b ^ (m + 1)\nhm : m ≠ 0\n⊢ log b n = m",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"instPowNat",
"Nat.log_eq_iff",
"Ne",
"instOfNatNat",
"LE.le",
"instLENat",
"instNatPowNat... | [] | · exact (log_eq_iff (Or.inl hm)).2 ⟨h₁, h₂⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 764,
"column": 2
} | {
"line": 766,
"column": 22
} | {
"line": 768,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_4}\n⊢ a % a = 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"False",
"instHDiv",
"HMul.hMul",
"eq_false",
"MulZeroClass.toMul",
"congrArg",
"Ordinal.div_self",
"HSub.hSub",
"Ordinal.sub_self",
"Ordinal.m... | [] | obtain rfl | ha := eq_or_ne a 0
· simp
· simp [mod_def, ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 764,
"column": 2
} | {
"line": 766,
"column": 22
} | {
"line": 768,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_4}\n⊢ a % a = 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"False",
"instHDiv",
"HMul.hMul",
"eq_false",
"MulZeroClass.toMul",
"congrArg",
"Ordinal.div_self",
"HSub.hSub",
"Ordinal.sub_self",
"Ordinal.m... | [] | obtain rfl | ha := eq_or_ne a 0
· simp
· simp [mod_def, ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 196,
"column": 10
} | {
"line": 196,
"column": 20
} | {
"line": 196,
"column": 20
} | [
{
"pp": "case inl.inl\na b : Ordinal.{u_1}\nb1 : 0 < b\na1 : a ≤ 1\na0 : a < 1\n⊢ a ^ 1 ≤ a ^ b",
"ppTerm": "?inl.inl",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"... | [
"case inl.inl\na b : Ordinal.{u_1}\nb1 : 0 < b\na1 : a ≤ 1\na0 : a = 0\n⊢ a ^ 1 ≤ a ^ b"
] | lt_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | {
"line": 89,
"column": 13
} | [
{
"pp": "ι : Type u_1\nf : ι → Ordinal.{u} → Ordinal.{u}\ninst✝¹ : Nonempty ι\ninst✝ : Small.{u, u_1} ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal.{u}\n⊢ (∃ i, nfpFamily f a ≤ f i b) ↔ nfpFamily f a ≤ b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exist... | [
"ι : Type u_1\nf : ι → Ordinal.{u} → Ordinal.{u}\ninst✝¹ : Nonempty ι\ninst✝ : Small.{u, u_1} ι\nH : ∀ (i : ι), IsNormal (f i)\na b : Ordinal.{u}\n⊢ (∀ (i : ι), f i b < nfpFamily f a) ↔ b < nfpFamily f a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 984,
"column": 4
} | {
"line": 990,
"column": 77
} | {
"line": 991,
"column": 2
} | [
{
"pp": "case refine_1\na : Ordinal.{u_4}\nl : IsSuccPrelimit a\n⊢ a ≤ ω * (a / ω)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"le_refl",
"Semigroup.toMul",
"Ordinal.instLinearOrder",
"Ordinal.instAddRightMono",
"Preord... | [] | refine l.le_iff_forall_le.2 fun x hx => le_of_lt ?_
rw [lt_mul_iff_div_lt omega0_ne_zero, ← succ_le_iff, ← mul_le_iff_le_div omega0_ne_zero,
mul_succ, add_le_iff_of_isSuccLimit isSuccLimit_omega0]
intro b hb
rcases lt_omega0.1 hb with ⟨n, rfl⟩
grw [mul_div_le]
exact (lt_sub.1 <| natCast_lt_of_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 984,
"column": 4
} | {
"line": 990,
"column": 77
} | {
"line": 991,
"column": 2
} | [
{
"pp": "case refine_1\na : Ordinal.{u_4}\nl : IsSuccPrelimit a\n⊢ a ≤ ω * (a / ω)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"le_refl",
"Semigroup.toMul",
"Ordinal.instLinearOrder",
"Ordinal.instAddRightMono",
"Preord... | [] | refine l.le_iff_forall_le.2 fun x hx => le_of_lt ?_
rw [lt_mul_iff_div_lt omega0_ne_zero, ← succ_le_iff, ← mul_le_iff_le_div omega0_ne_zero,
mul_succ, add_le_iff_of_isSuccLimit isSuccLimit_omega0]
intro b hb
rcases lt_omega0.1 hb with ⟨n, rfl⟩
grw [mul_div_le]
exact (lt_sub.1 <| natCast_lt_of_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 1015,
"column": 2
} | {
"line": 1015,
"column": 44
} | {
"line": 1016,
"column": 2
} | [
{
"pp": "case not_isMin\nc : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\n⊢ ¬IsMin c.ord",
"ppTerm": "?not_isMin",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"isMin_iff_eq_bot._simp_1",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"Cardinal... | [
"case isSuccPrelimit\nc : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\n⊢ IsSuccPrelimit c.ord"
] | · simpa using (aleph0_pos.trans_le hc).ne' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 397,
"column": 2
} | {
"line": 399,
"column": 46
} | {
"line": 401,
"column": 0
} | [
{
"pp": "case inr\nb x : Ordinal.{u_1}\nhx : x ≠ 0\n⊢ log b x ≤ x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.log_of_left_le_one",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroC... | [] | · obtain hb | hb := lt_or_ge 1 b
· exact (right_le_opow _ hb).trans (opow_log_le_self b hx)
· simp_rw [log_of_left_le_one hb, zero_le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 434,
"column": 44
} | {
"line": 434,
"column": 57
} | {
"line": 434,
"column": 57
} | [
{
"pp": "case left\nb u v w : Ordinal.{u_1}\nhb : 1 < b\nhv : v ≠ 0\nhw : w < b ^ u\n⊢ b ^ u * v ≤ b ^ u * v + w",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"le_refl",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"PartialOrder.toPreorder",
... | [
"case left\nb u v w : Ordinal.{u_1}\nhb : 1 < b\nhv : v ≠ 0\nhw : w < b ^ u\n⊢ b ^ u * v ≤ b ^ u * v"
] | ← le_self_add | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 369,
"column": 90
} | {
"line": 371,
"column": 33
} | {
"line": 373,
"column": 0
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nH : IsNormal f\n⊢ deriv f = enumOrd (fixedPoints f)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.iInter_const",
"Set.iInter",
"HEq.refl",
"Function.fixedPoints",
"instInhabitedPUnit",
"Eq.cas... | [] | by
convert! derivFamily_eq_enumOrd fun _ : Unit => H
exact (Set.iInter_const _).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 24
} | {
"line": 396,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ deriv 0 a = a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Pi.instZero",
"Ordinal.deriv_zero",
"Ordinal.deriv",
"Ordinal.zero",
"Zero.toOfNat0",
"Eq.refl",
"OfNat.ofNat"... | [] | rw [deriv_zero, id_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 24
} | {
"line": 396,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ deriv 0 a = a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Pi.instZero",
"Ordinal.deriv_zero",
"Ordinal.deriv",
"Ordinal.zero",
"Zero.toOfNat0",
"Eq.refl",
"OfNat.ofNat"... | [] | rw [deriv_zero, id_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 24
} | {
"line": 396,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ deriv 0 a = a",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Pi.instZero",
"Ordinal.deriv_zero",
"Ordinal.deriv",
"Ordinal.zero",
"Zero.toOfNat0",
"Eq.refl",
"OfNat.ofNat"... | [] | rw [deriv_zero, id_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 262,
"column": 4
} | {
"line": 262,
"column": 30
} | {
"line": 262,
"column": 30
} | [
{
"pp": "b c : Ordinal.{u}\nH : ∀ b_1 < b + c, ∀ c_1 < b + c, b_1 + c_1 ≠ b + c\nha : ¬IsPrincipal (fun x1 x2 ↦ x1 + x2) (b + c)\nhb : b < b + c\nhc : c < b + c\n⊢ False",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Ne.irrefl",
"instHAdd",
"HAdd.hAdd",
"Ordinal.a... | [] | exact (H b hb c hc).irrefl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 490,
"column": 4
} | {
"line": 490,
"column": 52
} | {
"line": 491,
"column": 4
} | [
{
"pp": "case a\na c b : Ordinal.{u_1}\nha : 0 < a\nhc : 0 < c\nhca : c ≤ a ^ ω\n⊢ nfp (fun x ↦ a * x) (a ^ ω * b + c) ≤ a ^ ω * succ b",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Ordinal.isNormal_mul_right",
"Ordinal.instLinearOrder",
"HMul.hMul",
"Order.succ",
... | [
"case a.ab\na c b : Ordinal.{u_1}\nha : 0 < a\nhc : 0 < c\nhca : c ≤ a ^ ω\n⊢ a ^ ω * b + c ≤ a ^ ω * succ b",
"case a.h\na c b : Ordinal.{u_1}\nha : 0 < a\nhc : 0 < c\nhca : c ≤ a ^ ω\n⊢ a * (a ^ ω * succ b) ≤ a ^ ω * succ b"
] | apply nfp_le_fp (isNormal_mul_right ha).monotone | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 14
} | {
"line": 173,
"column": 0
} | [
{
"pp": "a b : Cardinal.{u_1}\nh : b ≠ 0\n⊢ a ≤ b * a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Cardinal.instOne",
"Cardinal",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
... | [] | convert! mul_le_mul_left (Cardinal.one_le_iff_ne_zero.mpr h) a
rw [one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 14
} | {
"line": 173,
"column": 0
} | [
{
"pp": "a b : Cardinal.{u_1}\nh : b ≠ 0\n⊢ a ≤ b * a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Cardinal.instOne",
"Cardinal",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
... | [] | convert! mul_le_mul_left (Cardinal.one_le_iff_ne_zero.mpr h) a
rw [one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 438,
"column": 2
} | {
"line": 441,
"column": 22
} | {
"line": 443,
"column": 0
} | [
{
"pp": "case inr.inr\na b c : Ordinal.{u}\nc0 : 0 < c\nha : a < ω ^ c\nhb : b < ω\nl : IsSuccLimit c\n⊢ a * b < ω ^ c",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Ordinal.mulRightMono",
"Preorder.toLT",
"HMul.hMul",... | [] | · rcases ((isNormal_opow one_lt_omega0).lt_iff_exists_lt l).1 ha with ⟨x, hx, ax⟩
refine (mul_le_mul' (le_of_lt ax) (le_of_lt hb)).trans_lt ?_
rw [← opow_succ, opow_lt_opow_iff_right one_lt_omega0]
exact l.succ_lt hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 662,
"column": 4
} | {
"line": 662,
"column": 15
} | {
"line": 663,
"column": 4
} | [
{
"pp": "case refine_1\no : Ordinal.{u_1}\n⊢ (preBeth o).IsStrongPrelimit → IsSuccPrelimit o",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Order.IsSuccPrelimit",
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"Cardinal.preBeth",
... | [
"case refine_1\no : Ordinal.{u_1}\n⊢ ¬IsSuccPrelimit o → ¬(preBeth o).IsStrongPrelimit"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 91,
"column": 50
} | {
"line": 91,
"column": 68
} | {
"line": 92,
"column": 2
} | [
{
"pp": "a b o : Ordinal.{u_1}\nf : ↑(Iio a) → ↑(Iio o)\ng : ↑(Iio b) → ↑(Iio a)\nhf : IsFundamentalSeq f\nhg : IsFundamentalSeq g\n⊢ a.cof.ord ≤ a",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Ordinal.ord_cof_le"
],
"usedFVars": [
"a"
],
"usedGoals": []
}
... | [] | exact a.ord_cof_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 133,
"column": 11
} | {
"line": 133,
"column": 24
} | {
"line": 133,
"column": 24
} | [
{
"pp": "case add_one\na : Ordinal.{u_1}\nha : ω ≤ a\nb : Ordinal.{u_1}\nIH : (a ^ b).card ≤ max a.card b.card\n⊢ max a.card b.card ≤ max a.card (b.card + 1)",
"ppTerm": "?add_one",
"assigned": true,
"usedConstants": [
"le_refl",
"Lattice.toSemilatticeSup",
"sup_le_sup",
"Car... | [
"case add_one\na : Ordinal.{u_1}\nha : ω ≤ a\nb : Ordinal.{u_1}\nIH : (a ^ b).card ≤ max a.card b.card\n⊢ max a.card b.card ≤ max a.card b.card"
] | ← le_self_add | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 483,
"column": 18
} | {
"line": 483,
"column": 29
} | {
"line": 483,
"column": 30
} | [
{
"pp": "case mp\na b c : Cardinal.{u_1}\nhc : c < ℵ₀\n⊢ a + c < b + c → a < b",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"Cardinal.instAdd",... | [
"case mp\na b c : Cardinal.{u_1}\nhc : c < ℵ₀\n⊢ b ≤ a → b + c ≤ a + c"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 483,
"column": 18
} | {
"line": 483,
"column": 29
} | {
"line": 483,
"column": 30
} | [
{
"pp": "case mpr\na b c : Cardinal.{u_1}\nhc : c < ℵ₀\n⊢ a < b → a + c < b + c",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"LE.le",
"Cardinal.instAdd... | [
"case mpr\na b c : Cardinal.{u_1}\nhc : c < ℵ₀\n⊢ b + c ≤ a + c → b ≤ a"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 22
} | {
"line": 216,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\nh : o.IsInitial\nho : ω ≤ o\n⊢ IsPrincipal (fun x1 x2 ↦ x1 + x2) o",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.isPrincipal_add_ord",
"Ordinal.IsPrincipal",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Cardi... | [] | rw [← h.ord_card]
apply isPrincipal_add_ord
rwa [aleph0_le_card] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 212,
"column": 2
} | {
"line": 214,
"column": 22
} | {
"line": 216,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\nh : o.IsInitial\nho : ω ≤ o\n⊢ IsPrincipal (fun x1 x2 ↦ x1 + x2) o",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.isPrincipal_add_ord",
"Ordinal.IsPrincipal",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Cardi... | [] | rw [← h.ord_card]
apply isPrincipal_add_ord
rwa [aleph0_le_card] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 136,
"column": 9
} | {
"line": 136,
"column": 29
} | {
"line": 136,
"column": 30
} | [
{
"pp": "case le_cof_ord\nκ : Cardinal.{v}\nh₁ : ℵ₀ ≤ κ\nh₂ : κ ≤ κ.ord.cof\n⊢ Cardinal.lift.{u, v} κ ≤ (Cardinal.lift.{u, v} κ).ord.cof",
"ppTerm": "?le_cof_ord",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.lift",
"Ordinal.lift",
... | [
"case le_cof_ord\nκ : Cardinal.{v}\nh₁ : ℵ₀ ≤ κ\nh₂ : κ ≤ κ.ord.cof\n⊢ Cardinal.lift.{u, v} κ ≤ (Ordinal.lift.{u, v} κ.ord).cof"
] | ← Cardinal.lift_ord, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 199,
"column": 74
} | {
"line": 202,
"column": 41
} | {
"line": 204,
"column": 0
} | [
{
"pp": "c : Cardinal.{max u v}\nι : Type u\nf : ι → Cardinal.{max u v}\nhc : c.IsRegular\nhι : lift.{v, u} #ι < c\nhf : ∀ (i : ι), f i < c\n⊢ sum f < c",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"Cardinal",
"congrAr... | [] | by
apply (sum_le_lift_mk_mul_iSup _).trans_lt <|
mul_lt_of_lt hc.1 hι (lift_iSup_lt_of_lt_cof_ord _ hf)
rwa [lift_umax, c.lift_id', hc.cof_ord] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 666,
"column": 24
} | {
"line": 666,
"column": 55
} | {
"line": 666,
"column": 56
} | [
{
"pp": "α : Type u\ninst✝ : Infinite α\ne : α ≃ α × Bool\nthis : 2 ^ #α ≤ #(Perm (α × Bool))\n⊢ #α ^ #α ≤ #(α ≃ α)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.instPowCardinal",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"... | [
"α : Type u\ninst✝ : Infinite α\ne : α ≃ α × Bool\nthis : 2 ^ #α ≤ #(Perm (α × Bool))\n⊢ 2 ^ #α ≤ #(α ≃ α)"
] | power_self_eq (aleph0_le_mk α), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 354,
"column": 24
} | {
"line": 354,
"column": 50
} | {
"line": 354,
"column": 50
} | [
{
"pp": "case inr.inr\no : Ordinal.{u_1}\nho : IsSuccLimit o\n⊢ ℵ₀ ≤ ℵ_ o ∧ (ℵ_ o).ord.cof ≠ ℵ_ o ↔ IsSuccLimit o ∧ o.cof < ℵ_ o",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal.aleph",
"Ordinal.partialOrder",
"Cardinal",... | [
"case inr.inr\no : Ordinal.{u_1}\nho : IsSuccLimit o\n⊢ ℵ₀ ≤ ℵ_ o ∧ (ℵ_ o).ord.cof < ℵ_ o ↔ IsSuccLimit o ∧ o.cof < ℵ_ o"
] | ← (cof_ord_le _).lt_iff_ne | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.DFinsupp.Defs | {
"line": 490,
"column": 4
} | {
"line": 490,
"column": 24
} | {
"line": 491,
"column": 4
} | [
{
"pp": "case mpr.inl\nι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\ni : ι\nxi xj : β i\nhxi : xi ≍ xj\n⊢ single i xi = single i xj",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"DFinsupp.single",
"id",
... | [
"case mpr.inr\nι : Type u\nβ : ι → Type v\ninst✝¹ : (i : ι) → Zero (β i)\ninst✝ : DecidableEq ι\ni j : ι\nxi : β i\nxj : β j\nhi : xi = 0\nhj : xj = 0\n⊢ single i xi = single j xj"
] | · rw [eq_of_heq hxi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 10
} | {
"line": 96,
"column": 11
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : Π₀ (i : ι), β i\ng : (i : ι) → β i → γ\ns : Finset ι\nhs : f.support ⊆ s\nmap_zero : ∀ i ∈ s, g i 0 = 1\ni : ι\nhi : i ∈ s\nhi' : f ... | [
"ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → Zero (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : CommMonoid γ\nf : Π₀ (i : ι), β i\ng : (i : ι) → β i → γ\ns : Finset ι\nhs : f.support ⊆ s\nmap_zero : ∀ i ∈ s, g i 0 = 1\ni : ι\nhi : i ∈ s\nhi' : f i = 0\n⊢ g i... | hi', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 466,
"column": 71
} | {
"line": 470,
"column": 15
} | {
"line": 472,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set (Dual R M)\n⊢ ↑(span R s).dualCoannihilator = {x | ∀ f ∈ s, f x = 0}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"Submodule",
... | [] | by
ext x
have (φ : _) : x ∈ LinearMap.ker φ ↔ φ ∈ LinearMap.ker (Module.Dual.eval R M x) := by simp
simp only [SetLike.mem_coe, mem_dualCoannihilator, Set.mem_ofPred_eq, ← LinearMap.mem_ker, this]
exact span_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 260,
"column": 29
} | {
"line": 260,
"column": 48
} | {
"line": 260,
"column": 48
} | [
{
"pp": "case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\nh : ∀ (c d : R), c • f i + d • f j = 0 → c = 0 ∧ d = 0\nt : Finset ι\ng : ι → R\nht : ↑t ⊆ {i, j}\nhg0 : ∀ i ∉ t, g i = 0\nh0 : ∑ x ∈ {i, j}, g x • f x ... | [
"case refine_2\nι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\nh : ∀ (c d : R), c • f i + d • f j = 0 → c = 0 ∧ d = 0\nt : Finset ι\ng : ι → R\nht : ↑t ⊆ {i, j}\nhg0 : ∀ i ∉ t, g i = 0\nh0 : g i • f i + g j • f j = 0\nht' : t ⊆... | Finset.sum_pair hij | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 568,
"column": 2
} | {
"line": 568,
"column": 90
} | {
"line": 569,
"column": 2
} | [
{
"pp": "ι : Type u_1\nN : Type u_6\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommGroup N\np : ι → AddSubgroup N\nh : iSupIndep p\n⊢ Function.Injective ⇑(sumAddHom fun i ↦ (p i).subtype)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Submodule",
"AddSubgroup.instCompleteLattice",
... | [
"ι : Type u_1\nN : Type u_6\ninst✝¹ : DecidableEq ι\ninst✝ : AddCommGroup N\np : ι → AddSubgroup N\nh : iSupIndep (⇑AddSubgroup.toIntSubmodule ∘ p)\n⊢ Function.Injective ⇑(sumAddHom fun i ↦ (p i).subtype)"
] | rw [← iSupIndep_map_orderIso_iff (AddSubgroup.toIntSubmodule : AddSubgroup N ≃o _)] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Tactic.Module | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 49
} | {
"line": 87,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr₁ r₂ : R\nx : M\nl₁ l₂ l : NF R M\nh : l₁.eval + l₂.eval = l.eval\n⊢ ((r₁, x) ::ᵣ l₁).eval + ((r₂, x) ::ᵣ l₂).eval = ((r₁ + r₂, x) ::ᵣ l).eval",
"ppTerm": "?m.59",
"assigned": true,
"usedConstant... | [
"R : Type u_2\nM : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr₁ r₂ : R\nx : M\nl₁ l₂ l : NF R M\nh : l₁.eval + l₂.eval = l.eval\n⊢ r₁ • x + (l₁.eval + (r₂ • x + l₂.eval)) = r₁ • x + (r₂ • x + (l₁.eval + l₂.eval))"
] | simp only [← h, eval_cons, add_smul, add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 336,
"column": 20
} | {
"line": 336,
"column": 65
} | {
"line": 337,
"column": 4
} | [
{
"pp": "R : Type u_2\nM : Type u_4\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : Module R M\nx y : M\nS : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Module S R\ninst✝³ : Module S M\ninst✝² : SMulCommClass S R M\ninst✝¹ : IsScalarTower S R M\ninst✝ : IsTorsionFree S R\na b c d : S\nh : a... | [] | exact h (_root_.smul_left_injective S ht ‹_›) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.FreeAbelianGroup.Finsupp | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 13
} | [
{
"pp": "X : Type u_1\na : FreeAbelianGroup X\n⊢ a.support.Nonempty ↔ a ≠ 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"FreeAbelianGroup.support",
"congrArg",
"Finset",
"id",
"SubtractionMonoid.toSubNegZeroMonoid",
"SubNegZeroMonoid.t... | [
"X : Type u_1\na : FreeAbelianGroup X\n⊢ a.support = ∅ ↔ a = 0"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 664,
"column": 57
} | {
"line": 664,
"column": 83
} | {
"line": 666,
"column": 0
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx y : V\nhx : x ≠ 0\nhy : ∀ (a : K), a • x ≠ y\n⊢ y ∉ K ∙ x",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Submodule",
"instHSM... | [] | simpa [mem_span_singleton] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 664,
"column": 57
} | {
"line": 664,
"column": 83
} | {
"line": 666,
"column": 0
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx y : V\nhx : x ≠ 0\nhy : ∀ (a : K), a • x ≠ y\n⊢ y ∉ K ∙ x",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Submodule",
"instHSM... | [] | simpa [mem_span_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 664,
"column": 57
} | {
"line": 664,
"column": 83
} | {
"line": 666,
"column": 0
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nx y : V\nhx : x ≠ 0\nhy : ∀ (a : K), a • x ≠ y\n⊢ y ∉ K ∙ x",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"not_exists._simp_1",
"Submodule",
"instHSM... | [] | simpa [mem_span_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.NAry | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 27
} | {
"line": 77,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns s' : Finset α\nt t' : Finset β\nhs : s ⊆ s'\nht : t ⊆ t'\n⊢ image₂ f s t ⊆ image₂ f s' t'",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image2_subset",
"congrArg",
... | [] | rw [← coe_subset, coe_image₂, coe_image₂]
exact image2_subset hs ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NAry | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 27
} | {
"line": 77,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns s' : Finset α\nt t' : Finset β\nhs : s ⊆ s'\nht : t ⊆ t'\n⊢ image₂ f s t ⊆ image₂ f s' t'",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.image2_subset",
"congrArg",
... | [] | rw [← coe_subset, coe_image₂, coe_image₂]
exact image2_subset hs ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.NAry | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 64
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\nu : Finset γ\n⊢ image₂ f s t ⊆ u ↔ ∀ b ∈ t, image (fun a ↦ f a b) s ⊆ u",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
... | [] | simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_comm α] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Finset.NAry | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 64
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\nu : Finset γ\n⊢ image₂ f s t ⊆ u ↔ ∀ b ∈ t, image (fun a ↦ f a b) s ⊆ u",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
... | [] | simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_comm α] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NAry | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 64
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\nu : Finset γ\n⊢ image₂ f s t ⊆ u ↔ ∀ b ∈ t, image (fun a ↦ f a b) s ⊆ u",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
... | [] | simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_comm α] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.NAry | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 13
} | {
"line": 132,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ image₂ f s t = ∅ ↔ s = ∅ ∨ t = ∅",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mathlib.Tactic.Contrapose.contrapose_iff₁",
"congrArg",
... | [
"α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ (image₂ f s t).Nonempty ↔ s.Nonempty ∧ t.Nonempty"
] | contrapose! | Mathlib.Tactic.Contrapose._aux_Mathlib_Tactic_Contrapose___macroRules_Mathlib_Tactic_Contrapose_contrapose!_1 | Mathlib.Tactic.Contrapose.contrapose! |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 531,
"column": 2
} | {
"line": 542,
"column": 34
} | {
"line": 544,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Mul M\nx y : R[M]\nm : M\ns : Finset (M × M)\nhs : ∀ {p : M × M}, p ∈ s ↔ p.1 * p.2 = m\nF : M × M → R := fun p ↦ if p.1 * p.2 = m then x.coeff p.1 * y.coeff p.2 else 0\n⊢ (x * y).coeff m = ∑ p ∈ s, x.coeff p.1 * y.coeff p.2",
"ppTerm": "?m.8... | [] | calc
(x * y).coeff m = ∑ m₁ ∈ x.coeff.support, ∑ m₂ ∈ y.coeff.support, F (m₁, m₂) := coeff_mul ..
_ = ∑ p ∈ x.coeff.support ×ˢ y.coeff.support with p.1 * p.2 = m, x.coeff p.1 * y.coeff p.2 := by
rw [Finset.sum_filter, Finset.sum_product]
_ = ∑ p ∈ s with p.1 ∈ x.coeff.support ∧ p.2 ∈ y.coeff.support, ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 673,
"column": 79
} | {
"line": 674,
"column": 59
} | {
"line": 676,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : MulOneClass M\ninst✝ : Nontrivial R\na b : M\nh : (of R M) a = (of R M) b\n⊢ a = b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Algebra.MonoidAlgebra.Defs.0.MonoidAlgebra.of_injective._simp_1_1",
... | [] | by
simpa [← coeff_inj, Finsupp.single_eq_single_iff] using h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 738,
"column": 14
} | {
"line": 738,
"column": 54
} | {
"line": 740,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\ninst✝ : Monoid M\nmotive : R[M] → Prop\nx : R[M]\nof : ∀ (m : M), motive ((MonoidAlgebra.of R M) m)\nadd : ∀ (x y : R[M]), motive x → motive y → motive (x + y)\nsmul : ∀ (r : R) (x : R[M]), motive x → motive (r • x)\nm : M\nr : R\n⊢ motive (ofCoeff (Fins... | [] | by simpa using smul r (.of R M m) (of m) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 783,
"column": 66
} | {
"line": 783,
"column": 86
} | {
"line": 785,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nm : M\nn : N\nr : R\n⊢ curryAddEquiv (single (m, n) r) = single m (single n r)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Finsupp.instAdd... | [] | simp [curryAddEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 783,
"column": 66
} | {
"line": 783,
"column": 86
} | {
"line": 785,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nm : M\nn : N\nr : R\n⊢ curryAddEquiv (single (m, n) r) = single m (single n r)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Finsupp.instAdd... | [] | simp [curryAddEquiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 783,
"column": 66
} | {
"line": 783,
"column": 86
} | {
"line": 785,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nm : M\nn : N\nr : R\n⊢ curryAddEquiv (single (m, n) r) = single m (single n r)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Finsupp.instAdd... | [] | simp [curryAddEquiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 787,
"column": 74
} | {
"line": 787,
"column": 94
} | {
"line": 789,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nm : M\nn : N\nr : R\n⊢ curryAddEquiv.symm (single m (single n r)) = single (m, n) r",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Finsupp.in... | [] | simp [curryAddEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 787,
"column": 74
} | {
"line": 787,
"column": 94
} | {
"line": 789,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nm : M\nn : N\nr : R\n⊢ curryAddEquiv.symm (single m (single n r)) = single (m, n) r",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Finsupp.in... | [] | simp [curryAddEquiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 787,
"column": 74
} | {
"line": 787,
"column": 94
} | {
"line": 789,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\nN : Type u_5\ninst✝² : Semiring R\ninst✝¹ : Monoid M\ninst✝ : Monoid N\nm : M\nn : N\nr : R\n⊢ curryAddEquiv.symm (single m (single n r)) = single (m, n) r",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"MonoidAlgebra.semiring",
"Finsupp.in... | [] | simp [curryAddEquiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 843,
"column": 12
} | {
"line": 843,
"column": 78
} | {
"line": 845,
"column": 0
} | [
{
"pp": "R : Type u_1\nG : Type u_3\ninst✝¹ : Semiring R\ninst✝ : Group G\nx y : R[G]\ng : G\n⊢ (x * y).coeff g = x.coeff.sum fun h r ↦ r * y.coeff (h⁻¹ * g)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"instDecidableNot",
"Monoid... | [] | rw [coeff_mul]; gcongr; simp +contextual [← eq_inv_mul_iff_mul_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 843,
"column": 12
} | {
"line": 843,
"column": 78
} | {
"line": 845,
"column": 0
} | [
{
"pp": "R : Type u_1\nG : Type u_3\ninst✝¹ : Semiring R\ninst✝ : Group G\nx y : R[G]\ng : G\n⊢ (x * y).coeff g = x.coeff.sum fun h r ↦ r * y.coeff (h⁻¹ * g)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"instDecidableNot",
"Monoid... | [] | rw [coeff_mul]; gcongr; simp +contextual [← eq_inv_mul_iff_mul_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 985,
"column": 98
} | {
"line": 986,
"column": 59
} | {
"line": 988,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : AddZeroClass M\na b : Multiplicative M\nh : (of R M) a = (of R M) b\n⊢ a = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Monoid... | [] | by
simpa [← coeff_inj, Finsupp.single_eq_single_iff] using h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1003,
"column": 4
} | {
"line": 1006,
"column": 10
} | {
"line": 1007,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nn : ℤ\n⊢ s ^ n = ∅ → s = ∅ ∧ n ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Fi... | [] | contrapose! +distrib
rintro (hs | rfl)
· exact hs.zpow
· simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1003,
"column": 4
} | {
"line": 1006,
"column": 10
} | {
"line": 1007,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nn : ℤ\n⊢ s ^ n = ∅ → s = ∅ ∧ n ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Fi... | [] | contrapose! +distrib
rintro (hs | rfl)
· exact hs.zpow
· simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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