module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.Set.Card | {
"line": 1222,
"column": 2
} | {
"line": 1222,
"column": 69
} | {
"line": 1224,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nn : ℕ\nhns : n ≤ s.ncard\n⊢ ∃ t ⊆ s, t.ncard = n",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"congrArg",
"Set.exists_subsuperset_card_eq",
"zero_l... | [] | simpa using exists_subsuperset_card_eq s.empty_subset (by simp) hns | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coset.Basic | {
"line": 442,
"column": 27
} | {
"line": 442,
"column": 62
} | {
"line": 442,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝ : Group α\ns t : Subgroup α\nι : Type u_2\nf : ι → Subgroup α\nH : Subgroup α\n⊢ ∀ (a₁ a₂ : ↥H),\n (∀ (x : ι), quotientSubgroupOfMapOfLE H ⋯ (Quotient.mk'' a₁) = quotientSubgroupOfMapOfLE H ⋯ (Quotient.mk'' a₂)) →\n Quotient.mk'' a₁ = Quotient.mk'' a₂",
"ppTerm": "?m.38"... | [
"α : Type u_1\ninst✝ : Group α\ns t : Subgroup α\nι : Type u_2\nf : ι → Subgroup α\nH : Subgroup α\n⊢ ∀ (a₁ a₂ : ↥H), (∀ (x : ι), ↑a₁ = ↑a₂) → Quotient.mk'' a₁ = Quotient.mk'' a₂"
] | quotientSubgroupOfMapOfLE_apply_mk, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.GroupTheory.Congruence.Basic | {
"line": 223,
"column": 29
} | {
"line": 223,
"column": 54
} | {
"line": 223,
"column": 54
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : RightInverse g ⇑f\nx : P\n⊢ (kerLift f) ((toQuotient ∘ g) x) = x",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
... | [
"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : MulOneClass M\ninst✝¹ : MulOneClass N\ninst✝ : MulOneClass P\nc : Con M\nx✝ y : M\nf✝ f : M →* P\ng : P → M\nhf : RightInverse g ⇑f\nx : P\n⊢ (kerLift f) ((toQuotient ∘ g) x) = f (g x)"
] | (conv_rhs => rw [← hf x]) | Lean.Elab.Tactic.evalParen | Lean.Parser.Tactic.paren |
Mathlib.LinearAlgebra.Pi | {
"line": 212,
"column": 2
} | {
"line": 215,
"column": 30
} | {
"line": 217,
"column": 0
} | [
{
"pp": "R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nhd : Disjoint I J\nhu : Set.univ ⊆ I ∪ J\nhI : I.Finite\n⊢ ⨆ i ∈ I, (single R φ i).range = ⨅ i ∈ J, (proj i).ker",
"ppTerm": "?... | [] | refine le_antisymm (iSup_range_single_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_single R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Pi | {
"line": 212,
"column": 2
} | {
"line": 215,
"column": 30
} | {
"line": 217,
"column": 0
} | [
{
"pp": "R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nhd : Disjoint I J\nhu : Set.univ ⊆ I ∪ J\nhI : I.Finite\n⊢ ⨆ i ∈ I, (single R φ i).range = ⨅ i ∈ J, (proj i).ker",
"ppTerm": "?... | [] | refine le_antisymm (iSup_range_single_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_single R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Pi | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 40
} | {
"line": 236,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nb : (i : ι) → φ i\nhI : ∀ i ∈ Iᶜ, b i = 0\nhJ : ∀ i ∈ Jᶜ, b i = 0\ni : ι\nhiI : i ∈ I\nhiJ : i ∈ J\n⊢ b... | [] | exact (h.le_bot ⟨hiI, hiJ⟩).elim | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Pi | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 40
} | {
"line": 236,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nb : (i : ι) → φ i\nhI : ∀ i ∈ Iᶜ, b i = 0\nhJ : ∀ i ∈ Jᶜ, b i = 0\ni : ι\nhiI : i ∈ I\nhiJ : i ∈ J\n⊢ b... | [] | exact (h.le_bot ⟨hiI, hiJ⟩).elim | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Pi | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 40
} | {
"line": 236,
"column": 6
} | [
{
"pp": "case pos\nR : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nb : (i : ι) → φ i\nhI : ∀ i ∈ Iᶜ, b i = 0\nhJ : ∀ i ∈ Jᶜ, b i = 0\ni : ι\nhiI : i ∈ I\nhiJ : i ∈ J\n⊢ b... | [] | exact (h.le_bot ⟨hiI, hiJ⟩).elim | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.SuccPred | {
"line": 159,
"column": 37
} | {
"line": 159,
"column": 54
} | {
"line": 159,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : PredOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Icc a b",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Lattice.toSemilatticeSup",
"congrArg",
"Set.Ioc_insert_l... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : PredOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ Icc a b = Icc a b"
] | Ioc_insert_left h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 421,
"column": 73
} | {
"line": 421,
"column": 87
} | {
"line": 421,
"column": 87
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' P Q : Submodule R M\nhf : map (↑(LinearEquiv.refl R M)) P = Q\n⊢ P = Q",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 421,
"column": 73
} | {
"line": 421,
"column": 87
} | {
"line": 421,
"column": 87
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' P Q : Submodule R M\nhf : map (↑(LinearEquiv.refl R M)) P = Q\n⊢ P = Q",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 421,
"column": 73
} | {
"line": 421,
"column": 87
} | {
"line": 421,
"column": 87
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nr : R\nx y : M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\np p' P Q : Submodule R M\nhf : map (↑(LinearEquiv.refl R M)) P = Q\n⊢ P = Q",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjective.ids",
... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 213,
"column": 7
} | {
"line": 213,
"column": 74
} | {
"line": 215,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\nM₂ : Type w\nM₃ : Type y\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M₂\ninst✝³ : AddCommMonoid M₃\ninst✝² : Module R M\ninst✝¹ : Module R M₂\ninst✝ : Module R M₃\nf : M →ₗ[R] M₃\ng : M₂ →ₗ[R] M₃\nx✝ : M₂\n⊢ (f.coprod g ∘ₗ inr R M M₂) x✝ = g x✝",
"... | [] | simp only [map_zero, coprod_apply, inr_apply, zero_add, comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 70
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Order.succ",
"Order.succ_eq_add_one",
... | [] | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 70
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Order.succ",
"Order.succ_eq_add_one",
... | [] | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 70
} | {
"line": 58,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Add α\ninst✝ : SuccAddOrder α\na : α\nha : ¬IsMax a\nb : α\n⊢ Icc (a + 1) b = Ioc a b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Order.succ",
"Order.succ_eq_add_one",
... | [] | simpa [succ_eq_add_one] using Icc_succ_left_eq_Ioc_of_not_isMax ha b | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 31
} | {
"line": 134,
"column": 2
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : LinearOrder ι\ninst✝ : LocallyFiniteOrder ι\ni : ι\nhi : succFn i ≤ i\nj : ι\nx✝ : i ≤ j\nhij_lt : i < j\nh_succFn_eq : succFn i = i\nh_glb : IsGLB (↑(Finset.Ioc i j)) i\nhi_mem : i ∈ Finset.Ioc i j\n⊢ False",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\ninst✝¹ : LinearOrder ι\ninst✝ : LocallyFiniteOrder ι\ni : ι\nhi : succFn i ≤ i\nj : ι\nx✝ : i ≤ j\nhij_lt : i < j\nh_succFn_eq : succFn i = i\nh_glb : IsGLB (↑(Finset.Ioc i j)) i\nhi_mem : i < i ∧ i ≤ j\n⊢ False"
] | rw [Finset.mem_Ioc] at hi_mem | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Disjointed | {
"line": 238,
"column": 2
} | {
"line": 245,
"column": 42
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : SuccOrder ι\nf : ι → α\nhf : Monotone f\ni : ι\n⊢ disjointed f (succ i) ⊔ f i = f (succ i)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"par... | [] | by_cases h : IsMax i
· simpa only [succ_eq_iff_isMax.mpr h, sup_eq_right] using disjointed_le f i
· rw [disjointed_apply]
have : Iio (succ i) = Iic i := by
ext
simp only [mem_Iio, lt_succ_iff_eq_or_lt_of_not_isMax h, mem_Iic, le_iff_lt_or_eq, Or.comm]
rw [this, ← sup'_eq_sup nonempty_Iic, ← part... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Disjointed | {
"line": 238,
"column": 2
} | {
"line": 245,
"column": 42
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : SuccOrder ι\nf : ι → α\nhf : Monotone f\ni : ι\n⊢ disjointed f (succ i) ⊔ f i = f (succ i)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"par... | [] | by_cases h : IsMax i
· simpa only [succ_eq_iff_isMax.mpr h, sup_eq_right] using disjointed_le f i
· rw [disjointed_apply]
have : Iio (succ i) = Iic i := by
ext
simp only [mem_Iio, lt_succ_iff_eq_or_lt_of_not_isMax h, mem_Iic, le_iff_lt_or_eq, Or.comm]
rw [this, ← sup'_eq_sup nonempty_Iic, ← part... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 232,
"column": 2
} | {
"line": 232,
"column": 52
} | {
"line": 234,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i : ι\nhi : i0 ≤ i\n⊢ succ^[Nat.find ⋯] i0 = i",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Order.succ",
"LinearOrder.toDecidableEq",
"PartialOrd... | [] | exact Nat.find_spec (exists_succ_iterate_of_le hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 22
} | {
"line": 107,
"column": 22
} | [
{
"pp": "A : Type u_1\ninst✝¹ : AddCancelMonoid A\ninst✝ : HasAntidiagonal A\np q : A × A\nn : A\nhp : p ∈ antidiagonal n\nhq : q ∈ antidiagonal n\nh : p.1 = q.1\n⊢ q.1 + p.2 = q.1 + q.2",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
... | [
"A : Type u_1\ninst✝¹ : AddCancelMonoid A\ninst✝ : HasAntidiagonal A\np q : A × A\nn : A\nhp : p.1 + p.2 = n\nhq : q.1 + q.2 = n\nh : p.1 = q.1\n⊢ q.1 + p.2 = q.1 + q.2"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 158,
"column": 51
} | {
"line": 161,
"column": 70
} | {
"line": 162,
"column": 2
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : CanonicallyOrderedAdd A\ninst✝⁵ : Sub A\ninst✝⁴ : OrderedSub A\ninst✝³ : AddLeftReflectLE A\ninst✝² : HasAntidiagonal A\nn m : A\ninst✝¹ : DecidablePred fun x ↦ x = m\ninst✝ : Decidable (m ≤ n)\na b : A\nthis : a = m → (a + b = n... | [] | by
rw [mem_filter, mem_antidiagonal, apply_ite (fun n ↦ (a, b) ∈ n), mem_singleton,
Prod.mk_inj, ite_prop_iff_or]
simpa [← and_assoc, @and_right_comm _ (a = _), and_congr_left_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 30
} | {
"line": 171,
"column": 31
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : CanonicallyOrderedAdd A\ninst✝⁵ : Sub A\ninst✝⁴ : OrderedSub A\ninst✝³ : AddLeftReflectLE A\ninst✝² : HasAntidiagonal A\nn m : A\ninst✝¹ : DecidablePred fun x ↦ x = m\ninst✝ : Decidable (m ≤ n)\n⊢ {x ∈ antidiagonal n | x.2 = m} =... | [
"A : Type u_1\ninst✝⁸ : AddCommMonoid A\ninst✝⁷ : PartialOrder A\ninst✝⁶ : CanonicallyOrderedAdd A\ninst✝⁵ : Sub A\ninst✝⁴ : OrderedSub A\ninst✝³ : AddLeftReflectLE A\ninst✝² : HasAntidiagonal A\nn m : A\ninst✝¹ : DecidablePred fun x ↦ x = m\ninst✝ : Decidable (m ≤ n)\n⊢ {x ∈ map { toFun := Prod.swap, inj' := ⋯ } (... | ← map_swap_antidiagonal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 91
} | {
"line": 73,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nhab : a ≤ b\nf : ℕ → M\n⊢ ∏ k ∈ Ioc a (b + 1), f k = (∏ k ∈ Ioc a b, f k) * f (b + 1)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Ioc_succ_singleton",
"HMul.hMul",
"Finset.prod_singleton",
... | [] | rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b), Nat.Ioc_succ_singleton, prod_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 91
} | {
"line": 73,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nhab : a ≤ b\nf : ℕ → M\n⊢ ∏ k ∈ Ioc a (b + 1), f k = (∏ k ∈ Ioc a b, f k) * f (b + 1)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Ioc_succ_singleton",
"HMul.hMul",
"Finset.prod_singleton",
... | [] | rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b), Nat.Ioc_succ_singleton, prod_singleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Antidiag.Prod | {
"line": 185,
"column": 8
} | {
"line": 185,
"column": 24
} | {
"line": 185,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\nn k l : A\nh : (k, l) ∈ antidiagonal n\n⊢ (fun x ↦ ⟨x.1 + x.2, ⟨x, ⋯⟩⟩) ((fun x ↦ ↑x.snd) ⟨n, ⟨(k, l), h⟩⟩) = ⟨n, ⟨(k, l), h⟩⟩",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"co... | [
"A : Type u_1\ninst✝¹ : AddMonoid A\ninst✝ : HasAntidiagonal A\nn k l : A\nh✝ : (k, l) ∈ antidiagonal n\nh : (k, l).1 + (k, l).2 = n\n⊢ (fun x ↦ ⟨x.1 + x.2, ⟨x, ⋯⟩⟩) ((fun x ↦ ↑x.snd) ⟨n, ⟨(k, l), h✝⟩⟩) = ⟨n, ⟨(k, l), h✝⟩⟩"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 91
} | {
"line": 73,
"column": 0
} | [
{
"pp": "M : Type u_3\ninst✝ : CommMonoid M\na b : ℕ\nhab : a ≤ b\nf : ℕ → M\n⊢ ∏ k ∈ Ioc a (b + 1), f k = (∏ k ∈ Ioc a b, f k) * f (b + 1)",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.Ioc_succ_singleton",
"HMul.hMul",
"Finset.prod_singleton",
... | [] | rw [← prod_Ioc_consecutive _ hab (Nat.le_succ b), Nat.Ioc_succ_singleton, prod_singleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.NatAntidiagonal | {
"line": 37,
"column": 18
} | {
"line": 37,
"column": 42
} | {
"line": 37,
"column": 43
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p ∈ antidiagonal n, f p.swap",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Finset.map",
"Function.Embedding.mk",
"Finset.prod",
"Finset.Nat.instHasAntidiagona... | [
"M : Type u_1\ninst✝ : CommMonoid M\nn : ℕ\nf : ℕ × ℕ → M\n| ∏ p ∈ map { toFun := Prod.swap, inj' := ⋯ } (antidiagonal n), f p.swap"
] | ← map_swap_antidiagonal, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Data.Finset.NatAntidiagonal | {
"line": 49,
"column": 6
} | {
"line": 49,
"column": 30
} | {
"line": 49,
"column": 31
} | [
{
"pp": "n : ℕ\n⊢ antidiagonal n = map { toFun := fun i ↦ (n - i, i), inj' := ⋯ } (range (n + 1))",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"HSub.hSub",
"Nat.instAddMonoid",
"Finset.map",
"Function.Embedding.mk... | [
"n : ℕ\n⊢ map { toFun := Prod.swap, inj' := ⋯ } (antidiagonal n) =\n map { toFun := fun i ↦ (n - i, i), inj' := ⋯ } (range (n + 1))"
] | ← map_swap_antidiagonal, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Defs | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 41
} | {
"line": 81,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nI : Ideal α\nx y : α\nhy : IsUnit y\nh : y * x ∈ I\ny' : α\nhy' : y' * y = 1\nthis : y' * (y * x) ∈ I\n⊢ x ∈ I",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"Semiring.toModule",
"HMul.hMul... | [] | rwa [← mul_assoc, hy', one_mul] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Data.Finset.NatAntidiagonal | {
"line": 126,
"column": 11
} | {
"line": 126,
"column": 16
} | {
"line": 126,
"column": 17
} | [
{
"pp": "n k : ℕ\nh : k ≤ n\naux₁ : (fun a ↦ a.1 ≤ k) = (fun a ↦ a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm\naux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j\n⊢ {a ∈ map (Equiv.prodComm ℕ ℕ).toEmbedding (antidiagonal n) | a.1 ≤ k} =\n map ((Embedding.re... | [
"n k : ℕ\nh : k ≤ n\naux₁ : (fun a ↦ a.1 ≤ k) = (fun a ↦ a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm\naux₂ : ∀ (i j : ℕ), (∃ a b, a + b = k ∧ b = i ∧ a + (n - k) = j) ↔ ∃ a b, a + b = k ∧ a = i ∧ b + (n - k) = j\n⊢ filter ((fun a ↦ a.2 ≤ k) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm) (map (Equiv.prodComm ℕ ℕ).toEmbedding (antidiagonal... | aux₁, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.NatAntidiagonal | {
"line": 150,
"column": 11
} | {
"line": 150,
"column": 16
} | {
"line": 150,
"column": 17
} | [
{
"pp": "n k : ℕ\nh : k ≤ n\naux₁ : (fun a ↦ k ≤ a.2) = (fun a ↦ k ≤ a.1) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm\naux₂ : ∀ (i j : ℕ), (∃ a b, a + b = n - k ∧ b = i ∧ a + k = j) ↔ ∃ a b, a + b = n - k ∧ a = i ∧ b + k = j\n⊢ {a ∈ map (Equiv.prodComm ℕ ℕ).toEmbedding (antidiagonal n) | k ≤ a.2} =\n map ((Embedding.refl ℕ... | [
"n k : ℕ\nh : k ≤ n\naux₁ : (fun a ↦ k ≤ a.2) = (fun a ↦ k ≤ a.1) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm\naux₂ : ∀ (i j : ℕ), (∃ a b, a + b = n - k ∧ b = i ∧ a + k = j) ↔ ∃ a b, a + b = n - k ∧ a = i ∧ b + k = j\n⊢ filter ((fun a ↦ k ≤ a.1) ∘ ⇑(Equiv.prodComm ℕ ℕ).symm) (map (Equiv.prodComm ℕ ℕ).toEmbedding (antidiagonal n))... | aux₁, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Choose.Sum | {
"line": 52,
"column": 47
} | {
"line": 52,
"column": 56
} | {
"line": 52,
"column": 57
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\nh_eq : i = n\n⊢ x ^ n.succ * y ^ (n.succ... | [
"case pos\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\nh_eq : i = n\n⊢ 0 = x ^ n.succ * y * y ^ (n - n.succ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Idempotent | {
"line": 68,
"column": 40
} | {
"line": 68,
"column": 48
} | {
"line": 68,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\na b : R\nmul : a * b = 0\nadd : a + b = 1\n| a * (a + b)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"HMul.hMul",
"congrArg",
"MulOne.toMul",
"Distrib.toAdd",
"MulZeroOneClass.toM... | [
"R : Type u_1\ninst✝ : Semiring R\na b : R\nmul : a * b = 0\nadd : a + b = 1\n| a * a + a * b"
] | mul_add, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Algebra.Ring.Idempotent | {
"line": 96,
"column": 29
} | {
"line": 96,
"column": 37
} | {
"line": 96,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\na b : R\nha : IsIdempotentElem a\nhb : IsIdempotentElem b\nhab : a * b + b * a = 0\n⊢ (a + b) * (a + b) = a + b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocSemiring R\na b : R\nha : IsIdempotentElem a\nhb : IsIdempotentElem b\nhab : a * b + b * a = 0\n⊢ (a + b) * a + (a + b) * b = a + b"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Choose.Sum | {
"line": 64,
"column": 58
} | {
"line": 64,
"column": 67
} | {
"line": 64,
"column": 68
} | [
{
"pp": "case succ.e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nh_middle : ∀ (n i : ℕ), i ∈ range n.succ → t n.succ i.succ = x * t n i + y * t n i.succ\nn :... | [
"case succ.e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nh_middle : ∀ (n i : ℕ), i ∈ range n.succ → t n.succ i.succ = x * t n i + y * t n i.succ\nn : ℕ\nih : (x ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Idempotent | {
"line": 118,
"column": 54
} | {
"line": 118,
"column": 63
} | {
"line": 118,
"column": 64
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝¹ : NonUnitalSemiring R\ninst✝ : IsAddTorsionFree R\nha : IsIdempotentElem a\nhab : a * b + b * a = 0\n⊢ a * 0 * a = 0",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
... | [
"R : Type u_1\na b : R\ninst✝¹ : NonUnitalSemiring R\ninst✝ : IsAddTorsionFree R\nha : IsIdempotentElem a\nhab : a * b + b * a = 0\n⊢ 0 * a = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Idempotent | {
"line": 119,
"column": 13
} | {
"line": 119,
"column": 21
} | {
"line": 119,
"column": 22
} | [
{
"pp": "R : Type u_1\na b : R\ninst✝¹ : NonUnitalSemiring R\ninst✝ : IsAddTorsionFree R\nha : IsIdempotentElem a\nhab : a * b + b * a = 0\nthis : a * (a * b + b * a) * a = 0\n⊢ (1 + 1) • (a * b * a) = 0",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
... | [
"R : Type u_1\na b : R\ninst✝¹ : NonUnitalSemiring R\ninst✝ : IsAddTorsionFree R\nha : IsIdempotentElem a\nhab : a * b + b * a = 0\nthis : (a * (a * b) + a * (b * a)) * a = 0\n⊢ (1 + 1) • (a * b * a) = 0"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Choose.Sum | {
"line": 85,
"column": 2
} | {
"line": 87,
"column": 53
} | {
"line": 88,
"column": 2
} | [
{
"pp": "case a\nR : Type u_1\ninst✝ : CommRing R\nx y : R\nn m : ℕ\nhm : m ∈ range (n + 1)\n⊢ x ^ m * (-y) ^ (n - m) * ↑(n.choose m) = (-1) ^ (m + n) * x ^ m * y ^ (n - m) * ↑(n.choose m)",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"one_pow",
"NegZeroClass.toNeg",
"Mul... | [
"case a\nR : Type u_1\ninst✝ : CommRing R\nx y : R\nn m : ℕ\nhm : m ∈ range (n + 1)\nthis : (-1) ^ (n - m) = (-1) ^ (n + m)\n⊢ x ^ m * (-y) ^ (n - m) * ↑(n.choose m) = (-1) ^ (m + n) * x ^ m * y ^ (n - m) * ↑(n.choose m)"
] | have : (-1 : R) ^ (n - m) = (-1) ^ (n + m) := by
rw [mem_range] at hm
simp [show n + m = n - m + 2 * m by lia, pow_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Ring.Idempotent | {
"line": 142,
"column": 22
} | {
"line": 142,
"column": 30
} | {
"line": 142,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonUnitalRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsIdempotentElem p\nhq : IsIdempotentElem q\nh : p * (q - p) + (q - p) * p = 0\nhqp : p * q + q * p - p = p\nh2 : (p * q + q * p - p) * q = p * q\nh1 : q * (p * q + q * p) - q * p = q * p\n⊢ Commute p q",
"ppTerm": "?... | [
"R : Type u_1\ninst✝¹ : NonUnitalRing R\ninst✝ : IsAddTorsionFree R\np q : R\nhp : IsIdempotentElem p\nhq : IsIdempotentElem q\nh : p * (q - p) + (q - p) * p = 0\nhqp : p * q + q * p - p = p\nh2 : (p * q + q * p - p) * q = p * q\nh1 : q * (p * q) + q * (q * p) - q * p = q * p\n⊢ Commute p q"
] | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Nat.Choose.Sum | {
"line": 132,
"column": 39
} | {
"line": 132,
"column": 60
} | {
"line": 132,
"column": 61
} | [
{
"pp": "case inl\nn k : ℕ\nh : n < k\n⊢ ∑ m ∈ Icc k n, m.choose k = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"congrArg",
"Finset",
"Nat.instLocallyFiniteOrder",
"id",
"Finset.Icc_eq_empty_of_lt",
"instOfNatNat... | [
"case inl\nn k : ℕ\nh : n < k\n⊢ ∑ m ∈ ∅, m.choose k = 0"
] | Icc_eq_empty_of_lt h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Span | {
"line": 248,
"column": 39
} | {
"line": 248,
"column": 47
} | {
"line": 248,
"column": 48
} | [
{
"pp": "α : Type u\ninst✝ : Ring α\nx y x✝¹ : α\nx✝ : ∃ a b, a * x + b * y = x✝¹\na b : α\nh : a * x + b * y = x✝¹\n⊢ a * (x + y) + (b - a) * y = x✝¹",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne... | [
"α : Type u\ninst✝ : Ring α\nx y x✝¹ : α\nx✝ : ∃ a b, a * x + b * y = x✝¹\na b : α\nh : a * x + b * y = x✝¹\n⊢ a * x + a * y + (b - a) * y = x✝¹"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 161,
"column": 26
} | {
"line": 161,
"column": 34
} | {
"line": 161,
"column": 35
} | [
{
"pp": "α : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nH : I.IsMaximal\nx y : α\nhxy : x * y ∈ I\nhx : x ∉ I\nJ : Ideal α := Submodule.span α (insert x ↑I)\nIJ : I ≤ J\nxJ : x ∈ J\nleft✝ : 1 ∉ I\noJ : 1 ∈ J\na b : α\nh : b ∈ Submodule.span α ↑I\noe : 1 = a • x + b\nF : y * 1 = y * (a • x + b)\n⊢ y * (a • x +... | [
"α : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nH : I.IsMaximal\nx y : α\nhxy : x * y ∈ I\nhx : x ∉ I\nJ : Ideal α := Submodule.span α (insert x ↑I)\nIJ : I ≤ J\nxJ : x ∈ J\nleft✝ : 1 ∉ I\noJ : 1 ∈ J\na b : α\nh : b ∈ Submodule.span α ↑I\noe : 1 = a • x + b\nF : y * 1 = y * (a • x + b)\n⊢ y * a • x + y * b ∈ I"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 234,
"column": 4
} | {
"line": 235,
"column": 67
} | {
"line": 236,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝¹ : CommSemiring α\ninst✝ : IsPrincipalIdealRing α\nP : Ideal α\nhP : P ≠ ⊥\nh : P.IsPrime\n⊢ ∃ p, Prime p ∧ P = span {p}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Semiring.toModule",
"congrArg",
"CommSemi... | [] | obtain ⟨p, rfl⟩ := Submodule.IsPrincipal.principal P
exact ⟨p, (span_singleton_prime (by simp [·] at hP)).mp h, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 234,
"column": 4
} | {
"line": 235,
"column": 67
} | {
"line": 236,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝¹ : CommSemiring α\ninst✝ : IsPrincipalIdealRing α\nP : Ideal α\nhP : P ≠ ⊥\nh : P.IsPrime\n⊢ ∃ p, Prime p ∧ P = span {p}",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Submodule",
"False",
"Semiring.toModule",
"congrArg",
"CommSemi... | [] | obtain ⟨p, rfl⟩ := Submodule.IsPrincipal.principal P
exact ⟨p, (span_singleton_prime (by simp [·] at hP)).mp h, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Basic | {
"line": 466,
"column": 49
} | {
"line": 466,
"column": 94
} | {
"line": 468,
"column": 0
} | [
{
"pp": "α : Type u\nι : Sort x\nf : ι → Filter α\np : ι → Prop\nl : Filter α\nh : ∀ {s : Set α}, s ∈ l ↔ ∃ i, p i ∧ s ∈ f i\nx✝ : Set α\n⊢ x✝ ∈ l ↔ ∃ i, x✝ ∈ f ↑i",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.of_eq",
"congrArg",
"Mem... | [] | by simp only [Subtype.exists, h, exists_prop] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Basic | {
"line": 992,
"column": 15
} | {
"line": 994,
"column": 19
} | {
"line": 996,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nl : Filter α\nf f' : α → β\nhf : f =ᶠ[l] f'\ng g' : α → γ\nhg : g =ᶠ[l] g'\n⊢ ∀ (x : α), g x = g' x → f x = f' x → (fun x ↦ (f x, g x)) x = (fun x ↦ (f' x, g' x)) x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"congrArg",
"Prod.mk... | [] | by
intros
simp only [*] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Basic | {
"line": 1234,
"column": 4
} | {
"line": 1234,
"column": 67
} | {
"line": 1236,
"column": 0
} | [
{
"pp": "α : Type u\ns t : Set α\nl : Filter α\n⊢ t ∈ l ⊓ 𝓟 s ↔ l ⊓ 𝓟 s ≤ l ⊓ 𝓟 t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"CompleteLattice.toLattice",
"congrArg",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder"... | [] | simp only [le_inf_iff, inf_le_left, true_and, le_principal_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.Basic | {
"line": 1234,
"column": 4
} | {
"line": 1234,
"column": 67
} | {
"line": 1236,
"column": 0
} | [
{
"pp": "α : Type u\ns t : Set α\nl : Filter α\n⊢ t ∈ l ⊓ 𝓟 s ↔ l ⊓ 𝓟 s ≤ l ⊓ 𝓟 t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"CompleteLattice.toLattice",
"congrArg",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder"... | [] | simp only [le_inf_iff, inf_le_left, true_and, le_principal_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Basic | {
"line": 1234,
"column": 4
} | {
"line": 1234,
"column": 67
} | {
"line": 1236,
"column": 0
} | [
{
"pp": "α : Type u\ns t : Set α\nl : Filter α\n⊢ t ∈ l ⊓ 𝓟 s ↔ l ⊓ 𝓟 s ≤ l ⊓ 𝓟 t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"CompleteLattice.toLattice",
"congrArg",
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder"... | [] | simp only [le_inf_iff, inf_le_left, true_and, le_principal_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 613,
"column": 76
} | {
"line": 615,
"column": 65
} | {
"line": 617,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nl : Filter (α ⊕ β)\nm₁ : α → γ\nm₂ : β → γ\n⊢ map (Sum.elim m₁ m₂) l = map m₁ (comap inl l) ⊔ map m₂ (comap inr l)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"CompleteLattice.toL... | [] | by
rw [← map_comap_inl_sup_map_comap_inr l]
simp [map_sup, map_map, comap_sup, (gc_map_comap _).u_l_u_eq_u] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.AtTopBot.Disjoint | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 42
} | {
"line": 44,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Nontrivial α\n⊢ Disjoint atBot atTop",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Filter.instCompleteLatticeFilter",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disjoint",
"Exists",
"CompleteLat... | [
"α : Type u_3\ninst✝¹ : PartialOrder α\ninst✝ : Nontrivial α\nx y : α\nhne : x ≠ y\n⊢ Disjoint atBot atTop"
] | rcases exists_pair_ne α with ⟨x, y, hne⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.Filter.Map | {
"line": 700,
"column": 2
} | {
"line": 701,
"column": 87
} | {
"line": 702,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\n⊢ (∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A) → ∀ B ∈ F, f x ∈ B",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"_private.Mathlib.Order.Filter.Map.0.Filter.sInter_comap_sets.... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\n⊢ (∀ B ∈ F, f x ∈ B) → ∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A"
] | · intro h U U_in
simpa only [Subset.rfl, forall_prop_of_true, mem_preimage] using h (f ⁻¹' U) U U_in | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.Map | {
"line": 702,
"column": 4
} | {
"line": 702,
"column": 26
} | {
"line": 703,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\n⊢ (∀ B ∈ F, f x ∈ B) → ∀ (A : Set α), ∀ B ∈ F, f ⁻¹' B ⊆ A → x ∈ A",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Membership.mem",
"HasSubset.Subset",
"Set.preim... | [
"case mpr\nα : Type u_1\nβ : Type u_2\nf : α → β\nF : Filter β\nx : α\nh : ∀ B ∈ F, f x ∈ B\nV : Set α\nU : Set β\nU_in : U ∈ F\nf_U_V : f ⁻¹' U ⊆ V\n⊢ x ∈ V"
] | intro h V U U_in f_U_V | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.Filter.Map | {
"line": 885,
"column": 8
} | {
"line": 885,
"column": 23
} | {
"line": 885,
"column": 23
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nf : Filter (α → β)\na : α\ns : Set (α → β)\nhs : s ∈ f\n⊢ (fun g ↦ g a) '' s ∈ f.seq (pure a)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"F... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nf : Filter (α → β)\na : α\ns : Set (α → β)\nhs : s ∈ f\n⊢ s.seq {a} ∈ f.seq (pure a)"
] | ← seq_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Map | {
"line": 910,
"column": 4
} | {
"line": 910,
"column": 34
} | {
"line": 911,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nf : Filter α\ng : Filter β\ns : Set (α → α × β)\nhs✝ : s ∈ map (fun b a ↦ (a, b)) g\nt : Set α\nht : t ∈ f\nu : Set β\nhu : u ∈ g\nhs : (fun b a ↦ (a, b)) '' u ⊆ s\n⊢ ((fun b a ↦ (a, b)) '' u).seq t ∈ (map Prod.mk f).seq g",
"ppTerm": "?refine_1",
"ass... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nf : Filter α\ng : Filter β\ns : Set (α → α × β)\nhs✝ : s ∈ map (fun b a ↦ (a, b)) g\nt : Set α\nht : t ∈ f\nu : Set β\nhu : u ∈ g\nhs : (fun b a ↦ (a, b)) '' u ⊆ s\n⊢ (Prod.mk '' t).seq u ∈ (map Prod.mk f).seq g"
] | rw [← Set.prod_image_seq_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.TransferInstance | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 81
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Semiring R\ne : α ≃ β\ninst✝² : AddCommMonoid β\ninst✝¹ : Module R β\ninst✝ : Module.IsTorsionFree R β\n⊢ let this := e.addCommMonoid;\n let this_1 := Equiv.module R e;\n Module.IsTorsionFree R α",
"ppTerm": "?m.19",
"assigned": true,
"us... | [] | extract_lets; exact (e.linearEquiv R).injective.moduleIsTorsionFree _ (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.TransferInstance | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 81
} | {
"line": 75,
"column": 0
} | [
{
"pp": "R : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : Semiring R\ne : α ≃ β\ninst✝² : AddCommMonoid β\ninst✝¹ : Module R β\ninst✝ : Module.IsTorsionFree R β\n⊢ let this := e.addCommMonoid;\n let this_1 := Equiv.module R e;\n Module.IsTorsionFree R α",
"ppTerm": "?m.19",
"assigned": true,
"us... | [] | extract_lets; exact (e.linearEquiv R).injective.moduleIsTorsionFree _ (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.DirSupClosed | {
"line": 125,
"column": 2
} | {
"line": 128,
"column": 23
} | {
"line": 130,
"column": 0
} | [
{
"pp": "α : Type u_1\nD : Set (Set α)\ninst✝ : Preorder α\ns : Set (Set α)\nhs : ∀ x ∈ s, DirSupInaccOn D x\n⊢ DirSupInaccOn D (⋃₀ s)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DirSupInaccOn.compl",
"congrArg",
"Compl.compl",
"Set.sUnion",
... | [] | rw [← dirSupClosedOn_compl, Set.compl_sUnion]
apply DirSupClosedOn.sInter
rintro x ⟨x, hx, rfl⟩
exact (hs x hx).compl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.DirSupClosed | {
"line": 125,
"column": 2
} | {
"line": 128,
"column": 23
} | {
"line": 130,
"column": 0
} | [
{
"pp": "α : Type u_1\nD : Set (Set α)\ninst✝ : Preorder α\ns : Set (Set α)\nhs : ∀ x ∈ s, DirSupInaccOn D x\n⊢ DirSupInaccOn D (⋃₀ s)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DirSupInaccOn.compl",
"congrArg",
"Compl.compl",
"Set.sUnion",
... | [] | rw [← dirSupClosedOn_compl, Set.compl_sUnion]
apply DirSupClosedOn.sInter
rintro x ⟨x, hx, rfl⟩
exact (hs x hx).compl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.Defs | {
"line": 809,
"column": 93
} | {
"line": 810,
"column": 60
} | {
"line": 812,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nhb : 0 < b\nh : a < 1\n⊢ a • b < b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
... | [] | by
simpa only [one_smul] using smul_lt_smul_of_pos_right h hb | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Module.Defs | {
"line": 812,
"column": 93
} | {
"line": 813,
"column": 60
} | {
"line": 815,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na : α\nb : β\ninst✝⁵ : Monoid α\ninst✝⁴ : Zero β\ninst✝³ : MulAction α β\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : SMulPosStrictMono α β\nhb : 0 < b\nh : 1 < a\n⊢ b < a • b",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
... | [] | by
simpa only [one_smul] using smul_lt_smul_of_pos_right h hb | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 246,
"column": 12
} | {
"line": 246,
"column": 29
} | {
"line": 246,
"column": 29
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : IsSuccLimit (type r)\nx b : α\nhb : b ∈ {x}\n⊢ r x ((enum r) ⟨succ ((typein r).toRelEmbedding x), ⋯⟩)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Ord... | [
"α : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : IsSuccLimit (type r)\nx b : α\nhb : b ∈ {x}\n⊢ r ((enum r) ⟨(typein r).toRelEmbedding x, ⋯⟩) ((enum r) ⟨succ ((typein r).toRelEmbedding x), ⋯⟩)"
] | ← enum_typein r x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 752,
"column": 72
} | {
"line": 752,
"column": 83
} | {
"line": 752,
"column": 83
} | [
{
"pp": "o : Ordinal.{u}\n⊢ (typein LT.lt).toRelEmbedding o = typeLT ↑(Iio o)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"isWellOrder_lt",
"Ordinal.partialOrder",
"congrArg",
"Subtype.wellF... | [
"o : Ordinal.{u}\n⊢ (typein LT.lt).toRelEmbedding o = (typein LT.lt).toRelEmbedding o"
] | type_Iio_lt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 761,
"column": 2
} | {
"line": 762,
"column": 42
} | {
"line": 763,
"column": 2
} | [
{
"pp": "α✝ : Type u\nβ✝ : Type v\nγ : Type w\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\na : Ordinal.{v}\nb : Ordinal.{max u v}\nα : Type v\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type (max u v)\ns : β → β → Prop\nx✝ : IsWellOrder β s\nh : type s < (RelEmbedding.ofMonotone lift.{u, v} li... | [
"α✝ : Type u\nβ✝ : Type v\nγ : Type w\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\na : Ordinal.{v}\nb : Ordinal.{max u v}\nα : Type v\nr : α → α → Prop\nx✝¹ : IsWellOrder α r\nβ : Type (max u v)\ns : β → β → Prop\nx✝ : IsWellOrder β s\nh : Nonempty (s ≺i r)\n⊢ type s ∈ range ⇑(RelEmbedding.ofMonoton... | rw [RelEmbedding.ofMonotone_coe, ← lift_id'.{max u v} (type s),
← lift_umax.{v, u}, lift_type_lt] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 682,
"column": 28
} | {
"line": 682,
"column": 36
} | {
"line": 682,
"column": 37
} | [
{
"pp": "o : Ordinal.{u_4}\n⊢ o * (1 + 1) = o + o",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"MulZeroClass.toMul",
"congrArg",
"id",
"Ordinal.addMonoidWithOne",
"AddMonoidWithOne.toOne",... | [
"o : Ordinal.{u_4}\n⊢ o * 1 + o * 1 = o + o"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 751,
"column": 18
} | {
"line": 751,
"column": 26
} | {
"line": 751,
"column": 27
} | [
{
"pp": "case a\na b : Ordinal.{u_4}\nb0 : b ≠ 0\nc : Ordinal.{u_4}\n⊢ b * a + c < b * (a + c / b) + b",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"... | [
"case a\na b : Ordinal.{u_4}\nb0 : b ≠ 0\nc : Ordinal.{u_4}\n⊢ b * a + c < b * a + b * (c / b) + b"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 753,
"column": 32
} | {
"line": 753,
"column": 40
} | {
"line": 753,
"column": 41
} | [
{
"pp": "case a\na b : Ordinal.{u_4}\nb0 : b ≠ 0\nc : Ordinal.{u_4}\n⊢ b * (a + c / b) ≤ b * a + c",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"congrArg",
"PartialOr... | [
"case a\na b : Ordinal.{u_4}\nb0 : b ≠ 0\nc : Ordinal.{u_4}\n⊢ b * a + b * (c / b) ≤ b * a + c"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 767,
"column": 8
} | {
"line": 767,
"column": 17
} | {
"line": 767,
"column": 18
} | [
{
"pp": "case inl\na c : Ordinal.{u_4}\nhc : c < a\nb : Ordinal.{u_4}\n⊢ (a * b + c) / (a * 0) = b / 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"id",
"HDiv.hDiv",
"Mul... | [
"case inl\na c : Ordinal.{u_4}\nhc : c < a\nb : Ordinal.{u_4}\n⊢ (a * b + c) / 0 = b / 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 757,
"column": 2
} | {
"line": 757,
"column": 24
} | {
"line": 759,
"column": 0
} | [
{
"pp": "o : Ordinal.{u}\nf : (a : Ordinal.{u}) → a < o → Ordinal.{max u v}\n⊢ iSup (o.familyOfBFamily f) = lsub (o.familyOfBFamily f) ↔\n ∀ a < lsub (o.familyOfBFamily f), succ a < lsub (o.familyOfBFamily f)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Ordinal.familyOfBFamily... | [] | apply iSup_eq_lsub_iff | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 906,
"column": 31
} | {
"line": 906,
"column": 39
} | {
"line": 906,
"column": 40
} | [
{
"pp": "case inr\nx y z : Ordinal.{u_4}\nhx : x ≠ 0\n⊢ x * y + z - x * (y + z / x) = z % x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"HSub.hSub",
"Ordinal.mod",
"id",
... | [
"case inr\nx y z : Ordinal.{u_4}\nhx : x ≠ 0\n⊢ x * y + z - (x * y + x * (z / x)) = z % x",
"case inr.b0\nx y z : Ordinal.{u_4}\nhx : x ≠ 0\n⊢ x ≠ 0"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 952,
"column": 7
} | {
"line": 952,
"column": 30
} | {
"line": 952,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : ℕ\nh : ↑a = ↑b\n⊢ a = b",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Cardinal",
"congrArg",
"Eq.mp",
"AddMonoidWithOne.toNatCast",
"Nat.cast",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : ℕ\nh : (↑a).ord = ↑b\n⊢ a = b"
] | ← Cardinal.ord_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 952,
"column": 31
} | {
"line": 952,
"column": 54
} | {
"line": 952,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : ℕ\nh : (↑a).ord = ↑b\n⊢ a = b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Cardinal",
"congrArg",
"Eq.mp",
"AddMonoidWithOne.toNatCast",
"Nat.cas... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na b : ℕ\nh : (↑a).ord = (↑b).ord\n⊢ a = b"
] | ← Cardinal.ord_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 966,
"column": 65
} | {
"line": 966,
"column": 79
} | {
"line": 966,
"column": 80
} | [
{
"pp": "m n : ℕ\n⊢ ↑m * ↑n + ↑m = ↑m * ↑(n + 1)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"MulZeroClass.toMul",
"congrArg",
"id",
"AddMonoidWithOne.toNatCast",
"... | [
"m n : ℕ\n⊢ ↑m * ↑n + ↑m = ↑m * (↑n + 1)"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Log | {
"line": 202,
"column": 2
} | {
"line": 203,
"column": 30
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case inl\nb m n : ℕ\nh : m ≠ 0 ∨ 1 < b ∧ n ≠ 0\nhb : 1 < b\nhn : n ≠ 0\n⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"congrArg",
"Nat.log_lt_iff_lt_pow",
"Iff.rfl",
"PartialOr... | [
"case inr\nb m n : ℕ\nh : m ≠ 0 ∨ 1 < b ∧ n ≠ 0\nhbn : ¬(1 < b ∧ n ≠ 0)\n⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1)"
] | · rw [le_antisymm_iff, ← Nat.lt_succ_iff, le_log_iff_pow_le, log_lt_iff_lt_pow,
and_comm] <;> assumption | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Nat.Log | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 57
} | {
"line": 207,
"column": 58
} | [
{
"pp": "case inr.inl\nb m n : ℕ\nh : m ≠ 0 ∨ 1 < b ∧ n ≠ 0\nhm : m ≠ 0\nhb : b ≤ 1\n⊢ log b n = m ↔ b ^ m ≤ n ∧ n < b ^ (m + 1)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"instPowNat",
"Nat.le_one_iff_eq_zero_or_eq_one",
"PartialOrder.toPreorder",
"Preorder... | [
"case inr.inl.inl\nm n : ℕ\nhm : m ≠ 0\nh : m ≠ 0 ∨ 1 < 0 ∧ n ≠ 0\nhb : 0 ≤ 1\n⊢ log 0 n = m ↔ 0 ^ m ≤ n ∧ n < 0 ^ (m + 1)",
"case inr.inl.inr\nm n : ℕ\nhm : m ≠ 0\nh : m ≠ 0 ∨ 1 < 1 ∧ n ≠ 0\nhb : 1 ≤ 1\n⊢ log 1 n = m ↔ 1 ^ m ≤ n ∧ n < 1 ^ (m + 1)"
] | obtain rfl | rfl := le_one_iff_eq_zero_or_eq_one.1 hb | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 114,
"column": 21
} | {
"line": 114,
"column": 35
} | {
"line": 114,
"column": 36
} | [
{
"pp": "case succ\na : Ordinal.{u_1}\nn : ℕ\nIH : a ^ ↑n = a ^ n\n⊢ a ^ ↑(n + 1) = a ^ (n + 1)",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Ordinal.monoid",
"AddMonoid.toAddSemigroup",
"congrArg",
"id",
"AddMonoidWit... | [
"case succ\na : Ordinal.{u_1}\nn : ℕ\nIH : a ^ ↑n = a ^ n\n⊢ a ^ (↑n + 1) = a ^ (n + 1)"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 17
} | {
"line": 168,
"column": 17
} | [
{
"pp": "case inr.inl\na : Ordinal.{u_1}\nl : IsSuccLimit a\nb : Ordinal.{u_1}\nhb : succ b ≠ 0\n⊢ IsSuccLimit (a ^ succ b)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Order.succ",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
... | [
"case inr.inl\na : Ordinal.{u_1}\nl : IsSuccLimit a\nb : Ordinal.{u_1}\nhb : succ b ≠ 0\n⊢ IsSuccLimit (a ^ b * a)"
] | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 210,
"column": 17
} | {
"line": 210,
"column": 26
} | {
"line": 210,
"column": 26
} | [
{
"pp": "a b c : Ordinal.{u_1}\nab : a < b\n⊢ a ^ c * a < b ^ succ c",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"congrArg",
"PartialOrder... | [
"a b c : Ordinal.{u_1}\nab : a < b\n⊢ a ^ c * a < b ^ c * b"
] | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Log | {
"line": 434,
"column": 37
} | {
"line": 434,
"column": 59
} | {
"line": 434,
"column": 60
} | [
{
"pp": "b x : ℕ\nhb : 1 < b\nz : ℕ\n⊢ clog b (b ^ x) ≤ z ↔ x ≤ z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instPowNat",
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Nat.clog_le_iff_le_pow",
"id",
"LE.le",
"... | [
"b x : ℕ\nhb : 1 < b\nz : ℕ\n⊢ b ^ x ≤ b ^ z ↔ x ≤ z"
] | clog_le_iff_le_pow hb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 293,
"column": 2
} | {
"line": 294,
"column": 22
} | {
"line": 296,
"column": 0
} | [
{
"pp": "case inr\nx : Ordinal.{u_1}\nh : 1 ≤ 1\n⊢ log 1 x = 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Ordinal.instLinearOrder",
"Lattice.toSemilatticeSup",
"Ordinal.partialOrder",
"congrArg",
"Set.preimage_const",
... | [] | · simp_rw [log, one_opow, preimage_const]
split_ifs <;> simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 485,
"column": 33
} | {
"line": 485,
"column": 41
} | {
"line": 485,
"column": 42
} | [
{
"pp": "case inr.refine_1\na b : Ordinal.{u_1}\nha : 0 < a\nhab : a * (a ^ ω * (b / a ^ ω) + b % a ^ ω) = a ^ ω * (b / a ^ ω) + b % a ^ ω\n⊢ b % a ^ ω = 0",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Ordinal.omega0",
"MulZeroCla... | [
"case inr.refine_1\na b : Ordinal.{u_1}\nha : 0 < a\nhab : a * (a ^ ω * (b / a ^ ω)) + a * (b % a ^ ω) = a ^ ω * (b / a ^ ω) + b % a ^ ω\n⊢ b % a ^ ω = 0"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 429,
"column": 8
} | {
"line": 429,
"column": 17
} | {
"line": 429,
"column": 17
} | [
{
"pp": "case inr.inl\na b : Ordinal.{u}\nhb : b < ω\nc : Ordinal.{u}\nc0 : 0 < succ c\nha : a < ω ^ succ c\n⊢ a * b < ω ^ succ c",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"HMul.hMul",
"Order.succ",
"Ordinal.omega0",
"Ordinal.partialO... | [
"case inr.inl\na b : Ordinal.{u}\nhb : b < ω\nc : Ordinal.{u}\nc0 : 0 < succ c\nha : a < ω ^ c * ω\n⊢ a * b < ω ^ succ c"
] | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 627,
"column": 2
} | {
"line": 627,
"column": 73
} | {
"line": 629,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\nho : IsSuccLimit o\n⊢ ∀ (a : Cardinal.{u_1}), (∀ b < o, preBeth b ≤ a) → preBeth o ≤ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
... | [] | simp [preBeth_limit ho.isSuccPrelimit, ciSup_le_iff' bddAbove_of_small] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 500,
"column": 19
} | {
"line": 500,
"column": 28
} | {
"line": 500,
"column": 28
} | [
{
"pp": "case a\na b : Ordinal.{u}\nha : a ≠ 0\nhb : IsPrincipal (fun x1 x2 ↦ x1 * x2) b\nhb₂ : 2 < b\nhbl : IsSuccLimit b\nc : Ordinal.{u}\nhcb : c ∈ Set.Iio b\nhb₁ : 1 < b\nhbo₀ : b ^ log b a ≠ 0\n⊢ b ^ log b a * (succ (a / b ^ log b a) * c) ≤ b ^ succ (log b a)",
"ppTerm": "?a✝",
"assigned": true,
... | [
"case a\na b : Ordinal.{u}\nha : a ≠ 0\nhb : IsPrincipal (fun x1 x2 ↦ x1 * x2) b\nhb₂ : 2 < b\nhbl : IsSuccLimit b\nc : Ordinal.{u}\nhcb : c ∈ Set.Iio b\nhb₁ : 1 < b\nhbo₀ : b ^ log b a ≠ 0\n⊢ b ^ log b a * (succ (a / b ^ log b a) * c) ≤ b ^ log b a * b"
] | opow_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Principal | {
"line": 505,
"column": 2
} | {
"line": 505,
"column": 42
} | {
"line": 507,
"column": 0
} | [
{
"pp": "case a\na b : Ordinal.{u}\nha : a ≠ 0\nhb : IsPrincipal (fun x1 x2 ↦ x1 * x2) b\nhb₂ : 2 < b\n⊢ b ^ succ (log b a) ≤ a * b",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"Ordinal.mulRightMono",
"HMul.hMul",
"Order.succ",
"Ord... | [] | · grw [opow_succ, opow_log_le_self b ha] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 61,
"column": 53
} | {
"line": 61,
"column": 80
} | {
"line": 61,
"column": 80
} | [
{
"pp": "case neg\nι : Type u\nf : ι → Ordinal.{v}\nhf : BddAbove (range f)\n⊢ Cardinal.lift.{u, v} #(⨆ i, f i).ToType ≤ Cardinal.lift.{v, max v u} #((x : ι) × (f x).ToType)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"iSup",
... | [
"case neg\nι : Type u\nf : ι → Ordinal.{v}\nhf : BddAbove (range f)\n⊢ Cardinal.lift.{max v u, v} #(⨆ i, f i).ToType ≤ Cardinal.lift.{v, max v u} #((x : ι) × (f x).ToType)"
] | ← Cardinal.lift_umax.{v, u} | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 85,
"column": 24
} | {
"line": 85,
"column": 36
} | {
"line": 87,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ IsCofinal (range fun x ↦ ⟨o, ⋯⟩)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"isCofinal_singleton_iff._simp_1",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
... | [] | simp [IsTop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 85,
"column": 24
} | {
"line": 85,
"column": 36
} | {
"line": 87,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ IsCofinal (range fun x ↦ ⟨o, ⋯⟩)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"isCofinal_singleton_iff._simp_1",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
... | [] | simp [IsTop] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 85,
"column": 24
} | {
"line": 85,
"column": 36
} | {
"line": 87,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ IsCofinal (range fun x ↦ ⟨o, ⋯⟩)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"isCofinal_singleton_iff._simp_1",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
... | [] | simp [IsTop] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 110,
"column": 52
} | {
"line": 110,
"column": 84
} | {
"line": 111,
"column": 4
} | [
{
"pp": "o : Ordinal.{u}\nc : Cardinal.{v}\nf : ↑(Iio o) → Ordinal.{v}\nhι : Cardinal.lift.{v, u} o.card ≤ Cardinal.lift.{u, v} c\nhf : ∀ (i : ↑(Iio o)), (f i).card ≤ c\n⊢ Cardinal.lift.{max (u + 1) v, u} o.card ≤ Cardinal.lift.{u + 1, max u v} (Cardinal.lift.{u, v} c)",
"ppTerm": "?m.30",
"assigned": t... | [
"o : Ordinal.{u}\nc : Cardinal.{v}\nf : ↑(Iio o) → Ordinal.{v}\nhι : Cardinal.lift.{v, u} o.card ≤ Cardinal.lift.{u, v} c\nhf : ∀ (i : ↑(Iio o)), (f i).card ≤ c\n⊢ Cardinal.lift.{u + 1, max v u} (Cardinal.lift.{v, u} o.card) ≤ Cardinal.lift.{u + 1, max u v} (Cardinal.lift.{u, v} c)"
] | ← Cardinal.lift_lift.{v, u + 1}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 262,
"column": 2
} | {
"line": 277,
"column": 41
} | {
"line": 279,
"column": 0
} | [
{
"pp": "s : Set Ordinal.{u}\na : Ordinal.{u}\nha : #↑s < (lift.{u + 1, u} a).cof\nhs : ∀ i ∈ s, i < a\n⊢ sSup ((fun x ↦ x + 1) '' s) < a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.range_comp",
"Eq.mpr",
"Set.image_univ",
"False",
"OrderIso.range_eq"... | [] | let f := OrderIso.ofRelIsoLT (enum (α := s) (· < ·))
have : Small.{u} (Iio (typeLT s)) := by
refine small_of_injective (β := Iio a) (f := fun x ↦ ⟨f x, hs _ (f x).2⟩) fun _ ↦ ?_
simp [Subtype.val_inj]
have : range (fun i ↦ (f i).1 + 1) = (· + 1) '' s := by
convert! range_comp (· + 1) (fun i ↦ (f i).1)
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 262,
"column": 2
} | {
"line": 277,
"column": 41
} | {
"line": 279,
"column": 0
} | [
{
"pp": "s : Set Ordinal.{u}\na : Ordinal.{u}\nha : #↑s < (lift.{u + 1, u} a).cof\nhs : ∀ i ∈ s, i < a\n⊢ sSup ((fun x ↦ x + 1) '' s) < a",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Set.range_comp",
"Eq.mpr",
"Set.image_univ",
"False",
"OrderIso.range_eq"... | [] | let f := OrderIso.ofRelIsoLT (enum (α := s) (· < ·))
have : Small.{u} (Iio (typeLT s)) := by
refine small_of_injective (β := Iio a) (f := fun x ↦ ⟨f x, hs _ (f x).2⟩) fun _ ↦ ?_
simp [Subtype.val_inj]
have : range (fun i ↦ (f i).1 + 1) = (· + 1) '' s := by
convert! range_comp (· + 1) (fun i ↦ (f i).1)
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 47,
"column": 2
} | {
"line": 50,
"column": 58
} | {
"line": 52,
"column": 0
} | [
{
"pp": "β α : Type u\nf : β → α\nθ : Cardinal.{u}\nhθ : θ ≤ #β\nh₁ : ℵ₀ ≤ θ\nh₂ : #α < θ.ord.cof\n⊢ ∃ a, θ ≤ #↑(f ⁻¹' {a})",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"Cardinal.le_mk_iff_exists_set",
"Cardinal.infini... | [] | rcases le_mk_iff_exists_set.1 hθ with ⟨s, rfl⟩
obtain ⟨a, ha⟩ := infinite_pigeonhole (f ∘ Subtype.val : s → α) h₁ h₂
use a; rw [← ha, @preimage_comp _ _ _ Subtype.val f]
exact mk_preimage_of_injective _ _ Subtype.val_injective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 47,
"column": 2
} | {
"line": 50,
"column": 58
} | {
"line": 52,
"column": 0
} | [
{
"pp": "β α : Type u\nf : β → α\nθ : Cardinal.{u}\nhθ : θ ≤ #β\nh₁ : ℵ₀ ≤ θ\nh₂ : #α < θ.ord.cof\n⊢ ∃ a, θ ≤ #↑(f ⁻¹' {a})",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"Cardinal.le_mk_iff_exists_set",
"Cardinal.infini... | [] | rcases le_mk_iff_exists_set.1 hθ with ⟨s, rfl⟩
obtain ⟨a, ha⟩ := infinite_pigeonhole (f ∘ Subtype.val : s → α) h₁ h₂
use a; rw [← ha, @preimage_comp _ _ _ Subtype.val f]
exact mk_preimage_of_injective _ _ Subtype.val_injective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 55,
"column": 2
} | {
"line": 55,
"column": 57
} | {
"line": 56,
"column": 2
} | [
{
"pp": "β α : Type u\ns : Set β\nf : ↑s → α\nθ : Cardinal.{u}\nhθ : θ ≤ #↑s\nh₁ : ℵ₀ ≤ θ\nh₂ : #α < θ.ord.cof\n⊢ ∃ a t, ∃ (h : t ⊆ s), θ ≤ #↑t ∧ ∀ ⦃x : β⦄ (hx : x ∈ t), f ⟨x, ⋯⟩ = a",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Cardinal",
"Cardinal.mk",
"Membership.me... | [
"β α : Type u\ns : Set β\nf : ↑s → α\nθ : Cardinal.{u}\nhθ : θ ≤ #↑s\nh₁ : ℵ₀ ≤ θ\nh₂ : #α < θ.ord.cof\na : α\nha : θ ≤ #↑(f ⁻¹' {a})\n⊢ ∃ a t, ∃ (h : t ⊆ s), θ ≤ #↑t ∧ ∀ ⦃x : β⦄ (hx : x ∈ t), f ⟨x, ⋯⟩ = a"
] | obtain ⟨a, ha⟩ := infinite_pigeonhole_card f θ hθ h₁ h₂ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 74,
"column": 7
} | {
"line": 74,
"column": 40
} | {
"line": 74,
"column": 40
} | [
{
"pp": "β α : Type u\nf : β → α\nh : #α < #β\nhβ : ℵ₀ ≤ #β\nhα : #α < ℵ₀\n⊢ #α < ℵ₀.ord.cof",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.mk",
"Cardinal.IsRe... | [] | by rwa [isRegular_aleph0.cof_ord] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Pigeonhole | {
"line": 86,
"column": 7
} | {
"line": 86,
"column": 40
} | {
"line": 86,
"column": 40
} | [
{
"pp": "β α : Type u\nf : β → α\nh : #α < #β\ninst✝ : Infinite β\nhα : #α < ℵ₀\n⊢ #α < ℵ₀.ord.cof",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.mk",
"Cardina... | [] | by rwa [isRegular_aleph0.cof_ord] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 604,
"column": 6
} | {
"line": 605,
"column": 40
} | {
"line": 606,
"column": 4
} | [
{
"pp": "case a.refine_1\nα : Type u_1\nh : (#α).IsStrongPrelimit\nha : #α ≠ 0\nh' : (#α).IsStrongLimit\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : (#α).ord = type r\nthis : LinearOrder α := linearOrderOfSTO r\nx : α\n⊢ #↑{x} < (#α).ord.cof",
"ppTerm": "?a.refine_1✝",
"assigned": true,
"usedConsta... | [] | rw [mk_singleton, one_lt_cof_iff]
exact isSuccLimit_ord h'.aleph0_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 604,
"column": 6
} | {
"line": 605,
"column": 40
} | {
"line": 606,
"column": 4
} | [
{
"pp": "case a.refine_1\nα : Type u_1\nh : (#α).IsStrongPrelimit\nha : #α ≠ 0\nh' : (#α).IsStrongLimit\nr : α → α → Prop\nwo : IsWellOrder α r\nhr : (#α).ord = type r\nthis : LinearOrder α := linearOrderOfSTO r\nx : α\n⊢ #↑{x} < (#α).ord.cof",
"ppTerm": "?a.refine_1✝",
"assigned": true,
"usedConsta... | [] | rw [mk_singleton, one_lt_cof_iff]
exact isSuccLimit_ord h'.aleph0_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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