module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Data.Set.Card | {
"line": 1225,
"column": 2
} | {
"line": 1225,
"column": 50
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 2 < s.ncard ↔ ∃ a ∈ s, ∃ b ∈ s, ∃ c ∈ s, a ≠ b ∧ a ≠ c ∧ b ≠ c",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Ne",
"_private.Mathlib.Data.Set.Card.0.Set.two_lt_ncard._simp_1_1",
"instOfNatNat",
"iff_s... | simp only [two_lt_ncard_iff hs, exists_and_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 1225,
"column": 2
} | {
"line": 1225,
"column": 50
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ 2 < s.ncard ↔ ∃ a ∈ s, ∃ b ∈ s, ∃ c ∈ s, a ≠ b ∧ a ≠ c ∧ b ≠ c",
"usedConstants": [
"congrArg",
"Membership.mem",
"Exists",
"Ne",
"_private.Mathlib.Data.Set.Card.0.Set.two_lt_ncard._simp_1_1",
"instOfNatNat",
"iff_s... | simp only [two_lt_ncard_iff hs, exists_and_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Subgroup.Finite | {
"line": 204,
"column": 8
} | {
"line": 207,
"column": 30
} | [
{
"pp": "η : Type u_3\nf : η → Type u_4\ninst✝¹ : (i : η) → Group (f i)\ninst✝ : Finite η\ns : (i : η) → Set (f i)\nhs : ∀ (i : η), 1 ∈ s i\ni : η\n_x : f i\nhx : _x ∈ s i\nj : η\n⊢ (MonoidHom.mulSingle f i) _x j ∈ s j",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"MonoidHo... | by_cases H : j = i
· subst H
simpa
· simpa [H] using hs _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Subgroup.Finite | {
"line": 204,
"column": 8
} | {
"line": 207,
"column": 30
} | [
{
"pp": "η : Type u_3\nf : η → Type u_4\ninst✝¹ : (i : η) → Group (f i)\ninst✝ : Finite η\ns : (i : η) → Set (f i)\nhs : ∀ (i : η), 1 ∈ s i\ni : η\n_x : f i\nhx : _x ∈ s i\nj : η\n⊢ (MonoidHom.mulSingle f i) _x j ∈ s j",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"False",
"MonoidHo... | by_cases H : j = i
· subst H
simpa
· simpa [H] using hs _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.QuotientGroup.Basic | {
"line": 112,
"column": 65
} | {
"line": 115,
"column": 5
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\nH : Type v\ninst✝ : Group H\nφ : G →* H\n⊢ Surjective ⇑(rangeKerLift φ)",
"usedConstants": [
"MonoidHom.range",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneClass",
"QuotientGroup.mk",
"Subtype.casesOn",
"Membership.me... | by
rintro ⟨_, g, rfl⟩
use mk g
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Coset.Basic | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 62
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_3\ninst✝ : (i : ι) → Group (β i)\ns' : (i : ι) → Subgroup (β i)\nx y : (i : ι) → β i\n⊢ (leftRel (Subgroup.pi univ s')) x y ↔ piSetoid x y",
"usedConstants": [
"Semigroup.toMul",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",... | simp [Setoid.piSetoid_apply, leftRel_apply, Subgroup.mem_pi] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Graph | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 26
} | [
{
"pp": "G : Type u_1\nH : Type u_2\nI : Type u_3\ninst✝² : Monoid G\ninst✝¹ : Monoid H\ninst✝ : Monoid I\nf : G →* H × I\nhf₁ : Surjective (Prod.fst ∘ ⇑f)\nhf : ∀ (g₁ g₂ : G), (f g₁).1 = (f g₂).1 → (f g₁).2 = (f g₂).2\nf' : H → I\nhf' : ∀ (a : H) (b : I), (∃ y, f y = (a, b)) ↔ f' a = b\n⊢ ∀ (x y : H), f' (x * ... | simp_rw [hf₁.forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.Congruence.Basic | {
"line": 216,
"column": 46
} | {
"line": 216,
"column": 60
} | [
{
"pp": "R : Type u_3\ninst✝¹ : Add R\ninst✝ : Mul R\n⊢ ¬Nontrivial (RingCon R) ↔ ¬Nontrivial R",
"usedConstants": [
"Nontrivial",
"congrArg",
"RingCon",
"iff_self",
"Iff",
"True",
"of_eq_true",
"congrFun'",
"_private.Mathlib.RingTheory.Congruence.Basic.... | nontrivial_iff | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Pi | {
"line": 568,
"column": 6
} | {
"line": 568,
"column": 17
} | [
{
"pp": "R✝ : Type u\nK : Type u'\nM✝ : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝¹⁶ : Semiring R✝\nφ : ι → Type u_1\nψ : ι → Type u_2\nχ : ι → Type u_3\ninst✝¹⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝¹⁴ : (i : ι) → Module R✝ (φ i)\nins... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Pi | {
"line": 571,
"column": 6
} | {
"line": 571,
"column": 17
} | [
{
"pp": "R✝ : Type u\nK : Type u'\nM✝ : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nι' : Type x'\ninst✝¹⁶ : Semiring R✝\nφ : ι → Type u_1\nψ : ι → Type u_2\nχ : ι → Type u_3\ninst✝¹⁵ : (i : ι) → AddCommMonoid (φ i)\ninst✝¹⁴ : (i : ι) → Module R✝ (φ i)\nins... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Prod | {
"line": 457,
"column": 2
} | {
"line": 457,
"column": 16
} | [
{
"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] M₂\ng : M →ₗ[R] M₃\n⊢ ∀ ⦃x : M₂ × M₃⦄ (x_1 : M), Pi.prod (⇑f) (⇑g) x_1 = x →... | rintro _ x rfl | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 77
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Ioc (a - 1) b",
"usedConstants": [
"PredSubOrder.toPredOrder",
"LinearOrder.toDecidableEq",
"congr... | simpa [pred_eq_sub_one] using insert_Ioc_left_eq_Ioc_pred_of_not_isMin h ha | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 77
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Ioc (a - 1) b",
"usedConstants": [
"PredSubOrder.toPredOrder",
"LinearOrder.toDecidableEq",
"congr... | simpa [pred_eq_sub_one] using insert_Ioc_left_eq_Ioc_pred_of_not_isMin h ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Interval.Finset.SuccPred | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 77
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : One α\ninst✝² : LocallyFiniteOrder α\ninst✝¹ : Sub α\ninst✝ : PredSubOrder α\na b : α\nh : a ≤ b\nha : ¬IsMin a\n⊢ insert a (Ioc a b) = Ioc (a - 1) b",
"usedConstants": [
"PredSubOrder.toPredOrder",
"LinearOrder.toDecidableEq",
"congr... | simpa [pred_eq_sub_one] using insert_Ioc_left_eq_Ioc_pred_of_not_isMin h ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.BigOperators.Group.LocallyFinite | {
"line": 47,
"column": 16
} | {
"line": 47,
"column": 42
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝² : CommMonoid M\nf : α → M\na b : α\ninst✝¹ : PartialOrder α\ninst✝ : LocallyFiniteOrder α\nh : a ≤ b\n⊢ f a * ∏ x ∈ Ioc a b, f x = ∏ x ∈ Icc a b, f x",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommMonoid.toCommSemigroup",
"Monoid.toMulOne... | mul_prod_Ioc_eq_prod_Icc h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Disjointed | {
"line": 85,
"column": 53
} | {
"line": 85,
"column": 74
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : Preorder ι\ninst✝ : LocallyFiniteOrderBot ι\nf : ι → α\ni : ι\nhf : ∀ j < i, Disjoint (f j) (f i)\n⊢ f i ⊓ (Iio i).sup f = ⊥",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Ord... | sup_inf_distrib_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Interval.Finset.Basic | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 26
} | [
{
"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) = Ioo (a + c) (b + c)",
"usedConstants": [
"Eq.mpr",
"congrA... | ← map_add_right_Ioo, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 319,
"column": 6
} | {
"line": 321,
"column": 28
} | [
{
"pp": "case neg.inl.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find ⋯\nhm : succ^[m] i = j\nhj_eq : j = succ^[(toZ i0 i).toNat + m... | rw [hm0, Function.iterate_zero, id] at hm
rw [hm] at h
exact h (le_of_eq rfl) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 319,
"column": 6
} | {
"line": 321,
"column": 28
} | [
{
"pp": "case neg.inl.inl.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i0 ≤ i\nhj : i0 ≤ j\nm : ℕ := Nat.find ⋯\nhm : succ^[m] i = j\nhj_eq : j = succ^[(toZ i0 i).toNat + m... | rw [hm0, Function.iterate_zero, id] at hm
rw [hm] at h
exact h (le_of_eq rfl) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Disjointed | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 57
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝² : GeneralizedBooleanAlgebra α\ninst✝¹ : PartialOrder ι\ninst✝ : LocallyFiniteOrderBot ι\nf : ι → α\ni : ι\n⊢ (partialSups f) i \\ (f i \\ (Iio i).sup f) = (Iio i).sup f",
"usedConstants": [
"partialSups_apply",
"Eq.mpr",
"Lattice.toSemilatticeSup... | simp only [funext (partialSups_apply f), sup'_eq_sup] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 339,
"column": 6
} | {
"line": 341,
"column": 28
} | [
{
"pp": "case neg.inr.inr.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find ⋯\nhm : pred^[m] j = i\nhj_eq : i = pred^[(-toZ i0 j).toNat + ... | rw [hm0, Function.iterate_zero, id] at hm
rw [hm] at h
exact h (le_of_eq rfl) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 339,
"column": 6
} | {
"line": 341,
"column": 28
} | [
{
"pp": "case neg.inr.inr.inl\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 i j : ι\nh_le : i ≤ j\nhi_max : ¬IsMax i\nhj_min : ¬IsMin j\nhi : i < i0\nhj : j < i0\nm : ℕ := Nat.find ⋯\nhm : pred^[m] j = i\nhj_eq : i = pred^[(-toZ i0 j).toNat + ... | rw [hm0, Function.iterate_zero, id] at hm
rw [hm] at h
exact h (le_of_eq rfl) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Prime | {
"line": 90,
"column": 34
} | {
"line": 90,
"column": 87
} | [
{
"pp": "α : Type u\nβ : Type v\nF : Type w\ninst✝² : Semiring α\nI : Ideal α\na b : α\ninst✝¹ : Nontrivial α\ninst✝ : NoZeroDivisors α\nh : ⊥ = ⊤\n⊢ 1 = 0",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Semiring.toModule",
"congrArg",
"Submodule.m... | by rwa [Ideal.eq_top_iff_one, Submodule.mem_bot] at h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Idempotent | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 36
} | [
{
"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\nh1 : (fun x ↦ q * x) (p * q + q * p - p) = (fun x ↦ q * x) p\n⊢ Commute p q",
"usedConstants": [
"HMul... | have h2 := congr_arg (· * q) hqp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Nat.Choose.Sum | {
"line": 71,
"column": 29
} | {
"line": 71,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn : ℕ\n⊢ (x + y) ^ n = ∑ x_1 ∈ range n.succ, x ^ x_1 * y ^ (n - x_1) * ↑(n.choose x_1)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Nat.choose",
"HMul.hMul",
"congrArg",
"HSub.hSub",
... | h.add_pow | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.Span | {
"line": 137,
"column": 44
} | {
"line": 137,
"column": 95
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\n⊢ span 0 = ⊥",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Ideal.span_singleton_eq_bot",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Set.instSingletonSet",
"id",
"Bot.bot",
"Ideal",
"NonU... | by rw [← Set.singleton_zero, span_singleton_eq_bot] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Span | {
"line": 293,
"column": 28
} | {
"line": 293,
"column": 48
} | [
{
"pp": "α : Type u\ninst✝ : Ring α\nx y : α\n⊢ span {x, x - y - x} = span {x, y}",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"Set.instSingletonSet",
"sub_sub_cancel_lef... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Span | {
"line": 297,
"column": 29
} | {
"line": 297,
"column": 49
} | [
{
"pp": "α : Type u\ninst✝ : Ring α\nx y : α\n⊢ span {y - x - y, y} = span {x, y}",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"AddGroupWithOne.toAddGroup",
"congrArg",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"Set.instSingletonSet",
"sub_sub_cancel_lef... | sub_sub_cancel_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 12
} | [
{
"pp": "α : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nS : Submonoid α\ndisjoint : Disjoint ↑I ↑S\nmaximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S\n⊢ I ≠ ⊤",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Preorder.toLT",
"Semiring.toModule",
... | rintro rfl
have : 1 ∈ (S : Set α) := S.one_mem
simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 185,
"column": 4
} | {
"line": 187,
"column": 12
} | [
{
"pp": "α : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nS : Submonoid α\ndisjoint : Disjoint ↑I ↑S\nmaximally_disjoint : ∀ (J : Ideal α), I < J → ¬Disjoint ↑J ↑S\n⊢ I ≠ ⊤",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"Preorder.toLT",
"Semiring.toModule",
... | rintro rfl
have : 1 ∈ (S : Set α) := S.one_mem
simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 71
} | [
{
"pp": "case refine_1.inr\nα : Type u\ninst✝ : CommSemiring α\nI : Ideal α\nS : Submonoid α\ndisjoint : Disjoint ↑I ↑S\nc : Set (Ideal α)\nhc : c ⊆ {p | Disjoint ↑p ↑S}\nhc' : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nx : Ideal α\nhx : x ∈ c\nh✝ : Nonempty ↑c\n⊢ ∃ ub ∈ {p | Disjoint ↑p ↑S}, ∀ z ∈ c, z ≤ ub",
"usedC... | refine ⟨sSup c, Set.disjoint_left.mpr fun x hx ↦ ?_, fun _ ↦ le_sSup⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.Basic | {
"line": 252,
"column": 2
} | {
"line": 256,
"column": 25
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nhf : ¬IsField R\n⊢ ∃ x, ∃ (_ : x ≠ 0), ¬IsUnit x",
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"_priv... | have : ¬_ := fun h => hf ⟨exists_pair_ne R, mul_comm, h⟩
simp_rw [isUnit_iff_exists_inv]
push Not at this ⊢
obtain ⟨x, hx, not_unit⟩ := this
exact ⟨x, hx, not_unit⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Basic | {
"line": 252,
"column": 2
} | {
"line": 256,
"column": 25
} | [
{
"pp": "R : Type u_5\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nhf : ¬IsField R\n⊢ ∃ x, ∃ (_ : x ≠ 0), ¬IsUnit x",
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"_priv... | have : ¬_ := fun h => hf ⟨exists_pair_ne R, mul_comm, h⟩
simp_rw [isUnit_iff_exists_inv]
push Not at this ⊢
obtain ⟨x, hx, not_unit⟩ := this
exact ⟨x, hx, not_unit⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Quotient.Basic | {
"line": 39,
"column": 38
} | {
"line": 41,
"column": 34
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\ninst✝¹ : Ring R\nf : ι → R\ninst✝ : (span (range f)).IsTwoSided\ni : ι\n⊢ (mk (span (range f))) (f i) = 0",
"usedConstants": [
"Eq.mpr",
"Ideal.subset_span",
"Submodule.Quotient.instZeroQuotient",
"Semiring.toModule",
"congrArg",
"Idea... | by
rw [Ideal.Quotient.eq_zero_iff_mem]
exact Ideal.subset_span ⟨i, rfl⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Quotient.Basic | {
"line": 61,
"column": 11
} | {
"line": 61,
"column": 75
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝ : Ring R\nI J : Ideal R\na b x y : R\n⊢ ∀ (a : R ⧸ ⊤), a = default",
"usedConstants": [
"Iff.mpr",
"Inhabited.default",
"Submodule.Quotient.instZeroQuotient",
"Semiring.toModule",
"Ideal.Quotient.mk",
... | rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Quotient.Basic | {
"line": 61,
"column": 11
} | {
"line": 61,
"column": 75
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝ : Ring R\nI J : Ideal R\na b x y : R\n⊢ ∀ (a : R ⧸ ⊤), a = default",
"usedConstants": [
"Iff.mpr",
"Inhabited.default",
"Submodule.Quotient.instZeroQuotient",
"Semiring.toModule",
"Ideal.Quotient.mk",
... | rintro ⟨x⟩; exact Quotient.eq_zero_iff_mem.mpr Submodule.mem_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 973,
"column": 31
} | {
"line": 976,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf₁ f₂ : Filter α\ng₁ g₂ : α → Filter β\nhf : f₁ ≤ f₂\nhg : g₁ ≤ᶠ[f₁] g₂\n⊢ f₁.bind g₁ ≤ f₂.bind g₂",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Filter.map_mono",
"Filter.mem_map._simp_1",
"Filter.map",
"PartialOrder.toPreor... | by
refine le_trans (fun s hs => ?_) (join_mono <| map_mono hf)
simp only [mem_join, mem_bind', mem_map] at hs ⊢
filter_upwards [hg, hs] with _ hx hs using hx hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Noetherian.Defs | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 23
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ (∀ (a : Set (Submodule R M)), a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬M' < I) ↔ IsNoetherian R M",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Preorder.toLT",
"congrArg",
"PartialO... | isNoetherian_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 232,
"column": 26
} | {
"line": 232,
"column": 57
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : Finite M\nval✝ : Fintype M\n⊢ span R ↑Finset.univ = ⊤",
"usedConstants": [
"Eq.mpr",
"Submodul... | rw [Finset.coe_univ, span_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 232,
"column": 26
} | {
"line": 232,
"column": 57
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : Finite M\nval✝ : Fintype M\n⊢ span R ↑Finset.univ = ⊤",
"usedConstants": [
"Eq.mpr",
"Submodul... | rw [Finset.coe_univ, span_univ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 232,
"column": 26
} | {
"line": 232,
"column": 57
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\nN : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : Finite M\nval✝ : Fintype M\n⊢ span R ↑Finset.univ = ⊤",
"usedConstants": [
"Eq.mpr",
"Submodul... | rw [Finset.coe_univ, span_univ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Family | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 24
} | [
{
"pp": "case refine_1\nα : Type u_1\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\nf : ι → α\ni : Ordinal.{u}\nhi : i < type r\n⊢ bfamilyOfFamily' r f i hi ∈ range f",
"usedConstants": [
"Set.mem_range_self",
"Preorder.toLT",
"Ordinal.partialOrder",
"PartialOrder.toPreorder... | apply mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Family | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 24
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_4} α\ninst✝ : Small.{u, u_5} β\na : α\n⊢ (fun a ↦ f (Sum.inl a)) a ∈ range fun i ↦ f i",
"usedConstants": [
"Set.mem_range_self",
"Sum",
"Sum.inl",
"Ordinal"
]
}
] | apply mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Family | {
"line": 249,
"column": 4
} | {
"line": 249,
"column": 24
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_4} α\ninst✝ : Small.{u, u_5} β\na : β\n⊢ (fun b ↦ f (Sum.inr b)) a ∈ range fun i ↦ f i",
"usedConstants": [
"Set.mem_range_self",
"Sum",
"Sum.inr",
"Ordinal"
]
}
] | apply mem_range_self | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Family | {
"line": 242,
"column": 2
} | {
"line": 249,
"column": 24
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_4} α\ninst✝ : Small.{u, u_5} β\n⊢ iSup f = max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))",
"usedConstants": [
"Set.mem_range_self",
"small_sum",
"Ordinal.instLinearOrder",
"Ordinal.partialOrder",
... | apply (Ordinal.iSup_le _).antisymm (max_le _ _)
· rintro (i | i)
· exact le_max_of_le_left (Ordinal.le_iSup (fun x ↦ f (Sum.inl x)) i)
· exact le_max_of_le_right (Ordinal.le_iSup (fun x ↦ f (Sum.inr x)) i)
all_goals
apply csSup_le_csSup' (bddAbove_of_small _)
rintro i ⟨a, rfl⟩
apply mem_range_se... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Family | {
"line": 242,
"column": 2
} | {
"line": 249,
"column": 24
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nf : α ⊕ β → Ordinal.{u}\ninst✝¹ : Small.{u, u_4} α\ninst✝ : Small.{u, u_5} β\n⊢ iSup f = max (⨆ a, f (Sum.inl a)) (⨆ b, f (Sum.inr b))",
"usedConstants": [
"Set.mem_range_self",
"small_sum",
"Ordinal.instLinearOrder",
"Ordinal.partialOrder",
... | apply (Ordinal.iSup_le _).antisymm (max_le _ _)
· rintro (i | i)
· exact le_max_of_le_left (Ordinal.le_iSup (fun x ↦ f (Sum.inl x)) i)
· exact le_max_of_le_right (Ordinal.le_iSup (fun x ↦ f (Sum.inr x)) i)
all_goals
apply csSup_le_csSup' (bddAbove_of_small _)
rintro i ⟨a, rfl⟩
apply mem_range_se... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Enum | {
"line": 44,
"column": 97
} | {
"line": 46,
"column": 25
} | [
{
"pp": "o a : Ordinal.{u}\ns : Set Ordinal.{u}\nha : a ∈ s\nH : ∀ b < o, enumOrd s b < a\n⊢ enumOrd s o ≤ a",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",
"_private.Mathlib.SetTheory.Ordinal.Enum.0.Ordinal.enumOrd.eq_1",
"PartialOrder... | by
rw [enumOrd]
exact csInf_le' ⟨ha, H⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Family | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 92
} | [
{
"pp": "o : Ordinal.{u_4}\nf : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4}\n⊢ ⨆ a, f ↑a ⋯ = o.bsup f",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"Iff.of_eq",
"congrArg",
"iSup",
"Ordinal.familyOfBFamily",
"... | simp_rw [Iio, bsup, iSup, range_familyOfBFamily, brange, range, Subtype.exists, mem_setOf] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.SetTheory.Ordinal.Family | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 92
} | [
{
"pp": "o : Ordinal.{u_4}\nf : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4}\n⊢ ⨆ a, f ↑a ⋯ = o.bsup f",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"Iff.of_eq",
"congrArg",
"iSup",
"Ordinal.familyOfBFamily",
"... | simp_rw [Iio, bsup, iSup, range_familyOfBFamily, brange, range, Subtype.exists, mem_setOf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Family | {
"line": 492,
"column": 2
} | {
"line": 492,
"column": 92
} | [
{
"pp": "o : Ordinal.{u_4}\nf : (a : Ordinal.{u_4}) → a < o → Ordinal.{max u_5 u_4}\n⊢ ⨆ a, f ↑a ⋯ = o.bsup f",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"Iff.of_eq",
"congrArg",
"iSup",
"Ordinal.familyOfBFamily",
"... | simp_rw [Iio, bsup, iSup, range_familyOfBFamily, brange, range, Subtype.exists, mem_setOf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Family | {
"line": 561,
"column": 7
} | {
"line": 563,
"column": 30
} | [
{
"pp": "ι : Type u_4\nf : ι → Ordinal.{max u_5 u_4}\nhf : ∀ a < lsub f, succ a < lsub f\ni : ι\nhle : iSup f ≤ f i\nheq : succ (iSup f) = lsub f\n⊢ ?m.88 < lsub f",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Ordinal.partialOrder",
"congrArg",
"iSup",
... | by
rw [← heq]
exact lt_succ (iSup f) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 338,
"column": 6
} | {
"line": 338,
"column": 22
} | [
{
"pp": "o : Ordinal.{u_1}\n⊢ 0 < preAleph o ↔ 0 < o",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"OrderIso",
"Cardinal.instLE",
"Cardinal.pr... | ← preAleph_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 82,
"column": 58
} | {
"line": 83,
"column": 30
} | [
{
"pp": "a : Ordinal.{u_1}\n⊢ a ^ 1 = a",
"usedConstants": [
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"AddMonoid.toAddZeroClass",
"Eq.mp",
"zero_add",
"MulZeroOneClass.toMulOneClass",
"Ordinal.addMonoidWithOne",
"Ordinal.one",
"instHAdd",
... | by
simpa using opow_add_one a 0 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 41
} | [
{
"pp": "case limit\nι : Type u_1\nf : ι → Ordinal.{u} → Ordinal.{u}\ninst✝ : Small.{u, u_1} ι\nH : ∀ (i : ι), IsNormal (f i)\na : Ordinal.{u}\nha : ∀ (i : ι), f i a ≤ a\no : Ordinal.{u}\nl : IsSuccLimit o\nIH : ∀ o' < o, a ≤ derivFamily f o' → ∃ o, derivFamily f o = a\nh₁ : a ≤ derivFamily f o\n⊢ ∃ o, derivFam... | rcases eq_or_lt_of_le h₁ with h | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 776,
"column": 2
} | {
"line": 776,
"column": 44
} | [
{
"pp": "a b c : Ordinal.{u_4}\nb0 : b ≠ 0\n⊢ a / b ≤ c ↔ a < b * succ c",
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Order.succ",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"Ordinal.lt_mul_iff_d... | rw [← lt_succ_iff, ← lt_mul_iff_div_lt b0] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 776,
"column": 2
} | {
"line": 776,
"column": 44
} | [
{
"pp": "a b c : Ordinal.{u_4}\nb0 : b ≠ 0\n⊢ a / b ≤ c ↔ a < b * succ c",
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Order.succ",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"Ordinal.lt_mul_iff_d... | rw [← lt_succ_iff, ← lt_mul_iff_div_lt b0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 776,
"column": 2
} | {
"line": 776,
"column": 44
} | [
{
"pp": "a b c : Ordinal.{u_4}\nb0 : b ≠ 0\n⊢ a / b ≤ c ↔ a < b * succ c",
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"Order.succ",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"Ordinal.lt_mul_iff_d... | rw [← lt_succ_iff, ← lt_mul_iff_div_lt b0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 746,
"column": 2
} | {
"line": 751,
"column": 33
} | [
{
"pp": "α : Type u\ninst✝ : Infinite α\n⊢ #(Finset α) = #α",
"usedConstants": [
"Cardinal.mk_le_of_surjective",
"Trans.trans",
"Cardinal",
"Finset",
"Classical.propDecidable",
"Cardinal.mk",
"Cardinal.mk_list_eq_mk",
"List.toFinset_surjective",
"LE.le",... | classical
exact Eq.symm <|
le_antisymm (mk_le_of_injective fun _ _ => Finset.singleton_inj.1) <|
calc
#(Finset α) ≤ #(List α) := mk_le_of_surjective List.toFinset_surjective
_ = #α := mk_list_eq_mk α | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 746,
"column": 2
} | {
"line": 751,
"column": 33
} | [
{
"pp": "α : Type u\ninst✝ : Infinite α\n⊢ #(Finset α) = #α",
"usedConstants": [
"Cardinal.mk_le_of_surjective",
"Trans.trans",
"Cardinal",
"Finset",
"Classical.propDecidable",
"Cardinal.mk",
"Cardinal.mk_list_eq_mk",
"List.toFinset_surjective",
"LE.le",... | classical
exact Eq.symm <|
le_antisymm (mk_le_of_injective fun _ _ => Finset.singleton_inj.1) <|
calc
#(Finset α) ≤ #(List α) := mk_le_of_surjective List.toFinset_surjective
_ = #α := mk_list_eq_mk α | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 746,
"column": 2
} | {
"line": 751,
"column": 33
} | [
{
"pp": "α : Type u\ninst✝ : Infinite α\n⊢ #(Finset α) = #α",
"usedConstants": [
"Cardinal.mk_le_of_surjective",
"Trans.trans",
"Cardinal",
"Finset",
"Classical.propDecidable",
"Cardinal.mk",
"Cardinal.mk_list_eq_mk",
"List.toFinset_surjective",
"LE.le",... | classical
exact Eq.symm <|
le_antisymm (mk_le_of_injective fun _ _ => Finset.singleton_inj.1) <|
calc
#(Finset α) ≤ #(List α) := mk_le_of_surjective List.toFinset_surjective
_ = #α := mk_list_eq_mk α | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 74
} | [
{
"pp": "case inr\nb o : Ordinal.{u_1}\nho : o ≠ 0\nhb : b ≠ 0\n⊢ o % b ^ log b o < o",
"usedConstants": [
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"Ordinal.opow_log_le_self",
"Ordinal.opow_ne_zero",
"Ordinal.mod",
"Ordinal.mod_lt",
"instHMod",
"HMod... | exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 74
} | [
{
"pp": "case inr\nb o : Ordinal.{u_1}\nho : o ≠ 0\nhb : b ≠ 0\n⊢ o % b ^ log b o < o",
"usedConstants": [
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"Ordinal.opow_log_le_self",
"Ordinal.opow_ne_zero",
"Ordinal.mod",
"Ordinal.mod_lt",
"instHMod",
"HMod... | exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 401,
"column": 4
} | {
"line": 401,
"column": 74
} | [
{
"pp": "case inr\nb o : Ordinal.{u_1}\nho : o ≠ 0\nhb : b ≠ 0\n⊢ o % b ^ log b o < o",
"usedConstants": [
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"Ordinal.opow_log_le_self",
"Ordinal.opow_ne_zero",
"Ordinal.mod",
"Ordinal.mod_lt",
"instHMod",
"HMod... | exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 801,
"column": 2
} | {
"line": 801,
"column": 40
} | [
{
"pp": "α : Type u\ns : Set α\nc : Cardinal.{u}\n⊢ #{ t // t ⊆ s ∧ #↑t ≤ c } ≤ #{ t // #↑t ≤ c }",
"usedConstants": [
"Function.Embedding.codRestrict",
"Cardinal",
"Cardinal.mk",
"Set.Elem",
"Quot.lift",
"Subtype",
"Quotient.lift₂._proof_1",
"HasSubset.Subset... | refine ⟨Embedding.codRestrict _ ?_ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 858,
"column": 68
} | {
"line": 858,
"column": 84
} | [
{
"pp": "case inr\na b c : Ordinal.{u_4}\nha : a ≠ 0\nd : Ordinal.{u_4}\n⊢ b ≤ c + d / a ↔ b ≤ (a * c + d) / a",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"congrArg",
"Ordinal.mul_add_div",
"PartialOrde... | mul_add_div _ ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 470,
"column": 2
} | {
"line": 470,
"column": 74
} | [
{
"pp": "a b : Ordinal.{u_1}\nh : a < b\nn : ℕ\n⊢ ω ^ a * ↑n < ω ^ b",
"usedConstants": [
"Ordinal.opow_le_opow_right",
"HMul.hMul",
"Order.succ",
"Ordinal.omega0",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"PartialOrder.toPreorder",
"Ordinal.omega0_pos",
... | apply lt_of_lt_of_le _ (opow_le_opow_right omega0_pos (succ_le_of_lt h)) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 1026,
"column": 2
} | {
"line": 1026,
"column": 95
} | [
{
"pp": "case inl\nm n : ℕ\nh : m ≤ n\n⊢ ↑(m - n) = ↑m - ↑n",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.instOrderedSub",
"Ordinal.sub_eq_zero_iff_le",
"Ordinal.partialOrder",
"congrArg",
"AddMonoid.toAddZeroClass",
"Pa... | · rw [tsub_eq_zero_iff_le.2 h, Ordinal.sub_eq_zero_iff_le.2 (Nat.cast_le.2 h), Nat.cast_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Cofinality | {
"line": 378,
"column": 10
} | {
"line": 378,
"column": 49
} | [
{
"pp": "case refine_2.refine_2\nβ : Type v\ninst✝¹ : LinearOrder β\ninst✝ : Small.{u, v} β\nf : β → Ordinal.{u}\nhf : StrictMono f\nthis✝ : StrictMono fun i ↦ ⟨f i, ⋯⟩\nthis : Cardinal.lift.{u + 1, v} (Order.cof β) = Cardinal.lift.{v, u + 1} (Order.cof ↑(Iio (⨆ i, f i + 1)))\n⊢ Cardinal.lift.{v, u} (⨆ i, f i +... | ← Cardinal.lift_inj.{_, max (u + 1) v}, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.DFinsupp.Submonoid | {
"line": 76,
"column": 2
} | {
"line": 81,
"column": 15
} | [
{
"pp": "case a\nι : Type u\nγ : Type w\ninst✝² : DecidableEq ι\np : ι → Prop\ninst✝¹ : DecidablePred p\ninst✝ : AddCommMonoid γ\nS : ι → AddSubmonoid γ\n⊢ AddMonoidHom.mrange ((sumAddHom fun i ↦ (S i).subtype).comp (filterAddMonoidHom (fun i ↦ ↥(S i)) p)) ≤\n ⨆ i, ⨆ (_ : p i), S i",
"usedConstants": [
... | · rintro x ⟨v, rfl⟩
refine dfinsuppSumAddHom_mem _ _ _ fun i _ => ?_
refine AddSubmonoid.mem_iSup_of_mem i ?_
by_cases hp : p i
· simp [hp]
· simp [hp] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Fintype.Quotient | {
"line": 125,
"column": 40
} | {
"line": 125,
"column": 57
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nS : (i : ι) → Setoid (α i)\nf : (i : ι) → Quotient (S i)\na : (i : ι) → α i\n⊢ (finChoice fun x ↦ ⟦a x⟧).eval = fun x ↦ ⟦a x⟧",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Quotient.eval",
"Quotient.finC... | rw [finChoice_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fintype.Quotient | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 19
} | [
{
"pp": "case h.h\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nS : (i : ι) → Setoid (α i)\nβ : Sort u_3\na : (i : ι) → α i\nf : ((i : ι) → α i) → β\nh : ∀ (a b : (i : ι) → α i), (∀ (i : ι), a i ≈ b i) → f a = f b\n⊢ (finChoice fun x ↦ ⟦a x⟧).liftOn f h = f a",
"usedConstants":... | rw [finChoice_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fintype.Quotient | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 21
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nS : (i : ι) → Setoid (α i)\nβ : Sort u_3\nq : (i : ι) → Quotient (S i)\na : (i : ι) → α i\n⊢ (finChoice fun x ↦ ⟦a x⟧).eval = fun x ↦ ⟦a x⟧",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Quotient.eval",
... | rw [finChoice_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fintype.Quotient | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 19
} | [
{
"pp": "case h.h\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nα : ι → Sort u_2\nS : (i : ι) → Setoid (α i)\nC : ((i : ι) → Quotient (S i)) → Sort u_4\na : (i : ι) → α i\nf : (a : (i : ι) → α i) → C fun x ↦ ⟦a x⟧\nh : ∀ (a b : (i : ι) → α i), (∀ (i : ι), a i ≈ b i) → f a ≍ f b\n⊢ Quotient.hrecOn (f... | rw [finChoice_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.DFinsupp.Sigma | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 20
} | [
{
"pp": "case h.h.inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → DecidableEq (α i)\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\ni : ι\nj : α i\nx : δ ⟨i, j⟩.fst ⟨i, j⟩.snd\nj' : α i\nhj : j' ≠ j\n⊢ ⟨i, j'⟩ ≠ ⟨i, j⟩",
"usedConstants": [
"Eq... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.DFinsupp.Sigma | {
"line": 152,
"column": 6
} | {
"line": 152,
"column": 20
} | [
{
"pp": "case h.inl.inr\nι : Type u\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → (j : α i) → Zero (δ i j)\ninst✝ : (i : ι) → DecidableEq (α i)\ni : ι\nj : α i\nx : δ i j\nj' : α i\nhj : j' ≠ j\n⊢ ⟨i, j'⟩ ≠ ⟨i, j⟩",
"usedConstants": [
"Eq.mpr",
"congrA... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Data.DFinsupp.Sigma | {
"line": 166,
"column": 8
} | {
"line": 166,
"column": 25
} | [
{
"pp": "case h.h\nι : Type u\nγ : Type w\nβ : ι → Type v\nβ₁ : ι → Type v₁\nβ₂ : ι → Type v₂\nκ : Type u_1\nα : ι → Type u_2\nδ : (i : ι) → α i → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → (j : α i) → Zero (δ i j)\nf : Π₀ (i : ι) (j : α i), δ i j\ni : ι\nj : α i\n⊢ (f.sigmaUncurry.sigmaCurry i) j = (f i... | sigmaCurry_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.DFinsupp.Defs | {
"line": 129,
"column": 54
} | {
"line": 131,
"column": 5
} | [
{
"pp": "ι : Type u\nβ₁ : ι → Type v₁\ninst✝ : (i : ι) → Zero (β₁ i)\nh : ∀ (i : ι), id 0 = 0\ng : Π₀ (i : ι), β₁ i\n⊢ mapRange (fun i ↦ id) h g = g",
"usedConstants": [
"DFinsupp.ext",
"DFinsupp.instDFunLike",
"id",
"DFinsupp.mapRange",
"Eq.refl",
"DFinsupp",
"DFun... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.ToDFinsupp | {
"line": 270,
"column": 16
} | {
"line": 270,
"column": 36
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nη : ι → Type u_4\nN : Type u_5\ninst✝¹ : Semiring R\ninst✝ : Zero N\nf : (i : ι) × η i →₀ N\n⊢ (fun f ↦ onFinset (f.support.sigma fun j ↦ (f j).support) (fun ji ↦ (f ji.fst) ji.snd) ⋯)\n ((fun f ↦ { toFun := f.split, support' := Trunc.mk ⟨f.splitSupport.va... | by ext; simp [split] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.ToDFinsupp | {
"line": 271,
"column": 17
} | {
"line": 271,
"column": 37
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nM : Type u_3\nη : ι → Type u_4\nN : Type u_5\ninst✝¹ : Semiring R\ninst✝ : Zero N\nf : Π₀ (i : ι), η i →₀ N\n⊢ (fun f ↦ { toFun := f.split, support' := Trunc.mk ⟨f.splitSupport.val, ⋯⟩ })\n ((fun f ↦ onFinset (f.support.sigma fun j ↦ (f j).support) (fun ji ↦ (f ji.fst) ... | by ext; simp [split] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.ToDFinsupp | {
"line": 310,
"column": 90
} | {
"line": 312,
"column": 5
} | [
{
"pp": "ι : Type u_1\nη : ι → Type u_4\nN : Type u_5\ninst✝ : AddZeroClass N\nf g : (i : ι) × η i →₀ N\n⊢ sigmaFinsuppEquivDFinsupp (f + g) = sigmaFinsuppEquivDFinsupp f + sigmaFinsuppEquivDFinsupp g",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"DFinsupp.ext",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 21
} | [
{
"pp": "case neg\nι : 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\ni : ι\nb : β i\nh✝ : (i : ι) → β i → γ\nh_zero : h✝ i 0 = 1\nh : ¬b ≠ 0\n⊢ (single i b).prod h✝ = h✝ i b",
"usedConstants"... | rw [not_not] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Finsupp.ToDFinsupp | {
"line": 327,
"column": 75
} | {
"line": 329,
"column": 5
} | [
{
"pp": "ι : Type u_1\nη : ι → Type u_4\nN : Type u_5\nR : Type u_6\ninst✝² : Monoid R\ninst✝¹ : AddMonoid N\ninst✝ : DistribMulAction R N\nr : R\nf : (i : ι) × η i →₀ N\n⊢ sigmaFinsuppEquivDFinsupp (r • f) = r • sigmaFinsuppEquivDFinsupp f",
"usedConstants": [
"Finsupp.instFunLike",
"Finsupp.sm... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.Tuple.Reflection | {
"line": 49,
"column": 20
} | {
"line": 54,
"column": 11
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nn : ℕ\nf : Fin (n + 1) → α → β\nv : Fin (n + 1) → α\ni : Fin (n + 1)\n⊢ seq f v i = f i (v i)",
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Fin.succ",
"congrArg",
"FinVec.seq._proof_1",
"id",
"Fin.instOfNat",
"Matr... | by
simp_rw [seq, seq_eq]
refine i.cases ?_ fun i => ?_
· rfl
· rw [Matrix.cons_val_succ]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 395,
"column": 6
} | {
"line": 395,
"column": 23
} | [
{
"pp": "ι : Type u\nγ : Type w\nβ : ι → Type v\ninst✝³ : DecidableEq ι\ninst✝² : (i : ι) → AddGroup (β i)\ninst✝¹ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\ninst✝ : AddCommGroup γ\nf g : Π₀ (i : ι), β i\nh : (i : ι) → β i → γ\nh_sub : ∀ (i : ι) (b₁ b₂ : β i), h i (b₁ - b₂) = h i b₁ - h i b₂\nthis :\n (liftAdd... | liftAddHom_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.DFinsupp.BigOperators | {
"line": 421,
"column": 6
} | {
"line": 421,
"column": 23
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → AddCommMonoid (β i)\ninst✝ : (i : ι) → (x : β i) → Decidable (x ≠ 0)\nf : Π₀ (i : ι), β i\nthis : (liftAddHom (singleAddHom β)) f = (AddMonoidHom.id (Π₀ (i : ι), β i)) f\n⊢ f.sum single = f",
"usedConstants": [
"DFinsupp.l... | liftAddHom_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 48
} | [
{
"pp": "case succ\nn✝ : ℕ\ni : Fin (n✝ + 1 + 1)\n⊢ (finRotate (n✝ + 1 + 1)) i = i + 1",
"usedConstants": [
"Fin.eq_or_lt_of_le",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"Fin.last",
"instAddNat",
"OfNat.ofNat",
"Fin.le_last"
]
}
] | obtain rfl | h := Fin.eq_or_lt_of_le i.le_last | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 127,
"column": 86
} | {
"line": 129,
"column": 5
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\n⊢ id.dualMap = id",
"usedConstants": [
"LinearMap.id",
"Semiring.toModule",
"LinearMap.ext",
"CommSemiring.toSemiring",
"LinearMap.instFunLike",
"LinearMap.modul... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 157,
"column": 72
} | {
"line": 159,
"column": 5
} | [
{
"pp": "R : Type u_1\nM₁ : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R M₁\n⊢ (refl R M₁).dualMap = refl R (Dual R M₁)",
"usedConstants": [
"Semiring.toModule",
"LinearMap.ext",
"CommSemiring.toSemiring",
"LinearMap.instFunLike",
"LinearEquiv.... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 67
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nU V : Submodule R M\n⊢ U.dualAnnihilator ⊔ V.dualAnnihilator ≤ (U ⊓ V).dualAnnihilator",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Lattice.toSemilatticeSup",
"Semiring.toModul... | rw [le_dualAnnihilator_iff_le_dualCoannihilator, dualCoannihilator_sup_eq]
apply inf_le_inf <;> exact le_dualAnnihilator_dualCoannihilator _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dual.Defs | {
"line": 440,
"column": 2
} | {
"line": 441,
"column": 67
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nU V : Submodule R M\n⊢ U.dualAnnihilator ⊔ V.dualAnnihilator ≤ (U ⊓ V).dualAnnihilator",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Lattice.toSemilatticeSup",
"Semiring.toModul... | rw [le_dualAnnihilator_iff_le_dualCoannihilator, dualCoannihilator_sup_eq]
apply inf_le_inf <;> exact le_dualAnnihilator_dualCoannihilator _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 78
} | [
{
"pp": "case a\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : DecidableEq ι\np : ι → Prop\ninst✝ : DecidablePred p\nS : ι → Submodule R N\ni : ι\nhi : p i\ny : N\nhy : y ∈ S i\n⊢ (((lsum ℕ) fun i ↦ (S i).subtype) ∘ₗ filterLinearMap R (fun... | rw [LinearMap.comp_apply, filterLinearMap_apply, filter_single_pos _ _ hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 557,
"column": 2
} | {
"line": 557,
"column": 24
} | [
{
"pp": "case h\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝³ : DecidableEq ι\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\np : ι → Submodule R N\nh : ∀ (i : ι) (x : ↥(p i)) (v : Π₀ (i : ι), ↥(p i)), ((lsum ℕ) fun i ↦ (p i).subtype) (erase i v) = ↑x → x = 0\nm : Π₀ (i : ι), ↥(p i)\nhm : ((l... | refine h i (-m i) m ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Algebra.Opposite | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nr : R\nx... | simp only [RingHom.toOpposite_apply, Function.comp_apply, ← op_mul, Algebra.commutes] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Opposite | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nr : R\nx... | simp only [RingHom.toOpposite_apply, Function.comp_apply, ← op_mul, Algebra.commutes] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Opposite | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁹ : CommSemiring R\ninst✝⁸ : CommSemiring S\ninst✝⁷ : Semiring A\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Algebra S A\ninst✝¹ : SMulCommClass R S A\ninst✝ : IsScalarTower R S A\nr : R\nx... | simp only [RingHom.toOpposite_apply, Function.comp_apply, ← op_mul, Algebra.commutes] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.Module | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 39
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝¹ : AddMonoid M\ninst✝ : SMul R M\na₁ a₂ : R × M\nl₁ l₂ l : NF R M\nh : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval\n⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval",
"usedConstants": [
"instHSMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"ad... | simp only [eval_cons, ← h, add_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.Module | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 39
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝¹ : AddMonoid M\ninst✝ : SMul R M\na₁ a₂ : R × M\nl₁ l₂ l : NF R M\nh : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval\n⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval",
"usedConstants": [
"instHSMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"ad... | simp only [eval_cons, ← h, add_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.Module | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 39
} | [
{
"pp": "R : Type u_2\nM : Type u_3\ninst✝¹ : AddMonoid M\ninst✝ : SMul R M\na₁ a₂ : R × M\nl₁ l₂ l : NF R M\nh : l₁.eval + (a₂ ::ᵣ l₂).eval = l.eval\n⊢ (a₁ ::ᵣ l₁).eval + (a₂ ::ᵣ l₂).eval = (a₁ ::ᵣ l).eval",
"usedConstants": [
"instHSMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"ad... | simp only [eval_cons, ← h, add_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.AddChar | {
"line": 242,
"column": 60
} | {
"line": 243,
"column": 78
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝² : AddMonoid A\ninst✝¹ : AddMonoid B\ninst✝ : Monoid M\nf : A →+ B\nhf : Surjective ⇑f\n⊢ Injective fun ψ ↦ ψ.compAddMonoidHom f",
"usedConstants": [
"Eq.mpr",
"congrArg",
"DFunLike.ext'_iff",
"AddMonoid.toAddZeroClass",
... | by
rintro ψ χ h; rw [DFunLike.ext'_iff] at h ⊢; exact hf.injective_comp_right h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 34
} | [
{
"pp": "case a\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nη : Type u_6\nιs : η → Type u_7\nf : (j : η) → ιs j → M\nhindep : ∀ (j : η), LinearIndependent R (f j)\nhd : ∀ (i : η) (t : Set η), t.Finite → i ∉ t → Disjoint (span R (range (f i))) (⨆ i ∈ t, span R (rang... | rw [range_sigma_eq_iUnion_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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