module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.QuotientGroup.Basic | {
"line": 121,
"column": 55
} | {
"line": 121,
"column": 79
} | {
"line": 121,
"column": 80
} | [
{
"pp": "G : Type u\ninst✝¹ : Group G\nH : Type v\ninst✝ : Group H\nφ : G →* H\na✝ b✝ : G ⧸ φ.ker\na b : G\nh : φ.rangeRestrict a = φ.rangeRestrict b\n⊢ φ.rangeRestrict (a⁻¹ * b) = 1",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MonoidHom.rang... | [
"G : Type u\ninst✝¹ : Group G\nH : Type v\ninst✝ : Group H\nφ : G →* H\na✝ b✝ : G ⧸ φ.ker\na b : G\nh : φ.rangeRestrict a = φ.rangeRestrict b\n⊢ φ.rangeRestrict a⁻¹ * φ.rangeRestrict b = 1"
] | φ.rangeRestrict.map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Congruence.Basic | {
"line": 166,
"column": 31
} | {
"line": 166,
"column": 40
} | {
"line": 166,
"column": 40
} | [
{
"pp": "R : Type u_3\ninst✝¹ : Add R\ninst✝ : Mul R\nS : Set (RingCon R)\nx y : R\nh : (sInf S).toSetoid x y\nr : Setoid R\nx✝ : r ∈ (fun x ↦ x.toSetoid) '' S\nc : RingCon R\nhS : c ∈ S\nhr : (fun x ↦ x.toSetoid) c = r\n⊢ r x y",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"R : Type u_3\ninst✝¹ : Add R\ninst✝ : Mul R\nS : Set (RingCon R)\nx y : R\nh : (sInf S).toSetoid x y\nr : Setoid R\nx✝ : r ∈ (fun x ↦ x.toSetoid) '' S\nc : RingCon R\nhS : c ∈ S\nhr : (fun x ↦ x.toSetoid) c = r\n⊢ ((fun x ↦ x.toSetoid) c) x y"
] | rw [← hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Field.IsField | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 30
} | {
"line": 101,
"column": 2
} | [
{
"pp": "case hex\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃ x_1, x * x_1 = 1",
"ppTerm": "?hex",
"assigned": true,
"usedConstants": [
"Ring.toSemiring",
"IsField.mul_inv_cancel"
],
"usedFVars": [
"R",
"inst✝",
"hf",
"x",
"hx... | [] | exact hf.mul_inv_cancel hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Field.IsField | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 30
} | {
"line": 101,
"column": 2
} | [
{
"pp": "case hex\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃ x_1, x * x_1 = 1",
"ppTerm": "?hex",
"assigned": true,
"usedConstants": [
"Ring.toSemiring",
"IsField.mul_inv_cancel"
],
"usedFVars": [
"R",
"inst✝",
"hf",
"x",
"hx... | [] | exact hf.mul_inv_cancel hx | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.IsField | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 30
} | {
"line": 101,
"column": 2
} | [
{
"pp": "case hex\nR : Type u\ninst✝ : Ring R\nhf : IsField R\nx : R\nhx : x ≠ 0\n⊢ ∃ x_1, x * x_1 = 1",
"ppTerm": "?hex",
"assigned": true,
"usedConstants": [
"Ring.toSemiring",
"IsField.mul_inv_cancel"
],
"usedFVars": [
"R",
"inst✝",
"hf",
"x",
"hx... | [] | exact hf.mul_inv_cancel hx | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Quotient.Basic | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 32
} | {
"line": 312,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\ns : Set M₂\nh₀ : s.Nonempty\nh₁ : s ⊆ ↑f.range\n⊢... | [
"R : Type u_1\nM : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nR₂ : Type u_3\nM₂ : Type u_4\ninst✝³ : Ring R₂\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\ns : Set M₂\nh₀ : s.Nonempty\nh₁ : s ⊆ ↑f.range\ny : M₂ := Cla... | let y := Classical.choose h₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Data.Fintype.Pigeonhole | {
"line": 88,
"column": 2
} | {
"line": 94,
"column": 19
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Infinite α\ninst✝ : Finite β\nf : α → β\n⊢ ∃ y, Infinite ↑(f ⁻¹' {y})",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"False",
"Fintype.ofFinite",
"Finset.univ",
"congrArg",... | [] | classical
by_contra! hf
cases nonempty_fintype β
let key : Fintype α :=
{ elems := univ.biUnion fun y : β => (f ⁻¹' {y}).toFinset
complete := by simp }
exact key.false | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Data.Fintype.Pigeonhole | {
"line": 88,
"column": 2
} | {
"line": 94,
"column": 19
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Infinite α\ninst✝ : Finite β\nf : α → β\n⊢ ∃ y, Infinite ↑(f ⁻¹' {y})",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"False",
"Fintype.ofFinite",
"Finset.univ",
"congrArg",... | [] | classical
by_contra! hf
cases nonempty_fintype β
let key : Fintype α :=
{ elems := univ.biUnion fun y : β => (f ⁻¹' {y}).toFinset
complete := by simp }
exact key.false | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Fintype.Pigeonhole | {
"line": 88,
"column": 2
} | {
"line": 94,
"column": 19
} | {
"line": 95,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Infinite α\ninst✝ : Finite β\nf : α → β\n⊢ ∃ y, Infinite ↑(f ⁻¹' {y})",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"False",
"Fintype.ofFinite",
"Finset.univ",
"congrArg",... | [] | classical
by_contra! hf
cases nonempty_fintype β
let key : Fintype α :=
{ elems := univ.biUnion fun y : β => (f ⁻¹' {y}).toFinset
complete := by simp }
exact key.false | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Prod | {
"line": 610,
"column": 4
} | {
"line": 611,
"column": 21
} | {
"line": 612,
"column": 2
} | [
{
"pp": "case mp.right\nR : Type u\nM : Type v\nM₂ : Type w\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np₁ : Submodule R M\np₂ : Submodule R M₂\nq : Submodule R (M × M₂)\nh : q ≤ p₁.prod p₂\n⊢ map (LinearMap.snd R M M₂) q ≤ p₂",
"ppTer... | [] | · rintro x ⟨⟨y1, y2⟩, ⟨hy1, rfl⟩⟩
exact (h hy1).2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Prod | {
"line": 937,
"column": 26
} | {
"line": 937,
"column": 47
} | {
"line": 937,
"column": 47
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\ninst✝⁶ : RingHomSurjective σ\ninst✝⁵ : AddCommMonoid G\ninst✝⁴ : Module R G\ninst✝³ : AddCommMonoid H\ninst✝² : Module S H\ninst✝¹ : AddCommMonoid I\ninst✝ : Module S I\nf : G →ₛ... | [
"R : Type u_3\nS : Type u_4\nG : Type u_5\nH : Type u_6\nI : Type u_7\ninst✝⁸ : Semiring R\ninst✝⁷ : Semiring S\nσ : R →+* S\ninst✝⁶ : RingHomSurjective σ\ninst✝⁵ : AddCommMonoid G\ninst✝⁴ : Module R G\ninst✝³ : AddCommMonoid H\ninst✝² : Module S H\ninst✝¹ : AddCommMonoid I\ninst✝ : Module S I\nf : G →ₛₗ[σ] H × I\n... | ← LinearMap.mem_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Disjointed | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 78
} | {
"line": 172,
"column": 4
} | [
{
"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⊢ (Iic i).sup f \\ (f i \\ (Iio i).sup f) = (Iio i).sup f",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.to... | [
"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⊢ (Iic i).sup f \\ f i ≤ (Iio i).sup f"
] | rw [sdiff_sdiff_eq_sdiff_sup (sup_mono Iio_subset_Iic_self), sup_eq_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 80
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : PredOrder ι\ninst✝ : IsSuccArchimedean ι\nhι : Nonempty ι\n⊢ Countable ι",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Countable.of_equiv",
"instCountableInt",
"PartialOrder.toPreord... | [] | exact Countable.of_equiv _ orderIsoRangeToZOfLinearSuccPredArch.symm.toEquiv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 80
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : PredOrder ι\ninst✝ : IsSuccArchimedean ι\nhι : Nonempty ι\n⊢ Countable ι",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Countable.of_equiv",
"instCountableInt",
"PartialOrder.toPreord... | [] | exact Countable.of_equiv _ orderIsoRangeToZOfLinearSuccPredArch.symm.toEquiv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 351,
"column": 4
} | {
"line": 351,
"column": 80
} | {
"line": 353,
"column": 0
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : PredOrder ι\ninst✝ : IsSuccArchimedean ι\nhι : Nonempty ι\n⊢ Countable ι",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Countable.of_equiv",
"instCountableInt",
"PartialOrder.toPreord... | [] | exact Countable.of_equiv _ orderIsoRangeToZOfLinearSuccPredArch.symm.toEquiv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Multiset.NatAntidiagonal | {
"line": 70,
"column": 64
} | {
"line": 70,
"column": 78
} | {
"line": 70,
"column": 78
} | [
{
"pp": "n : ℕ\n⊢ (0, n + 1 + 1) ::ₘ\n Prod.map Nat.succ id (n + 1, 0) ::ₘ map (Prod.map Nat.succ id ∘ Prod.map id Nat.succ) (antidiagonal n) =\n (0, n + 2) ::ₘ (n + 2, 0) ::ₘ map (Prod.map Nat.succ Nat.succ) (antidiagonal n)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"E... | [
"n : ℕ\n⊢ (0, n + 1 + 1) ::ₘ ((n + 1).succ, id 0) ::ₘ map (Prod.map Nat.succ id ∘ Prod.map id Nat.succ) (antidiagonal n) =\n (0, n + 2) ::ₘ (n + 2, 0) ::ₘ map (Prod.map Nat.succ Nat.succ) (antidiagonal n)"
] | Prod.map_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 171,
"column": 4
} | {
"line": 172,
"column": 64
} | {
"line": 174,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∏ x ∈ Ico 1 (n + 1 + 1), x = (n + 1)!",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"HMul.hMul",
"Nat.succ_eq_add_one",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"... | [] | rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,
prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 171,
"column": 4
} | {
"line": 172,
"column": 64
} | {
"line": 174,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∏ x ∈ Ico 1 (n + 1 + 1), x = (n + 1)!",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"HMul.hMul",
"Nat.succ_eq_add_one",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"... | [] | rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,
prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Intervals | {
"line": 171,
"column": 4
} | {
"line": 172,
"column": 64
} | {
"line": 174,
"column": 0
} | [
{
"pp": "n : ℕ\n⊢ ∏ x ∈ Ico 1 (n + 1 + 1), x = (n + 1)!",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.zero_le",
"HMul.hMul",
"Nat.succ_eq_add_one",
"Monoid.toMulOneClass",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"... | [] | rw [prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n, Nat.factorial_succ,
prod_Ico_id_eq_factorial n, Nat.succ_eq_add_one, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Defs | {
"line": 79,
"column": 20
} | {
"line": 79,
"column": 24
} | {
"line": 79,
"column": 25
} | [
{
"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.51",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Semigroup.toMul",
"Semiring.toModule",
"HMul.hMul",... | [
"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 : 1 * x ∈ I\n⊢ x ∈ I"
] | hy', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maximal | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 16
} | {
"line": 94,
"column": 16
} | [
{
"pp": "α : Type u\ninst✝ : Semiring α\nM✝ : Ideal α\nhMmax : M✝.IsMaximal\n⊢ ¬⊤ ≤ M✝",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Submodule.instParti... | [
"α : Type u\ninst✝ : Semiring α\nM✝ : Ideal α\nhMmax : M✝.IsMaximal\n⊢ ¬M✝ = ⊤"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.IterateMapComap | {
"line": 59,
"column": 71
} | {
"line": 59,
"column": 85
} | {
"line": 59,
"column": 85
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf i : N →ₗ[R] M\nK : Submodule R N\nh : map f K ≤ map i K\nn : ℕ\nih : map f (f.iterateMapComap i n K) ≤ map i (f.iterateMapComap i n K)\n⊢ map i (... | [
"R : Type u_1\nN : Type u_2\nM : Type u_3\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nf i : N →ₗ[R] M\nK : Submodule R N\nh : map f K ≤ map i K\nn : ℕ\nih : map f (f.iterateMapComap i n K) ≤ map i (f.iterateMapComap i n K)\n⊢ map i (f.iterateMap... | ← le_comap_map | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Order.Filter.Bases.Basic | {
"line": 300,
"column": 52
} | {
"line": 301,
"column": 65
} | {
"line": 303,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nl : Filter α\np : ι → Prop\ns : ι → Set α\nhl : l.HasBasis p s\nq : α → Prop\n⊢ (∃ᶠ (x : α) in l, q x) ↔ ∀ (i : ι), p i → ∃ x ∈ s i, q x",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_forall_eq",
"Mathlib.Tactic.Pus... | [] | by
simp only [Filter.Frequently, hl.eventually_iff]; push Not; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Bases.Basic | {
"line": 331,
"column": 76
} | {
"line": 334,
"column": 79
} | {
"line": 336,
"column": 0
} | [
{
"pp": "α : Type u_1\nl : Filter α\nP : Set α → Prop\n⊢ l.HasBasis (fun s ↦ s ∈ l ∧ P s) id ↔ ∀ t ∈ l, ∃ r ∈ l, P r ∧ r ⊆ t",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Order.Filter.Bases.Basic.0.Filter.hasBasis_sel... | [] | by
simp only [hasBasis_iff, id, and_assoc]
exact forall_congr' fun s =>
⟨fun h => h.1, fun h => ⟨h, fun ⟨t, hl, _, hts⟩ => mem_of_superset hl hts⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Bases.Basic | {
"line": 388,
"column": 2
} | {
"line": 392,
"column": 17
} | {
"line": 394,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nι' : Sort u_5\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\np' : ι' → Prop\ns' : ι' → Set α\nhl : l.HasBasis p s\nhl' : l'.HasBasis p' s'\nh : ∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i\nh' : ∀ (i' : ι'), p' i' → ∃ i, p i ∧ s i ⊆ s' i'\n⊢ l = l'",
"ppTerm": "?m.16",
... | [] | apply le_antisymm
· rw [hl.le_basis_iff hl']
simpa using h'
· rw [hl'.le_basis_iff hl]
simpa using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Bases.Basic | {
"line": 388,
"column": 2
} | {
"line": 392,
"column": 17
} | {
"line": 394,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Sort u_4\nι' : Sort u_5\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\np' : ι' → Prop\ns' : ι' → Set α\nhl : l.HasBasis p s\nhl' : l'.HasBasis p' s'\nh : ∀ (i : ι), p i → ∃ i', p' i' ∧ s' i' ⊆ s i\nh' : ∀ (i' : ι'), p' i' → ∃ i, p i ∧ s i ⊆ s' i'\n⊢ l = l'",
"ppTerm": "?m.16",
... | [] | apply le_antisymm
· rw [hl.le_basis_iff hl']
simpa using h'
· rw [hl'.le_basis_iff hl]
simpa using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 631,
"column": 2
} | {
"line": 631,
"column": 66
} | {
"line": 632,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\nf : Filter β\nhα : IsEmpty α\n⊢ (comap Prod.snd f).NeBot ↔ Nonempty α ∧ f.NeBot",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"not_iff_not",
"Eq.mpr",
"congrArg",
"Filter.NeBot",
"id",
"Filter.not_neBot._si... | [
"case inr\nα : Type u_1\nβ : Type u_2\nf : Filter β\nhα : Nonempty α\n⊢ (comap Prod.snd f).NeBot ↔ Nonempty α ∧ f.NeBot"
] | · rw [filter_eq_bot_of_isEmpty (f.comap _), ← not_iff_not]; simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Interval.Set.Disjoint | {
"line": 228,
"column": 2
} | {
"line": 230,
"column": 27
} | {
"line": 232,
"column": 0
} | [
{
"pp": "ι : Sort u\nα : Type v\ninst✝ : LinearOrder α\nf : ι → α\nhf : ¬BddAbove (range f)\n⊢ ⋃ i, Iic (f i) = univ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.iUnion_mono''",
"Set.eq_univ_of_subset",
"Set.univ",
"PartialOrder.toPreorder",... | [] | refine Set.eq_univ_of_subset ?_ (iUnion_Iio_eq_univ_iff.mpr hf)
gcongr
exact Iio_subset_Iic_self | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.Disjoint | {
"line": 228,
"column": 2
} | {
"line": 230,
"column": 27
} | {
"line": 232,
"column": 0
} | [
{
"pp": "ι : Sort u\nα : Type v\ninst✝ : LinearOrder α\nf : ι → α\nhf : ¬BddAbove (range f)\n⊢ ⋃ i, Iic (f i) = univ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.iUnion_mono''",
"Set.eq_univ_of_subset",
"Set.univ",
"PartialOrder.toPreorder",... | [] | refine Set.eq_univ_of_subset ?_ (iUnion_Iio_eq_univ_iff.mpr hf)
gcongr
exact Iio_subset_Iic_self | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Tendsto | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 82
} | {
"line": 244,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\na : Filter α\nb : β\n⊢ Tendsto f a (pure b) ↔ ∀ᶠ (x : α) in a, f x = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"congrArg",
"Filter.map",
"PartialOrder.toPreorder",
... | [] | simp only [Tendsto, le_pure_iff, mem_map', mem_singleton_iff, Filter.Eventually] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.Tendsto | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 82
} | {
"line": 244,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\na : Filter α\nb : β\n⊢ Tendsto f a (pure b) ↔ ∀ᶠ (x : α) in a, f x = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"congrArg",
"Filter.map",
"PartialOrder.toPreorder",
... | [] | simp only [Tendsto, le_pure_iff, mem_map', mem_singleton_iff, Filter.Eventually] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Tendsto | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 82
} | {
"line": 244,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\na : Filter α\nb : β\n⊢ Tendsto f a (pure b) ↔ ∀ᶠ (x : α) in a, f x = b",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"congrArg",
"Filter.map",
"PartialOrder.toPreorder",
... | [] | simp only [Tendsto, le_pure_iff, mem_map', mem_singleton_iff, Filter.Eventually] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 813,
"column": 47
} | {
"line": 814,
"column": 82
} | {
"line": 816,
"column": 0
} | [
{
"pp": "α : Type u_1\nF : Filter α\ns : Set α\n⊢ F ⊓ 𝓟 s = ⊥ ↔ comap Subtype.val F = ⊥",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CompleteLattice.toLattice",
"Filter.push_pull'",
"congrArg",
"Filter.map",
"Filter.instCompleteLatticeFilt... | [] | by
rw [principal_eq_map_coe_top s, ← Filter.push_pull', inf_top_eq, map_eq_bot_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Action.TransferInstance | {
"line": 33,
"column": 17
} | {
"line": 33,
"column": 42
} | {
"line": 35,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nO : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid M\ne : α ≃ β\ninst✝ : MulAction M β\n⊢ ∀ (x y : M) (b : α), (x * y) • b = x • y • b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Equiv.apply_symm_apply",
... | [] | simp [smul_def, mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Group.Action.TransferInstance | {
"line": 33,
"column": 17
} | {
"line": 33,
"column": 42
} | {
"line": 35,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nO : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid M\ne : α ≃ β\ninst✝ : MulAction M β\n⊢ ∀ (x y : M) (b : α), (x * y) • b = x • y • b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Equiv.apply_symm_apply",
... | [] | simp [smul_def, mul_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Action.TransferInstance | {
"line": 33,
"column": 17
} | {
"line": 33,
"column": 42
} | {
"line": 35,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nO : Type u_3\nα : Type u_4\nβ : Type u_5\ninst✝¹ : Monoid M\ne : α ≃ β\ninst✝ : MulAction M β\n⊢ ∀ (x y : M) (b : α), (x * y) • b = x • y • b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Semigroup.toMul",
"Equiv.apply_symm_apply",
... | [] | simp [smul_def, mul_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GroupWithZero.Action.TransferInstance | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 3
} | {
"line": 67,
"column": 0
} | [
{
"pp": "M : Type u_1\nM₀ : Type u_2\nA : Type u_3\nB : Type u_4\ne : A ≃ B\ninst✝¹ : AddZeroClass B\ninst✝ : DistribSMul M B\n⊢ DistribSMul M A",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"SMulZeroClass",
"Equiv.smulZeroClass",
"Equiv.addZeroClass",
"AddZeroClas... | [] | letI := e.addZeroClass
exact {
e.smulZeroClass M with
smul_add := by simp [add_def, smul_def, smul_add]
} | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GroupWithZero.Action.TransferInstance | {
"line": 61,
"column": 2
} | {
"line": 65,
"column": 3
} | {
"line": 67,
"column": 0
} | [
{
"pp": "M : Type u_1\nM₀ : Type u_2\nA : Type u_3\nB : Type u_4\ne : A ≃ B\ninst✝¹ : AddZeroClass B\ninst✝ : DistribSMul M B\n⊢ DistribSMul M A",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"SMulZeroClass",
"Equiv.smulZeroClass",
"Equiv.addZeroClass",
"AddZeroClas... | [] | letI := e.addZeroClass
exact {
e.smulZeroClass M with
smul_add := by simp [add_def, smul_def, smul_add]
} | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 80
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : Preorder α\ninst✝¹ : IsDirectedOrder α\nF : Filter β\nu : α → β\ninst✝ : Nonempty α\n⊢ (F ⊓ map u atTop).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n, N ≤ n ∧ u n ∈ U",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.m... | [] | simp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 80
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝² : Preorder α\ninst✝¹ : IsDirectedOrder α\nF : Filter β\nu : α → β\ninst✝ : Nonempty α\n⊢ (F ⊓ map u atTop).NeBot ↔ ∀ U ∈ F, ∀ (N : α), ∃ n, N ≤ n ∧ u n ∈ U",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.m... | [] | simp_rw [inf_neBot_iff_frequently_left, frequently_map, frequently_atTop]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Noetherian.Defs | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 45
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ IsNoetherian R M ↔ WellFounded fun x1 x2 ↦ x1 > x2",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"isNoetherian_iff'",
"Eq.mpr",
"Submodule",
"Preorder.toLT",
... | [] | rw [isNoetherian_iff', ← isWellFounded_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Noetherian.Defs | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 45
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ IsNoetherian R M ↔ WellFounded fun x1 x2 ↦ x1 > x2",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"isNoetherian_iff'",
"Eq.mpr",
"Submodule",
"Preorder.toLT",
... | [] | rw [isNoetherian_iff', ← isWellFounded_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Noetherian.Defs | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 45
} | {
"line": 124,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\n⊢ IsNoetherian R M ↔ WellFounded fun x1 x2 ↦ x1 > x2",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"isNoetherian_iff'",
"Eq.mpr",
"Submodule",
"Preorder.toLT",
... | [] | rw [isNoetherian_iff', ← isWellFounded_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Congruence.Defs | {
"line": 145,
"column": 49
} | {
"line": 145,
"column": 72
} | {
"line": 147,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹ : SMul R M\ninst✝ : SMul S N\nφ : R → S\nf : M →ₑ[φ] N\nr : R\nx✝¹ x✝ : M\nh : f x✝¹ = f x✝\n⊢ f (r • x✝¹) = f (r • x✝)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instMulActionSemiHomClassMu... | [] | simp_rw [map_smulₛₗ, h] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.IsNormal | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 13
} | {
"line": 197,
"column": 4
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : ¬IsMax b\nIH : a < b → f a < f b\n⊢ a < succ b → f a... | [
"case succ\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : ¬IsMax b\nIH : a < b → f a < f b\nhab : a < succ b\n⊢ f a < f (s... | intro hab | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Order.IsNormal | {
"line": 196,
"column": 4
} | {
"line": 196,
"column": 13
} | {
"line": 197,
"column": 4
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : ¬IsMax b\nIH : a < b → f a < f b\n⊢ a < succ b → f a... | [
"case succ\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : ¬IsMax b\nIH : a < b → f a < f b\nhab : a < succ b\n⊢ f a < f (s... | intro hab | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Order.IsNormal | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 13
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case isSuccLimit\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : IsSuccLimit b\nIH : ∀ b_1 < b, a < b_1 → f a ... | [
"case isSuccLimit\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : IsSuccLimit b\nIH : ∀ b_1 < b, a < b_1 → f a < f b_1\nhab... | intro hab | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Order.IsNormal | {
"line": 201,
"column": 4
} | {
"line": 201,
"column": 13
} | {
"line": 202,
"column": 4
} | [
{
"pp": "case isSuccLimit\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : IsSuccLimit b\nIH : ∀ b_1 < b, a < b_1 → f a ... | [
"case isSuccLimit\nα : Type u_1\nβ : Type u_2\nf : α → β\ninst✝³ : LinearOrder α\ninst✝² : WellFoundedLT α\ninst✝¹ : SuccOrder α\ninst✝ : LinearOrder β\nhs : ∀ (a : α), f a < f (succ a)\nhl : ∀ {a : α}, IsSuccLimit a → IsLUB (f '' Iio a) (f a)\na b : α\nhb : IsSuccLimit b\nIH : ∀ b_1 < b, a < b_1 → f a < f b_1\nhab... | intro hab | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.Order.Module.Defs | {
"line": 977,
"column": 4
} | {
"line": 977,
"column": 54
} | {
"line": 978,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosReflectLT α β\n_b : βᵒᵈ\nhb : 0 ≤ _b\na₁ a₂ : α\nh : a₁ • _b < a₂ • _b\n⊢ a₁ < a... | [
"α : Type u_1\nβ : Type u_2\na a₁✝ a₂✝ : α\nb b₁ b₂ : β\ninst✝⁶ : Preorder α\ninst✝⁵ : Monoid α\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : DistribMulAction α β\ninst✝ : SMulPosReflectLT α β\n_b : βᵒᵈ\nhb : 0 ≤ _b\na₁ a₂ : α\nh : a₂ • -_b < a₁ • -_b\n⊢ a₁ < a₂"
] | rw [← neg_lt_neg_iff, ← smul_neg, ← smul_neg] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1249,
"column": 13
} | {
"line": 1249,
"column": 19
} | {
"line": 1249,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β✝\nι : Type u_3\nβ : ι → Type u_4\ninst✝⁵ : Zero α\ninst✝⁴ : (i : ι) → Zero (β i)\ninst✝³ : PartialOrder α\ninst✝² : (i : ι) → PartialOrder (β i)\ninst✝¹ : (i : ι) → SMulWithZero α (β i)\ninst✝ : ∀ (i : ι), PosSMulStrictMono α (β i)\n⊢ ∀ ⦃a : α⦄, 0 <... | [
"α : Type u_1\nβ✝ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β✝\nι : Type u_3\nβ : ι → Type u_4\ninst✝⁵ : Zero α\ninst✝⁴ : (i : ι) → Zero (β i)\ninst✝³ : PartialOrder α\ninst✝² : (i : ι) → PartialOrder (β i)\ninst✝¹ : (i : ι) → SMulWithZero α (β i)\ninst✝ : ∀ (i : ι), PosSMulStrictMono α (β i)\n⊢ ∀ ⦃a : α⦄,\n 0 < a → ∀ ... | lt_def | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1256,
"column": 13
} | {
"line": 1256,
"column": 19
} | {
"line": 1256,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β✝\nι : Type u_3\nβ : ι → Type u_4\ninst✝⁵ : Zero α\ninst✝⁴ : (i : ι) → Zero (β i)\ninst✝³ : PartialOrder α\ninst✝² : (i : ι) → PartialOrder (β i)\ninst✝¹ : (i : ι) → SMulWithZero α (β i)\ninst✝ : ∀ (i : ι), SMulPosStrictMono α (β i)\n⊢ ∀ ⦃b : (i : ι)... | [
"α : Type u_1\nβ✝ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β✝\nι : Type u_3\nβ : ι → Type u_4\ninst✝⁵ : Zero α\ninst✝⁴ : (i : ι) → Zero (β i)\ninst✝³ : PartialOrder α\ninst✝² : (i : ι) → PartialOrder (β i)\ninst✝¹ : (i : ι) → SMulWithZero α (β i)\ninst✝ : ∀ (i : ι), SMulPosStrictMono α (β i)\n⊢ ∀ ⦃b : (i : ι) → β i⦄,\n ... | lt_def | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1265,
"column": 13
} | {
"line": 1265,
"column": 19
} | {
"line": 1265,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ✝ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β✝\nι : Type u_3\nβ : ι → Type u_4\ninst✝⁵ : Zero α\ninst✝⁴ : (i : ι) → Zero (β i)\ninst✝³ : PartialOrder α\ninst✝² : (i : ι) → PartialOrder (β i)\ninst✝¹ : (i : ι) → SMulWithZero α (β i)\ninst✝ : ∀ (i : ι), SMulPosReflectLT α (β i)\n⊢ ∀ ⦃b : (i : ι) ... | [
"α : Type u_1\nβ✝ : Type u_2\na a₁ a₂ : α\nb b₁ b₂ : β✝\nι : Type u_3\nβ : ι → Type u_4\ninst✝⁵ : Zero α\ninst✝⁴ : (i : ι) → Zero (β i)\ninst✝³ : PartialOrder α\ninst✝² : (i : ι) → PartialOrder (β i)\ninst✝¹ : (i : ι) → SMulWithZero α (β i)\ninst✝ : ∀ (i : ι), SMulPosReflectLT α (β i)\n⊢ ∀ ⦃b : (i : ι) → β i⦄, 0 ≤ ... | lt_def | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 527,
"column": 64
} | {
"line": 529,
"column": 20
} | {
"line": 531,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\nh0 : 0 < o\na : o.ToType\n⊢ (enum fun x1 x2 ↦ x1 < x2) ⟨0, ⋯⟩ ≤ a",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"isWellOrder_lt",
"Ordinal.enum_zero_le",
"Ordinal.partialOrder",
"congrArg",
"... | [] | by
rw [← not_lt]
apply enum_zero_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Family | {
"line": 71,
"column": 48
} | {
"line": 71,
"column": 50
} | {
"line": 71,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal.{u}\nho : type r = o\nf : (a : Ordinal.{u}) → a < o → α\ni : Ordinal.{u}\nhi : i < o\n⊢ i ∈ Iio (type r)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.pa... | [
"α : Type u_1\nβ : Type u_2\nι : Type u\nr : ι → ι → Prop\ninst✝ : IsWellOrder ι r\no : Ordinal.{u}\nho : type r = o\nf : (a : Ordinal.{u}) → a < o → α\ni : Ordinal.{u}\nhi : i < o\n⊢ i ∈ Iio o"
] | ho | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 302,
"column": 32
} | {
"line": 302,
"column": 34
} | {
"line": 302,
"column": 34
} | [
{
"pp": "case inr\no : Ordinal.{u_4}\nho : IsSuccPrelimit o\n⊢ True ↔ IsSuccPrelimit o",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Order.IsSuccPrelimit",
"Ordinal.partialOrder",
"congrArg",
"PartialOrder.toPreorder",
"iff_self",
... | [] | ho | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.SetTheory.Ordinal.Family | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 54
} | {
"line": 295,
"column": 2
} | [
{
"pp": "ι : Type u\nf : ι → Ordinal.{max u v}\n⊢ #↑(range f) ≤ Cardinal.lift.{max v (u + 1) (v + 1), u} #ι",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.lift",
"Preorder.toLE",
... | [
"ι : Type u\nf : ι → Ordinal.{max u v}\n⊢ Cardinal.lift.{u, max u ((max u v) + 1)} #↑(range f) ≤ Cardinal.lift.{max v (u + 1) (v + 1), u} #ι"
] | rw [← Cardinal.lift_id'.{u, max u v + 1} #(range _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Family | {
"line": 560,
"column": 2
} | {
"line": 564,
"column": 31
} | {
"line": 565,
"column": 2
} | [
{
"pp": "case refine_2\nι : Type u_3\nf : ι → Ordinal.{max u_4 u_3}\nhf : ∀ a < lsub f, succ a < lsub f\ni : ι\nhle : iSup f ≤ f i\nheq : succ (iSup f) = lsub f\n⊢ False",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Or... | [
"case refine_2\nι : Type u_3\nf : ι → Ordinal.{max u_4 u_3}\nhf : ∀ a < lsub f, succ a < lsub f\ni : ι\nhle : iSup f ≤ f i\nheq : succ (iSup f) = lsub f\nthis : succ (iSup f) < lsub f\n⊢ False"
] | have :=
hf _
(by
rw [← heq]
exact lt_succ (iSup f)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1405,
"column": 11
} | {
"line": 1405,
"column": 24
} | {
"line": 1406,
"column": 2
} | [
{
"pp": "case nil\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nm : List α\no : Ordinal.{u}\nhm : m.SortedGT\nhl : [].SortedGT\nhlt : ∀ i ∈ [], (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRelEmbedding i < o\nhmltl : Lex (fun x1 x2 ↦ x1 < x2) m []\n⊢ ∀ i ∈ m, (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRel... | [] | simp at hmltl | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1405,
"column": 11
} | {
"line": 1405,
"column": 24
} | {
"line": 1406,
"column": 2
} | [
{
"pp": "case nil\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nm : List α\no : Ordinal.{u}\nhm : m.SortedGT\nhl : [].SortedGT\nhlt : ∀ i ∈ [], (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRelEmbedding i < o\nhmltl : Lex (fun x1 x2 ↦ x1 < x2) m []\n⊢ ∀ i ∈ m, (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRel... | [] | simp at hmltl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1405,
"column": 11
} | {
"line": 1405,
"column": 24
} | {
"line": 1406,
"column": 2
} | [
{
"pp": "case nil\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\nm : List α\no : Ordinal.{u}\nhm : m.SortedGT\nhl : [].SortedGT\nhlt : ∀ i ∈ [], (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRelEmbedding i < o\nhmltl : Lex (fun x1 x2 ↦ x1 < x2) m []\n⊢ ∀ i ∈ m, (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRel... | [] | simp at hmltl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 49,
"column": 72
} | {
"line": 50,
"column": 66
} | {
"line": 52,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_1}\na0 : a ≠ 0\n⊢ 0 ^ a = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.sub_eq_zero_iff_le",
"Ordinal.partialOrder",
"congrArg",
"instIsBotZeroClass",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreord... | [] | by
rwa [zero_opow', Ordinal.sub_eq_zero_iff_le, one_le_iff_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Family | {
"line": 882,
"column": 28
} | {
"line": 882,
"column": 30
} | {
"line": 882,
"column": 31
} | [
{
"pp": "case mp\nf : Ordinal.{u} → Ordinal.{max u v}\nh : ∀ (a : Ordinal.{u}), f a < f (succ a)\nH : ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o\no : Ordinal.{u}\n⊢ IsSuccLimit o → (o.blsub fun x x_1 ↦ f x) = f o",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
... | [
"case mp\nf : Ordinal.{u} → Ordinal.{max u v}\nh : ∀ (a : Ordinal.{u}), f a < f (succ a)\nH : ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o\no : Ordinal.{u}\nho : IsSuccLimit o\n⊢ (o.blsub fun x x_1 ↦ f x) = f o"
] | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.SetTheory.Ordinal.Family | {
"line": 882,
"column": 28
} | {
"line": 882,
"column": 30
} | {
"line": 882,
"column": 31
} | [
{
"pp": "case mpr\nf : Ordinal.{u} → Ordinal.{max u v}\nh : ∀ (a : Ordinal.{u}), f a < f (succ a)\nH : ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.blsub fun x x_1 ↦ f x) = f o\no : Ordinal.{u}\n⊢ IsSuccLimit o → (o.bsup fun x x_1 ↦ f x) = f o",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
... | [
"case mpr\nf : Ordinal.{u} → Ordinal.{max u v}\nh : ∀ (a : Ordinal.{u}), f a < f (succ a)\nH : ∀ (o : Ordinal.{u}), IsSuccLimit o → (o.blsub fun x x_1 ↦ f x) = f o\no : Ordinal.{u}\nho : IsSuccLimit o\n⊢ (o.bsup fun x x_1 ↦ f x) = f o"
] | ho | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 31
} | {
"line": 457,
"column": 0
} | [
{
"pp": "case inr\nb o : Ordinal.{u_1}\nho : o ≠ 0\nhb : 0 < b\n⊢ b ^ log b o ≤ o",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Ordinal.opow_log_le_self"
],
"usedFVars": [
"b",
"o",
"ho"
],
"usedGoals": []
}
] | [] | exact opow_log_le_self b ho | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 264,
"column": 54
} | {
"line": 266,
"column": 43
} | {
"line": 268,
"column": 0
} | [
{
"pp": "f : Ordinal.{u_1} → Ordinal.{u_1}\na : Ordinal.{u_1}\nn : ℕ\n⊢ f^[n] a ≤ nfp f a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"congrArg",
"iSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"... | [] | by
rw [← iSup_iterate_eq_nfp]
exact Ordinal.le_iSup (fun n ↦ f^[n] a) n | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 38
} | {
"line": 433,
"column": 38
} | [
{
"pp": "a b : Ordinal.{u_1}\n⊢ a + b ≤ b ↔ a * ω ≤ b",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Ordinal.omega0",
"Ordinal.partialOrder",
"MulZeroClass.toMul",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE... | [
"a b : Ordinal.{u_1}\n⊢ a + b ≤ b ↔ a + b = b"
] | ← add_eq_right_iff_mul_omega0_le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 484,
"column": 2
} | {
"line": 490,
"column": 27
} | {
"line": 491,
"column": 2
} | [
{
"pp": "case inr.refine_1\na b : Ordinal.{u_1}\nha : 0 < a\nhab : a * b = b\n⊢ a ^ ω ∣ b",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Preorder.toLT",
"Dvd.dvd",
"instHDiv",
"HMul.hMul",
"Ordinal.omega0",
... | [
"case inr.refine_2\na b : Ordinal.{u_1}\nha : 0 < a\nh : a ^ ω ∣ b\n⊢ a * b = b"
] | · rw [dvd_iff_mod_eq_zero]
rw [← div_add_mod b (a ^ ω), mul_add, ← mul_assoc, ← opow_one_add, one_add_omega0,
add_left_cancel_iff] at hab
rcases eq_zero_or_opow_omega0_le_of_mul_eq_right hab with hab | hab
· exact hab
refine (not_lt_of_ge hab (mod_lt b (opow_ne_zero ω ?_))).elim
rwa [← pos_iff... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 148,
"column": 20
} | {
"line": 148,
"column": 32
} | {
"line": 148,
"column": 32
} | [
{
"pp": "⊢ enumOrd {x | x.IsInitial} 0 = 0",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"setOf",
"Ordinal.enumOrd_zero",
"id",
"ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice",
"Ordinal.enumOrd",
"Ordi... | [
"⊢ sInf {x | x.IsInitial} = 0"
] | enumOrd_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 169,
"column": 49
} | {
"line": 176,
"column": 17
} | {
"line": 178,
"column": 0
} | [
{
"pp": "⊢ IsNormal ⇑preOmega",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Order.succ",
"StrictMono",
"Ordinal.isInitial_preOmega",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",... | [] | by
rw [isNormal_iff]
refine ⟨preOmega_strictMono, fun o ho a ha ↦
(preOmega_le_of_forall_lt (isInitial_ord _) fun b hb ↦ ?_).trans (ord_card_le a)⟩
rw [← (isInitial_ord _).card_lt_card, card_ord]
apply lt_of_lt_of_le _ (card_le_card <| ha _ (ho.succ_lt hb))
rw [(isInitial_preOmega _).card_lt_card, preOmeg... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Aleph | {
"line": 467,
"column": 70
} | {
"line": 467,
"column": 85
} | {
"line": 467,
"column": 85
} | [
{
"pp": "⊢ preAleph ω = ℵ₀",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.omega0",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder",
"Cardinal.preAleph_omega0",
"Preorder.toLE",
"id",
"... | [
"⊢ ℵ₀ = ℵ₀"
] | preAleph_omega0 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 65,
"column": 12
} | {
"line": 65,
"column": 18
} | {
"line": 65,
"column": 18
} | [
{
"pp": "case ind.mk\nα : Type u_1\nIH : ∀ y < #α, ℵ₀ ≤ y → y * y = y\nhc : ℵ₀ ≤ #α\nw✝¹ : LinearOrder α\nw✝ : WellFoundedLT α\nhα : (#α).ord = typeLT α\nthis✝ : NoMaxOrder α\ng : α × α → α := uncurry max\nf : α × α ↪ Lex (α × Lex (α × α)) := { toFun := fun p ↦ toLex (g p, toLex p), inj' := ⋯ }\ns : α × α → α ×... | [
"case ind.mk\nα : Type u_1\nIH : ∀ y < #α, ℵ₀ ≤ y → y * y = y\nhc : ℵ₀ ≤ #α\nw✝¹ : LinearOrder α\nw✝ : WellFoundedLT α\nhα : (#α).ord = typeLT α\nthis✝ : NoMaxOrder α\ng : α × α → α := uncurry max\nf : α × α ↪ Lex (α × Lex (α × α)) := { toFun := fun p ↦ toLex (g p, toLex p), inj' := ⋯ }\ns : α × α → α × α → Prop :=... | lt_ord | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 123,
"column": 88
} | {
"line": 126,
"column": 33
} | {
"line": 128,
"column": 0
} | [
{
"pp": "a b : Cardinal.{u_1}\nh : ℵ₀ ≤ a\n⊢ a * b ≤ max a b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"le_max_right",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"Cardinal",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"C... | [] | by
convert! mul_le_mul' (le_max_left a b) (le_max_right a b) using 1
rw [mul_eq_self]
exact h.trans (le_max_left a b) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 186,
"column": 2
} | {
"line": 186,
"column": 44
} | {
"line": 187,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\n⊢ (Order.cof α).ord.cof = Order.cof α",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"isWellOrder_lt",
"Cardinal",
"Subtype.wellFoundedLT",
"Ordinal.exists_ord_cof_eq",
... | [
"α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\ns : Set α\nhs : IsCofinal s\nhs' : typeLT ↑s = (Order.cof α).ord\n⊢ (Order.cof α).ord.cof = Order.cof α"
] | obtain ⟨s, hs, hs'⟩ := exists_ord_cof_eq α | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 212,
"column": 22
} | {
"line": 212,
"column": 38
} | {
"line": 213,
"column": 4
} | [
{
"pp": "n m : ℕ\nh2a : 1 ≤ n\nhb : 1 ≤ m\nh : n * m = n\nh2b✝ : 1 < m\n| n",
"ppTerm": "?m.341",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"Nat.instMulOneClass",
"MulOne.toMul",
"MulOneClass.toMulOne",
"Nat",
"One.t... | [
"n m : ℕ\nh2a : 1 ≤ n\nhb : 1 ≤ m\nh : n * m = n\nh2b✝ : 1 < m\n| n * 1"
] | rw [← mul_one n] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 308,
"column": 4
} | {
"line": 308,
"column": 23
} | {
"line": 310,
"column": 0
} | [
{
"pp": "case refine_2.inr\na b : Cardinal.{u_1}\nh3 : b = 0\n⊢ a + b = a",
"ppTerm": "?refine_2.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZero... | [] | · rw [h3, add_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 187,
"column": 4
} | {
"line": 189,
"column": 18
} | {
"line": 191,
"column": 0
} | [
{
"pp": "case refine_2\na o o' : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ng : (b : Ordinal.{u}) → b < o' → Ordinal.{u}\nhg : o.IsFundamentalSequence o' g\n⊢ (o'.blsub fun i hi ↦ f (g i hi) ⋯) = a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConst... | [] | rw [@blsub_comp.{u, u, u} o _ f (@IsFundamentalSequence.monotone _ _ f hf)]
· exact hf.2.2
· exact hg.2.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 187,
"column": 4
} | {
"line": 189,
"column": 18
} | {
"line": 191,
"column": 0
} | [
{
"pp": "case refine_2\na o o' : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ng : (b : Ordinal.{u}) → b < o' → Ordinal.{u}\nhg : o.IsFundamentalSequence o' g\n⊢ (o'.blsub fun i hi ↦ f (g i hi) ⋯) = a",
"ppTerm": "?refine_2",
"assigned": true,
"usedConst... | [] | rw [@blsub_comp.{u, u, u} o _ f (@IsFundamentalSequence.monotone _ _ f hf)]
· exact hf.2.2
· exact hg.2.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Regular | {
"line": 412,
"column": 2
} | {
"line": 412,
"column": 59
} | {
"line": 412,
"column": 59
} | [
{
"pp": "c : Cardinal.{u_1}\nhc : c.IsInaccessible\na : Ordinal.{u_1}\nha : a < c.ord\n⊢ preBeth a < c",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Ordinal.instLinearOrder",
"Preorder.toLT",
"Ordinal.partialOrder",
"Cardinal",
"PartialOrder.toPreorder",
... | [
"case ind\nc : Cardinal.{u_1}\nhc : c.IsInaccessible\na : Ordinal.{u_1}\nIH : ∀ y < a, y < c.ord → preBeth y < c\nha : a < c.ord\n⊢ preBeth a < c"
] | induction a using WellFoundedLT.induction with | ind a IH
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 15
} | {
"line": 581,
"column": 6
} | [
{
"pp": "case a.refine_1\nα : Type u_1\nh : (#α).IsStrongPrelimit\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : (#α).ord = type r\nha✝ : #α ≠ 0\nh' : (#α).IsStrongLimit\nha : ℵ₀ ≤ #α\nx : α\n⊢ IsSuccLimit (type r)",
"ppTerm": "?a.refine_1✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case a.refine_1\nα : Type u_1\nh : (#α).IsStrongPrelimit\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : (#α).ord = type r\nha✝ : #α ≠ 0\nh' : (#α).IsStrongLimit\nha : ℵ₀ ≤ #α\nx : α\n⊢ IsSuccLimit (#α).ord"
] | rw [← hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.DFinsupp.Ext | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 86
} | {
"line": 51,
"column": 2
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → AddZeroClass (β i)\nγ : Type w\ninst✝ : AddZeroClass γ\nf g : (Π₀ (i : ι), β i) →+ γ\nH : ∀ (i : ι) (y : β i), f (single i y) = g (single i y)\n⊢ f = g",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"DFi... | [
"ι : Type u\nβ : ι → Type v\ninst✝² : DecidableEq ι\ninst✝¹ : (i : ι) → AddZeroClass (β i)\nγ : Type w\ninst✝ : AddZeroClass γ\nf✝ g : (Π₀ (i : ι), β i) →+ γ\nH : ∀ (i : ι) (y : β i), f✝ (single i y) = g (single i y)\nf : Π₀ (i : ι), β i\nhf : f ∈ ⋃ i, Set.range (single i)\n⊢ f✝ f = g f"
] | refine AddMonoidHom.eq_of_eqOn_denseM add_closure_iUnion_range_single fun f hf => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 44
} | {
"line": 623,
"column": 2
} | [
{
"pp": "case mk\nα : Type u_1\nhc : ℵ₀ ≤ #α\nw✝¹ : LinearOrder α\nw✝ : WellFoundedLT α\nhα : (#α).ord = typeLT α\nthis : NoMaxOrder α\n⊢ #α < #α ^ (#α).ord.cof",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"isWellOrder_lt",
"Cardinal.instPowCardinal",
... | [
"case mk\nα : Type u_1\nhc : ℵ₀ ≤ #α\nw✝¹ : LinearOrder α\nw✝ : WellFoundedLT α\nhα : (#α).ord = typeLT α\nthis : NoMaxOrder α\ns : Set α\nhs : IsCofinal s\nhs' : typeLT ↑s = (Order.cof α).ord\n⊢ #α < #α ^ (#α).ord.cof"
] | obtain ⟨s, hs, hs'⟩ := exists_ord_cof_eq α | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Fintype.Quotient | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 34
} | {
"line": 160,
"column": 4
} | [
{
"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 : Quotient piSetoid\n⊢ finChoice q.eval = q",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Quotient.eval",
"Quotient.finChoice",
"Quotient.ind... | [
"case a\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\n⊢ finChoice ⟦a✝⟧.eval = ⟦a✝⟧"
] | induction q using Quotient.ind | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.DFinsupp.Defs | {
"line": 790,
"column": 15
} | {
"line": 790,
"column": 85
} | {
"line": 792,
"column": 0
} | [
{
"pp": "ι : Type u\nβ : ι → Type v\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddGroup (β i)\ns : Finset ι\nx y : (i : ↑↑s) → β ↑i\ni : ι\n⊢ (mk s (x - y)) i = (mk s x - mk s y) i",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"... | [] | by simp only [sub_apply, mk_apply]; split_ifs <;> [rfl; rw [sub_zero]] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.DFinsupp | {
"line": 459,
"column": 8
} | {
"line": 459,
"column": 25
} | {
"line": 460,
"column": 8
} | [
{
"pp": "case mpr.refine_1\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ns : Finset ι\np : ι → Submodule R N\na : N\nμ : (i : ι) → ↥(p i)\nhμ : ∑ i ∈ s, ↑(μ i) = a\ni : ι\nhi : i ∈ ↑s\n⊢ ↑(μ i) ∈ p ↑⟨i, hi⟩",
"ppTerm": "?mpr.refine_1",
"ass... | [
"case mpr.refine_1\nι : Type u_1\nR : Type u_3\nN : Type u_6\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ns : Finset ι\np : ι → Submodule R N\na : N\nμ : (i : ι) → ↥(p i)\nhμ : ∑ i ∈ s, ↑(μ i) = a\ni : ι\nhi : i ∈ ↑s\n⊢ ↑⟨i, hi⟩ ∈ s"
] | · exact coe_mem _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 253,
"column": 82
} | {
"line": 263,
"column": 24
} | {
"line": 265,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ni j : ι\nf : ι → M\nhij : i ≠ j\n⊢ LinearIndepOn R f {i, j} ↔ ∀ (c d : R), c • f i + d • f j = 0 → c = 0 ∧ d = 0",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
classical
rw [linearIndepOn_iff'']
refine ⟨fun h c d hcd ↦ ?_, fun h t g ht hg0 h0 ↦ ?_⟩
· specialize h {i, j} (Pi.single i c + Pi.single j d)
simpa +contextual [Finset.sum_pair, Pi.single_apply, hij, hij.symm, hcd] using h
have ht' : t ⊆ {i, j} := by simpa [← Finset.coe_subset]
rw [Finset.sum_subset... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 35
} | {
"line": 541,
"column": 36
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → M\nhf : ∀ (l : ι →₀ R), (linearCombination R f) l = 0 → l = 0\ni : ι\nm : M\nr : R\nhr : r ∈ nonZeroDivisors R\nl : ι →₀ R\nhl : l i ∈ nonZeroDivisors R\nhg : r • m... | [
"ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝³ : DecidableEq ι\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : ι → M\nhf : ∀ (l : ι →₀ R), (linearCombination R f) l = 0 → l = 0\ni : ι\nm : M\nr : R\nhr : r ∈ nonZeroDivisors R\nl : ι →₀ R\nhl : l i ∈ nonZeroDivisors R\nhg : r • m = (linearCo... | linearCombination_single_index, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 631,
"column": 2
} | {
"line": 633,
"column": 41
} | {
"line": 635,
"column": 0
} | [
{
"pp": "ι : Type u'\nK : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set ι\na : ι\nf : ι → V\nh : LinearIndepOn K f s\n⊢ f a ∈ span K (f '' s) ↔ LinearIndepOn K f (insert a s) → a ∈ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [] | by_cases has : a ∈ s
· exact iff_of_true (subset_span <| mem_image_of_mem f has) fun _ ↦ has
simp [linearIndepOn_insert_iff, h, has] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 631,
"column": 2
} | {
"line": 633,
"column": 41
} | {
"line": 635,
"column": 0
} | [
{
"pp": "ι : Type u'\nK : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set ι\na : ι\nf : ι → V\nh : LinearIndepOn K f s\n⊢ f a ∈ span K (f '' s) ↔ LinearIndepOn K f (insert a s) → a ∈ s",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [] | by_cases has : a ∈ s
· exact iff_of_true (subset_span <| mem_image_of_mem f has) fun _ ↦ has
simp [linearIndepOn_insert_iff, h, has] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 726,
"column": 35
} | {
"line": 726,
"column": 53
} | {
"line": 726,
"column": 53
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nn : ℕ\nv : Fin (n + 1) → V\n⊢ LinearIndependent K v ↔ LinearIndependent K (Fin.cons (v 0) (Fin.tail v))",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZero... | [
"K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nn : ℕ\nv : Fin (n + 1) → V\n⊢ LinearIndependent K v ↔ LinearIndependent K v"
] | Fin.cons_self_tail | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Multiset.Sort | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 69
} | {
"line": 67,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nr : α → α → Prop\ninst✝⁷ : DecidableRel r\ninst✝⁶ : IsTrans α r\ninst✝⁵ : Std.Antisymm r\ninst✝⁴ : Std.Total r\nr' : β → β → Prop\ninst✝³ : DecidableRel r'\ninst✝² : IsTrans β r'\ninst✝¹ : Std.Antisymm r'\ninst✝ : Std.Total r'\n⊢ ∀ (s : Multiset α),\n (∀ (a : α... | [] | exact Quot.ind fun l h => map_mergeSort (l := l) (by simpa using h) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 565,
"column": 26
} | {
"line": 565,
"column": 72
} | {
"line": 567,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nx : R[M]\nr : R\nm m₁ : M\ninst✝ : Mul M\nm₂ : M\nH : ∀ m' ∈ x.coeff.support, m' * m = m₁ ↔ m' = m₂\n⊢ (x.coeff.sum fun m' r' ↦ if m' = m₂ then r' * r else 0) = x.coeff m₂ * r",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"... | [] | by simp +contextual [Finsupp.sum_eq_single m₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 577,
"column": 26
} | {
"line": 577,
"column": 72
} | {
"line": 579,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nx : R[M]\nr : R\nm m₁ : M\ninst✝ : Mul M\nm₂ : M\nH : ∀ m' ∈ x.coeff.support, m * m' = m₁ ↔ m' = m₂\n⊢ (x.coeff.sum fun m' r' ↦ if m' = m₂ then r * r' else 0) = r * x.coeff m₂",
"ppTerm": "?m.124",
"assigned": true,
"usedConstants": [
"... | [] | by simp +contextual [Finsupp.sum_eq_single m₂] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 734,
"column": 4
} | {
"line": 734,
"column": 26
} | {
"line": 735,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝² : Semiring R\nx y : R[M]\nr r₁ r₂ : R\nm m' m₁ m₂ m₁' m₂' : M\ninst✝¹ : Monoid M\ninst✝ : Monoid N\n⊢ ∀ (a : R), (∃ b, singleOneRingHom a * b = 1 ∧ b * singleOneRingHom a = 1) → ∃ b, a * b = 1 ∧ b *... | [
"R : Type u_1\nS : Type u_2\nG : Type u_3\nM : Type u_4\nN : Type u_5\nO : Type u_6\nι : Type u_7\ninst✝² : Semiring R\nx✝ y : R[M]\nr r₁ r₂ : R\nm m' m₁ m₂ m₁' m₂' : M\ninst✝¹ : Monoid M\ninst✝ : Monoid N\na : R\nx : R[M]\nhax : singleOneRingHom a * x = 1\nhxa : x * singleOneRingHom a = 1\n⊢ ∃ b, a * b = 1 ∧ b * a... | rintro a ⟨x, hax, hxa⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1003,
"column": 2
} | {
"line": 1006,
"column": 10
} | {
"line": 1007,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nn : ℤ\n⊢ s ^ n = ∅ → s = ∅ ∧ n ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Fi... | [
"case mpr\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nn : ℤ\n⊢ s = ∅ ∧ n ≠ 0 → s ^ n = ∅"
] | · contrapose! +distrib
rintro (hs | rfl)
· exact hs.zpow
· simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 1007,
"column": 2
} | {
"line": 1008,
"column": 23
} | {
"line": 1010,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\nn : ℤ\n⊢ s = ∅ ∧ n ≠ 0 → s ^ n = ∅",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"Finset",... | [] | · rintro ⟨rfl, hn⟩
exact empty_zpow hn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finset.Sort | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 27
} | {
"line": 87,
"column": 27
} | [
{
"pp": "α : Type u_1\ns : Finset α\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : Std.Antisymm r\ninst✝ : Std.Total r\n⊢ s.sort r ~ s.toList",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Multiset",
"id",
"L... | [
"α : Type u_1\ns : Finset α\nr : α → α → Prop\ninst✝³ : DecidableRel r\ninst✝² : IsTrans α r\ninst✝¹ : Std.Antisymm r\ninst✝ : Std.Total r\n⊢ ↑(s.sort r) = ↑s.toList"
] | ← Multiset.coe_eq_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Sort | {
"line": 158,
"column": 29
} | {
"line": 158,
"column": 65
} | {
"line": 158,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\ns : Finset α\nh : s.Nonempty\n⊢ 0 < (s.sort fun a b ↦ a ≤ b).length",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"... | [] | rw [length_sort]; exact card_pos.2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Sort | {
"line": 158,
"column": 29
} | {
"line": 158,
"column": 65
} | {
"line": 158,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder α\ns : Finset α\nh : s.Nonempty\n⊢ 0 < (s.sort fun a b ↦ a ≤ b).length",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"... | [] | rw [length_sort]; exact card_pos.2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 81,
"column": 30
} | {
"line": 81,
"column": 47
} | {
"line": 81,
"column": 48
} | [
{
"pp": "case refine_1.single\nα : Type u_1\nR : Type u_5\ninst✝ : Semiring R\nβ : Type u_7\nu : α → β\ni : α\ns : R\n⊢ mapDomain u (single i s) ∈ span R (Set.range fun x ↦ single (u x) 1)",
"ppTerm": "?refine_1.single",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.to... | [
"case refine_1.single\nα : Type u_1\nR : Type u_5\ninst✝ : Semiring R\nβ : Type u_7\nu : α → β\ni : α\ns : R\n⊢ single (u i) s ∈ span R (Set.range fun x ↦ single (u x) 1)"
] | mapDomain_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 126,
"column": 2
} | {
"line": 127,
"column": 78
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Set (Submodule R M)\nm : M\n⊢ (∃ s, m ∈ ⨆ i ∈ s, ↑i) ↔ ∃ s, ↑s ⊆ S ∧ m ∈ ⨆ i ∈ s, i",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Submodule",
"iSup",
"Function.I... | [
"case refine_1\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Set (Submodule R M)\nm : M\nx✝ : ∃ s, m ∈ ⨆ i ∈ s, ↑i\ns : Finset (Subtype (Membership.mem S))\nhs : m ∈ ⨆ i ∈ s, ↑i\n⊢ ↑(Finset.map (Function.Embedding.subtype fun x ↦ x ∈ S) s) ⊆ S",
"case refine_2... | refine ⟨fun ⟨s, hs⟩ ↦ ⟨s.map (Function.Embedding.subtype (· ∈ S)), ?_, ?_⟩,
fun ⟨s, hsS, hs⟩ ↦ ⟨s.preimage (↑) Subtype.coe_injective.injOn, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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