module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.BigOperators.Pi | {
"line": 89,
"column": 4
} | {
"line": 90,
"column": 64
} | {
"line": 92,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nκ : Type u_2\nR : Type u_5\ninst✝ : CommSemiring R\ns : Finset ι\nf : ι → Set κ\ng : ι → κ → R\nj : κ\nhj : j ∉ ⋂ x ∈ s, f x\n⊢ ∏ i ∈ s, (f i).indicator (g i) j = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"congrArg",
"CommSemiring.toSemi... | [] | obtain ⟨i, hi, hj⟩ := by simpa using hj
exact Finset.prod_eq_zero hi <| Set.indicator_of_notMem hj _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 543,
"column": 97
} | {
"line": 545,
"column": 61
} | {
"line": 547,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\np p' : Submodule R M\nf : M →ₛₗ[τ₁₂] M₂\nhab : p ≤ p'\nh : p... | [] | by
simp_rw [← comap_map_eq] at h
exact lt_of_le_of_ne (map_mono hab) (ne_of_apply_ne _ h.ne) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 609,
"column": 2
} | {
"line": 611,
"column": 48
} | {
"line": 613,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nS : Submodule R M\nhf : Surjective ⇑f\n⊢ ... | [] | rw [← LinearMap.range_eq_top] at hf ⊢
rw [← hf]
exact LinearMap.range_domRestrict_eq_range_iff | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Span.Basic | {
"line": 609,
"column": 2
} | {
"line": 611,
"column": 48
} | {
"line": 613,
"column": 0
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nM : Type u_4\nM₂ : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring R₂\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup M₂\ninst✝¹ : Module R₂ M₂\nτ₁₂ : R →+* R₂\ninst✝ : RingHomSurjective τ₁₂\nf : M →ₛₗ[τ₁₂] M₂\nS : Submodule R M\nhf : Surjective ⇑f\n⊢ ... | [] | rw [← LinearMap.range_eq_top] at hf ⊢
rw [← hf]
exact LinearMap.range_domRestrict_eq_range_iff | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Finset | {
"line": 351,
"column": 91
} | {
"line": 356,
"column": 36
} | {
"line": 358,
"column": 0
} | [
{
"pp": "ι : Type u_5\ns : Finset ι\nf : ι → ℤ\nn : ℤ\nhf : ∀ i ∈ s, n ∣ f i\n⊢ (∑ i ∈ s, f i) / n = ∑ i ∈ s, f i / n",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"Int.instDiv",
"Int.ediv_mul_cancel",
"Dvd.dvd",
"ins... | [] | by
obtain rfl | hn := eq_or_ne n 0
· simp
rw [Int.ediv_eq_iff_eq_mul_left hn (dvd_sum hf), sum_mul]
refine sum_congr rfl fun s hs ↦ ?_
rw [Int.ediv_mul_cancel (hf _ hs)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Basic | {
"line": 55,
"column": 31
} | {
"line": 55,
"column": 54
} | {
"line": 55,
"column": 55
} | [
{
"pp": "case pos\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn ... | [
"case pos\nM : Type u_3\nM₂ : Type u_4\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid M₂\nF : Type u_5\ninst✝⁵ : FunLike F M M₂\ninst✝⁴ : AddMonoidHomClass F M M₂\nf : F\nR : Type u_6\nS : Type u_7\ninst✝³ : DivisionSemiring R\ninst✝² : DivisionSemiring S\ninst✝¹ : Module R M\ninst✝ : Module S M₂\nn : ℕ\nhR : ¬↑... | map_natCast_smul f R S, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.BigOperators.GroupWithZero.List | {
"line": 85,
"column": 6
} | {
"line": 88,
"column": 17
} | {
"line": 89,
"column": 6
} | [
{
"pp": "case h₂.h0\nR : Type u_1\ninst✝⁴ : CommMonoidWithZero R\ninst✝³ : PartialOrder R\ninst✝² : ZeroLEOneClass R\ninst✝¹ : PosMulStrictMono R\ninst✝ : NeZero 1\nι : Type u_2\ns✝ : List ι\nf g : ι → R\na : ι\ns : List ι\nhs : a :: s ≠ []\nh0 : ∀ (i : ι), i ∈ a :: s → 0 < f i\nh : ∀ (i : ι), i ∈ a :: s → f i ... | [
"case h\nR : Type u_1\ninst✝⁴ : CommMonoidWithZero R\ninst✝³ : PartialOrder R\ninst✝² : ZeroLEOneClass R\ninst✝¹ : PosMulStrictMono R\ninst✝ : NeZero 1\nι : Type u_2\ns✝ : List ι\nf g : ι → R\na : ι\ns : List ι\nhs : a :: s ≠ []\nh0 : ∀ (i : ι), i ∈ a :: s → 0 < f i\nh : ∀ (i : ι), i ∈ a :: s → f i < g i\nthis : Mu... | · intro i hi
apply le_of_lt
apply h0
simp [hi] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Finsupp.Basic | {
"line": 419,
"column": 60
} | {
"line": 423,
"column": 54
} | {
"line": 425,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\ninst✝ : AddCommMonoid M\nf : α → β\nhf : Injective f\n⊢ Injective (mapDomain f)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Finsupp.ext",
"congrArg",
"Finsupp.mapDomain",
... | [] | by
intro v₁ v₂ eq
ext a
have : mapDomain f v₁ (f a) = mapDomain f v₂ (f a) := by rw [eq]
rwa [mapDomain_apply hf, mapDomain_apply hf] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Finsupp.Basic | {
"line": 986,
"column": 6
} | {
"line": 986,
"column": 27
} | {
"line": 986,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_5\nN : Type u_6\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\n⊢ (f.curry.sum fun a f ↦ f.sum (g a)) = f.sum fun p c ↦ g p.1 p.2 c",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg"... | [
"α : Type u_1\nβ : Type u_2\nM : Type u_5\nN : Type u_6\ninst✝¹ : Zero M\ninst✝ : AddCommMonoid N\nf : α × β →₀ M\ng : α → β → M → N\n⊢ (f.curry.uncurry.sum fun p c ↦ g p.1 p.2 c) = f.sum fun p c ↦ g p.1 p.2 c"
] | ← sum_uncurry_index', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 173,
"column": 31
} | {
"line": 173,
"column": 57
} | {
"line": 173,
"column": 58
} | [
{
"pp": "M : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\n⊢ span R s = span R (Set.range Subtype.val)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"setOf",
"Membersh... | [
"M : Type u_2\nR : Type u_5\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\n⊢ span R s = span R {x | x ∈ s}"
] | Subtype.range_coe_subtype, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Finsupp.LinearCombination | {
"line": 475,
"column": 32
} | {
"line": 475,
"column": 90
} | {
"line": 477,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Finset M\nf : M → R\n⊢ ∑ a ∈ s, f a • a ∈ span R ↑s",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"Submodule",
"Submodule.addSubmonoidClass",
"instHSMul",
... | [] | exact sum_mem fun x hx ↦ smul_mem _ _ <| subset_span <| hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Fintype.Fin | {
"line": 59,
"column": 45
} | {
"line": 64,
"column": 67
} | {
"line": 66,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : ℕ\ninst✝ : DecidableEq α\na : α\nv : List.Vector α n\n⊢ #{i | v.get i = a} = List.count a v.toList",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"List.Vector.get",
"Eq.mpr",
"instNeZeroNatHAdd_1",
"Finset.univ",
"instLawfulBEq",
... | [] | by
induction v with
| nil => simp
| @cons n x xs hxs =>
simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Vector.get_cons_succ,
hxs, List.count_cons, add_comm (ite (x = a) 1 0), beq_iff_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Fin.Basic | {
"line": 363,
"column": 4
} | {
"line": 363,
"column": 67
} | {
"line": 364,
"column": 4
} | [
{
"pp": "m n✝ n : ℕ\ninst✝ : NeZero n\np : ℕ × Fin n\n⊢ (fun a ↦ (a / n, Fin.ofNat n a)) ((fun p ↦ p.1 * n + ↑p.2) p) = p",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"instHDiv",
"HMul.hMul",
"Fin.ofNat",
"Fin.is_lt",
"HDiv.hDiv",
"Prod.mk",
"in... | [
"m n✝ n : ℕ\ninst✝ : NeZero n\np : ℕ × Fin n\n⊢ ((fun a ↦ (a / n, Fin.ofNat n a)) ((fun p ↦ p.1 * n + ↑p.2) p)).1 = p.1"
] | refine Prod.ext ?_ (Fin.ext <| Nat.mul_add_mod_of_lt p.2.is_lt) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 131,
"column": 19
} | {
"line": 131,
"column": 53
} | {
"line": 132,
"column": 2
} | [
{
"pp": "n : ℕ\nk i : Fin n\n⊢ (fun i ↦ i - k) ((fun i ↦ i + k) i) = i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Fin.instSub",
"congrArg",
"Zero.ofOfNat0",
"HSub.hSub",
"Fin.pos",
"AddCommGroup.toAddGroup",
"instOfNatNat",
"NeZero.of_p... | [] | haveI := NeZero.of_pos k.pos; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 131,
"column": 19
} | {
"line": 131,
"column": 53
} | {
"line": 132,
"column": 2
} | [
{
"pp": "n : ℕ\nk i : Fin n\n⊢ (fun i ↦ i - k) ((fun i ↦ i + k) i) = i",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Fin.instSub",
"congrArg",
"Zero.ofOfNat0",
"HSub.hSub",
"Fin.pos",
"AddCommGroup.toAddGroup",
"instOfNatNat",
"NeZero.of_p... | [] | haveI := NeZero.of_pos k.pos; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 132,
"column": 20
} | {
"line": 132,
"column": 54
} | {
"line": 134,
"column": 0
} | [
{
"pp": "n : ℕ\nk i : Fin n\n⊢ (fun i ↦ i + k) ((fun i ↦ i - k) i) = i",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Fin.instSub",
"congrArg",
"Zero.ofOfNat0",
"HSub.hSub",
"Fin.pos",
"AddCommGroup.toAddGroup",
"instOfNatNat",
"NeZero.of_p... | [] | haveI := NeZero.of_pos k.pos; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Fin.Rotate | {
"line": 132,
"column": 20
} | {
"line": 132,
"column": 54
} | {
"line": 134,
"column": 0
} | [
{
"pp": "n : ℕ\nk i : Fin n\n⊢ (fun i ↦ i + k) ((fun i ↦ i - k) i) = i",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Fin.instSub",
"congrArg",
"Zero.ofOfNat0",
"HSub.hSub",
"Fin.pos",
"AddCommGroup.toAddGroup",
"instOfNatNat",
"NeZero.of_p... | [] | haveI := NeZero.of_pos k.pos; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENat.Pow | {
"line": 86,
"column": 2
} | {
"line": 89,
"column": 38
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case coe.top\nx : ℕ∞\nh : x ≠ 0\na✝ : ℕ\ny_z : ↑a✝ ≤ ⊤\n⊢ (fun y ↦ x ^ y) ↑a✝ ≤ (fun y ↦ x ^ y) ⊤",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"False",
"_private.Mathlib.Data.ENat.Pow.0.ENat.epow_right_mono._simp_1_1",
"Preorder.toLT",
"NeZero.one",
... | [
"case coe.coe\nx : ℕ∞\nh : x ≠ 0\na✝¹ a✝ : ℕ\ny_z : ↑a✝¹ ≤ ↑a✝\n⊢ (fun y ↦ x ^ y) ↑a✝¹ ≤ (fun y ↦ x ^ y) ↑a✝"
] | · rcases lt_trichotomy x 1 with x_0 | rfl | x_2
· exact (h (Order.lt_one_iff.1 x_0)).rec
· simp only [one_epow, le_refl]
· simp only [epow_top x_2, le_top] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Group.ModEq | {
"line": 125,
"column": 85
} | {
"line": 129,
"column": 82
} | {
"line": 131,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\na b p : M\nn : ℕ\nh : a ≡ b [PMOD p]\n⊢ n • a ≡ n • b [PMOD n • p]",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHSMul",
"HMul.hMul",
"congrArg",
"AddMonoid.toAddZeroC... | [] | by
rw [modEq_iff_nsmul] at *
rcases h with ⟨k, l, h⟩
use k, l
rw [← mul_nsmul, mul_nsmul', ← nsmul_add, h, nsmul_add, ← mul_nsmul, mul_nsmul'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.ModEq | {
"line": 328,
"column": 6
} | {
"line": 328,
"column": 29
} | {
"line": 328,
"column": 30
} | [
{
"pp": "G : Type u_1\ninst✝ : AddCommGroup G\np a b : G\n⊢ ¬a ≡ b [PMOD p] ↔ ∀ (z : ℤ), b ≠ a + z • p",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddCommGroup.ModEq",
"instHSMul",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddCommGroup.... | [
"G : Type u_1\ninst✝ : AddCommGroup G\np a b : G\n⊢ (¬∃ z, b = a + z • p) ↔ ∀ (z : ℤ), b ≠ a + z • p"
] | modEq_iff_eq_add_zsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.GCD.Basic | {
"line": 269,
"column": 2
} | {
"line": 269,
"column": 38
} | {
"line": 270,
"column": 2
} | [
{
"pp": "n m k : ℕ\nhkm : m ∣ k\nhkn : n ∣ k\nc : ℕ\nhmc : k / m ∣ c\nhnc : k / n ∣ c\n⊢ k / m.gcd n ∣ c",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"Dvd.dvd",
"instHDiv",
"HDiv.hDiv",
"instOfNatNat",
"Or.casesOn",
"GT.gt",
"Na... | [
"case inl\nn m k : ℕ\nhkm : m ∣ k\nhkn : n ∣ k\nc : ℕ\nhmc : k / m ∣ c\nhnc : k / n ∣ c\nhm : m = 0\n⊢ k / m.gcd n ∣ c",
"case inr\nn m k : ℕ\nhkm : m ∣ k\nhkn : n ∣ k\nc : ℕ\nhmc : k / m ∣ c\nhnc : k / n ∣ c\nhm : m > 0\n⊢ k / m.gcd n ∣ c"
] | rcases m.eq_zero_or_pos with hm | hm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.BigOperators.Fin | {
"line": 416,
"column": 45
} | {
"line": 417,
"column": 26
} | {
"line": 419,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝ : CommMonoid M\nn : ℕ\nf : Fin (n + 1) → M\na : Fin n\n⊢ ∏ i ∈ Ioi a.succ, f i = ∏ i ∈ Ioi a, f i.succ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Finset.Ioi",
"Fin.succ",
"congrArg",
"Finset",
"PartialOrder.toPreorder",
... | [] | by
simp [← map_succEmb_Ioi] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Fin | {
"line": 487,
"column": 69
} | {
"line": 487,
"column": 87
} | {
"line": 489,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\nn : ℕ\nf : Fin n → M\n⊢ partialProd f 0 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"MulOneClass.toMulOne",
"eq_self",
"of_eq_true",
"One.toOfNat1",
"OfNat.ofNa... | [] | simp [partialProd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Fin | {
"line": 487,
"column": 69
} | {
"line": 487,
"column": 87
} | {
"line": 489,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\nn : ℕ\nf : Fin n → M\n⊢ partialProd f 0 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"MulOneClass.toMulOne",
"eq_self",
"of_eq_true",
"One.toOfNat1",
"OfNat.ofNa... | [] | simp [partialProd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Fin | {
"line": 487,
"column": 69
} | {
"line": 487,
"column": 87
} | {
"line": 489,
"column": 0
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\nn : ℕ\nf : Fin n → M\n⊢ partialProd f 0 = 1",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"MulOneClass.toMulOne",
"eq_self",
"of_eq_true",
"One.toOfNat1",
"OfNat.ofNa... | [] | simp [partialProd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Fin | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 20
} | {
"line": 498,
"column": 2
} | [
{
"pp": "M : Type u_2\ninst✝ : Monoid M\nn : ℕ\nf : Fin (n + 1) → M\nj : Fin (n + 1)\n⊢ partialProd f j.succ = f 0 * partialProd (tail f) j",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instNeZeroNatHAdd_1",
"MulOne.toOne",
"HMul.hMul",
"Fin.succ"... | [
"M : Type u_2\ninst✝ : Monoid M\nn : ℕ\nf : Fin (n + 1) → M\nj : Fin (n + 1)\n⊢ f 0 * (List.take (↑j) (List.ofFn fun i ↦ f i.succ)).prod = f 0 * (List.take (↑j) (List.ofFn (tail f))).prod"
] | simp [partialProd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.ModEq | {
"line": 408,
"column": 2
} | {
"line": 408,
"column": 41
} | {
"line": 409,
"column": 2
} | [
{
"pp": "m a b c : ℕ\nhmc : m.gcd c = 1\nh : c * a ≡ c * b [MOD m]\n⊢ a ≡ b [MOD m]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nat.gcd",
"HMul.hMul",
"instMulNat",
"instOfNatNat",
"Or.casesOn",
"GT.gt",
"Nat.ModEq",
"Nat",
"Eq.ndr... | [
"case inl\na b c : ℕ\nhmc : gcd 0 c = 1\nh : c * a ≡ c * b [MOD 0]\n⊢ a ≡ b [MOD 0]",
"case inr\nm a b c : ℕ\nhmc : m.gcd c = 1\nh : c * a ≡ c * b [MOD m]\nhm : m > 0\n⊢ a ≡ b [MOD m]"
] | rcases m.eq_zero_or_pos with (rfl | hm) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.BigOperators.Fin | {
"line": 635,
"column": 6
} | {
"line": 635,
"column": 28
} | {
"line": 636,
"column": 6
} | [
{
"pp": "case succ.zero\nι : Type u_1\nM : Type u_2\nm : ℕ\nih : ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)), ∑ i, ↑(f i) * ∏ j, n (Fin.castLE ⋯ j) < ∏ i, n i\nn✝ : Fin (m + 1) → ℕ\nf✝ : (i : Fin (m + 1)) → Fin (n✝ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nfn : Fin 0\n⊢ ∑ i, ↑(f i) * ∏ j, n (Fin.castL... | [
"case succ.succ\nι : Type u_1\nM : Type u_2\nm : ℕ\nih : ∀ {n : Fin m → ℕ} (f : (i : Fin m) → Fin (n i)), ∑ i, ↑(f i) * ∏ j, n (Fin.castLE ⋯ j) < ∏ i, n i\nn✝¹ : Fin (m + 1) → ℕ\nf✝ : (i : Fin (m + 1)) → Fin (n✝¹ i)\nn : Fin m → ℕ\nf : (i : Fin m) → Fin (n i)\nn✝ : ℕ\nfn : Fin (n✝ + 1)\n⊢ ∑ i, ↑(f i) * ∏ j, n (Fin.... | · exact isEmptyElim fn | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 467,
"column": 52
} | {
"line": 467,
"column": 64
} | {
"line": 467,
"column": 64
} | [
{
"pp": "case inr\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nt : Set ι\nhs : LinearIndepOn R v s\nht : LinearIndepOn R v t\nhdj✝ : Disjoint (span R (v '' s)) (span R (v '' t))\na✝ : Nontrivial R\nhli : LinearIndependent R (Sum.el... | [
"case inr\nι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\nv : ι → M\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nt : Set ι\nhs : LinearIndepOn R v s\nht : LinearIndepOn R v t\nhdj✝ : Disjoint (span R (v '' s)) (span R (v '' t))\na✝ : Nontrivial R\nhli : LinearIndependent R (Sum.elim (fun x ↦ ... | Sum.elim_inr | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 565,
"column": 4
} | {
"line": 565,
"column": 76
} | {
"line": 566,
"column": 4
} | [
{
"pp": "case mp\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nκ : Type v\nw : κ → M\ni' : LinearIndependent R w\nj : ι → κ\ni : LinearIndependent R (w ∘ j)\np : i.Maximal\n⊢ Surjective j",
"ppTerm": "?mp",
"assigned": true... | [
"case mp\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nκ : Type v\nw : κ → M\ni' : LinearIndependent R w\nj : ι → κ\ni : LinearIndependent R (w ∘ j)\np : range (w ∘ j) = range w\n⊢ Surjective j"
] | specialize p (range w) i'.linearIndepOn_id (range_comp_subset_range _ _) | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 564,
"column": 2
} | {
"line": 567,
"column": 63
} | {
"line": 568,
"column": 2
} | [
{
"pp": "case mp\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\n⊢ i.Maximal → ∀ (κ : Type v) (w : κ → M), LinearIndependent R w → ∀ (j : ι → κ), w ∘ j = v → Surjective j",
"ppTerm": "?mp",
... | [
"case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\n⊢ (∀ (κ : Type v) (w : κ → M), LinearIndependent R w → ∀ (j : ι → κ), w ∘ j = v → Surjective j) → i.Maximal"
] | · rintro p κ w i' j rfl
specialize p (range w) i'.linearIndepOn_id (range_comp_subset_range _ _)
rw [range_comp, ← image_univ (f := w)] at p
exact range_eq_univ.mp (image_injective.mpr i'.injective p) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 575,
"column": 4
} | {
"line": 575,
"column": 39
} | {
"line": 576,
"column": 4
} | [
{
"pp": "case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v ≤ w\np : Surjective fun i ↦ ⟨v i, ⋯⟩\nq : (Subtype.val '' range fun ... | [
"case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ni' : LinearIndependent R Subtype.val\nh : range v ≤ w\np : Surjective fun i ↦ ⟨v i, ⋯⟩\nq : (fun x ↦ ↑⟨v x, ⋯⟩) '' univ = Subtype.... | rw [← image_univ, image_image] at q | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 568,
"column": 4
} | {
"line": 576,
"column": 17
} | {
"line": 578,
"column": 0
} | [
{
"pp": "case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\n⊢ (∀ (κ : Type v) (w : κ → M), LinearIndependent R w → ∀ (j : ι → κ), w ∘ j = v → Surjective j) → i.Maximal",
"ppTerm": "?mpr... | [] | intro p w i' h
specialize
p w ((↑) : w → M) i' (fun i => ⟨v i, range_subset_iff.mp h i⟩)
(by
ext
simp)
have q := congr_arg (fun s => ((↑) : w → M) '' s) p.range_eq
rw [← image_univ, image_image] at q
simpa using q | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 568,
"column": 4
} | {
"line": 576,
"column": 17
} | {
"line": 578,
"column": 0
} | [
{
"pp": "case mpr\nι : Type w\nR : Type u\ninst✝³ : Semiring R\ninst✝² : Nontrivial R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nv : ι → M\ni : LinearIndependent R v\n⊢ (∀ (κ : Type v) (w : κ → M), LinearIndependent R w → ∀ (j : ι → κ), w ∘ j = v → Surjective j) → i.Maximal",
"ppTerm": "?mpr... | [] | intro p w i' h
specialize
p w ((↑) : w → M) i' (fun i => ⟨v i, range_subset_iff.mp h i⟩)
(by
ext
simp)
have q := congr_arg (fun s => ((↑) : w → M) '' s) p.range_eq
rw [← image_univ, image_image] at q
simpa using q | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Basic | {
"line": 582,
"column": 67
} | {
"line": 582,
"column": 80
} | {
"line": 582,
"column": 80
} | [
{
"pp": "ι : Type u'\nR : Type u_2\nM : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.IsTorsionFree R M\ninst✝ : Subsingleton ι\nf : ι → M\nhe : Nonempty ι\n⊢ Subsingleton ι",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [],
... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Finrank | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 67
} | {
"line": 81,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nh : Module.rank R M ≤ ↑n\n⊢ finrank R M ≤ n",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroOneClass",
"Cardinal",
"congrArg",
"CommSemiring.to... | [
"case hc\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nh : Module.rank R M ≤ ↑n\n⊢ Module.rank R M < ℵ₀",
"case hd\nR : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nn : ℕ\nh : Module.rank R M ≤ ↑n\n⊢ ↑n < ℵ₀"
] | rwa [← Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.EuclideanDomain.Defs | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 23
} | {
"line": 140,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nm k : R\n⊢ m % k + m / k * k = m",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | rw [mul_comm]
exact mod_add_div _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.EuclideanDomain.Defs | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 23
} | {
"line": 140,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nm k : R\n⊢ m % k + m / k * k = m",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | [] | rw [mul_comm]
exact mod_add_div _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Card | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 51
} | {
"line": 252,
"column": 51
} | [
{
"pp": "α : Type u_1\ns t : Set α\nh : (s ∩ t).Finite\n⊢ (s \\ t).encard + (s ∩ t).encard < (t \\ s).encard + (s ∩ t).encard ↔ (s \\ t).encard < (t \\ s).encard",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"Eq.mpr",
... | [
"α : Type u_1\ns t : Set α\nh : (s ∩ t).Finite\n⊢ (s \\ t).encard < (t \\ s).encard ↔ (s \\ t).encard < (t \\ s).encard"
] | WithTop.add_lt_add_iff_right h.encard_lt_top.ne | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearIndependent.Defs | {
"line": 828,
"column": 93
} | {
"line": 829,
"column": 48
} | {
"line": 831,
"column": 0
} | [
{
"pp": "ι : Type u'\nR : Type u_2\ns : Set ι\nM : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : ι → M\n⊢ LinearIndepOn R v s ↔ Disjoint (Finsupp.supported R R s) (Finsupp.linearCombination R v).ker",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [] | by
rw [linearIndepOn_iff, LinearMap.disjoint_ker] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 80
} | {
"line": 681,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ s.ncard = 0 ↔ s = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.encard_eq_zero",
"Eq.mpr",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"congrArg",
... | [] | rw [← Nat.cast_inj (R := ℕ∞), hs.cast_ncard_eq, Nat.cast_zero, encard_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Card | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 80
} | {
"line": 681,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ s.ncard = 0 ↔ s = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.encard_eq_zero",
"Eq.mpr",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"congrArg",
... | [] | rw [← Nat.cast_inj (R := ℕ∞), hs.cast_ncard_eq, Nat.cast_zero, encard_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Card | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 80
} | {
"line": 681,
"column": 0
} | [
{
"pp": "α : Type u_1\ns : Set α\nhs : s.Finite\n⊢ s.ncard = 0 ↔ s = ∅",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Set.encard_eq_zero",
"Eq.mpr",
"Set.encard",
"instCharZeroENat",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"congrArg",
... | [] | rw [← Nat.cast_inj (R := ℕ∞), hs.cast_ncard_eq, Nat.cast_zero, encard_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.NatCard | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 28
} | {
"line": 117,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nhf : Function.Surjective f\nh : Nat.card β = 0\n⊢ Nat.card α = 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finite",
"finite_or_infinite",
"Nat.card",
"instOfNatNat",
"Or.casesOn",
"Nat",
"Eq.ref... | [
"case inl\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Function.Surjective f\nh : Nat.card β = 0\nh✝ : Finite β\n⊢ Nat.card α = 0",
"case inr\nα : Type u_1\nβ : Type u_2\nf : α → β\nhf : Function.Surjective f\nh : Nat.card β = 0\nh✝ : Infinite β\n⊢ Nat.card α = 0"
] | cases finite_or_infinite β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.GroupTheory.QuotientGroup.Basic | {
"line": 477,
"column": 16
} | {
"line": 477,
"column": 36
} | {
"line": 478,
"column": 4
} | [
{
"pp": "ι : Type u_1\nA : ι → Type u_2\ninst✝ : (i : ι) → CommGroup (A i)\nn : ℕ\n⊢ (fun x ↦ ↑(1 x)) = 1",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"MonoidHom.range",
"InvOneClass.toOne",
"CommMonoid.toCommSemigroup",
"DivInvOneMonoid.toInvOneClass",
"Gr... | [] | by simp [Pi.one_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Pi | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 25
} | {
"line": 193,
"column": 0
} | [
{
"pp": "R : Type u\nι : Type x\ninst✝³ : Semiring R\nφ : ι → Type i\ninst✝² : (i : ι) → AddCommMonoid (φ i)\ninst✝¹ : (i : ι) → Module R (φ i)\ninst✝ : DecidableEq ι\nI J : Set ι\nh : Disjoint I J\nj : ι\nhj : j ∈ J\nhi : j ∈ I\nb : φ j\n⊢ False",
"ppTerm": "?m.158",
"assigned": true,
"usedConstant... | [] | exact h.le_bot ⟨hi, hj⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.PartialSups | {
"line": 245,
"column": 2
} | {
"line": 249,
"column": 47
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : Preorder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : ConditionallyCompleteLinearOrder α\nf : ι → α\n⊢ ⨆ i, (partialSups f) i = ⨆ i, f i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice.toSemilatti... | [] | by_cases h : BddAbove (Set.range f)
· exact ciSup_partialSups_eq h
· rw [iSup, iSup, ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove _ h,
ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove _
(bddAbove_range_partialSups.not.mpr h)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.PartialSups | {
"line": 245,
"column": 2
} | {
"line": 249,
"column": 47
} | {
"line": 251,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝² : Preorder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : ConditionallyCompleteLinearOrder α\nf : ι → α\n⊢ ⨆ i, (partialSups f) i = ⨆ i, f i",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice.toSemilatti... | [] | by_cases h : BddAbove (Set.range f)
· exact ciSup_partialSups_eq h
· rw [iSup, iSup, ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove _ h,
ConditionallyCompleteLinearOrder.csSup_of_not_bddAbove _
(bddAbove_range_partialSups.not.mpr h)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SuccPred.LinearLocallyFinite | {
"line": 259,
"column": 8
} | {
"line": 259,
"column": 19
} | {
"line": 259,
"column": 19
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 : ι\nn : ℕ\nh : pred^[n] i0 < i0\n⊢ -↑n ≤ toZ i0 (pred^[n] i0)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrder.toDecidableE... | [
"case inr\nι : Type u_1\ninst✝³ : LinearOrder ι\ninst✝² : SuccOrder ι\ninst✝¹ : IsSuccArchimedean ι\ninst✝ : PredOrder ι\ni0 : ι\nn : ℕ\nh : pred^[n] i0 < i0\n⊢ -↑n ≤ -↑(Nat.find ⋯)"
] | toZ_of_lt h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Choose.Sum | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 18
} | {
"line": 47,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\n⊢ t n.succ i.succ = x * t n i + y * t n i.succ",
... | [
"R : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\n⊢ x ^ i.succ * y ^ (n.succ - i.succ) * ↑(n.succ.choose i.succ)... | dsimp only [t] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Data.Nat.Choose.Sum | {
"line": 50,
"column": 6
} | {
"line": 50,
"column": 69
} | {
"line": 51,
"column": 6
} | [
{
"pp": "case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\n⊢ x ^ i.succ * y ^ (n.succ - i.succ) * ↑... | [
"case e_a\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nn✝ : ℕ\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nn i : ℕ\nh_mem : i ∈ range n.succ\nh_le : i ≤ n\n⊢ x ^ i.succ * y ^ (n.succ - i.succ) * ↑(n.choose i.... | rw [← mul_assoc y, ← mul_assoc y, (h.symm.pow_right i.succ).eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Nat.Choose.Sum | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 18
} | {
"line": 59,
"column": 4
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nh_middle : ∀ (n i : ℕ), i ∈ range n.succ → t n.succ i.succ = x * t n i + y * t n i.succ\n⊢ 1 = t... | [
"case zero\nR : Type u_1\ninst✝ : Semiring R\nx y : R\nh : Commute x y\nt : ℕ → ℕ → R := fun n m ↦ x ^ m * y ^ (n - m) * ↑(n.choose m)\nh_first : ∀ (n : ℕ), t n 0 = y ^ n\nh_last : ∀ (n : ℕ), t n n.succ = 0\nh_middle : ∀ (n i : ℕ), i ∈ range n.succ → t n.succ i.succ = x * t n i + y * t n i.succ\n⊢ 1 = x ^ 0 * y ^ (... | dsimp only [t] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.RingTheory.Ideal.Quotient.Defs | {
"line": 59,
"column": 30
} | {
"line": 61,
"column": 33
} | {
"line": 63,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nI✝ J : Ideal R\na b : R\nS : Type v\nx y : R\nI : Ideal R\ninst✝ : I.IsTwoSided\na₁ b₁ a₂ b₂ : R\nh₁ : (QuotientAddGroup.con (Submodule.toAddSubgroup I)).toSetoid a₁ b₁\nh₂ : (QuotientAddGroup.con (Submodule.toAddSubgroup I)).toSetoid a₂ b₂\n⊢ (QuotientAddGroup.con (Submodu... | [] | by
rw [Submodule.quotientRel_def] at h₁ h₂ ⊢
exact mul_sub_mul_mem I h₁ h₂ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.Submodule.IterateMapComap | {
"line": 58,
"column": 6
} | {
"line": 60,
"column": 34
} | {
"line": 62,
"column": 0
} | [] | [] | _ ≤ (f.iterateMapComap i n K).map i := map_comap_le _ _
_ ≤ (((f.iterateMapComap i n K).map f).comap f).map i := by grw [← le_comap_map]
_ ≤ _ := by gcongr; exact ih | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Algebra.Order.SuccPred.PartialSups | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 61
} | {
"line": 44,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁶ : SemilatticeSup α\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : Add ι\ninst✝³ : One ι\ninst✝² : OrderBot ι\ninst✝¹ : LocallyFiniteOrder ι\ninst✝ : SuccAddOrder ι\nf : ι → α\ni : ι\n⊢ (partialSups f) (i + 1) = f ⊥ ⊔ (partialSups (f ∘ fun k ↦ k + 1)) i",
"ppTerm": "?m.40",
... | [] | simpa [← Order.succ_eq_add_one] using partialSups_succ' f i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.SuccPred.PartialSups | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 61
} | {
"line": 44,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁶ : SemilatticeSup α\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : Add ι\ninst✝³ : One ι\ninst✝² : OrderBot ι\ninst✝¹ : LocallyFiniteOrder ι\ninst✝ : SuccAddOrder ι\nf : ι → α\ni : ι\n⊢ (partialSups f) (i + 1) = f ⊥ ⊔ (partialSups (f ∘ fun k ↦ k + 1)) i",
"ppTerm": "?m.40",
... | [] | simpa [← Order.succ_eq_add_one] using partialSups_succ' f i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.SuccPred.PartialSups | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 61
} | {
"line": 44,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁶ : SemilatticeSup α\ninst✝⁵ : LinearOrder ι\ninst✝⁴ : Add ι\ninst✝³ : One ι\ninst✝² : OrderBot ι\ninst✝¹ : LocallyFiniteOrder ι\ninst✝ : SuccAddOrder ι\nf : ι → α\ni : ι\n⊢ (partialSups f) (i + 1) = f ⊥ ⊔ (partialSups (f ∘ fun k ↦ k + 1)) i",
"ppTerm": "?m.40",
... | [] | simpa [← Order.succ_eq_add_one] using partialSups_succ' f i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 68,
"column": 7
} | {
"line": 69,
"column": 38
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nα : Type u_5\ninst✝ : Unique α\nm : M\n⊢ ((finsuppUnique R M α).symm m) default = (single default m) default",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
... | [] | simp [LinearEquiv.finsuppUnique, Equiv.funUnique, single, Pi.single,
equivFunOnFinite, Function.update] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.Map | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 89
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ns : Set β\nF : Filter (α × β)\n⊢ s ∈ comap (Prod.mk x) F ↔ {p | p.1 = x → p.2 ∈ s} ∈ F",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Order.Filter.Map.0.Filter.mem_comap_prod... | [] | simp_rw [mem_comap', Prod.ext_iff, and_imp, @forall_comm β (_ = _), forall_eq, eq_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.Filter.Map | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 89
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ns : Set β\nF : Filter (α × β)\n⊢ s ∈ comap (Prod.mk x) F ↔ {p | p.1 = x → p.2 ∈ s} ∈ F",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Order.Filter.Map.0.Filter.mem_comap_prod... | [] | simp_rw [mem_comap', Prod.ext_iff, and_imp, @forall_comm β (_ = _), forall_eq, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Map | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 89
} | {
"line": 120,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx : α\ns : Set β\nF : Filter (α × β)\n⊢ s ∈ comap (Prod.mk x) F ↔ {p | p.1 = x → p.2 ∈ s} ∈ F",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"_private.Mathlib.Order.Filter.Map.0.Filter.mem_comap_prod... | [] | simp_rw [mem_comap', Prod.ext_iff, and_imp, @forall_comm β (_ = _), forall_eq, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Map | {
"line": 353,
"column": 35
} | {
"line": 353,
"column": 63
} | {
"line": 353,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nι : Sort u_5\nF : Filter α\nG : Filter β\nf✝ f₁ f₂ : Filter α\ng g₁ g₂ : Filter β\nm✝ : α → β\nm' : β → γ\ns : Set α\nt : Set β\nm : α → β\nf : Filter α\n⊢ kernImage m univ = univ",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants":... | [] | by simp [kernImage_eq_compl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 53
} | {
"line": 227,
"column": 0
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nα : Type u_5\np : α → Submodule R M\n⊢ ⨆ i, map (lsingle i) (p i) ≤ submodule p",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Submodule",
"RingHomSurjec... | [] | · simp [iSup_le_iff, Submodule.map_le_iff_le_comap] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 314,
"column": 75
} | {
"line": 316,
"column": 7
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nX : Type u_7\ninst✝ : Finite X\n⊢ linearMap R M _root_.id = LinearMap.id",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearMap.id",
"Pi.Function.module",
"Pi.addC... | [] | by
classical
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Finsupp.Pi | {
"line": 319,
"column": 75
} | {
"line": 321,
"column": 7
} | {
"line": 323,
"column": 0
} | [
{
"pp": "R : Type u_5\nM : Type u_6\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nX : Type u_7\nY : Type u_8\nZ : Type u_9\ninst✝² : Finite X\ninst✝¹ : Finite Y\ninst✝ : Finite Z\nf : X → Y\ng : Y → Z\n⊢ linearMap R M (g ∘ f) = linearMap R M g ∘ₗ linearMap R M f",
"ppTerm": "?m.53",
... | [] | by
classical
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Tendsto | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 68
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : Filter α\ns : Set β\n⊢ Tendsto f l (𝓟 s) ↔ ∀ᶠ (a : α) in l, f a ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"_private.Mathlib.Order.Filter.Tendsto.0.Filter.tendsto_principal._simp_1_3",
"... | [] | simp only [Tendsto, le_principal_iff, mem_map', Filter.Eventually] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.Tendsto | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 68
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : Filter α\ns : Set β\n⊢ Tendsto f l (𝓟 s) ↔ ∀ᶠ (a : α) in l, f a ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"_private.Mathlib.Order.Filter.Tendsto.0.Filter.tendsto_principal._simp_1_3",
"... | [] | simp only [Tendsto, le_principal_iff, mem_map', Filter.Eventually] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Tendsto | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 68
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : Filter α\ns : Set β\n⊢ Tendsto f l (𝓟 s) ↔ ∀ᶠ (a : α) in l, f a ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"_private.Mathlib.Order.Filter.Tendsto.0.Filter.tendsto_principal._simp_1_3",
"... | [] | simp only [Tendsto, le_principal_iff, mem_map', Filter.Eventually] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Tendsto | {
"line": 233,
"column": 46
} | {
"line": 234,
"column": 68
} | {
"line": 236,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nl : Filter α\ns : Set β\n⊢ Tendsto f l (𝓟 s) ↔ ∀ᶠ (a : α) in l, f a ∈ s",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"_private.Mathlib.Order.Filter.Tendsto.0.Filter.tendsto_principal._simp_1_3",
"... | [] | by
simp only [Tendsto, le_principal_iff, mem_map', Filter.Eventually] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Map | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 49
} | {
"line": 676,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Filter α\nm : α → β\n⊢ map m f = ⊥ → f = ⊥",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"congrArg",
"Filter.map",
"Membership.mem",
"id",
"Bot.bot",
"Filter.empty_... | [
"α : Type u_1\nβ : Type u_2\nf : Filter α\nm : α → β\n⊢ ∅ ∈ map m f → ∅ ∈ f"
] | rw [← empty_mem_iff_bot, ← empty_mem_iff_bot] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Map | {
"line": 850,
"column": 55
} | {
"line": 851,
"column": 37
} | {
"line": 853,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Filter (α → β)\ng : Filter α\ns : Set β\n⊢ s ∈ f.seq g ↔ ∃ u ∈ f, ∃ t ∈ g, u.seq t ⊆ s",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"congrArg",
"Filter.seq",
"Membership.mem",
"Exists",
... | [] | by
simp only [mem_seq_def, seq_subset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.AtTopBot.Basic | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 32
} | {
"line": 54,
"column": 2
} | [
{
"pp": "α : Type u_3\ninst✝² : Preorder α\ninst✝¹ : IsDirectedOrder α\ninst✝ : NoMaxOrder α\na : α\nthis : Nonempty α\nb : α\nx✝ : True\nc : α\nhac : a ≤ c\nhbc : b ≤ c\n⊢ ∃ i', a < i' ∧ Ioi i' ⊆ Ioi b",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Set.Ioi",
"Preorder.toLT",... | [
"α : Type u_3\ninst✝² : Preorder α\ninst✝¹ : IsDirectedOrder α\ninst✝ : NoMaxOrder α\na : α\nthis : Nonempty α\nb : α\nx✝ : True\nc : α\nhac : a ≤ c\nhbc : b ≤ c\nd : α\nhcd : c < d\n⊢ ∃ i', a < i' ∧ Ioi i' ⊆ Ioi b"
] | obtain ⟨d, hcd⟩ := exists_gt c | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.DirSupClosed | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 39
} | {
"line": 149,
"column": 0
} | [
{
"pp": "α : Type u_1\nD : Set (Set α)\ninst✝ : Preorder α\nι : Sort u_2\nf : ι → Set α\nhs : ∀ (i : ι), DirSupInaccOn D (f i)\n⊢ DirSupInaccOn D (⋃₀ range f)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Membership.mem",
"Exists",
"id",
"Set.mem_... | [] | exact DirSupInaccOn.sUnion (by simpa) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 29
} | {
"line": 475,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.Finite\nhf : f.Finite\n⊢ (g.comp f).Finite",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSem... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\ng : B →+* C\nf : A →+* B\nhg : g.Finite\nhf : f.Finite\nalgInst✝² : Algebra A B := f.toAlgebra\nalgInst✝¹ : Algebra B C := g.toAlgebra\nalgInst✝ : Algebra A C := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalar... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.RingTheory.Finiteness.Basic | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 29
} | {
"line": 479,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\ng : B →+* C\nh : (g.comp f).Finite\n⊢ g.Finite",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"I... | [
"A : Type u_1\nB : Type u_2\nC : Type u_3\ninst✝² : CommRing A\ninst✝¹ : CommRing B\ninst✝ : CommRing C\nf : A →+* B\ng : B →+* C\nh : (g.comp f).Finite\nalgInst✝² : Algebra A B := f.toAlgebra\nalgInst✝¹ : Algebra B C := g.toAlgebra\nalgInst✝ : Algebra A C := (g.comp f).toAlgebra\nscalarTowerInst✝ : IsScalarTower A... | algebraize [f, g, g.comp f] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 191,
"column": 12
} | {
"line": 191,
"column": 22
} | {
"line": 192,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nl : Ordinal.{?u.20}\nlLim : IsSuccLimit l\nmotive : ↑(Iio l) → Sort u_4\nzero : motive ⟨0, ⋯⟩\nsucc : (o : ↑(Iio l)) → motive o → motive ⟨Order.succ ↑o, ⋯⟩\nlimit : (o : ↑(Iio l)) → IsSuccLimit ↑o... | [] | exact zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 191,
"column": 12
} | {
"line": 191,
"column": 22
} | {
"line": 192,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nl : Ordinal.{?u.20}\nlLim : IsSuccLimit l\nmotive : ↑(Iio l) → Sort u_4\nzero : motive ⟨0, ⋯⟩\nsucc : (o : ↑(Iio l)) → motive o → motive ⟨Order.succ ↑o, ⋯⟩\nlimit : (o : ↑(Iio l)) → IsSuccLimit ↑o... | [] | exact zero | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 191,
"column": 12
} | {
"line": 191,
"column": 22
} | {
"line": 192,
"column": 2
} | [
{
"pp": "case zero\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nl : Ordinal.{?u.20}\nlLim : IsSuccLimit l\nmotive : ↑(Iio l) → Sort u_4\nzero : motive ⟨0, ⋯⟩\nsucc : (o : ↑(Iio l)) → motive o → motive ⟨Order.succ ↑o, ⋯⟩\nlimit : (o : ↑(Iio l)) → IsSuccLimit ↑o... | [] | exact zero | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 245,
"column": 6
} | {
"line": 245,
"column": 28
} | {
"line": 245,
"column": 28
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : IsSuccLimit (type r)\nx b : α\nhb : b ∈ {x}\n⊢ r b ((enum r) ⟨succ ((typein r).toRelEmbedding x), ⋯⟩)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Order.succ",
"Ord... | [
"α : Type u_1\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhr : IsSuccLimit (type r)\nx b : α\nhb : b ∈ {x}\n⊢ r x ((enum r) ⟨succ ((typein r).toRelEmbedding x), ⋯⟩)"
] | mem_singleton_iff.1 hb | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1255,
"column": 40
} | {
"line": 1255,
"column": 70
} | {
"line": 1255,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : Preorder α\ninst✝⁵ : Preorder β\ninst✝⁴ : Preorder γ\ninst✝³ : SMul α β\ninst✝² : SMul α γ\nf : β → γ\ninst✝¹ : Zero α\ninst✝ : PosSMulReflectLE α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nsmul : ∀ (a : α) (b : β), f (a • b) = a • f b\na : α\nha : ... | [] | simpa only [smul] using hf.2 h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1255,
"column": 40
} | {
"line": 1255,
"column": 70
} | {
"line": 1255,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : Preorder α\ninst✝⁵ : Preorder β\ninst✝⁴ : Preorder γ\ninst✝³ : SMul α β\ninst✝² : SMul α γ\nf : β → γ\ninst✝¹ : Zero α\ninst✝ : PosSMulReflectLE α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nsmul : ∀ (a : α) (b : β), f (a • b) = a • f b\na : α\nha : ... | [] | simpa only [smul] using hf.2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.Defs | {
"line": 1255,
"column": 40
} | {
"line": 1255,
"column": 70
} | {
"line": 1255,
"column": 70
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : Preorder α\ninst✝⁵ : Preorder β\ninst✝⁴ : Preorder γ\ninst✝³ : SMul α β\ninst✝² : SMul α γ\nf : β → γ\ninst✝¹ : Zero α\ninst✝ : PosSMulReflectLE α γ\nhf : ∀ {b₁ b₂ : β}, f b₁ ≤ f b₂ ↔ b₁ ≤ b₂\nsmul : ∀ (a : α) (b : β), f (a • b) = a • f b\na : α\nha : ... | [] | simpa only [smul] using hf.2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 29
} | {
"line": 526,
"column": 0
} | [
{
"pp": "α : Type u\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nh0 : 0 < type r\na : α\n⊢ ⟨0, h0⟩ ≤ ⟨(typein r).toRelEmbedding a, ⋯⟩",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"Ordinal.partialOrder",
"Ordinal.instOrderBot",
"PartialOrder.toPreo... | [] | exact bot_le (α := Ordinal) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.Univ | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 51
} | {
"line": 81,
"column": 51
} | [
{
"pp": "case mpr\nb : Ordinal.{max (u + 1) v}\nβ : Type (max (u + 1) v)\ns : β → β → Prop\ninst✝ : IsWellOrder β s\nh : lift.{max (u + 1) v, max (u + 1) v} (type s) < lift.{max (u + 1) v, u + 1} (typeLT Ordinal.{u})\nf : s ↪r fun x1 x2 ↦ x1 < x2\na : Ordinal.{u}\nhf : ∀ (b : Ordinal.{u}), b ∈ Set.range ⇑f ↔ b ... | [
"case mpr.type\nb : Ordinal.{max (u + 1) v}\nβ : Type (max (u + 1) v)\ns : β → β → Prop\ninst✝¹ : IsWellOrder β s\nh : lift.{max (u + 1) v, max (u + 1) v} (type s) < lift.{max (u + 1) v, u + 1} (typeLT Ordinal.{u})\nf : s ↪r fun x1 x2 ↦ x1 < x2\nα : Type u\nr : α → α → Prop\ninst✝ : IsWellOrder α r\nhf : ∀ (b : Ord... | induction a using inductionOn with | type α r
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 881,
"column": 4
} | {
"line": 883,
"column": 24
} | {
"line": 885,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc a b : Ordinal.{u}\n⊢ (fun x1 x2 ↦ x1 ≤ x2) a b →\n (fun x1 x2 ↦ x1 ≤ x2) (Function.swap (fun x1 x2 ↦ x1 + x2) c a) (Function.swap (fun x1 x2 ↦ x1 + x2) c b)",
"ppTerm": "?m.10",
"assigned": true,
... | [] | refine inductionOn₃ a b c fun α r _ β s _ γ t _ ⟨f⟩ ↦
(RelEmbedding.ofMonotone (Sum.recOn · (Sum.inl ∘ f) Sum.inr) ?_).ordinal_type_le
simp [f.map_rel_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 881,
"column": 4
} | {
"line": 883,
"column": 24
} | {
"line": 885,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nc a b : Ordinal.{u}\n⊢ (fun x1 x2 ↦ x1 ≤ x2) a b →\n (fun x1 x2 ↦ x1 ≤ x2) (Function.swap (fun x1 x2 ↦ x1 + x2) c a) (Function.swap (fun x1 x2 ↦ x1 + x2) c b)",
"ppTerm": "?m.10",
"assigned": true,
... | [] | refine inductionOn₃ a b c fun α r _ β s _ γ t _ ⟨f⟩ ↦
(RelEmbedding.ofMonotone (Sum.recOn · (Sum.inl ∘ f) Sum.inr) ?_).ordinal_type_le
simp [f.map_rel_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Enum | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 19
} | {
"line": 126,
"column": 4
} | [
{
"pp": "case mpr\ns : Set Ordinal.{u}\nf : Ordinal.{u} → Ordinal.{u}\nhs : ¬BddAbove s\n⊢ StrictMono f ∧ range f = s → enumOrd s = f",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"StrictMono",
"Ordinal.partialOrder",
"PartialOrder.toPreorder",
"Ordinal.enumOrd",
... | [
"case mpr\ns : Set Ordinal.{u}\nf : Ordinal.{u} → Ordinal.{u}\nhs : ¬BddAbove s\nh₁ : StrictMono f\nh₂ : range f = s\n⊢ enumOrd s = f"
] | rintro ⟨h₁, h₂⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1406,
"column": 6
} | {
"line": 1411,
"column": 37
} | {
"line": 1413,
"column": 0
} | [
{
"pp": "case cons.cons\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\no : Ordinal.{u}\na : α\nas : List α\nhl : (a :: as).SortedGT\nhlt : ∀ i ∈ a :: as, (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRelEmbedding i < o\nb : α\nbs : List α\nhm : (b :: bs).SortedGT\nhmltl : Lex (fun x1 x2 ↦ x1 < x2) (b ::... | [] | intro i hi
suffices h : i ≤ a by refine lt_of_le_of_lt ?_ (hlt a mem_cons_self); simpa
cases hi with
| head as => exact List.head_le_of_lt hmltl
| tail b hi => exact le_of_lt (lt_of_lt_of_le (List.rel_of_pairwise_cons hm.pairwise hi)
(List.head_le_of_lt hmltl)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Basic | {
"line": 1406,
"column": 6
} | {
"line": 1411,
"column": 37
} | {
"line": 1413,
"column": 0
} | [
{
"pp": "case cons.cons\nα : Type u\ninst✝¹ : LinearOrder α\ninst✝ : WellFoundedLT α\no : Ordinal.{u}\na : α\nas : List α\nhl : (a :: as).SortedGT\nhlt : ∀ i ∈ a :: as, (Ordinal.typein fun x1 x2 ↦ x1 < x2).toRelEmbedding i < o\nb : α\nbs : List α\nhm : (b :: bs).SortedGT\nhmltl : Lex (fun x1 x2 ↦ x1 < x2) (b ::... | [] | intro i hi
suffices h : i ≤ a by refine lt_of_le_of_lt ?_ (hlt a mem_cons_self); simpa
cases hi with
| head as => exact List.head_le_of_lt hmltl
| tail b hi => exact le_of_lt (lt_of_lt_of_le (List.rel_of_pairwise_cons hm.pairwise hi)
(List.head_le_of_lt hmltl)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Log | {
"line": 182,
"column": 2
} | {
"line": 182,
"column": 33
} | {
"line": 184,
"column": 0
} | [
{
"pp": "case inr\nb x y : ℕ\nhx : x ≠ 0\nhlt : y < b ^ x\nhy : y ≠ 0\n⊢ log b y < x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Nat.log_lt_of_lt_pow"
],
"usedFVars": [
"b",
"x",
"y",
"hy",
"hlt"
],
"usedGoals": []
}
] | [] | · exact log_lt_of_lt_pow hy hlt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Arithmetic | {
"line": 1128,
"column": 16
} | {
"line": 1128,
"column": 29
} | {
"line": 1128,
"column": 30
} | [
{
"pp": "c : Cardinal.{u_1}\nh : ℵ₀ ≤ c\n⊢ 1 + c = c",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardinal.instOne",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commSemiring",
"id",
"Cardinal.ord",
"Ordina... | [
"c : Cardinal.{u_1}\nh : ℵ₀ ≤ c\n⊢ 1 + c.ord.card = c.ord.card"
] | ← card_ord c, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Log | {
"line": 267,
"column": 2
} | {
"line": 269,
"column": 33
} | {
"line": 271,
"column": 0
} | [
{
"pp": "case inr\nb c n : ℕ\nhc : 1 < c\nhb : c ≤ b\nhn : n ≠ 0\n⊢ c ^ log b n ≤ n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"instPowNat",
"Trans.trans",
"LE.le",
"instLENat",
"instNatPowNat",
"HPow.hPow",
"Nat.instTransLe",
"Nat",
... | [] | calc
c ^ log b n ≤ b ^ log b n := Nat.pow_le_pow_left hb _
_ ≤ n := pow_log_le_self _ hn | Lean.Elab.Tactic.evalCalc | Lean.calcTactic |
Mathlib.SetTheory.Ordinal.Exponential | {
"line": 195,
"column": 2
} | {
"line": 199,
"column": 31
} | {
"line": 200,
"column": 2
} | [
{
"pp": "case inl\na b : Ordinal.{u_1}\nb1 : 0 < b\na1 : a ≤ 1\n⊢ a ^ 1 ≤ a ^ b",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"zero_le",
"Eq.mpr",
"le_refl",
"Ordinal.instLinearOrder",
"Preorder.toLT",
"lt_or_eq_of_le",
"Ordinal.partialOrder",
... | [
"case inr\na b : Ordinal.{u_1}\nb1 : 0 < b\na1 : 1 < a\n⊢ a ^ 1 ≤ a ^ b"
] | · rcases lt_or_eq_of_le a1 with a0 | a1
· rw [lt_one_iff] at a0
rw [a0, zero_opow one_ne_zero]
exact zero_le
rw [a1, one_opow, one_opow] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.FixedPoint | {
"line": 448,
"column": 74
} | {
"line": 450,
"column": 6
} | {
"line": 452,
"column": 0
} | [
{
"pp": "a : Ordinal.{u_1}\nha : 0 < a\n⊢ nfp (fun x ↦ a * x) 1 = a ^ ω",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Ordinal.iSup_pow_natCast",
"Eq.mpr",
"mul_left_iterate",
"HMul.hMul",
"Ordinal.monoid",
"Ordinal.omega0",
"MulZeroClass.toMul",... | [] | by
rw [← iSup_iterate_eq_nfp, ← iSup_pow_natCast ha]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Arithmetic | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 68
} | {
"line": 90,
"column": 69
} | [
{
"pp": "o₁ o₂ : Ordinal.{u_1}\n⊢ ℵ_ o₁ * ℵ_ o₂ = ℵ_ (max o₁ o₂)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"HMul.hMul",
"Cardinal.aleph",
"Ordinal.partialOrder",
"Cardinal",
"congrArg",
"PartialOrde... | [
"o₁ o₂ : Ordinal.{u_1}\n⊢ max (ℵ_ o₁) (ℵ_ o₂) = ℵ_ (max o₁ o₂)"
] | Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.FundamentalSequence | {
"line": 174,
"column": 52
} | {
"line": 177,
"column": 7
} | {
"line": 179,
"column": 0
} | [
{
"pp": "a o : Ordinal.{u}\nf : (b : Ordinal.{u}) → b < o → Ordinal.{u}\nhf : a.IsFundamentalSequence o f\ni j : Ordinal.{u}\nhi : i < o\nhj : j < o\nhij : i ≤ j\n⊢ f i hi ≤ f j hj",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"le_refl",
"Preorder.toLT",
"lt_or_eq_of_le... | [] | by
rcases lt_or_eq_of_le hij with (hij | rfl)
· exact (hf.2.1 hi hj hij).le
· rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 205,
"column": 14
} | {
"line": 205,
"column": 22
} | {
"line": 205,
"column": 22
} | [
{
"pp": "c : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\na b : Ordinal.{u_1}\nha : a.card < c\nhb : b.card < c\n⊢ ((fun x1 x2 ↦ x1 + x2) a b).card < c",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"congrArg",
"PartialOrder.toPreorder"... | [
"c : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\na b : Ordinal.{u_1}\nha : a.card < c\nhb : b.card < c\n⊢ a.card + b.card < c"
] | card_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Ordinal | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 29
} | {
"line": 208,
"column": 0
} | [
{
"pp": "c : Cardinal.{u_1}\nhc : ℵ₀ ≤ c\na b : Ordinal.{u_1}\nha : a.card < c\nhb : b.card < c\n⊢ a.card + b.card < c",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Cardinal.add_lt_of_lt",
"Ordinal.card"
],
"usedFVars": [
"a",
"b",
"c",
"hc",
... | [] | exact add_lt_of_lt hc ha hb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 428,
"column": 25
} | {
"line": 436,
"column": 34
} | {
"line": 438,
"column": 0
} | [
{
"pp": "ι : Type u\nf : ι → Ordinal.{max u v} → Ordinal.{max u v}\nc : Ordinal.{max u v}\nhc : ℵ₀ < c.cof\nhc' : Cardinal.lift.{v, u} #ι < c.cof\nhf : ∀ (i : ι), ∀ b < c, f i b < c\na : Ordinal.{max u v}\nha : a < c\n⊢ nfpFamily f a < c",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
... | [] | by
refine lift_iSup_lt_of_lt_cof ?_ (fun l ↦ ?_)
· rw [Cardinal.lift_umax, c.lift_id']
apply (Cardinal.lift_le.2 (mk_list_le_max _)).trans_lt
rw [Cardinal.lift_max]
apply max_lt <;> simpa
· induction l with
| nil => exact ha
| cons i l H => exact hf _ _ H | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Cardinal.Cofinality.Ordinal | {
"line": 543,
"column": 7
} | {
"line": 543,
"column": 36
} | {
"line": 543,
"column": 37
} | [
{
"pp": "α : Type u\nr : α → α → Prop\nH : IsWellOrder α r\nh : IsSuccLimit (type r)\nthis✝¹ : LinearOrder α := linearOrderOfSTO r\nthis✝ : WellFoundedLT α\nthis : NoMaxOrder α\ns : Set α\nhs : IsCofinal s\nhs' : #↑s = Order.cof α\n⊢ ∀ (a : α), ∃ b ∈ s, r a b",
"ppTerm": "?m.61",
"assigned": true,
"... | [
"α : Type u\nr : α → α → Prop\nH : IsWellOrder α r\nh : IsSuccLimit (type r)\nthis✝¹ : LinearOrder α := linearOrderOfSTO r\nthis✝ : WellFoundedLT α\nthis : NoMaxOrder α\ns : Set α\nhs : ¬BddAbove s\nhs' : #↑s = Order.cof α\n⊢ ∀ (a : α), ∃ b ∈ s, r a b"
] | ← not_bddAbove_iff_isCofinal, | Lean.Elab.Tactic.evalRewriteSeq | null |
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