module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Enumerative.Bell | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | [
{
"pp": "m : Multiset ℕ\n⊢ m.bell * (map (fun j ↦ j !) m).prod * ∏ j ∈ m.toFinset.erase 0, (count j m)! = m.sum !",
"usedConstants": [
"Multiset.sum",
"Multiset.toFinset",
"HMul.hMul",
"Multiset.map",
"Multiset.prod",
"Multiset.count",
"id",
"instMulNat",
... | unfold bell | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 34
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nA B : Finset G\na b : G\nha : a ∈ B\nhb : b ∈ B\nthis : #(a •> A ∪ b •> A) ≤ #(B * A)\n⊢ 2 * #A ≤ #(a •> A) + #(b •> A)",
"usedConstants": [
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"Monoid.toMulOneClass",
"instR... | simp [card_smul_finset, two_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 129,
"column": 76
} | {
"line": 139,
"column": 5
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\n⊢ U :: (↑p).dropLast.tail ++ [D] = ↑p",
"usedConstants": [
"List.head",
"Iff.mpr",
"List.getLast",
"Eq.mpr",
"instDecidableEqDyckStep",
"False",
"Nat.instMulZeroClass",
"DyckWord.getLast_eq_D",
"Nat.instOne",
"inst... | by
have h' := toList_ne_nil.mpr h
have : p.toList.dropLast.take 1 = [p.toList.head h'] := by
rcases p with - | ⟨s, ⟨- | ⟨t, r⟩⟩⟩
· tauto
· rename_i bal _
cases s <;> simp at bal
· tauto
nth_rw 2 [← p.toList.dropLast_append_getLast h', ← p.toList.dropLast.take_append_drop 1]
rw [getLast_eq_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 264,
"column": 29
} | {
"line": 264,
"column": 41
} | [
{
"pp": "p : DyckWord\nh : p ≠ 0\n⊢ ¬p.firstReturn = 0",
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"DyckStep.U",
"congrArg",
"id",
"instOfNatNat",
"List.range",
"instBEqOfDecidableEq",
"instHAdd",
"List.count",
"HAdd.hAdd",
... | firstReturn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 297,
"column": 6
} | {
"line": 297,
"column": 18
} | [
{
"pp": "case neg\np q : DyckWord\nh : ¬p = 0\nu : ↑(p + q) = ↑p ++ ↑q\n⊢ (p + q).firstReturn = p.firstReturn",
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"instAddDyckWord",
"DyckStep.U",
"congrArg",
"DyckWord",
"id",
"instOfNatNat",
"List.r... | firstReturn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Enumerative.DyckWord | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 18
} | [
{
"pp": "p : DyckWord\nu : ↑p.nest = U :: ↑p ++ [D]\n⊢ p.nest.firstReturn = (↑p).length + 1",
"usedConstants": [
"Eq.mpr",
"instDecidableEqDyckStep",
"DyckStep.U",
"congrArg",
"id",
"instOfNatNat",
"List.range",
"instBEqOfDecidableEq",
"instHAdd",
... | firstReturn, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 371,
"column": 2
} | {
"line": 372,
"column": 80
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : DecidableEq G\nH : Subgroup G\ninst✝ : Fintype ↥H\nZ : Finset G\nhZ : Set.InjOn (fun x ↦ ↑H <• x) ↑Z\nh₁ z₁ h₂ z₂ : G\nh : h₁ * z₁ = h₂ * z₂\nhh₁ : h₁ ∈ H\nhz₁ : z₁ ∈ Z\nhh₂ : h₂ ∈ H\nhz₂ : z₂ ∈ Z\n⊢ (h₁, z₁) = (h₂, z₂)",
"usedConstants": [
"Iff.mpr",
... | obtain rfl := hZ hz₁ hz₂ <| (rightCoset_eq_iff _).2 <| by
simpa [eq_inv_mul_iff_mul_eq.2 h, mul_assoc] using mul_mem (inv_mem hh₂) hh₁ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 276,
"column": 17
} | {
"line": 276,
"column": 66
} | [
{
"pp": "F : Type u_1\n𝕜 : Type u_2\n𝕝 : Type u_3\n𝕞 : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝⁵ : Preorder α\ninst✝⁴ : LocallyFiniteOrder α\ninst✝³ : DecidableEq α\ninst✝² : Semiring 𝕜\ninst✝¹ : Semiring 𝕝\ninst✝ : Module 𝕜 𝕝\nf : IncidenceAlgebra 𝕝 α\n⊢ 0 • f = 0",
"usedConstants": [
"Non... | by ext; exact sum_eq_zero fun x _ ↦ zero_smul _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Additive.VerySmallDoubling | {
"line": 525,
"column": 28
} | {
"line": 525,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : DecidableEq G\nK : ℝ\nA S : Finset G\nhK : K < 1\nhS : S.Nonempty\nhA : A.Nonempty\n⊢ 0 < ↑(#A)",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Preorder.toLT... | simp [hA] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 443,
"column": 98
} | {
"line": 444,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝⁴ : AddCommGroup 𝕜\ninst✝³ : One 𝕜\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\na b : α\n⊢ (mu' 𝕜) a b = if a = b then 1 else -∑ x ∈ Ioc a b, (mu' 𝕜) x b",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"cong... | by
rw [mu', coe_mk, muFun'_apply, sum_attach] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.IncidenceAlgebra | {
"line": 553,
"column": 10
} | {
"line": 553,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\nα : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder α\ninst✝² : OrderTop α\ninst✝¹ : LocallyFiniteOrder α\ninst✝ : DecidableEq α\nf g : α → 𝕜\nh : ∀ (x : α), g x = ∑ y ∈ Ici x, f y\nx : α\nthis : DecidableLE α := Classical.decRel LE.le\n⊢ ∑ x_1 ∈ (Ici x).sigma fun y ↦ Ici y, (mu 𝕜) x... | sum_sigma' (Ici x) fun z ↦ Icc x z | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Graph.Basic | {
"line": 168,
"column": 71
} | {
"line": 171,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nx y : α\ne : β\nG : Graph α β\nx' y' : α\nh : G.IsLink e x y\nh' : G.IsLink e x' y'\n⊢ x = x' ∧ y = y' ∨ x = y' ∧ y = x'",
"usedConstants": [
"congrArg",
"and_self",
"true_or",
"Or.casesOn",
"Graph.IsLink.symm",
"And",
"Graph.IsL... | by
obtain rfl | rfl := h.left_eq_or_eq h'
· simp [h.right_unique h']
simp [h'.symm.right_unique h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Minor.Delete | {
"line": 171,
"column": 6
} | {
"line": 171,
"column": 24
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI D X : Set α\nh : M.IsBasis I X\nhX : Disjoint X D\n⊢ (M ＼ D).IsBasis I X",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Disjoint",
"i... | delete_isBasis_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 85
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nK : Set α\nhJK : Disjoint K J\nhKJi : M.Indep (K ∪ J)\nhKX : K ⊆ X\nhIJK : I ⊆ K ∪ J\n⊢ K ⊆ I",
"usedConstants": [
"Set.subset_union_left._simp_1",
"congrArg",
"Set.instUnion",
"HasSubset.Subset.t... | simp [hIX.eq_of_subset_indep hKJi hIJK (union_subset hKX (hJI.trans hIX.subset))] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 85
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nK : Set α\nhJK : Disjoint K J\nhKJi : M.Indep (K ∪ J)\nhKX : K ⊆ X\nhIJK : I ⊆ K ∪ J\n⊢ K ⊆ I",
"usedConstants": [
"Set.subset_union_left._simp_1",
"congrArg",
"Set.instUnion",
"HasSubset.Subset.t... | simp [hIX.eq_of_subset_indep hKJi hIJK (union_subset hKX (hJI.trans hIX.subset))] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 85
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\nK : Set α\nhJK : Disjoint K J\nhKJi : M.Indep (K ∪ J)\nhKX : K ⊆ X\nhIJK : I ⊆ K ∪ J\n⊢ K ⊆ I",
"usedConstants": [
"Set.subset_union_left._simp_1",
"congrArg",
"Set.instUnion",
"HasSubset.Subset.t... | simp [hIX.eq_of_subset_indep hKJi hIJK (union_subset hKX (hJI.trans hIX.subset))] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 272,
"column": 39
} | {
"line": 272,
"column": 57
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ (M ／ J ＼ (C \\ J)).IsBasis (I \\ C) (X \\ C)",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Comple... | delete_isBasis_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 95
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis J C\nhdj : Disjoint C X\nh_ind : M.Indep (I ∪ J)\nh' : M.Indep (I \\ C ∪ J) → (M ／ C).IsBasis (I \\ C) (X \\ C)\n⊢ (M ／ C).IsBasis I X",
"usedConstants": [
"Disjoint.sdiff_eq_right",
"ChainCompletePartialOr... | rwa [(hdj.mono_right h.subset).sdiff_eq_right, hdj.sdiff_eq_right, imp_iff_right h_ind] at h' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 163,
"column": 6
} | {
"line": 163,
"column": 17
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nι : Type u_3\nb : ι → MvPolynomial σ R\nhb : ∀ (i : ι), IsUnit (m.leadingCoeff (b i))\nf : MvPolynomial σ R\nhb' : ∀ (i : ι), m.degree (b i) ≠ 0\nhf0 : ¬f = 0\ni : ι\nhf : m.degree (b i) ≤ m.degree f\nhf0' : m.degree f = 0\n⊢ False",
... | apply hb' i | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 80,
"column": 6
} | {
"line": 82,
"column": 92
} | [
{
"pp": "case of_equiv.Hdeg\nR : Type u_1\ninst✝² : CommRing R\nσ✝ : Type u_2\ninst✝¹ : Finite σ✝\ninst✝ : IsDomain R\nσ τ : Type u_2\ne : σ ≃ τ\nh :\n ∀ (P : MvPolynomial σ R) (S : σ → Finset R),\n (∀ (i : σ), degreeOf i P < #(S i)) → (∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0) → P = 0\nP : Mv... | classical
convert! Hdeg (e i)
conv_lhs => rw [← e.symm_apply_apply i, degreeOf_rename_of_injective e.symm.injective] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 298,
"column": 71
} | {
"line": 301,
"column": 70
} | [
{
"pp": "α : Type u_1\nM N : Matroid α\nh : Disjoint M.E N.E\nI : Set α\nhI : (M.disjointSum N h).Indep I\n⊢ ∃ IM IN, M.Indep IM ∧ N.Indep IN ∧ Disjoint IM IN ∧ I = IM ∪ IN",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.disjointSum",
"Compl... | by
rw [disjointSum_indep_iff] at hI
refine ⟨_, _, hI.1, hI.2.1, h.mono inter_subset_right inter_subset_right, ?_⟩
rw [← inter_union_distrib_left, inter_eq_self_of_subset_left hI.2.2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 19
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na : V\np : Path a a\n⊢ p.length = 0 ↔ p = nil",
"usedConstants": [
"Quiver.Hom",
"Quiver.Path.nil",
"HEq.refl",
"False.elim",
"noConfusion_of_Nat",
"instOfNatNat",
"Quiver.Path",
"Quiver.Path.ctorIdx",
"Quiver.Pat... | cases p <;> tauto | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 19
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na : V\np : Path a a\n⊢ p.length = 0 ↔ p = nil",
"usedConstants": [
"Quiver.Hom",
"Quiver.Path.nil",
"HEq.refl",
"False.elim",
"noConfusion_of_Nat",
"instOfNatNat",
"Quiver.Path",
"Quiver.Path.ctorIdx",
"Quiver.Pat... | cases p <;> tauto | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 19
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na : V\np : Path a a\n⊢ p.length = 0 ↔ p = nil",
"usedConstants": [
"Quiver.Hom",
"Quiver.Path.nil",
"HEq.refl",
"False.elim",
"noConfusion_of_Nat",
"instOfNatNat",
"Quiver.Path",
"Quiver.Path.ctorIdx",
"Quiver.Pat... | cases p <;> tauto | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 34
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\nha : a ∈ s\n⊢ s ∈ 𝓓 a 𝒜",
"usedConstants": [
"congrArg",
"Finset",
"Membership.mem",
"Eq.mp",
"Insert.insert",
"Finset.instInsert",
"Fins... | · rwa [insert_eq_of_mem ha] at h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\n𝒜 : Finset (Finset α)\ninst✝ : Nonempty α\nh𝒜₁ : 𝒜.Nonempty\nh𝒜₂ : univ ∉ 𝒜\n⊢ supSum 𝒜 = ↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹",
"usedConstants": [
"Finset",
"Nat",
"Finset.card"
]
}
] | set m := 𝒜.card with hm | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 47
} | [
{
"pp": "case ind.inr.succ.h𝒜\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 →... | exact card_image_le.trans_lt (lt_add_one _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 47
} | [
{
"pp": "case ind.inr.succ.h𝒜\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 →... | exact card_image_le.trans_lt (lt_add_one _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 424,
"column": 4
} | {
"line": 424,
"column": 47
} | [
{
"pp": "case ind.inr.succ.h𝒜\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\ns : Finset α\n𝒜 : Finset (Finset α)\nhs : s ∉ 𝒜\nh𝒜₁ : (insert s 𝒜).Nonempty\nh𝒜₂ : univ ∉ insert s 𝒜\nh𝒜₃ : (insert s 𝒜).Nontrivial\nih :\n ∀ (𝒜_1 : Finset (Finset α)),\n #𝒜_1 < #𝒜 + 1 →... | exact card_image_le.trans_lt (lt_add_one _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 323,
"column": 8
} | {
"line": 324,
"column": 31
} | [
{
"pp": "case neg.refine_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v : Finset α\nhuv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\nt : Finse... | rw [sup_eq_union, mem_sdiff, mem_union]
exact ⟨Or.inl hat, hav⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 323,
"column": 8
} | {
"line": 324,
"column": 31
} | [
{
"pp": "case neg.refine_1\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v : Finset α\nhuv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜\n𝒜' : Finset (Finset α) := 𝓒 u v 𝒜\nH : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \\ u ∈ ∂ 𝒜 ∧ (s ∪ v) \\ u ∉ ∂ 𝒜'\nt : Finse... | rw [sup_eq_union, mem_sdiff, mem_union]
exact ⟨Or.inl hat, hav⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Birkhoff | {
"line": 127,
"column": 78
} | {
"line": 128,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : Finite α\n⊢ Surjective ⇑supIrredLowerSet",
"usedConstants": [
"Lattice.toSemilatticeSup",
"LowerSet.completeLattice",
"congrArg",
"PartialOrder.toPreorder",
"Subtype.forall._simp_1",
"Preorder.toLE",
"Exists",
... | by
aesop (add simp Surjective) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Birkhoff | {
"line": 130,
"column": 78
} | {
"line": 131,
"column": 29
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ninst✝ : Finite α\n⊢ Surjective ⇑infIrredUpperSet",
"usedConstants": [
"UpperSet",
"congrArg",
"PartialOrder.toPreorder",
"UpperSet.infIrred_iff_of_finite._simp_1",
"Subtype.forall._simp_1",
"Preorder.toLE",
"Exists",
... | by
aesop (add simp Surjective) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Birkhoff | {
"line": 204,
"column": 10
} | {
"line": 204,
"column": 50
} | [
{
"pp": "case a.h.refine_1\nα : Type u_1\ninst✝³ : DistribLattice α\ninst✝² : Fintype α\ninst✝¹ : DecidablePred SupIrred\ninst✝ : OrderBot α\ns : LowerSet { a // SupIrred a }\na : { a // SupIrred a }\nha✝ : ↑a ≤ (↑s).toFinset.sup Subtype.val\ni : { a // SupIrred a }\nhi : i ∈ (↑s).toFinset\nha : ↑a ≤ ↑i\n⊢ a ∈ ... | exact s.lower ha (Set.mem_toFinset.1 hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Birkhoff | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 94
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝² : DistribLattice α\ninst✝¹ : Fintype α\ninst✝ : DecidablePred SupIrred\nh : Nonempty α\nthis : OrderBot α\n⊢ α ↪o Set { a // SupIrred a }",
"usedConstants": [
"RelEmbedding.mk",
"Lattice.toSemilatticeSup",
"OrderIso.toOrderEmbedding",
"Iff.rfl"... | exact OrderIso.lowerSetSupIrred.toOrderEmbedding.trans ⟨⟨_, SetLike.coe_injective⟩, Iff.rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 24
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nhs : (𝓓 a 𝒜).Shatters s\nu : Finset α\nht : s ∩ u ⊆ s\nhu : u ∈ 𝒜 ∧ u.erase a ∈ 𝒜\n⊢ ∃ u_1 ∈ 𝒜, s ∩ u_1 = s ∩ u",
"usedConstants": [
"Finset",
"Membership.mem",
"Inter.inter",
"F... | · exact ⟨u, hu.1, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝³ : Semifield 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜\n⊢ ∑ s ∈ 𝒜, (↑((Fintype.card α).choose #s))⁻¹ =\n ∑ r ∈ range (Fintype.card α + 1), ∑ s ∈ 𝒜 with #s =... | rw [sum_fiberwise_of_maps_to'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Setoid.Partition | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 24
} | [
{
"pp": "case h.mp\nα : Type u_1\nc : Set (Set α)\nH : ∀ (a : α), ∃! b, b ∈ c ∧ a ∈ b\ns : Set α\ny : α\nhs : s ∈ c\nhy : y ∈ s\nx : α\n⊢ x ∈ s → x ∈ {x | (mkClasses c H) x y}",
"usedConstants": [
"Membership.mem",
"Set.instMembership",
"Set"
]
}
] | intro hx _s' hs' hx' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 78
} | [
{
"pp": "n r : ℕ\n𝒜 : Finset (Finset (Fin n))\nh✝ : Set.Sized r ↑𝒜\nusable : Finset (Finset (Fin n) × Finset (Fin n)) := {t | UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜}\nhusable : usable.Nonempty\nU V : Finset (Fin n)\nhUV : (U, V) ∈ univ ∧ UsefulCompression (U, V).1 (U, V).2 ∧ ¬IsCompressed (U, V)... | exact hUcard.not_ge <| t ⟨U₁, V₁⟩ <| mem_filter.2 ⟨mem_univ _, huseful, h⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Setoid.Partition | {
"line": 276,
"column": 17
} | {
"line": 278,
"column": 7
} | [
{
"pp": "α : Type u_1\nC : Partitions α\n⊢ (fun r ↦ ⟨r.classes, ⋯⟩) ((fun C ↦ mkClasses ↑C ⋯) C) = C",
"usedConstants": [
"Eq.mpr",
"Setoid.classes_mkClasses",
"Setoid.Partitions.isPartition",
"Setoid.mkClasses",
"congrArg",
"Membership.mem",
"Setoid.classes_eqv_cla... | by
rw [Partitions.ext_iff, ← classes_mkClasses C.toSet C.isPartition]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 67
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nC✝ : G.Coloring α\ninst✝² : Nonempty (G.Coloring α)\ninst✝¹ : Infinite α\ninst✝ : Nonempty V\nC : G.Coloring α\nv : V\n⊢ Infinite (G.Coloring α)",
"usedConstants": [
"SimpleGraph.Iso.completeGraph",
"RelHom.... | let f c := (Iso.completeGraph <| Equiv.swap (C v) c).toHom.comp C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 328,
"column": 6
} | {
"line": 339,
"column": 40
} | [
{
"pp": "n r k i : ℕ\n𝒜 : Finset (Finset (Fin n))\nhir : i ≤ r\nhrk : r ≤ k\nhkn : k ≤ n\nh₁ : Set.Sized r ↑𝒜\nh₂ : k.choose r ≤ #𝒜\nrange'k : Finset (Fin n) := (range k).attachFin ⋯\n𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k\nthis : Set.Sized r ↑𝒞\n⊢ #(∂^[i] 𝒞) ≤ #(∂^[i] 𝒜)",
"usedConsta... | refine iterated_kk h₁ ?_ ⟨‹_›, ?_⟩
· rwa [card_powersetCard, card_attachFin, card_range]
simp_rw [𝒞, mem_powersetCard]
rintro A B hA ⟨HB₁, HB₂⟩
refine ⟨fun t ht ↦ ?_, ‹_›⟩
rw [mem_attachFin, mem_range]
have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [t... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 328,
"column": 6
} | {
"line": 339,
"column": 40
} | [
{
"pp": "n r k i : ℕ\n𝒜 : Finset (Finset (Fin n))\nhir : i ≤ r\nhrk : r ≤ k\nhkn : k ≤ n\nh₁ : Set.Sized r ↑𝒜\nh₂ : k.choose r ≤ #𝒜\nrange'k : Finset (Fin n) := (range k).attachFin ⋯\n𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k\nthis : Set.Sized r ↑𝒞\n⊢ #(∂^[i] 𝒞) ≤ #(∂^[i] 𝒜)",
"usedConsta... | refine iterated_kk h₁ ?_ ⟨‹_›, ?_⟩
· rwa [card_powersetCard, card_attachFin, card_range]
simp_rw [𝒞, mem_powersetCard]
rintro A B hA ⟨HB₁, HB₂⟩
refine ⟨fun t ht ↦ ?_, ‹_›⟩
rw [mem_attachFin, mem_range]
have : toColex (image Fin.val B) < toColex (image Fin.val A) := by
rwa [t... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 523,
"column": 6
} | {
"line": 524,
"column": 34
} | [
{
"pp": "V✝ : Type u\nG : SimpleGraph V✝\nn : ℕ\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nC : G.Coloring α\nV : Type u_4\nW : Type u_5\n⊢ ∀ {v w : V ⊕ W}, (completeBipartiteGraph V W).Adj v w → (fun v ↦ v.isRight) v ≠ (fun v ↦ v.isRight) w",
"usedConstants": [
"Sum.isRight",
"False",
"and... | intro v w
cases v <;> cases w <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.VertexColoring | {
"line": 523,
"column": 6
} | {
"line": 524,
"column": 34
} | [
{
"pp": "V✝ : Type u\nG : SimpleGraph V✝\nn : ℕ\nι : Type u_1\nα : Type u_2\nβ : Type u_3\nC : G.Coloring α\nV : Type u_4\nW : Type u_5\n⊢ ∀ {v w : V ⊕ W}, (completeBipartiteGraph V W).Adj v w → (fun v ↦ v.isRight) v ≠ (fun v ↦ v.isRight) w",
"usedConstants": [
"Sum.isRight",
"False",
"and... | intro v w
cases v <;> cases w <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 350,
"column": 35
} | {
"line": 389,
"column": 5
} | [
{
"pp": "n : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\n⊢ #𝒜 ≤ (n - 1).choose (r - 1)",
"usedConstants": [
"_private.Mathlib.Combinatorics.SetFamily.KruskalKatona.0.Finset.erdos_ko_rado._proof_1_5",
"IsRightCancelAdd.addRightStrictMon... | by
-- Take care of the r=0 case first: it's not very interesting.
rcases Nat.eq_zero_or_pos r with b | h1r
· convert! Nat.zero_le _
rw [Finset.card_eq_zero, eq_empty_iff_forall_notMem]
refine fun A HA ↦ h𝒜 HA HA ?_
rw [disjoint_self_iff_empty, ← Finset.card_eq_zero, ← b]
exact h₂ HA
refine le_o... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 211,
"column": 2
} | {
"line": 211,
"column": 38
} | [
{
"pp": "V : Type u_1\nv : V\nG : SimpleGraph V\ns t : Finset V\ninst✝ : Fintype ↑(G.neighborSet v)\nh : G.IsBipartiteWith ↑s ↑t\nhv : v ∈ s\n⊢ G.degree v ≤ #t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.neighborFinset",
"id",
"LE.le",
"instLENat",
"Simp... | rw [← card_neighborFinset_eq_degree] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 38
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ns t : Finset V\ninst✝ : Fintype ↑(G.neighborSet w)\nh : G.IsBipartiteWith ↑s ↑t\nhw : w ∈ t\n⊢ G.degree w ≤ #s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.neighborFinset",
"id",
"LE.le",
"instLENat",
"Simp... | rw [← card_neighborFinset_eq_degree] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 249,
"column": 11
} | {
"line": 249,
"column": 60
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ns t : Finset V\ninst✝ : G.LocallyFinite\nh : G.IsBipartiteWith ↑s ↑t\n⊢ ∑ v ∈ s.attach, #(bipartiteAbove G.Adj t ↑v) = ∑ w ∈ t.attach, #(bipartiteBelow G.Adj s ↑w)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.Adj",
... | sum_attach s fun w ↦ #(bipartiteAbove G.Adj t w), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 366,
"column": 4
} | {
"line": 368,
"column": 25
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\n⊢ completeBipartiteGraph α β ⊑ G →\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right",
"usedConstants": [
"SimpleGraph.IsContained",
"... | refine fun ⟨f⟩ ↦ ⟨univ.map ⟨f ∘ Sum.inl, f.injective.comp Sum.inl_injective⟩,
univ.map ⟨f ∘ Sum.inr, f.injective.comp Sum.inr_injective⟩, by simp, by simp,
fun _ hl _ hr ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 44
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nx✝ : ∃ C, C.toSubgraph = G'\nC : G.ConnectedComponent\nh : C.toSubgraph = G'\n⊢ Maximal Subgraph.Connected G'",
"usedConstants": [
"SimpleGraph.Subgraph.instPartialOrder",
"SimpleGraph.Subgraph",
"PartialOrder.toPreord... | · exact h ▸ maximal_connected_toSubgraph _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 572,
"column": 6
} | {
"line": 572,
"column": 25
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | rcases l with l | l | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.SimpleGraph.Metric | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 55
} | [
{
"pp": "case inr.inr\nV : Type u_1\nG : SimpleGraph V\nu v w : V\nhuv : G.edist u v ≠ ⊤\nhvw : G.edist v w ≠ ⊤\np : G.Walk u v\nhp : ↑p.length = G.edist u v\n⊢ G.edist u w ≤ G.edist u v + G.edist v w",
"usedConstants": [
"SimpleGraph.exists_walk_of_edist_ne_top"
]
}
] | obtain ⟨q, hq⟩ := exists_walk_of_edist_ne_top hvw | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 576,
"column": 6
} | {
"line": 576,
"column": 25
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | rcases l with l | l | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 578,
"column": 4
} | {
"line": 578,
"column": 23
} | [
{
"pp": "case refine_1\nV : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetwe... | rcases l with l | l | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 655,
"column": 4
} | {
"line": 657,
"column": 19
} | [
{
"pp": "case refine_2.hu\nV : Type u\nG : SimpleGraph V\nGpc : G.Preconnected\nt : Finset V\nu : V\nut : u ∈ t\n⊢ u ∈ ↑(t.biUnion fun v ↦ (Nonempty.some ⋯).support.toFinset)",
"usedConstants": [
"Eq.mpr",
"Finset.coe_biUnion",
"and_true",
"Iff.of_eq",
"congrArg",
"Finset... | · simp only [Finset.coe_biUnion, Finset.mem_coe, List.coe_toFinset, Set.mem_iUnion,
Set.mem_setOf_eq, Walk.start_mem_support, exists_prop, and_true]
exact ⟨u, ut⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 702,
"column": 25
} | {
"line": 702,
"column": 41
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nf : G'.coe →g G\nhpreconn : G''.Preconnected\nu' : V\nu : ↑G'.verts\nhu : u ∈ G''.verts\nhfu : f u = u'\n⊢ ∀ (v : ↑(Subgraph.map f G'').verts), (Subgraph.map f G'').coe.Reachable ⟨u', ⋯⟩ v",
"usedConstants": [
"Set.Elem",
... | ⟨v', v, hv, hfv⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 583,
"column": 58
} | {
"line": 583,
"column": 72
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 583,
"column": 58
} | {
"line": 583,
"column": 72
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 583,
"column": 58
} | {
"line": 583,
"column": 72
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 68
} | [
{
"pp": "case succ\nα : Type u_1\nV : Type u_2\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : Fintype V\ninst✝¹ : DecidableEq V\ninst✝ : Semiring α\nn : ℕ\nih : ∀ (u v : V), (adjMatrix α G ^ n) u v = ↑(#(G.finsetWalkLength n u v))\nu v : V\n⊢ (adjMatrix α G ^ (n + 1)) u v = ↑(#(G.finsetWalkLength (n ... | simp only [pow_succ', finsetWalkLength, ih, adjMatrix_mul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 591,
"column": 58
} | {
"line": 591,
"column": 72
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 591,
"column": 58
} | {
"line": 591,
"column": 72
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 591,
"column": 58
} | {
"line": 591,
"column": 72
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nα : Type u_2\nβ : Type u_3\ninst✝³ : Finite α\ninst✝² : Finite β\ninst✝¹ : Nonempty α\ninst✝ : Nonempty β\nthis✝ : Fintype α\nthis : Fintype β\nx✝ :\n ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 228,
"column": 8
} | {
"line": 228,
"column": 23
} | [
{
"pp": "case cons\nV : Type u_1\nG : SimpleGraph V\nv v✝ : V\nha : G.Adj v v✝\nc' : G.Walk v✝ v\nhc : (c'.IsTrail ∧ s(v, v✝) ∉ c'.edges) ∧ ¬cons ha c' = nil ∧ c'.support.Nodup\nh : ⟨c', ⋯⟩ = Path.singleton ⋯\n⊢ False",
"usedConstants": [
"SimpleGraph.Adj.symm",
"Sym2.mk",
"congrArg",
... | Path.singleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 18
} | [
{
"pp": "case mpr\nV : Type u_1\nG : SimpleGraph V\n⊢ (Nonempty V ∧ ∀ (v w : V), ∃! p, p.IsPath) → G.Connected ∧ ∀ ⦃v w : V⦄ (p q : G.Path v w), p = q",
"usedConstants": [
"SimpleGraph.Walk",
"And",
"ExistsUnique",
"Nonempty",
"SimpleGraph.Walk.IsPath"
]
}
] | rintro ⟨hV, h⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 286,
"column": 6
} | {
"line": 286,
"column": 28
} | [
{
"pp": "case mpr.refine_1\nV : Type u_1\nG : SimpleGraph V\nhV : Nonempty V\nh : ∀ (v w : V), ∃! p, p.IsPath\nv w : V\n⊢ G.Reachable v w",
"usedConstants": []
}
] | obtain ⟨p, _⟩ := h v w | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 444,
"column": 55
} | {
"line": 444,
"column": 96
} | [
{
"pp": "V : Type u_1\nG F : SimpleGraph V\nhle : F ≤ G\nhF : F.IsAcyclic\nh : F.Reachable = G.Reachable\nthis : ¬Maximal (fun F ↦ F ≤ G ∧ F.IsAcyclic) F\nH : SimpleGraph V\nhFH : F < H\nhHG : H ≤ G\nhH : H.IsAcyclic\ne : Sym2 V\nheH : e ∈ H.edgeSet\nheF : e ∉ F.edgeSet\n⊢ e ∈ (F ⊔ fromEdgeSet {e}).edgeSet",
... | by simp [H.not_isDiag_of_mem_edgeSet heH] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 108,
"column": 6
} | {
"line": 109,
"column": 99
} | [
{
"pp": "case intro\nV : Type u\nG : SimpleGraph V\ninst✝ : Finite V\nval✝ : Fintype V\n⊢ Odd G.oddComponents.ncard ↔ Odd #(univ.biUnion fun x ↦ x.supp.toFinset)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Finset.univ",
"congrArg",
"SimpleGraph.oddComponents",
"Finset",
... | Finset.card_biUnion
(fun x _ y _ hxy ↦ Set.disjoint_toFinset.mpr (pairwise_disjoint_supp_connectedComponent _ hxy)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 109,
"column": 2
} | {
"line": 113,
"column": 43
} | [
{
"pp": "case mpr\nα : Type u_1\nG : SimpleGraph α\nu : α\n⊢ (∀ (v : α), u ≠ v → G.Adj u v) → G.eccent u ≤ 1",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"instAddMonoidWithOneENat",
"Classical.or_iff_not_imp_right",
"congrArg",
"SimpleGraph.Adj",
"id",
"Ne",
... | · intro hall
rw [eccent_le_iff]
intro v
rw [edist_le_one_iff_adj_or_eq]
exact or_iff_not_imp_right.mpr (hall v) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 426,
"column": 48
} | {
"line": 426,
"column": 61
} | [
{
"pp": "case inl\nV : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\ncf : Fintype.card V = 0\n⊢ #G.edgeFinset ≤ 0 / 0 + (Fintype.card V).choose 2",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.choose",
"instHDiv",
"congrArg",
"S... | Nat.div_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 308,
"column": 2
} | {
"line": 308,
"column": 31
} | [
{
"pp": "α : Type u_1\n⊢ ⊥.diam = 0",
"usedConstants": [
"Eq.mpr",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
"id",
"SimpleGraph.ediam",
"instOfNatNat",
"Bot.bot",
"SimpleGraph",
"Nat",
"ENat",
"propext",
"instCommSemir... | rw [diam, ENat.toNat_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 13
} | [
{
"pp": "case h\nα : Type u_1\nG : SimpleGraph α\ninst✝¹ : Nonempty α\ninst✝ : Finite α\nw : α\nhw : G.eccent w = G.radius\nv : α\nhv : G.edist w v = G.eccent w\n⊢ G.edist w v = G.radius",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"SimpleGraph.edist",
"SimpleGraph.radius",... | rw [hv, hw] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Finsubgraph | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 49
} | [
{
"pp": "V : Type u\nW : Type v\nG : SimpleGraph V\nF : SimpleGraph W\nG' G'' : G.Finsubgraph\nh : G'' ≤ G'\nf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)",
... | refine ⟨fun ⟨v, hv⟩ => f.toFun ⟨v, h.1 hv⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 29
} | [
{
"pp": "case mpr.inl\nα : Type u_1\nt u : Set α\nhdtu : Disjoint t u\nhctu : #↑t = #↑u\nhfin : (t ∪ u).Finite\n⊢ Even (t ∪ u).ncard",
"usedConstants": [
"congrArg",
"Set.finite_union",
"Set.Finite",
"Set.instUnion",
"Eq.mp",
"And",
"propext",
"Union.union",
... | rw [finite_union] at hfin | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 86
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (r + 2)) G\nhnc : ¬G.IsCompleteMultipartite\n⊢ ∃ v w₁ w₂ s t, G.IsFiveWheelLike r (#(s ∩ t)) v w₁ w₂ s t",
"usedConstants": [
"SimpleGraph.exists_isPathGraph3Compl_of_not_isCompleteMultipartite"
... | obtain ⟨v, w₁, w₂, p3⟩ := exists_isPathGraph3Compl_of_not_isCompleteMultipartite hnc | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 24
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu : Set V\nhc : G.IsClique u\nhu : u.Finite\n⊢ (∃ M, M.verts = u ∧ M.IsMatching) → Even u.ncard",
"usedConstants": [
"SimpleGraph.Subgraph",
"Exists",
"And",
"Eq",
"SimpleGraph.Subgraph.IsMatching",
"SimpleGraph.Subgraph.verts",
... | rintro ⟨M, rfl, hMr⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Combinatorics.SimpleGraph.Trails | {
"line": 130,
"column": 61
} | {
"line": 130,
"column": 93
} | [
{
"pp": "case h.e'_1.h.e'_3\nV : Type u_1\nG : SimpleGraph V\ninst✝² : DecidableEq V\nx u v : V\np : G.Walk u v\nht : p.IsEulerian\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ G.degree x = (Multiset.filter (fun e ↦ x ∈ e) ↑p.edges).card",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Mem... | ← card_incidenceFinset_eq_degree | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 174,
"column": 42
} | {
"line": 174,
"column": 52
} | [
{
"pp": "V : Type u_1\na✝ : Nontrivial V\nn : ℕ\nhn : ↑n ≤ ENat.card V - 1\nhh : (completeGraph V).vertexCoverNum < ↑n\nthis : ↑n - 1 ≤ ENat.card V\nt : Set V\nht₁ : t.encard = ↑(n - 1)\nht₂ : (completeGraph V).IsVertexCover t\n⊢ t.encard ≠ ⊤",
"usedConstants": [
"ENat.coe_ne_top._simp_1",
"Fals... | simp [ht₁] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 174,
"column": 42
} | {
"line": 174,
"column": 52
} | [
{
"pp": "V : Type u_1\na✝ : Nontrivial V\nn : ℕ\nhn : ↑n ≤ ENat.card V - 1\nhh : (completeGraph V).vertexCoverNum < ↑n\nthis : ↑n - 1 ≤ ENat.card V\nt : Set V\nht₁ : t.encard = ↑(n - 1)\nht₂ : (completeGraph V).IsVertexCover t\n⊢ t.encard ≠ ⊤",
"usedConstants": [
"ENat.coe_ne_top._simp_1",
"Fals... | simp [ht₁] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 174,
"column": 42
} | {
"line": 174,
"column": 52
} | [
{
"pp": "V : Type u_1\na✝ : Nontrivial V\nn : ℕ\nhn : ↑n ≤ ENat.card V - 1\nhh : (completeGraph V).vertexCoverNum < ↑n\nthis : ↑n - 1 ≤ ENat.card V\nt : Set V\nht₁ : t.encard = ↑(n - 1)\nht₂ : (completeGraph V).IsVertexCover t\n⊢ t.encard ≠ ⊤",
"usedConstants": [
"ENat.coe_ne_top._simp_1",
"Fals... | simp [ht₁] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.Primrec.Basic | {
"line": 725,
"column": 27
} | {
"line": 725,
"column": 52
} | [
{
"pp": "this : PrimrecRel fun a b ↦ a.2 = 0 ∧ b = 0 ∨ 0 < a.2 ∧ b * a.2 ≤ a.1 ∧ a.1 < (b + 1) * a.2\na k q : ℕ\nH : ¬k = 0\n⊢ q * k ≤ a ∧ a < (q + 1) * k ↔ q ≤ a / k ∧ a / k ≤ q",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
... | ← (@Nat.lt_succ_iff _ q), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.Primrec.List | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 51
} | [
{
"pp": "n : ℕ\n⊢ Nat.rec [] (fun n_1 IH ↦ (n, n_1, IH).2.2 ++ [(n, n_1, IH).2.1]) (id n) = List.range n",
"usedConstants": [
"Nat.recAux",
"congrArg",
"Nat.rec",
"id",
"Prod.mk",
"instOfNatNat",
"List.range",
"Prod.fst",
"List.cons",
"instHAppendO... | simp; induction n <;> simp [*, List.range_succ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Primrec.List | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 51
} | [
{
"pp": "n : ℕ\n⊢ Nat.rec [] (fun n_1 IH ↦ (n, n_1, IH).2.2 ++ [(n, n_1, IH).2.1]) (id n) = List.range n",
"usedConstants": [
"Nat.recAux",
"congrArg",
"Nat.rec",
"id",
"Prod.mk",
"instOfNatNat",
"List.range",
"Prod.fst",
"List.cons",
"instHAppendO... | simp; induction n <;> simp [*, List.range_succ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 35
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nx a b c : V\nM1 : (G ⊔ edge x b).Subgraph\nM2 : (G ⊔ edge a c).Subgraph\nhxa : G.Adj x a\nhab : G.Adj a b\nhnGxb : ¬G.Adj x b\nhnGac : ¬G.Adj a c\nhnxb : x ≠ b\nhnxc : x ≠ c\nhnac : a ≠ c\nhnbc : b ≠ c\nhM1 : M1.IsPerfectMatching\nhM2 : M2.IsPerfectMat... | apply Walk.toSubgraph_adj_snd | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 85
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nx a b c : V\nM1 : (G ⊔ edge x b).Subgraph\nM2 : (G ⊔ edge a c).Subgraph\nhxa : G.Adj x a\nhab : G.Adj a b\nhnGxb : ¬G.Adj x b\nhnGac : ¬G.Adj a c\nhnxb : x ≠ b\nhnxc : x ≠ c\nhnac : a ≠ c\nhnbc : b ≠ c\nhM1 : M1.IsPerfectMatching\nhM2 : M2.IsPerfectMat... | rw [← sdiff_edge _ (by simpa : ¬p.toSubgraph.spanningCoe.Adj x b), sdiff_le_iff'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.Partrec | {
"line": 71,
"column": 6
} | {
"line": 77,
"column": 43
} | [
{
"pp": "p : ℕ →. Bool\nH : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom\nm : ℕ\nIH : (y : ℕ) → lbp p y m → (∀ n < y, false ∈ p n) → { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }\nal : ∀ n < m, false ∈ p n\n⊢ { n // true ∈ p n ∧ ∀ m < n, false ∈ p m }",
"usedConstants": [
"Part",
"Eq.mpr",
"Preorder.... | have pm : (p m).Dom := by
rcases H with ⟨n, h₁, h₂⟩
rcases lt_trichotomy m n with (h₃ | h₃ | h₃)
· exact h₂ _ h₃
· rw [h₃]
exact h₁.fst
· injection mem_unique h₁ (al _ h₃) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Computability.Ackermann | {
"line": 104,
"column": 16
} | {
"line": 106,
"column": 17
} | [
{
"pp": "m : ℕ\n⊢ 0 < ack (m + 1) 0",
"usedConstants": [
"Eq.mpr",
"ack",
"congrArg",
"ack_succ_zero",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"LT.lt",
"instAddNat",
"instLTNat",
"OfNat.ofNat",
"Eq"
]
}
] | by
rw [ack_succ_zero]
apply ack_pos | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 28
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nh : ∀ (M : G.Subgraph), ¬M.IsPerfectMatching\nhvEven : Even (Nat.card V)\nval✝ : Fintype V\nGmax : SimpleGraph V\nhSubgraph : G ≤ Gmax\nhMatchingFree : Gmax.IsMatchingFree\nhMaximal : ∀ G' > Gmax, ∃ M, M.IsPerfectMatching\nh' : ∀ (K : Gmax.de... | exact hMatchingFree M hM | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 67
} | [
{
"pp": "f : ℝ → ℝ\nhf : GrowsPolynomially f\nhf' : ∃ᶠ (x : ℝ) in atTop, f x = 0\n⊢ ∀ᶠ (x : ℝ) in atTop, f x = 0",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"instHDiv",
"Mathlib.Meta.NormNum.IsNat.to_isNNRat",
"GroupWithZero.to... | obtain ⟨c₁, hc₁_mem, c₂, hc₂_mem, hf⟩ := hf (1 / 2) (by norm_num) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Computability.PartrecCode | {
"line": 561,
"column": 26
} | {
"line": 561,
"column": 42
} | [
{
"pp": "case refine_2.comp\ncf cg : Code\npf : Nat.Partrec cf.eval\npg : Nat.Partrec cg.eval\n⊢ Nat.Partrec (cf.comp cg).eval",
"usedConstants": [
"Nat.Partrec.comp",
"Nat.Partrec.Code.eval"
]
}
] | exact pf.comp pg | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.PartrecCode | {
"line": 561,
"column": 26
} | {
"line": 561,
"column": 42
} | [
{
"pp": "case refine_2.comp\ncf cg : Code\npf : Nat.Partrec cf.eval\npg : Nat.Partrec cg.eval\n⊢ Nat.Partrec (cf.comp cg).eval",
"usedConstants": [
"Nat.Partrec.comp",
"Nat.Partrec.Code.eval"
]
}
] | exact pf.comp pg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.PartrecCode | {
"line": 561,
"column": 26
} | {
"line": 561,
"column": 42
} | [
{
"pp": "case refine_2.comp\ncf cg : Code\npf : Nat.Partrec cf.eval\npg : Nat.Partrec cg.eval\n⊢ Nat.Partrec (cf.comp cg).eval",
"usedConstants": [
"Nat.Partrec.comp",
"Nat.Partrec.Code.eval"
]
}
] | exact pf.comp pg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 476,
"column": 57
} | {
"line": 476,
"column": 85
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\n⊢ (fun n ↦ -log (b i) * 1 / log ↑n ^ 2) = fun n ↦ -log (b i) * (1 / log ↑n ^ 2)",
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
... | by simp_rw [← mul_div_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 585,
"column": 8
} | {
"line": 585,
"column": 82
} | [
{
"pp": "case hbc.bc\nα : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nn : ℕ\nhn : 0 < n\nthis : ∀ x ≥ 0, 0 ≤ g x\n⊢ 0 ≤ ∑ u ∈ range n, g ↑u / ↑u ^ (p a b + 1)",
"usedConstants": [
"Real.instIsOrderedRing",
... | aesop (add safe Real.rpow_nonneg, safe div_nonneg, safe Finset.sum_nonneg) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\n⊢ ∀ (i : α), ∀ᶠ (x : ℕ) in atTop, 0 < asympBound g a b (r i x)",
"usedConstants": [
"AkraBazziRecurrence.asympBound",
"Real",
"Real.instZero",... | exact fun i => (R.tendsto_atTop_r i).eventually R.eventually_asympBound_pos | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Computability.PartrecCode | {
"line": 692,
"column": 2
} | {
"line": 692,
"column": 51
} | [
{
"pp": "c : Code\nn x : ℕ\n⊢ x ∈ c.eval n ↔ ∃ k, x ∈ evaln k c n",
"usedConstants": [
"Part",
"_private.Mathlib.Computability.PartrecCode.0.Nat.Partrec.Code.evaln_complete.match_1_1",
"Nat.Partrec.Code.evaln",
"Option.instMembership",
"Membership.mem",
"Exists",
"P... | refine ⟨fun h => ?_, fun ⟨k, h⟩ => evaln_sound h⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 595,
"column": 85
} | {
"line": 668,
"column": 84
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\n⊢ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c * g ↑n",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
... | by
obtain ⟨c₁, hc₁_mem, hc₁⟩ := R.exists_eventually_const_mul_le_r
obtain ⟨c₂, hc₂_mem, hc₂⟩ := R.g_grows_poly.eventually_atTop_le_nat hc₁_mem
have hc₁_pos : 0 < c₁ := hc₁_mem.1
refine ⟨max c₂ (c₂ / c₁ ^ (p a b + 1)), by positivity, ?_⟩
filter_upwards [hc₁, hc₂, R.eventually_r_pos, R.eventually_r_lt_n, eventu... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 560,
"column": 2
} | {
"line": 560,
"column": 21
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := b (min_bi b) / 2\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c... | refine ⟨2 * c₁, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.Language | {
"line": 322,
"column": 8
} | {
"line": 322,
"column": 40
} | [
{
"pp": "case a.zero\nα : Type u_1\nl m n : Language α\nhm : [] ∉ m\nh : l = m * l + n\n⊢ m ^ 0 * n ≤ m * l + n",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instCompleteAtomicBooleanAlgebraLanguage",
"HMul.hMul",
"Language.instAdd",
"Monoid.toMulOneClass",
"congr... | rw [pow_zero, one_mul, add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.