module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 43
} | {
"line": 603,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nD C D' : Set α\n⊢ M ＼ D ／ C ＼ D' = M ／ (C \\ D) ＼ (D ∪ D')",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"SDiff.sdiff",
"Matroid.delete_contract_comm'",
"Eq.refl",... | [] | rw [delete_contract_comm', delete_delete] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 43
} | {
"line": 603,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nD C D' : Set α\n⊢ M ＼ D ／ C ＼ D' = M ／ (C \\ D) ＼ (D ∪ D')",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"SDiff.sdiff",
"Matroid.delete_contract_comm'",
"Eq.refl",... | [] | rw [delete_contract_comm', delete_delete] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 601,
"column": 2
} | {
"line": 601,
"column": 43
} | {
"line": 603,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nD C D' : Set α\n⊢ M ＼ D ／ C ＼ D' = M ／ (C \\ D) ＼ (D ∪ D')",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instUnion",
"id",
"SDiff.sdiff",
"Matroid.delete_contract_comm'",
"Eq.refl",... | [] | rw [delete_contract_comm', delete_delete] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 99,
"column": 4
} | {
"line": 143,
"column": 77
} | {
"line": 145,
"column": 0
} | [
{
"pp": "case h_option\nR : Type u_1\ninst✝³ : CommRing R\nσ✝ : Type u_2\ninst✝² : Finite σ✝\ninst✝¹ : IsDomain R\nσ : Type u_2\ninst✝ : Fintype σ\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 : ... | [] | set Q := optionEquivLeft R σ P with hQ
suffices Q = 0 by
rw [← AlgEquiv.symm_apply_apply (optionEquivLeft R σ) P, ← hQ, this, map_zero]
have Heval' (x : σ → R) (hx : ∀ i, x i ∈ S (some i)) : Polynomial.map (eval x) Q = 0 := by
apply Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero' _ (S none)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 99,
"column": 4
} | {
"line": 143,
"column": 77
} | {
"line": 145,
"column": 0
} | [
{
"pp": "case h_option\nR : Type u_1\ninst✝³ : CommRing R\nσ✝ : Type u_2\ninst✝² : Finite σ✝\ninst✝¹ : IsDomain R\nσ : Type u_2\ninst✝ : Fintype σ\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 : ... | [] | set Q := optionEquivLeft R σ P with hQ
suffices Q = 0 by
rw [← AlgEquiv.symm_apply_apply (optionEquivLeft R σ) P, ← hQ, this, map_zero]
have Heval' (x : σ → R) (hx : ∀ i, x i ∈ S (some i)) : Polynomial.map (eval x) Q = 0 := by
apply Polynomial.eq_zero_of_natDegree_lt_card_of_eval_eq_zero' _ (S none)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 183,
"column": 2
} | {
"line": 188,
"column": 15
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommRing R\nι : Type u_3\ni : ι\nS : Finset R\nm : ι →₀ ℕ\nhP : P S i = (rename fun x ↦ i) (P S ())\ne : Unit →₀ ℕ\nhe : ¬coeff e (P S ()) = 0\nhm : mapDomain (fun x ↦ i) e = m\nthis : Nontrivial R\n⊢ e () ≤ #S",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_2\nR : Type u_1\ninst✝ : CommRing R\nι : Type u_3\ni : ι\nS : Finset R\nm : ι →₀ ℕ\nhP : P S i = (rename fun x ↦ i) (P S ())\ne : Unit →₀ ℕ\nhe : ¬coeff e (P S ()) = 0\nhm : mapDomain (fun x ↦ i) e = m\nthis : Nontrivial R\n⊢ m = single i (e ())"
] | · suffices e ≼[lex] single () #S by
simpa [MonomialOrder.lex_le_iff_of_unique] using this
rw [← Alon.degree_P]
apply MonomialOrder.le_degree
rw [mem_support_iff]
convert! he | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 197,
"column": 4
} | {
"line": 202,
"column": 37
} | {
"line": 203,
"column": 2
} | [
{
"pp": "case nil.cons\nU : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : ∀ (u : U), Injective (φ.star u)\nu v₁ y₂ b✝ : U\np₂ : Path u b✝\ne₂ : b✝ ⟶ y₂\n⊢ (fun p ↦ ⟨φ.obj p.fst, φ.mapPath p.snd⟩) ⟨u, Path.nil⟩ = (fun p ↦ ⟨φ.obj p.fst, φ.mapPath p.snd⟩) ⟨y₂, p₂.cons e₂⟩ →\n ⟨u,... | [] | · intro h
simp only [mapPath_cons, Sigma.mk.inj_iff] at h
exfalso
obtain ⟨h, h'⟩ := h
rw [← Path.eq_cast_iff_heq rfl h.symm, Path.cast_cons] at h'
exact (Path.nil_ne_cons _ _) h' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 265,
"column": 38
} | {
"line": 266,
"column": 20
} | {
"line": 266,
"column": 20
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ : Type u_2\ninst✝¹ : Finite σ\ninst✝ : IsDomain R\nf : MvPolynomial σ R\nt : σ →₀ ℕ\nht : coeff t f ≠ 0\nht' : f.totalDegree = Finsupp.degree t\nS : σ → Finset R\nhtS : ∀ (i : σ), t i < #(S i)\nx✝ : LinearOrder σ := IsWellOrder.linearOrder WellOrderingRel\nHeval : ∀... | [] | by simp only [(Alon.monic_P ..).leadingCoeff_eq_one,
isRegular_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 23
} | {
"line": 248,
"column": 4
} | [
{
"pp": "case cons\nU : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : ∀ (u : U), Surjective (φ.star u)\nu : U\nv : V\nu' : U\nq' : Path u u'\nu'' : U\neu : u' ⟶ u''\n⊢ ∃ a, (fun p ↦ ⟨φ.obj p.fst, φ.mapPath p.snd⟩) a = ⟨φ.obj u'', (φ.mapPath q').cons (φ.map eu)⟩",
"ppTerm": "?... | [
"case h\nU : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : ∀ (u : U), Surjective (φ.star u)\nu : U\nv : V\nu' : U\nq' : Path u u'\nu'' : U\neu : u' ⟶ u''\n⊢ (fun p ↦ ⟨φ.obj p.fst, φ.mapPath p.snd⟩) ⟨u'', q'.cons eu⟩ = ⟨φ.obj u'', (φ.mapPath q').cons (φ.map eu)⟩"
] | use ⟨_, q'.cons eu⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Combinatorics.Schnirelmann | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 16
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case mp.inr\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhA : 0 ∈ A\nh : {0}ᶜ ⊆ A\nx : ℕ\nhx : x ≠ 0\n⊢ x ∈ A",
"ppTerm": "?mp.inr",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h",
"x",
"hx"
],
"usedGoals": []
}
] | [] | · exact h hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 69
} | {
"line": 105,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\na : α\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : s ∈ 𝒜\nha : a ∉ s\n⊢ insert a (s.erase a) ∈ 𝒜 ∧ a ∉ s.erase a ∨ s.erase a ∈ 𝒜 ∧ a ∉ s.erase a",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | exact Or.inr ⟨by rwa [erase_eq_of_notMem ha], notMem_erase _ _⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 69
} | {
"line": 105,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\na : α\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : s ∈ 𝒜\nha : a ∉ s\n⊢ insert a (s.erase a) ∈ 𝒜 ∧ a ∉ s.erase a ∨ s.erase a ∈ 𝒜 ∧ a ∉ s.erase a",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | exact Or.inr ⟨by rwa [erase_eq_of_notMem ha], notMem_erase _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 103,
"column": 6
} | {
"line": 103,
"column": 69
} | {
"line": 105,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝ : DecidableEq α\na : α\n𝒜 : Finset (Finset α)\ns : Finset α\nhs : s ∈ 𝒜\nha : a ∉ s\n⊢ insert a (s.erase a) ∈ 𝒜 ∧ a ∉ s.erase a ∨ s.erase a ∈ 𝒜 ∧ a ∉ s.erase a",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [] | exact Or.inr ⟨by rwa [erase_eq_of_notMem ha], notMem_erase _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 248,
"column": 2
} | {
"line": 251,
"column": 52
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\n⊢ s ∈ 𝓓 a 𝒜",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"Eq.mp",
"id",
"Ins... | [] | by_cases ha : a ∈ s
· rwa [insert_eq_of_mem ha] at h
· rw [← erase_insert ha]
exact erase_mem_compression_of_mem_compression h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.Down | {
"line": 248,
"column": 2
} | {
"line": 251,
"column": 52
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns : Finset α\na : α\nh : insert a s ∈ 𝓓 a 𝒜\n⊢ s ∈ 𝓓 a 𝒜",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"Eq.mp",
"id",
"Ins... | [] | by_cases ha : a ∈ s
· rwa [insert_eq_of_mem ha] at h
· rw [← erase_insert ha]
exact erase_mem_compression_of_mem_compression h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 243,
"column": 10
} | {
"line": 243,
"column": 27
} | {
"line": 243,
"column": 27
} | [
{
"pp": "case succ.mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nt : Finset α\na : α\nu : Finset α\nhau : a ∉ u\nih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[#u] 𝒜 ↔ ∃ u_1, #u_1 = #u ∧ u_1 ⊆ t ∧ t \\ u_1 ∈ 𝒜\nhut : insert a u ⊆ t\nhtu : t \\ insert a u ∈ 𝒜\n⊢ ∃ a ∈ t, ∃ u_1, #u_1 =... | [
"case succ.mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nt : Finset α\na : α\nu : Finset α\nhau : a ∉ u\nih : ∀ {t : Finset α}, t ∈ ∂⁺ ^[#u] 𝒜 ↔ ∃ u_1, #u_1 = #u ∧ u_1 ⊆ t ∧ t \\ u_1 ∈ 𝒜\nhut : a ∈ t ∧ u ⊆ t\nhtu : t \\ insert a u ∈ 𝒜\n⊢ ∃ a ∈ t, ∃ u_1, #u_1 = #u ∧ (u_1 ⊆ ... | insert_subset_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.Shadow | {
"line": 297,
"column": 4
} | {
"line": 302,
"column": 9
} | {
"line": 304,
"column": 0
} | [
{
"pp": "case succ.mpr\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nk : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ (∃ t ∈ 𝒜, t ⊆ s ∧ #t + (k + 1) = #s) → ∃ t ∈ ∂⁺ 𝒜, t ⊆ s ∧ #t + k = #s",
"ppTerm": "?succ.mpr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | [] | · rintro ⟨t, ht, hts, hcard⟩
obtain ⟨u, htu, hus, hu⟩ := Finset.exists_subsuperset_card_eq hts (Nat.le_add_right _ 1)
(by lia)
refine ⟨u, mem_upShadow_iff_exists_mem_card_add_one.2 ⟨t, ht, htu, hu⟩, hus, ?_⟩
rw [hu, ← hcard, add_right_comm]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 223,
"column": 4
} | {
"line": 224,
"column": 29
} | {
"line": 225,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\nha : a ∉ s\nb : α\nhb : b ∈ s\nh : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ (a ⊔ v) \\ u ∈ s",
"ppTerm": "?pos✝",
"assigne... | [] | rwa [← hba, sdiff_sup_cancel (le_sup_of_le_left h.2), sup_sdiff_right_self,
h.1.symm.sdiff_eq_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 223,
"column": 4
} | {
"line": 224,
"column": 29
} | {
"line": 225,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\nha : a ∉ s\nb : α\nhb : b ∈ s\nh : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ (a ⊔ v) \\ u ∈ s",
"ppTerm": "?pos✝",
"assigne... | [] | rwa [← hba, sdiff_sup_cancel (le_sup_of_le_left h.2), sup_sdiff_right_self,
h.1.symm.sdiff_eq_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 223,
"column": 4
} | {
"line": 224,
"column": 29
} | {
"line": 225,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : GeneralizedBooleanAlgebra α\ninst✝² : DecidableRel Disjoint\ninst✝¹ : DecidableLE α\ns : Finset α\nu v a : α\ninst✝ : DecidableEq α\nha : a ∉ s\nb : α\nhb : b ∈ s\nh : Disjoint u b ∧ v ≤ b\nhba : (b ⊔ u) \\ v = a\n⊢ (a ⊔ v) \\ u ∈ s",
"ppTerm": "?pos✝",
"assigne... | [] | rwa [← hba, sdiff_sup_cancel (le_sup_of_le_left h.2), sup_sdiff_right_self,
h.1.symm.sdiff_eq_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 378,
"column": 6
} | {
"line": 378,
"column": 40
} | {
"line": 378,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜\nh𝒜₀ : ∅ ∉ 𝒜\ns : Finset α\nhs : s ∈ 𝒜\n⊢ (↑((card α).choose #s))⁻¹ = ↑(#(𝒜.truncatedInf s)) / (↑(#s) * ↑((card α).choose #s))",
"ppTerm": "?m.85",
"assigned": true,
... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜\nh𝒜₀ : ∅ ∉ 𝒜\ns : Finset α\nhs : s ∈ 𝒜\n⊢ (↑((card α).choose #s))⁻¹ = ↑(#s) / (↑(#s) * ↑((card α).choose #s))"
] | truncatedInf_of_isAntichain h𝒜 hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.HarrisKleitman | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 97
} | {
"line": 74,
"column": 2
} | [
{
"pp": "case insert\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nhs : a ∉ s\nih :\n ∀ {𝒜 ℬ : Finset (Finset α)},\n IsLowerSet ↑𝒜 → IsLowerSet ↑ℬ → (∀ t ∈ 𝒜, t ⊆ s) → (∀ t ∈ ℬ, t ⊆ s) → #𝒜 * #ℬ ≤ 2 ^ #s * #(𝒜 ∩ ℬ)\n𝒜 ℬ : Finset (Finset α)\nh𝒜 : IsLowerSet ↑𝒜\nhℬ : IsLowerSet ↑ℬ\nh𝒜s :... | [
"case insert\nα : Type u_1\ninst✝ : DecidableEq α\na : α\ns : Finset α\nhs : a ∉ s\nih :\n ∀ {𝒜 ℬ : Finset (Finset α)},\n IsLowerSet ↑𝒜 → IsLowerSet ↑ℬ → (∀ t ∈ 𝒜, t ⊆ s) → (∀ t ∈ ℬ, t ⊆ s) → #𝒜 * #ℬ ≤ 2 ^ #s * #(𝒜 ∩ ℬ)\n𝒜 ℬ : Finset (Finset α)\nh𝒜 : IsLowerSet ↑𝒜\nhℬ : IsLowerSet ↑ℬ\nh𝒜s : ∀ t ∈ 𝒜, t... | grw [mul_add_mul_le_mul_add_mul
(card_le_card h𝒜.memberSubfamily_subset_nonMemberSubfamily) <|
card_le_card hℬ.memberSubfamily_subset_nonMemberSubfamily, ← two_mul, pow_succ', mul_assoc] | Mathlib.Tactic.GRewrite._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_GRewrite_grwSeq_1 | Mathlib.Tactic.GRewrite.grwSeq |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 63,
"column": 57
} | {
"line": 63,
"column": 69
} | {
"line": 63,
"column": 70
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nx✝ : 𝒜.Nonempty\nt : Finset α\nht : t ⊆ ∅\ns : Finset α\nhs : s ∈ 𝒜\n⊢ ∅ ∩ s = t",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finset.empty_inter",
"Eq.mpr",
"congrArg",
"Finset",
"id",... | [
"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nx✝ : 𝒜.Nonempty\nt : Finset α\nht : t ⊆ ∅\ns : Finset α\nhs : s ∈ 𝒜\n⊢ ∅ = t"
] | empty_inter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 279,
"column": 17
} | {
"line": 279,
"column": 56
} | {
"line": 280,
"column": 2
} | [
{
"pp": "n r : ℕ\n𝒜 𝒞 : Finset (Finset (Fin n))\nh𝒜r : Set.Sized r ↑𝒜\nh𝒞𝒜 : #𝒞 ≤ #𝒜\nh𝒞 : IsInitSeg 𝒞 r\n𝒜' : Finset (Finset (Fin n))\nh𝒜 : 𝒜' ⊆ 𝒜\nh𝒜𝒞 : #𝒜' = #𝒞\nℬ : Finset (Finset (Fin n))\nhℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')\nh𝒜ℬ : #𝒜' = #ℬ\nhℬr : Set.Sized r ↑ℬ\nhℬ : ∀ (U V : Finset (Fin n)), UV.... | [] | by subst 𝒞; exact hℬ𝒜.trans (by gcongr) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 510,
"column": 83
} | {
"line": 513,
"column": 23
} | {
"line": 515,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G.chromaticNumber = 1 ↔ G = ⊥ ∧ Nonempty V",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"NeZero.one",
"instAddMonoidWithOneENat",
"ChainCompletePartialO... | [] | by
rw [eq_iff_le_not_lt, Order.lt_one_iff_nonpos, ← not_isEmpty_iff, ← Nat.cast_one, ← Nat.cast_zero,
chromaticNumber_le_iff_colorable, chromaticNumber_le_iff_colorable, colorable_one_iff,
colorable_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 517,
"column": 4
} | {
"line": 520,
"column": 66
} | {
"line": 521,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nh : 2 ≤ G.chromaticNumber\n⊢ G ≠ ⊥",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"instCharZeroENat",
"instAddMonoidWithOneEN... | [] | contrapose! h
by_cases h' : IsEmpty V
· simp [chromaticNumber_eq_zero_of_isEmpty]
· simp [chromaticNumber_eq_one_iff.mpr ⟨h, by simpa using h'⟩] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 517,
"column": 4
} | {
"line": 520,
"column": 66
} | {
"line": 521,
"column": 2
} | [
{
"pp": "case refine_1\nV : Type u\nG : SimpleGraph V\nh : 2 ≤ G.chromaticNumber\n⊢ G ≠ ⊥",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"instCharZeroENat",
"instAddMonoidWithOneEN... | [] | contrapose! h
by_cases h' : IsEmpty V
· simp [chromaticNumber_eq_zero_of_isEmpty]
· simp [chromaticNumber_eq_one_iff.mpr ⟨h, by simpa using h'⟩] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 178,
"column": 2
} | {
"line": 178,
"column": 61
} | {
"line": 179,
"column": 2
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ns t : Finset V\ninst✝¹ : Fintype ↑(G.neighborSet w)\ninst✝ : DecidableRel G.Adj\nh : G.IsBipartiteWith ↑s ↑t\nhw : w ∈ t\nv : V\n⊢ v ∈ G.neighborFinset w ↔ v ∈ {v ∈ s | G.Adj v w}",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"V : Type u_1\nw : V\nG : SimpleGraph V\ns t : Finset V\ninst✝¹ : Fintype ↑(G.neighborSet w)\ninst✝ : DecidableRel G.Adj\nh : G.IsBipartiteWith ↑s ↑t\nhw : w ∈ t\nv : V\n⊢ G.Adj v w → v ∈ s"
] | rw [mem_neighborFinset, adj_comm, mem_filter, iff_and_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 249,
"column": 2
} | {
"line": 250,
"column": 53
} | {
"line": 251,
"column": 2
} | [
{
"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)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"V : Type u_1\nG : SimpleGraph V\ns t : Finset V\ninst✝ : G.LocallyFinite\nh : G.IsBipartiteWith ↑s ↑t\n⊢ ∑ x ∈ s, #(bipartiteAbove G.Adj t x) = ∑ x ∈ t, #(bipartiteBelow G.Adj s x)"
] | simp_rw [sum_attach s fun w ↦ #(bipartiteAbove G.Adj t w),
sum_attach t fun v ↦ #(bipartiteBelow G.Adj s v)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 65,
"column": 66
} | {
"line": 65,
"column": 91
} | {
"line": 67,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\n⊢ G.IsEdgeReachable 0 u v",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"SimpleGraph.deleteEdges",
"not_lt_zero._simp_1",
"False",
"Set.encard",
"instAddMonoidWithOneENat",
"ENat.instNatCast",
"co... | [] | by simp [IsEdgeReachable] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 95
} | {
"line": 458,
"column": 0
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ G.neighborFinset w ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset w ∪ sᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set... | [] | simpa [neighborFinset_def] using G.neighborSet_subset_between_union_compl (by simpa using hw) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 95
} | {
"line": 458,
"column": 0
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ G.neighborFinset w ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset w ∪ sᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set... | [] | simpa [neighborFinset_def] using G.neighborSet_subset_between_union_compl (by simpa using hw) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 456,
"column": 2
} | {
"line": 456,
"column": 95
} | {
"line": 458,
"column": 0
} | [
{
"pp": "V : Type u_1\nw : V\nG : SimpleGraph V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ns : Finset V\ninst✝ : DecidableRel G.Adj\nhw : w ∈ sᶜ\n⊢ G.neighborFinset w ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset w ∪ sᶜ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set... | [] | simpa [neighborFinset_def] using G.neighborSet_subset_between_union_compl (by simpa using hw) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 53
} | {
"line": 320,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u' v' : V\np : G.Walk u v\n⊢ p.toSubgraph.Adj u' v' ↔ s(u', v') ∈ p.edges",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.Subgraph.mem_edgeSet",
"Iff.rfl",
"Members... | [] | rw [← p.mem_edges_toSubgraph, Subgraph.mem_edgeSet] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 53
} | {
"line": 320,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u' v' : V\np : G.Walk u v\n⊢ p.toSubgraph.Adj u' v' ↔ s(u', v') ∈ p.edges",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.Subgraph.mem_edgeSet",
"Iff.rfl",
"Members... | [] | rw [← p.mem_edges_toSubgraph, Subgraph.mem_edgeSet] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 53
} | {
"line": 320,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u' v' : V\np : G.Walk u v\n⊢ p.toSubgraph.Adj u' v' ↔ s(u', v') ∈ p.edges",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.Subgraph.mem_edgeSet",
"Iff.rfl",
"Members... | [] | rw [← p.mem_edges_toSubgraph, Subgraph.mem_edgeSet] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 546,
"column": 62
} | {
"line": 555,
"column": 49
} | {
"line": 557,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\n⊢ #G.bipartiteDoubleCover.edgeFinset = 2 * #G.edgeFinset",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Adj.symm",
"SimpleGraph.bipartiteDoubleCover",
"... | [] | by
rw [two_mul_card_edgeFinset, eq_comm]
apply card_bij (fun (v, w) _ ↦ s(.inl v, .inr w))
(fun _ h ↦ by simpa using h) (by grind) (fun e he ↦ ?_)
induction e with | _ v w
rw [mem_edgeFinset, mem_edgeSet] at he
match v, w with
| .inl _, .inr _ => simpa using he
| .inr _, .inl _ => simpa using he.symm
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 94
} | {
"line": 412,
"column": 4
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv u : V\ni : ℕ\np : G.Walk u v\nhp : p.IsPath\nh✝ : i ≠ 0\nh' : i < p.length\nh : p.getVert (i - 1) = p.getVert (i + 1)\n⊢ False",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.length",
... | [
"V : Type u\nG : SimpleGraph V\nv u : V\ni : ℕ\np : G.Walk u v\nhp : p.IsPath\nh✝ : i ≠ 0\nh' : i < p.length\nh : p.getVert (i - 1) = p.getVert (i + 1)\nthis : i - 1 = i + 1\n⊢ False"
] | have := hp.getVert_injOn (by rw [Set.mem_setOf_eq]; lia) (by rw [Set.mem_setOf_eq]; lia) h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 567,
"column": 4
} | {
"line": 596,
"column": 25
} | {
"line": 597,
"column": 2
} | [
{
"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... | [] | simp_rw [← card_left, ← card_right]
obtain ⟨l, hl⟩ : left.Nonempty := card_pos.mp <| card_pos.trans_le card_left.ge
obtain ⟨r, hr⟩ : right.Nonempty := card_pos.mp <| card_pos.trans_le card_right.ge
have hmem_left {l'} (hl' : l' ∈ left) :
(l.isLeft → l'.isLeft) ∧ (l.isRight → l'.isRight) := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 567,
"column": 4
} | {
"line": 596,
"column": 25
} | {
"line": 597,
"column": 2
} | [
{
"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... | [] | simp_rw [← card_left, ← card_right]
obtain ⟨l, hl⟩ : left.Nonempty := card_pos.mp <| card_pos.trans_le card_left.ge
obtain ⟨r, hr⟩ : right.Nonempty := card_pos.mp <| card_pos.trans_le card_right.ge
have hmem_left {l'} (hl' : l' ∈ left) :
(l.isLeft → l'.isLeft) ∧ (l.isRight → l'.isRight) := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 51
} | {
"line": 185,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Sym2.mk",
"congrArg",
"SimpleGraph.IsAcyclic",
"SimpleGraph.Adj",
"Membership.mem",
"_private.Mathlib.Co... | [] | simp [isAcyclic_iff_forall_isBridge, Sym2.forall] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 51
} | {
"line": 185,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Sym2.mk",
"congrArg",
"SimpleGraph.IsAcyclic",
"SimpleGraph.Adj",
"Membership.mem",
"_private.Mathlib.Co... | [] | simp [isAcyclic_iff_forall_isBridge, Sym2.forall] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 51
} | {
"line": 185,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsAcyclic ↔ ∀ ⦃v w : V⦄, G.Adj v w → G.IsBridge s(v, w)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Sym2.mk",
"congrArg",
"SimpleGraph.IsAcyclic",
"SimpleGraph.Adj",
"Membership.mem",
"_private.Mathlib.Co... | [] | simp [isAcyclic_iff_forall_isBridge, Sym2.forall] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 366,
"column": 66
} | {
"line": 369,
"column": 65
} | {
"line": 371,
"column": 0
} | [
{
"pp": "α : Type u_1\nV : Type u_2\nG : SimpleGraph V\ninst✝² : DecidableRel G.Adj\ninst✝¹ : Fintype V\ninst✝ : NonAssocSemiring α\nv : V\nvec : V → α\n⊢ (vec ᵥ* adjMatrix α G) v = ∑ u ∈ G.neighborFinset v, vec u",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAs... | [] | by
simp only [← dotProduct_adjMatrix, vecMul]
refine congr rfl ?_; ext x
rw [← transpose_apply (adjMatrix α G) x v, transpose_adjMatrix] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 478,
"column": 2
} | {
"line": 479,
"column": 51
} | {
"line": 481,
"column": 0
} | [
{
"pp": "α : Type u_1\nV : Type u_2\ninst✝² : MulZeroOneClass α\ninst✝¹ : Nontrivial α\nA : Matrix V V α\nh : A.IsAdjMatrix\ninst✝ : DecidableEq α\n⊢ adjMatrix α h.toGraph = A",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Matrix.IsAdjMatrix.toGraph_adj",
"MulOne.toOne",
... | [] | ext i j
obtain h' | h' := h.zero_or_one i j <;> simp [h'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.AdjMatrix | {
"line": 478,
"column": 2
} | {
"line": 479,
"column": 51
} | {
"line": 481,
"column": 0
} | [
{
"pp": "α : Type u_1\nV : Type u_2\ninst✝² : MulZeroOneClass α\ninst✝¹ : Nontrivial α\nA : Matrix V V α\nh : A.IsAdjMatrix\ninst✝ : DecidableEq α\n⊢ adjMatrix α h.toGraph = A",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Matrix.IsAdjMatrix.toGraph_adj",
"MulOne.toOne",
... | [] | ext i j
obtain h' | h' := h.zero_or_one i j <;> simp [h'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 21
} | {
"line": 46,
"column": 4
} | [
{
"pp": "n : ℕ\nh : 2 ≤ n\nv w : Fin 2\n⊢ (fun v ↦ ⟨↑v, ⋯⟩) v = (fun v ↦ ⟨↑v, ⋯⟩) w → v = w",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fin.mk.injEq",
"Trans.trans",
"congrArg",
"Fin.isLt",
"Fin.mk",
"id",
"Nat.instTransLtLe",
... | [
"n : ℕ\nh : 2 ≤ n\nv w : Fin 2\n⊢ ↑v = ↑w → v = w"
] | rw [Fin.mk.injEq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 429,
"column": 6
} | {
"line": 429,
"column": 62
} | {
"line": 430,
"column": 6
} | [
{
"pp": "case refine_2\nV : Type u_1\nG T : SimpleGraph V\nhG : G.Connected\nhT : T ≤ G\nthis : Nonempty V\nhT' : T.IsTree\n⊢ T.Reachable = G.Reachable",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"SimpleGraph.Connected.preconnected",
"Eq.mpr",
"congrArg",
"S... | [
"case refine_2\nV : Type u_1\nG T : SimpleGraph V\nhG : G.Connected\nhT : T ≤ G\nthis : Nonempty V\nhT' : T.IsTree\n⊢ ⊤ = G.Reachable"
] | T.preconnected_iff_reachable_eq_top.mp hT'.preconnected, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 65
} | {
"line": 257,
"column": 2
} | [
{
"pp": "V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\n⊢ Nonempty (G ≃g turanGraph (Fintype.card V) r)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"SimpleGraph.turanGraph",
"Finse... | [
"V : Type u_1\ninst✝¹ : Fintype V\nG : SimpleGraph V\ninst✝ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\nzm : ↥univ ≃ Fin #univ\nzp :\n ∀ (a b : ↥univ),\n h.finpartition.part ↑a = h.finpartition.part ↑b ↔ ↑(zm a) % #h.finpartition.parts = ↑(zm b) % #h.finpartition.parts\n⊢ Nonempty (G ≃g turanGraph (Fin... | obtain ⟨zm, zp⟩ := h.isEquipartition.exists_partPreservingEquiv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 175,
"column": 33
} | {
"line": 175,
"column": 43
} | {
"line": 175,
"column": 44
} | [
{
"pp": "α : Type u\nG : SimpleGraph α\ns : Set α\nv w₁ w₂ : α\nh : G.IsPathGraph3Compl v w₁ w₂\na✝ b✝ : Fin 3\n⊢ G.Adj\n (match a✝ with\n | 0 => w₁\n | 1 => v\n | 2 => w₂)\n (match b✝ with\n | 0 => w₁\n | 1 => v\n | 2 => w₂) ↔\n (pathGraph 3)ᶜ.Adj a✝ b✝",
"ppTerm"... | [
"α : Type u\nG : SimpleGraph α\ns : Set α\nv w₁ w₂ : α\nh : G.IsPathGraph3Compl v w₁ w₂\na✝ b✝ : Fin 3\n⊢ G.Adj\n (match a✝ with\n | 0 => w₁\n | 1 => v\n | 2 => w₂)\n (match b✝ with\n | 0 => w₁\n | 1 => v\n | 2 => w₂) ↔\n a✝ ≠ b✝ ∧ ¬(pathGraph 3).Adj a✝ b✝"
] | compl_adj, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Finite | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 30
} | {
"line": 105,
"column": 30
} | [
{
"pp": "case intro\nV : Type u\nG : SimpleGraph V\ninst✝ : Finite V\nval✝ : Fintype V\n⊢ Odd G.oddComponents.ncard ↔ Odd (Nat.card V)",
"ppTerm": "?intro",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.oddComponents",
"Odd",
"Fintype.card",
... | [
"case intro\nV : Type u\nG : SimpleGraph V\ninst✝ : Finite V\nval✝ : Fintype V\n⊢ Odd G.oddComponents.ncard ↔ Odd (Fintype.card V)"
] | Nat.card_eq_fintype_card | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 298,
"column": 46
} | {
"line": 298,
"column": 63
} | {
"line": 298,
"column": 64
} | [
{
"pp": "r t : ℕ\nv : Fin r × Fin t\n⊢ (Fintype.card (Fin r) - 1) * #univ = (r - 1) * t",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"HMul.hMul",
"Finset.univ",
"congrArg",
"HSub.hSub",
"Fintype.card",
"id",
... | [
"r t : ℕ\nv : Fin r × Fin t\n⊢ (r - 1) * #univ = (r - 1) * t"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 306,
"column": 52
} | {
"line": 306,
"column": 69
} | {
"line": 306,
"column": 70
} | [
{
"pp": "r t : ℕ\n⊢ Fintype.card (Fin r) * Fintype.card (Fin t) * ((r - 1) * t) = 2 * (r.choose 2 * t ^ 2)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"Nat.choose",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"HSub.... | [
"r t : ℕ\n⊢ r * Fintype.card (Fin t) * ((r - 1) * t) = 2 * (r.choose 2 * t ^ 2)"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 393,
"column": 8
} | {
"line": 393,
"column": 25
} | {
"line": 393,
"column": 26
} | [
{
"pp": "α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nht : ¬t = 0\n⊢ Fintype.card (Fin r) ≤ Fintype.card ↥K.parts",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"... | [
"α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nht : ¬t = 0\n⊢ r ≤ Fintype.card ↥K.parts"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 396,
"column": 42
} | {
"line": 396,
"column": 59
} | {
"line": 396,
"column": 60
} | [
{
"pp": "α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nht : ¬t = 0\nthis : Nonempty (Fin r ↪ ↥K.parts)\nfᵣ : Fin r ↪ ↥K.parts := Classical.arbitrary (Fin r ↪ ↥K.parts)\np : ↥K.parts\n⊢ Fintype.card (Fin t) ≤ Fintype.card ↥↑p",
"pp... | [
"α : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nht : ¬t = 0\nthis : Nonempty (Fin r ↪ ↥K.parts)\nfᵣ : Fin r ↪ ↥K.parts := Classical.arbitrary (Fin r ↪ ↥K.parts)\np : ↥K.parts\n⊢ t ≤ Fintype.card ↥↑p"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Ends.Defs | {
"line": 206,
"column": 4
} | {
"line": 208,
"column": 40
} | {
"line": 209,
"column": 4
} | [
{
"pp": "case mp\nV : Type u\nG : SimpleGraph V\nK : Finset V\nC : G.ComponentCompl ↑K\n⊢ C.supp.Infinite → ∀ (L : Finset V) (h : K ⊆ L), ∃ D, hom h D = C",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Finset",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Set.Infini... | [
"case mpr\nV : Type u\nG : SimpleGraph V\nK : Finset V\nC : G.ComponentCompl ↑K\n⊢ (∀ (L : Finset V) (h : K ⊆ L), ∃ D, hom h D = C) → C.supp.Infinite"
] | · rintro Cinf L h
obtain ⟨v, ⟨vK, rfl⟩, vL⟩ := Set.Infinite.nonempty (Set.Infinite.sdiff Cinf L.finite_toSet)
exact ⟨componentComplMk _ vL, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Extremal.TuranDensity | {
"line": 69,
"column": 41
} | {
"line": 69,
"column": 58
} | {
"line": 69,
"column": 59
} | [
{
"pp": "case hn\nW : Type u_1\nH : SimpleGraph W\nn : ℕ\nhn : n ≥ 2\nG : SimpleGraph (Fin (n + 1))\ninst✝ : DecidableRel G.Adj\nh : H.Free G\ne : Sym2 (Fin (n + 1))\nhe : e ∈ G.edgeFinset\n⊢ n - 1 ≤ ↑(Fintype.card (Fin (n + 1)) - #e.toFinset)",
"ppTerm": "?hn",
"assigned": true,
"usedConstants": [
... | [
"case hn\nW : Type u_1\nH : SimpleGraph W\nn : ℕ\nhn : n ≥ 2\nG : SimpleGraph (Fin (n + 1))\ninst✝ : DecidableRel G.Adj\nh : H.Free G\ne : Sym2 (Fin (n + 1))\nhe : e ∈ G.edgeFinset\n⊢ n - 1 ≤ ↑(n + 1 - #e.toFinset)"
] | Fintype.card_fin, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 124,
"column": 30
} | {
"line": 124,
"column": 52
} | {
"line": 125,
"column": 6
} | [
{
"pp": "case pos\nα : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\nhs : t.PairwiseDisjoint s\nh : ∀ i ∈ t, (s i).Finite\nthis : (⋃ i ∈ t, s i).Finite\n⊢ ↑(⋃ i ∈ t, s i).ncard = ∑ᶠ (i : ι) (_ : i ∈ t), (s i).encard",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"... | [
"case pos\nα : Type u_1\nι : Type u_2\nt : Set ι\nht : t.Finite\ns : ι → Set α\nhs : t.PairwiseDisjoint s\nh : ∀ i ∈ t, (s i).Finite\nthis : (⋃ i ∈ t, s i).Finite\n⊢ ↑(∑ᶠ (i : ι) (_ : i ∈ t), (s i).ncard) = ∑ᶠ (i : ι) (_ : i ∈ t), (s i).encard"
] | ncard_biUnion ht h hs, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 116,
"column": 47
} | {
"line": 116,
"column": 64
} | {
"line": 116,
"column": 65
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\n⊢ (↑(Fintype.card (Fin n)) - ↑(#K.verts) - ↑(#(filter K t))) * (↑(#K.verts) - ↑t' + ↑t) +\n ∑ x ∈ filter K t, ↑(#K.verts) =\n (↑n - ↑(#K.verts)) * (↑... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\n⊢ (↑n - ↑(#K.verts) - ↑(#(filter K t))) * (↑(#K.verts) - ↑t' + ↑t) + ∑ x ∈ filter K t, ↑(#K.verts) =\n (↑n - ↑(#K.verts)) * (↑(#K.verts) - (↑t' - ↑t)) + ↑(#(filter K ... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 133,
"column": 20
} | {
"line": 137,
"column": 95
} | {
"line": 138,
"column": 4
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑(#K.verts) * ((1 - 1 / ↑r + ε) * ↑n... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑r * ↑t' * ε - ↑t) - ↑r * ↑t' * (↑t' - ↑t)... | rw [sub_eq_add_neg, ← neg_mul, neg_sub, sub_mul, mul_sub, ← add_sub_assoc,
mul_sub, ← add_sub_assoc, sub_add_cancel, sub_right_comm, ← mul_assoc, ← mul_rotate,
mul_assoc, ← mul_sub, mul_add, mul_sub (#K.verts : ℝ) _ _, mul_one,
sub_add_eq_add_sub, add_sub_assoc, add_sub_sub_cancel, K.card_... | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 133,
"column": 20
} | {
"line": 137,
"column": 95
} | {
"line": 138,
"column": 4
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑(#K.verts) * ((1 - 1 / ↑r + ε) * ↑n... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑r * ↑t' * ε - ↑t) - ↑r * ↑t' * (↑t' - ↑t)... | rw [sub_eq_add_neg, ← neg_mul, neg_sub, sub_mul, mul_sub, ← add_sub_assoc,
mul_sub, ← add_sub_assoc, sub_add_cancel, sub_right_comm, ← mul_assoc, ← mul_rotate,
mul_assoc, ← mul_sub, mul_add, mul_sub (#K.verts : ℝ) _ _, mul_one,
sub_add_eq_add_sub, add_sub_assoc, add_sub_sub_cancel, K.card_... | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 133,
"column": 20
} | {
"line": 137,
"column": 95
} | {
"line": 138,
"column": 4
} | [
{
"pp": "n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑(#K.verts) * ((1 - 1 / ↑r + ε) * ↑n... | [
"n : ℕ\nG : SimpleGraph (Fin n)\ninst✝ : DecidableRel G.Adj\nε : ℝ\nr t t' : ℕ\nK : G.CompleteEquipartiteSubgraph r t'\nhr_pos : 0 < r\nht'_pos : 0 < t'\nhδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\nN : ℕ\nhN : (↑N + ↑r * ↑t') * (↑t' - ↑t) ≤ ↑n * (↑r * ↑t' * ε - ↑t)\n| ↑n * (↑r * ↑t' * ε - ↑t) - ↑r * ↑t' * (↑t' - ↑t)... | rw [sub_eq_add_neg, ← neg_mul, neg_sub, sub_mul, mul_sub, ← add_sub_assoc,
mul_sub, ← add_sub_assoc, sub_add_cancel, sub_right_comm, ← mul_assoc, ← mul_rotate,
mul_assoc, ← mul_sub, mul_add, mul_sub (#K.verts : ℝ) _ _, mul_one,
sub_add_eq_add_sub, add_sub_assoc, add_sub_sub_cancel, K.card_... | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 57
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case h\nV : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\nb : ↑p₁ → ↑p₂\nhb₁ : Bijective b\nhb₂ : ∀ (a : ↑p₁), G.Adj ↑a ↑(b a)\nthis : (p₁ ∪ Set.range fun v ↦ ↑(b v)) = Set.univ\n⊢ (hall... | [
"case h\nV : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\nb : ↑p₁ → ↑p₂\nhb₁ : Bijective b\nhb₂ : ∀ (a : ↑p₁), G.Adj ↑a ↑(b a)\nthis : (p₁ ∪ Set.range fun v ↦ ↑(b v)) = Set.univ\nv : V\nx✝ : v ∈ (ha... | refine ⟨fun v _ ↦ ?_, Subgraph.isSpanning_iff.mpr this⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 205,
"column": 17
} | {
"line": 205,
"column": 34
} | {
"line": 205,
"column": 35
} | [
{
"pp": "case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nh0 : r + 1 ≤ 1 ∨ t = 0\nn : ℕ\nhn : (r + 1) * t ≤ n\nG : SimpleGraph (Fin n)\nx✝¹ : DecidableRel G.Adj\nx✝ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\n⊢ Fintype.card (Fin (r + 1)) * Fintype.card (Fin t) ≤ Fintype.card (Fin n)",
"ppTerm": "?inl",
"assigned": tru... | [
"case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nh0 : r + 1 ≤ 1 ∨ t = 0\nn : ℕ\nhn : (r + 1) * t ≤ n\nG : SimpleGraph (Fin n)\nx✝¹ : DecidableRel G.Adj\nx✝ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\n⊢ (r + 1) * Fintype.card (Fin t) ≤ Fintype.card (Fin n)"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 205,
"column": 35
} | {
"line": 205,
"column": 52
} | {
"line": 205,
"column": 53
} | [
{
"pp": "case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nh0 : r + 1 ≤ 1 ∨ t = 0\nn : ℕ\nhn : (r + 1) * t ≤ n\nG : SimpleGraph (Fin n)\nx✝¹ : DecidableRel G.Adj\nx✝ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\n⊢ (r + 1) * Fintype.card (Fin t) ≤ Fintype.card (Fin n)",
"ppTerm": "?inl",
"assigned": true,
"usedConstan... | [
"case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nh0 : r + 1 ≤ 1 ∨ t = 0\nn : ℕ\nhn : (r + 1) * t ≤ n\nG : SimpleGraph (Fin n)\nx✝¹ : DecidableRel G.Adj\nx✝ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n\n⊢ (r + 1) * t ≤ Fintype.card (Fin n)"
] | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Hamiltonian | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 92
} | {
"line": 192,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ p.length = Fintype.card α",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.Walk.length_tail_add_one",
"SimpleGrap... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\nG : SimpleGraph α\na : α\np : G.Walk a a\ninst✝ : Fintype α\nhp : p.IsHamiltonianCycle\n⊢ 1 ≤ Fintype.card α"
] | rw [← length_tail_add_one hp.not_nil, hp.isHamiltonian_tail.length_eq, Nat.sub_add_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 94
} | {
"line": 164,
"column": 2
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nG' : G.Subgraph\nM : G'.coe.Subgraph\nhM : M.IsMatching\nv✝ : V\nhv : v✝ ∈ (Subgraph.coeSubgraph M).verts\n⊢ ∃! w, (Subgraph.coeSubgraph M).Adj v✝ w",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SimpleGraph.Subgraph.verts_coeSubgraph",
... | [
"V : Type u_1\nG : SimpleGraph V\nG' : G.Subgraph\nM : G'.coe.Subgraph\nhM : M.IsMatching\nv✝ : V\nhv : v✝ ∈ (Subgraph.coeSubgraph M).verts\nw : ↑G'.verts\nhw : (fun w ↦ M.Adj ⟨v✝, ⋯⟩ w) w ∧ ∀ (y : ↑G'.verts), (fun w ↦ M.Adj ⟨v✝, ⋯⟩ w) y → y = w\n⊢ ∃! w, (Subgraph.coeSubgraph M).Adj v✝ w"
] | obtain ⟨w, hw⟩ := hM <| Set.mem_of_mem_image_val <| (Subgraph.verts_coeSubgraph M).symm ▸ hv | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 245,
"column": 21
} | {
"line": 245,
"column": 38
} | {
"line": 245,
"column": 39
} | [
{
"pp": "case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : 0 < r\nht_pos : 0 < t\nε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε\nhε' : 0 < ε'\nt' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1\nht_lt_rt'ε : ↑t < ↑r * ↑t' * ε\nht'_pos : 0 < t'\nN' : ℕ\nih :\n ∀ (b : ℕ),\n N' ≤ b →\n ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj... | [
"case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : 0 < r\nht_pos : 0 < t\nε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε\nhε' : 0 < ε'\nt' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1\nht_lt_rt'ε : ↑t < ↑r * ↑t' * ε\nht'_pos : 0 < t'\nN' : ℕ\nih :\n ∀ (b : ℕ),\n N' ≤ b →\n ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],\n ... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 245,
"column": 39
} | {
"line": 245,
"column": 56
} | {
"line": 245,
"column": 57
} | [
{
"pp": "case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : 0 < r\nht_pos : 0 < t\nε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε\nhε' : 0 < ε'\nt' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1\nht_lt_rt'ε : ↑t < ↑r * ↑t' * ε\nht'_pos : 0 < t'\nN' : ℕ\nih :\n ∀ (b : ℕ),\n N' ≤ b →\n ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj... | [
"case inl\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : 0 < r\nht_pos : 0 < t\nε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε\nhε' : 0 < ε'\nt' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1\nht_lt_rt'ε : ↑t < ↑r * ↑t' * ε\nht'_pos : 0 < t'\nN' : ℕ\nih :\n ∀ (b : ℕ),\n N' ≤ b →\n ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],\n ... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.LapMatrix | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 9
} | {
"line": 120,
"column": 0
} | [
{
"pp": "case e_a.e_f.e_f\nV : Type u_1\nR : Type u_2\ninst✝⁴ : Fintype V\nG : SimpleGraph V\ninst✝³ : DecidableRel G.Adj\ninst✝² : DecidableEq V\ninst✝¹ : Field R\ninst✝ : CharZero R\nx : V → R\ni j : V\n⊢ (if G.Adj i j then x i * x i - x i * x j + (x j * x j - x j * x i) else 0 + 0) =\n if G.Adj i j then (... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits | {
"line": 295,
"column": 8
} | {
"line": 296,
"column": 70
} | {
"line": 297,
"column": 8
} | [
{
"pp": "case refine_3\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : 0 < r\nht_pos : 0 < t\nε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε\nhε' : 0 < ε'\nt' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1\nht_lt_rt'ε : ↑t < ↑r * ↑t' * ε\nht'_pos : 0 < t'\nN' : ℕ\nN : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ... | [
"case refine_3\nε : ℝ\nhε : 0 < ε\nr t : ℕ\nhr_pos : 0 < r\nht_pos : 0 < t\nε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε\nhε' : 0 < ε'\nt' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1\nht_lt_rt'ε : ↑t < ↑r * ↑t' * ε\nht'_pos : 0 < t'\nN' : ℕ\nN : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊\nn... | simp_rw [univ_eq_attach, Finset.mem_map, mem_attach,
Function.Embedding.coeFn_mk, true_and, Subtype.exists] at hp | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 85,
"column": 52
} | {
"line": 86,
"column": 38
} | {
"line": 88,
"column": 0
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G ≃g H\nc : Set V\n⊢ H.IsVertexCover (⇑f '' c) ↔ G.IsVertexCover c",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"SimpleGraph.isVertexCover_preimage_iso._simp_1",
"SimpleGraph.Iso",
"congrArg"... | [] | by
simp [RelIso.image_eq_preimage_symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 398,
"column": 8
} | {
"line": 410,
"column": 11
} | {
"line": 412,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | [] | by_cases hk : k = 0 -- so `s ∩ t = ∅` and hence `Xᶜ = ∅`
· have Xu : X = univ := by
rw [← hw.card_inter, card_eq_zero] at hk
exact eq_univ_of_forall fun _ ↦ by simp [X, hk]
subst k
rw [add_zero] at Wc
simp [Xu, Wc, mul_comm]
have w3 : 3 ≤ #W := two_l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 398,
"column": 8
} | {
"line": 410,
"column": 11
} | {
"line": 412,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | [] | by_cases hk : k = 0 -- so `s ∩ t = ∅` and hence `Xᶜ = ∅`
· have Xu : X = univ := by
rw [← hw.card_inter, card_eq_zero] at hk
exact eq_univ_of_forall fun _ ↦ by simp [X, hk]
subst k
rw [add_zero] at Wc
simp [Xu, Wc, mul_comm]
have w3 : 3 ≤ #W := two_l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Tiling.Tile | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 35
} | {
"line": 218,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\npt : PlacedTile ps\n⊢ (↑pt).Nonempty ↔ (↑(↑ps pt.index)).Nonempty",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"DiscreteTiling.Protoset.tiles",
"Subgroup.map",
... | [
"G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\nindex : ιₚ\ngroupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries\n⊢ (↑{ index := index, groupElts := groupElts }).Nonempty ↔\n (↑(↑ps { index := index, groupElts... | rcases pt with ⟨index, groupElts⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Combinatorics.Tiling.Tile | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 35
} | {
"line": 229,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\npt : PlacedTile ps\n⊢ (↑pt).Finite ↔ (↑(↑ps pt.index)).Finite",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"DiscreteTiling.Protoset.tiles",
"Subgroup.map",
... | [
"G : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\nindex : ιₚ\ngroupElts : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps index)).subtype (↑ps index).symmetries\n⊢ (↑{ index := index, groupElts := groupElts }).Finite ↔\n (↑(↑ps { index := index, groupElts :... | rcases pt with ⟨index, groupElts⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Computability.Primrec.List | {
"line": 260,
"column": 81
} | {
"line": 264,
"column": 64
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Primcodable α\np : α → Prop\ninst✝ : DecidablePred p\nhf : PrimrecPred p\n⊢ Primrec fun L ↦ List.filter (fun x ↦ decide (p x)) L",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Option.guard",
"congrArg",
"P... | [] | by
rw [← List.filterMap_eq_filter]
apply listFilterMap .id
simp only [Primrec₂, Option.guard, decide_eq_true_eq]
exact ite (hf.comp snd) (option_some_iff.mpr snd) (const none) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Ackermann | {
"line": 163,
"column": 69
} | {
"line": 163,
"column": 76
} | {
"line": 163,
"column": 76
} | [
{
"pp": "m n : ℕ\n⊢ m + n + 2 = m + 1 + n + 1",
"ppTerm": "?m.169",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Mathlib.Tactic.RingNF.add_assoc_rev",
"HMul.hMul",
"Mathlib.Tactic.Ring... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 149,
"column": 4
} | {
"line": 153,
"column": 36
} | {
"line": 154,
"column": 4
} | [
{
"pp": "f : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nx : ℝ\nhx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f ... | [
"f : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhf' : ∀ (a : ℝ), ∃ b, a ≤ b ∧ f b = 0\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nx : ℝ\nhx : ∀ (y : ℝ), x ≤ y → ∀ u ∈ Set.Icc (1 / 2 * y) y, f u ∈ Set.Icc (c₁ * f y) (c₂ * f y... | have : 0 ≤ -logb 2 (x / x₀) := by
rw [neg_nonneg]
refine logb_nonpos (by norm_num) (by positivity) ?_
rw [div_le_one x₀_pos]
exact le_of_max_le_left hx₀_ge | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 368,
"column": 60
} | {
"line": 368,
"column": 67
} | {
"line": 368,
"column": 67
} | [
{
"pp": "⊢ (fun x ↦ (-(x * log x ^ 2))⁻¹) = fun x ↦ (-x * log x ^ 2)⁻¹",
"ppTerm": "?m.148",
"assigned": true,
"usedConstants": [
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"DivisionCommMonoid.toDivisionMonoid",... | [] | neg_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 323,
"column": 45
} | {
"line": 323,
"column": 76
} | {
"line": 323,
"column": 76
} | [
{
"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\nq : ℝ → ℝ := fun x ↦ x ^ p a b * (1 - ε x)\nh_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)\nh_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)\nh_main_norm :... | [
"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\nq : ℝ → ℝ := fun x ↦ x ^ p a b * (1 - ε x)\nh_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)\nh_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)\nh_main_norm : (fun n ↦ ‖q... | mul_inv_cancel₀ (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 308,
"column": 22
} | {
"line": 308,
"column": 40
} | {
"line": 308,
"column": 40
} | [
{
"pp": "f g : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhg✝ : GrowsPolynomially g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf :\n ∀ᶠ (x : ℝ) in atTop,\n ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)\nc₃ : ℝ\nhc₃_mem :... | [
"f g : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhg✝ : GrowsPolynomially g\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf :\n ∀ᶠ (x : ℝ) in atTop,\n ∀ u ∈ Set.Icc (b * x) x, (fun x ↦ |f x|) u ∈ Set.Icc (c₁ * (fun x ↦ |f x|) x) (c₂ * (fun x ↦ |f x|) x)\nc₃ : ℝ\nhc₃_mem : c₃ > 0\nc₄ ... | change 0 < c₁ * c₃ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 419,
"column": 45
} | {
"line": 419,
"column": 76
} | {
"line": 419,
"column": 76
} | [
{
"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\nq : ℝ → ℝ := fun x ↦ x ^ p a b * (1 + ε x)\nh_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)\nh_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)\nh_main_norm :... | [
"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\nq : ℝ → ℝ := fun x ↦ x ^ p a b * (1 + ε x)\nh_diff_q : DifferentiableOn ℝ q (Set.Ioi 1)\nh_deriv_q : deriv q =O[atTop] fun x ↦ x ^ (p a b - 1)\nh_main_norm : (fun n ↦ ‖q... | mul_inv_cancel₀ (by positivity) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 27
} | {
"line": 375,
"column": 2
} | [
{
"pp": "f g : ℝ → ℝ\nhf : GrowsPolynomially f\nhfg : g =o[atTop] f\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_ub : b < 1\n⊢ ∃ c₁ > 0,\n ∃ c₂ > 0,\n ∀ᶠ (x : ℝ) in atTop,\n ∀ u ∈ Set.Icc (b * x) x,\n (fun x ↦ f x + g x) u ∈ Set.Icc (c₁ * (fun x ↦ f x + g x) x) (c₂ * (fun x ↦ f x + g x) x)",
"pp... | [
"f g : ℝ → ℝ\nhf : GrowsPolynomially f\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : ℝ) in atTop, ‖g x‖ ≤ c * ‖f x‖\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_ub : b < 1\n⊢ ∃ c₁ > 0,\n ∃ c₂ > 0,\n ∀ᶠ (x : ℝ) in atTop,\n ∀ u ∈ Set.Icc (b * x) x,\n (fun x ↦ f x + g x) u ∈ Set.Icc (c₁ * (fun x ↦ f x + g x) x) (c₂ * ... | rw [isLittleO_iff] at hfg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 514,
"column": 30
} | {
"line": 514,
"column": 42
} | {
"line": 514,
"column": 42
} | [
{
"pp": "f : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0\nhf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0\nhf✝ : GrowsPolynomially fun x ↦ |f x|\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_pos : 0 < b\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf :\n ∀ᶠ (x... | [
"f : ℝ → ℝ\nhf✝¹ : GrowsPolynomially f\nhf_pos_or_neg : (∀ᶠ (x : ℝ) in atTop, 0 < f x) ∨ ∀ᶠ (x : ℝ) in atTop, f x < 0\nhf' : ∀ᶠ (x : ℝ) in atTop, f x ≠ 0\nhf✝ : GrowsPolynomially fun x ↦ |f x|\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhb_pos : 0 < b\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf :\n ∀ᶠ (x : ℝ) in atT... | rw [abs_pos] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.PartrecCode | {
"line": 952,
"column": 6
} | {
"line": 956,
"column": 34
} | {
"line": 957,
"column": 6
} | [
{
"pp": "case succ.pair\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (cf.pair cg)) →\n lup\n ... | [
"case succ.comp\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf cg : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode (cf.comp cg)) →\n lup\n (List.... | · obtain ⟨lf, lg⟩ := encode_lt_pair cf cg
rw [hg (Nat.pair_lt_pair_right _ lf), hg (Nat.pair_lt_pair_right _ lg)]
cases evaln k cf n
· rfl
cases evaln k cg n <;> rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Num.Lemmas | {
"line": 75,
"column": 12
} | {
"line": 75,
"column": 52
} | {
"line": 76,
"column": 2
} | [
{
"pp": "a : PosNum\n⊢ ↑(a + 1) = ↑a + ↑1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"castPosNum",
"Nat.instOne",
"congrArg",
"PosNum.cast_one",
"PosNum.instAdd",
"PosNum.add_one",
"id",
"instOfNatNat",
"instOnePosN... | [] | by rw [add_one a, succ_to_nat, cast_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.List.ReduceOption | {
"line": 119,
"column": 28
} | {
"line": 119,
"column": 69
} | {
"line": 119,
"column": 70
} | [
{
"pp": "α : Type u_1\nl : List (Option α)\n⊢ l.reduceOption.length ≤ l.length ∧ l.reduceOption.length ≠ l.length ↔ none ∈ l",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"List.reduceOption_length_le",
"congrArg",
"Membership.mem",
"id",
"Ne"... | [
"α : Type u_1\nl : List (Option α)\n⊢ l.reduceOption.length ≠ l.length ↔ none ∈ l"
] | and_iff_right (reduceOption_length_le l), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.RegularExpressions | {
"line": 285,
"column": 8
} | {
"line": 285,
"column": 34
} | {
"line": 286,
"column": 8
} | [
{
"pp": "case mpr.cons.nil\nα : Type u_1\ninst✝ : DecidableEq α\nP : RegularExpression α\na : α\nx : List α\nIH :\n ∀ (t : List α),\n t.length < (a :: x).length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)\nhelem : ∀ t ∈ [], t ≠ [] ∧ P.rmatch t = true\nhsum : a :: x ... | [
"case mpr.cons.cons\nα : Type u_1\ninst✝ : DecidableEq α\nP : RegularExpression α\na : α\nx : List α\nIH :\n ∀ (t : List α),\n t.length < (a :: x).length → (P.star.rmatch t = true ↔ ∃ S, t = S.flatten ∧ ∀ t ∈ S, t ≠ [] ∧ P.rmatch t = true)\nt' : List α\nU : List (List α)\nhelem : ∀ t ∈ t' :: U, t ≠ [] ∧ P.rmatc... | · exact ⟨[], [], by tauto⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Computability.RegularExpressions | {
"line": 311,
"column": 34
} | {
"line": 311,
"column": 38
} | {
"line": 311,
"column": 39
} | [
{
"pp": "case plus\nα : Type u_1\ninst✝ : DecidableEq α\na✝¹ a✝ : RegularExpression α\nih₁ : ∀ (x : List α), a✝¹.rmatch x = true ↔ x ∈ a✝¹.matches'\nih₂ : ∀ (x : List α), a✝.rmatch x = true ↔ x ∈ a✝.matches'\nx : List α\n⊢ a✝¹.rmatch x = true ∨ a✝.rmatch x = true ↔ x ∈ (a✝¹ + a✝).matches'",
"ppTerm": "?plus... | [
"case plus\nα : Type u_1\ninst✝ : DecidableEq α\na✝¹ a✝ : RegularExpression α\nih₁ : ∀ (x : List α), a✝¹.rmatch x = true ↔ x ∈ a✝¹.matches'\nih₂ : ∀ (x : List α), a✝.rmatch x = true ↔ x ∈ a✝.matches'\nx : List α\n⊢ x ∈ a✝¹.matches' ∨ a✝.rmatch x = true ↔ x ∈ (a✝¹ + a✝).matches'"
] | ih₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Num.Lemmas | {
"line": 788,
"column": 4
} | {
"line": 788,
"column": 85
} | {
"line": 789,
"column": 4
} | [
{
"pp": "case pos.pos.one.bit1\nf : Num → Num → Num\ng : Bool → Bool → Bool\np : PosNum → PosNum → Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : ∀ (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : ∀ (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : ∀ (m n : Pos... | [
"case pos.pos.bit1.one\nf : Num → Num → Num\ng : Bool → Bool → Bool\np : PosNum → PosNum → Num\ngff : g false false = false\nf00 : f 0 0 = 0\nf0n : ∀ (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0\nfn0 : ∀ (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0\nfnn : ∀ (m n : PosNum), f (pos... | any_goals rw [Nat.bitwise_zero_left, ← Bool.cond_eq_ite, this, ← bit_to_nat, p1b] | Lean.Elab.Tactic.evalAnyGoals | Lean.Parser.Tactic.anyGoals |
Mathlib.Computability.TuringMachine.StackTuringMachine | {
"line": 560,
"column": 8
} | {
"line": 562,
"column": 36
} | {
"line": 563,
"column": 6
} | [
{
"pp": "case pos\nK : Type u_1\nΓ : K → Type u_2\nΛ : Type u_3\nσ : Type u_4\ninst✝ : DecidableEq K\nk : K\nq : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse\nf : σ... | [] | simp only [List.length_append, List.length_reverse, List.length_map, ← h,
Nat.sub_self, List.length_singleton, List.getElem_singleton,
le_refl, Nat.lt_succ_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.TuringMachine.StackTuringMachine | {
"line": 560,
"column": 8
} | {
"line": 562,
"column": 36
} | {
"line": 563,
"column": 6
} | [
{
"pp": "case pos.h₁\nK : Type u_1\nΓ : K → Type u_2\nΛ : Type u_3\nσ : Type u_4\ninst✝ : DecidableEq K\nk : K\nq : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse\nf ... | [] | simp only [List.length_append, List.length_reverse, List.length_map, ← h,
Nat.sub_self, List.length_singleton, List.getElem_singleton,
le_refl, Nat.lt_succ_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.TuringMachine.StackTuringMachine | {
"line": 560,
"column": 8
} | {
"line": 562,
"column": 36
} | {
"line": 563,
"column": 6
} | [
{
"pp": "case pos\nK : Type u_1\nΓ : K → Type u_2\nΛ : Type u_3\nσ : Type u_4\ninst✝ : DecidableEq K\nk : K\nq : TM1.Stmt (Γ' K Γ) (Λ' K Γ Λ σ) σ\nv : σ\nS : (k : K) → List (Γ k)\nL : ListBlank ((k : K) → Option (Γ k))\nhL : ∀ (k : K), ListBlank.map (proj k) L = ListBlank.mk (List.map some (S k)).reverse\nf : σ... | [] | simp only [List.length_append, List.length_reverse, List.length_map, ← h,
Nat.sub_self, List.length_singleton, List.getElem_singleton,
le_refl, Nat.lt_succ_self] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.TuringMachine.PostTuringMachine | {
"line": 1002,
"column": 6
} | {
"line": 1002,
"column": 56
} | {
"line": 1003,
"column": 4
} | [
{
"pp": "case none\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : TM0.Cfg Γ Λ\nq : Λ\nT : Tape Γ\ne : M q T.head = none\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })",
"ppTerm": "?none",
"ass... | [] | simp only [TM0.step, trCfg, e]; exact Eq.refl none | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.TuringMachine.PostTuringMachine | {
"line": 1002,
"column": 6
} | {
"line": 1002,
"column": 56
} | {
"line": 1003,
"column": 4
} | [
{
"pp": "case none\nΓ : Type u_1\ninst✝¹ : Inhabited Γ\nΛ : Type u_2\ninst✝ : Inhabited Λ\nM : TM0.Machine Γ Λ\nx✝ : TM0.Cfg Γ Λ\nq : Λ\nT : Tape Γ\ne : M q T.head = none\n⊢ FRespects (TM1.step (tr M)) (trCfg M) (trCfg M { q := q, Tape := T }) (TM0.step M { q := q, Tape := T })",
"ppTerm": "?none",
"ass... | [] | simp only [TM0.step, trCfg, e]; exact Eq.refl none | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.TuringMachine.Config | {
"line": 342,
"column": 6
} | {
"line": 342,
"column": 14
} | {
"line": 343,
"column": 6
} | [
{
"pp": "case rfind.mp\nn✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\nn : ℕ\nf : List.Vector ℕ (n + 1) → ℕ\na✝ : Nat.Partrec' ↑f\ncf : Code\nv : List.Vector ℕ n\nhf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]\nv' : List ℕ\nh1 :\n v' ∈\n PFun.fix\n (fun v ↦\n (cf.eval v).bind fun y ↦\n ... | [
"case rfind.mp\nn✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\nn : ℕ\nf : List.Vector ℕ (n + 1) → ℕ\na✝ : Nat.Partrec' ↑f\ncf : Code\nv : List.Vector ℕ n\nhf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]\nv' : List ℕ\n⊢ ∀ (v₁ : List ℕ),\n v' ∈\n PFun.fix\n (fun v ↦\n (cf.eval v).bind f... | clear h1 | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.Computability.TuringMachine.Config | {
"line": 345,
"column": 6
} | {
"line": 345,
"column": 14
} | {
"line": 346,
"column": 6
} | [
{
"pp": "case rfind.mp\nn✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\nn : ℕ\nf : List.Vector ℕ (n + 1) → ℕ\na✝ : Nat.Partrec' ↑f\ncf : Code\nv : List.Vector ℕ n\nhf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]\nv' v₀ : List ℕ\nh1 :\n v' ∈\n PFun.fix\n (fun v ↦\n (cf.eval v).bind fun y ↦\n ... | [
"case rfind.mp\nn✝ : ℕ\nf✝ : List.Vector ℕ n✝ →. ℕ\nn : ℕ\nf : List.Vector ℕ (n + 1) → ℕ\na✝ : Nat.Partrec' ↑f\ncf : Code\nv : List.Vector ℕ n\nhf : ∀ (a : ℕ), cf.eval (a :: ↑v) = Part.some [f (a ::ᵥ v)]\nv' v₀ v₁ : List ℕ\nh2 :\n v' ∈\n PFun.fix\n (fun v ↦\n (cf.eval v).bind fun y ↦\n Part... | clear h1 | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.Computability.TuringMachine.Config | {
"line": 527,
"column": 4
} | {
"line": 528,
"column": 55
} | {
"line": 530,
"column": 0
} | [
{
"pp": "case fix\nk' : Cont\na✝¹ : Code\na✝ : Cont\nk_ih : ∀ {v : List ℕ}, (a✝.then k').eval v = a✝.eval v >>= k'.eval\nv : List ℕ\n⊢ ((fix a✝¹ a✝).then k').eval v = (fix a✝¹ a✝).eval v >>= k'.eval",
"ppTerm": "?fix",
"assigned": true,
"usedConstants": [
"Part",
"Eq.mpr",
"PFun",
... | [] | simp only [Cont.eval, Cont.then, *]
split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.TuringMachine.Config | {
"line": 527,
"column": 4
} | {
"line": 528,
"column": 55
} | {
"line": 530,
"column": 0
} | [
{
"pp": "case fix\nk' : Cont\na✝¹ : Code\na✝ : Cont\nk_ih : ∀ {v : List ℕ}, (a✝.then k').eval v = a✝.eval v >>= k'.eval\nv : List ℕ\n⊢ ((fix a✝¹ a✝).then k').eval v = (fix a✝¹ a✝).eval v >>= k'.eval",
"ppTerm": "?fix",
"assigned": true,
"usedConstants": [
"Part",
"Eq.mpr",
"PFun",
... | [] | simp only [Cont.eval, Cont.then, *]
split_ifs <;> [rfl; simp only [← k_ih, bind_assoc]] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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