module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 24
} | {
"line": 164,
"column": 25
} | [
{
"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⊢ m.degree... | [
"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⊢ m.degree (b i) = 0"
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 134,
"column": 21
} | {
"line": 134,
"column": 32
} | {
"line": 134,
"column": 33
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝ :\n I ⊆ X ∧\n (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧\n X ⊆ univ.sigma fun i ↦ (M i).E\nhIX : ... | [
"ι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝ :\n I ⊆ X ∧\n (∀ ⦃t : Set ((i : ι) × α i)⦄, (∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' t)) → t ⊆ X → I ⊆ t → I = t) ∧\n X ⊆ univ.sigma fun i ↦ (M i).E\nhIX : I ⊆ X\nh : ∀... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 135,
"column": 29
} | {
"line": 135,
"column": 40
} | {
"line": 135,
"column": 41
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝¹ :\n (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧\n (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻... | [
"ι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝¹ :\n (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧\n (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t) ∧\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | {
"line": 137,
"column": 16
} | [
{
"pp": "case refine_4\nι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝¹ :\n (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧\n (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t... | [
"case refine_4\nι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝¹ :\n (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧\n (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 138,
"column": 26
} | {
"line": 138,
"column": 37
} | {
"line": 138,
"column": 38
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝¹ :\n (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧\n (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻... | [
"ι : Type u_1\nα : ι → Type u_2\nM : (i : ι) → Matroid (α i)\nI X : Set ((i : ι) × α i)\nhI : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nx✝¹ :\n (∀ (x : ι), Sigma.mk x ⁻¹' I ⊆ Sigma.mk x ⁻¹' X) ∧\n (∀ (x : ι) ⦃t : Set (α x)⦄, (M x).Indep t → t ⊆ Sigma.mk x ⁻¹' X → Sigma.mk x ⁻¹' I ⊆ t → Sigma.mk x ⁻¹' I = t) ∧\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 68
} | {
"line": 143,
"column": 69
} | [
{
"pp": "case h_option.Heval\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... | [
"case h_option.Heval\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 : MvPoly... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 54
} | {
"line": 73,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nσ : Type u_2\ninst✝¹ : Finite σ\ninst✝ : IsDomain R\nP : MvPolynomial σ R\nS : σ → Finset R\nHdeg : ∀ (i : σ), degreeOf i P < #(S i)\nHeval : ∀ (x : σ → R), (∀ (i : σ), x i ∈ S i) → (eval x) P = 0\n⊢ P = 0",
"ppTerm": "?m.24",
"assigned": true,
"usedConsta... | [] | induction σ using Finite.induction_empty_option with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 294,
"column": 9
} | {
"line": 294,
"column": 33
} | {
"line": 294,
"column": 34
} | [
{
"pp": "case h\nα : Type u_1\nM N : Matroid α\nh : Disjoint M.E N.E\nI✝ : Set α\na✝ : I✝ ⊆ (M.disjointSum N h).E\n⊢ (M.disjointSum N h).Indep I✝ ↔ (N.disjointSum M ⋯).Indep I✝",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLatti... | [
"case h\nα : Type u_1\nM N : Matroid α\nh : Disjoint M.E N.E\nI✝ : Set α\na✝ : I✝ ⊆ (M.disjointSum N h).E\n⊢ M.Indep (I✝ ∩ M.E) ∧ N.Indep (I✝ ∩ N.E) ∧ I✝ ⊆ M.E ∪ N.E ↔ N.Indep (I✝ ∩ N.E) ∧ M.Indep (I✝ ∩ M.E) ∧ I✝ ⊆ M.E ∪ N.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Arborescence | {
"line": 71,
"column": 16
} | {
"line": 71,
"column": 55
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case zero\nV : Type u\ninst✝ : Quiver V\nr : V\nheight : V → ℕ\nheight_lt : ∀ ⦃a b : V⦄ (a_1 : a ⟶ b), height a < height b\nunique_arrow : ∀ ⦃a b c : V⦄ (e : a ⟶ c) (f : b ⟶ c), a = b ∧ e ≍ f\nroot_or_arrow : ∀ (b : V), b = r ∨ ∃ a, Nonempty (a ⟶ b)\nb : V\nhn : height b < 0\n⊢ Nonempty (Path r b)",
... | [] | exact False.elim (Nat.not_lt_zero _ hn) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Quiver.Arborescence | {
"line": 71,
"column": 16
} | {
"line": 71,
"column": 55
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case zero\nV : Type u\ninst✝ : Quiver V\nr : V\nheight : V → ℕ\nheight_lt : ∀ ⦃a b : V⦄ (a_1 : a ⟶ b), height a < height b\nunique_arrow : ∀ ⦃a b c : V⦄ (e : a ⟶ c) (f : b ⟶ c), a = b ∧ e ≍ f\nroot_or_arrow : ∀ (b : V), b = r ∨ ∃ a, Nonempty (a ⟶ b)\nb : V\nhn : height b < 0\n⊢ Nonempty (Path r b)",
... | [] | exact False.elim (Nat.not_lt_zero _ hn) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Arborescence | {
"line": 71,
"column": 16
} | {
"line": 71,
"column": 55
} | {
"line": 72,
"column": 6
} | [
{
"pp": "case zero\nV : Type u\ninst✝ : Quiver V\nr : V\nheight : V → ℕ\nheight_lt : ∀ ⦃a b : V⦄ (a_1 : a ⟶ b), height a < height b\nunique_arrow : ∀ ⦃a b c : V⦄ (e : a ⟶ c) (f : b ⟶ c), a = b ∧ e ≍ f\nroot_or_arrow : ∀ (b : V), b = r ∨ ∃ a, Nonempty (a ⟶ b)\nb : V\nhn : height b < 0\n⊢ Nonempty (Path r b)",
... | [] | exact False.elim (Nat.not_lt_zero _ hn) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Quiver.ConnectedComponent | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 85
} | {
"line": 201,
"column": 6
} | [
{
"pp": "V : Type u_2\ninst✝ : Quiver V\nh_sc : IsStronglyConnected V\ni₀ j₀ : V\ne₀ : i₀ ⟶ j₀\ni j : V\np₁ : Path i i₀\np₂ : Path j₀ j\np : Path i j := p₁.comp (e₀.toPath.comp p₂)\n⊢ 0 < p.length",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",... | [
"V : Type u_2\ninst✝ : Quiver V\nh_sc : IsStronglyConnected V\ni₀ j₀ : V\ne₀ : i₀ ⟶ j₀\ni j : V\np₁ : Path i i₀\np₂ : Path j₀ j\np : Path i j := p₁.comp (e₀.toPath.comp p₂)\n⊢ 0 < 1 + (p₂.length + p₁.length)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 54
} | {
"line": 184,
"column": 55
} | [
{
"pp": "R : 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\nthis : lex.toSyn e ≤ lex.toSyn (single () #S)\n⊢ e () ≤ #S",
"ppTerm": "?m.11... | [
"R : 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\nthis : lex.toSyn e ≤ lex.toSyn (single () #S)\n⊢ e () ≤ #S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 176,
"column": 2
} | {
"line": 194,
"column": 55
} | {
"line": 196,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\ni : ι\nS : Finset R\nm : ι →₀ ℕ\nhm : m ∈ (P S i).support\n⊢ ∃ e ≤ #S, m = single i e",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"MonomialOrder... | [] | classical
have hP : Alon.P S i = .rename (fun _ ↦ i) (Alon.P S ()) := by simp [Alon.P]
rw [hP, support_rename_of_injective (Function.injective_of_subsingleton _)] at hm
simp only [Finset.mem_image, mem_support_iff, ne_eq] at hm
obtain ⟨e, he, hm⟩ := hm
haveI : Nontrivial R := nontrivial_of_ne _ _ he
refine ... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 176,
"column": 2
} | {
"line": 194,
"column": 55
} | {
"line": 196,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\ni : ι\nS : Finset R\nm : ι →₀ ℕ\nhm : m ∈ (P S i).support\n⊢ ∃ e ≤ #S, m = single i e",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"MonomialOrder... | [] | classical
have hP : Alon.P S i = .rename (fun _ ↦ i) (Alon.P S ()) := by simp [Alon.P]
rw [hP, support_rename_of_injective (Function.injective_of_subsingleton _)] at hm
simp only [Finset.mem_image, mem_support_iff, ne_eq] at hm
obtain ⟨e, he, hm⟩ := hm
haveI : Nontrivial R := nontrivial_of_ne _ _ he
refine ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 176,
"column": 2
} | {
"line": 194,
"column": 55
} | {
"line": 196,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nι : Type u_3\ni : ι\nS : Finset R\nm : ι →₀ ℕ\nhm : m ∈ (P S i).support\n⊢ ∃ e ≤ #S, m = single i e",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"MonomialOrder... | [] | classical
have hP : Alon.P S i = .rename (fun _ ↦ i) (Alon.P S ()) := by simp [Alon.P]
rw [hP, support_rename_of_injective (Function.injective_of_subsingleton _)] at hm
simp only [Finset.mem_image, mem_support_iff, ne_eq] at hm
obtain ⟨e, he, hm⟩ := hm
haveI : Nontrivial R := nontrivial_of_ne _ _ he
refine ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 205,
"column": 10
} | {
"line": 205,
"column": 49
} | {
"line": 205,
"column": 50
} | [
{
"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\nhf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f\ng' : ι →₀ MvPolynomial σ R\n... | [
"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\nhf : ∀ (i : ι), ¬m.degree (b i) ≤ m.degree f\ng' : ι →₀ MvPolynomial σ R\nr' : MvPolyn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 109,
"column": 73
} | {
"line": 109,
"column": 84
} | {
"line": 109,
"column": 85
} | [
{
"pp": "U : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f ↦ φ.map f) f = (fun f ↦ φ.map f) g\n⊢ φ.star u (Star.mk f) = φ.star u (Star.mk g)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"U : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f ↦ φ.map f) f = (fun f ↦ φ.map f) g\n⊢ φ.map f = φ.map g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 14
} | [
{
"pp": "U : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f ↦ φ.map f) f = (fun f ↦ φ.map f) g\nthis : φ.star u (Star.mk f) = φ.star u (Star.mk g)\n⊢ f = g",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants": [],
... | [
"U : Type u_1\ninst✝¹ : Quiver U\nV : Type u_2\ninst✝ : Quiver V\nφ : U ⥤q V\nhφ : φ.IsCovering\nu v : U\nf g : u ⟶ v\nhe : (fun f ↦ φ.map f) f = (fun f ↦ φ.map f) g\nthis : φ.star u (Star.mk f) = φ.star u (Star.mk g)\n⊢ f = g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Weight | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 31
} | {
"line": 105,
"column": 32
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝³ : Quiver V\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nw : {i j : V} → (i ⟶ j) → R\nhw : ∀ {i j : V} (e : i ⟶ j), 0 < w e\ni j b✝ c✝ : V\np : Path i b✝\ne : b✝ ⟶ c✝\nih : 0 < weight (fun {i j} ↦ w) p\nhe : 0 < w e\n⊢ 0 < wei... | [
"case cons\nV : Type u_1\ninst✝³ : Quiver V\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nw : {i j : V} → (i ⟶ j) → R\nhw : ∀ {i j : V} (e : i ⟶ j), 0 < w e\ni j b✝ c✝ : V\np : Path i b✝\ne : b✝ ⟶ c✝\nih : 0 < weight (fun {i j} ↦ w) p\nhe : 0 < w e\n⊢ 0 < weight (fun {i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 15
} | {
"line": 141,
"column": 16
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\n⊢ (p.comp q).length = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
"instHAdd... | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\n⊢ p.length = 0 ∧ q.length = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 29
} | {
"line": 143,
"column": 30
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\nhlen : (p.comp q).length = 0\n⊢ p.length + q.length = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
... | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\nhlen : (p.comp q).length = 0\n⊢ p.length = 0 ∧ q.length = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Weight | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 31
} | {
"line": 116,
"column": 32
} | [
{
"pp": "case cons\nV : Type u_1\ninst✝³ : Quiver V\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nw : {i j : V} → (i ⟶ j) → R\nhw : ∀ {i j : V} (e : i ⟶ j), 0 ≤ w e\ni j b✝ c✝ : V\np : Path i b✝\ne : b✝ ⟶ c✝\nih : 0 ≤ weight (fun {i j} ↦ w) p\nhe : 0 ≤ w e\n⊢ 0 ≤ wei... | [
"case cons\nV : Type u_1\ninst✝³ : Quiver V\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nw : {i j : V} → (i ⟶ j) → R\nhw : ∀ {i j : V} (e : i ⟶ j), 0 ≤ w e\ni j b✝ c✝ : V\np : Path i b✝\ne : b✝ ⟶ c✝\nih : 0 ≤ weight (fun {i j} ↦ w) p\nhe : 0 ≤ w e\n⊢ 0 ≤ weight (fun {i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 15
} | {
"line": 150,
"column": 16
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\n⊢ (p.comp q).length = 0",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
"instHAdd... | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\n⊢ p.length = 0 ∧ q.length = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 29
} | {
"line": 152,
"column": 30
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\nhlen : (p.comp q).length = 0\n⊢ p.length + q.length = 0",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
... | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nq : Path b a\nh : p.comp q = nil\nhlen : (p.comp q).length = 0\n⊢ p.length = 0 ∧ q.length = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 159,
"column": 11
} | {
"line": 159,
"column": 22
} | {
"line": 159,
"column": 23
} | [
{
"pp": "case nil\nV : Type u_1\ninst✝ : Quiver V\na b : V\nq : Path a a\nx✝ : nil.length = 0 ∧ q.length = 0\nhp : nil.length = 0\nhq : q.length = 0\n⊢ nil.comp q = nil",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Quiver.Path.nil",
"congrArg",
"id",
... | [
"case nil\nV : Type u_1\ninst✝ : Quiver V\na b : V\nq : Path a a\nx✝ : nil.length = 0 ∧ q.length = 0\nhp : nil.length = 0\nhq : q.length = 0\n⊢ q = nil"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 18
} | {
"line": 167,
"column": 19
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nh₁ : p.vertices.getLast ⋯ = b\nh₂ : p.vertices.getLast ⋯ ∈ p.vertices\n⊢ b ∈ p.vertices",
"ppTerm": "?m.41",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nh₁ : p.vertices.getLast ⋯ = b\nh₂ : p.vertices.getLast ⋯ ∈ p.vertices\n⊢ b ∈ p.vertices"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 42
} | {
"line": 183,
"column": 43
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nn : ℕ\nhn : n ≤ nil.length\n⊢ n = 0",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nn : ℕ\nhn : n ≤ nil.length\n⊢ n = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 52
} | {
"line": 219,
"column": 53
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ nil.vertices\n⊢ v = a",
"ppTerm": "?m.48",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ nil.vertices\n⊢ v = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 76
} | {
"line": 222,
"column": 76
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\nb v : V\np : Path v b\nhv : v ∈ nil.vertices\n⊢ ¬v ∈ nil.vertices.tail",
"ppTerm": "?m.82",
"assigned": true,
"usedConstants": [
"False",
"Quiver.Path.nil",
"congrArg",
"Membership.mem",
"List.tail",
"List",
"_private... | [] | by simp only [vertices_nil, tail_cons, not_mem_nil, not_false_eq_true] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 225,
"column": 6
} | {
"line": 225,
"column": 17
} | {
"line": 225,
"column": 18
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv b✝ c✝ : V\npPrev : Path a b✝\ne : b✝ ⟶ c✝\nih : v ∈ pPrev.vertices → ∃ p₁ p₂, pPrev = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail\nhv : v ∈ (pPrev.cons e).vertices\n⊢ v ∈ pPrev.vertices ∨ v = (pPrev.cons e).end",
"ppTerm": "?m.104",
"assigned": tr... | [
"V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv b✝ c✝ : V\npPrev : Path a b✝\ne : b✝ ⟶ c✝\nih : v ∈ pPrev.vertices → ∃ p₁ p₂, pPrev = p₁.comp p₂ ∧ ¬v ∈ p₂.vertices.tail\nhv : v ∈ (pPrev.cons e).vertices\n⊢ v ∈ pPrev.vertices ∨ v = c✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 15
} | {
"line": 100,
"column": 16
} | [
{
"pp": "case inl\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhk : 0 ∉ A\n⊢ schnirelmannDensity A ≤ 1 - (↑0)⁻¹",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"congrArg",
"Real.instIn... | [
"case inl\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhk : 0 ∉ A\n⊢ schnirelmannDensity A ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 16
} | {
"line": 102,
"column": 17
} | [
{
"pp": "case inr\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nk : ℕ\nhk : k ∉ A\nhk' : k > 0\n⊢ ↑(#({a ∈ Ioc 0 k | a ∈ A})) / ↑k ≤ 1 - (↑k)⁻¹",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.instLE",
"Real",
"DivInvMonoid.toI... | [
"case inr\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nk : ℕ\nhk : k ∉ A\nhk' : k > 0\n⊢ ↑(#({a ∈ Ioc 0 k | a ∈ A})) / ↑k ≤ 1 - 1 / ↑k"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Sups | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 28
} | {
"line": 159,
"column": 29
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq β\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : F\nhf : Injective ⇑f\ns t : Finset α\n⊢ map { toFun := ⇑f, inj' := hf } (s ⊻ t) = map { toFun := ⇑f, inj... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq β\ninst✝³ : SemilatticeSup α\ninst✝² : SemilatticeSup β\ninst✝¹ : FunLike F α β\ninst✝ : SupHomClass F α β\nf : F\nhf : Injective ⇑f\ns t : Finset α\n⊢ image (⇑f) (s ⊻ t) = image (⇑f) s ⊻ image (⇑f) t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 28
} | {
"line": 304,
"column": 29
} | [
{
"pp": "F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq β\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nf : F\nhf : Injective ⇑f\ns t : Finset α\n⊢ map { toFun := ⇑f, inj' := hf } (s ⊼ t) = map { toFun := ⇑f, inj... | [
"F : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁵ : DecidableEq α\ninst✝⁴ : DecidableEq β\ninst✝³ : SemilatticeInf α\ninst✝² : SemilatticeInf β\ninst✝¹ : FunLike F α β\ninst✝ : InfHomClass F α β\nf : F\nhf : Injective ⇑f\ns t : Finset α\n⊢ image (⇑f) (s ⊼ t) = image (⇑f) s ⊼ image (⇑f) t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 212,
"column": 36
} | {
"line": 212,
"column": 47
} | {
"line": 212,
"column": 48
} | [
{
"pp": "A : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhA : A.Finite\n⊢ schnirelmannDensity A = 0",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"A : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhA : A.Finite\n⊢ schnirelmannDensity A = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 18
} | {
"line": 235,
"column": 19
} | [
{
"pp": "m : ℕ\nhm : m ≠ 1\nhm' : m > 0\n⊢ ↑(#({a ∈ Ioc 0 m | a ∈ {n | n % m = 1}})) / ↑m ≤ (↑m)⁻¹",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneCl... | [
"m : ℕ\nhm : m ≠ 1\nhm' : m > 0\n⊢ ↑(#({a ∈ Ioc 0 m | a ∈ {n | n % m = 1}})) / ↑m ≤ 1 / ↑m"
] | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Sups | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 66
} | {
"line": 466,
"column": 67
} | [
{
"pp": "α : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns₁ s₂ t : Finset α\n⊢ (s₁ ∩ s₂) ○ t ⊆ s₁ ○ t ∩ s₂ ○ t",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqProd",
"SProd.spro... | [
"α : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns₁ s₂ t : Finset α\n⊢ image (fun ab ↦ ab.1 ⊔ ab.2) ({ab ∈ s₁ ×ˢ t | Disjoint ab.1 ab.2} ∩ {ab ∈ s₂ ×ˢ t | Disjoint ab.1 ab.2}) ⊆\n image (fun ab ↦ ab.1 ⊔ ab.2) ({ab ∈ s₁ ×ˢ t | Disjoint ab.1 ab.2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 66
} | {
"line": 469,
"column": 67
} | [
{
"pp": "α : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns t₁ t₂ : Finset α\n⊢ s ○ (t₁ ∩ t₂) ⊆ s ○ t₁ ∩ s ○ t₂",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instDecidableEqProd",
"SProd.spro... | [
"α : Type u_2\ninst✝³ : DecidableEq α\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ninst✝ : DecidableRel Disjoint\ns t₁ t₂ : Finset α\n⊢ image (fun ab ↦ ab.1 ⊔ ab.2) ({ab ∈ s ×ˢ t₁ | Disjoint ab.1 ab.2} ∩ {ab ∈ s ×ˢ t₂ | Disjoint ab.1 ab.2}) ⊆\n image (fun ab ↦ ab.1 ⊔ ab.2) ({ab ∈ s ×ˢ t₁ | Disjoint ab.1 ab.2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 70
} | {
"line": 272,
"column": 2
} | [
{
"pp": "⊢ schnirelmannDensity (setOf Odd) = 2⁻¹",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Set.ext",
"Odd",
"setOf",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"HMod.hMod",
"Nat",
"Nat.instSemiring",
"OfNat.ofNat",
"E... | [
"h : setOf Odd = {n | n % 2 = 1}\n⊢ schnirelmannDensity (setOf Odd) = 2⁻¹"
] | have h : setOf Odd = {n | n % 2 = 1} := Set.ext fun _ => Nat.odd_iff | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Finset.Sups | {
"line": 660,
"column": 14
} | {
"line": 660,
"column": 25
} | {
"line": 660,
"column": 26
} | [
{
"pp": "α : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nn : ℕ\nhn : n ≤ Fintype.card α\nh𝒜 : Set.Sized n ↑𝒜ᶜˢ\n⊢ Set.Sized (Fintype.card α - n) ↑𝒜",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nn : ℕ\nhn : n ≤ Fintype.card α\nh𝒜 : Set.Sized n ↑𝒜ᶜˢ\n⊢ Set.Sized (Fintype.card α - n) ↑𝒜"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 661,
"column": 15
} | {
"line": 661,
"column": 53
} | {
"line": 661,
"column": 54
} | [
{
"pp": "α : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nn : ℕ\nhn : n ≤ Fintype.card α\nh𝒜 : Set.Sized (Fintype.card α - n) ↑𝒜\n⊢ Set.Sized n ↑𝒜ᶜˢ",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_4\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nn : ℕ\nhn : n ≤ Fintype.card α\nh𝒜 : Set.Sized (Fintype.card α - n) ↑𝒜\n⊢ Set.Sized n ↑𝒜ᶜˢ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 87
} | {
"line": 169,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝³ : SemilatticeSup α\ns t : Finset α\na : α\ninst✝² : DecidableLE α\ninst✝¹ : OrderTop α\ninst✝ : DecidableEq α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ (s ∪ t).truncatedSup a = s.truncatedSup a ⊔ t.truncatedSup a",
"ppTerm": "?m.33",
"assigned": true,
"used... | [
"α : Type u_1\ninst✝³ : SemilatticeSup α\ns t : Finset α\na : α\ninst✝² : DecidableLE α\ninst✝¹ : OrderTop α\ninst✝ : DecidableEq α\nhs : a ∈ lowerClosure ↑s\nht : a ∈ lowerClosure ↑t\n⊢ (({b ∈ s | a ≤ b} ∪ {b ∈ t | a ≤ b}).sup' ⋯ fun x ↦ id x) = {b ∈ s | a ≤ b}.sup' ⋯ id ⊔ {b ∈ t | a ≤ b}.sup' ⋯ id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 87
} | {
"line": 246,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝³ : SemilatticeInf α\ns t : Finset α\na : α\ninst✝² : DecidableLE α\ninst✝¹ : BoundedOrder α\ninst✝ : DecidableEq α\nhs : a ∈ upperClosure ↑s\nht : a ∈ upperClosure ↑t\n⊢ (s ∪ t).truncatedInf a = s.truncatedInf a ⊓ t.truncatedInf a",
"ppTerm": "?m.34",
"assigned": true,
"... | [
"α : Type u_1\ninst✝³ : SemilatticeInf α\ns t : Finset α\na : α\ninst✝² : DecidableLE α\ninst✝¹ : BoundedOrder α\ninst✝ : DecidableEq α\nhs : a ∈ upperClosure ↑s\nht : a ∈ upperClosure ↑t\n⊢ (({b ∈ s | b ≤ a} ∪ {b ∈ t | b ≤ a}).inf' ⋯ fun x ↦ id x) = {b ∈ s | b ≤ a}.inf' ⋯ id ⊓ {b ∈ t | b ≤ a}.inf' ⋯ id"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 79,
"column": 35
} | {
"line": 79,
"column": 53
} | {
"line": 79,
"column": 54
} | [
{
"pp": "α : Type u_2\ninst✝ : SemilatticeSup α\na : α\nh : ∀ ⦃b c : α⦄, a ≤ b ⊔ c → a ≤ b ∨ a ≤ c\nb c : α\nha : b ⊔ c = a\n⊢ b = a ∨ c = a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Semi... | [
"α : Type u_2\ninst✝ : SemilatticeSup α\na : α\nh : ∀ ⦃b c : α⦄, a ≤ b ⊔ c → a ≤ b ∨ a ≤ c\nb c : α\nha : b ⊔ c = a\n⊢ c ≤ b ∨ b ≤ c"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 101,
"column": 13
} | {
"line": 101,
"column": 36
} | {
"line": 101,
"column": 37
} | [
{
"pp": "case empty\nι : Type u_1\nα : Type u_2\ninst✝¹ : SemilatticeSup α\na : α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\nha : SupIrred a\nh : ∅.sup f = a\n⊢ ∃ i ∈ ∅, f i = a",
"ppTerm": "?empty",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Fins... | [
"case empty\nι : Type u_1\nα : Type u_2\ninst✝¹ : SemilatticeSup α\na : α\ninst✝ : OrderBot α\ns : Finset ι\nf : ι → α\nha : SupIrred a\nh : ∅.sup f = a\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 162,
"column": 35
} | {
"line": 162,
"column": 53
} | {
"line": 162,
"column": 54
} | [
{
"pp": "α : Type u_2\ninst✝ : SemilatticeInf α\na : α\nh : ∀ ⦃b c : α⦄, b ⊓ c ≤ a → b ≤ a ∨ c ≤ a\nb c : α\nha : b ⊓ c = a\n⊢ b = a ∨ c = a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"left_eq_inf._simp_1",
"PartialOrder.toPreorder",
... | [
"α : Type u_2\ninst✝ : SemilatticeInf α\na : α\nh : ∀ ⦃b c : α⦄, b ⊓ c ≤ a → b ≤ a ∨ c ≤ a\nb c : α\nha : b ⊓ c = a\n⊢ b ≤ c ∨ c ≤ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 286,
"column": 29
} | {
"line": 286,
"column": 58
} | {
"line": 286,
"column": 59
} | [
{
"pp": "α : Type u_2\ninst✝ : LinearOrder α\na x✝¹ x✝ : α\n⊢ max x✝¹ x✝ = a → x✝¹ = a ∨ x✝ = a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeSup.toMax",
"_pri... | [
"α : Type u_2\ninst✝ : LinearOrder α\na x✝¹ x✝ : α\n⊢ x✝¹ = a ∧ x✝ ≤ x✝¹ ∨ x✝ = a ∧ x✝¹ ≤ x✝ → x✝¹ = a ∨ x✝ = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 290,
"column": 29
} | {
"line": 290,
"column": 58
} | {
"line": 290,
"column": 59
} | [
{
"pp": "α : Type u_2\ninst✝ : LinearOrder α\na x✝¹ x✝ : α\n⊢ min x✝¹ x✝ = a → x✝¹ = a ∨ x✝ = a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
... | [
"α : Type u_2\ninst✝ : LinearOrder α\na x✝¹ x✝ : α\n⊢ x✝¹ = a ∧ x✝¹ ≤ x✝ ∨ x✝ = a ∧ x✝ ≤ x✝¹ → x✝¹ = a ∨ x✝ = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Birkhoff | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 89
} | {
"line": 285,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝¹ : Finite α\ninst✝ : DistribLattice α\n⊢ ∃ β x x_1 f, Injective ⇑f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset",
"Classical.propDecidable",
"Exists",
"Subtype.fintype",
"inferInstance",
... | [] | cases nonempty_fintype α
exact ⟨{a : α // SupIrred a}, _, inferInstance, _, LatticeHom.birkhoffFinset_injective⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Birkhoff | {
"line": 282,
"column": 2
} | {
"line": 283,
"column": 89
} | {
"line": 285,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝¹ : Finite α\ninst✝ : DistribLattice α\n⊢ ∃ β x x_1 f, Injective ⇑f",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset",
"Classical.propDecidable",
"Exists",
"Subtype.fintype",
"inferInstance",
... | [] | cases nonempty_fintype α
exact ⟨{a : α // SupIrred a}, _, inferInstance, _, LatticeHom.birkhoffFinset_injective⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 16
} | {
"line": 255,
"column": 2
} | [
{
"pp": "case inr\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\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : a ∉ s\nb : α\nhb : b ∈ s\nh : (if Disjoint u b ∧ v ≤ b then (b ⊔ u) \\ v else b) = a\n⊢ a ∈ s",... | [
"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\nhva : v ≤ a\nhvu : v = ⊥ → u = ⊥\nleft✝ : a ∉ s\nb : α\nhb : b ∈ s\nh✝ : Disjoint u b ∧ v ≤ b\nh : (b ⊔ u) \\ v = a\n⊢ a ∈ s",
"case neg\nα : Type ... | split_ifs at h | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 417,
"column": 4
} | {
"line": 417,
"column": 25
} | {
"line": 417,
"column": 26
} | [
{
"pp": "case ind.inl\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nm : ℕ\nih : ∀ (𝒜 : Finset (Finset α)), #𝒜 < m → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹\na : Finset α\nh𝒜₁ : {a}.Nonempty\nh𝒜₂ : univ ∉ {a}\nhm : m = #{a}\n⊢ a ≠ univ",... | [
"case ind.inl\nα : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : Nonempty α\nm : ℕ\nih : ∀ (𝒜 : Finset (Finset α)), #𝒜 < m → 𝒜.Nonempty → univ ∉ 𝒜 → supSum 𝒜 = ↑(card α) * ∑ k ∈ range (card α), (↑k)⁻¹\na : Finset α\nh𝒜₁ : {a}.Nonempty\nh𝒜₂ : univ ∉ {a}\nhm : m = #{a}\n⊢ ¬a = univ"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 15
} | {
"line": 426,
"column": 16
} | [
{
"pp": "case ind.inr.succ.h𝒜₂\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 ... | [
"case ind.inr.succ.h𝒜₂\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 → 𝒜_1.Nonem... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 22
} | {
"line": 68,
"column": 23
} | [
{
"pp": "case mp.right\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\na : α\nha : a ∉ t\nhst : #s = #(insert a t)\nhts : toColex (insert a t) ≤ toColex s\n⊢ toColex t ≤ toColex (s.erase (s.min' hs))",
"ppTerm": "?mp.right",
"assigned": false,
"... | [
"case mp.right\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\na : α\nha : a ∉ t\nhst : #s = #(insert a t)\nhts : toColex (insert a t) ≤ toColex s\n⊢ toColex t ≤ toColex (s.erase (s.min' hs))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 34
} | {
"line": 91,
"column": 35
} | [
{
"pp": "case mpr.inr.inl\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\ncards' : #(s.erase (s.min' hs)) = #t\nk : α\nhks : k ∈ s.erase (s.min' hs)\nhkt : k ∉ t\nz : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))\nj : α := tᶜ.min' ⋯\nhjk✝ : j ≤ k\nth... | [
"case mpr.inr.inl\nα : Type u_1\ninst✝¹ : LinearOrder α\ns : Finset α\ninst✝ : Fintype α\nhs : s.Nonempty\nt : Finset α\ncards' : #(s.erase (s.min' hs)) = #t\nk : α\nhks : k ∈ s.erase (s.min' hs)\nhkt : k ∉ t\nz : ∀ ⦃a : α⦄, k < a → (a ∈ t ↔ a ∈ s.erase (s.min' hs))\nj : α := tᶜ.min' ⋯\nhjk✝ : j ≤ k\nthis : j ∉ t\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 55
} | {
"line": 106,
"column": 56
} | [
{
"pp": "case e'_3\n𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : #𝒜 * (r + 1) ≤ #(∂ 𝒜) * (Fintype.card α - r)\n⊢ #𝒜 * ... | [
"case e'_3\n𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Semifield 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nr : ℕ\nhr : r + 1 ≠ 0\nhr' : r + 1 ≤ Fintype.card α\nh𝒜 : #𝒜 * (r + 1) ≤ #(∂ 𝒜) * (Fintype.card α - r)\n⊢ (Fintype.card α).c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 24
} | {
"line": 88,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\nh : 𝒜 ⊆ ℬ\nx✝ : Finset α\n⊢ x✝ ∈ 𝒜.shatterer → x✝ ∈ ℬ.shatterer",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.mem_shatterer._simp_1",
"Finset",
"Membership.mem",
"id"... | [
"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\nh : 𝒜 ⊆ ℬ\nx✝ : Finset α\n⊢ 𝒜.Shatters x✝ → ℬ.Shatters x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 94,
"column": 62
} | {
"line": 94,
"column": 73
} | {
"line": 94,
"column": 74
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns t : Finset α\n⊢ t ⊆ s → s ∈ ↑𝒜.shatterer → t ∈ ↑𝒜.shatterer",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset.mem_shatterer._simp_1",
"Finset",
... | [
"α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\ns t : Finset α\n⊢ t ⊆ s → 𝒜.Shatters s → 𝒜.Shatters t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 174,
"column": 8
} | {
"line": 174,
"column": 23
} | {
"line": 174,
"column": 24
} | [
{
"pp": "case pos.refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Fi... | [
"case pos.refine_1\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 175,
"column": 8
} | {
"line": 175,
"column": 23
} | {
"line": 175,
"column": 24
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Fi... | [
"case pos.refine_2\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 23
} | {
"line": 176,
"column": 24
} | [
{
"pp": "case pos.refine_3\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Fi... | [
"case pos.refine_3\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : DecidableEq α\ninst✝³ : CommSemiring β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\na : α\nf₁ f₂ f₃ f₄ : Finset α → β\nu : Finset α\ninst✝ : ExistsAddOfLE β\nhu : a ∉ u\nh₁ : 0 ≤ f₁\nh₂ : 0 ≤ f₂\nh₃ : 0 ≤ f₃\nh₄ : 0 ≤ f₄\nh : ∀ ⦃s : Finset α⦄, s ⊆... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 78
} | {
"line": 209,
"column": 8
} | [
{
"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⊢ #(falling (Fintype.card α - Fintype.card α) 𝒜) ≤ 1 * (Fintype.card α).choose (Fintype.card α - Fintype... | [
"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⊢ #(𝒜.sup (powersetCard 0)) ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 37
} | {
"line": 245,
"column": 38
} | [
{
"pp": "α : Type u_2\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜\nthis : 0 < ↑((Fintype.card α).choose (Fintype.card α / 2))\nh : ∑ s ∈ 𝒜, (↑((Fintype.card α).choose (Fintype.card α / 2)))⁻¹ ≤ 1\n⊢ #𝒜 ≤ (Fintype.card α).choose (Fintype.card α / 2)",
"ppTerm": "... | [
"α : Type u_2\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜\nthis : 0 < ↑((Fintype.card α).choose (Fintype.card α / 2))\nh : ∑ s ∈ 𝒜, (↑((Fintype.card α).choose (Fintype.card α / 2)))⁻¹ ≤ 1\n⊢ #𝒜 ≤ (Fintype.card α).choose (Fintype.card α / 2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 117,
"column": 2
} | {
"line": 117,
"column": 36
} | {
"line": 117,
"column": 37
} | [
{
"pp": "V : Type u\ns : Set (Sym2 V)\nx✝¹ x✝ : V\n⊢ Relation.ReflGen (fromEdgeSet s).Adj x✝¹ x✝ ↔ Relation.ReflGen (Sym2.ToRel s) x✝¹ x✝",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.fromEdgeSet",
"SimpleGraph.A... | [
"V : Type u\ns : Set (Sym2 V)\nx✝¹ x✝ : V\n⊢ x✝ = x✝¹ ∨ s(x✝¹, x✝) ∈ s ∧ ¬x✝¹ = x✝ ↔ x✝ = x✝¹ ∨ s(x✝¹, x✝) ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 448,
"column": 35
} | {
"line": 451,
"column": 23
} | {
"line": 453,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs : IndexedPartition s\nx✝ : hs.Quotient\nx : α\n⊢ hs.proj ⁻¹' {Quotient.mk'' x} = s (hs.equivQuotient.symm (Quotient.mk'' x))",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"IndexedPartition.setoi... | [] | by
ext y
simp only [Set.mem_preimage, Set.mem_singleton_iff, hs.mem_iff_index_eq]
exact Quotient.eq'' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Setoid.Partition | {
"line": 474,
"column": 4
} | {
"line": 474,
"column": 44
} | {
"line": 474,
"column": 45
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nh_injOn : ∀ (i : ι), InjOn (f i) (s i)\nh_disjoint : univ.PairwiseDisjoint fun i ↦ f i '' s i\nx y : α\nh : hs.piecewise f x = hs.piecewise f y\nthis : hs.index x = hs.index y\n⊢ f (hs.index x) x = f (hs.in... | [
"ι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nh_injOn : ∀ (i : ι), InjOn (f i) (s i)\nh_disjoint : univ.PairwiseDisjoint fun i ↦ f i '' s i\nx y : α\nh : hs.piecewise f x = hs.piecewise f y\nthis : hs.index x = hs.index y\n⊢ f (hs.index y) x = f (hs.index y) y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 516,
"column": 22
} | {
"line": 516,
"column": 68
} | {
"line": 516,
"column": 69
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nx : β\nx✝ : x ∈ range (hs.piecewise f)\ny : α\nhy : hs.piecewise f y = x\n⊢ x ∈ ⋃ i, range (f i)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"ι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nx : β\nx✝ : x ∈ range (hs.piecewise f)\ny : α\nhy : hs.piecewise f y = x\n⊢ ∃ i y, f i y = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 537,
"column": 6
} | {
"line": 537,
"column": 17
} | {
"line": 537,
"column": 18
} | [
{
"pp": "case left\nι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs✝ : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nhs : IndexedPartition s\nκ : Type u_4\ng : ι → κ\nhg : Surjective g\nk : κ\n⊢ g ⋯.some = k",
"ppTerm": "?left",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"used... | [
"case left\nι : Type u_1\nα : Type u_2\ns : ι → Set α\nhs✝ : IndexedPartition s\nβ : Type u_3\nf : ι → α → β\nhs : IndexedPartition s\nκ : Type u_4\ng : ι → κ\nhg : Surjective g\nk : κ\n⊢ g ⋯.some = k"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 784,
"column": 4
} | {
"line": 784,
"column": 15
} | {
"line": 784,
"column": 16
} | [
{
"pp": "case h\nV : Type u\nG : SimpleGraph V\nv w v' w' : V\np : (G.deleteEdges {s(v, w)}).Walk v' w'\nh : s(v, w) ∈ p.edges\n⊢ False",
"ppTerm": "?h",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h\nV : Type u\nG : SimpleGraph V\nv w v' w' : V\np : (G.deleteEdges {s(v, w)}).Walk v' w'\nh : s(v, w) ∈ p.edges\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 35
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case map_rel'\nV : Type u\nG : SimpleGraph V\nn : ℕ\nβ : Type u_3\nf : V ↪ β\ninst✝ : NeZero n\nC : G.Coloring (Fin n)\n⊢ ∀ {a b : β},\n (SimpleGraph.map (⇑f) G).Adj a b →\n (completeGraph (Fin n)).Adj (extend (⇑f) (⇑C) (const β default) a) (extend (⇑f) (⇑C) (const β default) b)",
"ppTerm":... | [
"case map_rel'\nV : Type u\nG : SimpleGraph V\nn : ℕ\nβ : Type u_3\nf : V ↪ β\ninst✝ : NeZero n\nC : G.Coloring (Fin n)\na b : β\nleft✝ : a ≠ b\nw✝¹ w✝ : V\nhadj : G.Adj w✝¹ w✝\nha : f w✝¹ = a\nhb : f w✝ = b\n⊢ (completeGraph (Fin n)).Adj (extend (⇑f) (⇑C) (const β default) a) (extend (⇑f) (⇑C) (const β default) b)... | intro a b ⟨_, _, _, hadj, ha, hb⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 13
} | {
"line": 208,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\nι : Type u_1\nf : ι → V\nhf : Pairwise fun i j ↦ G.Adj (f i) (f j)\nC : G.Coloring (Fin n)\n⊢ Nat.card ι ≤ n",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nn : ℕ\nι : Type u_1\nf : ι → V\nhf : Pairwise fun i j ↦ G.Adj (f i) (f j)\nC : G.Coloring (Fin n)\n⊢ Nat.card ι ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 358,
"column": 2
} | {
"line": 361,
"column": 48
} | {
"line": 363,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable G.chromaticNumber.toNat",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"SimpleGraph.Colorable.chromaticNumber_eq_sInf",
"setOf",
"C... | [] | classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 358,
"column": 2
} | {
"line": 361,
"column": 48
} | {
"line": 363,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable G.chromaticNumber.toNat",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"SimpleGraph.Colorable.chromaticNumber_eq_sInf",
"setOf",
"C... | [] | classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 358,
"column": 2
} | {
"line": 361,
"column": 48
} | {
"line": 363,
"column": 0
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nm : ℕ\nhc : G.Colorable m\n⊢ G.Colorable G.chromaticNumber.toNat",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"SimpleGraph.Colorable.chromaticNumber_eq_sInf",
"setOf",
"C... | [] | classical
rw [hc.chromaticNumber_eq_sInf, Nat.sInf_def]
· apply Nat.find_spec
· exact colorable_set_nonempty_of_colorable hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 844,
"column": 6
} | {
"line": 844,
"column": 32
} | {
"line": 844,
"column": 33
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\nc : G.Walk u u\nhc : c.IsCycle\nhe : s(v, w) ∈ c.edges\nhb : ∀ (p : G.Walk v w), s(v, w) ∈ p.edges\np : G.Walk w v\n⊢ s(w, v) ∈ p.edges",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
... | [
"V : Type u\nG : SimpleGraph V\nv w u : V\nc : G.Walk u u\nhc : c.IsCycle\nhe : s(v, w) ∈ c.edges\nhb : ∀ (p : G.Walk v w), s(v, w) ∈ p.edges\np : G.Walk w v\n⊢ s(v, w) ∈ p.edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 430,
"column": 34
} | {
"line": 430,
"column": 45
} | {
"line": 430,
"column": 46
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nα : Type u_2\ninst✝ : Fintype α\ni : ℕ\nC : G.Coloring (Fin i)\nh : i < card α\n⊢ card (Fin i) ≤ card α",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"congrArg",
"Fintype.card",
"id",
... | [
"V : Type u\nG : SimpleGraph V\nα : Type u_2\ninst✝ : Fintype α\ni : ℕ\nC : G.Coloring (Fin i)\nh : i < card α\n⊢ i ≤ card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 26
} | {
"line": 432,
"column": 27
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nα : Type u_2\ninst✝ : Fintype α\ni : ℕ\nC : G.Coloring (Fin i)\nh : i < card α\nhC : Surjective (⇑⋯.some ∘ ⇑C)\n⊢ False",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nV : Type u\nG : SimpleGraph V\nα : Type u_2\ninst✝ : Fintype α\ni : ℕ\nC : G.Coloring (Fin i)\nh : i < card α\nhC : Surjective (⇑⋯.some ∘ ⇑C)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 479,
"column": 49
} | {
"line": 479,
"column": 60
} | {
"line": 479,
"column": 61
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : Fintype V\nh : G.chromaticNumber = ↑(card V)\nhh : G ≠ ⊤\na b : V\nhne : a ≠ b\nright✝ : ¬G.Adj a b\nthis : G.Coloring ↥(Finset.univ.erase b)\n⊢ G.Colorable (card V - 1)",
"ppTerm": "?m.72",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"V : Type u\nG : SimpleGraph V\ninst✝ : Fintype V\nh : G.chromaticNumber = ↑(card V)\nhh : G ≠ ⊤\na b : V\nhne : a ≠ b\nright✝ : ¬G.Adj a b\nthis : G.Coloring ↥(Finset.univ.erase b)\n⊢ G.Colorable (card V - 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 520,
"column": 50
} | {
"line": 520,
"column": 61
} | {
"line": 520,
"column": 62
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nh : G = ⊥\nh' : ¬IsEmpty V\n⊢ Nonempty V",
"ppTerm": "?m.39",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nh : G = ⊥\nh' : ¬IsEmpty V\n⊢ Nonempty V"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 339,
"column": 6
} | {
"line": 339,
"column": 32
} | {
"line": 339,
"column": 33
} | [
{
"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\nA B : Finset (Fin n)\nhA : A ⊆ range'k ∧ #A ... | [
"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\nA B : Finset (Fin n)\nhA : A ⊆ range'k ∧ #A = r\nHB₁ : t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 535,
"column": 8
} | {
"line": 535,
"column": 19
} | {
"line": 535,
"column": 20
} | [
{
"pp": "V : Type u_4\nW : Type u_5\ninst✝¹ : Nonempty V\ninst✝ : Nonempty W\n⊢ (completeBipartiteGraph V W).Colorable 2",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_4\nW : Type u_5\ninst✝¹ : Nonempty V\ninst✝ : Nonempty W\n⊢ (completeBipartiteGraph V W).Colorable 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 544,
"column": 4
} | {
"line": 544,
"column": 15
} | {
"line": 544,
"column": 16
} | [
{
"pp": "case neg\nV : Type u_4\nW : Type u_5\ninst✝¹ : Nonempty V\ninst✝ : Nonempty W\nC : (completeBipartiteGraph V W).Coloring (Fin 2)\nb : Fin 2\nv : V\nw : W\nh : (completeBipartiteGraph V W).Adj (Sum.inl v) (Sum.inr w)\nhe : ¬C (Sum.inl v) = b\nhe' : ¬C (Sum.inr w) = b\n⊢ ∃ a, C a = b",
"ppTerm": "?ne... | [
"case neg\nV : Type u_4\nW : Type u_5\ninst✝¹ : Nonempty V\ninst✝ : Nonempty W\nC : (completeBipartiteGraph V W).Coloring (Fin 2)\nb : Fin 2\nv : V\nw : W\nh : (completeBipartiteGraph V W).Adj (Sum.inl v) (Sum.inr w)\nhe : ¬C (Sum.inl v) = b\nhe' : ¬C (Sum.inr w) = b\n⊢ (∃ a, C (Sum.inl a) = b) ∨ ∃ b_1, C (Sum.inr ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 937,
"column": 2
} | {
"line": 937,
"column": 13
} | {
"line": 937,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nw : G.Walk u v\ne : Sym2 V\nhuv : ¬(G.deleteEdges {e}).Reachable u v\n⊢ e ∈ w.edges",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\nw : G.Walk u v\ne : Sym2 V\nhuv : ¬(G.deleteEdges {e}).Reachable u v\n⊢ e ∈ w.edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 947,
"column": 8
} | {
"line": 947,
"column": 19
} | {
"line": 947,
"column": 20
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v x y : V\nw : G.Walk u v\nhw : w.IsTrail\nhuy : ¬(G.deleteEdges {s(x, y)}).Reachable u y\nhvy : ¬(G.deleteEdges {s(x, y)}).Reachable v y\nhxy : s(x, y) ∈ w.edges\n⊢ s(x, y) ∈ (w.dropUntil y ⋯).edges",
"ppTerm": "?m.54",
"assigned": false,
"usedConstants": [... | [
"V : Type u\nG : SimpleGraph V\nu v x y : V\nw : G.Walk u v\nhw : w.IsTrail\nhuy : ¬(G.deleteEdges {s(x, y)}).Reachable u y\nhvy : ¬(G.deleteEdges {s(x, y)}).Reachable v y\nhxy : s(x, y) ∈ w.edges\n⊢ s(x, y) ∈ (w.dropUntil y ⋯).edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 13
} | {
"line": 578,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nn : ℕ\nhc : ↑G.chromaticNumber.toNat < ↑n\nhne : ↑G.chromaticNumber.toNat = G.chromaticNumber\nm : ℕ\nhc' : G.Colorable m\nthis : G.Colorable G.chromaticNumber.toNat\n⊢ G.chromaticNumber.toNat < n",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants": [],
... | [
"V : Type u\nG : SimpleGraph V\nn : ℕ\nhc : ↑G.chromaticNumber.toNat < ↑n\nhne : ↑G.chromaticNumber.toNat = G.chromaticNumber\nm : ℕ\nhc' : G.Colorable m\nthis : G.Colorable G.chromaticNumber.toNat\n⊢ G.chromaticNumber.toNat < n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 965,
"column": 6
} | {
"line": 965,
"column": 17
} | {
"line": 965,
"column": 18
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhxu : x ≠ u\nhxv : x ≠ v\nhx : (G.neighborSet x).Subsingleton\nhxw : x ∈ w.support\ny : V\nhxy : G.Adj x y\np : (G.deleteEdges {s(y, x)}).Walk u x\n⊢ G.Adj x p.penultimate ∧ ?m.117 ∧ ¬p.penultimate = y",
"ppTerm": "?m.124",
... | [
"V : Type u\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhxu : x ≠ u\nhxv : x ≠ v\nhx : (G.neighborSet x).Subsingleton\nhxw : x ∈ w.support\ny : V\nhxy : G.Adj x y\np : (G.deleteEdges {s(y, x)}).Walk u x\n⊢ G.Adj x p.penultimate ∧ ?m.117 ∧ ¬p.penultimate = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 371,
"column": 64
} | {
"line": 371,
"column": 75
} | {
"line": 371,
"column": 76
} | [
{
"pp": "n : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nh1r : r > 0\nsize : (n - 1).choose (r - 1) < #𝒜\nthis✝¹ : Disjoint 𝒜 (∂^[n - 2 * r] 𝒜ᶜˢ)\nthis✝ : r ≤ n\nthis : 1 ≤ n\nz : (n - 1).choose (n - r) < #𝒜ᶜˢ\n⊢ Set.Sized (n - r) ↑𝒜ᶜˢ",
"ppTe... | [
"n : ℕ\n𝒜 : Finset (Finset (Fin n))\nr : ℕ\nh𝒜 : (↑𝒜).Intersecting\nh₂ : Set.Sized r ↑𝒜\nh₃ : r ≤ n / 2\nh1r : r > 0\nsize : (n - 1).choose (r - 1) < #𝒜\nthis✝¹ : Disjoint 𝒜 (∂^[n - 2 * r] 𝒜ᶜˢ)\nthis✝ : r ≤ n\nthis : 1 ≤ n\nz : (n - 1).choose (n - r) < #𝒜ᶜˢ\n⊢ Set.Sized (n - r) (compl '' ↑𝒜)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 965,
"column": 6
} | {
"line": 965,
"column": 17
} | {
"line": 965,
"column": 18
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhxu : x ≠ u\nhxv : x ≠ v\nhx : (G.neighborSet x).Subsingleton\nhxw : x ∈ w.support\ny : V\nhxy : G.Adj x y\np : (G.deleteEdges {s(y, x)}).Walk v x\n⊢ G.Adj x p.penultimate ∧ ?m.183 ∧ ¬p.penultimate = y",
"ppTerm": "?m.190",
... | [
"V : Type u\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhxu : x ≠ u\nhxv : x ≠ v\nhx : (G.neighborSet x).Subsingleton\nhxw : x ∈ w.support\ny : V\nhxy : G.Adj x y\np : (G.deleteEdges {s(y, x)}).Walk v x\n⊢ G.Adj x p.penultimate ∧ ?m.183 ∧ ¬p.penultimate = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 102,
"column": 2
} | {
"line": 102,
"column": 23
} | {
"line": 102,
"column": 24
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhv : v ∈ s\nhadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s\nnhv : v ∉ t\n⊢ w ∈ t",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhv : v ∈ s\nhadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s\nnhv : v ∉ t\n⊢ w ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 23
} | {
"line": 128,
"column": 24
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhw : w ∈ t\nhadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s\nnhw : w ∉ s\n⊢ v ∈ s",
"ppTerm": "?m.34",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nh : G.IsBipartiteWith s t\nhw : w ∈ t\nhadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s\nnhw : w ∉ s\n⊢ v ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 13
} | {
"line": 101,
"column": 14
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\ninst✝ : Fintype ↑(G.neighborSet u)\nh : G.IsEdgeReachable k u v\nhuv : u ≠ v\nhh : (G.incidenceSet u).encard < ↑k\nw : (G.deleteEdges (G.incidenceSet u)).Walk u v\nh✝ : w.IsPath\n⊢ False",
"ppTerm": "?m.57",
"assigned": false,
"usedConstants"... | [
"V : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\ninst✝ : Fintype ↑(G.neighborSet u)\nh : G.IsEdgeReachable k u v\nhuv : u ≠ v\nhh : (G.incidenceSet u).encard < ↑k\nw : (G.deleteEdges (G.incidenceSet u)).Walk u v\nh✝ : w.IsPath\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 15
} | {
"line": 116,
"column": 16
} | [
{
"pp": "case refine_2.inl\nV : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\nhk : k ≠ 0\nh : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v\nhs : ∅.encard < ↑(k + 1)\n⊢ (G.deleteEdges ∅).Reachable u v",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"SimpleGraph.del... | [
"case refine_2.inl\nV : Type u_1\nG : SimpleGraph V\nk : ℕ\nu v : V\nhk : k ≠ 0\nh : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v\nhs : ∅.encard < ↑(k + 1)\n⊢ G.Reachable u v"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 122,
"column": 78
} | {
"line": 123,
"column": 79
} | {
"line": 125,
"column": 0
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nk : ℕ\nhk : k ≠ 0\n⊢ G.IsEdgeConnected (k + 1) ↔ ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeConnected k",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsEdgeReachable",
"SimpleGraph.deleteEdges",
"SimpleGraph.isEdgeRe... | [] | by
simp [IsEdgeConnected, isEdgeReachable_add_one hk, forall_comm (α := Sym2 _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 17
} | {
"line": 344,
"column": 18
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nleft right : Finset V\ncard_left : #left = Fintype.card α\ncard_right : #right = Fintype.card β\nh : G.IsCompleteBetween ↑left ↑right\nthis✝ : Nonempty (α ↪ ↥left)\nfα : α ↪ ↥left :... | [
"V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nleft right : Finset V\ncard_left : #left = Fintype.card α\ncard_right : #right = Fintype.card β\nh : G.IsCompleteBetween ↑left ↑right\nthis✝ : Nonempty (α ↪ ↥left)\nfα : α ↪ ↥left := Classical.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Bipartite | {
"line": 346,
"column": 6
} | {
"line": 346,
"column": 17
} | {
"line": 346,
"column": 18
} | [
{
"pp": "V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nleft right : Finset V\ncard_left : #left = Fintype.card α\ncard_right : #right = Fintype.card β\nh : G.IsCompleteBetween ↑left ↑right\nthis✝ : Nonempty (α ↪ ↥left)\nfα : α ↪ ↥left :... | [
"V : Type u_1\nv w : V\nG : SimpleGraph V\ns t : Set V\nα : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nleft right : Finset V\ncard_left : #left = Fintype.card α\ncard_right : #right = Fintype.card β\nh : G.IsCompleteBetween ↑left ↑right\nthis✝ : Nonempty (α ↪ ↥left)\nfα : α ↪ ↥left := Classical.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 15
} | {
"line": 165,
"column": 16
} | [
{
"pp": "case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhne : u ≠ v\nh : G.IsEdgeReachable 2 u v\nw : G.Walk u v\nhw : w.IsPath\nthis✝ : G.Adj u w.snd\nhs : {s(u, w.snd)}.encard < ↑2\nhh : s(u, w.snd) ∈ w.tail.edges\nthis : u = w.getVert 2\n⊢ False",
"ppTerm": "?pos✝",
"assigned": false,
"usedC... | [
"case pos\nV : Type u_1\nG : SimpleGraph V\nu v : V\nhne : u ≠ v\nh : G.IsEdgeReachable 2 u v\nw : G.Walk u v\nhw : w.IsPath\nthis✝ : G.Adj u w.snd\nhs : {s(u, w.snd)}.encard < ↑2\nhh : s(u, w.snd) ∈ w.tail.edges\nthis : u = w.getVert 2\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 184,
"column": 23
} | {
"line": 184,
"column": 56
} | {
"line": 184,
"column": 57
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhuv : G.IsEdgeReachable 2 u v\nhuy : ¬G.IsEdgeReachable 2 u x\n⊢ ?m.25",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u_1\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhuv : G.IsEdgeReachable 2 u v\nhuy : ¬G.IsEdgeReachable 2 u x\n⊢ ?m.25"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity | {
"line": 189,
"column": 8
} | {
"line": 189,
"column": 19
} | {
"line": 189,
"column": 20
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhuv : G.IsEdgeReachable 2 u v\nhuy : ¬G.IsEdgeReachable 2 u x\ne : Sym2 V\nhe : ¬(G.deleteEdges {e}).Reachable u x\nhe' : ¬(G.deleteEdges {e}).Reachable v x\nhy : x ∈ w.support\n⊢ e ∈ (w.dropUntil x hy).edges",
"ppTerm": "?... | [
"V : Type u_1\nG : SimpleGraph V\nu v x : V\nw : G.Walk u v\nhw : w.IsTrail\nhuv : G.IsEdgeReachable 2 u v\nhuy : ¬G.IsEdgeReachable 2 u x\ne : Sym2 V\nhe : ¬(G.deleteEdges {e}).Reachable u x\nhe' : ¬(G.deleteEdges {e}).Reachable v x\nhy : x ∈ w.support\n⊢ e ∈ (w.dropUntil x hy).edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 36
} | {
"line": 83,
"column": 2
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\nhuv : G.Adj u v\n⊢ (⊤.induce {u, v}).Connected",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
"Set.instSingletonSet",
"id",
"Insert.insert",
"SimpleGrap... | [
"V : Type u\nG : SimpleGraph V\nu v : V\nhuv : G.Adj u v\n⊢ (G.subgraphOfAdj huv).Connected"
] | rw [← subgraphOfAdj_eq_induce huv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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