module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.HalesJewett | {
"line": 202,
"column": 30
} | {
"line": 202,
"column": 68
} | [
{
"pp": "case idxFun.h.a.refine_2.none\nα : Type u_2\nι : Type u_3\ninst✝ : Nontrivial α\nl m : Line α ι\ni : ι\na b : α\nhba : b ≠ a\nhlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1\nh : m.idxFun i = some a\nhi : l.idxFun i = none\n⊢ none = some a",
"usedConstants": [
"Eq.mpr",
"False",
"O... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.HalesJewett | {
"line": 202,
"column": 30
} | {
"line": 202,
"column": 68
} | [
{
"pp": "case idxFun.h.a.refine_2.some\nα : Type u_2\nι : Type u_3\ninst✝ : Nontrivial α\nl m : Line α ι\ni : ι\na b : α\nhba : b ≠ a\nhlm : ∀ (x : α) (x_1 : ι), ↑l x x_1 = ↑m x x_1\nh : m.idxFun i = some a\nval✝ : α\nhi : l.idxFun i = some val✝\n⊢ some val✝ = some a",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hindman | {
"line": 170,
"column": 4
} | {
"line": 170,
"column": 50
} | [
{
"pp": "case h.a\nM : Type u_1\ninst✝ : Semigroup M\na : Stream' M\nS : Set (Ultrafilter M) := ⋯\nU : Ultrafilter M\nhU : ∀ (i : ℕ), U ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}\nV : Ultrafilter M\nhV : ∀ (i : ℕ), V ∈ {U | ∀ᶠ (m : M) in ↑U, m ∈ FP (Stream'.drop i a)}\nn : ℕ\nm : M\nhm : m ∈ FP (Stream... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Hindman | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 51
} | [
{
"pp": "case h.cons'\nM : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty\nelem : { s // s ∈ U } → M := fun p ↦ ⋯.some\nsucc : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * m ∈ ↑p}, ⋯⟩... | have := Set.inter_subset_right (ih (succ p) ?_) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Hindman | {
"line": 206,
"column": 6
} | {
"line": 206,
"column": 22
} | [
{
"pp": "case h.cons'.refine_2\nM : Type u_1\ninst✝ : Semigroup M\nU : Ultrafilter M\nU_idem : U * U = U\nexists_elem : ∀ {s : Set M}, s ∈ U → (s ∩ {m | ∀ᶠ (m' : M) in ↑U, m * m' ∈ s}).Nonempty\nelem : { s // s ∈ U } → M := fun p ↦ ⋯.some\nsucc : { s // s ∈ U } → { s // s ∈ U } := fun p ↦ ⟨↑p ∩ {m | elem p * m ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.HalesJewett | {
"line": 512,
"column": 2
} | {
"line": 512,
"column": 13
} | [
{
"pp": "case h\nα : Type u_5\nκ : Type u_6\nη : Type u_7\ninst✝² : Finite α\ninst✝¹ : Finite κ\ninst✝ : Finite η\nι : Type\nιfin : Fintype ι\nhι : ∀ (C : (ι → α) → κ), ∃ l, IsMono C l\nC : (Fin (Fintype.card ι) → α) → κ\nl : Subspace η α ι\nc : κ\ncl : ∀ (x : η → α), (fun v ↦ C (v ∘ ⇑(Fintype.equivFin ι).symm)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.KatonaCircle | {
"line": 58,
"column": 41
} | {
"line": 58,
"column": 52
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\nf✝ : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nf : ↥(prefixed s)\nn : Fin (Fintype.card ↥s)\n⊢ Fintype.card ↥s ≤ Fintype.card X",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Preorder.toLE",
"Membership.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.KatonaCircle | {
"line": 59,
"column": 35
} | {
"line": 59,
"column": 46
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\nf✝ : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nf : ↥(prefixed s)\nn : Fin (Fintype.card ↥s)\n⊢ ↑(↑f ((Equiv.symm ↑f) ⟨↑n, ⋯⟩)) < #s",
"usedConstants": [
"Eq.mpr",
"Equiv.apply_symm_apply",
"Equiv.instEquivLike",
"con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.KatonaCircle | {
"line": 67,
"column": 54
} | {
"line": 67,
"column": 65
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\nf✝ : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nf : ↥(prefixed s)\nn : Fin (Fintype.card ↥sᶜ)\n⊢ ↑n < Fintype.card X - #s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.KatonaCircle | {
"line": 82,
"column": 24
} | {
"line": 82,
"column": 35
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\nf : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nx✝ : Numbering ↥s × Numbering ↥sᶜ\ng : Numbering ↥s\ng' : Numbering ↥sᶜ\nn : Fin (Fintype.card X)\nhn : ↑n < #s\n⊢ ↑n < Fintype.card ↥s",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.KatonaCircle | {
"line": 87,
"column": 38
} | {
"line": 87,
"column": 49
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\nf : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nx✝ : Numbering ↥s × Numbering ↥sᶜ\ng : Numbering ↥s\ng' : Numbering ↥sᶜ\nx : X\nhx : x ∈ s\n⊢ ↑(g ⟨x, hx⟩) < #s",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 95,
"column": 26
} | {
"line": 95,
"column": 37
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nC : Set α\n⊢ (M✶ ＼ C)✶ = M ↔ Disjoint C M.E",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"congrArg",
"Matroid.E",
"Disjoint",
"Matroid.dual"... | ← dual_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.KatonaCircle | {
"line": 97,
"column": 10
} | {
"line": 97,
"column": 38
} | [
{
"pp": "case mp\nX : Type u_1\ninst✝¹ : Fintype X\nf : Numbering X\ns✝ t : Finset X\ninst✝ : DecidableEq X\ns : Finset X\nx✝ : Numbering ↥s × Numbering ↥sᶜ\ng : Numbering ↥s\ng' : Numbering ↥sᶜ\nx : X\nhx : x ∈ s\n⊢ ↑({ toFun := fun x ↦ if hx : x ∈ s then Fin.castLE ⋯ (g ⟨x, hx⟩) else Fin.cast ⋯ ((g' ⟨x, ⋯⟩).a... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 170,
"column": 46
} | {
"line": 170,
"column": 57
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\n⊢ M ／ (X \\ I ∪ I) = M ／ I ＼ (X \\ I)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.dual",
"Set.instUnion",
"id",
"SDiff.sdiff",
"propext",
"Union.union",
"Eq.symm",
"Eq",... | ← dual_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 175,
"column": 41
} | {
"line": 175,
"column": 52
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhI : M.IsBasis I X\nJ : Set α\nhJ : J ⊆ (M✶ ＼ I ＼ (X \\ I)).E\ne : α\nhe : e ∈ X \\ I\n⊢ Disjoint {e} I",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 199,
"column": 2
} | {
"line": 200,
"column": 85
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\n⊢ ∀ ⦃K : Set α⦄, Disjoint K J → M.Indep (K ∪ J) → K ⊆ X → I ⊆ K ∪ J → K ⊆ I",
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"Se... | exact fun K hJK hKJi hKX hIJK ↦ by
simp [hIX.eq_of_subset_indep hKJi hIJK (union_subset hKX (hJI.trans hIX.subset))] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 95
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nhIX : M.IsBasis' I X\nhJI : J ⊆ I\n⊢ (M ／ J).IsBasis' (I \\ J) X",
"usedConstants": [
"Eq.mpr",
"Matroid.E",
"Matroid.IsBasis'",
"id",
"Set.instInter",
"Inter.inter",
"SDiff.sdiff",
"Matroid.IsBasis",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 217,
"column": 26
} | {
"line": 217,
"column": 41
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nI J X Y : Set α\nhIX : M.IsBasis I X\nhJY : M.IsBasis J Y\nhIJ : I ⊆ J\n⊢ Disjoint (J \\ (J ∩ X)) X",
"usedConstants": [
"Eq.mpr",
"Set.diff_self_inter",
"congrArg",
"Disjoint",
"SemilatticeInf.toPartialOrder",
"id",
... | diff_self_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 52
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI J X : Set α\nh : M.IsBasis' (J ∪ I) (X ∪ I)\nhJI : Disjoint J I\nhXI : Disjoint X I\n⊢ (M ／ I).IsBasis' J X",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 277,
"column": 4
} | {
"line": 278,
"column": 80
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Matroid α\nI J X C : Set α\nh : M.IsBasis I X\nhJC : M.IsBasis' J C\nh_ind : M.Indep (I \\ C ∪ J)\nhIX : I ⊆ X\nhJCss : J ⊆ C\n⊢ X \\ C ∪ J ⊆ M.closure (X ∪ C)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Set.diff_subset",
"_private.Mathlib.Com... | exact subset_closure_of_subset' _ (by tauto_set)
(union_subset (diff_subset.trans h.subset_ground) hJC.indep.subset_ground) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.KatonaCircle | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 29
} | [
{
"pp": "X : Type u_1\ninst✝¹ : Fintype X\ninst✝ : DecidableEq X\ns : Finset X\n⊢ #(prefixed s) = (#s)! * (Fintype.card X - #s)!",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 345,
"column": 6
} | {
"line": 345,
"column": 17
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.coloops\n⊢ M ／ X = M ＼ X",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.dual",
"id",
"propext",
"Eq.symm",
"Eq",
"Matroid",
"Matroid.contract",
"Matroid.dual_inj",
"Matroid.dele... | ← dual_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 381,
"column": 6
} | {
"line": 381,
"column": 22
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI X : Set α\nhIX : M.IsBasis I X\n⊢ X \\ I ⊆ (M ／ I).loops",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.diff_subset_iff",
"Set.instUnion",
"id",
"HasSubset.Subset",
"SDiff.sdiff",
"propext",
"Union.union",
"... | diff_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 49
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nK C : Set α\nhC : M.IsCircuit C\nhK : K.Nonempty\nhKC : K ⊆ C\n⊢ (M ／ (C \\ K)).IsCircuit K",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 67,
"column": 38
} | {
"line": 67,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nM✝ M : (i : ι) → Matroid (α i)\nI : Set ((i : ι) × α i)\nh : ∀ (i : ι), (M i).Indep (Sigma.mk i ⁻¹' I)\nBs : (i : ι) → Set (α i)\nhBs : ∀ (i : ι), (M i).IsBase (Bs i) ∧ Sigma.mk i ⁻¹' I ⊆ Bs i\ni : ι\n⊢ (M i).IsBase (Sigma.mk i ⁻¹' univ.sigma Bs)",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 51
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nh : Disjoint C R\nI : Set α\nhI : I ⊆ R\n⊢ (M ／ C ↾ R).Indep I ↔ ((M ↾ (R ∪ C)) ／ C).Indep I",
"usedConstants": [
"Set.instUnion",
"Union.union",
"Matroid.restrict",
"Matroid.exists_isBasis'",
"Set"
]
}
] | obtain ⟨J, hJ⟩ := (M ↾ (R ∪ C)).exists_isBasis' C | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Minor.Contract | {
"line": 510,
"column": 4
} | {
"line": 510,
"column": 65
} | [
{
"pp": "α : Type u_1\nR C : Set α\nM : Matroid α\nh : Disjoint C R\nI : Set α\nhI : I ⊆ R\nJ : Set α\nhJ : (M ↾ (R ∪ C)).IsBasis' J C\n⊢ M.IsBasis' J C",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 134,
"column": 21
} | {
"line": 134,
"column": 32
} | [
{
"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 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 135,
"column": 29
} | {
"line": 135,
"column": 40
} | [
{
"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 ⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Groebner | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 24
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 138,
"column": 26
} | {
"line": 138,
"column": 37
} | [
{
"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 ⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Sum | {
"line": 294,
"column": 9
} | {
"line": 294,
"column": 33
} | [
{
"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✝",
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Matroid.disjointSum",
"C... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Arborescence | {
"line": 71,
"column": 16
} | {
"line": 71,
"column": 55
} | [
{
"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
} | [
{
"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
} | [
{
"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.RingTheory.MvPolynomial.Groebner | {
"line": 205,
"column": 10
} | {
"line": 205,
"column": 49
} | [
{
"pp": "case h.e'_1.h.e'_4\nσ : 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 →... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.ConnectedComponent | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 85
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 109,
"column": 73
} | {
"line": 109,
"column": 84
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"Quiver.Hom",
"congrArg",
"heq_e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Covering | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 143,
"column": 6
} | {
"line": 143,
"column": 68
} | [
{
"pp": "case h_option.a.a.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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 54
} | [
{
"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",
"usedConstants": [
"not_le",
"Iff.mpr",
... | induction σ using Finite.induction_empty_option with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 54
} | [
{
"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",
"usedConstants":... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Nullstellensatz | {
"line": 176,
"column": 2
} | {
"line": 194,
"column": 55
} | [
{
"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",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Unit.unit",
"Inhabited.default",
"False",
... | 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
} | [
{
"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",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Unit.unit",
"Inhabited.default",
"False",
... | 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
} | [
{
"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",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Unit.unit",
"Inhabited.default",
"False",
... | 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.Combinatorics.Quiver.Path.Weight | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Weight | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 31
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 15
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
"instHAdd",
"And",
"HAdd.hAdd",
"Nat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 29
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
"instHAdd",
"And",
"HAdd.hAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 150,
"column": 4
} | {
"line": 150,
"column": 15
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
"instHAdd",
"And",
"HAdd.hAdd",
"Nat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 29
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Nat.add_eq_zero_iff._simp_1",
"id",
"instOfNatNat",
"instHAdd",
"And",
"HAdd.hAd... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 159,
"column": 11
} | {
"line": 159,
"column": 22
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Quiver.Path.nil",
"congrArg",
"id",
"Quiver.Path.nil_comp",
"Quiver.Pat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 18
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 183,
"column": 31
} | {
"line": 183,
"column": 42
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nn : ℕ\nhn : n ≤ nil.length\n⊢ n = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 52
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\na b : V\np : Path a b\nv : V\nhv : v ∈ nil.vertices\n⊢ v = a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Quiver.Path.Vertices | {
"line": 222,
"column": 6
} | {
"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",
"usedConstants": [
"False",
"Quiver.Path.nil",
"congrArg",
"Membership.mem",
"List.tail",
"List",
"_private.Mathlib.Combinatorics.Quiver.Path.Vertices.0... | 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
} | [
{
"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",
"usedConstants": [
"Membership.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 15
} | [
{
"pp": "case inl\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhk : 0 ∉ A\n⊢ schnirelmannDensity A ≤ 1 - (↑0)⁻¹",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"congrArg",
"Real.instInv",
"sub_zero",
"Real.instSub",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 16
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOn... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 87
} | [
{
"pp": "case mp\nA : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nx : ℕ\nhx : x ∈ {0}ᶜ\nhx' : x ∉ A\n⊢ 1 - (↑x)⁻¹ < 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWith... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 209,
"column": 36
} | {
"line": 209,
"column": 47
} | [
{
"pp": "A : Set ℕ\ninst✝ : DecidablePred fun x ↦ x ∈ A\nhA : A.Finite\n⊢ schnirelmannDensity A = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 232,
"column": 8
} | {
"line": 232,
"column": 18
} | [
{
"pp": "m : ℕ\nhm : m ≠ 1\nhm' : m > 0\n⊢ ↑(#({a ∈ Ioc 0 m | a ∈ {n | n % m = 1}})) / ↑m ≤ (↑m)⁻¹",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.instLE",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Schnirelmann | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 70
} | [
{
"pp": "⊢ schnirelmannDensity (setOf Odd) = 2⁻¹",
"usedConstants": [
"Set.ext",
"Odd",
"setOf",
"Nat.instMod",
"instHMod",
"instOfNatNat",
"HMod.hMod",
"Nat",
"Nat.instSemiring",
"OfNat.ofNat",
"Eq",
"Nat.odd_iff",
"Set"
]
... | 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.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 87
} | [
{
"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",
"usedConstants": [
"Iff.mpr",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 87
} | [
{
"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",
"usedConstants": [
"Iff.mpr",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 28
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 28
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 466,
"column": 2
} | {
"line": 466,
"column": 66
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"instDecidableEqProd",
"SProd.sprod",
"congrArg",
"Finset",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 469,
"column": 2
} | {
"line": 469,
"column": 66
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"instDecidableEqProd",
"SProd.sprod",
"congrArg",
"Finset.product_i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.Compression.UV | {
"line": 254,
"column": 2
} | {
"line": 254,
"column": 16
} | [
{
"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",... | 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
} | [
{
"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",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.AhlswedeZhang | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 15
} | [
{
"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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 658,
"column": 14
} | {
"line": 658,
"column": 25
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finset.Sups | {
"line": 659,
"column": 15
} | {
"line": 659,
"column": 53
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 79,
"column": 35
} | {
"line": 79,
"column": 53
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeSup.toMax",
"id",
"LE.le",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 101,
"column": 13
} | {
"line": 101,
"column": 36
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"Finset",
"false_and",
"Membership.mem"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 162,
"column": 35
} | {
"line": 162,
"column": 53
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"congrArg",
"left_eq_inf._simp_1",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPar... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 286,
"column": 29
} | {
"line": 286,
"column": 58
} | [
{
"pp": "α : Type u_2\ninst✝ : LinearOrder α\na x✝¹ x✝ : α\n⊢ max x✝¹ x✝ = a → x✝¹ = a ∨ x✝ = a",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeSup.toMax",
"_private.Mathlib.Order.Irreducible.0.supIrred_if... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Irreducible | {
"line": 290,
"column": 29
} | {
"line": 290,
"column": 58
} | [
{
"pp": "α : Type u_2\ninst✝ : LinearOrder α\na x✝¹ x✝ : α\n⊢ min x✝¹ x✝ = a → x✝¹ = a ∨ x✝ = a",
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"SemilatticeInf.toMin",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Birkhoff | {
"line": 285,
"column": 2
} | {
"line": 286,
"column": 89
} | [
{
"pp": "α : Type u\ninst✝¹ : Finite α\ninst✝ : DistribLattice α\n⊢ ∃ β x x_1 f, Injective ⇑f",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset",
"Classical.propDecidable",
"Exists",
"Subtype.fintype",
"inferInstance",
"DistribLattice.toLattice",
"Subty... | 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": 285,
"column": 2
} | {
"line": 286,
"column": 89
} | [
{
"pp": "α : Type u\ninst✝¹ : Finite α\ninst✝ : DistribLattice α\n⊢ ∃ β x x_1 f, Injective ⇑f",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Finset",
"Classical.propDecidable",
"Exists",
"Subtype.fintype",
"inferInstance",
"DistribLattice.toLattice",
"Subty... | cases nonempty_fintype α
exact ⟨{a : α // SupIrred a}, _, inferInstance, _, LatticeHom.birkhoffFinset_injective⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 88,
"column": 13
} | {
"line": 88,
"column": 24
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Finset.mem_shatterer._simp_1",
"Finset",
"Membership.mem",
"id",
"Finset.instSetLike",
"implies_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.Shatter | {
"line": 94,
"column": 62
} | {
"line": 94,
"column": 73
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset.mem_shatterer._simp_1",
"Finset",
"PartialOrder.toPreorder",
"Preorder.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 68,
"column": 6
} | {
"line": 68,
"column": 22
} | [
{
"pp": "case h.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))",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.KruskalKatona | {
"line": 90,
"column": 4
} | {
"line": 91,
"column": 34
} | [
{
"pp": "case h.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\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 55
} | [
{
"pp": "case h.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 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 174,
"column": 8
} | {
"line": 174,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 175,
"column": 8
} | {
"line": 175,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 208,
"column": 6
} | {
"line": 208,
"column": 78
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.FourFunctions | {
"line": 176,
"column": 8
} | {
"line": 176,
"column": 23
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SetFamily.LYM | {
"line": 245,
"column": 2
} | {
"line": 245,
"column": 37
} | [
{
"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)",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 36
} | [
{
"pp": "case e_r.h.h.a\nV : 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✝",
"usedConstants": [
"Eq.mpr",
"Sym2.mk",
"congrArg",
"SimpleGraph.fromEdgeSet",
"SimpleGraph.Adj",
"Membership.mem",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 448,
"column": 35
} | {
"line": 451,
"column": 23
} | [
{
"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))",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"IndexedPartition.setoid",
"Equiv.instEquivLike",
"Index... | 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
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 622,
"column": 2
} | {
"line": 622,
"column": 38
} | [
{
"pp": "case h\nV : Type u\nw v : V\n⊢ v ∈ supp (Quot.mk ⊤.Reachable w) ↔ v ∈ Set.univ",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.connectedComponentMk",
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"iff_true",
"Membership.mem",
"id",
"SimpleGraph.Co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 513,
"column": 4
} | {
"line": 513,
"column": 44
} | [
{
"pp": "case refine_2\nι : 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 ∈ ⋃ i, f i '' s i\nt : Set β\ni : ι\nhi : (fun i ↦ f i '' s i) i = t\na : α\nha2 : f i a = x\nha1 : hs.index a = i\n⊢ hs.piecewise f a = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 516,
"column": 22
} | {
"line": 516,
"column": 68
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mem_iUnion._simp_1",
"Membership.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Setoid.Partition | {
"line": 537,
"column": 6
} | {
"line": 537,
"column": 17
} | [
{
"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",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected | {
"line": 711,
"column": 77
} | {
"line": 711,
"column": 88
} | [
{
"pp": "V : Type u\nV' : Type v\nV'' : Type w\nG✝ : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nG : SimpleGraph V\nH : SimpleGraph V'\nC : (c : G.ConnectedComponent) → c.toSimpleGraph →g H\na✝ b✝ : V\nhab : G.Adj a✝ b✝\n⊢ (G.connectedComponentMk a✝).toSimpleGraph.Adj ⟨a✝, ⋯⟩ ⟨b✝, ⋯⟩",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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