diff options
| -rw-r--r-- | Nominal/Basic.lean | 124 |
1 files changed, 115 insertions, 9 deletions
diff --git a/Nominal/Basic.lean b/Nominal/Basic.lean index dde26fc..66f43b1 100644 --- a/Nominal/Basic.lean +++ b/Nominal/Basic.lean @@ -86,14 +86,120 @@ lemma support_subset {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S ⊆ T intro a ha exact hπ a (hsub ha) -#check Set.Finite.exists_injOn_of_encard_le -#check Set.injOn_iff_injective -#check Function.partialInv_of_injective - -lemma support_inter {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S.Finite → HasSupport 𝔸 x T → T.Finite → HasSupport 𝔸 x (S ∩ T) := by - sorry +/-- For two finite supports, the intersection is also a support. -/ +lemma support_inter [Infinite 𝔸] {x : X} {S T : Set 𝔸} : HasSupport 𝔸 x S → S.Finite → HasSupport 𝔸 x T → T.Finite → HasSupport 𝔸 x (S ∩ T) := by + intro hsuppS hfinS hsuppT hfinT π hfixπ + have h_cod_inf : ((π.val.supp ∪ S ∪ T)ᶜ).Infinite := by + apply Set.Finite.infinite_compl + simp; exact ⟨⟨π.val_prop, hfinS⟩, hfinT⟩ + have h_dom_fin : (S \ T).Finite := Set.Finite.diff hfinS + have hle : (S \ T).encard ≤ ((π.val.supp ∪ S ∪ T)ᶜ).encard := by + rw [Set.Infinite.encard_eq h_cod_inf, Set.Finite.encard_eq_coe h_dom_fin] + simp + obtain ⟨σ, ⟨hpreσ, hinjσ⟩⟩ := Set.Finite.exists_injOn_of_encard_le h_dom_fin hle + have himσ {a : 𝔸} : a ∈ S \ T → σ a ∉ S \ T := by + intro ha + suffices : σ a ∈ (π.val.supp ∪ S ∪ T)ᶜ + · have h : S \ T ⊆ π.val.supp ∪ S ∪ T := by calc + S \ T ⊆ S := Set.diff_subset + _ ⊆ π.val.supp ∪ S := Set.subset_union_right + _ ⊆ π.val.supp ∪ S ∪ T := Set.subset_union_left + exact Set.notMem_subset h (hpreσ ha) + · rw [← Set.image_subset_iff] at hpreσ + exact hpreσ (Set.mem_image_of_mem σ ha) + have himσ' {a : 𝔸} : a ∈ S \ T → σ a ∉ π.val.supp := by + intro ha + suffices : σ a ∈ (π.val.supp ∪ S ∪ T)ᶜ + · have h : π.val.supp ⊆ π.val.supp ∪ S ∪ T := by calc + π.val.supp ⊆ π.val.supp ∪ S := Set.subset_union_left + _ ⊆ π.val.supp ∪ S ∪ T := Set.subset_union_left + exact Set.notMem_subset h (hpreσ ha) + · rw [← Set.image_subset_iff] at hpreσ + exact hpreσ (Set.mem_image_of_mem σ ha) + rw [Set.injOn_iff_injective] at hinjσ + obtain hσinv := Function.partialInv_of_injective hinjσ + have {a : 𝔸} : Decidable (a ∈ S \ T) := Classical.propDecidable (a ∈ S \ T) + let τ : 𝔸 → 𝔸 := fun a => + if h : a ∈ S \ T then (S \ T).restrict σ ⟨a, h⟩ + else if let .some b := ((S \ T).restrict σ).partialInv a then b + else a + have hselfinvτ : ∀ a, τ (τ a) = a := by + intro a + by_cases h₁ : a ∈ S \ T + · have h₂ : ((S \ T).restrict σ).partialInv (((S \ T).restrict σ) ⟨a, h₁⟩) = .some ⟨a, h₁⟩ := + Function.partialInv_left hinjσ ⟨a, h₁⟩ + simp at h₂ + simp [τ, h₁, himσ h₁, h₂] + · cases h₂ : ((S \ T).restrict σ).partialInv a with + | some b => simp [τ, h₂, h₁] + rwa [← Set.restrict_apply σ (S \ T) b, ← hσinv] + | none => simp [τ, h₂, h₁] + have hfinsuppτ : {a | τ a ≠ a}.Finite := by + apply Set.Finite.subset + · exact Set.finite_union.mpr ⟨h_dom_fin, Set.Finite.image σ h_dom_fin⟩ + · intro a ha₁; by_contra ha₂ + rw [← Set.mem_compl_iff, Set.compl_union, Set.mem_inter_iff] at ha₂ + suffices τ a = a by contradiction + have ha₃ : a ∉ S \ T := by rw [← Set.mem_compl_iff]; exact ha₂.left + cases ha₄ : ((S \ T).restrict σ).partialInv a with + | some b => rw [hσinv, Set.restrict_apply] at ha₄ + have : a ∈ (σ '' (S \ T)) := by rw [← ha₄, Set.mem_image]; use b; simp + have : a ∉ (σ '' (S \ T)) := Set.notMem_of_mem_diff ha₂ + contradiction + | none => simp [τ, ha₃, ha₄] + let τ' : FPerm 𝔸 := ⟨⟨τ, τ, hselfinvτ, hselfinvτ⟩, hfinsuppτ⟩ + have h₁ : τ' ∈ fix 𝔸 T := by + intro a haT; show τ a = a + have ha₁ : a ∉ S \ T := Set.notMem_diff_of_mem haT + have h₁ : σ '' (S \ T) ⊆ (π.val.supp ∪ S ∪ T)ᶜ := by rwa [Set.image_subset_iff] + have h₂ : a ∉ (π.val.supp ∪ S ∪ T)ᶜ := by + rw [Set.notMem_compl_iff] + apply Set.mem_union_right; assumption + cases ha₂ : ((S \ T).restrict σ).partialInv a with + | some b => rw [hσinv, Set.restrict_apply] at ha₂ + have : a ∈ (σ '' (S \ T)) := by rw [← ha₂, Set.mem_image]; use b; simp + have : a ∉ (σ '' (S \ T)) := fun x => h₂ (h₁ x) + contradiction + | none => simp [τ, ha₁, ha₂] + have h₂ : (τ' * π * τ') ∈ fix 𝔸 S := by + intro a haS; show τ (π (τ a)) = a + by_cases haT : a ∈ T + · have ht : τ a = a := by + -- XXX: copied from above + have ha₁ : a ∉ S \ T := Set.notMem_diff_of_mem haT + have h₁ : σ '' (S \ T) ⊆ (π.val.supp ∪ S ∪ T)ᶜ := by rwa [Set.image_subset_iff] + have h₂ : a ∉ (π.val.supp ∪ S ∪ T)ᶜ := by + rw [Set.notMem_compl_iff] + apply Set.mem_union_right; assumption + cases ha₂ : ((S \ T).restrict σ).partialInv a with + | some b => rw [hσinv, Set.restrict_apply] at ha₂ + have : a ∈ (σ '' (S \ T)) := by rw [← ha₂, Set.mem_image]; use b; simp + have : a ∉ (σ '' (S \ T)) := fun x => h₂ (h₁ x) + contradiction + | none => simp [τ, ha₁, ha₂] + calc + τ (π (τ a)) = τ (π a) := by rw [ht] + _ = τ (π • a) := rfl + _ = τ a := by rw [hfixπ a (Set.mem_inter haS haT)] + _ = a := by rw [ht] + · have h₁ : a ∈ S \ T := Set.mem_diff_of_mem haS haT + have h₂ : π (τ a) = τ a := by + apply Equiv.Perm.supp_notin + rw [show τ a = σ a by simp [τ, h₁]] + exact himσ' h₁ + rw [h₂] + exact hselfinvτ a + calc + π • x = (1 * π * 1) • x := rfl + _ = ((τ' * τ'⁻¹) * π * (τ' * τ'⁻¹)) • x := by rw [@mul_inv_cancel] + _ = ((τ' * τ' * π * τ') * τ') • x := rfl + _ = (τ' * τ' * π * τ') • x := by rw [@mul_smul, hsuppT h₁] + _ = (τ' * (τ' * π * τ')) • x := rfl + _ = τ' • x := by rw [@mul_smul, hsuppS h₂] + _ = x := by rw [hsuppT h₁] -lemma support_sInter {x : X} {C : Set (Set 𝔸)}: C.Finite → C.Nonempty → (∀ S ∈ C, HasSupport 𝔸 x S ∧ S.Finite) → HasSupport 𝔸 x (⋂₀ C) := by +/-- For a finite number of finite supports, the intersection is also a support. -/ +lemma support_sInter [Infinite 𝔸] {x : X} {C : Set (Set 𝔸)}: C.Finite → C.Nonempty → (∀ S ∈ C, HasSupport 𝔸 x S ∧ S.Finite) → HasSupport 𝔸 x (⋂₀ C) := by suffices (C : Finset (Set 𝔸)) : C.Nonempty → (∀ S ∈ C, HasSupport 𝔸 x S ∧ S.Finite) → HasSupport 𝔸 x (⋂₀ C.toSet) ∧ (⋂₀ C.toSet).Finite · intro hfin rw [show C = hfin.toFinset.toSet by simp] @@ -106,7 +212,7 @@ lemma support_sInter {x : X} {C : Set (Set 𝔸)}: C.Finite → C.Nonempty → ( apply hS simp · intro S C' hS hC' hsing - rw [show (⋂₀ (insert S C' : Finset (Set 𝔸))) = S ∩ (⋂₀ C') by aesop] + rw [Finset.coe_insert S C', Set.sInter_insert S ↑C'] have hother : (∀ S ∈ C', HasSupport 𝔸 x S ∧ S.Finite) := by intro S' hS' apply hsing @@ -152,7 +258,7 @@ lemma supp_finite {x : X} : (supp 𝔸 x).Finite := by · grind /-- The least finite support is itself a support. -/ -lemma supp_is_support {x : X} : PermSet.HasSupport 𝔸 x (supp 𝔸 x) := by +lemma supp_is_support [Infinite 𝔸] {x : X} : PermSet.HasSupport 𝔸 x (supp 𝔸 x) := by obtain ⟨S, ⟨hsuppS, hfinS⟩⟩ := nom.finsupp x have suppf : (supp 𝔸 x = ⋂₀ { T : Set 𝔸 | T ⊆ S ∧ PermSet.HasSupport 𝔸 x T ∧ T.Finite }) := by unfold supp; ext a |