finish name abstraction proofs

1 files changed, 61 insertions, 18 deletions

diff --git a/Nominal/Instances/Abstraction.lean b/Nominal/Instances/Abstraction.lean
index 660c2ab..bb654ed 100644
--- a/Nominal/Instances/Abstraction.lean
+++ b/Nominal/Instances/Abstraction.lean
@@ -133,15 +133,19 @@ lemma alpha_eq_iff_all {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ ∀ c,
 
 For ⟪a⟫ x = ⟪b⟫ y, we either have (a, x) = (b, y) or x is just (a b) y, requiring that a is not in the support of y. -/
 lemma alpha_eq_iff_cases {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ (x.atom = y.atom ∧ x.body = y.body) ∨ (x.atom ≠ y.atom ∧ x.atom # y.body ∧ x.body = ⸨x.atom, y.atom⸩ • y.body) := by
-  by_cases h_eq : x.atom = y.atom
-  · constructor
-    · sorry
-    · sorry
-  · constructor
-    · rw [alpha_eq_iff_exists]
-      intro h; right
-      obtain ⟨c, ⟨hcx, ⟨hcy, hc⟩⟩⟩ := h
-      constructor
+  constructor
+  · rw [alpha_eq_iff_exists]
+    intro h
+    obtain ⟨c, ⟨hcx, ⟨hcy, hc⟩⟩⟩ := h
+    by_cases h_eq : x.atom = y.atom
+    · left; constructor
+      · exact h_eq
+      · calc
+          x.body = ⸨c, x.atom⸩ • ⸨c, x.atom⸩ • x.body := by rw [← mul_smul, FPerm.swap_mul_self, one_smul]
+          _      = ⸨c, x.atom⸩ • ⸨c, y.atom⸩ • y.body := by rw [hc]
+          _      = ⸨c, y.atom⸩ • ⸨c, y.atom⸩ • y.body := by rw [h_eq]
+          _      = y.body                             := by rw [← mul_smul, FPerm.swap_mul_self, one_smul]
+    · right; constructor
       · exact h_eq
       · simp at hcx
         simp at hcy
@@ -160,24 +164,63 @@ lemma alpha_eq_iff_cases {x y : NameAbstraction' 𝔸 X} : AlphaEq x y ↔ (x.at
             _      = (⸨c, x.atom⸩ * ⸨c, y.atom⸩) • y.body    := by rw [mul_smul, hc]
             _      = ⸨x.atom, y.atom⸩ • ⸨c, x.atom⸩ • y.body := by rw [← mul_smul, h_swap]
             _      = ⸨x.atom, y.atom⸩ • y.body               := by rw [NominalSet.fresh_swap_fix 𝔸 hcy.right h_atom]
-    · rw [alpha_eq_iff_all]
-      intro h c ⟨hcx, hcy⟩
+  · rw [alpha_eq_iff_all]
+    intro h c ⟨hcx, hcy⟩
+    cases h with
+    | inl h =>
+      rw [h.left, h.right]
+    | inr h =>
       simp at hcx
       simp at hcy
-      simp only [h_eq, false_and, ne_eq, not_false_eq_true, true_and, false_or] at h
       have h_swap : ⸨c, x.atom⸩ * ⸨x.atom, y.atom⸩ = ⸨c, y.atom⸩ * ⸨c, x.atom⸩ := by
         ext a
         by_cases a = c <;> by_cases a = x.atom <;> by_cases a = y.atom
-        all_goals simp_all [FPerm.swap_apply_of_ne_of_ne, hcy.left, hcx.left, Ne.symm hcy.left, Ne.symm hcx.left, Ne.symm h_eq]
-      rw [h.right, ← mul_smul, h_swap, mul_smul, NominalSet.fresh_swap_fix 𝔸 hcy.right h.left]
+        all_goals simp_all [FPerm.swap_apply_of_ne_of_ne, hcy.left, hcx.left, Ne.symm hcy.left, Ne.symm hcx.left, Ne.symm h.left]
+      rw [h.right.right, ← mul_smul, h_swap, mul_smul, NominalSet.fresh_swap_fix 𝔸 hcy.right h.right.left]
 
-private lemma alpha_eq_refl (x : NameAbstraction' 𝔸 X) : AlphaEq x x := by sorry
+private lemma alpha_eq_refl (x : NameAbstraction' 𝔸 X) : AlphaEq x x := by
+  rw [alpha_eq_iff_cases]; left
+  trivial
 
-private lemma alpha_eq_symm {x y : NameAbstraction' 𝔸 X} : AlphaEq x y → AlphaEq y x := by sorry
+private lemma alpha_eq_symm {x y : NameAbstraction' 𝔸 X} : AlphaEq x y → AlphaEq y x := by
+  rw [alpha_eq_iff_cases, alpha_eq_iff_cases]
+  intro h
+  cases h with
+  | inl h =>
+    left; exact ⟨h.left.symm, h.right.symm⟩
+  | inr h =>
+    right; constructor
+    · exact h.left.symm
+    · constructor
+      · have h : ⸨x.atom, y.atom⸩ • x.atom # x.body := by
+          rw [h.right.right]
+          exact NominalSet.fresh_equivariant 𝔸 _ _ _ h.right.left
+        simp at h; exact h
+      · rw [h.right.right, ← mul_smul, FPerm.swap_comm, FPerm.swap_mul_self, one_smul]
 
-private lemma alpha_eq_trans {x y z : NameAbstraction' 𝔸 X} : AlphaEq x y → AlphaEq y z → AlphaEq x z := by sorry
+private lemma alpha_eq_trans {x y z : NameAbstraction' 𝔸 X} : AlphaEq x y → AlphaEq y z → AlphaEq x z := by
+  intro hxy hyz
+  have h : AlphaEq x y ∧ AlphaEq y z := ⟨hxy, hyz⟩
+  simp only [AlphaEq, AlphaEq, NominalSet.for_fresh_and] at h
+  apply NominalSet.for_fresh_imp
+  · exact h
+  · intro c ⟨hxy, hyz⟩
+    rw [hxy, hyz]
 
-lemma alpha_eq_equivariant : EquivariantRelation 𝔸 (AlphaEq : NameAbstraction' 𝔸 X → NameAbstraction' 𝔸 X → Prop) := by sorry
+/-- Alpha-equivalence is an equivariant relation. -/
+lemma alpha_eq_equivariant : EquivariantRelation 𝔸 (AlphaEq : NameAbstraction' 𝔸 X → NameAbstraction' 𝔸 X → Prop) := by
+  intro x y π h
+  rw [alpha_eq_iff_exists] at *
+  obtain ⟨c, ⟨hcx, ⟨hcy, hc⟩⟩⟩ := h
+  use π • c
+  constructor
+  · apply NominalSet.fresh_equivariant; exact hcx
+  · constructor
+    · apply NominalSet.fresh_equivariant; exact hcy
+    · -- XXX: This is largely similar to alpha_eq_pred...
+      have h' : π • ⸨c, x.atom⸩ • x.body = π • ⸨c, y.atom⸩ • y.body := by congr
+      simp only [smul_smul, FPerm.mul_swap_eq_swap_mul] at h'
+      simpa [smul_smul]
 
 end NameAbstraction