Prove that it's a least fixed point

Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>

Lattice.agda

@@ -498,7 +498,7 @@ module FixedPoint {a} {A : Set a}
 
         -- using 'g', for gas, here helps make sure the function terminates.
         -- since A forms a fixed-height lattice, we must find a solution after
-        -- enough 'h' steps at most. Gas is set up such that as soon as it runs
+        -- 'h' steps at most. Gas is set up such that as soon as it runs
         -- out, we have exceeded h steps, which shouldn't be possible.
 
         doStep : ∀ (g hᶜ : ℕ) (a₁ a₂ : A) (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ ≡ suc h) (a₂≼fa₂ : a₂ ≼ f a₂) → Σ A (λ a → a ≈ f a)
@@ -516,3 +516,24 @@ module FixedPoint {a} {A : Set a}
 
         fix : Σ A (λ a → a ≈ f a)
         fix = doStep (suc h) 0 ⊥ᴬ ⊥ᴬ (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))
+
+    aᶠ : A
+    aᶠ = proj₁ fix
+
+    aᶠ≈faᶠ : aᶠ ≈ f aᶠ
+    aᶠ≈faᶠ = proj₂ fix
+
+    private
+        stepPreservesLess : ∀ (g hᶜ : ℕ) (a₁ a₂ a : A) (a≈fa : a ≈ f a) (a₂≼a : a₂ ≼ a)
+                         (c : ChainA.Chain a₁ a₂ hᶜ) (g+hᶜ≡h : g + hᶜ ≡ suc h)
+                         (a₂≼fa₂ : a₂ ≼ f a₂) →
+                         proj₁ (doStep g hᶜ a₁ a₂ c g+hᶜ≡h a₂≼fa₂) ≼ a
+        stepPreservesLess 0 _ _ _ _ _ _ c g+hᶜ≡sh _ rewrite g+hᶜ≡sh = absurd (ChainA.Bounded-suc-n (proj₂ fixedHeight) c)
+        stepPreservesLess (suc g') hᶜ a₁ a₂ a a≈fa a₂≼a c g+hᶜ≡sh a₂≼fa₂ rewrite sym (+-suc g' hᶜ)
+            with ≈-dec a₂ (f a₂)
+        ...   | yes _ = a₂≼a
+        ...   | no  _ = stepPreservesLess g' _ _ _ a a≈fa (≼-cong ≈-refl (≈-sym a≈fa) (Monotonicᶠ a₂≼a)) _ _ _
+
+    aᶠ≼ : ∀ (a : A) → a ≈ f a → aᶠ ≼ a
+    aᶠ≼ a a≈fa = stepPreservesLess (suc h) 0 ⊥ᴬ ⊥ᴬ a a≈fa (⊥ᴬ≼ a) (ChainA.done ≈-refl) (+-comm (suc h) 0) (⊥ᴬ≼ (f ⊥ᴬ))
+