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+{-# OPTIONS --prop --rewriting #-}
+
+module Examples.Sequence.DerivedFormsRBT where
+
+open import Algebra.Cost
+
+parCostMonoid = ℕ²-ParCostMonoid
+open ParCostMonoid parCostMonoid
+
+open import Calf costMonoid
+open import Calf.Parallel parCostMonoid
+
+open import Calf.Data.Nat
+open import Calf.Data.List
+open import Calf.Data.Product
+-- open import Calf.Data.Sum
+open import Calf.Data.IsBounded costMonoid
+open import Calf.Data.IsBoundedG costMonoid
+-- open import Data.Product hiding (map)
+-- open import Data.List as List hiding (sum; map)
+-- import Data.List.Properties as List
+open import Data.Nat as Nat using (_+_; _*_; _<_; _>_; _≤ᵇ_; _<ᵇ_; ⌊_/2⌋; _≡ᵇ_; _∸_)
+import Data.Nat.Properties as Nat
+-- open import Data.Nat.Logarithm
+
+open import Level using (0ℓ)
+open import Function using (_$_)
+open import Relation.Nullary
+open import Relation.Binary
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; _≢_; module ≡-Reasoning; ≢-sym)
+
+open import Examples.Sequence.RedBlackTree
+
+
+module MapReduce {A B : tp⁺} where
+  mapreduce : cmp $
+    Π color λ y → Π nat λ n → Π (list A) λ l → Π (irbt A y n l) λ _ →
+    Π (U (Π A λ _ → F B)) λ _ →
+    Π (U (Π B λ _ → Π B λ _ → F B)) λ _ →
+    Π B λ _ →
+    F B
+  mapreduce .black .zero .[] leaf f g z = ret z
+  mapreduce .red n l (red t₁ a t₂) f g z =
+    bind (F B)
+      ((mapreduce _ _ _ t₁ f g z) ∥ (mapreduce _ _ _ t₂ f g z)) λ s →
+        bind (F B) (f a) (λ b →
+          bind (F B) (g b (proj₂ s)) (λ s₃ →
+            bind (F B) (g (proj₁ s) s₃) ret))
+  mapreduce .black n@(suc n') l (black t₁ a t₂) f g z =
+    bind (F B)
+      ((mapreduce _ _ _ t₁ f g z) ∥ (mapreduce _ _ _ t₂ f g z)) λ s →
+        bind (F B) (f a) (λ b →
+          bind (F B) (g b (proj₂ s)) (λ s₃ →
+            bind (F B) (g (proj₁ s) s₃) ret))
+
+  mapreduce/cost/work : val color → val nat → val nat
+  mapreduce/cost/work red n = 12 * (4 ^ (n ∸ 1)) ∸ 3
+  mapreduce/cost/work black n = 6 * (4 ^ (n ∸ 1)) ∸ 3
+
+  mapreduce/cost/work' : val nat → val nat
+  mapreduce/cost/work' n = 12 * (4 ^ (n ∸ 1)) ∸ 3
+
+  mapreduce/cost/work≤mapreduce/cost/work' : (y : val color) → (n : val nat) → mapreduce/cost/work y n Nat.≤ mapreduce/cost/work' n
+  mapreduce/cost/work≤mapreduce/cost/work' red n = Nat.≤-refl
+  mapreduce/cost/work≤mapreduce/cost/work' black n =
+    Nat.∸-monoˡ-≤ {n = 12 * (4 ^ (n ∸ 1))} 3
+      (Nat.*-monoˡ-≤ (4 ^ (n ∸ 1)) {y = 12} (Nat.s≤s (Nat.s≤s (Nat.s≤s (Nat.s≤s (Nat.s≤s (Nat.s≤s Nat.z≤n)))))))
+
+  mapreduce/cost/span : val color → val nat → val nat
+  mapreduce/cost/span red n = 6 * n
+  mapreduce/cost/span black n = 6 * n ∸ 3
+
+  mapreduce/cost/span' : val nat → val nat
+  mapreduce/cost/span' n = 6 * n
+
+  mapreduce/cost/span≤mapreduce/cost/span' : (y : val color) → (n : val nat) → mapreduce/cost/span y n Nat.≤ mapreduce/cost/span' n
+  mapreduce/cost/span≤mapreduce/cost/span' red n = Nat.≤-refl
+  mapreduce/cost/span≤mapreduce/cost/span' black n = Nat.m∸n≤m (6 * n) 3
+
+  mapreduce/is-bounded :
+    (f : cmp (Π A λ _ → F B)) →
+    ({x : val A} → IsBounded B (f x) (1 , 1)) →
+    (g : cmp (Π B λ _ → Π B λ _ → F B)) →
+    ({x y : val B} → IsBounded B (g x y) (1 , 1)) →
+    (z : val B) →
+    (y : val color) → (n : val nat) → (l : val (list A)) → (t : val (irbt A y n l)) →
+    IsBounded B (mapreduce y n l t f g z) (mapreduce/cost/work y n , mapreduce/cost/span y n)
+  mapreduce/is-bounded f f-bound g g-bound z .black .zero .[] leaf =
+    step⋆-mono-≤⁻ {c' = 3 , 0} (Nat.z≤n , Nat.z≤n)
+  mapreduce/is-bounded f f-bound g g-bound z .red n l (red t₁ a t₂) =
+    let open ≤⁻-Reasoning cost in
+      begin
+        bind cost (
+          bind (F B)
+            ((mapreduce _ _ _ t₁ f g z) ∥ (mapreduce _ _ _ t₂ f g z)) λ _ →
+              bind (F B) (f a) (λ _ →
+                bind (F B) (g _ _) (λ _ →
+                  bind (F B) (g _ _) ret)))
+          (λ _ → ret triv)
+      ≡⟨⟩
+        (
+          bind cost
+            ((mapreduce _ _ _ t₁ f g z) ∥ (mapreduce _ _ _ t₂ f g z)) λ (s₁ , s₂) →
+              bind cost (f a) λ b →
+                bind cost (g b s₂) λ s₃ →
+                  bind cost (g s₁ s₃) λ _ →
+                    ret triv
+        )
+      ≤⟨ ≤⁻-mono (λ e →
+          bind cost
+            ((mapreduce _ _ _ t₁ f g z) ∥ (mapreduce _ _ _ t₂ f g z)) λ (s₁ , s₂) →
+              bind cost (f a) λ b →
+                bind cost (g b s₂) λ s₃ →
+                  bind cost e λ _ →
+                    ret triv) {! g-bound  !} ⟩
+       (
+          bind cost
+            ((mapreduce _ _ _ t₁ f g z) ∥ (mapreduce _ _ _ t₂ f g z)) λ (s₁ , s₂) →
+              bind cost (f a) λ b →
+                bind cost (g b s₂) λ s₃ →
+                  step⋆ (1 , 1)
+        )
+      ≤⟨ {!   !} ⟩
+        {!   !}
+      ∎
+  mapreduce/is-bounded f f-bound g g-bound z .black n@(suc n') l (black t₁ a t₂) =
+    let open ≤⁻-Reasoning cost in
+      begin
+        {!   !}
+      ≤⟨ {!   !} ⟩
+        {!   !}
+      ∎
+
+  mapreduce/is-bounded' :
+    (f : cmp (Π A λ _ → F B)) →
+    ({x : val A} → IsBounded B (f x) (1 , 1)) →
+    (g : cmp (Π B λ _ → Π B λ _ → F B)) →
+    ({x y : val B} → IsBounded B (g x y) (1 , 1)) →
+    (z : val B) →
+    (y : val color) → (n : val nat) → (l : val (list A)) → (t : val (irbt A y n l)) →
+    IsBounded B (mapreduce y n l t f g z) (mapreduce/cost/work' n , mapreduce/cost/span' n)
+  mapreduce/is-bounded' f f-bound g g-bound z y n l t =
+    let open ≤⁻-Reasoning cost in
+      begin
+        bind cost (mapreduce y n l t f g z) (λ _ → ret triv)
+      ≤⟨ mapreduce/is-bounded f f-bound g g-bound z y n l t ⟩
+        step⋆ (mapreduce/cost/work y n , mapreduce/cost/span y n)
+      ≤⟨ step⋆-mono-≤⁻ (mapreduce/cost/work≤mapreduce/cost/work' y n , mapreduce/cost/span≤mapreduce/cost/span' y n) ⟩
+        step⋆ (mapreduce/cost/work' n , mapreduce/cost/span' n)
+      ∎