Start working on proving 'sufficiency'

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

Language.agda

@@ -3,6 +3,7 @@ module Language where
 open import Language.Base public
 open import Language.Semantics public
 open import Language.Graphs public
+open import Language.Properties public
 
 open import Data.Fin using (Fin; suc; zero)
 open import Data.Fin.Properties as FinProp using (suc-injective)

Language/Graphs.agda

@@ -1,7 +1,6 @@
 module Language.Graphs where
 
 open import Language.Base
-open import Language.Semantics
 
 open import Data.Fin as Fin using (Fin; suc; zero; _↑ˡ_; _↑ʳ_)
 open import Data.Fin.Properties as FinProp using (suc-injective)
@@ -12,8 +11,8 @@ open import Data.Nat as Nat using (ℕ; suc)
 open import Data.Nat.Properties using (+-assoc; +-comm)
 open import Data.Product using (_×_; Σ; _,_)
 open import Data.Vec using (Vec; []; _∷_; lookup; cast; _++_)
-open import Data.Vec.Properties using (cast-is-id; ++-assoc; lookup-++ˡ; cast-sym; ++-identityʳ)
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; refl; subst)
+open import Data.Vec.Properties using (cast-is-id; ++-assoc; lookup-++ˡ; cast-sym; ++-identityʳ; lookup-++ʳ)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; sym; refl; subst; trans)
 
 open import Lattice
 open import Utils using (x∈xs⇒fx∈fxs; _⊗_; _,_)
@@ -57,38 +56,39 @@ record _⊆_ (g₁ g₂ : Graph) : Set where
                     e ListMem.∈ (Graph.edges g₁) →
                     (↑ˡ-Edge e n) ListMem.∈ (subst (λ m → List (Fin m × Fin m)) sg₂≡sg₁+n (Graph.edges g₂))
 
-castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
-castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
+private
+    castᵉ : ∀ {n m : ℕ} .(p : n ≡ m) → (Fin n × Fin n) → (Fin m × Fin m)
+    castᵉ p (idx₁ , idx₂) = (Fin.cast p idx₁ , Fin.cast p idx₂)
 
-↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
-           f ↑ˡ n₁ ↑ˡ n₂ ≡ Fin.cast p (f ↑ˡ (n₁ Nat.+ n₂))
-↑ˡ-assoc zero p = refl
-↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
+    ↑ˡ-assoc : ∀ {s n₁ n₂} (f : Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
+               f ↑ˡ n₁ ↑ˡ n₂ ≡ Fin.cast p (f ↑ˡ (n₁ Nat.+ n₂))
+    ↑ˡ-assoc zero p = refl
+    ↑ˡ-assoc {suc s'} {n₁} {n₂} (suc f') p rewrite ↑ˡ-assoc f' (sym (+-assoc s' n₁ n₂)) = refl
 
-↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
-                ↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ Nat.+ n₂))
-↑ˡ-Edge-assoc (idx₁ , idx₂) p
-    rewrite ↑ˡ-assoc idx₁ p
-    rewrite ↑ˡ-assoc idx₂ p = refl
+    ↑ˡ-Edge-assoc : ∀ {s n₁ n₂} (e : Fin s × Fin s) (p : s Nat.+ (n₁ Nat.+ n₂) ≡ s Nat.+ n₁ Nat.+ n₂) →
+                    ↑ˡ-Edge (↑ˡ-Edge e n₁) n₂ ≡ castᵉ p (↑ˡ-Edge e (n₁ Nat.+ n₂))
+    ↑ˡ-Edge-assoc (idx₁ , idx₂) p
+        rewrite ↑ˡ-assoc idx₁ p
+        rewrite ↑ˡ-assoc idx₂ p = refl
 
-↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s Nat.+ 0 ≡ s) →
-           f ≡ Fin.cast p (f ↑ˡ 0)
-↑ˡ-identityʳ zero p = refl
-↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
+    ↑ˡ-identityʳ : ∀ {s} (f : Fin s) (p : s Nat.+ 0 ≡ s) →
+               f ≡ Fin.cast p (f ↑ˡ 0)
+    ↑ˡ-identityʳ zero p = refl
+    ↑ˡ-identityʳ {suc s'} (suc f') p rewrite sym (↑ˡ-identityʳ f' (+-comm s' 0)) = refl
 
-↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s Nat.+ 0 ≡ s) →
-                    e ≡ castᵉ p (↑ˡ-Edge e 0)
-↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
-    rewrite sym (↑ˡ-identityʳ idx₁ p)
-    rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
+    ↑ˡ-Edge-identityʳ : ∀ {s} (e : Fin s × Fin s) (p : s Nat.+ 0 ≡ s) →
+                        e ≡ castᵉ p (↑ˡ-Edge e 0)
+    ↑ˡ-Edge-identityʳ (idx₁ , idx₂) p
+        rewrite sym (↑ˡ-identityʳ idx₁ p)
+        rewrite sym (↑ˡ-identityʳ idx₂ p) = refl
 
-cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
-               (e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
-               castᵉ p e ListMem.∈ es →
-               e ListMem.∈ subst (λ m → List (Fin m × Fin m)) q es
-cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
-    rewrite FinProp.cast-is-id refl idx₁
-    rewrite FinProp.cast-is-id refl idx₂ = e∈es
+    cast∈⇒∈subst : ∀ {n m : ℕ} (p : n ≡ m) (q : m ≡ n)
+                   (e : Fin n × Fin n) (es : List (Fin m × Fin m)) →
+                   castᵉ p e ListMem.∈ es →
+                   e ListMem.∈ subst (λ m → List (Fin m × Fin m)) q es
+    cast∈⇒∈subst refl refl (idx₁ , idx₂) es e∈es
+        rewrite FinProp.cast-is-id refl idx₁
+        rewrite FinProp.cast-is-id refl idx₂ = e∈es
 
 ⊆-trans : ∀ {g₁ g₂ g₃ : Graph} → g₁ ⊆ g₂ → g₂ ⊆ g₃ → g₁ ⊆ g₃
 ⊆-trans {MkGraph s₁ ns₁ es₁} {MkGraph s₂ ns₂ es₂} {MkGraph s₃ ns₃ es₃}
@@ -128,12 +128,6 @@ instance
 
 open Relaxable {{...}}
 
-relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
-                      g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
-relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
-    rewrite cast-is-id refl (Graph.nodes g₂)
-    with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
-
 pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
 pushBasicBlock bss g =
     ( record
@@ -152,6 +146,9 @@ pushBasicBlock bss g =
       )
     )
 
+pushEmptyBlock : MonotonicGraphFunction Graph.Index
+pushEmptyBlock = pushBasicBlock []
+
 addEdges : ∀ (g : Graph) → List (Graph.Edge g) → Σ Graph (λ g' → g ⊆ g')
 addEdges (MkGraph s ns es) es' =
     ( record
@@ -172,9 +169,6 @@ addEdges (MkGraph s ns es) es' =
         }
     )
 
-pushEmptyBlock : MonotonicGraphFunction Graph.Index
-pushEmptyBlock = pushBasicBlock []
-
 buildCfg : Stmt → MonotonicGraphFunction (Graph.Index ⊗ Graph.Index)
 buildCfg ⟨ bs₁ ⟩ = pushBasicBlock (bs₁ ∷ []) map (λ g idx → (idx , idx))
 buildCfg (s₁ then s₂) =

Language/Properties.agda

@@ -0,0 +1,52 @@
+module Language.Properties where
+
+open import Language.Base
+open import Language.Semantics
+open import Language.Graphs
+open import Language.Traces
+
+open import MonotonicState _⊆_ ⊆-trans renaming (MonotonicState to MonotonicGraphFunction)
+open import Utils using (_⊗_; _,_)
+open Relaxable {{...}}
+
+open import Data.Fin using (zero)
+open import Data.List using (List; _∷_; [])
+open import Data.Vec using (_∷_; [])
+open import Data.Vec.Properties using (cast-is-id; lookup-++ˡ; lookup-++ʳ)
+open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; sym; trans; subst)
+
+relax-preserves-[]≡ : ∀ (g₁ g₂ : Graph) (g₁⊆g₂ : g₁ ⊆ g₂) (idx : Graph.Index g₁) →
+                      g₁ [ idx ] ≡ g₂ [ relax g₁⊆g₂ idx ]
+relax-preserves-[]≡ g₁ g₂ (Mk-⊆ n refl newNodes nsg₂≡nsg₁++newNodes _) idx
+    rewrite cast-is-id refl (Graph.nodes g₂)
+    with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
+
+instance
+    NodeEqualsMonotonic : ∀ {bss : List BasicStmt} →
+                          MonotonicPredicate (λ g n → g [ n ] ≡ bss)
+    NodeEqualsMonotonic = record
+        { relaxPredicate = λ g₁ g₂ idx g₁⊆g₂ g₁[idx]≡bss →
+            trans (sym (relax-preserves-[]≡ g₁ g₂ g₁⊆g₂ idx)) g₁[idx]≡bss
+        }
+
+pushBasicBlock-works : ∀ (bss : List BasicStmt) → Always (λ g idx → g [ idx ] ≡ bss) (pushBasicBlock bss)
+pushBasicBlock-works bss = MkAlways (λ g → lookup-++ʳ (Graph.nodes g) (bss ∷ []) zero)
+
+TransformsEnv : ∀ (ρ₁ ρ₂ : Env) → DependentPredicate (Graph.Index ⊗ Graph.Index)
+TransformsEnv ρ₁ ρ₂ g (idx₁ , idx₂) = Trace {g} idx₁ idx₂ ρ₁ ρ₂
+
+instance
+    TransformsEnvMonotonic : ∀ {ρ₁ ρ₂ : Env} → MonotonicPredicate (TransformsEnv ρ₁ ρ₂)
+    TransformsEnvMonotonic = {!!}
+
+buildCfg-sufficient : ∀ {ρ₁ ρ₂ : Env} {s : Stmt} → ρ₁ , s ⇒ˢ ρ₂ → Always (TransformsEnv ρ₁ ρ₂) (buildCfg s)
+buildCfg-sufficient {ρ₁} {ρ₂} {⟨ bs ⟩} (⇒ˢ-⟨⟩ ρ₁ ρ₂ bs ρ₁,bs⇒ρ₂) =
+    pushBasicBlock-works (bs ∷ [])
+    map-reason
+        (λ g idx g[idx]≡[bs] → Trace-single (subst (ρ₁ ,_⇒ᵇˢ ρ₂)
+                                                   (sym g[idx]≡[bs])
+                                                   (ρ₁,bs⇒ρ₂ ∷ [])))
+buildCfg-sufficient {ρ₁} {ρ₂} {s₁ then s₂} (⇒ˢ-then ρ₁ ρ ρ₂ s₁ s₂ ρ₁,s₁⇒ρ₂ ρ₂,s₂⇒ρ₃) =
+    (buildCfg-sufficient ρ₁,s₁⇒ρ₂ ⟨⊗⟩-reason buildCfg-sufficient ρ₂,s₂⇒ρ₃)
+    update-reason {!!}
+    map-reason {!!}

Language/Semantics.agda

@@ -5,7 +5,7 @@ open import Language.Base
 open import Data.Integer using (ℤ; +_) renaming (_+_ to _+ᶻ_; _-_ to _-ᶻ_)
 open import Data.Product using (_×_; _,_)
 open import Data.String using (String)
-open import Data.List using (List; _∷_)
+open import Data.List as List using (List)
 open import Data.Nat using (ℕ)
 open import Relation.Nullary using (¬_)
 open import Relation.Binary.PropositionalEquality using (_≡_)
@@ -17,8 +17,8 @@ Env : Set
 Env = List (String × Value)
 
 data _∈_ : (String × Value) → Env → Set where
-    here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) ∷ ρ)
-    there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') ∷ ρ)
+    here : ∀ (s : String) (v : Value) (ρ : Env) → (s , v) ∈ ((s , v) List.∷ ρ)
+    there : ∀ (s s' : String) (v v' : Value) (ρ : Env) → ¬ (s ≡ s') → (s , v) ∈ ρ → (s , v) ∈ ((s' , v') List.∷ ρ)
 
 data _,_⇒ᵉ_ : Env → Expr → Value → Set where
     ⇒ᵉ-ℕ : ∀ (ρ : Env) (n : ℕ) → ρ , (# n) ⇒ᵉ (↑ᶻ (+ n))
@@ -33,7 +33,12 @@ data _,_⇒ᵉ_ : Env → Expr → Value → Set where
 data _,_⇒ᵇ_ : Env → BasicStmt → Env → Set where
     ⇒ᵇ-noop : ∀ (ρ : Env) → ρ , noop ⇒ᵇ ρ
     ⇒ᵇ-← : ∀ (ρ : Env) (x : String) (e : Expr) (v : Value) →
-           ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) ∷ ρ)
+           ρ , e ⇒ᵉ v → ρ , (x ← e) ⇒ᵇ ((x , v) List.∷ ρ)
+
+data _,_⇒ᵇˢ_ : Env → List BasicStmt → Env → Set where
+    [] : ∀ {ρ : Env} → ρ , List.[] ⇒ᵇˢ ρ
+    _∷_ : ∀ {ρ₁ ρ₂ ρ₃ : Env} {bs : BasicStmt} {bss : List BasicStmt} →
+          ρ₁ , bs ⇒ᵇ ρ₂ → ρ₂ , bss ⇒ᵇˢ ρ₃ → ρ₁ , (bs List.∷ bss) ⇒ᵇˢ ρ₃
 
 data _,_⇒ˢ_ : Env → Stmt → Env → Set where
     ⇒ˢ-⟨⟩ : ∀ (ρ₁ ρ₂ : Env) (bs : BasicStmt) →

Language/Traces.agda

@@ -0,0 +1,24 @@
+module Language.Traces where
+
+open import Language.Base public
+open import Language.Semantics public
+open import Language.Graphs public
+
+open import Data.Product using (_,_)
+open import Data.List.Membership.Propositional as MemProp using ()
+
+module _ {g : Graph} where
+    open Graph g using (Index; edges)
+
+    data Trace : Index → Index → Env → Env → Set where
+        Trace-single : ∀ {ρ₁ ρ₂ : Env} {idx : Index} →
+                       ρ₁ , (g [ idx ]) ⇒ᵇˢ ρ₂ → Trace idx idx ρ₁ ρ₂
+        Trace-edge : ∀ {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : Index} →
+                     ρ₁ , (g [ idx₁ ]) ⇒ᵇˢ ρ₂ → (idx₁ , idx₂) MemProp.∈ edges →
+                     Trace idx₂ idx₃ ρ₂ ρ₃ → Trace idx₁ idx₃ ρ₁ ρ₃
+
+    _++⟨_⟩_ : ∀ {idx₁ idx₂ idx₃ idx₄ : Index} {ρ₁ ρ₂ ρ₃ : Env} →
+           Trace idx₁ idx₂ ρ₁ ρ₂ → (idx₂ , idx₃) MemProp.∈ edges →
+           Trace idx₃ idx₄ ρ₂ ρ₃ → Trace idx₁ idx₄ ρ₁ ρ₃
+    _++⟨_⟩_ (Trace-single ρ₁⇒ρ₂) idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₂→idx₃ tr
+    _++⟨_⟩_ (Trace-edge ρ₁⇒ρ₂ idx₁→idx' tr') idx₂→idx₃ tr = Trace-edge ρ₁⇒ρ₂ idx₁→idx' (tr' ++⟨ idx₂→idx₃ ⟩ tr)

MonotonicState.agda

@@ -83,9 +83,15 @@ _map_ f fn s = let (s' , (t₁ , s≼s')) = f s in (s' , (fn s' t₁ , s≼s'))
 DependentPredicate : (S → Set s) → Set (lsuc s)
 DependentPredicate T = ∀ (s₁ : S) → T s₁ → Set s
 
-Both : {T₁ T₂ : S → Set s} → DependentPredicate T₁ → DependentPredicate T₂ →
-       DependentPredicate (T₁ ⊗ T₂)
-Both P Q = (λ { s (t₁ , t₂) → (P s t₁ × Q s t₂) })
+data Both {T₁ T₂ : S → Set s}
+          (P : DependentPredicate T₁)
+          (Q : DependentPredicate T₂) : DependentPredicate (T₁ ⊗ T₂) where
+    MkBoth : ∀ {s : S} {t₁ : T₁ s} {t₂ : T₂ s} → P s t₁ → Q s t₂ → Both P Q s (t₁ , t₂)
+
+data And {T : S → Set s}
+         (P : DependentPredicate T)
+         (Q : DependentPredicate T) : DependentPredicate T where
+    MkAnd : ∀ {s : S} {t : T s} → P s t → Q s t → And P Q s t
 
 -- Since monotnic functions keep adding on to the state, proofs of
 -- predicates over their outputs go stale fast (they describe old values of
@@ -97,20 +103,82 @@ record MonotonicPredicate {T : S → Set s} {{ r : Relaxable T }} (P : Dependent
     field relaxPredicate : ∀ (s₁ s₂ : S) (t₁ : T s₁) (s₁≼s₂ : s₁ ≼ s₂) →
                            P s₁ t₁ → P s₂ (Relaxable.relax r s₁≼s₂ t₁)
 
+instance
+    BothMonotonic : ∀ {T₁ : S → Set s} {T₂ : S → Set s}
+                    {{ _ : Relaxable T₁ }} {{ _ : Relaxable T₂ }}
+                    {P : DependentPredicate T₁} {Q : DependentPredicate T₂}
+                    {{_ : MonotonicPredicate P}} {{_ : MonotonicPredicate Q}} →
+                    MonotonicPredicate (Both P Q)
+    BothMonotonic {{_}} {{_}} {{P-Mono}} {{Q-Mono}} = record
+        { relaxPredicate = (λ { s₁ s₂ (t₁ , t₂) s₁≼s₂ (MkBoth p q) →
+            MkBoth (MonotonicPredicate.relaxPredicate P-Mono s₁ s₂ t₁ s₁≼s₂ p)
+                   (MonotonicPredicate.relaxPredicate Q-Mono s₁ s₂ t₂ s₁≼s₂ q)})
+        }
+
+    AndMonotonic : ∀ {T : S → Set s} {{ _ : Relaxable T }}
+                   {P : DependentPredicate T} {Q : DependentPredicate T}
+                   {{_ : MonotonicPredicate P}} {{_ : MonotonicPredicate Q}} →
+                   MonotonicPredicate (And P Q)
+    AndMonotonic {{_}} {{P-Mono}} {{Q-Mono}} = record
+        { relaxPredicate = (λ { s₁ s₂ t s₁≼s₂ (MkAnd p q) →
+            MkAnd (MonotonicPredicate.relaxPredicate P-Mono s₁ s₂ t s₁≼s₂ p)
+                   (MonotonicPredicate.relaxPredicate Q-Mono s₁ s₂ t s₁≼s₂ q)})
+        }
+
 -- A MonotonicState "monad" m has a certain property if its ouputs satisfy that
 -- property for all inputs.
 
-always : ∀ {T : S → Set s} → DependentPredicate T → MonotonicState T → Set s
-always P m = ∀ s₁ → let (s₂ , t , _) = m s₁ in P s₂ t
+data Always {T : S → Set s} (P : DependentPredicate T) (m : MonotonicState T) : Set s where
+    MkAlways : (∀ s₁ → let (s₂ , t , _) = m s₁ in P s₂ t) → Always P m
 
-⟨⊗⟩-reason : ∀ {T₁ T₂ : S → Set s} {{ _ : Relaxable T₁ }}
+infixr 4 _⟨⊗⟩-reason_
+_⟨⊗⟩-reason_ : ∀ {T₁ T₂ : S → Set s} {{ _ : Relaxable T₁ }}
              {P : DependentPredicate T₁} {Q : DependentPredicate T₂}
              {{P-Mono : MonotonicPredicate P}}
              {m₁ : MonotonicState T₁} {m₂ : MonotonicState T₂} →
-             always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂)
-⟨⊗⟩-reason {{P-Mono = P-Mono}} {m₁ = m₁} {m₂ = m₂} aP aQ s
-    with p ← aP s
-    with (s' , (t₁ , s≼s')) ← m₁ s
-    with q ← aQ s'
-    with (s'' , (t₂ , s'≼s'')) ← m₂ s' =
-    (MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p , q)
+             Always P m₁ → Always Q m₂ → Always (Both P Q) (m₁ ⟨⊗⟩ m₂)
+_⟨⊗⟩-reason_ {P = P} {Q = Q} {{P-Mono = P-Mono}} {m₁ = m₁} {m₂ = m₂} (MkAlways aP) (MkAlways aQ) =
+    MkAlways impl
+    where
+        impl : ∀ s₁ → let (s₂ , t , _) = (m₁ ⟨⊗⟩ m₂) s₁ in (Both P Q) s₂ t
+        impl s
+            with p ← aP s
+            with (s' , (t₁ , s≼s')) ← m₁ s
+            with q ← aQ s'
+            with (s'' , (t₂ , s'≼s'')) ← m₂ s' =
+            MkBoth (MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p) q
+
+infixl 4 _update-reason_
+_update-reason_ : ∀ {T : S → Set s} {{ r : Relaxable T }} →
+                  {P : DependentPredicate T} {Q : DependentPredicate T}
+                  {{P-Mono : MonotonicPredicate P}}
+                  {m : MonotonicState T} {mod : ∀ (s : S) → T s → Σ S (λ s' → s ≼ s')} →
+                  Always P m → (∀ (s : S) (t : T s) →
+                                let (s' , s≼s') = mod s t
+                                in P s t → Q s' (Relaxable.relax r s≼s' t)) →
+                  Always (And P Q) (m update mod)
+_update-reason_ {{r = r}} {P = P} {Q = Q} {{P-Mono = P-Mono}} {m = m} {mod = mod} (MkAlways aP) modQ =
+    MkAlways impl
+    where
+        impl : ∀ s₁ → let (s₂ , t , _) = (m update mod) s₁ in (And P Q) s₂ t
+        impl s
+            with p ← aP s
+            with (s' , (t , s≼s')) ← m s
+            with q ← modQ s' t p
+            with (s'' , s'≼s'') ← mod s' t =
+            MkAnd (MonotonicPredicate.relaxPredicate P-Mono _ _ _ s'≼s'' p) q
+
+infixl 4 _map-reason_
+_map-reason_ : ∀ {T₁ T₂ : S → Set s}
+               {P : DependentPredicate T₁} {Q : DependentPredicate T₂}
+               {m : MonotonicState T₁}
+               {f : ∀ (s : S) → T₁ s → T₂ s} →
+               Always P m → (∀ (s : S) (t₁ : T₁ s) → P s t₁ → Q s (f s t₁)) →
+               Always Q (m map f)
+_map-reason_ {P = P} {Q = Q} {m = m} {f = f} (MkAlways aP) P⇒Q =
+    MkAlways impl
+    where
+        impl : ∀ s₁ → let (s₂ , t , _) = (m map f) s₁ in Q s₂ t
+        impl s
+            with p ← aP s
+            with (s' , (t₁ , s≼s')) ← m s = P⇒Q s' t₁ p