Start formalizing monotonic function predicates
Signed-off-by: Danila Fedorin <danila.fedorin@gmail.com>
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@ -209,9 +209,16 @@ module Graphs where
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rewrite cast-is-id refl (Graph.nodes g₂)
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with refl ← nsg₂≡nsg₁++newNodes = sym (lookup-++ˡ (Graph.nodes g₁) _ _)
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-- Tools for graph construction. The most important is a 'monotonic function':
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-- one that takes a graph, and produces another graph, such that the
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-- new graph includes all the information from the old one.
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MonotonicGraphFunction : (Graph → Set) → Set
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MonotonicGraphFunction T = (g₁ : Graph) → Σ Graph (λ g₂ → T g₂ × g₁ ⊆ g₂)
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-- Now, define some operations on monotonic functions; these are useful
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-- to save the work of threading intermediate graphs in and out of operations.
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infixr 2 _⟨⊗⟩_
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_⟨⊗⟩_ : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }} →
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MonotonicGraphFunction T₁ → MonotonicGraphFunction T₂ →
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@ -236,6 +243,43 @@ module Graphs where
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MonotonicGraphFunction T₂
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_map_ f fn g = let (g' , (t₁ , g⊆g')) = f g in (g' , (fn g' t₁ , g⊆g'))
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-- To reason about monotonic functions and what we do, we need a way
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-- to describe values they produce. A 'graph-value predicate' is
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-- just a predicate for some (dependent) value.
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GraphValuePredicate : (Graph → Set) → Set₁
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GraphValuePredicate T = ∀ (g : Graph) → T g → Set
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Both : {T₁ T₂ : Graph → Set} → GraphValuePredicate T₁ → GraphValuePredicate T₂ →
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GraphValuePredicate (T₁ ⊗ T₂)
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Both P Q = (λ { g (t₁ , t₂) → (P g t₁ × Q g t₂) })
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-- Since monotnic functions keep adding on to a function, proofs of
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-- graph-value predicates go stale fast (they describe old values of
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-- the graph). To keep propagating them through, we need them to still
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-- on 'bigger graphs'. We call such predicates monotonic as well, since
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-- they respect the ordering of graphs.
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MonotonicPredicate : ∀ {T : Graph → Set} {{ TRelaxable : Relaxable T }} →
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GraphValuePredicate T → Set
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MonotonicPredicate {T} P = ∀ (g₁ g₂ : Graph) (t₁ : T g₁) (g₁⊆g₂ : g₁ ⊆ g₂) →
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P g₁ t₁ → P g₂ (relax g₁⊆g₂ t₁)
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-- A 'map' has a certain property if its ouputs satisfy that property
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-- for all inputs.
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always : ∀ {T : Graph → Set} → GraphValuePredicate T → MonotonicGraphFunction T → Set
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always P m = ∀ g₁ → let (g₂ , t , _) = m g₁ in P g₂ t
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⟨⊗⟩-reason : ∀ {T₁ T₂ : Graph → Set} {{ T₁Relaxable : Relaxable T₁ }}
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{P : GraphValuePredicate T₁} {Q : GraphValuePredicate T₂}
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{P-Mono : MonotonicPredicate P}
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{m₁ : MonotonicGraphFunction T₁} {m₂ : MonotonicGraphFunction T₂} →
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always P m₁ → always Q m₂ → always (Both P Q) (m₁ ⟨⊗⟩ m₂)
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⟨⊗⟩-reason = {!!}
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module Construction where
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pushBasicBlock : List BasicStmt → MonotonicGraphFunction Graph.Index
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pushBasicBlock bss g =
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