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2 changed files with 100 additions and 16 deletions
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@ -50,7 +50,7 @@ subst (x ∙ x₁) v = (subst x v) ∙ (subst x₁ v)
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data value-rel : type → term → Set where
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v-`true : value-rel bool `true
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v-`false : value-rel bool `false
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v-`λ[_]_ : ∀ {τ e} → value-rel τ (`λ[ τ ] e)
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v-`λ[_]_ : ∀ {τ₁ τ₂ e} → value-rel (τ₁ -→ τ₂) (`λ[ τ₁ ] e)
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data good-subst : ctx → sub → Set where
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nil : good-subst nil nil
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@ -60,14 +60,20 @@ data good-subst : ctx → sub → Set where
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→ good-subst (cons ctx τ) γ
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data step : term → term → Set where
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step-if-e : ∀ {e e' e₁ e₂} → step e e' → step (`if e then e₁ else e₂) (`if e' then e₁ else e₂)
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step-if-1 : ∀ {e₁ e₂} → step (`if `true then e₁ else e₂) e₁
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step-if-2 : ∀ {e₁ e₂} → step (`if `false then e₁ else e₂) e₂
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step-`λ : ∀ {τ e v} → step ((`λ[ τ ] e) ∙ v) (subst e v)
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step-`λ-1 : ∀ {e₁ e₁' e₂} → step e₁ e₁' → step (e₁ ∙ e₂) (e₁' ∙ e₂)
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step-`λ-2 : ∀ {τ e₁ e₂ e₂'} → step e₂ e₂' → step ((`λ[ τ ] e₁) ∙ e₂) ((`λ[ τ ] e₁) ∙ e₂')
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step-`λ-β : ∀ {τ e v} → step ((`λ[ τ ] e) ∙ v) (subst e v)
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data steps : ℕ → term → term → Set where
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zero : ∀ {e} → steps zero e e
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suc : ∀ {e e' e''} → (n : ℕ) → step e e' → steps n e' e'' → steps (suc n) e e''
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safe : (τ : type) → (e : term) → Set
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safe τ e = ∀ {n} → (e' : term) → steps n e e' → (value-rel τ e') ⊎ (∃ λ e'' → step e' e'')
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data _⊢_∶_ : ctx → term → type → Set where
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type-`true : ∀ {ctx} → ctx ⊢ `true ∶ bool
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type-`false : ∀ {ctx} → ctx ⊢ `false ∶ bool
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@ -79,9 +85,9 @@ data _⊢_∶_ : ctx → term → type → Set where
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type-`x : ∀ {ctx x}
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→ (p : Is-just (lookup ctx x))
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→ ctx ⊢ (` x) ∶ (to-witness p)
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type-`λ : ∀ {ctx τ τ₂ e}
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→ (cons ctx τ) ⊢ e ∶ τ₂
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→ ctx ⊢ (`λ[ τ ] e) ∶ (τ -→ τ₂)
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type-`λ : ∀ {ctx τ₁ τ₂ e}
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→ (cons ctx τ₁) ⊢ e ∶ τ₂
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→ ctx ⊢ (`λ[ τ₁ ] e) ∶ (τ₁ -→ τ₂)
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type-∙ : ∀ {ctx τ₁ τ₂ e₁ e₂}
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→ ctx ⊢ e₁ ∶ (τ₁ -→ τ₂)
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→ ctx ⊢ e₂ ∶ τ₂
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@ -90,19 +96,81 @@ data _⊢_∶_ : ctx → term → type → Set where
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irreducible : term → Set
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irreducible e = ¬ (∃ λ e' → step e e')
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data term-relation : type → term → Set where
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data term-rel : type → term → Set where
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e-term : ∀ {τ e}
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→ (∀ {n} → (e' : term) → steps n e e' → irreducible e' → value-rel τ e')
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→ term-relation τ e
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→ term-rel τ e
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type-sound : ∀ {Γ e τ} → Γ ⊢ e ∶ τ → Set
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type-sound {Γ} {e} {τ} s = ∀ {n} → (e' : term) → steps n e e' → value-rel τ e' ⊎ ∃ λ e'' → step e' e''
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type-soundness : ∀ {Γ e τ} → Γ ⊢ e ∶ τ → Set
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type-soundness {Γ} {e} {τ} s = ∀ {n} → (e' : term) → steps n e e' → (value-rel τ e') ⊎ (∃ λ e'' → step e' e'')
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type-sound : (e : term) → (τ : type) → (p : nil ⊢ e ∶ τ) → type-soundness p
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type-sound .`true .bool type-`true .`true zero = inj₁ v-`true
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type-sound .`false .bool type-`false .`false zero = inj₁ v-`false
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type-sound .(`if _ then _ else _) τ (type-`ifthenelse p p₁ p₂) e' steps = {! !}
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type-sound .(`λ[ _ ] _) .(_ -→ _) (type-`λ p) e' steps = {! !}
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type-sound .(_ ∙ _) τ (type-∙ p p₁) e' steps = {! !}
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_⊨_∶_ : (Γ : ctx) → (e : term) → (τ : type) → Set
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_⊨_∶_ Γ e τ = (γ : sub) → (good-subst Γ γ) → term-relation τ e
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_⊨_∶_ Γ e τ = (γ : sub) → (good-subst Γ γ) → term-rel τ e
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fundamental : ∀ {Γ e τ} → (well-typed : Γ ⊢ e ∶ τ) → type-sound well-typed → Γ ⊨ e ∶ τ
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fundamental {Γ} {e} {τ} well-typed type-sound γ good-sub = e-term f
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fundamental : ∀ {Γ e τ} → (well-typed : Γ ⊢ e ∶ τ) → type-soundness well-typed → Γ ⊨ e ∶ τ
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fundamental {Γ} {e} {τ} well-typed tsound γ good-sub = e-term f
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where
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f : {n : ℕ} (e' : term) → steps n e e' → irreducible e' → value-rel τ e'
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f e' steps irred = [ id , (λ exists → ⊥-elim (irred exists)) ] (type-sound e' steps)
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f e' steps irred = [ id , (λ exists → ⊥-elim (irred exists)) ] (tsound e' steps)
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module semantic-type-soundness where
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part1 : (e : term) → (τ : type) → nil ⊢ e ∶ τ → term-rel τ e
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part1 (` x) _ (type-`x ())
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part1 `true bool p = e-term f
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where
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f : {n : ℕ} (e' : term) → steps n `true e' → irreducible e' → value-rel bool e'
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f {.zero} .`true zero irred = v-`true
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part1 `false bool p = e-term f
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where
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f : {n : ℕ} (e' : term) → steps n `false e' → irreducible e' → value-rel bool e'
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f {zero} .`false zero irred = v-`false
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part1 (`λ[ τ₁ ] e) (.τ₁ -→ τ₂) (type-`λ p) = e-term f
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where
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f : {n : ℕ} (e' : term) → steps n (`λ[ τ₁ ] e) e' → irreducible e' → value-rel (τ₁ -→ τ₂) e'
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f .(`λ[ τ₁ ] e) zero irred = v-`λ[_]_
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part1 (`if e then e₁ else e₂) τ (type-`ifthenelse p p₁ p₂) = e-term f
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where
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v : ∀ {n} → (e' : term) → steps n e e' → value-rel bool e' → ∃ (step (`if e then e₁ else e₂))
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v .`true zero v-`true = e₁ , step-if-1
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v .`false zero v-`false = e₂ , step-if-2
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v e' (suc {_} {e''} n step steps) val =
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(`if e'' then e₁ else e₂) , step-if-e step
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f : {n : ℕ} (e' : term) → steps n (`if e then e₁ else e₂) e' → irreducible e' → value-rel τ e'
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f .(`if e then e₁ else e₂) zero irred with part1 e bool p
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f .(`if e then e₁ else e₂) zero irred | e-term f' =
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let
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wtf = v e zero {! !}
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wtf3 = {! irred ? !}
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-- wtf2 = v e wtf
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-- wtf3 = irred wtf2
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-- in ⊥-elim wtf3
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in {! !}
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f e' (suc n (step-if-e step) steps) irred = {! !}
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f e' (suc n step-if-1 steps) irred with part1 e₁ τ p₁
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f e' (suc n step-if-1 steps) irred | e-term f' = f' e' steps irred
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f e' (suc n step-if-2 steps) irred with part1 e₂ τ p₂
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f e' (suc n step-if-2 steps) irred | e-term f' = f' e' steps irred
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part1 (e₁ ∙ e₂) τ₂ (type-∙ {_} {τ₁} p₁ p₂) = e-term f
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where
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f : {n : ℕ} (e' : term) → steps n (e₁ ∙ e₂) e' → irreducible e' → value-rel τ₂ e'
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f .(e₁ ∙ e₂) zero irred = {! !}
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f e' (suc n (step-`λ-1 step) steps) irred with part1 e₁ (τ₁ -→ τ₂) p₁
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f e' (suc _ (step-`λ-1 step) steps) irred | e-term f' = {! !}
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f e' (suc n (step-`λ-2 step) steps) irred = {! !}
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f e' (suc n step-`λ-β steps) irred = {! !}
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part2 : (e : term) → (τ : type) → term-rel τ e → safe τ e
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part2 e τ (e-term p) .e zero =
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inj₁ (p e zero λ q → {! !})
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where
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v : ⊥
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v = {! !}
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part2 e τ (e-term p) e' (suc {f} {g} n step steps) = inj₂ ({! !} , {! step !})
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@ -196,3 +196,19 @@ ret z$
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You can draw a search tree of probabilities. Add up the probabilities to get the probability that a program returns a specific value.
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You can share $z$ since it doesn't depend directly on $x$. This builds a *binary decision diagram*.
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=== Knowledge compilation
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https://en.wikipedia.org/wiki/Knowledge_compilation
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Relationship between hardness of propositional reasoning tasks and its syntax of the formula
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SAT for DNF is easy. What kinds of structure enables efficient reasoning?
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==== Succinctness
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$cal(L)_1$ is more succinct than $cal(L)_2$ if it's efficient (polynomial-time) to translate (in a semantics-preserving way) programs written in $cal(L)_2$ to programs written in $cal(L)_1$
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==== Canonicity of BDDs
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There is only 1 structural BDD for any particular formula.
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