Mercurial > hg > Members > kono > Proof > FirstOrder
view first-order.agda @ 3:9633bb018116
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 13 Aug 2020 11:07:35 +0900 |
parents | 08f8256a6b11 |
children | 8e4a4d27c621 |
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module first-order where open import Data.List hiding (all ; and ; or ) open import Data.Maybe open import Data.Bool hiding ( not ) open import Relation.Nullary hiding (¬_) open import Relation.Binary.PropositionalEquality hiding ( [_] ) data Term : Set where X : Term Y : Term Z : Term a : Term b : Term c : Term f : Term g : Term h : Term p : Term q : Term r : Term _⟨_⟩ : Term → Term → Term _,_ : Term → Term → Term _/\_ : Term → Term → Term _\/_ : Term → Term → Term ¬ : Term → Term all_=>_ : Term → Term → Term ∃_=>_ : Term → Term → Term module Logic (Const Func Var Pred : Term → Set) where data Expr : Term → Set where var : {v : Term} → Var v → Expr v func : {x args : Term} → (f : Func x) → Expr args → Expr ( x ⟨ args ⟩ ) const : {c : Term} → Const c → Expr c args : {x y : Term} → (ex : Expr x) → (ey : Expr y) → Expr ( x , y ) data Statement : Term → Set where atom : {x : Term } → Pred x → Statement ( x ) predicate : {p args : Term } → Pred p → Expr args → Statement ( p ⟨ args ⟩ ) and : {x y : Term } → Statement x → Statement y → Statement ( x /\ y ) or : {x y : Term } → Statement x → Statement y → Statement ( x \/ y ) not : {x : Term } → Statement x → Statement ( ¬ x ) All : {x t : Term} → Var x → Statement t → Statement ( all x => t ) Exist : {x t : Term} → Var x → Statement t → Statement ( ∃ x => t ) data Kind : Set where pred : Kind const : Kind var : Kind func : Kind conn : Kind arg : Kind args : Kind data Kinds : Term → Kind → Set where kX : Kinds X var kY : Kinds Y var kZ : Kinds Z var ka : Kinds a const kb : Kinds b const kc : Kinds c const kf : Kinds f func kg : Kinds g func kh : Kinds h func kp : Kinds p pred kq : Kinds q pred kr : Kinds r pred karg : (x y : Term ) → Kinds (x ⟨ y ⟩ ) arg kargs : (x y : Term ) → Kinds (x , y ) args kand : (x y : Term ) → Kinds (x /\ y ) conn kor : (x y : Term ) → Kinds (x \/ y ) conn knot : (x : Term ) → Kinds (¬ x ) conn kall_ : (x y : Term ) → Kinds (all x => y ) conn kexit : (x y : Term ) → Kinds (∃ x => y ) conn kindOf : Term → Kind kindOf X = var kindOf Y = var kindOf Z = var kindOf a = const kindOf b = const kindOf c = const kindOf f = func kindOf g = func kindOf h = func kindOf p = pred kindOf q = pred kindOf r = pred kindOf (t ⟨ t₁ ⟩) = arg kindOf (t , t₁) = args kindOf (t /\ t₁) = conn kindOf (t \/ t₁) = conn kindOf (¬ t) = conn kindOf (all t => t₁) = conn kindOf (∃ t => t₁) = conn Const : Term → Set Const x = Kinds x const Func : Term → Set Func x = Kinds x func Var : Term → Set Var x = Kinds x var Pred : Term → Set Pred x = Kinds x pred ex1 : Term ex1 = ¬ ( p /\ ( all X => ( p ⟨ f ⟨ X ⟩ ⟩ ))) open Logic Const Func Var Pred parse-arg : (t : Term ) → Maybe (Expr t ) parse-arg X = just (var kX) parse-arg Y = just (var kY) parse-arg Z = just (var kZ) parse-arg a = just (const ka) parse-arg b = just (const kb) parse-arg c = just (const kc) parse-arg (f ⟨ x ⟩) with parse-arg x parse-arg (f ⟨ x ⟩) | just pt = just ( func kf pt ) parse-arg (f ⟨ x ⟩) | nothing = nothing parse-arg (g ⟨ x ⟩) with parse-arg x parse-arg (g ⟨ x ⟩) | just pt = just ( func kg pt ) parse-arg (g ⟨ x ⟩) | nothing = nothing parse-arg (h ⟨ x ⟩) with parse-arg x parse-arg (h ⟨ x ⟩) | just pt = just ( func kh pt ) parse-arg (h ⟨ x ⟩) | nothing = nothing parse-arg (_ ⟨ x ⟩) = nothing parse-arg (t , t₁) with parse-arg t | parse-arg t₁ ... | just x | just y = just ( args x y ) ... | _ | _ = nothing parse-arg _ = nothing parse : (t : Term ) → Maybe (Statement t ) parse p = just ( atom kp ) parse q = just ( atom kq ) parse r = just ( atom kr ) parse (p ⟨ x ⟩) with parse-arg x parse (p ⟨ x ⟩) | just y = just ( predicate kp y ) parse (p ⟨ x ⟩) | nothing = nothing parse (q ⟨ x ⟩) with parse-arg x parse (q ⟨ x ⟩) | just y = just ( predicate kq y ) parse (q ⟨ x ⟩) | nothing = nothing parse (r ⟨ x ⟩) with parse-arg x parse (r ⟨ x ⟩) | just y = just ( predicate kr y ) parse (r ⟨ x ⟩) | nothing = nothing parse (_ ⟨ x ⟩) = nothing parse (t /\ t₁) with parse t | parse t₁ parse (t /\ t₁) | just p₁ | just p₂ = just ( and p₁ p₂ ) parse (t /\ t₁) | _ | _ = nothing parse (t \/ t₁) with parse t | parse t₁ parse (t \/ t₁) | just p₁ | just p₂ = just ( or p₁ p₂ ) parse (t \/ t₁) | _ | _ = nothing parse (¬ t) with parse t parse (¬ t) | just tx = just ( not tx ) parse (¬ t) | _ = nothing parse (all X => t₁) with parse t₁ ... | just tx = just ( All kX tx ) ... | _ = nothing parse (all Y => t₁) with parse t₁ ... | just tx = just ( All kY tx ) ... | _ = nothing parse (all Z => t₁) with parse t₁ ... | just tx = just ( All kZ tx ) ... | _ = nothing parse (∃ X => t₁) with parse t₁ ... | just tx = just ( Exist kX tx ) ... | _ = nothing parse (∃ Y => t₁) with parse t₁ ... | just tx = just ( Exist kY tx ) ... | _ = nothing parse (∃ Z => t₁) with parse t₁ ... | just tx = just ( Exist kZ tx ) ... | _ = nothing parse _ = nothing ex2 = parse ex1