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Some stuff on L&S (#22)
This commit is contained in:
Patrick Stevens
GitHub
16
ExampleSheets/LogicAndSets/Sheet1.agda
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16
ExampleSheets/LogicAndSets/Sheet1.agda
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{-# OPTIONS --safe --warning=error #-}
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Logic.PropositionalLogic
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module ExampleSheets.LogicAndSets.Sheet1 where
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q1i : {a : _} {A : Set a} → (p1 p2 p3 : Propositions A) → Tautology (implies (implies p1 (implies p2 p3)) (implies p2 (implies p1 p3)))
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Tautology.isTaut (q1i p1 p2 p3) {v} with inspect (Valuation.v v p3)
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Tautology.isTaut (q1i p1 p2 p3) {v} | BoolTrue with≡ p3T = Valuation.vImplicationT v (Valuation.vImplicationT v (Valuation.vImplicationT v p3T))
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Tautology.isTaut (q1i p1 p2 p3) {v} | BoolFalse with≡ p3F with inspect (Valuation.v v p2)
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Tautology.isTaut (q1i p1 p2 p3) {v} | BoolFalse with≡ p3F | BoolTrue with≡ p2T with inspect (Valuation.v v p1)
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Tautology.isTaut (q1i p1 p2 p3) {v} | BoolFalse with≡ p3F | BoolTrue with≡ p2T | BoolTrue with≡ p1T = Valuation.vImplicationVacuous v (Valuation.vImplicationF v p1T (Valuation.vImplicationF v p2T p3F))
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Tautology.isTaut (q1i p1 p2 p3) {v} | BoolFalse with≡ p3F | BoolTrue with≡ p2T | BoolFalse with≡ p1F = Valuation.vImplicationT v (Valuation.vImplicationT v (Valuation.vImplicationVacuous v p1F))
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Tautology.isTaut (q1i p1 p2 p3) {v} | BoolFalse with≡ p3F | BoolFalse with≡ p2F = Valuation.vImplicationT v (Valuation.vImplicationVacuous v p2F)
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171
Logic/PropositionalLogic.agda
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171
Logic/PropositionalLogic.agda
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{-# OPTIONS --safe --warning=error #-}
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Functions
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open import Numbers.Naturals
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open import Vectors
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module Logic.PropositionalLogic where
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data Propositions {a : _} (primitives : Set a) : Set a where
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ofPrimitive : primitives → Propositions primitives
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false : Propositions primitives
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implies : (a b : Propositions primitives) → Propositions primitives
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prNot : {a : _} {pr : Set a} → Propositions pr → Propositions pr
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prNot p = implies p false
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impliesIsBigger : {a : _} {pr : Set a} {P Q : Propositions pr} → Q ≡ implies P Q → False
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impliesIsBigger {P = P} {Q} ()
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impliesInjectiveL : {a : _} {A : Set a} → {p q r : Propositions A} → implies p q ≡ implies r q → p ≡ r
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impliesInjectiveL refl = refl
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impliesInjectiveR : {a : _} {A : Set a} → {p q r : Propositions A} → implies p q ≡ implies p r → q ≡ r
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impliesInjectiveR refl = refl
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impliesInjective : {a : _} {A : Set a} → {p q r s : Propositions A} → implies p q ≡ implies r s → (p ≡ r) && (q ≡ s)
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impliesInjective refl = refl ,, refl
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record Valuation {a : _} (primitives : Set a) : Set a where
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field
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v : Propositions primitives → Bool
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vFalse : v false ≡ BoolFalse
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vImplicationF : {p q : Propositions primitives} → v p ≡ BoolTrue → v q ≡ BoolFalse → v (implies p q) ≡ BoolFalse
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vImplicationVacuous : {p q : Propositions primitives} → v p ≡ BoolFalse → v (implies p q) ≡ BoolTrue
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vImplicationT : {p q : Propositions primitives} → v q ≡ BoolTrue → v (implies p q) ≡ BoolTrue
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-- Proposition 1a
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valuationIsDetermined : {a : _} {pr : Set a} → (v1 v2 : Valuation pr) → ({x : pr} → Valuation.v v1 (ofPrimitive x) ≡ Valuation.v v2 (ofPrimitive x)) → {x : Propositions pr} → Valuation.v v1 x ≡ Valuation.v v2 x
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valuationIsDetermined v1 v2 pr {ofPrimitive x} = pr
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valuationIsDetermined v1 v2 pr {false} rewrite Valuation.vFalse v1 | Valuation.vFalse v2 = refl
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valuationIsDetermined v1 v2 pr {implies x y} with valuationIsDetermined v1 v2 pr {x}
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valuationIsDetermined v1 v2 pr {implies x y} | eqX with valuationIsDetermined v1 v2 pr {y}
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... | eqY with inspect (Valuation.v v1 x)
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valuationIsDetermined v1 v2 pr {implies x y} | eqX | eqY | BoolTrue with≡ p with inspect (Valuation.v v1 y)
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valuationIsDetermined v1 v2 pr {implies x y} | eqX | eqY | BoolTrue with≡ p | BoolTrue with≡ q rewrite p | q | Valuation.vImplicationT v2 {p = x} {q = y} (equalityCommutative eqY) | Valuation.vImplicationT v1 {p = x} {q = y} q = refl
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valuationIsDetermined v1 v2 pr {implies x y} | eqX | eqY | BoolTrue with≡ p | BoolFalse with≡ q rewrite p | q | Valuation.vImplicationF v1 p q | Valuation.vImplicationF v2 (equalityCommutative eqX) (equalityCommutative eqY) = refl
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valuationIsDetermined v1 v2 pr {implies x y} | eqX | eqY | BoolFalse with≡ p rewrite p | Valuation.vImplicationVacuous v1 {q = y} p | Valuation.vImplicationVacuous v2 {q = y} (equalityCommutative eqX) = refl
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extendValuation : {a : _} {pr : Set a} → (w : pr → Bool) → Valuation pr
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Valuation.v (extendValuation w) (ofPrimitive x) = w x
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Valuation.v (extendValuation w) false = BoolFalse
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Valuation.v (extendValuation w) (implies x y) with Valuation.v (extendValuation w) x
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Valuation.v (extendValuation w) (implies x y) | BoolTrue with Valuation.v (extendValuation w) y
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Valuation.v (extendValuation w) (implies x y) | BoolTrue | BoolTrue = BoolTrue
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Valuation.v (extendValuation w) (implies x y) | BoolTrue | BoolFalse = BoolFalse
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Valuation.v (extendValuation w) (implies x y) | BoolFalse = BoolTrue
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Valuation.vFalse (extendValuation w) = refl
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Valuation.vImplicationF (extendValuation w) {p} {q} pT qF with Valuation.v (extendValuation w) p
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Valuation.vImplicationF (extendValuation w) {p} {q} refl qF | BoolTrue with Valuation.v (extendValuation w) q
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Valuation.vImplicationF (extendValuation w) {p} {q} refl () | BoolTrue | BoolTrue
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Valuation.vImplicationF (extendValuation w) {p} {q} refl refl | BoolTrue | BoolFalse = refl
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Valuation.vImplicationF (extendValuation w) {p} {q} () qF | BoolFalse
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Valuation.vImplicationVacuous (extendValuation w) {p} {q} pF with Valuation.v (extendValuation w) p
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Valuation.vImplicationVacuous (extendValuation w) {p} {q} () | BoolTrue
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Valuation.vImplicationVacuous (extendValuation w) {p} {q} refl | BoolFalse = refl
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Valuation.vImplicationT (extendValuation w) {p} {q} qT with Valuation.v (extendValuation w) p
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Valuation.vImplicationT (extendValuation w) {p} {q} qT | BoolTrue with Valuation.v (extendValuation w) q
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Valuation.vImplicationT (extendValuation w) {p} {q} refl | BoolTrue | BoolTrue = refl
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Valuation.vImplicationT (extendValuation w) {p} {q} () | BoolTrue | BoolFalse
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Valuation.vImplicationT (extendValuation w) {p} {q} qT | BoolFalse = refl
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-- Proposition 1b
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valuationsAreFree : {a : _} {pr : Set a} → (w : pr → Bool) → {x : pr} → Valuation.v (extendValuation w) (ofPrimitive x) ≡ w x
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valuationsAreFree w = refl
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record Tautology {a : _} {pr : Set a} (prop : Propositions pr) : Set a where
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field
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isTaut : {v : Valuation pr} → Valuation.v v prop ≡ BoolTrue
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record IsSubset {a b : _} (sub : Set a) (super : Set b) : Set (a ⊔ b) where
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field
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ofElt : sub → super
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inj : Injection ofElt
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mapProp : {a b : _} {pr1 : Set a} {pr2 : Set b} → (pr1 → pr2) → Propositions pr1 → Propositions pr2
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mapProp f (ofPrimitive x) = ofPrimitive (f x)
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mapProp f false = false
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mapProp f (implies p q) = implies (mapProp f p) (mapProp f q)
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inheritedValuation : {a b : _} {sub : Set a} {super : Set b} → (IsSubset sub super) → Valuation super → Valuation sub
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Valuation.v (inheritedValuation isSub v) prop = Valuation.v v (mapProp (IsSubset.ofElt isSub) prop)
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Valuation.vFalse (inheritedValuation isSub v) = Valuation.vFalse v
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Valuation.vImplicationF (inheritedValuation isSub v) pT qF = Valuation.vImplicationF v pT qF
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Valuation.vImplicationVacuous (inheritedValuation isSub v) pF = Valuation.vImplicationVacuous v pF
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Valuation.vImplicationT (inheritedValuation isSub v) qT = Valuation.vImplicationT v qT
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inheritedValuation' : {a b : _} {sub : Set a} {super : Set b} → (IsSubset sub (Propositions super)) → Valuation super → (x : sub) → Bool
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inheritedValuation' subset v x = Valuation.v v (IsSubset.ofElt subset x)
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record Entails {a b : _} {sub : Set a} {super : Set b} (S : IsSubset sub (Propositions super)) (P : Propositions super) : Set (a ⊔ b) where
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field
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entails : {v : Valuation super} → ({s : sub} → inheritedValuation' S v s ≡ BoolTrue) → Valuation.v v P ≡ BoolTrue
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data ThreeElements : Set where
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One : ThreeElements
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Two : ThreeElements
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Three : ThreeElements
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indexAxiom : {a : _} (A : Set a) → ThreeElements → Set a
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indexAxiom A One = Propositions A && Propositions A
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indexAxiom A Two = Propositions A & Propositions A & Propositions A
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indexAxiom A Three = Propositions A
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indexPropositionalAxioms : {a : _} {A : Set a} → Set a
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indexPropositionalAxioms {A = A} = Sg ThreeElements (indexAxiom A)
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-- An axiom system is simply a subset of a set of propositions.
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propositionalAxioms : {a : _} {A : Set a} → IsSubset (indexPropositionalAxioms {A = A}) (Propositions A)
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IsSubset.ofElt propositionalAxioms (One , (p ,, q)) = implies p (implies q p)
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IsSubset.ofElt propositionalAxioms (Two , record { one = p ; two = q ; three = r }) = implies (implies p (implies q r)) (implies (implies p q) (implies p r))
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IsSubset.ofElt propositionalAxioms (Three , p) = implies (prNot (prNot p)) p
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {One , (a ,, b)} pr with impliesInjective pr
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {One , (a ,, b)} pr | fst ,, snd with impliesInjective snd
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {One , (a ,, b)} pr | p=a ,, _ | q=b ,, _ rewrite p=a | q=b = refl
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {Two , record { one = a ; two = b ; three = c }} pr with impliesInjective pr
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {Two , record { one = a ; two = b ; three = c }} pr | fst ,, snd with impliesInjective snd
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {Two , record { one = a ; two = b ; three = c }} pr | p=a-b-c ,, _ | q=a-b ,, p=a-c with impliesInjective (transitivity (equalityCommutative p=a-c) p=a-b-c)
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {Two , record { one = a ; two = b ; three = c }} pr | p=a-b-c ,, _ | q=a-b ,, p=a-c | _ ,, c=b->c = exFalso (impliesIsBigger c=b->c)
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {Three , b} pr with impliesInjective pr
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Injection.property (IsSubset.inj propositionalAxioms) {One , (p ,, q)} {Three , b} pr | p=nnb ,, q-p=b rewrite equalityCommutative q-p=b = exFalso (d pr)
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where
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d : _ → _
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d ()
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Injection.property (IsSubset.inj propositionalAxioms) {Two , record { one = p ; two = q ; three = r }} {One , (fst ,, snd)} ()
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Injection.property (IsSubset.inj propositionalAxioms) {Two , record { one = p ; two = q ; three = r }} {Two , record { one = one ; two = two ; three = three }} pr with impliesInjective pr
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Injection.property (IsSubset.inj propositionalAxioms) {Two , record { one = p ; two = q ; three = r }} {Two , record { one = one ; two = two ; three = three }} pr | fst ,, snd with impliesInjective fst
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Injection.property (IsSubset.inj propositionalAxioms) {Two , record { one = p ; two = q ; three = r }} {Two , record { one = one ; two = two ; three = three }} pr | _ ,, snd | p=one ,, snd1 with impliesInjective snd1
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Injection.property (IsSubset.inj propositionalAxioms) {Two , record { one = p ; two = q ; three = r }} {Two , record { one = one ; two = two ; three = three }} pr | _ ,, snd | p=one ,, _ | q=two ,, r=three rewrite p=one | q=two | r=three = refl
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Injection.property (IsSubset.inj propositionalAxioms) {Two , record { one = p ; two = q ; three = r }} {Three , b} ()
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Injection.property (IsSubset.inj propositionalAxioms) {Three , p} {One , (a ,, b)} pr with impliesInjective pr
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Injection.property (IsSubset.inj propositionalAxioms) {Three , p} {One , (a ,, b)} () | nnp=a ,, p=b-a
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Injection.property (IsSubset.inj propositionalAxioms) {Three , p} {Two , record { one = a ; two = b ; three = c }} ()
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Injection.property (IsSubset.inj propositionalAxioms) {Three , p} {Three , b} pr rewrite _&&_.snd (impliesInjective pr) = refl
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record Selection {a : _} {A : Set a} {n : ℕ} (l : Vec A n) : Set a where
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field
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element : A
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position : ℕ
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pos<N : position <N n
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elementIsAt : vecIndex l position pos<N ≡ element
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data Proof {a b c : _} {A : Set a} {axioms : Set b} (axiomsSubset : IsSubset axioms (Propositions A)) {givens : Set c} (givensSubset : IsSubset givens (Propositions A)) : (n : ℕ) → Set (a ⊔ b ⊔ c)
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data ProofStep {a b c : _} {A : Set a} {axioms : Set b} (axiomsSubset : IsSubset axioms (Propositions A)) {givens : Set c} (givensSubset : IsSubset givens (Propositions A)) {n : ℕ} (proofSoFar : Proof {a} {b} {c} {A} {axioms} axiomsSubset {givens} givensSubset n) : Set (a ⊔ b ⊔ c)
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toSteps : {a b c : _} {A : Set a} {axioms : Set b} {axiomsSubset : IsSubset axioms (Propositions A)} {givens : Set c} {givensSubset : IsSubset givens (Propositions A)} {n : ℕ} (pr : Proof {axioms = axioms} axiomsSubset {givens = givens} givensSubset n) → Vec (Propositions A) n
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data ProofStep {a} {b} {c} {A} {axioms} axiomsSubset {givens} givensSubset proofSoFar where
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axiom : axioms → ProofStep axiomsSubset givensSubset proofSoFar
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given : givens → ProofStep axiomsSubset givensSubset proofSoFar
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modusPonens : (implication : Selection (toSteps proofSoFar)) → (argument : Selection (toSteps proofSoFar)) → (conclusion : Propositions A) → (Selection.element implication ≡ implies (Selection.element argument) conclusion) → ProofStep axiomsSubset givensSubset proofSoFar
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data Proof {a} {b} {c} {A} {axioms} axiomsSubset {givens} givensSubset where
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empty : Proof axiomsSubset givensSubset 0
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nextStep : (n : ℕ) → (previous : Proof {axioms = axioms} axiomsSubset {givens = givens} givensSubset n) → ProofStep axiomsSubset givensSubset previous → Proof axiomsSubset givensSubset (succ n)
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toSteps empty = []
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toSteps {axiomsSubset = axiomsSubset} (nextStep n pr (axiom x)) = (IsSubset.ofElt axiomsSubset x) ,- toSteps pr
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toSteps {givensSubset = givensSubset} (nextStep n pr (given x)) = IsSubset.ofElt givensSubset x ,- toSteps pr
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toSteps (nextStep n pr (modusPonens implication argument conclusion x)) = conclusion ,- toSteps pr
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99
Logic/PropositionalLogicExamples.agda
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99
Logic/PropositionalLogicExamples.agda
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@@ -0,0 +1,99 @@
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{-# OPTIONS --safe --warning=error #-}
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open import Agda.Primitive using (Level; lzero; lsuc; _⊔_)
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open import LogicalFormulae
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open import Logic.PropositionalLogic
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open import Functions
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open import Numbers.Naturals
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open import Vectors
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module Logic.PropositionalLogicExamples where
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axiomK : {a : _} {A : Set a} {P Q : Propositions A} → Tautology (implies P (implies Q P))
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Tautology.isTaut (axiomK {P = P} {Q}) {v = v} with inspect (Valuation.v v P)
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Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolTrue with≡ pT with inspect (Valuation.v v Q)
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Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolTrue with≡ pT | BoolTrue with≡ qT = Valuation.vImplicationT v (Valuation.vImplicationT v pT)
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Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolTrue with≡ pT | BoolFalse with≡ qF = Valuation.vImplicationT v (Valuation.vImplicationVacuous v qF)
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Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolFalse with≡ pF = Valuation.vImplicationVacuous v pF
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excludedMiddle : {a : _} {A : Set a} {P : Propositions A} → Tautology (implies (prNot (prNot P)) P)
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Tautology.isTaut (excludedMiddle {P = P}) {v} with inspect (Valuation.v v P)
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Tautology.isTaut (excludedMiddle {P = P}) {v} | BoolTrue with≡ pT = Valuation.vImplicationT v pT
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Tautology.isTaut (excludedMiddle {P = P}) {v} | BoolFalse with≡ pF = Valuation.vImplicationVacuous v (Valuation.vImplicationF v (Valuation.vImplicationVacuous v pF) (Valuation.vFalse v))
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axiomS : {a : _} {A : Set a} {P Q R : Propositions A} → Tautology (implies (implies P (implies Q R)) (implies (implies P Q) (implies P R)))
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} with inspect (Valuation.v v P)
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT with inspect (Valuation.v v Q)
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT | BoolTrue with≡ qT with inspect (Valuation.v v R)
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT | BoolTrue with≡ qT | BoolTrue with≡ rT = Valuation.vImplicationT v (Valuation.vImplicationT v (Valuation.vImplicationT v rT))
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT | BoolTrue with≡ qT | BoolFalse with≡ rF = Valuation.vImplicationVacuous v (Valuation.vImplicationF v pT (Valuation.vImplicationF v qT rF))
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT | BoolFalse with≡ qF = Valuation.vImplicationT v (Valuation.vImplicationVacuous v (Valuation.vImplicationF v pT qF))
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Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolFalse with≡ pF = Valuation.vImplicationT v (Valuation.vImplicationT v (Valuation.vImplicationVacuous v pF))
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emptySubset : {a : _} (A : Set a) → IsSubset False A
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emptySubset A = record { ofElt = λ () ; inj = record { property = λ {x} → exFalso x } }
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emptyEntailment : {a b : _} {A : Set a} {P : Propositions A} → Entails (emptySubset (Propositions A)) P → Tautology P
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Tautology.isTaut (emptyEntailment {a} {b} {A} {P} record { entails = entails }) {v} = entails {v} λ {s} → exFalso s
|
||||
|
||||
emptyEntailment' : {a b : _} {A : Set a} {P : Propositions A} → Tautology P → Entails (emptySubset (Propositions A)) P
|
||||
Entails.entails (emptyEntailment' record { isTaut = isTaut }) {v} _ = isTaut {v}
|
||||
|
||||
data TwoElements : Set where
|
||||
One : TwoElements
|
||||
Two : TwoElements
|
||||
|
||||
twoElementSubset : {a : _} {A : Set a} → {P Q R : Propositions A} → TwoElements → Propositions A
|
||||
twoElementSubset {P = P} {Q} {R} One = implies P Q
|
||||
twoElementSubset {P = P} {Q} {R} Two = implies Q R
|
||||
|
||||
twoElementSubsetInj : {a : _} {A : Set a} {P Q R : Propositions A} → (P ≡ Q → False) → Injection (twoElementSubset {P = P} {Q} {R})
|
||||
Injection.property (twoElementSubsetInj {P = P} {Q} {R} p!=q) {One} {One} refl = refl
|
||||
Injection.property (twoElementSubsetInj {P = P} {Q} {R} p!=q) {One} {Two} pr with impliesInjective pr
|
||||
Injection.property (twoElementSubsetInj {P = P} {Q} {R} p!=q) {One} {Two} pr | fst ,, snd = exFalso (p!=q fst)
|
||||
Injection.property (twoElementSubsetInj {P = P} {Q} {R} p!=q) {Two} {One} pr with impliesInjective pr
|
||||
Injection.property (twoElementSubsetInj {P = P} {Q} {R} p!=q) {Two} {One} pr | fst ,, snd = exFalso (p!=q (equalityCommutative fst))
|
||||
Injection.property (twoElementSubsetInj {P = P} {Q} {R} p!=q) {Two} {Two} refl = refl
|
||||
|
||||
badBool : BoolFalse ≡ BoolTrue → False
|
||||
badBool ()
|
||||
|
||||
semanticEntailmentTransitive : {a : _} {A : Set a} {P Q R : Propositions A} → (p!=q : P ≡ Q → False) → Entails record { ofElt = twoElementSubset ; inj = twoElementSubsetInj {R = R} p!=q } (implies P R)
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr with inspect (Valuation.v v P)
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr | BoolTrue with≡ pT with inspect (Valuation.v v Q)
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr | BoolTrue with≡ pT | BoolTrue with≡ qT with inspect (Valuation.v v R)
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr | BoolTrue with≡ pT | BoolTrue with≡ qT | BoolTrue with≡ rT = Valuation.vImplicationT v rT
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr | BoolTrue with≡ pT | BoolTrue with≡ qT | BoolFalse with≡ rF = exFalso (badBool (transitivity (equalityCommutative (Valuation.vImplicationF v qT rF)) (pr {Two})))
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr | BoolTrue with≡ pT | BoolFalse with≡ qF = exFalso (badBool (transitivity (equalityCommutative (Valuation.vImplicationF v pT qF)) (pr {One})))
|
||||
Entails.entails (semanticEntailmentTransitive {P = P} {Q} {R} p!=q) {v} pr | BoolFalse with≡ pF = Valuation.vImplicationVacuous v pF
|
||||
|
||||
-- Subset {p -> q, q -> r}
|
||||
pQQR : {a : _} {A : Set a} {P Q R : Propositions A} → TwoElements → Propositions A
|
||||
pQQR {P = P} {Q} {R} One = implies P Q
|
||||
pQQR {P = P} {Q} {R} Two = implies Q R
|
||||
|
||||
pQQRSubsetInj : {a : _} {A : Set a} {P Q R : Propositions A} → (P ≡ Q → False) → Injection (pQQR {P = P} {Q} {R})
|
||||
Injection.property (pQQRSubsetInj {P = P} {Q} {R} p!=q) {One} {One} refl = refl
|
||||
Injection.property (pQQRSubsetInj {P = P} {Q} {R} p!=q) {One} {Two} pr with impliesInjective pr
|
||||
Injection.property (pQQRSubsetInj {P = P} {Q} {R} p!=q) {One} {Two} pr | p=q ,, _ = exFalso (p!=q p=q)
|
||||
Injection.property (pQQRSubsetInj {P = P} {Q} {R} p!=q) {Two} {One} pr with impliesInjective pr
|
||||
Injection.property (pQQRSubsetInj {P = P} {Q} {R} p!=q) {Two} {One} pr | q=p ,, _ = exFalso (p!=q (equalityCommutative q=p))
|
||||
Injection.property (pQQRSubsetInj {P = P} {Q} {R} p!=q) {Two} {Two} refl = refl
|
||||
|
||||
_|>_ : {a b : _} {A : Set a} {B : Set b} (a : A) (f : A → B) → B
|
||||
a |> f = f a
|
||||
infix 1 _|>_
|
||||
|
||||
_||>_!_ : {a b c : _} {A : Set a} {B : Set b} {C : Set c} (a : A) (f : A → B → C) (b : B) → C
|
||||
b ||> f ! a = f b a
|
||||
infix 1 _||>_!_
|
||||
|
||||
f : ℕ → ℕ → (ℕ && ℕ)
|
||||
f a b = a ,, b
|
||||
|
||||
e : ℕ && ℕ
|
||||
e = 5
|
||||
||> f ! 3
|
||||
|
||||
syntacticEntailmentExample : {a : _} {A : Set a} {P Q R : Propositions A} → (p!=q : P ≡ Q → False) → Proof propositionalAxioms (record { ofElt = pQQR {P = P} {Q} {R} ; inj = pQQRSubsetInj p!=q }) 7
|
||||
syntacticEntailmentExample {P = P} {Q} {R} p!=q = nextStep 6 (nextStep 5 (nextStep 4 (nextStep 3 (nextStep 2 (nextStep 1 (nextStep 0 empty (axiom (Two , record { one = P ; two = Q ; three = R }))) (given Two)) (axiom (One , ((implies Q R) ,, P)))) (modusPonens (record { element = implies (implies Q R) (implies P (implies Q R)) ; position = 0 ; pos<N = succIsPositive _ ; elementIsAt = refl }) (record { element = implies Q R ; position = 1 ; elementIsAt = refl ; pos<N = succPreservesInequality (succIsPositive _) }) (implies P (implies Q R)) refl)) (modusPonens (record { element = implies (implies P (implies Q R)) (implies (implies P Q) (implies P R)) ; position = 3 ; pos<N = le 0 refl ; elementIsAt = refl }) (record { element = implies P (implies Q R) ; position = 0 ; pos<N = succIsPositive _ ; elementIsAt = refl }) (implies (implies P Q) (implies P R)) refl)) (given One)) (modusPonens (record { element = implies (implies P Q) (implies P R) ; position = 1 ; pos<N = succPreservesInequality (succIsPositive _) ; elementIsAt = refl }) (record { element = implies P Q ; position = 0 ; pos<N = succIsPositive _ ; elementIsAt = refl }) (implies P R) refl)
|
||||
Reference in New Issue
Block a user