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Tweak tautology (#100)

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This commit is contained in:
Patrick Stevens
2020-02-23 12:35:05 +00:00
committed by GitHub
parent f6b13d677a
commit e0e627514e
2 changed files with 19 additions and 20 deletions
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5
Logic/PropositionalLogic.agda
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@@ -77,9 +77,8 @@ Valuation.vImplicationT (extendValuation w) {p} {q} qT | BoolFalse = refl
valuationsAreFree : {a : _} {pr : Set a} → (w : pr → Bool) → {x : pr} → Valuation.v (extendValuation w) (ofPrimitive x) ≡ w x
valuationsAreFree w = refl
record Tautology {a : _} {pr : Set a} (prop : Propositions pr) : Set a where
field
isTaut : {v : Valuation pr} → Valuation.v v prop ≡ BoolTrue
Tautology : {a : _} {pr : Set a} (prop : Propositions pr) → Set a
Tautology {pr = pr} prop = {v : Valuation pr} → Valuation.v v prop ≡ BoolTrue
record IsSubset {a b : _} (sub : Set a) (super : Set b) : Set (a ⊔ b) where
field

34
Logic/PropositionalLogicExamples.agda
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@@ -10,34 +10,34 @@ open import Vectors
module Logic.PropositionalLogicExamples where
axiomK : {a : _} {A : Set a} {P Q : Propositions A} → Tautology (implies P (implies Q P))
Tautology.isTaut (axiomK {P = P} {Q}) {v = v} with inspect (Valuation.v v P)
Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolTrue with≡ pT with inspect (Valuation.v v Q)
Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolTrue with≡ pT | BoolTrue with≡ qT = Valuation.vImplicationT v (Valuation.vImplicationT v pT)
Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolTrue with≡ pT | BoolFalse with≡ qF = Valuation.vImplicationT v (Valuation.vImplicationVacuous v qF)
Tautology.isTaut (axiomK {P = P} {Q}) {v} | BoolFalse with≡ pF = Valuation.vImplicationVacuous v pF
axiomK {P = P} {Q} {v = v} with inspect (Valuation.v v P)
axiomK {P = P} {Q} {v} | BoolTrue with≡ pT with inspect (Valuation.v v Q)
axiomK {P = P} {Q} {v} | BoolTrue with≡ pT | BoolTrue with≡ qT = Valuation.vImplicationT v (Valuation.vImplicationT v pT)
axiomK {P = P} {Q} {v} | BoolTrue with≡ pT | BoolFalse with≡ qF = Valuation.vImplicationT v (Valuation.vImplicationVacuous v qF)
axiomK {P = P} {Q} {v} | BoolFalse with≡ pF = Valuation.vImplicationVacuous v pF
excludedMiddle : {a : _} {A : Set a} {P : Propositions A} → Tautology (implies (prNot (prNot P)) P)
Tautology.isTaut (excludedMiddle {P = P}) {v} with inspect (Valuation.v v P)
Tautology.isTaut (excludedMiddle {P = P}) {v} | BoolTrue with≡ pT = Valuation.vImplicationT v pT
Tautology.isTaut (excludedMiddle {P = P}) {v} | BoolFalse with≡ pF = Valuation.vImplicationVacuous v (Valuation.vImplicationF v (Valuation.vImplicationVacuous v pF) (Valuation.vFalse v))
excludedMiddle {P = P} {v} with inspect (Valuation.v v P)
excludedMiddle {P = P} {v} | BoolTrue with≡ pT = Valuation.vImplicationT v pT
excludedMiddle {P = P} {v} | BoolFalse with≡ pF = Valuation.vImplicationVacuous v (Valuation.vImplicationF v (Valuation.vImplicationVacuous v pF) (Valuation.vFalse v))
axiomS : {a : _} {A : Set a} {P Q R : Propositions A} → Tautology (implies (implies P (implies Q R)) (implies (implies P Q) (implies P R)))
Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} with inspect (Valuation.v v P)
Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT with inspect (Valuation.v v Q)
Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolTrue with≡ pT | BoolTrue with≡ qT with inspect (Valuation.v v R)
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))
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))
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))
Tautology.isTaut (axiomS {P = P} {Q} {R}) {v} | BoolFalse with≡ pF = Valuation.vImplicationT v (Valuation.vImplicationT v (Valuation.vImplicationVacuous v pF))
axiomS {P = P} {Q} {R} {v} with inspect (Valuation.v v P)
axiomS {P = P} {Q} {R} {v} | BoolTrue with≡ pT with inspect (Valuation.v v Q)
axiomS {P = P} {Q} {R} {v} | BoolTrue with≡ pT | BoolTrue with≡ qT with inspect (Valuation.v v R)
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))
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))
axiomS {P = P} {Q} {R} {v} | BoolTrue with≡ pT | BoolFalse with≡ qF = Valuation.vImplicationT v (Valuation.vImplicationVacuous v (Valuation.vImplicationF v pT qF))
axiomS {P = P} {Q} {R} {v} | BoolFalse with≡ pF = Valuation.vImplicationT v (Valuation.vImplicationT v (Valuation.vImplicationVacuous v pF))
emptySubset : {a : _} (A : Set a) → IsSubset False A
emptySubset A = record { ofElt = λ () }
emptyEntailment : {a b : _} {A : Set a} {P : Propositions A} → Entails (emptySubset (Propositions A)) P → Tautology P
Tautology.isTaut (emptyEntailment {a} {b} {A} {P} record { entails = entails }) {v} = entails {v} λ {s} → exFalso s
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}
Entails.entails (emptyEntailment' isTaut) {v} _ = isTaut {v}
data TwoElements : Set where
One : TwoElements