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61 lines
2.1 KiB
Plaintext
61 lines
2.1 KiB
Plaintext
-- many-sorted predicate calculus
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-- AR 1999, revised 2001 and 2006
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abstract Logic = {
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cat
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Prop ; -- proposition
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Dom ; -- domain of quantification
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Elem Dom ; -- individual element of a domain
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Proof Prop ; -- proof of a proposition
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Hypo Prop ; -- hypothesis of a proposition
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Text ; -- theorem with proof etc.
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fun
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-- texts
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Statement : Prop -> Text ;
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ThmWithProof : (A : Prop) -> Proof A -> Text ;
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ThmWithTrivialProof : (A : Prop) -> Proof A -> Text ;
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-- logically complex propositions
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Disj : (A,B : Prop) -> Prop ;
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Conj : (A,B : Prop) -> Prop ;
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Impl : (A,B : Prop) -> Prop ;
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Abs : Prop ;
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Neg : Prop -> Prop ;
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Univ : (A : Dom) -> (Elem A -> Prop) -> Prop ;
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Exist : (A : Dom) -> (Elem A -> Prop) -> Prop ;
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-- inference rules
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ConjI : (A,B : Prop) -> Proof A -> Proof B -> Proof (Conj A B) ;
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ConjEl : (A,B : Prop) -> Proof (Conj A B) -> Proof A ;
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ConjEr : (A,B : Prop) -> Proof (Conj A B) -> Proof B ;
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DisjIl : (A,B : Prop) -> Proof A -> Proof (Disj A B) ;
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DisjIr : (A,B : Prop) -> Proof B -> Proof (Disj A B) ;
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DisjE : (A,B,C : Prop) -> Proof (Disj A B) ->
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(Hypo A -> Proof C) -> (Hypo B -> Proof C) -> Proof C ;
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ImplI : (A,B : Prop) -> (Hypo A -> Proof B) -> Proof (Impl A B) ;
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ImplE : (A,B : Prop) -> Proof (Impl A B) -> Proof A -> Proof B ;
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NegI : (A : Prop) -> (Hypo A -> Proof Abs) -> Proof (Neg A) ;
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NegE : (A : Prop) -> Proof (Neg A) -> Proof A -> Proof Abs ;
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AbsE : (C : Prop) -> Proof Abs -> Proof C ;
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UnivI : (A : Dom) -> (B : Elem A -> Prop) ->
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((x : Elem A) -> Proof (B x)) -> Proof (Univ A B) ;
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UnivE : (A : Dom) -> (B : Elem A -> Prop) ->
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Proof (Univ A B) -> (a : Elem A) -> Proof (B a) ;
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ExistI : (A : Dom) -> (B : Elem A -> Prop) ->
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(a : Elem A) -> Proof (B a) -> Proof (Exist A B) ;
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ExistE : (A : Dom) -> (B : Elem A -> Prop) -> (C : Prop) ->
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Proof (Exist A B) -> ((x : Elem A) -> Proof (B x) -> Proof C) ->
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Proof C ;
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-- use a hypothesis
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Hypoth : (A : Prop) -> Hypo A -> Proof A ;
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-- pronoun
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Pron : (A : Dom) -> Elem A -> Elem A ;
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} ;
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