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|
; $Id: quant.scm 2240 2008-04-18 06:22:33Z schwicht $
(load "~/minlog/init.scm")
(set! DOT-NOTATION #f)
(add-var-name "x" "y" "z" (py "alpha"))
(add-pvar-name "P" (make-arity))
(add-pvar-name "Q" (make-arity (py "alpha")))
(add-pvar-name "R" (make-arity (py "alpha") (py "alpha")))
; "AllImp"
(set-goal (pf "all x(Q1 x -> Q2 x) -> all x Q1 x -> all x Q2 x"))
(assume 1 2 "x")
(use 1)
(use 2)
; Proof finished.
(set-goal (pf "all x(Q1 x -> Q2 x) -> all x Q1 x -> all x Q2 x"))
(search)
; Proof finished.
; "AllAndOne"
(set-goal (pf "all x(Q1 x & Q2 x) -> all x Q1 x & all x Q2 x"))
(assume 1)
(split)
(assume "x")
(use 1)
(assume "x")
(use 1)
; Proof finished.
(set-goal (pf "all x(Q1 x & Q2 x) -> all x Q1 x & all x Q2 x"))
(search)
; Proof finished.
; "AllAndTwo"
(set-goal (pf " all x Q1 x & all x Q2 x -> all x(Q1 x & Q2 x)"))
(assume 1 "x")
(split)
(use 1)
(use 1)
; Proof finished.
(set-goal (pf " all x Q1 x & all x Q2 x -> all x(Q1 x & Q2 x)"))
(search)
; Proof finished.
; "AllExca"
(set-goal (pf "all x Q x -> exca x Q x"))
(assume 1 2)
(use 2 (pt "(Inhab alpha)"))
(use 1)
; Proof finished.
(set-goal (pf "all x Q x -> exca x Q x"))
(search)
; Proof finished.
(set-goal (pf "all x,y(R x y -> R y x) ->
all x,y,z(R x y -> R y z -> R x z) ->
all x,y(R x y -> R x x)"))
(assume "Symm" "Trans" "x" "y" 1)
(use "Trans" (pt "y"))
(use 3)
(use "Symm")
(use 3)
; Proof finished.
(set-goal (pf "all x,y(R x y -> R y x) ->
all x,y,z(R x y -> R y z -> R x z) ->
all x,y(R x y -> R x x)"))
(search)
; Proof finished.
; Now we treat somewhat systematically how in classical logic one can
; deal with quantifiers in implications. - Below we shall do the same
; for the constructive existential quantifier.
; qf1m is obtained from the formula (all x Q x -> P) -> ex x(Q x -> P)
; by translating "ex" into "not all not" and adding stability of Q:
(define qf1m (pf "all x(((Q x -> F) -> F) -> Q x)
-> (all x Q x -> P)
-> all x((Q x -> P) -> F)
-> F"))
(set-goal qf1m)
(assume 1 2 3)
(use 3 (pt "x"))
(assume 4)
(use 2)
(assume "x1")
(use 1)
(assume 5)
(use 3 (pt "x1"))
(assume 6)
(use 2)
(assume "x2")
(use 1)
(assume 7)
(use 5)
(use 6)
; Proof finished.
(set-goal qf1m)
(search)
; Proof finished.
(define qf2 (pf "(P -> all y Q y) -> all y(P -> Q y)"))
(set-goal qf2)
(assume 1 "y" 2)
(use 1)
(use 2)
; Proof finished.
(set-goal qf2)
(search)
; Proof finished.
; qf3 is obtained from the formula (ex x Q x -> P) -> all x(Q x -> P)
; by translating "ex" into "not all not":
(define qf3 (pf "((all x(Q x -> F) -> F) -> P) -> all x(Q x -> P)"))
(set-goal qf3)
(assume 1 "x" 2)
(use 1)
(assume 3)
(use 3 (pt "x"))
(use 2)
; Proof finished.
(set-goal qf3)
(search)
; Proof finished.
; qf4m is obtained from the formula (P -> ex y Q y) -> ex y(P -> Q y)
; by translating "ex" into "not all not" and adding ef-falso for Q:
(define qf4m (pf "all y(F -> Q y)
-> (P -> all y(Q y -> F) -> F)
-> all y((P -> Q y) -> F)
-> F"))
(set-goal qf4m)
(assume 1 2 3)
(use 3 (pt "y"))
(assume 4)
(use 1)
(use 2)
(use 4)
(assume "y1" 5)
(use 3 (pt "y1"))
(assume 6)
(use 5)
; Proof finished.
(set-goal qf4m)
(search)
; Proof finished.
; qf5m is obtained from the formula ex x(Q x -> P) -> all x Q x -> P
; by translating "ex" into "not all not" and adding stability of P:
(define qf5m (pf "(((P -> F) -> F) -> P)
-> (all x((Q x -> P) -> F) -> F)
-> all x Q x
-> P"))
(set-goal qf5m)
(assume 1 2 3)
(use 1)
(assume 4)
(use 2)
(assume "x" 5)
(use 4)
(use 5)
(use 3)
; Proof finished.
(set-goal qf5m)
(search)
; Proof finished.
(define qf6 (pf "all y( P -> Q y) -> P -> all y Q y"))
(set-goal qf6)
(assume 1 2 "y")
(use 1)
(use 2)
; Proof finished.
(set-goal qf6)
(search)
; Proof finished.
; qf7m is obtained from the formula all x(Q x -> P) -> ex x Q x -> P
; by translating "ex" into "not all not" and adding stability of P:
(define qf7m (pf "(((P -> F) -> F) -> P)
-> all x(Q x -> P)
-> (all x(Q x -> F) -> F)
-> P"))
(set-goal qf7m)
(assume 1 2 3)
(use 1)
(assume 4)
(use 3)
(assume "x" 5)
(use 4)
(use 2 (pt "x"))
(use 5)
; Proof finished.
(set-goal qf7m)
(search)
; Proof finished.
; qf8 is obtained from the formula ex y(P -> Q y) -> P -> ex y Q y
; by translating "ex" into "not all not":
(define qf8 (pf "(all y((P -> Q y) -> F) -> F)
-> P
-> all y(Q y -> F)
-> F"))
(set-goal qf8)
(assume 1 2 3)
(use 1)
(assume "y" 4)
(use 3 (pt "y"))
(use 4)
(use 2)
; Proof finished.
(set-goal qf8)
(search)
; Proof finished.
; Some more examples involving classical existence
; "Drinker"
(set-goal (pf "all y(((Q y -> F) -> F) -> Q y) -> exca x(Q x -> all y Q y)"))
(assume 1 2)
(use 2 (pt "(Inhab alpha)"))
(assume 3 "y")
(use 1)
(assume 4)
(use 2 (pt "y"))
(assume 5 "z")
(use 1)
(assume 6)
(use 4)
(use 5)
; Proof finished.
(set-goal (pf "all y(((Q y -> F) -> F) -> Q y) -> exca x(Q x -> all y Q y)"))
(search)
; Proof finished.
; Now we treat the constructive existential quantifier
; "ExAndOne"
(set-goal (pf "ex x(Q1 x & Q2 x) -> ex x Q1 x & ex x Q2 x"))
(assume 1)
(split)
(ex-elim 1)
(assume "x" 2)
(ex-intro (pt "x"))
(use 2)
(ex-elim 1)
(assume "x" 2)
(ex-intro (pt "x"))
(use 2)
; Proof finished.
(set-goal (pf "ex x(Q^1 x & Q^2 x) -> ex x Q^1 x & ex x Q^2 x"))
(search)
; Proof finished.
; Normalized extracted term:
(dnet) ;"[x0]x0@x0"
; ExAndTwo"
(set-goal (pf "ex x Q x & P -> ex x(Q x & P)"))
(assume 1)
(inst-with 1 'left)
(ex-elim 2)
(assume "x" 3)
(ex-intro (pt "x"))
(split)
(use 3)
(use 1)
; Proof finished.
(set-goal (pf "ex x Q^ x & P^ -> ex x(Q^ x & P^)"))
(search)
; Proof finished.
; Normalized extracted term:
(dnet) ;"[x0]x0"
; "AllEx"
(set-goal (pf "all x Q^ x -> ex x Q^ x"))
(assume 1)
(ex-intro (pt "(Inhab alpha)"))
(use 1)
; Proof finished.
; Normalized extracted term:
(dnet) ;"(Inhab alpha)"
(set-goal (pf "all x Q^ x -> ex x Q^ x"))
(search)
; Proof finished.
; Normalized extracted term:
(dnet) ;"x"
; qf1 is the formula (all x Q x -> P) -> ex x(Q x -> P)
; It is not provable in minimal logic.
; (define qf2 (pf "(P -> all y Q y) -> all y(P -> Q y)")) see above
(define qf3 (pf "(ex x Q x -> P) -> all x(Q x -> P)"))
(set-goal qf3)
(assume 1 "x" 2)
(use 1)
(ex-intro (pt "x"))
(use 2)
; Proof finished.
(set-goal qf3)
(search)
; Proof finished.
; qf4 is the formula (P -> ex y Q y) -> ex y(P -> Q y)
; It is not provable in minimal logic.
(define qf5 (pf "ex x(Q x -> P) -> all x Q x -> P"))
(set-goal qf5)
(assume 1 2)
(ex-elim 1)
(assume "x" 3)
(use 3)
(use 2)
; Proof finished.
(set-goal qf5)
(assume 1 2)
(ex-elim 1)
(search)
; Proof finished.
; (define qf6 (pf "all y(P -> Q y) -> P -> all y Q y")) see above
(define qf7 (pf "all x(Q x -> P) -> ex x Q x -> P"))
(set-goal qf7)
(assume 1 2)
(ex-elim 2)
(use 1)
; Proof finished.
(set-goal qf7)
(assume 1 2)
(ex-elim 2)
(search)
; Proof finished.
(define qf8 (pf "ex y(P^ -> Q^ y) -> P^ -> ex y Q^ y"))
(set-goal qf8)
(assume 1 2)
(ex-elim 1)
(assume "x" 3)
(ex-intro (pt "x"))
(use 3)
(use 2)
; Proof finished.
(set-goal qf8)
(search)
; Proof finished.
; Normalized extracted term:
(dnet) ;"[x0]x0"
|
