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(* Title: ZF/Resid/Redex.thy
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ID: $Id$
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Author: Ole Rasmussen
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Copyright 1995 University of Cambridge
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Logic Image: ZF
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*)
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Redex = Main +
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consts
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redexes :: i
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datatype
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"redexes" = Var ("n: nat")
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| Fun ("t: redexes")
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| App ("b:bool" ,"f:redexes" , "a:redexes")
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consts
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Ssub,Scomp,Sreg :: i
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"<==","~" :: [i,i]=>o (infixl 70)
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un :: [i,i]=>i (infixl 70)
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union_aux :: i=>i
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regular :: i=>o
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translations
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"a<==b" == "<a,b>:Ssub"
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"a ~ b" == "<a,b>:Scomp"
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"regular(a)" == "a:Sreg"
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primrec (*explicit lambda is required because both arguments of "un" vary*)
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"union_aux(Var(n)) =
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(lam t:redexes. redexes_case(%j. Var(n), %x. 0, %b x y.0, t))"
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"union_aux(Fun(u)) =
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(lam t:redexes. redexes_case(%j. 0, %y. Fun(union_aux(u)`y),
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%b y z. 0, t))"
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"union_aux(App(b,f,a)) =
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(lam t:redexes.
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redexes_case(%j. 0, %y. 0,
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%c z u. App(b or c, union_aux(f)`z, union_aux(a)`u), t))"
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defs
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union_def "u un v == union_aux(u)`v"
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inductive
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domains "Ssub" <= "redexes*redexes"
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intrs
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Sub_Var "n:nat ==> Var(n)<== Var(n)"
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Sub_Fun "[|u<== v|]==> Fun(u)<== Fun(v)"
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Sub_App1 "[|u1<== v1; u2<== v2; b:bool|]==>
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App(0,u1,u2)<== App(b,v1,v2)"
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Sub_App2 "[|u1<== v1; u2<== v2|]==>
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App(1,u1,u2)<== App(1,v1,v2)"
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type_intrs "redexes.intrs@bool_typechecks"
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inductive
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domains "Scomp" <= "redexes*redexes"
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intrs
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Comp_Var "n:nat ==> Var(n) ~ Var(n)"
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Comp_Fun "[|u ~ v|]==> Fun(u) ~ Fun(v)"
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Comp_App "[|u1 ~ v1; u2 ~ v2; b1:bool; b2:bool|]==>
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App(b1,u1,u2) ~ App(b2,v1,v2)"
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type_intrs "redexes.intrs@bool_typechecks"
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inductive
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domains "Sreg" <= "redexes"
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intrs
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Reg_Var "n:nat ==> regular(Var(n))"
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Reg_Fun "[|regular(u)|]==> regular(Fun(u))"
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Reg_App1 "[|regular(Fun(u)); regular(v)
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|]==>regular(App(1,Fun(u),v))"
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Reg_App2 "[|regular(u); regular(v)
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|]==>regular(App(0,u,v))"
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type_intrs "redexes.intrs@bool_typechecks"
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end
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