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val expand_all_PROD = prove_goal LCF.thy


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"(ALL p. P(p)) <> (ALL x y. P(<x,y>))"


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(fn _ => [rtac iffI 1, fast_tac FOL_cs 1, rtac allI 1,


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rtac (surj_pairing RS subst) 1, fast_tac FOL_cs 1]);


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local


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val ppair = read_instantiate [("z","p::'a*'b")] surj_pairing;


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val qpair = read_instantiate [("z","q::'a*'b")] surj_pairing;


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in


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val PROD_less = prove_goal LCF.thy


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"p::'a*'b << q <> FST(p) << FST(q) & SND(p) << SND(q)"


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(fn _ => [EVERY1[rtac iffI,


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rtac conjI, etac less_ap_term, etac less_ap_term,


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rtac (ppair RS subst), rtac (qpair RS subst),


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etac conjE, rtac mono, etac less_ap_term, atac]]);


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end;


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val PROD_eq = prove_goal LCF.thy "p=q <> FST(p)=FST(q) & SND(p)=SND(q)"


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(fn _ => [rtac iffI 1, asm_simp_tac LCF_ss 1,


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rewrite_goals_tac [eq_def],


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asm_simp_tac (LCF_ss addsimps [PROD_less]) 1]);


22 


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val PAIR_less = prove_goal LCF.thy "<a,b> << <c,d> <> a<<c & b<<d"


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(fn _ => [simp_tac (LCF_ss addsimps [PROD_less])1]);


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val PAIR_eq = prove_goal LCF.thy "<a,b> = <c,d> <> a=c & b=d"


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(fn _ => [simp_tac (LCF_ss addsimps [PROD_eq])1]);


28 


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val UU_is_UU_UU = prove_goal LCF.thy "<UU,UU> << UU"


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(fn _ => [simp_tac (LCF_ss addsimps [PROD_less]) 1])


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RS less_UU RS sym;


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val LCF_ss = LCF_ss addsimps [PAIR_less,PAIR_eq,UU_is_UU_UU];
