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(* Title: FOLP/ex/nat.ML

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ID: $Id$

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Author: Lawrence C Paulson, Cambridge University Computer Laboratory

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Copyright 1992 University of Cambridge


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*)


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Goal "?p : ~ (Suc(k) = k)";

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by (res_inst_tac [("n","k")] induct 1);


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by (rtac notI 1);


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by (etac Suc_neq_0 1);


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by (rtac notI 1);


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by (etac notE 1);


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by (etac Suc_inject 1);


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val Suc_n_not_n = result();


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Goal "?p : (k+m)+n = k+(m+n)";

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prths ([induct] RL [topthm()]); (*prints all 14 next states!*)


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by (rtac induct 1);


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back();


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back();


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back();


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back();


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back();


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back();


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Goalw [add_def] "?p : 0+n = n";

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by (rtac rec_0 1);


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val add_0 = result();


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Goalw [add_def] "?p : Suc(m)+n = Suc(m+n)";

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by (rtac rec_Suc 1);


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val add_Suc = result();


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(*

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val nat_congs = mk_congs (the_context ()) ["Suc", "op +"];

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prths nat_congs;


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*)

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val prems = goal (the_context ()) "p: x=y ==> ?p : Suc(x) = Suc(y)";

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by (resolve_tac (prems RL [subst]) 1);


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by (rtac refl 1);


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val Suc_cong = result();


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val prems = goal (the_context ()) "[ p : a=x; q: b=y ] ==> ?p : a+b=x+y";


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by (resolve_tac (prems RL [subst]) 1 THEN resolve_tac (prems RL [subst]) 1 THEN

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rtac refl 1);


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val Plus_cong = result();


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val nat_congs = [Suc_cong,Plus_cong];


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val add_ss = FOLP_ss addcongs nat_congs

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addrews [add_0, add_Suc];

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Goal "?p : (k+m)+n = k+(m+n)";

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by (res_inst_tac [("n","k")] induct 1);


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by (SIMP_TAC add_ss 1);


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by (ASM_SIMP_TAC add_ss 1);


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val add_assoc = result();


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Goal "?p : m+0 = m";

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by (res_inst_tac [("n","m")] induct 1);


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by (SIMP_TAC add_ss 1);


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by (ASM_SIMP_TAC add_ss 1);


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val add_0_right = result();


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Goal "?p : m+Suc(n) = Suc(m+n)";

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by (res_inst_tac [("n","m")] induct 1);


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by (ALLGOALS (ASM_SIMP_TAC add_ss));


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val add_Suc_right = result();


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(*mk_typed_congs appears not to work with FOLP's version of subst*)
