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(* Title: Sequents/LK/Nat

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ID: $Id$


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Author: Lawrence C Paulson, Cambridge University Computer Laboratory

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Copyright 1999 University of Cambridge

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Theory of the natural numbers: Peano's axioms, primitive recursion

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*)


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Addsimps [Suc_neq_0];


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Add_safes [Suc_inject RS L_of_imp];


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Goal " Suc(k) ~= k";


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


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by (Simp_tac 1);


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by (Fast_tac 1);


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qed "Suc_n_not_n";


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Goalw [add_def] " 0+n = n";


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by (rtac rec_0 1);


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qed "add_0";


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


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by (rtac rec_Suc 1);


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qed "add_Suc";


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Addsimps [add_0, add_Suc];


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Goal " (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 1);


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by (Asm_simp_tac 1);


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qed "add_assoc";


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Goal " m+0 = m";


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


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by (Simp_tac 1);


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by (Asm_simp_tac 1);


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qed "add_0_right";


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Goal " 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));


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qed "add_Suc_right";


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(*Example used in Reference Manual, Doc/Ref/simplifier.tex*)


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val [prem] = Goal "(!!n.  f(Suc(n)) = Suc(f(n))) ==>  f(i+j) = i+f(j)";


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


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by (Simp_tac 1);


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by (simp_tac (simpset() addsimps [prem]) 1);


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result();
