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theory Compiler = Natural:
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datatype instr = ASIN loc aexp | JMPF bexp nat | JMPB nat
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consts stepa1 :: "instr list => ((state*nat) * (state*nat))set"
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syntax
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"@stepa1" :: "[instr list,state,nat,state,nat] => bool"
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("_ |- <_,_>/ -1-> <_,_>" [50,0,0,0,0] 50)
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"@stepa" :: "[instr list,state,nat,state,nat] => bool"
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("_ |-/ <_,_>/ -*-> <_,_>" [50,0,0,0,0] 50)
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translations "P |- <s,m> -1-> <t,n>" == "((s,m),t,n) : stepa1 P"
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"P |- <s,m> -*-> <t,n>" == "((s,m),t,n) : ((stepa1 P)^*)"
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inductive "stepa1 P"
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intros
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ASIN: "P!n = ASIN x a ==> P |- <s,n> -1-> <s[x::= a s],Suc n>"
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JMPFT: "[| P!n = JMPF b i; b s |] ==> P |- <s,n> -1-> <s,Suc n>"
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JMPFF: "[| P!n = JMPF b i; ~b s; m=n+i |] ==> P |- <s,n> -1-> <s,m>"
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JMPB: "[| P!n = JMB i |] ==> P |- <s,n> -1-> <s,n-i>"
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consts compile :: "com => instr list"
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primrec
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"compile SKIP = []"
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"compile (x:==a) = [ASIN x a]"
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"compile (c1;c2) = compile c1 @ compile c2"
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"compile (IF b THEN c1 ELSE c2) =
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[JMPF b (length(compile c1)+2)] @ compile c1 @
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[JMPF (%x. False) (length(compile c2)+1)] @ compile c2"
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"compile (WHILE b DO c) = [JMPF b (length(compile c)+2)] @ compile c @
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[JMPB (length(compile c)+1)]"
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declare nth_append[simp];
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lemma nth_tl[simp]: "tl(xs @ y # ys) ! (length xs + z) = ys!z";
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apply(induct_tac xs);
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by(auto);
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theorem "<c,s> -c-> t ==>
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!a z. a@compile c@z |- <s,length a> -*-> <t,length a + length(compile c)>";
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apply(erule evalc.induct);
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apply simp;
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apply(force intro!: ASIN);
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apply(intro strip);
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apply(erule_tac x = a in allE);
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apply(erule_tac x = "a@compile c0" in allE);
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apply(erule_tac x = "compile c1@z" in allE);
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apply(erule_tac x = z in allE);
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apply(simp add:add_assoc[THEN sym]);
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apply(blast intro:rtrancl_trans);
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(* IF b THEN c0 ELSE c1; case b is true *)
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apply(intro strip);
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(* instantiate assumption sufficiently for later: *)
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apply(erule_tac x = "a@[?I]" in allE);
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apply(simp);
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(* execute JMPF: *)
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apply(rule rtrancl_into_rtrancl2);
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apply(rule JMPFT);
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apply(simp);
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apply(blast);
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apply assumption;
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(* execute compile c0: *)
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apply(rule rtrancl_trans);
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apply(erule allE);
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apply assumption;
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(* execute JMPF: *)
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apply(rule r_into_rtrancl);
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apply(rule JMPFF);
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apply(simp);
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apply(blast);
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apply(blast);
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apply(simp);
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(* end of case b is true *)
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apply(intro strip);
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apply(erule_tac x = "a@[?I]@compile c0@[?J]" in allE);
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apply(simp add:add_assoc);
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apply(rule rtrancl_into_rtrancl2);
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apply(rule JMPFF);
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apply(simp);
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apply(blast);
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apply assumption;
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apply(simp);
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apply(blast);
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apply(force intro: JMPFF);
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apply(intro strip);
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apply(erule_tac x = "a@[?I]" in allE);
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apply(erule_tac x = a in allE);
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apply(simp);
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apply(rule rtrancl_into_rtrancl2);
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apply(rule JMPFT);
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apply(simp);
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apply(blast);
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apply assumption;
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apply(rule rtrancl_trans);
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apply(erule allE);
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apply assumption;
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apply(rule rtrancl_into_rtrancl2);
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apply(rule JMPB);
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apply(simp);
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apply(simp);
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done
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(* Missing: the other direction! *)
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end
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