author | wenzelm |
Tue, 07 Dec 1999 12:13:09 +0100 | |
changeset 8051 | 5724bea1da53 |
parent 8031 | 826c6222cca2 |
child 8052 | 6ae3ca78a558 |
permissions | -rw-r--r-- |
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(* Title: HOL/Isar_examples/ExprCompiler.thy |
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ID: $Id$ |
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Author: Markus Wenzel, TU Muenchen |
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Correctness of a simple expression/stack-machine compiler. |
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*) |
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header {* Correctness of a simple expression compiler *}; |
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theory ExprCompiler = Main:; |
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text {* |
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This is a (rather trivial) example of program verification. We model |
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a compiler for translating expressions to stack machine instructions, |
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and prove its correctness wrt.\ some evaluation semantics. |
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*}; |
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subsection {* Binary operations *}; |
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text {* |
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Binary operations are just functions over some type of values. This |
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is both for abstract syntax and semantics, i.e.\ we use a ``shallow |
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embedding'' here. |
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*}; |
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types |
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'val binop = "'val => 'val => 'val"; |
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subsection {* Expressions *}; |
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text {* |
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The language of expressions is defined as an inductive type, |
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consisting of variables, constants, and binary operations on |
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expressions. |
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*}; |
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datatype ('adr, 'val) expr = |
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Variable 'adr | |
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Constant 'val | |
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Binop "'val binop" "('adr, 'val) expr" "('adr, 'val) expr"; |
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text {* |
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Evaluation (wrt.\ some environment of variable assignments) is |
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defined by primitive recursion over the structure of expressions. |
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*}; |
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consts |
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eval :: "('adr, 'val) expr => ('adr => 'val) => 'val"; |
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primrec |
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"eval (Variable x) env = env x" |
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"eval (Constant c) env = c" |
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"eval (Binop f e1 e2) env = f (eval e1 env) (eval e2 env)"; |
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subsection {* Machine *}; |
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text {* |
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Next we model a simple stack machine, with three instructions. |
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*}; |
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datatype ('adr, 'val) instr = |
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Const 'val | |
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Load 'adr | |
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Apply "'val binop"; |
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text {* |
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Execution of a list of stack machine instructions is easily defined |
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as follows. |
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*}; |
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consts |
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exec :: "(('adr, 'val) instr) list |
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=> 'val list => ('adr => 'val) => 'val list"; |
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primrec |
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"exec [] stack env = stack" |
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"exec (instr # instrs) stack env = |
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(case instr of |
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Const c => exec instrs (c # stack) env |
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| Load x => exec instrs (env x # stack) env |
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| Apply f => exec instrs (f (hd stack) (hd (tl stack)) |
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# (tl (tl stack))) env)"; |
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constdefs |
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execute :: "(('adr, 'val) instr) list => ('adr => 'val) => 'val" |
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"execute instrs env == hd (exec instrs [] env)"; |
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subsection {* Compiler *}; |
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text {* |
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We are ready to define the compilation function of expressions to |
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lists of stack machine instructions. |
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*}; |
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consts |
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compile :: "('adr, 'val) expr => (('adr, 'val) instr) list"; |
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primrec |
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"compile (Variable x) = [Load x]" |
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"compile (Constant c) = [Const c]" |
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"compile (Binop f e1 e2) = compile e2 @ compile e1 @ [Apply f]"; |
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text {* |
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The main result of this development is the correctness theorem for |
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$\idt{compile}$. We first establish a lemma about $\idt{exec}$ and |
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list append. |
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*}; |
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lemma exec_append: |
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"ALL stack. exec (xs @ ys) stack env = |
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exec ys (exec xs stack env) env" (is "?P xs"); |
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proof (induct ?P xs type: list); |
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show "?P []"; by simp; |
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fix x xs; assume "?P xs"; |
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show "?P (x # xs)" (is "?Q x"); |
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proof (induct ?Q x type: instr); |
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fix val; show "?Q (Const val)"; by (simp!); |
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next; |
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fix adr; show "?Q (Load adr)"; by (simp!); |
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next; |
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fix fun; show "?Q (Apply fun)"; by (simp!); |
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qed; |
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qed; |
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theorem correctness: "execute (compile e) env = eval e env"; |
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proof -; |
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have "ALL stack. exec (compile e) stack env = |
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eval e env # stack" (is "?P e"); |
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proof (induct ?P e type: expr); |
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fix adr; show "?P (Variable adr)"; by (simp!); |
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next; |
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fix val; show "?P (Constant val)"; by (simp!); |
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next; |
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fix fun e1 e2; assume "?P e1" "?P e2"; show "?P (Binop fun e1 e2)"; |
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by (simp! add: exec_append); |
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qed; |
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thus ?thesis; by (simp add: execute_def); |
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qed; |
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text {* |
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\bigskip In the proofs above, the \name{simp} method does quite a lot |
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of work behind the scenes (mostly ``functional program execution''). |
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Subsequently, the same reasoning is elaborated in detail --- at most |
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one recursive function definition is used at a time. Thus we get a |
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better idea of what is actually going on. |
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*}; |
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lemma exec_append: |
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"ALL stack. exec (xs @ ys) stack env |
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= exec ys (exec xs stack env) env" (is "?P xs"); |
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proof (induct ?P xs); |
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show "?P []" (is "ALL s. ?Q s"); |
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proof; |
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fix s; have "exec ([] @ ys) s env = exec ys s env"; by simp; |
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also; have "... = exec ys (exec [] s env) env"; by simp; |
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finally; show "?Q s"; .; |
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qed; |
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fix x xs; assume hyp: "?P xs"; |
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show "?P (x # xs)"; |
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proof (induct x); |
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fix val; |
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show "?P (Const val # xs)" (is "ALL s. ?Q s"); |
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proof; |
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fix s; |
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have "exec ((Const val # xs) @ ys) s env = exec (Const val # xs @ ys) s env"; |
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by simp; |
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also; have "... = exec (xs @ ys) (val # s) env"; by simp; |
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also; from hyp; have "... = exec ys (exec xs (val # s) env) env"; ..; |
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also; have "... = exec ys (exec (Const val # xs) s env) env"; by simp; |
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finally; show "?Q s"; .; |
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qed; |
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next; |
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fix adr; from hyp; show "?P (Load adr # xs)"; by simp -- {* same as above *}; |
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next; |
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fix fun; |
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show "?P (Apply fun # xs)" (is "ALL s. ?Q s"); |
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proof; |
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fix s; |
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have "exec ((Apply fun # xs) @ ys) s env = exec (Apply fun # xs @ ys) s env"; |
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by simp; |
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also; have "... = exec (xs @ ys) (fun (hd s) (hd (tl s)) # (tl (tl s))) env"; |
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by simp; |
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also; from hyp; have "... |
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= exec ys (exec xs (fun (hd s) (hd (tl s)) # tl (tl s)) env) env"; ..; |
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also; have "... = exec ys (exec (Apply fun # xs) s env) env"; by simp; |
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finally; show "?Q s"; .; |
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qed; |
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qed; |
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qed; |
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theorem correctness: "execute (compile e) env = eval e env"; |
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proof -; |
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have exec_compile: |
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"ALL stack. exec (compile e) stack env = eval e env # stack" (is "?P e"); |
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proof (induct e); |
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fix adr; show "?P (Variable adr)" (is "ALL s. ?Q s"); |
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proof; |
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fix s; |
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have "exec (compile (Variable adr)) s env = exec [Load adr] s env"; by simp; |
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also; have "... = env adr # s"; by simp; |
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also; have "env adr = eval (Variable adr) env"; by simp; |
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finally; show "?Q s"; .; |
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qed; |
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next; |
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fix val; show "?P (Constant val)"; by simp -- {* same as above *}; |
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next; |
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fix fun e1 e2; assume hyp1: "?P e1" and hyp2: "?P e2"; |
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show "?P (Binop fun e1 e2)" (is "ALL s. ?Q s"); |
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proof; |
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fix s; have "exec (compile (Binop fun e1 e2)) s env |
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= exec (compile e2 @ compile e1 @ [Apply fun]) s env"; by simp; |
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also; have "... |
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= exec [Apply fun] (exec (compile e1) (exec (compile e2) s env) env) env"; |
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by (simp only: exec_append); |
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also; from hyp2; have "exec (compile e2) s env = eval e2 env # s"; ..; |
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also; from hyp1; have "exec (compile e1) ... env = eval e1 env # ..."; ..; |
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also; have "exec [Apply fun] ... env |
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= fun (hd ...) (hd (tl ...)) # (tl (tl ...))"; by simp; |
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also; have "... = fun (eval e1 env) (eval e2 env) # s"; by simp; |
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also; have "fun (eval e1 env) (eval e2 env) = eval (Binop fun e1 e2) env"; |
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by simp; |
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finally; show "?Q s"; .; |
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qed; |
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qed; |
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have "execute (compile e) env = hd (exec (compile e) [] env)"; |
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by (simp add: execute_def); |
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also; from exec_compile; have "exec (compile e) [] env = [eval e env]"; ..; |
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also; have "hd ... = eval e env"; by simp; |
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finally; show ?thesis; .; |
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qed; |
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end; |