Theory Compiler0

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theory Compiler0
imports Natural

(*  Title:      HOL/IMP/Compiler.thy
ID: $Id$
Author: Tobias Nipkow, TUM
Copyright 1996 TUM

This is an early version of the compiler, where the abstract machine
has an explicit pc. This turned out to be awkward, and a second
development was started. See Machines.thy and Compiler.thy.
*)


header "A Simple Compiler"

theory Compiler0 imports Natural begin

subsection "An abstract, simplistic machine"

text {* There are only three instructions: *}
datatype instr = ASIN loc aexp | JMPF bexp nat | JMPB nat

text {* We describe execution of programs in the machine by
an operational (small step) semantics:
*}


inductive_set
stepa1 :: "instr list => ((state×nat) × (state×nat))set"
and stepa1' :: "[instr list,state,nat,state,nat] => bool"
("_ \<turnstile> (3⟨_,_⟩/ -1-> ⟨_,_⟩)" [50,0,0,0,0] 50)
for P :: "instr list"
where
"P \<turnstile> ⟨s,m⟩ -1-> ⟨t,n⟩ == ((s,m),t,n) : stepa1 P"
| ASIN[simp]:
"[| n<size P; P!n = ASIN x a |] ==> P \<turnstile> ⟨s,n⟩ -1-> ⟨s[x\<mapsto> a s],Suc n⟩"
| JMPFT[simp,intro]:
"[| n<size P; P!n = JMPF b i; b s |] ==> P \<turnstile> ⟨s,n⟩ -1-> ⟨s,Suc n⟩"
| JMPFF[simp,intro]:
"[| n<size P; P!n = JMPF b i; ~b s; m=n+i |] ==> P \<turnstile> ⟨s,n⟩ -1-> ⟨s,m⟩"
| JMPB[simp]:
"[| n<size P; P!n = JMPB i; i <= n; j = n-i |] ==> P \<turnstile> ⟨s,n⟩ -1-> ⟨s,j⟩"


abbreviation
stepa :: "[instr list,state,nat,state,nat] => bool"
("_ \<turnstile>/ (3⟨_,_⟩/ -*-> ⟨_,_⟩)" [50,0,0,0,0] 50) where
"P \<turnstile> ⟨s,m⟩ -*-> ⟨t,n⟩ == ((s,m),t,n) : ((stepa1 P)^*)"


abbreviation
stepan :: "[instr list,state,nat,nat,state,nat] => bool"
("_ \<turnstile>/ (3⟨_,_⟩/ -(_)-> ⟨_,_⟩)" [50,0,0,0,0,0] 50) where
"P \<turnstile> ⟨s,m⟩ -(i)-> ⟨t,n⟩ == ((s,m),t,n) : (stepa1 P ^^ i)"


subsection "The compiler"

consts compile :: "com => instr list"
primrec
"compile \<SKIP> = []"
"compile (x:==a) = [ASIN x a]"
"compile (c1;c2) = compile c1 @ compile c2"
"compile (\<IF> b \<THEN> c1 \<ELSE> c2) =
[JMPF b (length(compile c1) + 2)] @ compile c1 @
[JMPF (%x. False) (length(compile c2)+1)] @ compile c2"

"compile (\<WHILE> b \<DO> c) = [JMPF b (length(compile c) + 2)] @ compile c @
[JMPB (length(compile c)+1)]"


declare nth_append[simp]

subsection "Context lifting lemmas"

text {*
Some lemmas for lifting an execution into a prefix and suffix
of instructions; only needed for the first proof.
*}

lemma app_right_1:
assumes "is1 \<turnstile> ⟨s1,i1⟩ -1-> ⟨s2,i2⟩"
shows "is1 @ is2 \<turnstile> ⟨s1,i1⟩ -1-> ⟨s2,i2⟩"

using assms
by induct auto

lemma app_left_1:
assumes "is2 \<turnstile> ⟨s1,i1⟩ -1-> ⟨s2,i2⟩"
shows "is1 @ is2 \<turnstile> ⟨s1,size is1+i1⟩ -1-> ⟨s2,size is1+i2⟩"

using assms
by induct auto

declare rtrancl_induct2 [induct set: rtrancl]

lemma app_right:
assumes "is1 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s2,i2⟩"
shows "is1 @ is2 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s2,i2⟩"

using assms
proof induct
show "is1 @ is2 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s1,i1⟩" by simp
next
fix s1' i1' s2 i2
assume "is1 @ is2 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s1',i1'⟩"
and "is1 \<turnstile> ⟨s1',i1'⟩ -1-> ⟨s2,i2⟩"

thus "is1 @ is2 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s2,i2⟩"
by (blast intro: app_right_1 rtrancl_trans)
qed

lemma app_left:
assumes "is2 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s2,i2⟩"
shows "is1 @ is2 \<turnstile> ⟨s1,size is1+i1⟩ -*-> ⟨s2,size is1+i2⟩"

using assms
proof induct
show "is1 @ is2 \<turnstile> ⟨s1,length is1 + i1⟩ -*-> ⟨s1,length is1 + i1⟩" by simp
next
fix s1' i1' s2 i2
assume "is1 @ is2 \<turnstile> ⟨s1,length is1 + i1⟩ -*-> ⟨s1',length is1 + i1'⟩"
and "is2 \<turnstile> ⟨s1',i1'⟩ -1-> ⟨s2,i2⟩"

thus "is1 @ is2 \<turnstile> ⟨s1,length is1 + i1⟩ -*-> ⟨s2,length is1 + i2⟩"
by (blast intro: app_left_1 rtrancl_trans)
qed

lemma app_left2:
"[| is2 \<turnstile> ⟨s1,i1⟩ -*-> ⟨s2,i2⟩; j1 = size is1+i1; j2 = size is1+i2 |] ==>
is1 @ is2 \<turnstile> ⟨s1,j1⟩ -*-> ⟨s2,j2⟩"

by (simp add: app_left)

lemma app1_left:
assumes "is \<turnstile> ⟨s1,i1⟩ -*-> ⟨s2,i2⟩"
shows "instr # is \<turnstile> ⟨s1,Suc i1⟩ -*-> ⟨s2,Suc i2⟩"

proof -
from app_left [OF assms, of "[instr]"]
show ?thesis by simp
qed

subsection "Compiler correctness"

declare rtrancl_into_rtrancl[trans]
converse_rtrancl_into_rtrancl[trans]
rtrancl_trans[trans]


text {*
The first proof; The statement is very intuitive,
but application of induction hypothesis requires the above lifting lemmas
*}

theorem
assumes "⟨c,s⟩ -->c t"
shows "compile c \<turnstile> ⟨s,0⟩ -*-> ⟨t,length(compile c)⟩" (is "?P c s t")

using assms
proof induct
show "!!s. ?P \<SKIP> s s" by simp
next
show "!!a s x. ?P (x :== a) s (s[x\<mapsto> a s])" by force
next
fix c0 c1 s0 s1 s2
assume "?P c0 s0 s1"
hence "compile c0 @ compile c1 \<turnstile> ⟨s0,0⟩ -*-> ⟨s1,length(compile c0)⟩"
by (rule app_right)
moreover assume "?P c1 s1 s2"
hence "compile c0 @ compile c1 \<turnstile> ⟨s1,length(compile c0)⟩ -*->
⟨s2,length(compile c0)+length(compile c1)⟩"

proof -
show "!!is1 is2 s1 s2 i2.
is2 \<turnstile> ⟨s1,0⟩ -*-> ⟨s2,i2⟩ ==>
is1 @ is2 \<turnstile> ⟨s1,size is1⟩ -*-> ⟨s2,size is1+i2⟩"

using app_left[of _ 0] by simp
qed
ultimately have "compile c0 @ compile c1 \<turnstile> ⟨s0,0⟩ -*->
⟨s2,length(compile c0)+length(compile c1)⟩"

by (rule rtrancl_trans)
thus "?P (c0; c1) s0 s2" by simp
next
fix b c0 c1 s0 s1
let ?comp = "compile(\<IF> b \<THEN> c0 \<ELSE> c1)"
assume "b s0" and IH: "?P c0 s0 s1"
hence "?comp \<turnstile> ⟨s0,0⟩ -1-> ⟨s0,1⟩" by auto
also from IH
have "?comp \<turnstile> ⟨s0,1⟩ -*-> ⟨s1,length(compile c0)+1⟩"
by(auto intro:app1_left app_right)
also have "?comp \<turnstile> ⟨s1,length(compile c0)+1⟩ -1-> ⟨s1,length ?comp⟩"
by(auto)
finally show "?P (\<IF> b \<THEN> c0 \<ELSE> c1) s0 s1" .
next
fix b c0 c1 s0 s1
let ?comp = "compile(\<IF> b \<THEN> c0 \<ELSE> c1)"
assume "¬b s0" and IH: "?P c1 s0 s1"
hence "?comp \<turnstile> ⟨s0,0⟩ -1-> ⟨s0,length(compile c0) + 2⟩" by auto
also from IH
have "?comp \<turnstile> ⟨s0,length(compile c0)+2⟩ -*-> ⟨s1,length ?comp⟩"
by (force intro!: app_left2 app1_left)
finally show "?P (\<IF> b \<THEN> c0 \<ELSE> c1) s0 s1" .
next
fix b c and s::state
assume "¬b s"
thus "?P (\<WHILE> b \<DO> c) s s" by force
next
fix b c and s0::state and s1 s2
let ?comp = "compile(\<WHILE> b \<DO> c)"
assume "b s0" and
IHc: "?P c s0 s1" and IHw: "?P (\<WHILE> b \<DO> c) s1 s2"

hence "?comp \<turnstile> ⟨s0,0⟩ -1-> ⟨s0,1⟩" by auto
also from IHc
have "?comp \<turnstile> ⟨s0,1⟩ -*-> ⟨s1,length(compile c)+1⟩"
by (auto intro: app1_left app_right)
also have "?comp \<turnstile> ⟨s1,length(compile c)+1⟩ -1-> ⟨s1,0⟩" by simp
also note IHw
finally show "?P (\<WHILE> b \<DO> c) s0 s2".
qed

text {*
Second proof; statement is generalized to cater for prefixes and suffixes;
needs none of the lifting lemmas, but instantiations of pre/suffix.
*}

(*
theorem assumes A: "⟨c,s⟩ -->c t"
shows "!!a z. a@compile c@z \<turnstile> ⟨s,size a⟩ -*-> ⟨t,size a + size(compile c)⟩"
(is "!!a z. ?P c s t a z")
proof -
from A show "!!a z. ?thesis a z"
proof induct
case Skip thus ?case by simp
next
case Assign thus ?case by (force intro!: ASIN)
next
fix c1 c2 s s' s'' a z
assume IH1: "!!a z. ?P c1 s s' a z" and IH2: "!!a z. ?P c2 s' s'' a z"
from IH1 IH2[of "a@compile c1"]
show "?P (c1;c2) s s'' a z"
by(simp add:add_assoc[THEN sym])(blast intro:rtrancl_trans)
next
(* at this point I gave up converting to structured proofs *)
(* \<IF> b \<THEN> c0 \<ELSE> c1; case b is true *)
apply(intro strip)
(* instantiate assumption sufficiently for later: *)
apply(erule_tac x = "a@[?I]" in allE)
apply(simp)
(* execute JMPF: *)
apply(rule converse_rtrancl_into_rtrancl)
apply(force intro!: JMPFT)
(* execute compile c0: *)
apply(rule rtrancl_trans)
apply(erule allE)
apply assumption
(* execute JMPF: *)
apply(rule r_into_rtrancl)
apply(force intro!: JMPFF)
(* end of case b is true *)
apply(intro strip)
apply(erule_tac x = "a@[?I]@compile c0@[?J]" in allE)
apply(simp add:add_assoc)
apply(rule converse_rtrancl_into_rtrancl)
apply(force intro!: JMPFF)
apply(blast)
apply(force intro: JMPFF)
apply(intro strip)
apply(erule_tac x = "a@[?I]" in allE)
apply(erule_tac x = a in allE)
apply(simp)
apply(rule converse_rtrancl_into_rtrancl)
apply(force intro!: JMPFT)
apply(rule rtrancl_trans)
apply(erule allE)
apply assumption
apply(rule converse_rtrancl_into_rtrancl)
apply(force intro!: JMPB)
apply(simp)
done
*)

text {* Missing: the other direction! I did much of it, and although
the main lemma is very similar to the one in the new development, the
lemmas surrounding it seemed much more complicated. In the end I gave
up. *}


end