--- a/thys/Recursive.thy Thu May 02 12:49:15 2013 +0100
+++ b/thys/Recursive.thy Thu May 02 12:49:33 2013 +0100
@@ -6,7 +6,6 @@
where
"addition m n p = [Dec m 4, Inc n, Inc p, Goto 0, Dec p 7, Inc m, Goto 4]"
-(* moves register m to register n *)
fun mv_box :: "nat \<Rightarrow> nat \<Rightarrow> abc_prog"
where
"mv_box m n = [Dec m 3, Inc n, Goto 0]"
@@ -19,7 +18,7 @@
text {* The compilation of @{text "s"}-operator. *}
definition rec_ci_s :: "abc_inst list"
where
- "rec_ci_s \<equiv> addition 0 1 2 [+] [Inc 1]"
+ "rec_ci_s \<equiv> (addition 0 1 2 [+] [Inc 1])"
text {* The compilation of @{text "id i j"}-operator *}
@@ -37,7 +36,8 @@
"empty_boxes 0 = []" |
"empty_boxes (Suc n) = empty_boxes n [+] [Dec n 2, Goto 0]"
-fun cn_merge_gs :: "(abc_inst list \<times> nat \<times> nat) list \<Rightarrow> nat \<Rightarrow> abc_inst list"
+fun cn_merge_gs ::
+ "(abc_inst list \<times> nat \<times> nat) list \<Rightarrow> nat \<Rightarrow> abc_inst list"
where
"cn_merge_gs [] p = []" |
"cn_merge_gs (g # gs) p =
@@ -91,7 +91,20 @@
declare abc_steps_l.simps[simp del] abc_fetch.simps[simp del]
abc_step_l.simps[simp del]
-fun addition_inv :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> bool"
+inductive_cases terminate_pr_reverse: "terminate (Pr n f g) xs"
+
+inductive_cases terminate_z_reverse[elim!]: "terminate z xs"
+
+inductive_cases terminate_s_reverse[elim!]: "terminate s xs"
+
+inductive_cases terminate_id_reverse[elim!]: "terminate (id m n) xs"
+
+inductive_cases terminate_cn_reverse[elim!]: "terminate (Cn n f gs) xs"
+
+inductive_cases terminate_mn_reverse[elim!]:"terminate (Mn n f) xs"
+
+fun addition_inv :: "nat \<times> nat list \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow>
+ nat list \<Rightarrow> bool"
where
"addition_inv (as, lm') m n p lm =
(let sn = lm ! n in
@@ -140,18 +153,21 @@
else if as = 4 then 0
else 0)"
-fun addition_measure :: "((nat \<times> nat list) \<times> nat \<times> nat) \<Rightarrow> (nat \<times> nat \<times> nat)"
+fun addition_measure :: "((nat \<times> nat list) \<times> nat \<times> nat) \<Rightarrow>
+ (nat \<times> nat \<times> nat)"
where
"addition_measure ((as, lm), m, p) =
(addition_stage1 (as, lm) m p,
addition_stage2 (as, lm) m p,
addition_stage3 (as, lm) m p)"
-definition addition_LE :: "(((nat \<times> nat list) \<times> nat \<times> nat) \<times> ((nat \<times> nat list) \<times> nat \<times> nat)) set"
+definition addition_LE :: "(((nat \<times> nat list) \<times> nat \<times> nat) \<times>
+ ((nat \<times> nat list) \<times> nat \<times> nat)) set"
where "addition_LE \<equiv> (inv_image lex_triple addition_measure)"
lemma [simp]: "wf addition_LE"
-by(auto simp: wf_inv_image addition_LE_def lex_triple_def lex_pair_def)
+by(auto simp: wf_inv_image addition_LE_def lex_triple_def
+ lex_pair_def)
declare addition_inv.simps[simp del]
@@ -313,23 +329,17 @@
qed
lemma compile_s_correct':
- "{\<lambda>nl. nl = n # 0 \<up> 2 @ anything}
- addition 0 (Suc 0) 2 [+] [Inc (Suc 0)]
- {\<lambda>nl. nl = n # Suc n # 0 # anything}"
+ "{\<lambda>nl. nl = n # 0 \<up> 2 @ anything} addition 0 (Suc 0) 2 [+] [Inc (Suc 0)] {\<lambda>nl. nl = n # Suc n # 0 # anything}"
proof(rule_tac abc_Hoare_plus_halt)
- show "{\<lambda>nl. nl = n # 0 \<up> 2 @ anything}
- addition 0 (Suc 0) 2
- {\<lambda>nl. addition_inv (7, nl) 0 (Suc 0) 2 (n # 0 \<up> 2 @ anything)}"
+ show "{\<lambda>nl. nl = n # 0 \<up> 2 @ anything} addition 0 (Suc 0) 2 {\<lambda> nl. addition_inv (7, nl) 0 (Suc 0) 2 (n # 0 \<up> 2 @ anything)}"
by(rule_tac addition_correct, auto simp: numeral_2_eq_2)
next
- show "{\<lambda>nl. addition_inv (7, nl) 0 (Suc 0) 2 (n # 0 \<up> 2 @ anything)}
- [Inc (Suc 0)]
- {\<lambda>nl. nl = n # Suc n # 0 # anything}"
+ show "{\<lambda>nl. addition_inv (7, nl) 0 (Suc 0) 2 (n # 0 \<up> 2 @ anything)} [Inc (Suc 0)] {\<lambda>nl. nl = n # Suc n # 0 # anything}"
by(rule_tac abc_Hoare_haltI, rule_tac x = 1 in exI, auto simp: addition_inv.simps
abc_steps_l.simps abc_step_l.simps abc_fetch.simps numeral_2_eq_2 abc_lm_s.simps abc_lm_v.simps)
qed
-declare rec_eval.simps[simp del]
+declare rec_exec.simps[simp del]
lemma abc_comp_commute: "(A [+] B) [+] C = A [+] (B [+] C)"
apply(auto simp: abc_comp.simps abc_shift.simps)
@@ -339,24 +349,24 @@
lemma compile_z_correct:
- "\<lbrakk>rec_ci z = (ap, arity, fp); rec_eval z [n] = r\<rbrakk> \<Longrightarrow>
+ "\<lbrakk>rec_ci z = (ap, arity, fp); rec_exec z [n] = r\<rbrakk> \<Longrightarrow>
{\<lambda>nl. nl = n # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = n # r # 0 \<up> (fp - Suc arity) @ anything}"
apply(rule_tac abc_Hoare_haltI)
apply(rule_tac x = 1 in exI)
apply(auto simp: abc_steps_l.simps rec_ci.simps rec_ci_z_def
- numeral_2_eq_2 abc_fetch.simps abc_step_l.simps rec_eval.simps)
+ numeral_2_eq_2 abc_fetch.simps abc_step_l.simps rec_exec.simps)
done
lemma compile_s_correct:
- "\<lbrakk>rec_ci s = (ap, arity, fp); rec_eval s [n] = r\<rbrakk> \<Longrightarrow>
+ "\<lbrakk>rec_ci s = (ap, arity, fp); rec_exec s [n] = r\<rbrakk> \<Longrightarrow>
{\<lambda>nl. nl = n # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = n # r # 0 \<up> (fp - Suc arity) @ anything}"
-apply(auto simp: rec_ci.simps rec_ci_s_def compile_s_correct' rec_eval.simps)
+apply(auto simp: rec_ci.simps rec_ci_s_def compile_s_correct' rec_exec.simps)
done
lemma compile_id_correct':
assumes "n < length args"
shows "{\<lambda>nl. nl = args @ 0 \<up> 2 @ anything} addition n (length args) (Suc (length args))
- {\<lambda>nl. nl = args @ rec_eval (recf.id (length args) n) args # 0 # anything}"
+ {\<lambda>nl. nl = args @ rec_exec (recf.id (length args) n) args # 0 # anything}"
proof -
have "{\<lambda>nl. nl = args @ 0 \<up> 2 @ anything} addition n (length args) (Suc (length args))
{\<lambda>nl. addition_inv (7, nl) n (length args) (Suc (length args)) (args @ 0 \<up> 2 @ anything)}"
@@ -364,12 +374,12 @@
by(rule_tac addition_correct, auto simp: numeral_2_eq_2 nth_append)
thus "?thesis"
using assms
- by(simp add: addition_inv.simps rec_eval.simps
+ by(simp add: addition_inv.simps rec_exec.simps
nth_append numeral_2_eq_2 list_update_append)
qed
lemma compile_id_correct:
- "\<lbrakk>n < m; length xs = m; rec_ci (recf.id m n) = (ap, arity, fp); rec_eval (recf.id m n) xs = r\<rbrakk>
+ "\<lbrakk>n < m; length xs = m; rec_ci (recf.id m n) = (ap, arity, fp); rec_exec (recf.id m n) xs = r\<rbrakk>
\<Longrightarrow> {\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ r # 0 \<up> (fp - Suc arity) @ anything}"
apply(auto simp: rec_ci.simps rec_ci_id.simps compile_id_correct')
done
@@ -391,8 +401,8 @@
done
lemma param_pattern:
- "\<lbrakk>terminates f xs; rec_ci f = (p, arity, fp)\<rbrakk> \<Longrightarrow> length xs = arity"
-apply(induct arbitrary: p arity fp rule: terminates.induct)
+ "\<lbrakk>terminate f xs; rec_ci f = (p, arity, fp)\<rbrakk> \<Longrightarrow> length xs = arity"
+apply(induct arbitrary: p arity fp rule: terminate.induct)
apply(auto simp: rec_ci.simps)
apply(case_tac "rec_ci f", simp)
apply(case_tac "rec_ci f", case_tac "rec_ci g", simp)
@@ -590,11 +600,11 @@
lemma [simp]:
"\<lbrakk>length xs < gf; gf \<le> ft; n < length gs\<rbrakk>
- \<Longrightarrow> (rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything)
- [ft + n - length xs := rec_eval (gs ! n) xs, 0 := 0] =
- 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ rec_eval (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything"
-using list_update_append[of "rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_eval i xs) (take n gs)"
- "0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything" "ft + n - length xs" "rec_eval (gs ! n) xs"]
+ \<Longrightarrow> (rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything)
+ [ft + n - length xs := rec_exec (gs ! n) xs, 0 := 0] =
+ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ rec_exec (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything"
+using list_update_append[of "rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_exec i xs) (take n gs)"
+ "0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything" "ft + n - length xs" "rec_exec (gs ! n) xs"]
apply(auto)
apply(case_tac "length gs - n", simp, simp add: list_update.simps replicate_Suc_iff_anywhere Suc_diff_Suc del: replicate_Suc)
apply(simp add: list_update.simps)
@@ -602,14 +612,14 @@
lemma compile_cn_gs_correct':
assumes
- g_cond: "\<forall>g\<in>set (take n gs). terminates g xs \<and>
- (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_eval g xs # 0 \<up> (xb - Suc xa) @ xc}))"
+ g_cond: "\<forall>g\<in>set (take n gs). terminate g xs \<and>
+ (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
and ft: "ft = max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
shows
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything}
cn_merge_gs (map rec_ci (take n gs)) ft
{\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0\<up>(length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0\<up>(length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
using g_cond
proof(induct n)
case 0
@@ -624,17 +634,17 @@
next
case (Suc n)
have ind': "\<forall>g\<in>set (take n gs).
- terminates g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow>
- (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_eval g xs # 0 \<up> (xb - Suc xa) @ xc})) \<Longrightarrow>
+ terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow>
+ (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc})) \<Longrightarrow>
{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
by fact
have g_newcond: "\<forall>g\<in>set (take (Suc n) gs).
- terminates g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_eval g xs # 0 \<up> (xb - Suc xa) @ xc}))"
+ terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
by fact
from g_newcond have ind:
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
apply(rule_tac ind', rule_tac ballI, erule_tac x = g in ballE, simp_all add: take_Suc)
by(case_tac "n < length gs", simp add:take_Suc_conv_app_nth, simp)
show "?case"
@@ -650,17 +660,17 @@
moreover have
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything}
cn_merge_gs (map rec_ci (take n gs)) ft [+] (gp [+] mv_box ga (ft + n))
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @
- rec_eval (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @
+ rec_exec (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
proof(rule_tac abc_Hoare_plus_halt)
show "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci (take n gs)) ft
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}"
using ind by simp
next
have x: "gs!n \<in> set (take (Suc n) gs)"
using h
by(simp add: take_Suc_conv_app_nth)
- have b: "terminates (gs!n) xs"
+ have b: "terminate (gs!n) xs"
using a g_newcond h x
by(erule_tac x = "gs!n" in ballE, simp_all)
hence c: "length xs = ga"
@@ -672,18 +682,18 @@
rule_tac f = "(\<lambda>(aprog, p, n). n)" and x = "rec_ci (gs!n)" in image_eqI, simp,
rule_tac x = "gs!n" in image_eqI, simp, simp)
show "{\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything} gp [+] mv_box ga (ft + n)
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs)
- (take n gs) @ rec_eval (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything} gp [+] mv_box ga (ft + n)
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs)
+ (take n gs) @ rec_exec (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
proof(rule_tac abc_Hoare_plus_halt)
- show "{\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything} gp
- {\<lambda>nl. nl = xs @ (rec_eval (gs!n) xs) # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_eval i xs)
+ show "{\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything} gp
+ {\<lambda>nl. nl = xs @ (rec_exec (gs!n) xs) # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_exec i xs)
(take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything}"
proof -
have
- "({\<lambda>nl. nl = xs @ 0 \<up> (gf - ga) @ 0\<up>(ft - gf)@map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}
- gp {\<lambda>nl. nl = xs @ (rec_eval (gs!n) xs) # 0 \<up> (gf - Suc ga) @
- 0\<up>(ft - gf)@map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything})"
+ "({\<lambda>nl. nl = xs @ 0 \<up> (gf - ga) @ 0\<up>(ft - gf)@map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything}
+ gp {\<lambda>nl. nl = xs @ (rec_exec (gs!n) xs) # 0 \<up> (gf - Suc ga) @
+ 0\<up>(ft - gf)@map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 \<up> Suc (length xs) @ anything})"
using a g_newcond h x
apply(erule_tac x = "gs!n" in ballE)
apply(simp, simp)
@@ -694,33 +704,33 @@
qed
next
show
- "{\<lambda>nl. nl = xs @ rec_eval (gs ! n) xs #
- 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything}
+ "{\<lambda>nl. nl = xs @ rec_exec (gs ! n) xs #
+ 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything}
mv_box ga (ft + n)
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @
- rec_eval (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @
+ rec_exec (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
proof -
- have "{\<lambda>nl. nl = xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything}
- mv_box ga (ft + n) {\<lambda>nl. nl = (xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything)
- [ft + n := (xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_eval i xs) (take n gs) @
+ have "{\<lambda>nl. nl = xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything}
+ mv_box ga (ft + n) {\<lambda>nl. nl = (xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything)
+ [ft + n := (xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_exec i xs) (take n gs) @
0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything) ! ga +
- (xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything) !
+ (xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything) !
(ft + n), ga := 0]}"
using a c d e h
apply(rule_tac mv_box_correct)
apply(simp, arith, arith)
done
- moreover have "(xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything)
- [ft + n := (xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_eval i xs) (take n gs) @
+ moreover have "(xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything)
+ [ft + n := (xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @ map (\<lambda>i. rec_exec i xs) (take n gs) @
0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything) ! ga +
- (xs @ rec_eval (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
- map (\<lambda>i. rec_eval i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything) !
+ (xs @ rec_exec (gs ! n) xs # 0 \<up> (ft - Suc (length xs)) @
+ map (\<lambda>i. rec_exec i xs) (take n gs) @ 0 \<up> (length gs - n) @ 0 # 0 \<up> length xs @ anything) !
(ft + n), ga := 0]=
- xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ rec_eval (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything"
+ xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ rec_exec (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything"
using a c d e h
by(simp add: list_update_append nth_append length_replicate split: if_splits del: list_update.simps(2), auto)
ultimately show "?thesis"
@@ -732,7 +742,7 @@
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything}
cn_merge_gs (map rec_ci (take n gs)) ft [+] (case rec_ci (gs ! n) of (gprog, gpara, gn) \<Rightarrow>
gprog [+] mv_box gpara (ft + min (length gs) n))
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) (take n gs) @ rec_eval (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) (take n gs) @ rec_exec (gs ! n) xs # 0 \<up> (length gs - Suc n) @ 0 # 0 \<up> length xs @ anything}"
by simp
qed
next
@@ -740,7 +750,7 @@
have h: "\<not> n < length gs" by fact
hence ind':
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything} cn_merge_gs (map rec_ci gs) ft
- {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_eval i xs) gs @ 0 \<up> Suc (length xs) @ anything}"
+ {\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @ map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> Suc (length xs) @ anything}"
using ind
by simp
thus "?thesis"
@@ -751,14 +761,14 @@
lemma compile_cn_gs_correct:
assumes
- g_cond: "\<forall>g\<in>set gs. terminates g xs \<and>
- (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_eval g xs # 0 \<up> (xb - Suc xa) @ xc}))"
+ g_cond: "\<forall>g\<in>set gs. terminate g xs \<and>
+ (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
and ft: "ft = max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
shows
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (ft + length gs) @ anything}
cn_merge_gs (map rec_ci gs) ft
{\<lambda>nl. nl = xs @ 0 \<up> (ft - length xs) @
- map (\<lambda>i. rec_eval i xs) gs @ 0 \<up> Suc (length xs) @ anything}"
+ map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> Suc (length xs) @ anything}"
using assms
using compile_cn_gs_correct'[of "length gs" gs xs ft ffp anything ]
apply(auto)
@@ -957,14 +967,14 @@
lemma save_paras:
"{\<lambda>nl. nl = xs @ 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length xs) @
- map (\<lambda>i. rec_eval i xs) gs @ 0 \<up> Suc (length xs) @ anything}
+ map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> Suc (length xs) @ anything}
mv_boxes 0 (Suc (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs)) (length xs)
- {\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_eval i xs) gs @ 0 # xs @ anything}"
+ {\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_exec i xs) gs @ 0 # xs @ anything}"
proof -
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
- have "{\<lambda>nl. nl = [] @ xs @ (0\<up>(?ft - length xs) @ map (\<lambda>i. rec_eval i xs) gs @ [0]) @
+ have "{\<lambda>nl. nl = [] @ xs @ (0\<up>(?ft - length xs) @ map (\<lambda>i. rec_exec i xs) gs @ [0]) @
0 \<up> (length xs) @ anything} mv_boxes 0 (Suc ?ft + length gs) (length xs)
- {\<lambda>nl. nl = [] @ 0 \<up> (length xs) @ (0\<up>(?ft - length xs) @ map (\<lambda>i. rec_eval i xs) gs @ [0]) @ xs @ anything}"
+ {\<lambda>nl. nl = [] @ 0 \<up> (length xs) @ (0\<up>(?ft - length xs) @ map (\<lambda>i. rec_exec i xs) gs @ [0]) @ xs @ anything}"
by(rule_tac mv_boxes_correct, auto)
thus "?thesis"
by(simp add: replicate_merge_anywhere)
@@ -978,15 +988,15 @@
lemma restore_new_paras:
"ffp \<ge> length gs
- \<Longrightarrow> {\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_eval i xs) gs @ 0 # xs @ anything}
+ \<Longrightarrow> {\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ map (\<lambda>i. rec_exec i xs) gs @ 0 # xs @ anything}
mv_boxes (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) 0 (length gs)
- {\<lambda>nl. nl = map (\<lambda>i. rec_eval i xs) gs @ 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ 0 # xs @ anything}"
+ {\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ 0 # xs @ anything}"
proof -
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
assume j: "ffp \<ge> length gs"
- hence "{\<lambda> nl. nl = [] @ 0\<up>length gs @ 0\<up>(?ft - length gs) @ map (\<lambda>i. rec_eval i xs) gs @ ((0 # xs) @ anything)}
+ hence "{\<lambda> nl. nl = [] @ 0\<up>length gs @ 0\<up>(?ft - length gs) @ map (\<lambda>i. rec_exec i xs) gs @ ((0 # xs) @ anything)}
mv_boxes ?ft 0 (length gs)
- {\<lambda> nl. nl = [] @ map (\<lambda>i. rec_eval i xs) gs @ 0\<up>(?ft - length gs) @ 0\<up>length gs @ ((0 # xs) @ anything)}"
+ {\<lambda> nl. nl = [] @ map (\<lambda>i. rec_exec i xs) gs @ 0\<up>(?ft - length gs) @ 0\<up>length gs @ ((0 # xs) @ anything)}"
by(rule_tac mv_boxes_correct2, auto)
moreover have "?ft \<ge> length gs"
using j
@@ -1007,29 +1017,29 @@
"\<lbrakk>far = length gs;
ffp \<le> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)));
far < ffp\<rbrakk>
-\<Longrightarrow> {\<lambda>nl. nl = map (\<lambda>i. rec_eval i xs) gs @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> max (Suc (length xs))
+\<Longrightarrow> {\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> max (Suc (length xs))
(Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ xs @ anything}
mv_box far (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))))
- {\<lambda>nl. nl = map (\<lambda>i. rec_eval i xs) gs @
+ {\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @
0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length gs) @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}"
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}"
proof -
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
thm mv_box_correct
- let ?lm= " map (\<lambda>i. rec_eval i xs) gs @ rec_eval (Cn (length xs) f gs) xs # 0 \<up> ?ft @ xs @ anything"
+ let ?lm= " map (\<lambda>i. rec_exec i xs) gs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> ?ft @ xs @ anything"
assume h: "far = length gs" "ffp \<le> ?ft" "far < ffp"
hence "{\<lambda> nl. nl = ?lm} mv_box far ?ft {\<lambda> nl. nl = ?lm[?ft := ?lm!far + ?lm!?ft, far := 0]}"
apply(rule_tac mv_box_correct)
by(case_tac "rec_ci a", auto)
moreover have "?lm[?ft := ?lm!far + ?lm!?ft, far := 0]
- = map (\<lambda>i. rec_eval i xs) gs @
+ = map (\<lambda>i. rec_exec i xs) gs @
0 \<up> (?ft - length gs) @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
using h
apply(simp add: nth_append)
- using list_update_length[of "map (\<lambda>i. rec_eval i xs) gs @ rec_eval (Cn (length xs) f gs) xs #
- 0 \<up> (?ft - Suc (length gs))" 0 "0 \<up> length gs @ xs @ anything" "rec_eval (Cn (length xs) f gs) xs"]
+ using list_update_length[of "map (\<lambda>i. rec_exec i xs) gs @ rec_exec (Cn (length xs) f gs) xs #
+ 0 \<up> (?ft - Suc (length gs))" 0 "0 \<up> length gs @ xs @ anything" "rec_exec (Cn (length xs) f gs) xs"]
apply(simp add: replicate_merge_anywhere replicate_Suc_iff_anywhere del: replicate_Suc)
by(simp add: list_update_append list_update.simps replicate_Suc_iff_anywhere del: replicate_Suc)
ultimately show "?thesis"
@@ -1108,21 +1118,21 @@
lemma clean_paras:
"ffp \<ge> length gs \<Longrightarrow>
- {\<lambda>nl. nl = map (\<lambda>i. rec_eval i xs) gs @
+ {\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @
0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length gs) @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}
empty_boxes (length gs)
{\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}"
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}"
proof-
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
assume h: "length gs \<le> ffp"
- let ?lm = "map (\<lambda>i. rec_eval i xs) gs @ 0 \<up> (?ft - length gs) @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
+ let ?lm = "map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> (?ft - length gs) @
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
have "{\<lambda> nl. nl = ?lm} empty_boxes (length gs) {\<lambda> nl. nl = 0\<up>length gs @ drop (length gs) ?lm}"
by(rule_tac empty_boxes_correct, simp)
moreover have "0\<up>length gs @ drop (length gs) ?lm
- = 0 \<up> ?ft @ rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
+ = 0 \<up> ?ft @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
using h
by(simp add: replicate_merge_anywhere)
ultimately show "?thesis"
@@ -1132,20 +1142,20 @@
lemma restore_rs:
"{\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @
- rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}
+ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}
mv_box (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) (length xs)
{\<lambda>nl. nl = 0 \<up> length xs @
- rec_eval (Cn (length xs) f gs) xs #
+ rec_exec (Cn (length xs) f gs) xs #
0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - (length xs)) @
0 \<up> length gs @ xs @ anything}"
proof -
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
- let ?lm = "0\<up>(length xs) @ 0\<up>(?ft - (length xs)) @ rec_eval (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
+ let ?lm = "0\<up>(length xs) @ 0\<up>(?ft - (length xs)) @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything"
thm mv_box_correct
have "{\<lambda> nl. nl = ?lm} mv_box ?ft (length xs) {\<lambda> nl. nl = ?lm[length xs := ?lm!?ft + ?lm!(length xs), ?ft := 0]}"
by(rule_tac mv_box_correct, simp, simp)
moreover have "?lm[length xs := ?lm!?ft + ?lm!(length xs), ?ft := 0]
- = 0 \<up> length xs @ rec_eval (Cn (length xs) f gs) xs # 0 \<up> (?ft - (length xs)) @ 0 \<up> length gs @ xs @ anything"
+ = 0 \<up> length xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (?ft - (length xs)) @ 0 \<up> length gs @ xs @ anything"
apply(auto simp: list_update_append nth_append)
apply(case_tac ?ft, simp_all add: Suc_diff_le list_update.simps)
apply(simp add: exp_suc replicate_Suc[THEN sym] del: replicate_Suc)
@@ -1156,17 +1166,17 @@
lemma restore_orgin_paras:
"{\<lambda>nl. nl = 0 \<up> length xs @
- rec_eval (Cn (length xs) f gs) xs #
+ rec_exec (Cn (length xs) f gs) xs #
0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length xs) @ 0 \<up> length gs @ xs @ anything}
mv_boxes (Suc (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs)) 0 (length xs)
- {\<lambda>nl. nl = xs @ rec_eval (Cn (length xs) f gs) xs # 0 \<up>
+ {\<lambda>nl. nl = xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up>
(max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs) @ anything}"
proof -
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
thm mv_boxes_correct2
- have "{\<lambda> nl. nl = [] @ 0\<up>(length xs) @ (rec_eval (Cn (length xs) f gs) xs # 0 \<up> (?ft - length xs) @ 0 \<up> length gs) @ xs @ anything}
+ have "{\<lambda> nl. nl = [] @ 0\<up>(length xs) @ (rec_exec (Cn (length xs) f gs) xs # 0 \<up> (?ft - length xs) @ 0 \<up> length gs) @ xs @ anything}
mv_boxes (Suc ?ft + length gs) 0 (length xs)
- {\<lambda> nl. nl = [] @ xs @ (rec_eval (Cn (length xs) f gs) xs # 0 \<up> (?ft - length xs) @ 0 \<up> length gs) @ 0\<up>length xs @ anything}"
+ {\<lambda> nl. nl = [] @ xs @ (rec_exec (Cn (length xs) f gs) xs # 0 \<up> (?ft - length xs) @ 0 \<up> length gs) @ 0\<up>length xs @ anything}"
by(rule_tac mv_boxes_correct2, auto)
thus "?thesis"
by(simp add: replicate_merge_anywhere)
@@ -1174,14 +1184,14 @@
lemma compile_cn_correct':
assumes f_ind:
- "\<And> anything r. rec_eval f (map (\<lambda>g. rec_eval g xs) gs) = rec_eval (Cn (length xs) f gs) xs \<Longrightarrow>
- {\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ 0 \<up> (ffp - far) @ anything} fap
- {\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ rec_eval (Cn (length xs) f gs) xs # 0 \<up> (ffp - Suc far) @ anything}"
+ "\<And> anything r. rec_exec f (map (\<lambda>g. rec_exec g xs) gs) = rec_exec (Cn (length xs) f gs) xs \<Longrightarrow>
+ {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (ffp - far) @ anything} fap
+ {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (ffp - Suc far) @ anything}"
and compile: "rec_ci f = (fap, far, ffp)"
- and term_f: "terminates f (map (\<lambda>g. rec_eval g xs) gs)"
- and g_cond: "\<forall>g\<in>set gs. terminates g xs \<and>
+ and term_f: "terminate f (map (\<lambda>g. rec_exec g xs) gs)"
+ and g_cond: "\<forall>g\<in>set gs. terminate g xs \<and>
(\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow>
- (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_eval g xs # 0 \<up> (xb - Suc xa) @ xc}))"
+ (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
shows
"{\<lambda>nl. nl = xs @ 0 # 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs) @ anything}
cn_merge_gs (map rec_ci gs) (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) [+]
@@ -1191,7 +1201,7 @@
(empty_boxes (length gs) [+]
(mv_box (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) (length xs) [+]
mv_boxes (Suc (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs)) 0 (length xs)))))))
- {\<lambda>nl. nl = xs @ rec_eval (Cn (length xs) f gs) xs #
+ {\<lambda>nl. nl = xs @ rec_exec (Cn (length xs) f gs) xs #
0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs) @ anything}"
proof -
let ?ft = "max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
@@ -1204,8 +1214,8 @@
let ?G = "mv_box ?ft (length xs)"
let ?H = "mv_boxes (Suc (?ft + length gs)) 0 (length xs)"
let ?P1 = "\<lambda>nl. nl = xs @ 0 # 0 \<up> (?ft + length gs) @ anything"
- let ?S = "\<lambda>nl. nl = xs @ rec_eval (Cn (length xs) f gs) xs # 0 \<up> (?ft + length gs) @ anything"
- let ?Q1 = "\<lambda> nl. nl = xs @ 0\<up>(?ft - length xs) @ map (\<lambda> i. rec_eval i xs) gs @ 0\<up>(Suc (length xs)) @ anything"
+ let ?S = "\<lambda>nl. nl = xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (?ft + length gs) @ anything"
+ let ?Q1 = "\<lambda> nl. nl = xs @ 0\<up>(?ft - length xs) @ map (\<lambda> i. rec_exec i xs) gs @ 0\<up>(Suc (length xs)) @ anything"
show "{?P1} (?A [+] (?B [+] (?C [+] (?D [+] (?E [+] (?F [+] (?G [+] ?H))))))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
show "{?P1} ?A {?Q1}"
@@ -1213,13 +1223,13 @@
by(rule_tac compile_cn_gs_correct, auto)
next
let ?Q2 = "\<lambda>nl. nl = 0 \<up> ?ft @
- map (\<lambda>i. rec_eval i xs) gs @ 0 # xs @ anything"
+ map (\<lambda>i. rec_exec i xs) gs @ 0 # xs @ anything"
show "{?Q1} (?B [+] (?C [+] (?D [+] (?E [+] (?F [+] (?G [+] ?H)))))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
show "{?Q1} ?B {?Q2}"
by(rule_tac save_paras)
next
- let ?Q3 = "\<lambda> nl. nl = map (\<lambda>i. rec_eval i xs) gs @ 0\<up>?ft @ 0 # xs @ anything"
+ let ?Q3 = "\<lambda> nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0\<up>?ft @ 0 # xs @ anything"
show "{?Q2} (?C [+] (?D [+] (?E [+] (?F [+] (?G [+] ?H))))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
have "ffp \<ge> length gs"
@@ -1230,7 +1240,7 @@
thus "{?Q2} ?C {?Q3}"
by(erule_tac restore_new_paras)
next
- let ?Q4 = "\<lambda> nl. nl = map (\<lambda>i. rec_eval i xs) gs @ rec_eval (Cn (length xs) f gs) xs # 0\<up>?ft @ xs @ anything"
+ let ?Q4 = "\<lambda> nl. nl = map (\<lambda>i. rec_exec i xs) gs @ rec_exec (Cn (length xs) f gs) xs # 0\<up>?ft @ xs @ anything"
have a: "far = length gs"
using compile term_f
by(drule_tac param_pattern, auto)
@@ -1241,23 +1251,23 @@
by(erule_tac footprint_ge)
show "{?Q3} (?D [+] (?E [+] (?F [+] (?G [+] ?H)))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
- have "{\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ 0 \<up> (ffp - far) @ 0\<up>(?ft - ffp + far) @ 0 # xs @ anything} fap
- {\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ rec_eval (Cn (length xs) f gs) xs #
+ have "{\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (ffp - far) @ 0\<up>(?ft - ffp + far) @ 0 # xs @ anything} fap
+ {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec (Cn (length xs) f gs) xs #
0 \<up> (ffp - Suc far) @ 0\<up>(?ft - ffp + far) @ 0 # xs @ anything}"
- by(rule_tac f_ind, simp add: rec_eval.simps)
+ by(rule_tac f_ind, simp add: rec_exec.simps)
thus "{?Q3} fap {?Q4}"
using a b c
by(simp add: replicate_merge_anywhere,
case_tac "?ft", simp_all add: exp_suc del: replicate_Suc)
next
- let ?Q5 = "\<lambda>nl. nl = map (\<lambda>i. rec_eval i xs) gs @
- 0\<up>(?ft - length gs) @ rec_eval (Cn (length xs) f gs) xs # 0\<up>(length gs)@ xs @ anything"
+ let ?Q5 = "\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @
+ 0\<up>(?ft - length gs) @ rec_exec (Cn (length xs) f gs) xs # 0\<up>(length gs)@ xs @ anything"
show "{?Q4} (?E [+] (?F [+] (?G [+] ?H))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
from a b c show "{?Q4} ?E {?Q5}"
by(erule_tac save_rs, simp_all)
next
- let ?Q6 = "\<lambda>nl. nl = 0\<up>?ft @ rec_eval (Cn (length xs) f gs) xs # 0\<up>(length gs)@ xs @ anything"
+ let ?Q6 = "\<lambda>nl. nl = 0\<up>?ft @ rec_exec (Cn (length xs) f gs) xs # 0\<up>(length gs)@ xs @ anything"
show "{?Q5} (?F [+] (?G [+] ?H)) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
have "length gs \<le> ffp" using a b c
@@ -1265,7 +1275,7 @@
thus "{?Q5} ?F {?Q6}"
by(erule_tac clean_paras)
next
- let ?Q7 = "\<lambda>nl. nl = 0\<up>length xs @ rec_eval (Cn (length xs) f gs) xs # 0\<up>(?ft - (length xs)) @ 0\<up>(length gs)@ xs @ anything"
+ let ?Q7 = "\<lambda>nl. nl = 0\<up>length xs @ rec_exec (Cn (length xs) f gs) xs # 0\<up>(?ft - (length xs)) @ 0\<up>(length gs)@ xs @ anything"
show "{?Q6} (?G [+] ?H) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
show "{?Q6} ?G {?Q7}"
@@ -1283,24 +1293,24 @@
qed
lemma compile_cn_correct:
- assumes termi_f: "terminates f (map (\<lambda>g. rec_eval g xs) gs)"
+ assumes termi_f: "terminate f (map (\<lambda>g. rec_exec g xs) gs)"
and f_ind: "\<And>ap arity fp anything.
rec_ci f = (ap, arity, fp)
- \<Longrightarrow> {\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ 0 \<up> (fp - arity) @ anything} ap
- {\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ rec_eval f (map (\<lambda>g. rec_eval g xs) gs) # 0 \<up> (fp - Suc arity) @ anything}"
+ \<Longrightarrow> {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (fp - arity) @ anything} ap
+ {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec f (map (\<lambda>g. rec_exec g xs) gs) # 0 \<up> (fp - Suc arity) @ anything}"
and g_cond:
- "\<forall>g\<in>set gs. terminates g xs \<and>
- (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_eval g xs # 0 \<up> (xb - Suc xa) @ xc}))"
+ "\<forall>g\<in>set gs. terminate g xs \<and>
+ (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))"
and compile: "rec_ci (Cn n f gs) = (ap, arity, fp)"
and len: "length xs = n"
- shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_eval (Cn n f gs) xs # 0 \<up> (fp - Suc arity) @ anything}"
+ shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec (Cn n f gs) xs # 0 \<up> (fp - Suc arity) @ anything}"
proof(case_tac "rec_ci f")
fix fap far ffp
assume h: "rec_ci f = (fap, far, ffp)"
- then have f_newind: "\<And> anything .{\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ 0 \<up> (ffp - far) @ anything} fap
- {\<lambda>nl. nl = map (\<lambda>g. rec_eval g xs) gs @ rec_eval f (map (\<lambda>g. rec_eval g xs) gs) # 0 \<up> (ffp - Suc far) @ anything}"
+ then have f_newind: "\<And> anything .{\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (ffp - far) @ anything} fap
+ {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec f (map (\<lambda>g. rec_exec g xs) gs) # 0 \<up> (ffp - Suc far) @ anything}"
by(rule_tac f_ind, simp_all)
- thus "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_eval (Cn n f gs) xs # 0 \<up> (fp - Suc arity) @ anything}"
+ thus "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec (Cn n f gs) xs # 0 \<up> (fp - Suc arity) @ anything}"
using compile len h termi_f g_cond
apply(auto simp: rec_ci.simps abc_comp_commute)
apply(rule_tac compile_cn_correct', simp_all)
@@ -1319,9 +1329,9 @@
lemma save_init_rs:
"\<lbrakk>length xs = n; ft = max (n+3) (max fft gft)\<rbrakk>
- \<Longrightarrow> {\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (ft - n) @ anything} mv_box n (Suc n)
- {\<lambda>nl. nl = xs @ 0 # rec_eval f xs # 0 \<up> (ft - Suc n) @ anything}"
-using mv_box_correct[of n "Suc n" "xs @ rec_eval f xs # 0 \<up> (ft - n) @ anything"]
+ \<Longrightarrow> {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (ft - n) @ anything} mv_box n (Suc n)
+ {\<lambda>nl. nl = xs @ 0 # rec_exec f xs # 0 \<up> (ft - Suc n) @ anything}"
+using mv_box_correct[of n "Suc n" "xs @ rec_exec f xs # 0 \<up> (ft - n) @ anything"]
apply(auto simp: list_update_append list_update.simps nth_append split: if_splits)
apply(case_tac "(max (length xs + 3) (max fft gft))", simp_all add: list_update.simps Suc_diff_le)
done
@@ -1371,9 +1381,9 @@
lemma [simp]:
"length xs = n
- \<Longrightarrow> {\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything} mv_box n (Suc n)
- {\<lambda>nl. nl = xs @ 0 # rec_eval f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ x # anything}"
-using mv_box_correct[of n "Suc n" "xs @ rec_eval f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything"]
+ \<Longrightarrow> {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything} mv_box n (Suc n)
+ {\<lambda>nl. nl = xs @ 0 # rec_exec f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ x # anything}"
+using mv_box_correct[of n "Suc n" "xs @ rec_exec f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything"]
apply(simp add: nth_append list_update_append list_update.simps)
apply(case_tac "max (n + 3) (max fft gft)", simp_all)
apply(case_tac nat, simp_all add: Suc_diff_le list_update.simps)
@@ -1457,14 +1467,14 @@
by arith
lemma [simp]: "\<lbrakk>ft = max (n + 3) (max fft gft); length xs = n\<rbrakk> \<Longrightarrow>
- {\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything}
+ {\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything}
[Dec ft (length gap + 7)]
- {\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything}"
+ {\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything}"
apply(simp add: abc_Hoare_halt_def)
apply(rule_tac x = 1 in exI)
apply(auto simp: abc_steps_l.simps abc_step_l.simps abc_fetch.simps nth_append abc_lm_v.simps abc_lm_s.simps list_update_append)
using list_update_length
-[of "(x - Suc y) # rec_eval (Pr (length xs) f g) (xs @ [x - Suc y]) #
+[of "(x - Suc y) # rec_exec (Pr (length xs) f g) (xs @ [x - Suc y]) #
0 \<up> (max (length xs + 3) (max fft gft) - Suc (Suc (length xs)))" "Suc y" anything y]
apply(simp)
done
@@ -1516,7 +1526,7 @@
using adjust_paras'[of xs n x y anything]
by(simp add: abc_comp.simps abc_shift.simps numeral_2_eq_2 numeral_3_eq_3)
-lemma [simp]: "\<lbrakk>rec_ci g = (gap, gar, gft); \<forall>y<x. terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]);
+lemma [simp]: "\<lbrakk>rec_ci g = (gap, gar, gft); \<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]);
length xs = n; Suc y\<le>x\<rbrakk> \<Longrightarrow> gar = Suc (Suc n)"
apply(erule_tac x = y in allE, simp)
apply(drule_tac param_pattern, auto)
@@ -1563,14 +1573,14 @@
apply(simp add: abc_steps_l.simps abc_step_l.simps abc_fetch.simps nth_append abc_lm_v.simps abc_lm_s.simps)
done
-lemma rec_eval_pr_0_simps: "rec_eval (Pr n f g) (xs @ [0]) = rec_eval f xs"
- by(simp add: rec_eval.simps)
+lemma rec_exec_pr_0_simps: "rec_exec (Pr n f g) (xs @ [0]) = rec_exec f xs"
+ by(simp add: rec_exec.simps)
-lemma rec_eval_pr_Suc_simps: "rec_eval (Pr n f g) (xs @ [Suc y])
- = rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])])"
+lemma rec_exec_pr_Suc_simps: "rec_exec (Pr n f g) (xs @ [Suc y])
+ = rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
apply(induct y)
-apply(simp add: rec_eval.simps)
-apply(simp add: rec_eval.simps)
+apply(simp add: rec_exec.simps)
+apply(simp add: rec_exec.simps)
done
lemma [simp]: "Suc (max (n + 3) (max fft gft) - Suc (Suc (Suc n))) = max (n + 3) (max fft gft) - Suc (Suc n)"
@@ -1580,44 +1590,44 @@
assumes code: "code = ([Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)])) @
[Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]"
and len: "length xs = n"
- and g_ind: "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
- {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
+ and g_ind: "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
+ {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
and compile_g: "rec_ci g = (gap, gar, gft)"
- and termi_g: "\<forall> y<x. terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])])"
+ and termi_g: "\<forall> y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
and ft: "ft = max (n + 3) (max fft gft)"
and less: "Suc y \<le> x"
shows
- "\<exists>stp. abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
- code stp = (0, xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything)"
+ "\<exists>stp. abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
+ code stp = (0, xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything)"
proof -
let ?A = "[Dec ft (length gap + 7)]"
let ?B = "gap"
let ?C = "[Inc n, Dec (Suc n) 3, Goto (Suc 0)]"
let ?D = "[Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]"
- have "\<exists> stp. abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
+ have "\<exists> stp. abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
((?A [+] (?B [+] ?C)) @ ?D) stp = (length (?A [+] (?B [+] ?C)),
- xs @ (x - y) # 0 # rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])])
+ xs @ (x - y) # 0 # rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])])
# 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)"
proof -
- have "\<exists> stp. abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
+ have "\<exists> stp. abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
((?A [+] (?B [+] ?C))) stp = (length (?A [+] (?B [+] ?C)), xs @ (x - y) # 0 #
- rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)"
+ rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)"
proof -
- have "{\<lambda> nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything}
+ have "{\<lambda> nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything}
(?A [+] (?B [+] ?C))
{\<lambda> nl. nl = xs @ (x - y) # 0 #
- rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
+ rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
proof(rule_tac abc_Hoare_plus_halt)
- show "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything}
+ show "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything}
[Dec ft (length gap + 7)]
- {\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything}"
+ {\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything}"
using ft len
by(simp)
next
show
- "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything}
+ "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything}
?B [+] ?C
- {\<lambda>nl. nl = xs @ (x - y) # 0 # rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
+ {\<lambda>nl. nl = xs @ (x - y) # 0 # rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
proof(rule_tac abc_Hoare_plus_halt)
have a: "gar = Suc (Suc n)"
using compile_g termi_g len less
@@ -1625,14 +1635,14 @@
have b: "gft > gar"
using compile_g
by(erule_tac footprint_ge)
- show "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything} gap
- {\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) #
- rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
+ show "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ y # anything} gap
+ {\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) #
+ rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
proof -
have
- "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (gft - gar) @ 0\<up>(ft - gft) @ y # anything} gap
- {\<lambda>nl. nl = xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) #
- rec_eval g (xs @ [(x - Suc y), rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (gft - Suc gar) @ 0\<up>(ft - gft) @ y # anything}"
+ "{\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (gft - gar) @ 0\<up>(ft - gft) @ y # anything} gap
+ {\<lambda>nl. nl = xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) #
+ rec_exec g (xs @ [(x - Suc y), rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (gft - Suc gar) @ 0\<up>(ft - gft) @ y # anything}"
using g_ind less by simp
thus "?thesis"
using a b ft
@@ -1640,41 +1650,41 @@
qed
next
show "{\<lambda>nl. nl = xs @ (x - Suc y) #
- rec_eval (Pr n f g) (xs @ [x - Suc y]) #
- rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}
+ rec_exec (Pr n f g) (xs @ [x - Suc y]) #
+ rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}
[Inc n, Dec (Suc n) 3, Goto (Suc 0)]
- {\<lambda>nl. nl = xs @ (x - y) # 0 # rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g)
+ {\<lambda>nl. nl = xs @ (x - y) # 0 # rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g)
(xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything}"
using len less
- using adjust_paras[of xs n "x - Suc y" " rec_eval (Pr n f g) (xs @ [x - Suc y])"
- " rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) #
+ using adjust_paras[of xs n "x - Suc y" " rec_exec (Pr n f g) (xs @ [x - Suc y])"
+ " rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) #
0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything"]
by(simp add: Suc_diff_Suc)
qed
qed
thus "?thesis"
- by(simp add: abc_Hoare_halt_def, auto, rule_tac x = na in exI, case_tac "abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) #
+ by(simp add: abc_Hoare_halt_def, auto, rule_tac x = na in exI, case_tac "abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) #
0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
([Dec ft (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)])) na", simp)
qed
- then obtain stpa where "abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
+ then obtain stpa where "abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (ft - Suc (Suc n)) @ Suc y # anything)
((?A [+] (?B [+] ?C))) stpa = (length (?A [+] (?B [+] ?C)),
- xs @ (x - y) # 0 # rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])])
+ xs @ (x - y) # 0 # rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])])
# 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)" ..
thus "?thesis"
by(erule_tac abc_append_frist_steps_halt_eq)
qed
moreover have
"\<exists> stp. abc_steps_l (length (?A [+] (?B [+] ?C)),
- xs @ (x - y) # 0 # rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)
- ((?A [+] (?B [+] ?C)) @ ?D) stp = (0, xs @ (x - y) # rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) #
+ xs @ (x - y) # 0 # rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)
+ ((?A [+] (?B [+] ?C)) @ ?D) stp = (0, xs @ (x - y) # rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) #
0 # 0 \<up> (ft - Suc (Suc (Suc n))) @ y # anything)"
using len
by(rule_tac loop_back, simp_all)
- moreover have "rec_eval g (xs @ [x - Suc y, rec_eval (Pr n f g) (xs @ [x - Suc y])]) = rec_eval (Pr n f g) (xs @ [x - y])"
+ moreover have "rec_exec g (xs @ [x - Suc y, rec_exec (Pr n f g) (xs @ [x - Suc y])]) = rec_exec (Pr n f g) (xs @ [x - y])"
using less
- thm rec_eval.simps
- apply(case_tac "x - y", simp_all add: rec_eval_pr_Suc_simps)
+ thm rec_exec.simps
+ apply(case_tac "x - y", simp_all add: rec_exec_pr_Suc_simps)
by(subgoal_tac "nat = x - Suc y", simp, arith)
ultimately show "?thesis"
using code ft
@@ -1682,21 +1692,21 @@
qed
lemma [simp]:
- "length xs = n \<Longrightarrow> abc_lm_s (xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3)
+ "length xs = n \<Longrightarrow> abc_lm_s (xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3)
(max fft gft) - Suc (Suc n)) @ 0 # anything) (max (n + 3) (max fft gft)) 0 =
- xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything"
+ xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything"
apply(simp add: abc_lm_s.simps)
-using list_update_length[of "xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n))"
+using list_update_length[of "xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n))"
0 anything 0]
apply(auto simp: Suc_diff_Suc)
apply(simp add: exp_suc[THEN sym] Suc_diff_Suc del: replicate_Suc)
done
lemma [simp]:
- "(xs @ x # rec_eval (Pr (length xs) f g) (xs @ [x]) # 0 \<up> (max (length xs + 3)
+ "(xs @ x # rec_exec (Pr (length xs) f g) (xs @ [x]) # 0 \<up> (max (length xs + 3)
(max fft gft) - Suc (Suc (length xs))) @ 0 # anything) !
max (length xs + 3) (max fft gft) = 0"
-using nth_append_length[of "xs @ x # rec_eval (Pr (length xs) f g) (xs @ [x]) #
+using nth_append_length[of "xs @ x # rec_exec (Pr (length xs) f g) (xs @ [x]) #
0 \<up> (max (length xs + 3) (max fft gft) - Suc (Suc (length xs)))" 0 anything]
by(simp)
@@ -1704,12 +1714,12 @@
assumes less: "y \<le> x"
and len: "length xs = n"
and compile_g: "rec_ci g = (gap, gar, gft)"
- and termi_g: "\<forall> y<x. terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])])"
- and g_ind: "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
- {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
- shows "{\<lambda>nl. nl = xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ y # anything}
+ and termi_g: "\<forall> y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
+ and g_ind: "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
+ {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
+ shows "{\<lambda>nl. nl = xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (max (n + 3) (max fft gft) - Suc (Suc n)) @ y # anything}
([Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)])) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]
- {\<lambda>nl. nl = xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
+ {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
using less
proof(induct y)
case 0
@@ -1725,37 +1735,37 @@
let ?C = "[Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+]
[Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @ [Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]"
have ind: "y \<le> x \<Longrightarrow>
- {\<lambda>nl. nl = xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything}
- ?C {\<lambda>nl. nl = xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything}" by fact
+ {\<lambda>nl. nl = xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything}
+ ?C {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything}" by fact
have less: "Suc y \<le> x" by fact
have stp1:
- "\<exists> stp. abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (?ft - Suc (Suc n)) @ Suc y # anything)
- ?C stp = (0, xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)"
+ "\<exists> stp. abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (?ft - Suc (Suc n)) @ Suc y # anything)
+ ?C stp = (0, xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)"
using assms less
by(rule_tac pr_loop, auto)
then obtain stp1 where a:
- "abc_steps_l (0, xs @ (x - Suc y) # rec_eval (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (?ft - Suc (Suc n)) @ Suc y # anything)
- ?C stp1 = (0, xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)" ..
+ "abc_steps_l (0, xs @ (x - Suc y) # rec_exec (Pr n f g) (xs @ [x - Suc y]) # 0 \<up> (?ft - Suc (Suc n)) @ Suc y # anything)
+ ?C stp1 = (0, xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)" ..
moreover have
- "\<exists> stp. abc_steps_l (0, xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)
- ?C stp = (length ?C, xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything)"
+ "\<exists> stp. abc_steps_l (0, xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)
+ ?C stp = (length ?C, xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything)"
using ind less
- by(auto simp: abc_Hoare_halt_def, case_tac "abc_steps_l (0, xs @ (x - y) # rec_eval (Pr n f g)
+ by(auto simp: abc_Hoare_halt_def, case_tac "abc_steps_l (0, xs @ (x - y) # rec_exec (Pr n f g)
(xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything) ?C na", rule_tac x = na in exI, simp)
then obtain stp2 where b:
- "abc_steps_l (0, xs @ (x - y) # rec_eval (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)
- ?C stp2 = (length ?C, xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything)" ..
+ "abc_steps_l (0, xs @ (x - y) # rec_exec (Pr n f g) (xs @ [x - y]) # 0 \<up> (?ft - Suc (Suc n)) @ y # anything)
+ ?C stp2 = (length ?C, xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything)" ..
from a b show "?case"
by(simp add: abc_Hoare_halt_def, rule_tac x = "stp1 + stp2" in exI, simp add: abc_steps_add)
qed
lemma compile_pr_correct':
- assumes termi_g: "\<forall> y<x. terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])])"
+ assumes termi_g: "\<forall> y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
and g_ind:
- "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
- {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
- and termi_f: "terminates f xs"
- and f_ind: "\<And> anything. {\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (fft - Suc far) @ anything}"
+ "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
+ {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
+ and termi_f: "terminate f xs"
+ and f_ind: "\<And> anything. {\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fft - Suc far) @ anything}"
and len: "length xs = n"
and compile1: "rec_ci f = (fap, far, fft)"
and compile2: "rec_ci g = (gap, gar, gft)"
@@ -1765,7 +1775,7 @@
(fap [+] (mv_box n (Suc n) [+]
([Dec (max (n + 3) (max fft gft)) (length gap + 7)] [+] (gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]) @
[Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)])))
- {\<lambda>nl. nl = xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
+ {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ anything}"
proof -
let ?ft = "max (n+3) (max fft gft)"
let ?A = "mv_box n ?ft"
@@ -1775,14 +1785,14 @@
let ?E = "gap [+] [Inc n, Dec (Suc n) 3, Goto (Suc 0)]"
let ?F = "[Dec (Suc (Suc n)) 0, Inc (Suc n), Goto (length gap + 4)]"
let ?P = "\<lambda>nl. nl = xs @ x # 0 \<up> (?ft - n) @ anything"
- let ?S = "\<lambda>nl. nl = xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything"
+ let ?S = "\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (?ft - Suc n) @ anything"
let ?Q1 = "\<lambda>nl. nl = xs @ 0 \<up> (?ft - n) @ x # anything"
show "{?P} (?A [+] (?B [+] (?C [+] (?D [+] ?E @ ?F)))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
show "{?P} ?A {?Q1}"
using len by simp
next
- let ?Q2 = "\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (?ft - Suc n) @ x # anything"
+ let ?Q2 = "\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (?ft - Suc n) @ x # anything"
have a: "?ft \<ge> fft"
by arith
have b: "far = n"
@@ -1794,7 +1804,7 @@
show "{?Q1} (?B [+] (?C [+] (?D [+] ?E @ ?F))) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
have "{\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ 0\<up>(?ft - fft) @ x # anything} fap
- {\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (fft - Suc far) @ 0\<up>(?ft - fft) @ x # anything}"
+ {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fft - Suc far) @ 0\<up>(?ft - fft) @ x # anything}"
by(rule_tac f_ind)
moreover have "fft - far + ?ft - fft = ?ft - far"
using a b c by arith
@@ -1804,50 +1814,50 @@
using b
by(simp add: replicate_merge_anywhere)
next
- let ?Q3 = "\<lambda> nl. nl = xs @ 0 # rec_eval f xs # 0\<up>(?ft - Suc (Suc n)) @ x # anything"
+ let ?Q3 = "\<lambda> nl. nl = xs @ 0 # rec_exec f xs # 0\<up>(?ft - Suc (Suc n)) @ x # anything"
show "{?Q2} (?C [+] (?D [+] ?E @ ?F)) {?S}"
proof(rule_tac abc_Hoare_plus_halt)
show "{?Q2} (?C) {?Q3}"
- using mv_box_correct[of n "Suc n" "xs @ rec_eval f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything"]
+ using mv_box_correct[of n "Suc n" "xs @ rec_exec f xs # 0 \<up> (max (n + 3) (max fft gft) - Suc n) @ x # anything"]
using len
by(auto)
next
show "{?Q3} (?D [+] ?E @ ?F) {?S}"
using pr_loop_correct[of x x xs n g gap gar gft f fft anything] assms
- by(simp add: rec_eval_pr_0_simps)
+ by(simp add: rec_exec_pr_0_simps)
qed
qed
qed
qed
lemma compile_pr_correct:
- assumes g_ind: "\<forall>y<x. terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) \<and>
+ assumes g_ind: "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \<and>
(\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow>
- (\<forall>xc. {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # 0 \<up> (xb - xa) @ xc} x
- {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) # 0 \<up> (xb - Suc xa) @ xc}))"
- and termi_f: "terminates f xs"
+ (\<forall>xc. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (xb - xa) @ xc} x
+ {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (xb - Suc xa) @ xc}))"
+ and termi_f: "terminate f xs"
and f_ind:
"\<And>ap arity fp anything.
- rec_ci f = (ap, arity, fp) \<Longrightarrow> {\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (fp - Suc arity) @ anything}"
+ rec_ci f = (ap, arity, fp) \<Longrightarrow> {\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fp - Suc arity) @ anything}"
and len: "length xs = n"
and compile: "rec_ci (Pr n f g) = (ap, arity, fp)"
- shows "{\<lambda>nl. nl = xs @ x # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ x # rec_eval (Pr n f g) (xs @ [x]) # 0 \<up> (fp - Suc arity) @ anything}"
+ shows "{\<lambda>nl. nl = xs @ x # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (fp - Suc arity) @ anything}"
proof(case_tac "rec_ci f", case_tac "rec_ci g")
fix fap far fft gap gar gft
assume h: "rec_ci f = (fap, far, fft)" "rec_ci g = (gap, gar, gft)"
have g:
- "\<forall>y<x. (terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) \<and>
- (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
- {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}))"
+ "\<forall>y<x. (terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \<and>
+ (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
+ {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything}))"
using g_ind h
by(auto)
- hence termi_g: "\<forall> y<x. terminates g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])])"
+ hence termi_g: "\<forall> y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])])"
by simp
from g have g_newind:
- "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
- {\<lambda>nl. nl = xs @ y # rec_eval (Pr n f g) (xs @ [y]) # rec_eval g (xs @ [y, rec_eval (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
+ "\<forall> y<x. (\<forall>anything. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (gft - gar) @ anything} gap
+ {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (gft - Suc gar) @ anything})"
by auto
- have f_newind: "\<And> anything. {\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_eval f xs # 0 \<up> (fft - Suc far) @ anything}"
+ have f_newind: "\<And> anything. {\<lambda>nl. nl = xs @ 0 \<up> (fft - far) @ anything} fap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fft - Suc far) @ anything}"
using h
by(rule_tac f_ind, simp)
show "?thesis"
@@ -1984,20 +1994,20 @@
and len: "length xs = arity"
and far: "far = Suc arity"
and ind: " (\<forall>xc. ({\<lambda>nl. nl = xs @ x # 0 \<up> (fft - far) @ xc} fap
- {\<lambda>nl. nl = xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (fft - Suc far) @ xc}))"
- and exec_less: "rec_eval f (xs @ [x]) > 0"
+ {\<lambda>nl. nl = xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (fft - Suc far) @ xc}))"
+ and exec_less: "rec_exec f (xs @ [x]) > 0"
and compile: "rec_ci f = (fap, far, fft)"
shows "\<exists> stp > 0. abc_steps_l (0, xs @ x # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
(0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)"
proof -
have "\<exists> stp. abc_steps_l (0, xs @ x # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
- (length fap, xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
+ (length fap, xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
proof -
have "\<exists> stp. abc_steps_l (0, xs @ x # 0 \<up> (ft - Suc arity) @ anything) fap stp =
- (length fap, xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
+ (length fap, xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
proof -
have "{\<lambda>nl. nl = xs @ x # 0 \<up> (fft - far) @ 0\<up>(ft - fft) @ anything} fap
- {\<lambda>nl. nl = xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (fft - Suc far) @ 0\<up>(ft - fft) @ anything}"
+ {\<lambda>nl. nl = xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (fft - Suc far) @ 0\<up>(ft - fft) @ anything}"
using ind by simp
moreover have "fft > far"
using compile
@@ -2008,17 +2018,17 @@
by(simp add: replicate_merge_anywhere max_absorb2)
qed
then obtain stp where "abc_steps_l (0, xs @ x # 0 \<up> (ft - Suc arity) @ anything) fap stp =
- (length fap, xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)" ..
+ (length fap, xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)" ..
thus "?thesis"
by(erule_tac abc_append_frist_steps_halt_eq)
qed
moreover have
- "\<exists> stp > 0. abc_steps_l (length fap, xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything) (fap @ B) stp =
+ "\<exists> stp > 0. abc_steps_l (length fap, xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything) (fap @ B) stp =
(0, xs @ Suc x # 0 # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
- using mn_correct[of f fap far fft "rec_eval f (xs @ [x])" xs arity B
- "(\<lambda>stp. (abc_steps_l (length fap, xs @ x # rec_eval f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything) (fap @ B) stp, length fap, arity))"
+ using mn_correct[of f fap far fft "rec_exec f (xs @ [x])" xs arity B
+ "(\<lambda>stp. (abc_steps_l (length fap, xs @ x # rec_exec f (xs @ [x]) # 0 \<up> (ft - Suc (Suc arity)) @ anything) (fap @ B) stp, length fap, arity))"
x "0 \<up> (ft - Suc (Suc arity)) @ anything" "(\<lambda>((as, lm), ss, arity). as = 0)"
- "(\<lambda>((as, lm), ss, aritya). mn_ind_inv (as, lm) (length fap) x (rec_eval f (xs @ [x])) (0 \<up> (ft - Suc (Suc arity)) @ anything) xs) "]
+ "(\<lambda>((as, lm), ss, aritya). mn_ind_inv (as, lm) (length fap) x (rec_exec f (xs @ [x])) (0 \<up> (ft - Suc (Suc arity)) @ anything) xs) "]
B compile exec_less len
apply(subgoal_tac "fap \<noteq> []", auto)
apply(rule_tac x = stp in exI, auto simp: mn_ind_inv.simps)
@@ -2038,8 +2048,8 @@
and ft: "ft = max (Suc arity) fft"
and len: "length xs = arity"
and ind_all: "\<forall>i\<le>x. (\<forall>xc. ({\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap
- {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
- and exec_ge: "\<forall> i\<le>x. rec_eval f (xs @ [i]) > 0"
+ {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
+ and exec_ge: "\<forall> i\<le>x. rec_exec f (xs @ [i]) > 0"
and compile: "rec_ci f = (fap, far, fft)"
and far: "far = Suc arity"
shows "\<exists> stp > x. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
@@ -2052,12 +2062,12 @@
by(rule_tac mn_loop, simp_all)
next
case (Suc x)
- have ind': "\<lbrakk>\<forall>i\<le>x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc};
- \<forall>i\<le>x. 0 < rec_eval f (xs @ [i])\<rbrakk> \<Longrightarrow>
+ have ind': "\<lbrakk>\<forall>i\<le>x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc};
+ \<forall>i\<le>x. 0 < rec_exec f (xs @ [i])\<rbrakk> \<Longrightarrow>
\<exists>stp > x. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp = (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)" by fact
- have exec_ge: "\<forall>i\<le>Suc x. 0 < rec_eval f (xs @ [i])" by fact
+ have exec_ge: "\<forall>i\<le>Suc x. 0 < rec_exec f (xs @ [i])" by fact
have ind_all: "\<forall>i\<le>Suc x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap
- {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}" by fact
+ {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}" by fact
have ind: "\<exists>stp > x. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
(0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)" using ind' exec_ge ind_all by simp
have stp: "\<exists> stp > 0. abc_steps_l (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
@@ -2074,8 +2084,8 @@
and ft: "ft = max (Suc arity) fft"
and len: "length xs = arity"
and ind_all: "\<forall>i\<le>x. (\<forall>xc. ({\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap
- {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
- and exec_ge: "\<forall> i\<le>x. rec_eval f (xs @ [i]) > 0"
+ {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
+ and exec_ge: "\<forall> i\<le>x. rec_exec f (xs @ [i]) > 0"
and compile: "rec_ci f = (fap, far, fft)"
and far: "far = Suc arity"
shows "\<exists> stp. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
@@ -2092,17 +2102,17 @@
assumes B: "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"
and ft: "ft = max (Suc arity) fft"
and len: "length xs = arity"
- and termi_f: "terminates f (xs @ [r])"
+ and termi_f: "terminate f (xs @ [r])"
and f_ind: "\<And>anything. {\<lambda>nl. nl = xs @ r # 0 \<up> (fft - far) @ anything} fap
{\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fft - Suc far) @ anything}"
and ind_all: "\<forall>i < r. (\<forall>xc. ({\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap
- {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
- and exec_less: "\<forall> i<r. rec_eval f (xs @ [i]) > 0"
- and exec: "rec_eval f (xs @ [r]) = 0"
+ {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
+ and exec_less: "\<forall> i<r. rec_exec f (xs @ [i]) > 0"
+ and exec: "rec_exec f (xs @ [r]) = 0"
and compile: "rec_ci f = (fap, far, fft)"
shows "{\<lambda>nl. nl = xs @ 0 \<up> (max (Suc arity) fft - arity) @ anything}
fap @ B
- {\<lambda>nl. nl = xs @ rec_eval (Mn arity f) xs # 0 \<up> (max (Suc arity) fft - Suc arity) @ anything}"
+ {\<lambda>nl. nl = xs @ rec_exec (Mn arity f) xs # 0 \<up> (max (Suc arity) fft - Suc arity) @ anything}"
proof -
have a: "far = Suc arity"
using len compile termi_f
@@ -2117,13 +2127,13 @@
by(rule_tac mn_loop_correct, auto)
moreover have
"\<exists> stp. abc_steps_l (0, xs @ r # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp =
- (length fap, xs @ r # rec_eval f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
+ (length fap, xs @ r # rec_exec f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
proof -
have "\<exists> stp. abc_steps_l (0, xs @ r # 0 \<up> (ft - Suc arity) @ anything) fap stp =
- (length fap, xs @ r # rec_eval f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
+ (length fap, xs @ r # rec_exec f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
proof -
have "{\<lambda>nl. nl = xs @ r # 0 \<up> (fft - far) @ 0\<up>(ft - fft) @ anything} fap
- {\<lambda>nl. nl = xs @ r # rec_eval f (xs @ [r]) # 0 \<up> (fft - Suc far) @ 0\<up>(ft - fft) @ anything}"
+ {\<lambda>nl. nl = xs @ r # rec_exec f (xs @ [r]) # 0 \<up> (fft - Suc far) @ 0\<up>(ft - fft) @ anything}"
using f_ind exec by simp
thus "?thesis"
using ft a b
@@ -2131,20 +2141,20 @@
by(simp add: replicate_merge_anywhere max_absorb2)
qed
then obtain stp where "abc_steps_l (0, xs @ r # 0 \<up> (ft - Suc arity) @ anything) fap stp =
- (length fap, xs @ r # rec_eval f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)" ..
+ (length fap, xs @ r # rec_exec f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)" ..
thus "?thesis"
by(erule_tac abc_append_frist_steps_halt_eq)
qed
moreover have
- "\<exists> stp. abc_steps_l (length fap, xs @ r # rec_eval f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything) (fap @ B) stp =
- (length fap + 5, xs @ r # rec_eval f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
+ "\<exists> stp. abc_steps_l (length fap, xs @ r # rec_exec f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything) (fap @ B) stp =
+ (length fap + 5, xs @ r # rec_exec f (xs @ [r]) # 0 \<up> (ft - Suc (Suc arity)) @ anything)"
using ft a b len B exec
apply(rule_tac x = 1 in exI, auto)
by(auto simp: abc_steps_l.simps B abc_step_l.simps
abc_fetch.simps nth_append max_absorb2 abc_lm_v.simps abc_lm_s.simps)
- moreover have "rec_eval (Mn (length xs) f) xs = r"
+ moreover have "rec_exec (Mn (length xs) f) xs = r"
using exec exec_less
- apply(auto simp: rec_eval.simps Least_def)
+ apply(auto simp: rec_exec.simps Least_def)
thm the_equality
apply(rule_tac the_equality, auto)
apply(metis exec_less less_not_refl3 linorder_not_less)
@@ -2158,18 +2168,18 @@
lemma compile_mn_correct:
assumes len: "length xs = n"
- and termi_f: "terminates f (xs @ [r])"
+ and termi_f: "terminate f (xs @ [r])"
and f_ind: "\<And>ap arity fp anything. rec_ci f = (ap, arity, fp) \<Longrightarrow>
{\<lambda>nl. nl = xs @ r # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ r # 0 # 0 \<up> (fp - Suc arity) @ anything}"
- and exec: "rec_eval f (xs @ [r]) = 0"
+ and exec: "rec_exec f (xs @ [r]) = 0"
and ind_all:
- "\<forall>i<r. terminates f (xs @ [i]) \<and>
+ "\<forall>i<r. terminate f (xs @ [i]) \<and>
(\<forall>x xa xb. rec_ci f = (x, xa, xb) \<longrightarrow>
- (\<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (xb - Suc xa) @ xc})) \<and>
- 0 < rec_eval f (xs @ [i])"
+ (\<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (xb - Suc xa) @ xc})) \<and>
+ 0 < rec_exec f (xs @ [i])"
and compile: "rec_ci (Mn n f) = (ap, arity, fp)"
shows "{\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap
- {\<lambda>nl. nl = xs @ rec_eval (Mn n f) xs # 0 \<up> (fp - Suc arity) @ anything}"
+ {\<lambda>nl. nl = xs @ rec_exec (Mn n f) xs # 0 \<up> (fp - Suc arity) @ anything}"
proof(case_tac "rec_ci f")
fix fap far fft
assume h: "rec_ci f = (fap, far, fft)"
@@ -2179,10 +2189,10 @@
by(rule_tac f_ind, simp)
have newind_all:
"\<forall>i < r. (\<forall>xc. ({\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap
- {\<lambda>nl. nl = xs @ i # rec_eval f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
+ {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}))"
using ind_all h
by(auto)
- have all_less: "\<forall> i<r. rec_eval f (xs @ [i]) > 0"
+ have all_less: "\<forall> i<r. rec_exec f (xs @ [i]) > 0"
using ind_all by auto
show "?thesis"
using h compile f_newind newind_all all_less len termi_f exec
@@ -2191,10 +2201,10 @@
qed
lemma recursive_compile_correct:
- "\<lbrakk>terminates recf args; rec_ci recf = (ap, arity, fp)\<rbrakk>
+ "\<lbrakk>terminate recf args; rec_ci recf = (ap, arity, fp)\<rbrakk>
\<Longrightarrow> {\<lambda> nl. nl = args @ 0\<up>(fp - arity) @ anything} ap
- {\<lambda> nl. nl = args@ rec_eval recf args # 0\<up>(fp - Suc arity) @ anything}"
-apply(induct arbitrary: ap arity fp anything r rule: terminates.induct)
+ {\<lambda> nl. nl = args@ rec_exec recf args # 0\<up>(fp - Suc arity) @ anything}"
+apply(induct arbitrary: ap arity fp anything r rule: terminate.induct)
apply(simp_all add: compile_s_correct compile_z_correct compile_id_correct
compile_cn_correct compile_pr_correct compile_mn_correct)
done
@@ -2323,8 +2333,8 @@
lemma recursive_compile_correct_norm':
"\<lbrakk>rec_ci f = (ap, arity, ft);
- terminates f args\<rbrakk>
- \<Longrightarrow> \<exists> stp rl. (abc_steps_l (0, args) ap stp) = (length ap, rl) \<and> abc_list_crsp (args @ [rec_eval f args]) rl"
+ terminate f args\<rbrakk>
+ \<Longrightarrow> \<exists> stp rl. (abc_steps_l (0, args) ap stp) = (length ap, rl) \<and> abc_list_crsp (args @ [rec_exec f args]) rl"
using recursive_compile_correct[of f args ap arity ft "[]"]
apply(auto simp: abc_Hoare_halt_def)
apply(rule_tac x = n in exI)
@@ -2334,9 +2344,9 @@
done
lemma [simp]:
- assumes a: "args @ [rec_eval f args] = lm @ 0 \<up> n"
+ assumes a: "args @ [rec_exec f args] = lm @ 0 \<up> n"
and b: "length args < length lm"
- shows "\<exists>m. lm = args @ rec_eval f args # 0 \<up> m"
+ shows "\<exists>m. lm = args @ rec_exec f args # 0 \<up> m"
using assms
apply(case_tac n, simp)
apply(rule_tac x = 0 in exI, simp)
@@ -2344,9 +2354,9 @@
done
lemma [simp]:
-"\<lbrakk>args @ [rec_eval f args] = lm @ 0 \<up> n; \<not> length args < length lm\<rbrakk>
+"\<lbrakk>args @ [rec_exec f args] = lm @ 0 \<up> n; \<not> length args < length lm\<rbrakk>
\<Longrightarrow> \<exists>m. (lm @ 0 \<up> (length args - length lm) @ [Suc 0])[length args := 0] =
- args @ rec_eval f args # 0 \<up> m"
+ args @ rec_exec f args # 0 \<up> m"
apply(case_tac n, simp_all add: exp_suc list_update_append list_update.simps del: replicate_Suc)
apply(subgoal_tac "length args = Suc (length lm)", simp)
apply(rule_tac x = "Suc (Suc 0)" in exI, simp)
@@ -2355,16 +2365,16 @@
lemma compile_append_dummy_correct:
assumes compile: "rec_ci f = (ap, ary, fp)"
- and termi: "terminates f args"
- shows "{\<lambda> nl. nl = args} (ap [+] dummy_abc (length args)) {\<lambda> nl. (\<exists> m. nl = args @ rec_eval f args # 0\<up>m)}"
+ and termi: "terminate f args"
+ shows "{\<lambda> nl. nl = args} (ap [+] dummy_abc (length args)) {\<lambda> nl. (\<exists> m. nl = args @ rec_exec f args # 0\<up>m)}"
proof(rule_tac abc_Hoare_plus_halt)
- show "{\<lambda>nl. nl = args} ap {\<lambda> nl. abc_list_crsp (args @ [rec_eval f args]) nl}"
+ show "{\<lambda>nl. nl = args} ap {\<lambda> nl. abc_list_crsp (args @ [rec_exec f args]) nl}"
using compile termi recursive_compile_correct_norm'[of f ap ary fp args]
apply(auto simp: abc_Hoare_halt_def)
by(rule_tac x = stp in exI, simp)
next
- show "{abc_list_crsp (args @ [rec_eval f args])} dummy_abc (length args)
- {\<lambda>nl. \<exists>m. nl = args @ rec_eval f args # 0 \<up> m}"
+ show "{abc_list_crsp (args @ [rec_exec f args])} dummy_abc (length args)
+ {\<lambda>nl. \<exists>m. nl = args @ rec_exec f args # 0 \<up> m}"
apply(auto simp: dummy_abc_def abc_Hoare_halt_def)
apply(rule_tac x = 3 in exI)
by(auto simp: abc_steps_l.simps abc_list_crsp_def abc_step_l.simps numeral_3_eq_3 abc_fetch.simps
@@ -2386,15 +2396,15 @@
and compile2: "rec_ci (gs!i) = (gap, gar, gft) \<and> gar = length args"
and g_unhalt: "\<And> anything. {\<lambda> nl. nl = args @ 0\<up>(gft - gar) @ anything} gap \<up>"
and g_ind: "\<And> apj arj ftj j anything. \<lbrakk>j < i; rec_ci (gs!j) = (apj, arj, ftj)\<rbrakk>
- \<Longrightarrow> {\<lambda> nl. nl = args @ 0\<up>(ftj - arj) @ anything} apj {\<lambda> nl. nl = args @ rec_eval (gs!j) args # 0\<up>(ftj - Suc arj) @ anything}"
- and all_termi: "\<forall> j<i. terminates (gs!j) args"
+ \<Longrightarrow> {\<lambda> nl. nl = args @ 0\<up>(ftj - arj) @ anything} apj {\<lambda> nl. nl = args @ rec_exec (gs!j) args # 0\<up>(ftj - Suc arj) @ anything}"
+ and all_termi: "\<forall> j<i. terminate (gs!j) args"
shows "{\<lambda> nl. nl = args @ 0\<up>(ft - ar) @ anything} ap \<up>"
using compile1
apply(case_tac "rec_ci f", auto simp: rec_ci.simps abc_comp_commute)
proof(rule_tac abc_Hoare_plus_unhalt1)
fix fap far fft
let ?ft = "max (Suc (length args)) (Max (insert fft ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))"
- let ?Q = "\<lambda>nl. nl = args @ 0\<up> (?ft - length args) @ map (\<lambda>i. rec_eval i args) (take i gs) @
+ let ?Q = "\<lambda>nl. nl = args @ 0\<up> (?ft - length args) @ map (\<lambda>i. rec_exec i args) (take i gs) @
0\<up>(length gs - i) @ 0\<up> Suc (length args) @ anything"
have "cn_merge_gs (map rec_ci gs) ?ft =
cn_merge_gs (map rec_ci (take i gs)) ?ft [+] (gap [+]
@@ -2403,11 +2413,11 @@
thus "{\<lambda>nl. nl = args @ 0 # 0 \<up> (?ft + length gs) @ anything} (cn_merge_gs (map rec_ci gs) ?ft) \<up>"
proof(simp, rule_tac abc_Hoare_plus_unhalt1, rule_tac abc_Hoare_plus_unhalt2,
rule_tac abc_Hoare_plus_unhalt1)
- let ?Q_tmp = "\<lambda>nl. nl = args @ 0\<up> (gft - gar) @ 0\<up>(?ft - (length args) - (gft -gar)) @ map (\<lambda>i. rec_eval i args) (take i gs) @
+ let ?Q_tmp = "\<lambda>nl. nl = args @ 0\<up> (gft - gar) @ 0\<up>(?ft - (length args) - (gft -gar)) @ map (\<lambda>i. rec_exec i args) (take i gs) @
0\<up>(length gs - i) @ 0\<up> Suc (length args) @ anything"
have a: "{?Q_tmp} gap \<up>"
using g_unhalt[of "0 \<up> (?ft - (length args) - (gft - gar)) @
- map (\<lambda>i. rec_eval i args) (take i gs) @ 0 \<up> (length gs - i) @ 0 \<up> Suc (length args) @ anything"]
+ map (\<lambda>i. rec_exec i args) (take i gs) @ 0 \<up> (length gs - i) @ 0 \<up> Suc (length args) @ anything"]
by simp
moreover have "?ft \<ge> gft"
using g compile2
@@ -2424,7 +2434,7 @@
show "{\<lambda>nl. nl = args @ 0 # 0 \<up> (?ft + length gs) @ anything}
cn_merge_gs (map rec_ci (take i gs)) ?ft
{\<lambda>nl. nl = args @ 0 \<up> (?ft - length args) @
- map (\<lambda>i. rec_eval i args) (take i gs) @ 0 \<up> (length gs - i) @ 0 \<up> Suc (length args) @ anything}"
+ map (\<lambda>i. rec_exec i args) (take i gs) @ 0 \<up> (length gs - i) @ 0 \<up> Suc (length args) @ anything}"
using all_termi
apply(rule_tac compile_cn_gs_correct', auto simp: set_conv_nth)
by(drule_tac apj = x and arj = xa and ftj = xb and j = ia and anything = xc in g_ind, auto)
@@ -2435,7 +2445,7 @@
lemma mn_unhalt_case':
assumes compile: "rec_ci f = (a, b, c)"
- and all_termi: "\<forall>i. terminates f (args @ [i]) \<and> 0 < rec_eval f (args @ [i])"
+ and all_termi: "\<forall>i. terminate f (args @ [i]) \<and> 0 < rec_exec f (args @ [i])"
and B: "B = [Dec (Suc (length args)) (length a + 5), Dec (Suc (length args)) (length a + 3),
Goto (Suc (length a)), Inc (length args), Goto 0]"
shows "{\<lambda>nl. nl = args @ 0 \<up> (max (Suc (length args)) c - length args) @ anything}
@@ -2455,7 +2465,7 @@
proof(rule_tac mn_loop_correct', auto)
fix i xc
show "{\<lambda>nl. nl = args @ i # 0 \<up> (c - Suc (length args)) @ xc} a
- {\<lambda>nl. nl = args @ i # rec_eval f (args @ [i]) # 0 \<up> (c - Suc (Suc (length args))) @ xc}"
+ {\<lambda>nl. nl = args @ i # rec_exec f (args @ [i]) # 0 \<up> (c - Suc (Suc (length args))) @ xc}"
using all_termi recursive_compile_correct[of f "args @ [i]" a b c xc] compile a
by(simp)
qed
@@ -2477,7 +2487,7 @@
lemma mn_unhalt_case:
assumes compile: "rec_ci (Mn n f) = (ap, ar, ft) \<and> length args = ar"
- and all_term: "\<forall> i. terminates f (args @ [i]) \<and> rec_eval f (args @ [i]) > 0"
+ and all_term: "\<forall> i. terminate f (args @ [i]) \<and> rec_exec f (args @ [i]) > 0"
shows "{\<lambda> nl. nl = args @ 0\<up>(ft - ar) @ anything} ap \<up> "
using assms
apply(case_tac "rec_ci f", auto simp: rec_ci.simps abc_comp_commute)
@@ -2490,34 +2500,34 @@
lemma recursive_compile_to_tm_correct1:
assumes compile: "rec_ci recf = (ap, ary, fp)"
- and termi: " terminates recf args"
+ and termi: " terminate recf args"
and tp: "tp = tm_of (ap [+] dummy_abc (length args))"
shows "\<exists> stp m l. steps0 (Suc 0, Bk # Bk # ires, <args> @ Bk\<up>rn)
- (tp @ shift (mopup (length args)) (length tp div 2)) stp = (0, Bk\<up>m @ Bk # Bk # ires, Oc\<up>Suc (rec_eval recf args) @ Bk\<up>l)"
+ (tp @ shift (mopup (length args)) (length tp div 2)) stp = (0, Bk\<up>m @ Bk # Bk # ires, Oc\<up>Suc (rec_exec recf args) @ Bk\<up>l)"
proof -
- have "{\<lambda>nl. nl = args} ap [+] dummy_abc (length args) {\<lambda>nl. \<exists>m. nl = args @ rec_eval recf args # 0 \<up> m}"
+ have "{\<lambda>nl. nl = args} ap [+] dummy_abc (length args) {\<lambda>nl. \<exists>m. nl = args @ rec_exec recf args # 0 \<up> m}"
using compile termi compile
by(rule_tac compile_append_dummy_correct, auto)
then obtain stp m where h: "abc_steps_l (0, args) (ap [+] dummy_abc (length args)) stp =
- (length (ap [+] dummy_abc (length args)), args @ rec_eval recf args # 0\<up>m) "
+ (length (ap [+] dummy_abc (length args)), args @ rec_exec recf args # 0\<up>m) "
apply(simp add: abc_Hoare_halt_def, auto)
by(case_tac "abc_steps_l (0, args) (ap [+] dummy_abc (length args)) n", auto)
thus "?thesis"
using assms tp
- by(rule_tac lm = args and stp = stp and am = "args @ rec_eval recf args # 0 \<up> m"
+ by(rule_tac lm = args and stp = stp and am = "args @ rec_exec recf args # 0 \<up> m"
in compile_correct_halt, auto simp: crsp.simps start_of.simps length_abc_comp abc_lm_v.simps)
qed
lemma recursive_compile_to_tm_correct2:
- assumes termi: " terminates recf args"
+ assumes termi: " terminate recf args"
shows "\<exists> stp m l. steps0 (Suc 0, [Bk, Bk], <args>) (tm_of_rec recf) stp =
- (0, Bk\<up>Suc (Suc m), Oc\<up>Suc (rec_eval recf args) @ Bk\<up>l)"
+ (0, Bk\<up>Suc (Suc m), Oc\<up>Suc (rec_exec recf args) @ Bk\<up>l)"
proof(case_tac "rec_ci recf", simp add: tm_of_rec.simps)
fix ap ar fp
assume "rec_ci recf = (ap, ar, fp)"
thus "\<exists>stp m l. steps0 (Suc 0, [Bk, Bk], <args>)
(tm_of (ap [+] dummy_abc ar) @ shift (mopup ar) (listsum (layout_of (ap [+] dummy_abc ar)))) stp =
- (0, Bk # Bk # Bk \<up> m, Oc # Oc \<up> rec_eval recf args @ Bk \<up> l)"
+ (0, Bk # Bk # Bk \<up> m, Oc # Oc \<up> rec_exec recf args @ Bk \<up> l)"
using recursive_compile_to_tm_correct1[of recf ap ar fp args "tm_of (ap [+] dummy_abc (length args))" "[]" 0]
assms param_pattern[of recf args ap ar fp]
by(simp add: replicate_Suc[THEN sym] replicate_Suc_iff_anywhere del: replicate_Suc tm_of_rec_def,
@@ -2525,9 +2535,9 @@
qed
lemma recursive_compile_to_tm_correct3:
- assumes termi: "terminates recf args"
+ assumes termi: "terminate recf args"
shows "{\<lambda> tp. tp =([Bk, Bk], <args>)} (tm_of_rec recf)
- {\<lambda> tp. \<exists> k l. tp = (Bk\<up> k, <rec_eval recf args> @ Bk \<up> l)}"
+ {\<lambda> tp. \<exists> k l. tp = (Bk\<up> k, <rec_exec recf args> @ Bk \<up> l)}"
using recursive_compile_to_tm_correct2[OF assms]
apply(auto simp add: Hoare_halt_def)
apply(rule_tac x = stp in exI)