--- a/Paper/Paper.thy Wed Jan 30 02:29:47 2013 +0000
+++ b/Paper/Paper.thy Wed Jan 30 03:33:05 2013 +0000
@@ -118,10 +118,10 @@
done
lemmas HR1 =
- Hoare_plus_halt_simple[where ?P1.0="P" and ?P2.0="Q" and ?P3.0="R\<iota>" and ?A="p\<^isub>1" and B="p\<^isub>2"]
+ Hoare_plus_halt[where ?S.0="R\<iota>" and ?A="p\<^isub>1" and B="p\<^isub>2"]
lemmas HR2 =
- Hoare_plus_unhalt_simple[where ?P1.0="P" and ?P2.0="Q" and ?A="p\<^isub>1" and B="p\<^isub>2"]
+ Hoare_plus_unhalt[where ?A="p\<^isub>1" and B="p\<^isub>2"]
lemma inv_begin01:
assumes "n > 1"
@@ -765,6 +765,14 @@
\end{center}
\noindent
+ For our Hoare-triples we can easily prove the following consequence rule
+
+ \begin{equation}
+ @{thm[mode=Rule] Hoare_consequence}
+ \end{equation}
+
+
+ \noindent
Like Asperti and Ricciotti with their notion of realisability, we
have set up our Hoare-rules so that we can deal explicitly
with total correctness and non-terminantion, rather than have
@@ -928,7 +936,7 @@
\begin{center}
- @{thm haltP_def[where lm="ns"]}
+ @{thm haltP_def}
\end{center}
Binary file paper.pdf has changed
--- a/thys/turing_hoare.thy Wed Jan 30 02:29:47 2013 +0000
+++ b/thys/turing_hoare.thy Wed Jan 30 03:33:05 2013 +0000
@@ -10,6 +10,10 @@
where
"P \<mapsto> Q \<equiv> \<forall>l r. P (l, r) \<longrightarrow> Q (l, r)"
+lemma [intro, simp]:
+ "P \<mapsto> P"
+unfolding assert_imp_def by simp
+
fun
holds_for :: "(tape \<Rightarrow> bool) \<Rightarrow> config \<Rightarrow> bool" ("_ holds'_for _" [100, 99] 100)
where
@@ -57,24 +61,23 @@
text {*
- {P1} A {Q1} {P2} B {Q2} Q1 \<mapsto> P2 A well-formed
- ---------------------------------------------------
- {P1} A |+| B {Q2}
+ {P} A {Q} {Q} B {S} A well-formed
+ -----------------------------------------
+ {P} A |+| B {S}
*}
-lemma Hoare_plus_halt [case_names A_halt B_halt Imp A_wf]:
- assumes A_halt : "{P1} A {Q1}"
- and B_halt : "{P2} B {Q2}"
- and a_imp: "Q1 \<mapsto> P2"
+lemma Hoare_plus_halt [case_names A_halt B_halt A_wf]:
+ assumes A_halt : "{P} A {Q}"
+ and B_halt : "{Q} B {S}"
and A_wf : "tm_wf (A, 0)"
- shows "{P1} A |+| B {Q2}"
+ shows "{P} A |+| B {S}"
proof(rule Hoare_haltI)
fix l r
- assume h: "P1 (l, r)"
+ assume h: "P (l, r)"
then obtain n1 l' r'
where "is_final (steps0 (1, l, r) A n1)"
- and a1: "Q1 holds_for (steps0 (1, l, r) A n1)"
+ and a1: "Q holds_for (steps0 (1, l, r) A n1)"
and a2: "steps0 (1, l, r) A n1 = (0, l', r')"
using A_halt unfolding Hoare_halt_def
by (metis is_final_eq surj_pair)
@@ -82,48 +85,37 @@
where "steps0 (1, l, r) (A |+| B) n2 = (Suc (length A div 2), l', r')"
using A_wf by (rule_tac tm_comp_pre_halt_same)
moreover
- from a1 a2 a_imp have "P2 (l', r')" by (simp add: assert_imp_def)
+ from a1 a2 have "Q (l', r')" by (simp)
then obtain n3 l'' r''
where "is_final (steps0 (1, l', r') B n3)"
- and b1: "Q2 holds_for (steps0 (1, l', r') B n3)"
+ and b1: "S holds_for (steps0 (1, l', r') B n3)"
and b2: "steps0 (1, l', r') B n3 = (0, l'', r'')"
using B_halt unfolding Hoare_halt_def
by (metis is_final_eq surj_pair)
then have "steps0 (Suc (length A div 2), l', r') (A |+| B) n3 = (0, l'', r'')"
using A_wf by (rule_tac tm_comp_second_halt_same)
ultimately show
- "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> Q2 holds_for (steps0 (1, l, r) (A |+| B) n)"
+ "\<exists>n. is_final (steps0 (1, l, r) (A |+| B) n) \<and> S holds_for (steps0 (1, l, r) (A |+| B) n)"
using b1 b2 by (rule_tac x = "n2 + n3" in exI) (simp)
qed
-lemma Hoare_plus_halt_simple [case_names A_halt B_halt A_wf]:
- assumes A_halt : "{P1} A {P2}"
- and B_halt : "{P2} B {P3}"
- and A_wf : "tm_wf (A, 0)"
- shows "{P1} A |+| B {P3}"
-by (rule Hoare_plus_halt[OF A_halt B_halt _ A_wf])
- (simp add: assert_imp_def)
-
-
-
text {*
- {P1} A {Q1} {P2} B loops Q1 \<mapsto> P2 A well-formed
- ------------------------------------------------------
- {P1} A |+| B loops
+ {P} A {Q} {Q} B loops A well-formed
+ ------------------------------------------
+ {P} A |+| B loops
*}
-lemma Hoare_plus_unhalt [case_names A_halt B_unhalt Imp A_wf]:
- assumes A_halt: "{P1} A {Q1}"
- and B_uhalt: "{P2} B \<up>"
- and a_imp: "Q1 \<mapsto> P2"
+lemma Hoare_plus_unhalt [case_names A_halt B_unhalt A_wf]:
+ assumes A_halt: "{P} A {Q}"
+ and B_uhalt: "{Q} B \<up>"
and A_wf : "tm_wf (A, 0)"
- shows "{P1} (A |+| B) \<up>"
+ shows "{P} (A |+| B) \<up>"
proof(rule_tac Hoare_unhaltI)
fix n l r
- assume h: "P1 (l, r)"
+ assume h: "P (l, r)"
then obtain n1 l' r'
where a: "is_final (steps0 (1, l, r) A n1)"
- and b: "Q1 holds_for (steps0 (1, l, r) A n1)"
+ and b: "Q holds_for (steps0 (1, l, r) A n1)"
and c: "steps0 (1, l, r) A n1 = (0, l', r')"
using A_halt unfolding Hoare_halt_def
by (metis is_final_eq surj_pair)
@@ -132,12 +124,12 @@
then show "\<not> is_final (steps0 (1, l, r) (A |+| B) n)"
proof(cases "n2 \<le> n")
case True
- from b c a_imp have "P2 (l', r')" by (simp add: assert_imp_def)
+ from b c have "Q (l', r')" by simp
then have "\<forall> n. \<not> is_final (steps0 (1, l', r') B n) "
using B_uhalt unfolding Hoare_unhalt_def by simp
- then have "\<not> is_final (steps0 (Suc 0, l', r') B (n - n2))" by auto
+ then have "\<not> is_final (steps0 (1, l', r') B (n - n2))" by auto
then obtain s'' l'' r''
- where "steps0 (Suc 0, l', r') B (n - n2) = (s'', l'', r'')"
+ where "steps0 (1, l', r') B (n - n2) = (s'', l'', r'')"
and "\<not> is_final (s'', l'', r'')" by (metis surj_pair)
then have "steps0 (Suc (length A div 2), l', r') (A |+| B) (n - n2) = (s''+ length A div 2, l'', r'')"
using A_wf by (auto dest: tm_comp_second_same simp del: tm_wf.simps)
@@ -155,19 +147,8 @@
qed
qed
-lemma Hoare_plus_unhalt_simple [case_names A_halt B_unhalt A_wf]:
- assumes A_halt: "{P1} A {P2}"
- and B_uhalt: "{P2} B \<up>"
- and A_wf : "tm_wf (A, 0)"
- shows "{P1} (A |+| B) \<up>"
-by (rule Hoare_plus_unhalt[OF A_halt B_uhalt _ A_wf])
- (simp add: assert_imp_def)
-
-
-lemma Hoare_weaken:
- assumes a: "{P} p {Q}"
- and b: "P' \<mapsto> P"
- and c: "Q \<mapsto> Q'"
+lemma Hoare_consequence:
+ assumes "P' \<mapsto> P" "{P} p {Q}" "Q \<mapsto> Q'"
shows "{P'} p {Q'}"
using assms
unfolding Hoare_halt_def assert_imp_def
--- a/thys/uncomputable.thy Wed Jan 30 02:29:47 2013 +0000
+++ b/thys/uncomputable.thy Wed Jan 30 03:33:05 2013 +0000
@@ -895,16 +895,18 @@
proof -
have "{inv_begin1 x} tcopy_begin {inv_begin0 x}"
by (metis assms begin_correct)
- moreover
- have "{inv_loop1 x} tcopy_loop {inv_loop0 x}"
- by (metis assms loop_correct)
- moreover
+ moreover
have "inv_begin0 x \<mapsto> inv_loop1 x"
unfolding assert_imp_def
unfolding inv_begin0.simps inv_loop1.simps
unfolding inv_loop1_loop.simps inv_loop1_exit.simps
apply(auto simp add: numeral Cons_eq_append_conv)
by (rule_tac x = "Suc 0" in exI, auto)
+ ultimately have "{inv_begin1 x} tcopy_begin {inv_loop1 x}"
+ by (rule_tac Hoare_consequence) (auto)
+ moreover
+ have "{inv_loop1 x} tcopy_loop {inv_loop0 x}"
+ by (metis assms loop_correct)
ultimately
have "{inv_begin1 x} (tcopy_begin |+| tcopy_loop) {inv_loop0 x}"
by (rule_tac Hoare_plus_halt) (auto)
@@ -912,7 +914,7 @@
have "{inv_end1 x} tcopy_end {inv_end0 x}"
by (metis assms end_correct)
moreover
- have "inv_loop0 x \<mapsto> inv_end1 x"
+ have "inv_loop0 x = inv_end1 x"
by(auto simp: inv_end1.simps inv_loop1.simps assert_imp_def)
ultimately
show "{inv_begin1 x} tcopy {inv_end0 x}"
@@ -955,10 +957,10 @@
(* invariants of dither *)
abbreviation (input)
- "dither_halt_inv \<equiv> \<lambda>tp. (\<exists>n. tp = (Bk \<up> n, <1::nat>))"
+ "dither_halt_inv \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>)"
abbreviation (input)
- "dither_unhalt_inv \<equiv> \<lambda>tp. (\<exists>n. tp = (Bk \<up> n, <0::nat>))"
+ "dither_unhalt_inv \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>)"
lemma dither_loops_aux:
"(steps0 (1, Bk \<up> m, [Oc]) dither stp = (1, Bk \<up> m, [Oc])) \<or>
@@ -974,7 +976,6 @@
apply(auto simp add: numeral tape_of_nat_abv)
by (metis Suc_neq_Zero is_final_eq)
-
lemma dither_halts_aux:
shows "steps0 (1, Bk \<up> m, [Oc, Oc]) dither 2 = (0, Bk \<up> m, [Oc, Oc])"
unfolding dither_def
@@ -988,7 +989,6 @@
by (metis (lifting, mono_tags) holds_for.simps is_final_eq prod.cases)
-
section {* The diagnal argument below shows the undecidability of Halting problem *}
text {*
@@ -998,7 +998,7 @@
definition haltP :: "tprog0 \<Rightarrow> nat list \<Rightarrow> bool"
where
- "haltP p lm \<equiv> {(\<lambda>tp. tp = ([], <lm>))} p {(\<lambda>tp. (\<exists>k n. tp = (Bk \<up> k, <n::nat>)))}"
+ "haltP p ns \<equiv> {(\<lambda>tp. tp = ([], <ns>))} p {(\<lambda>tp. (\<exists>k n. tp = (Bk \<up> k, <n::nat>)))}"
lemma [intro, simp]: "tm_wf0 tcopy"
by (auto simp: tcopy_def)
@@ -1026,9 +1026,9 @@
The following two assumptions specifies that @{text "H"} does solve the Halting problem.
*}
and h_case:
- "\<And> M lm. haltP M lm \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <code M#lm>))} H {(\<lambda>tp. \<exists>n. tp = (Bk \<up> n, <0::nat>))}"
+ "\<And> M ns. haltP M ns \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <code M#ns>))} H {(\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>))}"
and nh_case:
- "\<And> M lm. \<not> haltP M lm \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <code M#lm>))} H {(\<lambda>tp. \<exists>n. tp = (Bk \<up> n, <1::nat>))}"
+ "\<And> M ns. \<not> haltP M ns \<Longrightarrow> {(\<lambda>tp. tp = ([Bk], <code M#ns>))} H {(\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>))}"
begin
(* invariants for H *)
@@ -1036,10 +1036,10 @@
"pre_H_inv M n \<equiv> \<lambda>tp. tp = ([Bk], <[code M, n]>)"
abbreviation
- "post_H_halt_inv \<equiv> \<lambda>tp. \<exists>n. tp = (Bk \<up> n, <1::nat>)"
+ "post_H_halt_inv \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>)"
abbreviation
- "post_H_unhalt_inv \<equiv> \<lambda>tp. \<exists>n. tp = (Bk \<up> n, <0::nat>)"
+ "post_H_unhalt_inv \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>)"
lemma H_halt_inv:
@@ -1067,7 +1067,7 @@
(* invariants *)
def P1 \<equiv> "\<lambda>tp. tp = ([]::cell list, <[code_tcontra]>)"
def P2 \<equiv> "\<lambda>tp. tp = ([Bk], <[code_tcontra, code_tcontra]>)"
- def P3 \<equiv> "\<lambda>tp. \<exists>n. tp = (Bk \<up> n, <1::nat>)"
+ def P3 \<equiv> "\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1::nat>)"
(*
{P1} tcopy {P2} {P2} H {P3}
@@ -1081,7 +1081,7 @@
(* {P1} (tcopy |+| H) {P3} *)
have first: "{P1} (tcopy |+| H) {P3}"
- proof (cases rule: Hoare_plus_halt_simple)
+ proof (cases rule: Hoare_plus_halt)
case A_halt (* of tcopy *)
show "{P1} tcopy {P2}" unfolding P1_def P2_def
by (rule tcopy_correct)
@@ -1099,7 +1099,7 @@
(* {P1} tcontra {P3} *)
have "{P1} tcontra {P3}"
unfolding tcontra_def
- by (rule Hoare_plus_halt_simple[OF first second H_wf])
+ by (rule Hoare_plus_halt[OF first second H_wf])
with assms show "False"
unfolding P1_def P3_def
@@ -1120,7 +1120,7 @@
(* invariants *)
def P1 \<equiv> "\<lambda>tp. tp = ([]::cell list, <[code_tcontra]>)"
def P2 \<equiv> "\<lambda>tp. tp = ([Bk], <[code_tcontra, code_tcontra]>)"
- def Q3 \<equiv> "\<lambda>tp. \<exists>n. tp = (Bk \<up> n, <0::nat>)"
+ def Q3 \<equiv> "\<lambda>tp. \<exists>k. tp = (Bk \<up> k, <0::nat>)"
(*
{P1} tcopy {P2} {P2} H {Q3}
@@ -1134,7 +1134,7 @@
(* {P1} (tcopy |+| H) {Q3} *)
have first: "{P1} (tcopy |+| H) {Q3}"
- proof (cases rule: Hoare_plus_halt_simple)
+ proof (cases rule: Hoare_plus_halt)
case A_halt (* of tcopy *)
show "{P1} tcopy {P2}" unfolding P1_def P2_def
by (rule tcopy_correct)
@@ -1152,7 +1152,7 @@
(* {P1} tcontra loops *)
have "{P1} tcontra \<up>"
unfolding tcontra_def
- by (rule Hoare_plus_unhalt_simple[OF first second H_wf])
+ by (rule Hoare_plus_unhalt[OF first second H_wf])
with assms show "False"
unfolding P1_def