# HG changeset patch
# User Christian Urban <christian dot urban at kcl dot ac dot uk>
# Date 1359516785 0
# Node ID fe7a257bdff4fe022d089e03a03cdb4293905146
# Parent  860f05037c36e704d8131f37d948f49bc56984dc
updated paper

diff -r 860f05037c36 -r fe7a257bdff4 Paper/Paper.thy
--- 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}
 
 
diff -r 860f05037c36 -r fe7a257bdff4 paper.pdf
Binary file paper.pdf has changed
diff -r 860f05037c36 -r fe7a257bdff4 thys/turing_hoare.thy
--- 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
diff -r 860f05037c36 -r fe7a257bdff4 thys/uncomputable.thy
--- 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