1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/HOLCF/ex/Loop.thy	Sat Nov 27 16:08:10 2010 -0800
     1.3 @@ -0,0 +1,200 @@
     1.4 +(*  Title:      HOLCF/ex/Loop.thy
     1.5 +    Author:     Franz Regensburger
     1.6 +*)
     1.7 +
     1.8 +header {* Theory for a loop primitive like while *}
     1.9 +
    1.10 +theory Loop
    1.11 +imports HOLCF
    1.12 +begin
    1.13 +
    1.14 +definition
    1.15 +  step  :: "('a -> tr)->('a -> 'a)->'a->'a" where
    1.16 +  "step = (LAM b g x. If b$x then g$x else x)"
    1.17 +
    1.18 +definition
    1.19 +  while :: "('a -> tr)->('a -> 'a)->'a->'a" where
    1.20 +  "while = (LAM b g. fix$(LAM f x. If b$x then f$(g$x) else x))"
    1.21 +
    1.22 +(* ------------------------------------------------------------------------- *)
    1.23 +(* access to definitions                                                     *)
    1.24 +(* ------------------------------------------------------------------------- *)
    1.25 +
    1.26 +
    1.27 +lemma step_def2: "step$b$g$x = If b$x then g$x else x"
    1.28 +apply (unfold step_def)
    1.29 +apply simp
    1.30 +done
    1.31 +
    1.32 +lemma while_def2: "while$b$g = fix$(LAM f x. If b$x then f$(g$x) else x)"
    1.33 +apply (unfold while_def)
    1.34 +apply simp
    1.35 +done
    1.36 +
    1.37 +
    1.38 +(* ------------------------------------------------------------------------- *)
    1.39 +(* rekursive properties of while                                             *)
    1.40 +(* ------------------------------------------------------------------------- *)
    1.41 +
    1.42 +lemma while_unfold: "while$b$g$x = If b$x then while$b$g$(g$x) else x"
    1.43 +apply (rule trans)
    1.44 +apply (rule while_def2 [THEN fix_eq5])
    1.45 +apply simp
    1.46 +done
    1.47 +
    1.48 +lemma while_unfold2: "ALL x. while$b$g$x = while$b$g$(iterate k$(step$b$g)$x)"
    1.49 +apply (induct_tac k)
    1.50 +apply simp
    1.51 +apply (rule allI)
    1.52 +apply (rule trans)
    1.53 +apply (rule while_unfold)
    1.54 +apply (subst iterate_Suc2)
    1.55 +apply (rule trans)
    1.56 +apply (erule_tac [2] spec)
    1.57 +apply (subst step_def2)
    1.58 +apply (rule_tac p = "b$x" in trE)
    1.59 +apply simp
    1.60 +apply (subst while_unfold)
    1.61 +apply (rule_tac s = "UU" and t = "b$UU" in ssubst)
    1.62 +apply (erule strictI)
    1.63 +apply simp
    1.64 +apply simp
    1.65 +apply simp
    1.66 +apply (subst while_unfold)
    1.67 +apply simp
    1.68 +done
    1.69 +
    1.70 +lemma while_unfold3: "while$b$g$x = while$b$g$(step$b$g$x)"
    1.71 +apply (rule_tac s = "while$b$g$ (iterate (Suc 0) $ (step$b$g) $x) " in trans)
    1.72 +apply (rule while_unfold2 [THEN spec])
    1.73 +apply simp
    1.74 +done
    1.75 +
    1.76 +
    1.77 +(* ------------------------------------------------------------------------- *)
    1.78 +(* properties of while and iterations                                        *)
    1.79 +(* ------------------------------------------------------------------------- *)
    1.80 +
    1.81 +lemma loop_lemma1: "[| EX y. b$y=FF; iterate k$(step$b$g)$x = UU |]
    1.82 +     ==>iterate(Suc k)$(step$b$g)$x=UU"
    1.83 +apply (simp (no_asm))
    1.84 +apply (rule trans)
    1.85 +apply (rule step_def2)
    1.86 +apply simp
    1.87 +apply (erule exE)
    1.88 +apply (erule flat_codom [THEN disjE])
    1.89 +apply simp_all
    1.90 +done
    1.91 +
    1.92 +lemma loop_lemma2: "[|EX y. b$y=FF;iterate (Suc k)$(step$b$g)$x ~=UU |]==>
    1.93 +      iterate k$(step$b$g)$x ~=UU"
    1.94 +apply (blast intro: loop_lemma1)
    1.95 +done
    1.96 +
    1.97 +lemma loop_lemma3 [rule_format (no_asm)]:
    1.98 +  "[| ALL x. INV x & b$x=TT & g$x~=UU --> INV (g$x);
    1.99 +         EX y. b$y=FF; INV x |]
   1.100 +      ==> iterate k$(step$b$g)$x ~=UU --> INV (iterate k$(step$b$g)$x)"
   1.101 +apply (induct_tac "k")
   1.102 +apply (simp (no_asm_simp))
   1.103 +apply (intro strip)
   1.104 +apply (simp (no_asm) add: step_def2)
   1.105 +apply (rule_tac p = "b$ (iterate n$ (step$b$g) $x) " in trE)
   1.106 +apply (erule notE)
   1.107 +apply (simp add: step_def2)
   1.108 +apply (simp (no_asm_simp))
   1.109 +apply (rule mp)
   1.110 +apply (erule spec)
   1.111 +apply (simp (no_asm_simp) del: iterate_Suc add: loop_lemma2)
   1.112 +apply (rule_tac s = "iterate (Suc n) $ (step$b$g) $x"
   1.113 +  and t = "g$ (iterate n$ (step$b$g) $x) " in ssubst)
   1.114 +prefer 2 apply (assumption)
   1.115 +apply (simp add: step_def2)
   1.116 +apply (drule (1) loop_lemma2, simp)
   1.117 +done
   1.118 +
   1.119 +lemma loop_lemma4 [rule_format]:
   1.120 +  "ALL x. b$(iterate k$(step$b$g)$x)=FF --> while$b$g$x= iterate k$(step$b$g)$x"
   1.121 +apply (induct_tac k)
   1.122 +apply (simp (no_asm))
   1.123 +apply (intro strip)
   1.124 +apply (simplesubst while_unfold)
   1.125 +apply simp
   1.126 +apply (rule allI)
   1.127 +apply (simplesubst iterate_Suc2)
   1.128 +apply (intro strip)
   1.129 +apply (rule trans)
   1.130 +apply (rule while_unfold3)
   1.131 +apply simp
   1.132 +done
   1.133 +
   1.134 +lemma loop_lemma5 [rule_format (no_asm)]:
   1.135 +  "ALL k. b$(iterate k$(step$b$g)$x) ~= FF ==>
   1.136 +    ALL m. while$b$g$(iterate m$(step$b$g)$x)=UU"
   1.137 +apply (simplesubst while_def2)
   1.138 +apply (rule fix_ind)
   1.139 +apply simp
   1.140 +apply simp
   1.141 +apply (rule allI)
   1.142 +apply (simp (no_asm))
   1.143 +apply (rule_tac p = "b$ (iterate m$ (step$b$g) $x) " in trE)
   1.144 +apply (simp (no_asm_simp))
   1.145 +apply (simp (no_asm_simp))
   1.146 +apply (rule_tac s = "xa$ (iterate (Suc m) $ (step$b$g) $x) " in trans)
   1.147 +apply (erule_tac [2] spec)
   1.148 +apply (rule cfun_arg_cong)
   1.149 +apply (rule trans)
   1.150 +apply (rule_tac [2] iterate_Suc [symmetric])
   1.151 +apply (simp add: step_def2)
   1.152 +apply blast
   1.153 +done
   1.154 +
   1.155 +lemma loop_lemma6: "ALL k. b$(iterate k$(step$b$g)$x) ~= FF ==> while$b$g$x=UU"
   1.156 +apply (rule_tac t = "x" in iterate_0 [THEN subst])
   1.157 +apply (erule loop_lemma5)
   1.158 +done
   1.159 +
   1.160 +lemma loop_lemma7: "while$b$g$x ~= UU ==> EX k. b$(iterate k$(step$b$g)$x) = FF"
   1.161 +apply (blast intro: loop_lemma6)
   1.162 +done
   1.163 +
   1.164 +
   1.165 +(* ------------------------------------------------------------------------- *)
   1.166 +(* an invariant rule for loops                                               *)
   1.167 +(* ------------------------------------------------------------------------- *)
   1.168 +
   1.169 +lemma loop_inv2:
   1.170 +"[| (ALL y. INV y & b$y=TT & g$y ~= UU --> INV (g$y));
   1.171 +    (ALL y. INV y & b$y=FF --> Q y);
   1.172 +    INV x; while$b$g$x~=UU |] ==> Q (while$b$g$x)"
   1.173 +apply (rule_tac P = "%k. b$ (iterate k$ (step$b$g) $x) =FF" in exE)
   1.174 +apply (erule loop_lemma7)
   1.175 +apply (simplesubst loop_lemma4)
   1.176 +apply assumption
   1.177 +apply (drule spec, erule mp)
   1.178 +apply (rule conjI)
   1.179 +prefer 2 apply (assumption)
   1.180 +apply (rule loop_lemma3)
   1.181 +apply assumption
   1.182 +apply (blast intro: loop_lemma6)
   1.183 +apply assumption
   1.184 +apply (rotate_tac -1)
   1.185 +apply (simp add: loop_lemma4)
   1.186 +done
   1.187 +
   1.188 +lemma loop_inv:
   1.189 +  assumes premP: "P(x)"
   1.190 +    and premI: "!!y. P y ==> INV y"
   1.191 +    and premTT: "!!y. [| INV y; b$y=TT; g$y~=UU|] ==> INV (g$y)"
   1.192 +    and premFF: "!!y. [| INV y; b$y=FF|] ==> Q y"
   1.193 +    and premW: "while$b$g$x ~= UU"
   1.194 +  shows "Q (while$b$g$x)"
   1.195 +apply (rule loop_inv2)
   1.196 +apply (rule_tac [3] premP [THEN premI])
   1.197 +apply (rule_tac [3] premW)
   1.198 +apply (blast intro: premTT)
   1.199 +apply (blast intro: premFF)
   1.200 +done
   1.201 +
   1.202 +end
   1.203 +