author | paulson |
Wed, 25 Nov 1998 15:52:45 +0100 | |
changeset 5968 | 06f9dbfff032 |
parent 5804 | 8e0a4c4fd67b |
child 6012 | 1894bfc4aee9 |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/Comp.thy |
2 |
ID: $Id$ |
|
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
|
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Copyright 1998 University of Cambridge |
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||
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Composition |
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||
8 |
From Chandy and Sanders, "Reasoning About Program Composition" |
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*) |
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10 |
||
11 |
(*split_all_tac causes a big blow-up*) |
|
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claset_ref() := claset() delSWrapper record_split_name; |
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|
14 |
Delsimps [split_paired_All]; |
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15 |
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16 |
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(*** component ***) |
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Goalw [component_def] "component SKIP F"; |
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by (blast_tac (claset() addIs [Join_SKIP_left]) 1); |
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21 |
qed "component_SKIP"; |
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5597 | 23 |
Goalw [component_def] "component F F"; |
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by (blast_tac (claset() addIs [Join_SKIP_right]) 1); |
5597 | 25 |
qed "component_refl"; |
26 |
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AddIffs [component_SKIP, component_refl]; |
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|
5968 | 29 |
Goalw [component_def] "component F (F Join G)"; |
30 |
by (Blast_tac 1); |
|
31 |
qed "component_Join1"; |
|
32 |
||
33 |
Goalw [component_def] "component G (F Join G)"; |
|
34 |
by (simp_tac (simpset() addsimps [Join_commute]) 1); |
|
35 |
by (Blast_tac 1); |
|
36 |
qed "component_Join2"; |
|
37 |
||
38 |
Goalw [component_def] "i : I ==> component (F i) (JN i:I. (F i))"; |
|
39 |
by (blast_tac (claset() addIs [JN_absorb]) 1); |
|
40 |
qed "component_JN"; |
|
41 |
||
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Goalw [component_def] "[| component F G; component G H |] ==> component F H"; |
43 |
by (blast_tac (claset() addIs [Join_assoc RS sym]) 1); |
|
44 |
qed "component_trans"; |
|
45 |
||
46 |
Goalw [component_def,Join_def] "component F G ==> Acts F <= Acts G"; |
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by Auto_tac; |
5620 | 48 |
qed "component_Acts"; |
5597 | 49 |
|
50 |
Goalw [component_def,Join_def] "component F G ==> Init G <= Init F"; |
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by Auto_tac; |
5620 | 52 |
qed "component_Init"; |
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|
54 |
Goal "[| component F G; component G F |] ==> F=G"; |
|
55 |
by (asm_simp_tac (simpset() addsimps [program_equalityI, equalityI, |
|
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component_Acts, component_Init]) 1); |
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qed "component_anti_sym"; |
58 |
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Goalw [component_def] |
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"component F H = (EX G. F Join G = H & Disjoint F G)"; |
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by (blast_tac (claset() addSIs [Diff_Disjoint, Join_Diff2]) 1); |
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qed "component_eq"; |
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|
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(*** existential properties ***) |
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65 |
||
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Goalw [ex_prop_def] |
|
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"[| ex_prop X; finite GG |] ==> GG Int X ~= {} --> (JN G:GG. G) : X"; |
|
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by (etac finite_induct 1); |
|
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by (auto_tac (claset(), simpset() addsimps [Int_insert_left])); |
|
70 |
qed_spec_mp "ex1"; |
|
71 |
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Goalw [ex_prop_def] |
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"ALL GG. finite GG & GG Int X ~= {} --> (JN G:GG. G) : X ==> ex_prop X"; |
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by (Clarify_tac 1); |
75 |
by (dres_inst_tac [("x", "{F,G}")] spec 1); |
|
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76 |
by Auto_tac; |
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qed "ex2"; |
78 |
||
79 |
(*Chandy & Sanders take this as a definition*) |
|
80 |
Goal "ex_prop X = (ALL GG. finite GG & GG Int X ~= {} --> (JN G:GG. G) : X)"; |
|
81 |
by (blast_tac (claset() addIs [ex1,ex2]) 1); |
|
82 |
qed "ex_prop_finite"; |
|
83 |
||
84 |
(*Their "equivalent definition" given at the end of section 3*) |
|
85 |
Goal "ex_prop X = (ALL G. G:X = (ALL H. component G H --> H: X))"; |
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86 |
by Auto_tac; |
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|
87 |
by (rewrite_goals_tac [ex_prop_def, component_def]); |
5597 | 88 |
by (Blast_tac 1); |
89 |
by Safe_tac; |
|
90 |
by (stac Join_commute 2); |
|
91 |
by (ALLGOALS Blast_tac); |
|
92 |
qed "ex_prop_equiv"; |
|
93 |
||
94 |
||
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(*** universal properties ***) |
|
96 |
||
97 |
Goalw [uv_prop_def] |
|
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"[| uv_prop X; finite GG |] ==> GG <= X --> (JN G:GG. G) : X"; |
|
99 |
by (etac finite_induct 1); |
|
100 |
by (auto_tac (claset(), simpset() addsimps [Int_insert_left])); |
|
101 |
qed_spec_mp "uv1"; |
|
102 |
||
103 |
Goalw [uv_prop_def] |
|
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"ALL GG. finite GG & GG <= X --> (JN G:GG. G) : X ==> uv_prop X"; |
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105 |
by (rtac conjI 1); |
5597 | 106 |
by (Clarify_tac 2); |
107 |
by (dres_inst_tac [("x", "{F,G}")] spec 2); |
|
108 |
by (dres_inst_tac [("x", "{}")] spec 1); |
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by Auto_tac; |
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qed "uv2"; |
111 |
||
112 |
(*Chandy & Sanders take this as a definition*) |
|
113 |
Goal "uv_prop X = (ALL GG. finite GG & GG <= X --> (JN G:GG. G) : X)"; |
|
114 |
by (blast_tac (claset() addIs [uv1,uv2]) 1); |
|
115 |
qed "uv_prop_finite"; |
|
116 |
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117 |
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118 |
(*** guarantees ***) |
|
119 |
||
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(*This equation is more intuitive than the official definition*) |
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Goal "(F : X guarantees Y) = \ |
122 |
\ (ALL G. F Join G : X & Disjoint F G --> F Join G : Y)"; |
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by (simp_tac (simpset() addsimps [guarantees_def, component_eq]) 1); |
5668 | 124 |
by (Blast_tac 1); |
125 |
qed "guarantees_eq"; |
|
126 |
||
5597 | 127 |
Goalw [guarantees_def] "X <= Y ==> X guarantees Y = UNIV"; |
128 |
by (Blast_tac 1); |
|
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qed "subset_imp_guarantees"; |
|
130 |
||
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(*Remark at end of section 4.1*) |
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Goalw [guarantees_def] "ex_prop Y = (Y = UNIV guarantees Y)"; |
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by (simp_tac (simpset() addsimps [ex_prop_equiv]) 1); |
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by (blast_tac (claset() addEs [equalityE]) 1); |
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qed "ex_prop_equiv2"; |
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136 |
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Goalw [guarantees_def] |
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"(INT X:XX. X guarantees Y) = (UN X:XX. X) guarantees Y"; |
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by (Blast_tac 1); |
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qed "INT_guarantees_left"; |
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141 |
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Goalw [guarantees_def] |
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"(INT Y:YY. X guarantees Y) = X guarantees (INT Y:YY. Y)"; |
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by (Blast_tac 1); |
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qed "INT_guarantees_right"; |
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146 |
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Goalw [guarantees_def] "(X guarantees Y) = (UNIV guarantees (-X Un Y))"; |
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by (Blast_tac 1); |
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qed "shunting"; |
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150 |
|
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Goalw [guarantees_def] "(X guarantees Y) = -Y guarantees -X"; |
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152 |
by (Blast_tac 1); |
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153 |
qed "contrapositive"; |
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154 |
|
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155 |
Goalw [guarantees_def] |
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"V guarantees X Int ((X Int Y) guarantees Z) <= (V Int Y) guarantees Z"; |
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157 |
by (Blast_tac 1); |
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158 |
qed "combining1"; |
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159 |
|
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160 |
Goalw [guarantees_def] |
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"V guarantees (X Un Y) Int (Y guarantees Z) <= V guarantees (X Un Z)"; |
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162 |
by (Blast_tac 1); |
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163 |
qed "combining2"; |
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164 |
|
5630 | 165 |
Goalw [guarantees_def] |
5968 | 166 |
"ALL i:I. F : X guarantees (Y i) ==> F : X guarantees (INT i:I. Y i)"; |
5630 | 167 |
by (Blast_tac 1); |
168 |
qed "all_guarantees"; |
|
169 |
||
170 |
Goalw [guarantees_def] |
|
5968 | 171 |
"EX i:I. F : X guarantees (Y i) ==> F : X guarantees (UN i:I. Y i)"; |
5630 | 172 |
by (Blast_tac 1); |
173 |
qed "ex_guarantees"; |
|
174 |
||
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175 |
val prems = Goal |
5968 | 176 |
"(!!G. [| F Join G : X; Disjoint F G |] ==> F Join G : Y) \ |
177 |
\ ==> F : X guarantees Y"; |
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178 |
by (simp_tac (simpset() addsimps [guarantees_def, component_eq]) 1); |
5630 | 179 |
by (blast_tac (claset() addIs prems) 1); |
180 |
qed "guaranteesI"; |
|
181 |
||
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182 |
Goalw [guarantees_def, component_def] |
5968 | 183 |
"[| F : X guarantees Y; F Join G : X |] ==> F Join G : Y"; |
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184 |
by (Blast_tac 1); |
5637 | 185 |
qed "guaranteesD"; |
186 |
||
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187 |
|
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188 |
(*** well-definedness ***) |
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189 |
|
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190 |
Goalw [welldef_def] "F Join G: welldef ==> F: welldef"; |
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191 |
by Auto_tac; |
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192 |
qed "Join_welldef_D1"; |
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193 |
|
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194 |
Goalw [welldef_def] "F Join G: welldef ==> G: welldef"; |
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195 |
by Auto_tac; |
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196 |
qed "Join_welldef_D2"; |
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197 |
|
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198 |
(*** refinement ***) |
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199 |
|
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200 |
Goalw [refines_def] "F refines F wrt X"; |
5597 | 201 |
by (Blast_tac 1); |
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202 |
qed "refines_refl"; |
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203 |
|
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204 |
Goalw [refines_def] |
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205 |
"[| H refines G wrt X; G refines F wrt X |] ==> H refines F wrt X"; |
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206 |
by (Blast_tac 1); |
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207 |
qed "refines_trans"; |
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208 |
|
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209 |
Goalw [strict_ex_prop_def] |
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210 |
"strict_ex_prop X \ |
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211 |
\ ==> (ALL H. F Join H : X --> G Join H : X) = (F:X --> G:X)"; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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212 |
by (Blast_tac 1); |
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213 |
qed "strict_ex_refine_lemma"; |
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214 |
|
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215 |
Goalw [strict_ex_prop_def] |
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216 |
"strict_ex_prop X \ |
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217 |
\ ==> (ALL H. F Join H : welldef & F Join H : X --> G Join H : X) = \ |
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218 |
\ (F: welldef Int X --> G:X)"; |
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219 |
by Safe_tac; |
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220 |
by (eres_inst_tac [("x","SKIP"), ("P", "%H. ?PP H --> ?RR H")] allE 1); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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221 |
by (auto_tac (claset() addDs [Join_welldef_D1, Join_welldef_D2], simpset())); |
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222 |
qed "strict_ex_refine_lemma_v"; |
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223 |
|
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224 |
Goal "[| strict_ex_prop X; \ |
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225 |
\ ALL H. F Join H : welldef Int X --> G Join H : welldef |] \ |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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226 |
\ ==> (G refines F wrt X) = (G iso_refines F wrt X)"; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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227 |
by (res_inst_tac [("x","SKIP")] allE 1 |
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228 |
THEN assume_tac 1); |
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229 |
by (asm_full_simp_tac (simpset() addsimps [refines_def, iso_refines_def, |
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230 |
strict_ex_refine_lemma_v]) 1); |
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231 |
qed "ex_refinement_thm"; |
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232 |
|
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233 |
Goalw [strict_uv_prop_def] |
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234 |
"strict_uv_prop X \ |
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|
235 |
\ ==> (ALL H. F Join H : X --> G Join H : X) = (F:X --> G:X)"; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
parents:
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changeset
|
236 |
by (Blast_tac 1); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
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changeset
|
237 |
qed "strict_uv_refine_lemma"; |
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|
238 |
|
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|
239 |
Goalw [strict_uv_prop_def] |
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|
240 |
"strict_uv_prop X \ |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
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changeset
|
241 |
\ ==> (ALL H. F Join H : welldef & F Join H : X --> G Join H : X) = \ |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
parents:
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changeset
|
242 |
\ (F: welldef Int X --> G:X)"; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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|
243 |
by Safe_tac; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
parents:
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changeset
|
244 |
by (eres_inst_tac [("x","SKIP"), ("P", "%H. ?PP H --> ?RR H")] allE 1); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
parents:
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changeset
|
245 |
by (auto_tac (claset() addDs [Join_welldef_D1, Join_welldef_D2], |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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|
246 |
simpset())); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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changeset
|
247 |
qed "strict_uv_refine_lemma_v"; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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|
248 |
|
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|
249 |
Goal "[| strict_uv_prop X; \ |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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|
250 |
\ ALL H. F Join H : welldef Int X --> G Join H : welldef |] \ |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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changeset
|
251 |
\ ==> (G refines F wrt X) = (G iso_refines F wrt X)"; |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
parents:
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changeset
|
252 |
by (res_inst_tac [("x","SKIP")] allE 1 |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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|
253 |
THEN assume_tac 1); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
paulson
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|
254 |
by (asm_full_simp_tac (simpset() addsimps [refines_def, iso_refines_def, |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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|
255 |
strict_uv_refine_lemma_v]) 1); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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changeset
|
256 |
qed "uv_refinement_thm"; |