author | paulson |
Fri, 18 Feb 2000 15:20:44 +0100 | |
changeset 8251 | 9be357df93d4 |
parent 8216 | e4b3192dfefa |
child 9403 | aad13b59b8d9 |
permissions | -rw-r--r-- |
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(* Title: HOL/UNITY/Comp.thy |
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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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Composition |
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From Chandy and Sanders, "Reasoning About Program Composition" |
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*) |
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(*** component ***) |
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Goalw [component_def] |
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"H <= F | H <= G ==> H <= (F Join G)"; |
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by Auto_tac; |
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by (res_inst_tac [("x", "G Join Ga")] exI 1); |
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by (res_inst_tac [("x", "G Join F")] exI 2); |
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by (auto_tac (claset(), simpset() addsimps Join_ac)); |
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qed "componentI"; |
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Goalw [component_def] |
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"(F <= G) = (Init G <= Init F & Acts F <= Acts G)"; |
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by (force_tac (claset() addSIs [exI, program_equalityI], |
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simpset()) 1); |
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qed "component_eq_subset"; |
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Goalw [component_def] "SKIP <= F"; |
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by (force_tac (claset() addIs [Join_SKIP_left], simpset()) 1); |
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Finished proofs to end of section 5.1 of Chandy and Sanders
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qed "component_SKIP"; |
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Goalw [component_def] "F <= (F :: 'a program)"; |
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by (blast_tac (claset() addIs [Join_SKIP_right]) 1); |
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qed "component_refl"; |
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AddIffs [component_SKIP, component_refl]; |
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Goal "F <= SKIP ==> F = SKIP"; |
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by (auto_tac (claset() addSIs [program_equalityI], |
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simpset() addsimps [component_eq_subset])); |
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qed "SKIP_minimal"; |
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Goalw [component_def] "F <= (F Join G)"; |
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by (Blast_tac 1); |
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qed "component_Join1"; |
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Goalw [component_def] "G <= (F Join G)"; |
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by (simp_tac (simpset() addsimps [Join_commute]) 1); |
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by (Blast_tac 1); |
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qed "component_Join2"; |
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Goal "F<=G ==> F Join G = G"; |
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by (auto_tac (claset(), simpset() addsimps [component_def, Join_left_absorb])); |
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qed "Join_absorb1"; |
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Goal "G<=F ==> F Join G = F"; |
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by (auto_tac (claset(), simpset() addsimps Join_ac@[component_def])); |
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qed "Join_absorb2"; |
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Goal "((JOIN I F) <= H) = (ALL i: I. F i <= H)"; |
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by (simp_tac (simpset() addsimps [component_eq_subset]) 1); |
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by (Blast_tac 1); |
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qed "JN_component_iff"; |
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Goalw [component_def] "i : I ==> (F i) <= (JN i:I. (F i))"; |
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by (blast_tac (claset() addIs [JN_absorb]) 1); |
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qed "component_JN"; |
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Goalw [component_def] "[| F <= G; G <= H |] ==> F <= (H :: 'a program)"; |
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by (blast_tac (claset() addIs [Join_assoc RS sym]) 1); |
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qed "component_trans"; |
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Goal "[| F <= G; G <= F |] ==> F = (G :: 'a program)"; |
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by (full_simp_tac (simpset() addsimps [component_eq_subset]) 1); |
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by (blast_tac (claset() addSIs [program_equalityI]) 1); |
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qed "component_antisym"; |
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Goal "((F Join G) <= H) = (F <= H & G <= H)"; |
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by (simp_tac (simpset() addsimps [component_eq_subset]) 1); |
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by (Blast_tac 1); |
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qed "Join_component_iff"; |
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Goal "[| F <= G; G : A co B |] ==> F : A co B"; |
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by (auto_tac (claset(), |
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simpset() addsimps [constrains_def, component_eq_subset])); |
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qed "component_constrains"; |
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(*Used in Guar.thy to show that programs are partially ordered*) |
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bind_thm ("program_less_le", strict_component_def RS meta_eq_to_obj_eq); |
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(*** preserves ***) |
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val prems = |
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Goalw [preserves_def] "(!!z. F : stable {s. v s = z}) ==> F : preserves v"; |
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by (blast_tac (claset() addIs prems) 1); |
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qed "preservesI"; |
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Goalw [preserves_def, stable_def, constrains_def] |
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"[| F : preserves v; act : Acts F; (s,s') : act |] ==> v s = v s'"; |
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by (Force_tac 1); |
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qed "preserves_imp_eq"; |
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Goal "(F Join G : preserves v) = (F : preserves v & G : preserves v)"; |
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by (simp_tac (simpset() addsimps [Join_stable, preserves_def]) 1); |
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by (Blast_tac 1); |
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qed "Join_preserves"; |
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Goal "(JOIN I F : preserves v) = (ALL i:I. F i : preserves v)"; |
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by (simp_tac (simpset() addsimps [JN_stable, preserves_def]) 1); |
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by (Blast_tac 1); |
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qed "JN_preserves"; |
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AddIffs [Join_preserves, JN_preserves]; |
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Goalw [funPair_def] "(funPair f g) x = (f x, g x)"; |
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by (Simp_tac 1); |
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qed "funPair_apply"; |
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Addsimps [funPair_apply]; |
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Goal "preserves (funPair v w) = preserves v Int preserves w"; |
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by (auto_tac (claset(), |
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simpset() addsimps [preserves_def, stable_def, constrains_def])); |
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by (Blast_tac 1); |
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qed "preserves_funPair"; |
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(* (F : preserves (funPair v w)) = (F : preserves v Int preserves w) *) |
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AddIffs [preserves_funPair RS eqset_imp_iff]; |
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Goal "(funPair f g) o h = funPair (f o h) (g o h)"; |
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by (simp_tac (simpset() addsimps [funPair_def, o_def]) 1); |
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qed "funPair_o_distrib"; |
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Goal "fst o (funPair f g) = f"; |
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by (simp_tac (simpset() addsimps [funPair_def, o_def]) 1); |
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qed "fst_o_funPair"; |
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Goal "snd o (funPair f g) = g"; |
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by (simp_tac (simpset() addsimps [funPair_def, o_def]) 1); |
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qed "snd_o_funPair"; |
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Addsimps [fst_o_funPair, snd_o_funPair]; |
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Goal "preserves v <= preserves (w o v)"; |
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by (force_tac (claset(), |
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simpset() addsimps [preserves_def, |
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stable_def, constrains_def]) 1); |
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qed "subset_preserves_o"; |
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Goal "preserves v <= stable {s. P (v s)}"; |
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by (auto_tac (claset(), |
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simpset() addsimps [preserves_def, stable_def, constrains_def])); |
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by (rename_tac "s' s" 1); |
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by (subgoal_tac "v s = v s'" 1); |
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156 |
by (ALLGOALS Force_tac); |
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157 |
qed "preserves_subset_stable"; |
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158 |
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Goal "preserves v <= increasing v"; |
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by (auto_tac (claset(), |
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simpset() addsimps [impOfSubs preserves_subset_stable, |
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increasing_def])); |
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qed "preserves_subset_increasing"; |
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164 |
||
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Goal "preserves id <= stable A"; |
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166 |
by (force_tac (claset(), |
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167 |
simpset() addsimps [preserves_def, stable_def, constrains_def]) 1); |
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168 |
qed "preserves_id_subset_stable"; |
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169 |
|
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170 |
|
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171 |
(** Some lemmas used only in Client.ML **) |
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173 |
Goal "[| F : stable {s. P (v s) (w s)}; \ |
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174 |
\ G : preserves v; G : preserves w |] \ |
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175 |
\ ==> F Join G : stable {s. P (v s) (w s)}"; |
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176 |
by (asm_simp_tac (simpset() addsimps [Join_stable]) 1); |
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177 |
by (subgoal_tac "G: preserves (funPair v w)" 1); |
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178 |
by (Asm_simp_tac 2); |
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179 |
by (dres_inst_tac [("P1", "split ?Q")] |
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180 |
(impOfSubs preserves_subset_stable) 1); |
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181 |
by Auto_tac; |
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182 |
qed "stable_localTo_stable2"; |
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183 |
|
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184 |
Goal "[| F : stable {s. v s <= w s}; G : preserves v; \ |
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185 |
\ F Join G : Increasing w |] \ |
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186 |
\ ==> F Join G : Stable {s. v s <= w s}"; |
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187 |
by (auto_tac (claset(), |
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188 |
simpset() addsimps [stable_def, Stable_def, Increasing_def, |
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189 |
Constrains_def, Join_constrains, all_conj_distrib])); |
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190 |
by (blast_tac (claset() addIs [constrains_weaken]) 1); |
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191 |
(*The G case remains*) |
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192 |
by (auto_tac (claset(), |
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193 |
simpset() addsimps [preserves_def, stable_def, constrains_def])); |
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194 |
by (case_tac "act: Acts F" 1); |
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195 |
by (Blast_tac 1); |
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196 |
(*We have a G-action, so delete assumptions about F-actions*) |
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197 |
by (thin_tac "ALL act:Acts F. ?P act" 1); |
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198 |
by (thin_tac "ALL z. ALL act:Acts F. ?P z act" 1); |
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199 |
by (subgoal_tac "v x = v xa" 1); |
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200 |
by (Blast_tac 2); |
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201 |
by Auto_tac; |
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202 |
by (etac order_trans 1); |
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203 |
by (Blast_tac 1); |
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204 |
qed "Increasing_preserves_Stable"; |
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205 |