# HG changeset patch # User paulson # Date 902307445 -7200 # Node ID 82a5ca6290aa37a192b2bb98ee821c1d91244ed0 # Parent 1b0f14d111429d02b7a481fb90c0dd7aafd34185 New record type of programs diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Channel.ML --- a/src/HOL/UNITY/Channel.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Channel.ML Wed Aug 05 10:57:25 1998 +0200 @@ -26,8 +26,7 @@ by (Blast_tac 1); qed_spec_mp "minSet_nonempty"; -Goal - "leadsTo Acts (minSet -`` {Some x}) (minSet -`` (Some``greaterThan x))"; +Goal "leadsTo acts (minSet -`` {Some x}) (minSet -`` (Some``greaterThan x))"; by (rtac leadsTo_weaken 1); by (rtac ([UC2, UC1] MRS PSP) 1); by (ALLGOALS Asm_simp_tac); @@ -39,7 +38,7 @@ (*The induction*) -Goal "leadsTo Acts (UNIV-{{}}) (minSet -`` (Some``atLeast y))"; +Goal "leadsTo acts (UNIV-{{}}) (minSet -`` (Some``atLeast y))"; by (rtac leadsTo_weaken_R 1); by (res_inst_tac [("l", "y"), ("f", "the o minSet"), ("B", "{}")] greaterThan_bounded_induct 1); @@ -55,7 +54,7 @@ val lemma = result(); -Goal "!!y::nat. leadsTo Acts (UNIV-{{}}) {s. y ~: s}"; +Goal "!!y::nat. leadsTo acts (UNIV-{{}}) {s. y ~: s}"; by (rtac (lemma RS leadsTo_weaken_R) 1); by (Clarify_tac 1); by (forward_tac [minSet_nonempty] 1); diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Channel.thy --- a/src/HOL/UNITY/Channel.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Channel.thy Wed Aug 05 10:57:25 1998 +0200 @@ -18,12 +18,12 @@ rules - skip "id: Acts" + skip "id: acts" - UC1 "constrains Acts (minSet -`` {Some x}) (minSet -`` (Some``atLeast x))" + UC1 "constrains acts (minSet -`` {Some x}) (minSet -`` (Some``atLeast x))" - (* UC1 "constrains Acts {s. minSet s = x} {s. x <= minSet s}" *) + (* UC1 "constrains acts {s. minSet s = x} {s. x <= minSet s}" *) - UC2 "leadsTo Acts (minSet -`` {Some x}) {s. x ~: s}" + UC2 "leadsTo acts (minSet -`` {Some x}) {s. x ~: s}" end diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Common.ML --- a/src/HOL/UNITY/Common.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Common.ML Wed Aug 05 10:57:25 1998 +0200 @@ -10,12 +10,9 @@ From Misra, "A Logic for Concurrent Programming" (1994), sections 5.1 and 13.1. *) - -open Common; - (*Misra's property CMT4: t exceeds no common meeting time*) -Goal "[| ALL m. constrains Acts {m} (maxfg m); n: common |] \ -\ ==> stable Acts (atMost n)"; +Goal "[| ALL m. constrains acts {m} (maxfg m); n: common |] \ +\ ==> stable acts (atMost n)"; by (dres_inst_tac [("P", "%t. t<=n")] elimination_sing 1); by (asm_full_simp_tac (simpset() addsimps [atMost_def, stable_def, common_def, maxfg_def, @@ -27,8 +24,8 @@ addIs [order_eq_refl, fmono, gmono, le_trans]) 1); qed "common_stable"; -Goal "[| ALL m. constrains Acts {m} (maxfg m); n: common |] \ -\ ==> invariant ({0},Acts) (atMost n)"; +Goal "[| ALL m. constrains acts {m} (maxfg m); n: common |] \ +\ ==> invariant (|Init={0}, Acts=acts|) (atMost n)"; by (rtac invariantI 1); by (asm_simp_tac (simpset() addsimps [common_stable]) 2); by (simp_tac (simpset() addsimps [atMost_def]) 1); @@ -75,10 +72,10 @@ Addsimps [atMost_Int_atLeast]; Goal - "[| ALL m. constrains Acts {m} (maxfg m); \ -\ ALL m: lessThan n. leadsTo Acts {m} (greaterThan m); \ -\ n: common; id: Acts |] \ -\ ==> leadsTo Acts (atMost n) common"; + "[| ALL m. constrains acts {m} (maxfg m); \ +\ ALL m: lessThan n. leadsTo acts {m} (greaterThan m); \ +\ n: common; id: acts |] \ +\ ==> leadsTo acts (atMost n) common"; by (rtac leadsTo_weaken_R 1); by (res_inst_tac [("f","%x. x"), ("l", "n")] greaterThan_bounded_induct 1); by (ALLGOALS Asm_simp_tac); @@ -89,10 +86,10 @@ (*The "ALL m: Compl common" form echoes CMT6.*) Goal - "[| ALL m. constrains Acts {m} (maxfg m); \ -\ ALL m: Compl common. leadsTo Acts {m} (greaterThan m); \ -\ n: common; id: Acts |] \ -\ ==> leadsTo Acts (atMost (LEAST n. n: common)) common"; + "[| ALL m. constrains acts {m} (maxfg m); \ +\ ALL m: Compl common. leadsTo acts {m} (greaterThan m); \ +\ n: common; id: acts |] \ +\ ==> leadsTo acts (atMost (LEAST n. n: common)) common"; by (rtac lemma 1); by (ALLGOALS Asm_simp_tac); by (etac LeastI 2); diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Mutex.ML --- a/src/HOL/UNITY/Mutex.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Mutex.ML Wed Aug 05 10:57:25 1998 +0200 @@ -9,14 +9,14 @@ (*split_all_tac causes a big blow-up*) claset_ref() := claset() delSWrapper "split_all_tac"; -val cmd_defs = [mutex_def, +val cmd_defs = [Mprg_def, cmd0U_def, cmd1U_def, cmd2U_def, cmd3U_def, cmd4U_def, cmd0V_def, cmd1V_def, cmd2V_def, cmd3V_def, cmd4V_def]; -Goalw [mutex_def] "id : mutex"; +Goalw [Mprg_def] "id : Acts Mprg"; by (Simp_tac 1); -qed "id_in_mutex"; -AddIffs [id_in_mutex]; +qed "id_in_Acts"; +AddIffs [id_in_Acts]; (*Simplification for records*) @@ -30,30 +30,30 @@ Addsimps [invariantU_def, invariantV_def]; -Goalw [MInit_def] "invariant (MInit,mutex) invariantU"; +Goalw [Mprg_def] "invariant Mprg invariantU"; by (rtac invariantI 1); by (constrains_tac cmd_defs 2); by Auto_tac; qed "invariantU"; -Goalw [MInit_def] "invariant (MInit,mutex) invariantV"; +Goalw [Mprg_def] "invariant Mprg invariantV"; by (rtac invariantI 1); by (constrains_tac cmd_defs 2); by Auto_tac; qed "invariantV"; -val mutex_invariant = invariant_Int_rule [invariantU, invariantV]; +val invariantUV = invariant_Int_rule [invariantU, invariantV]; (*The safety property: mutual exclusion*) -Goal "disjoint (reachable (MInit,mutex)) {s. MM s = 3 & NN s = 3}"; -by (cut_facts_tac [mutex_invariant RS invariant_includes_reachable] 1); +Goal "(reachable Mprg) Int {s. MM s = 3 & NN s = 3} = {}"; +by (cut_facts_tac [invariantUV RS invariant_includes_reachable] 1); by Auto_tac; qed "mutual_exclusion"; (*The bad invariant FAILS in cmd1V*) -Goalw [bad_invariantU_def] "stable mutex bad_invariantU"; +Goalw [bad_invariantU_def] "stable (Acts Mprg) bad_invariantU"; by (constrains_tac cmd_defs 1); by (REPEAT (trans_tac 1)); by (safe_tac (claset() addSEs [le_SucE])); @@ -67,49 +67,48 @@ (*** Progress for U ***) -Goalw [unless_def] "unless mutex {s. MM s=2} {s. MM s=3}"; +Goalw [unless_def] "unless (Acts Mprg) {s. MM s=2} {s. MM s=3}"; by (constrains_tac cmd_defs 1); qed "U_F0"; -Goal "LeadsTo(MInit,mutex) {s. MM s=1} {s. PP s = VV s & MM s = 2}"; +Goal "LeadsTo Mprg {s. MM s=1} {s. PP s = VV s & MM s = 2}"; by (ensures_tac cmd_defs "cmd1U" 1); qed "U_F1"; -Goal "LeadsTo(MInit,mutex) {s. ~ PP s & MM s = 2} {s. MM s = 3}"; -by (cut_facts_tac [mutex_invariant] 1); +Goal "LeadsTo Mprg {s. ~ PP s & MM s = 2} {s. MM s = 3}"; +by (cut_facts_tac [invariantUV] 1); +bw Mprg_def; by (ensures_tac cmd_defs "cmd2U" 1); qed "U_F2"; -Goalw [mutex_def] "LeadsTo(MInit,mutex) {s. MM s = 3} {s. PP s}"; +Goal "LeadsTo Mprg {s. MM s = 3} {s. PP s}"; by (rtac leadsTo_imp_LeadsTo 1); by (res_inst_tac [("B", "{s. MM s = 4}")] leadsTo_Trans 1); by (ensures_tac cmd_defs "cmd4U" 2); by (ensures_tac cmd_defs "cmd3U" 1); qed "U_F3"; -Goal "LeadsTo(MInit,mutex) {s. MM s = 2} {s. PP s}"; +Goal "LeadsTo Mprg {s. MM s = 2} {s. PP s}"; by (rtac ([LeadsTo_weaken_L, subset_refl RS subset_imp_LeadsTo] MRS LeadsTo_Diff) 1); by (rtac ([U_F2, U_F3] MRS LeadsTo_Trans) 1); by (auto_tac (claset() addSEs [less_SucE], simpset())); val U_lemma2 = result(); -Goal "LeadsTo(MInit,mutex) {s. MM s = 1} {s. PP s}"; +Goal "LeadsTo Mprg {s. MM s = 1} {s. PP s}"; by (rtac ([U_F1 RS LeadsTo_weaken_R, U_lemma2] MRS LeadsTo_Trans) 1); by (Blast_tac 1); val U_lemma1 = result(); - -Goal "LeadsTo(MInit,mutex) {s. 1 <= MM s & MM s <= 3} {s. PP s}"; +Goal "LeadsTo Mprg {s. 1 <= MM s & MM s <= 3} {s. PP s}"; by (simp_tac (simpset() addsimps [le_Suc_eq, conj_disj_distribL] addcongs [rev_conj_cong]) 1); by (simp_tac (simpset() addsimps [Collect_disj_eq, LeadsTo_Un_distrib, U_lemma1, U_lemma2, U_F3] ) 1); val U_lemma123 = result(); - (*Misra's F4*) -Goal "LeadsTo(MInit,mutex) {s. UU s} {s. PP s}"; +Goal "LeadsTo Mprg {s. UU s} {s. PP s}"; by (rtac ([invariantU, U_lemma123] MRS invariant_LeadsTo_weaken) 1); by Auto_tac; qed "u_leadsto_p"; @@ -118,39 +117,39 @@ (*** Progress for V ***) -Goalw [unless_def] "unless mutex {s. NN s=2} {s. NN s=3}"; +Goalw [unless_def] "unless (Acts Mprg) {s. NN s=2} {s. NN s=3}"; by (constrains_tac cmd_defs 1); qed "V_F0"; -Goal "LeadsTo(MInit,mutex) {s. NN s=1} {s. PP s = (~ UU s) & NN s = 2}"; +Goal "LeadsTo Mprg {s. NN s=1} {s. PP s = (~ UU s) & NN s = 2}"; by (ensures_tac cmd_defs "cmd1V" 1); qed "V_F1"; -Goal "LeadsTo(MInit,mutex) {s. PP s & NN s = 2} {s. NN s = 3}"; -by (cut_facts_tac [mutex_invariant] 1); +Goal "LeadsTo Mprg {s. PP s & NN s = 2} {s. NN s = 3}"; +by (cut_facts_tac [invariantUV] 1); by (ensures_tac cmd_defs "cmd2V" 1); qed "V_F2"; -Goalw [mutex_def] "LeadsTo(MInit,mutex) {s. NN s = 3} {s. ~ PP s}"; +Goal "LeadsTo Mprg {s. NN s = 3} {s. ~ PP s}"; by (rtac leadsTo_imp_LeadsTo 1); by (res_inst_tac [("B", "{s. NN s = 4}")] leadsTo_Trans 1); by (ensures_tac cmd_defs "cmd4V" 2); by (ensures_tac cmd_defs "cmd3V" 1); qed "V_F3"; -Goal "LeadsTo(MInit,mutex) {s. NN s = 2} {s. ~ PP s}"; +Goal "LeadsTo Mprg {s. NN s = 2} {s. ~ PP s}"; by (rtac ([LeadsTo_weaken_L, subset_refl RS subset_imp_LeadsTo] MRS LeadsTo_Diff) 1); by (rtac ([V_F2, V_F3] MRS LeadsTo_Trans) 1); by (auto_tac (claset() addSEs [less_SucE], simpset())); val V_lemma2 = result(); -Goal "LeadsTo(MInit,mutex) {s. NN s = 1} {s. ~ PP s}"; +Goal "LeadsTo Mprg {s. NN s = 1} {s. ~ PP s}"; by (rtac ([V_F1 RS LeadsTo_weaken_R, V_lemma2] MRS LeadsTo_Trans) 1); by (Blast_tac 1); val V_lemma1 = result(); -Goal "LeadsTo(MInit,mutex) {s. 1 <= NN s & NN s <= 3} {s. ~ PP s}"; +Goal "LeadsTo Mprg {s. 1 <= NN s & NN s <= 3} {s. ~ PP s}"; by (simp_tac (simpset() addsimps [le_Suc_eq, conj_disj_distribL] addcongs [rev_conj_cong]) 1); by (simp_tac (simpset() addsimps [Collect_disj_eq, LeadsTo_Un_distrib, @@ -159,7 +158,7 @@ (*Misra's F4*) -Goal "LeadsTo(MInit,mutex) {s. VV s} {s. ~ PP s}"; +Goal "LeadsTo Mprg {s. VV s} {s. ~ PP s}"; by (rtac ([invariantV, V_lemma123] MRS invariant_LeadsTo_weaken) 1); by Auto_tac; qed "v_leadsto_not_p"; @@ -168,7 +167,7 @@ (** Absence of starvation **) (*Misra's F6*) -Goal "LeadsTo(MInit,mutex) {s. MM s = 1} {s. MM s = 3}"; +Goal "LeadsTo Mprg {s. MM s = 1} {s. MM s = 3}"; by (rtac LeadsTo_Un_duplicate 1); by (rtac LeadsTo_cancel2 1); by (rtac U_F2 2); @@ -182,7 +181,7 @@ (*The same for V*) -Goal "LeadsTo(MInit,mutex) {s. NN s = 1} {s. NN s = 3}"; +Goal "LeadsTo Mprg {s. NN s = 1} {s. NN s = 3}"; by (rtac LeadsTo_Un_duplicate 1); by (rtac LeadsTo_cancel2 1); by (rtac V_F2 2); diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Mutex.thy --- a/src/HOL/UNITY/Mutex.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Mutex.thy Wed Aug 05 10:57:25 1998 +0200 @@ -26,6 +26,9 @@ VV :: bool constdefs + + (** The program for process U **) + cmd0U :: "(state*state) set" "cmd0U == {(s,s'). s' = s (|UU:=True, MM:=1|) & MM s = 0}" @@ -41,6 +44,8 @@ cmd4U :: "(state*state) set" "cmd4U == {(s,s'). s' = s (|PP:=True, MM:=0|) & MM s = 4}" + (** The program for process V **) + cmd0V :: "(state*state) set" "cmd0V == {(s,s'). s' = s (|VV:=True, NN:=1|) & NN s = 0}" @@ -56,10 +61,12 @@ cmd4V :: "(state*state) set" "cmd4V == {(s,s'). s' = s (|PP:=False, NN:=0|) & NN s = 4}" - mutex :: "(state*state) set set" - "mutex == {id, - cmd0U, cmd1U, cmd2U, cmd3U, cmd4U, - cmd0V, cmd1V, cmd2V, cmd3V, cmd4V}" + Mprg :: state program + "Mprg == (|Init = {s. ~ UU s & ~ VV s & MM s = 0 & NN s = 0}, + Acts = {id, + cmd0U, cmd1U, cmd2U, cmd3U, cmd4U, + cmd0V, cmd1V, cmd2V, cmd3V, cmd4V}|)" + (** The correct invariants **) @@ -77,7 +84,4 @@ "bad_invariantU == {s. (UU s = (1 <= MM s & MM s <= 3)) & (3 <= MM s & MM s <= 4 --> ~ PP s)}" - MInit :: "state set" - "MInit == {s. ~ UU s & ~ VV s & MM s = 0 & NN s = 0}" - end diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/ROOT.ML --- a/src/HOL/UNITY/ROOT.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/ROOT.ML Wed Aug 05 10:57:25 1998 +0200 @@ -19,3 +19,4 @@ time_use_thy "Mutex"; time_use_thy "FP"; time_use_thy "Reach"; +time_use_thy "Handshake"; diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Reach.ML --- a/src/HOL/UNITY/Reach.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Reach.ML Wed Aug 05 10:57:25 1998 +0200 @@ -19,22 +19,22 @@ AddSEs [ifE]; -val cmd_defs = [racts_def, asgt_def, fun_upd_def]; +val cmd_defs = [Rprg_def, asgt_def, fun_upd_def]; -Goalw [racts_def] "id : racts"; +Goalw [Rprg_def] "id : (Acts Rprg)"; by (Simp_tac 1); -qed "id_in_racts"; -AddIffs [id_in_racts]; +qed "id_in_Acts"; +AddIffs [id_in_Acts]; (*All vertex sets are finite*) AddIffs [[subset_UNIV, finite_graph] MRS finite_subset]; Addsimps [reach_invariant_def]; -Goalw [rinit_def] "invariant (rinit,racts) reach_invariant"; +Goalw [Rprg_def] "invariant Rprg reach_invariant"; by (rtac invariantI 1); by Auto_tac; (*for the initial state; proof of stability remains*) -by (constrains_tac [racts_def, asgt_def] 1); +by (constrains_tac [Rprg_def, asgt_def] 1); by (blast_tac (claset() addIs [r_into_rtrancl,rtrancl_trans]) 1); qed "reach_invariant"; @@ -52,7 +52,7 @@ qed "fixedpoint_invariant_correct"; Goalw (cmd_defs @ [FP_def, fixedpoint_def, stable_def, constrains_def]) - "FP racts <= fixedpoint"; + "FP (Acts Rprg) <= fixedpoint"; by Auto_tac; by (dtac bspec 1); by (Blast_tac 1); @@ -62,11 +62,11 @@ val lemma1 = result(); Goalw (cmd_defs @ [FP_def, fixedpoint_def, stable_def, constrains_def]) - "fixedpoint <= FP racts"; + "fixedpoint <= FP (Acts Rprg)"; by (auto_tac (claset() addIs [ext], simpset())); val lemma2 = result(); -Goal "FP racts = fixedpoint"; +Goal "FP (Acts Rprg) = fixedpoint"; by (rtac ([lemma1,lemma2] MRS equalityI) 1); qed "FP_fixedpoint"; @@ -80,7 +80,7 @@ Goal "Compl fixedpoint = (UN (u,v): edges. {s. s u & ~ s v})"; by (simp_tac (simpset() addsimps ([Compl_FP, UN_UN_flatten, FP_fixedpoint RS sym, - racts_def, asgt_def])) 1); + Rprg_def, asgt_def])) 1); by Safe_tac; by (rtac fun_upd_idem 1); by (Blast_tac 1); @@ -118,34 +118,34 @@ qed "metric_le"; Goal "(u,v): edges ==> \ -\ ensures racts ((metric-``{m}) Int {s. s u & ~ s v}) \ +\ ensures (Acts Rprg) ((metric-``{m}) Int {s. s u & ~ s v}) \ \ (metric-``(lessThan m))"; -by (ensures_tac [racts_def, asgt_def] "asgt u v" 1); +by (ensures_tac [Rprg_def, asgt_def] "asgt u v" 1); by (cut_facts_tac [metric_le] 1); by (fast_tac (claset() addSDs [le_imp_less_or_eq]) 1); qed "edges_ensures"; -Goal "leadsTo racts ((metric-``{m}) - fixedpoint) (metric-``(lessThan m))"; +Goal "leadsTo (Acts Rprg) ((metric-``{m}) - fixedpoint) (metric-``(lessThan m))"; by (simp_tac (simpset() addsimps [Diff_fixedpoint]) 1); by (rtac leadsTo_UN 1); by (split_all_tac 1); by (asm_simp_tac (simpset() addsimps [edges_ensures RS leadsTo_Basis]) 1); qed "leadsTo_Diff_fixedpoint"; -Goal "leadsTo racts (metric-``{m}) (metric-``(lessThan m) Un fixedpoint)"; +Goal "leadsTo (Acts Rprg) (metric-``{m}) (metric-``(lessThan m) Un fixedpoint)"; by (rtac (leadsTo_Diff_fixedpoint RS leadsTo_weaken_R RS leadsTo_Diff) 1); by (ALLGOALS (blast_tac (claset() addIs [subset_imp_leadsTo]))); qed "leadsTo_Un_fixedpoint"; (*Execution in any state leads to a fixedpoint (i.e. can terminate)*) -Goal "leadsTo racts UNIV fixedpoint"; +Goal "leadsTo (Acts Rprg) UNIV fixedpoint"; by (rtac lessThan_induct 1); by Auto_tac; by (rtac leadsTo_Un_fixedpoint 1); qed "leadsTo_fixedpoint"; -Goal "LeadsTo(rinit,racts) UNIV { %v. (init, v) : edges^* }"; +Goal "LeadsTo Rprg UNIV { %v. (init, v) : edges^* }"; by (stac (fixedpoint_invariant_correct RS sym) 1); by (rtac ([reach_invariant, leadsTo_fixedpoint RS leadsTo_imp_LeadsTo] diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Reach.thy --- a/src/HOL/UNITY/Reach.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Reach.thy Wed Aug 05 10:57:25 1998 +0200 @@ -23,11 +23,9 @@ asgt :: "[vertex,vertex] => (state*state) set" "asgt u v == {(s,s'). s' = s(v:= s u | s v)}" - racts :: "(state*state) set set" - "racts == insert id (UN (u,v): edges. {asgt u v})" - - rinit :: "state set" - "rinit == {%v. v=init}" + Rprg :: state program + "Rprg == (|Init = {%v. v=init}, + Acts = insert id (UN (u,v): edges. {asgt u v})|)" reach_invariant :: state set "reach_invariant == {s. (ALL v. s v --> (init, v) : edges^*) & s init}" diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/SubstAx.ML --- a/src/HOL/UNITY/SubstAx.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/SubstAx.ML Wed Aug 05 10:57:25 1998 +0200 @@ -16,31 +16,27 @@ (*** Introduction rules: Basis, Trans, Union ***) -Goal "leadsTo Acts A B ==> LeadsTo(Init,Acts) A B"; +Goal "leadsTo (Acts prg) A B ==> LeadsTo prg A B"; by (simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [PSP_stable2, stable_reachable]) 1); qed "leadsTo_imp_LeadsTo"; -Goal "[| constrains Acts (reachable(Init,Acts) Int (A - A')) \ -\ (A Un A'); \ -\ transient Acts (reachable(Init,Acts) Int (A-A')) |] \ -\ ==> LeadsTo(Init,Acts) A A'"; -by (simp_tac (simpset() addsimps [LeadsTo_def]) 1); +Goal "ensures (Acts prg) (reachable prg Int A) (reachable prg Int A') \ +\ ==> LeadsTo prg A A'"; +by (full_simp_tac (simpset() addsimps [ensures_def, LeadsTo_def]) 1); by (rtac (stable_reachable RS stable_ensures_Int RS leadsTo_Basis) 1); -by (assume_tac 2); -by (asm_simp_tac - (simpset() addsimps [Int_Un_distrib RS sym, Diff_Int_distrib RS sym, - stable_constrains_Int]) 1); +by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [Diff_Int_distrib]))); +by (blast_tac (claset() addIs [constrains_weaken]) 1); qed "LeadsTo_Basis"; -Goal "[| LeadsTo(Init,Acts) A B; LeadsTo(Init,Acts) B C |] \ -\ ==> LeadsTo(Init,Acts) A C"; +Goal "[| LeadsTo prg A B; LeadsTo prg B C |] \ +\ ==> LeadsTo prg A C"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [leadsTo_Trans]) 1); qed "LeadsTo_Trans"; val [prem] = goalw thy [LeadsTo_def] - "(!!A. A : S ==> LeadsTo(Init,Acts) A B) ==> LeadsTo(Init,Acts) (Union S) B"; + "(!!A. A : S ==> LeadsTo prg A B) ==> LeadsTo prg (Union S) B"; by (Simp_tac 1); by (stac Int_Union 1); by (blast_tac (claset() addIs [leadsTo_UN, @@ -50,42 +46,42 @@ (*** Derived rules ***) -Goal "id: Acts ==> LeadsTo(Init,Acts) A UNIV"; +Goal "id: (Acts prg) ==> LeadsTo prg A UNIV"; by (asm_simp_tac (simpset() addsimps [LeadsTo_def, Int_lower1 RS subset_imp_leadsTo]) 1); qed "LeadsTo_UNIV"; Addsimps [LeadsTo_UNIV]; (*Useful with cancellation, disjunction*) -Goal "LeadsTo(Init,Acts) A (A' Un A') ==> LeadsTo(Init,Acts) A A'"; +Goal "LeadsTo prg A (A' Un A') ==> LeadsTo prg A A'"; by (asm_full_simp_tac (simpset() addsimps Un_ac) 1); qed "LeadsTo_Un_duplicate"; -Goal "LeadsTo(Init,Acts) A (A' Un C Un C) ==> LeadsTo(Init,Acts) A (A' Un C)"; +Goal "LeadsTo prg A (A' Un C Un C) ==> LeadsTo prg A (A' Un C)"; by (asm_full_simp_tac (simpset() addsimps Un_ac) 1); qed "LeadsTo_Un_duplicate2"; val prems = goal thy - "(!!i. i : I ==> LeadsTo(Init,Acts) (A i) B) \ -\ ==> LeadsTo(Init,Acts) (UN i:I. A i) B"; + "(!!i. i : I ==> LeadsTo prg (A i) B) \ +\ ==> LeadsTo prg (UN i:I. A i) B"; by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1); by (blast_tac (claset() addIs (LeadsTo_Union::prems)) 1); qed "LeadsTo_UN"; (*Binary union introduction rule*) -Goal "[| LeadsTo(Init,Acts) A C; LeadsTo(Init,Acts) B C |] ==> LeadsTo(Init,Acts) (A Un B) C"; +Goal "[| LeadsTo prg A C; LeadsTo prg B C |] ==> LeadsTo prg (A Un B) C"; by (stac Un_eq_Union 1); by (blast_tac (claset() addIs [LeadsTo_Union]) 1); qed "LeadsTo_Un"; -Goal "[| reachable(Init,Acts) Int A <= B; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A B"; +Goal "[| reachable prg Int A <= B; id: (Acts prg) |] \ +\ ==> LeadsTo prg A B"; by (simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1); qed "Int_subset_imp_LeadsTo"; -Goal "[| A <= B; id: Acts |] ==> LeadsTo(Init,Acts) A B"; +Goal "[| A <= B; id: (Acts prg) |] ==> LeadsTo prg A B"; by (simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1); qed "subset_imp_LeadsTo"; @@ -93,61 +89,62 @@ bind_thm ("empty_LeadsTo", empty_subsetI RS subset_imp_LeadsTo); Addsimps [empty_LeadsTo]; -Goal "[| reachable(Init,Acts) Int A = {}; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A B"; +Goal "[| reachable prg Int A = {}; id: (Acts prg) |] \ +\ ==> LeadsTo prg A B"; by (asm_simp_tac (simpset() addsimps [Int_subset_imp_LeadsTo]) 1); qed "Int_empty_LeadsTo"; -Goal "[| LeadsTo(Init,Acts) A A'; \ -\ reachable(Init,Acts) Int A' <= B' |] \ -\ ==> LeadsTo(Init,Acts) A B'"; +Goal "[| LeadsTo prg A A'; \ +\ reachable prg Int A' <= B' |] \ +\ ==> LeadsTo prg A B'"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [leadsTo_weaken_R]) 1); qed_spec_mp "LeadsTo_weaken_R"; -Goal "[| LeadsTo(Init,Acts) A A'; \ -\ reachable(Init,Acts) Int B <= A; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) B A'"; +Goal "[| LeadsTo prg A A'; \ +\ reachable prg Int B <= A; id: (Acts prg) |] \ +\ ==> LeadsTo prg B A'"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [leadsTo_weaken_L]) 1); qed_spec_mp "LeadsTo_weaken_L"; (*Distributes over binary unions*) -Goal "id: Acts ==> \ -\ LeadsTo(Init,Acts) (A Un B) C = \ -\ (LeadsTo(Init,Acts) A C & LeadsTo(Init,Acts) B C)"; +Goal "id: (Acts prg) ==> \ +\ LeadsTo prg (A Un B) C = \ +\ (LeadsTo prg A C & LeadsTo prg B C)"; by (blast_tac (claset() addIs [LeadsTo_Un, LeadsTo_weaken_L]) 1); qed "LeadsTo_Un_distrib"; -Goal "id: Acts ==> \ -\ LeadsTo(Init,Acts) (UN i:I. A i) B = \ -\ (ALL i : I. LeadsTo(Init,Acts) (A i) B)"; +Goal "id: (Acts prg) ==> \ +\ LeadsTo prg (UN i:I. A i) B = \ +\ (ALL i : I. LeadsTo prg (A i) B)"; by (blast_tac (claset() addIs [LeadsTo_UN, LeadsTo_weaken_L]) 1); qed "LeadsTo_UN_distrib"; -Goal "id: Acts ==> \ -\ LeadsTo(Init,Acts) (Union S) B = \ -\ (ALL A : S. LeadsTo(Init,Acts) A B)"; +Goal "id: (Acts prg) ==> \ +\ LeadsTo prg (Union S) B = \ +\ (ALL A : S. LeadsTo prg A B)"; by (blast_tac (claset() addIs [LeadsTo_Union, LeadsTo_weaken_L]) 1); qed "LeadsTo_Union_distrib"; -Goal "[| LeadsTo(Init,Acts) A A'; id: Acts; \ -\ reachable(Init,Acts) Int B <= A; \ -\ reachable(Init,Acts) Int A' <= B' |] \ -\ ==> LeadsTo(Init,Acts) B B'"; +Goal "[| LeadsTo prg A A'; id: (Acts prg); \ +\ reachable prg Int B <= A; \ +\ reachable prg Int A' <= B' |] \ +\ ==> LeadsTo prg B B'"; (*PROOF FAILED: why?*) by (blast_tac (claset() addIs [LeadsTo_Trans, LeadsTo_weaken_R, LeadsTo_weaken_L]) 1); qed "LeadsTo_weaken"; -(*Set difference: maybe combine with leadsTo_weaken_L??*) -Goal "[| LeadsTo(Init,Acts) (A-B) C; LeadsTo(Init,Acts) B C; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A C"; +(*Set difference: maybe combine with leadsTo_weaken_L?? + This is the most useful form of the "disjunction" rule*) +Goal "[| LeadsTo prg (A-B) C; LeadsTo prg B C; id: (Acts prg) |] \ +\ ==> LeadsTo prg A C"; by (blast_tac (claset() addIs [LeadsTo_Un, LeadsTo_weaken]) 1); qed "LeadsTo_Diff"; @@ -156,8 +153,8 @@ see ball_constrains_UN in UNITY.ML***) val prems = goal thy - "(!! i. i:I ==> LeadsTo(Init,Acts) (A i) (A' i)) \ -\ ==> LeadsTo(Init,Acts) (UN i:I. A i) (UN i:I. A' i)"; + "(!! i. i:I ==> LeadsTo prg (A i) (A' i)) \ +\ ==> LeadsTo prg (UN i:I. A i) (UN i:I. A' i)"; by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1); by (blast_tac (claset() addIs [LeadsTo_Union, LeadsTo_weaken_R] addIs prems) 1); @@ -166,22 +163,22 @@ (*Version with no index set*) val prems = goal thy - "(!! i. LeadsTo(Init,Acts) (A i) (A' i)) \ -\ ==> LeadsTo(Init,Acts) (UN i. A i) (UN i. A' i)"; + "(!! i. LeadsTo prg (A i) (A' i)) \ +\ ==> LeadsTo prg (UN i. A i) (UN i. A' i)"; by (blast_tac (claset() addIs [LeadsTo_UN_UN] addIs prems) 1); qed "LeadsTo_UN_UN_noindex"; (*Version with no index set*) -Goal "ALL i. LeadsTo(Init,Acts) (A i) (A' i) \ -\ ==> LeadsTo(Init,Acts) (UN i. A i) (UN i. A' i)"; +Goal "ALL i. LeadsTo prg (A i) (A' i) \ +\ ==> LeadsTo prg (UN i. A i) (UN i. A' i)"; by (blast_tac (claset() addIs [LeadsTo_UN_UN]) 1); qed "all_LeadsTo_UN_UN"; (*Binary union version*) -Goal "[| LeadsTo(Init,Acts) A A'; LeadsTo(Init,Acts) B B' |] \ -\ ==> LeadsTo(Init,Acts) (A Un B) (A' Un B')"; +Goal "[| LeadsTo prg A A'; LeadsTo prg B B' |] \ +\ ==> LeadsTo prg (A Un B) (A' Un B')"; by (blast_tac (claset() addIs [LeadsTo_Un, LeadsTo_weaken_R]) 1); qed "LeadsTo_Un_Un"; @@ -189,28 +186,28 @@ (** The cancellation law **) -Goal "[| LeadsTo(Init,Acts) A (A' Un B); LeadsTo(Init,Acts) B B'; \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A (A' Un B')"; +Goal "[| LeadsTo prg A (A' Un B); LeadsTo prg B B'; \ +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg A (A' Un B')"; by (blast_tac (claset() addIs [LeadsTo_Un_Un, subset_imp_LeadsTo, LeadsTo_Trans]) 1); qed "LeadsTo_cancel2"; -Goal "[| LeadsTo(Init,Acts) A (A' Un B); LeadsTo(Init,Acts) (B-A') B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A (A' Un B')"; +Goal "[| LeadsTo prg A (A' Un B); LeadsTo prg (B-A') B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg A (A' Un B')"; by (rtac LeadsTo_cancel2 1); by (assume_tac 2); by (ALLGOALS Asm_simp_tac); qed "LeadsTo_cancel_Diff2"; -Goal "[| LeadsTo(Init,Acts) A (B Un A'); LeadsTo(Init,Acts) B B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A (B' Un A')"; +Goal "[| LeadsTo prg A (B Un A'); LeadsTo prg B B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg A (B' Un A')"; by (asm_full_simp_tac (simpset() addsimps [Un_commute]) 1); by (blast_tac (claset() addSIs [LeadsTo_cancel2]) 1); qed "LeadsTo_cancel1"; -Goal "[| LeadsTo(Init,Acts) A (B Un A'); LeadsTo(Init,Acts) (B-A') B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A (B' Un A')"; +Goal "[| LeadsTo prg A (B Un A'); LeadsTo prg (B-A') B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg A (B' Un A')"; by (rtac LeadsTo_cancel1 1); by (assume_tac 2); by (ALLGOALS Asm_simp_tac); @@ -220,7 +217,7 @@ (** The impossibility law **) -Goal "LeadsTo(Init,Acts) A {} ==> reachable(Init,Acts) Int A = {}"; +Goal "LeadsTo prg A {} ==> reachable prg Int A = {}"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (etac leadsTo_empty 1); qed "LeadsTo_empty"; @@ -229,20 +226,20 @@ (** PSP: Progress-Safety-Progress **) (*Special case of PSP: Misra's "stable conjunction". Doesn't need id:Acts. *) -Goal "[| LeadsTo(Init,Acts) A A'; stable Acts B |] \ -\ ==> LeadsTo(Init,Acts) (A Int B) (A' Int B)"; +Goal "[| LeadsTo prg A A'; stable (Acts prg) B |] \ +\ ==> LeadsTo prg (A Int B) (A' Int B)"; by (asm_full_simp_tac (simpset() addsimps [LeadsTo_def, Int_assoc RS sym, PSP_stable]) 1); qed "R_PSP_stable"; -Goal "[| LeadsTo(Init,Acts) A A'; stable Acts B |] \ -\ ==> LeadsTo(Init,Acts) (B Int A) (B Int A')"; +Goal "[| LeadsTo prg A A'; stable (Acts prg) B |] \ +\ ==> LeadsTo prg (B Int A) (B Int A')"; by (asm_simp_tac (simpset() addsimps (R_PSP_stable::Int_ac)) 1); qed "R_PSP_stable2"; -Goal "[| LeadsTo(Init,Acts) A A'; constrains Acts B B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) (A Int B) ((A' Int B) Un (B' - B))"; +Goal "[| LeadsTo prg A A'; constrains (Acts prg) B B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg (A Int B) ((A' Int B) Un (B' - B))"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (dtac PSP 1); by (etac constrains_reachable 1); @@ -250,14 +247,14 @@ by (ALLGOALS Blast_tac); qed "R_PSP"; -Goal "[| LeadsTo(Init,Acts) A A'; constrains Acts B B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) (B Int A) ((B Int A') Un (B' - B))"; +Goal "[| LeadsTo prg A A'; constrains (Acts prg) B B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg (B Int A) ((B Int A') Un (B' - B))"; by (asm_simp_tac (simpset() addsimps (R_PSP::Int_ac)) 1); qed "R_PSP2"; Goalw [unless_def] - "[| LeadsTo(Init,Acts) A A'; unless Acts B B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) (A Int B) ((A' Int B) Un B')"; + "[| LeadsTo prg A A'; unless (Acts prg) B B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg (A Int B) ((A' Int B) Un B')"; by (dtac R_PSP 1); by (assume_tac 1); by (asm_full_simp_tac (simpset() addsimps [Un_Diff_Diff, Int_Diff_Un]) 2); @@ -272,10 +269,10 @@ (** Meta or object quantifier ????? **) Goal "[| wf r; \ -\ ALL m. LeadsTo(Init,Acts) (A Int f-``{m}) \ +\ ALL m. LeadsTo prg (A Int f-``{m}) \ \ ((A Int f-``(r^-1 ^^ {m})) Un B); \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A B"; +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg A B"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (etac leadsTo_wf_induct 1); by (assume_tac 2); @@ -284,10 +281,10 @@ Goal "[| wf r; \ -\ ALL m:I. LeadsTo(Init,Acts) (A Int f-``{m}) \ +\ ALL m:I. LeadsTo prg (A Int f-``{m}) \ \ ((A Int f-``(r^-1 ^^ {m})) Un B); \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A ((A - (f-``I)) Un B)"; +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg A ((A - (f-``I)) Un B)"; by (etac LeadsTo_wf_induct 1); by Safe_tac; by (case_tac "m:I" 1); @@ -296,29 +293,29 @@ qed "R_bounded_induct"; -Goal "[| ALL m. LeadsTo(Init,Acts) (A Int f-``{m}) \ -\ ((A Int f-``(lessThan m)) Un B); \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A B"; +Goal "[| ALL m. LeadsTo prg (A Int f-``{m}) \ +\ ((A Int f-``(lessThan m)) Un B); \ +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg A B"; by (rtac (wf_less_than RS LeadsTo_wf_induct) 1); by (assume_tac 2); by (Asm_simp_tac 1); qed "R_lessThan_induct"; -Goal "[| ALL m:(greaterThan l). LeadsTo(Init,Acts) (A Int f-``{m}) \ +Goal "[| ALL m:(greaterThan l). LeadsTo prg (A Int f-``{m}) \ \ ((A Int f-``(lessThan m)) Un B); \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A ((A Int (f-``(atMost l))) Un B)"; +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg A ((A Int (f-``(atMost l))) Un B)"; by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, Compl_greaterThan RS sym]) 1); by (rtac (wf_less_than RS R_bounded_induct) 1); by (assume_tac 2); by (Asm_simp_tac 1); qed "R_lessThan_bounded_induct"; -Goal "[| ALL m:(lessThan l). LeadsTo(Init,Acts) (A Int f-``{m}) \ +Goal "[| ALL m:(lessThan l). LeadsTo prg (A Int f-``{m}) \ \ ((A Int f-``(greaterThan m)) Un B); \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) A ((A Int (f-``(atLeast l))) Un B)"; +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg A ((A Int (f-``(atLeast l))) Un B)"; by (res_inst_tac [("f","f"),("f1", "%k. l - k")] (wf_less_than RS wf_inv_image RS LeadsTo_wf_induct) 1); by (assume_tac 2); @@ -333,19 +330,19 @@ (*** Completion: Binary and General Finite versions ***) -Goal "[| LeadsTo(Init,Acts) A A'; stable Acts A'; \ -\ LeadsTo(Init,Acts) B B'; stable Acts B'; id: Acts |] \ -\ ==> LeadsTo(Init,Acts) (A Int B) (A' Int B')"; +Goal "[| LeadsTo prg A A'; stable (Acts prg) A'; \ +\ LeadsTo prg B B'; stable (Acts prg) B'; id: (Acts prg) |] \ +\ ==> LeadsTo prg (A Int B) (A' Int B')"; by (full_simp_tac (simpset() addsimps [LeadsTo_def]) 1); by (blast_tac (claset() addIs [stable_completion RS leadsTo_weaken] addSIs [stable_Int, stable_reachable]) 1); qed "R_stable_completion"; -Goal "[| finite I; id: Acts |] \ -\ ==> (ALL i:I. LeadsTo(Init,Acts) (A i) (A' i)) --> \ -\ (ALL i:I. stable Acts (A' i)) --> \ -\ LeadsTo(Init,Acts) (INT i:I. A i) (INT i:I. A' i)"; +Goal "[| finite I; id: (Acts prg) |] \ +\ ==> (ALL i:I. LeadsTo prg (A i) (A' i)) --> \ +\ (ALL i:I. stable (Acts prg) (A' i)) --> \ +\ LeadsTo prg (INT i:I. A i) (INT i:I. A' i)"; by (etac finite_induct 1); by (Asm_simp_tac 1); by (asm_simp_tac @@ -354,10 +351,10 @@ qed_spec_mp "R_finite_stable_completion"; -Goal "[| LeadsTo(Init,Acts) A (A' Un C); constrains Acts A' (A' Un C); \ -\ LeadsTo(Init,Acts) B (B' Un C); constrains Acts B' (B' Un C); \ -\ id: Acts |] \ -\ ==> LeadsTo(Init,Acts) (A Int B) ((A' Int B') Un C)"; +Goal "[| LeadsTo prg A (A' Un C); constrains (Acts prg) A' (A' Un C); \ +\ LeadsTo prg B (B' Un C); constrains (Acts prg) B' (B' Un C); \ +\ id: (Acts prg) |] \ +\ ==> LeadsTo prg (A Int B) ((A' Int B') Un C)"; by (full_simp_tac (simpset() addsimps [LeadsTo_def, Int_Un_distrib]) 1); by (dtac completion 1); by (assume_tac 2); @@ -368,10 +365,10 @@ qed "R_completion"; -Goal "[| finite I; id: Acts |] \ -\ ==> (ALL i:I. LeadsTo(Init,Acts) (A i) (A' i Un C)) --> \ -\ (ALL i:I. constrains Acts (A' i) (A' i Un C)) --> \ -\ LeadsTo(Init,Acts) (INT i:I. A i) ((INT i:I. A' i) Un C)"; +Goal "[| finite I; id: (Acts prg) |] \ +\ ==> (ALL i:I. LeadsTo prg (A i) (A' i Un C)) --> \ +\ (ALL i:I. constrains (Acts prg) (A' i) (A' i Un C)) --> \ +\ LeadsTo prg (INT i:I. A i) ((INT i:I. A' i) Un C)"; by (etac finite_induct 1); by (ALLGOALS Asm_simp_tac); by (Clarify_tac 1); @@ -384,32 +381,37 @@ (*** Specialized laws for handling invariants ***) Goalw [transient_def] - "[| reachable(Init,Acts) <= INV; transient Acts (INV Int A) |] \ -\ ==> transient Acts (reachable(Init,Acts) Int A)"; + "[| reachable prg <= INV; transient (Acts prg) (INV Int A) |] \ +\ ==> transient (Acts prg) (reachable prg Int A)"; by (Clarify_tac 1); by (rtac bexI 1); by (assume_tac 2); by (Blast_tac 1); qed "reachable_transient"; -(*Uses the premise "invariant (Init,Acts)". Raw theorem-proving on this +(*Uses the premise "invariant prg". Raw theorem-proving on this inclusion is slow: the term contains a copy of the program.*) -Goal "[| invariant (Init,Acts) INV; \ -\ constrains Acts (INV Int (A - A')) (A Un A'); \ -\ transient Acts (INV Int (A-A')) |] \ -\ ==> LeadsTo(Init,Acts) A A'"; +Goal "[| invariant prg INV; \ +\ constrains (Acts prg) (INV Int (A-A')) (A Un A'); \ +\ transient (Acts prg) (INV Int (A-A')) |] \ +\ ==> ensures (Acts prg) (reachable prg Int A) (reachable prg Int A')"; bd invariant_includes_reachable 1; -by (rtac LeadsTo_Basis 1); +by (rtac ensuresI 1); +by (ALLGOALS + (full_simp_tac (simpset() addsimps [Int_Un_distrib RS sym, + Diff_Int_distrib RS sym]))); by (blast_tac (claset() addSIs [reachable_transient]) 2); -by (rewtac constrains_def); -by (Blast_tac 1); -qed "invariant_LeadsTo_Basis"; +br (stable_reachable RS stable_constrains_Int) 1; +by (blast_tac (claset() addIs [constrains_weaken]) 1); +qed "invariant_ensuresI"; + +bind_thm ("invariant_LeadsTo_Basis", invariant_ensuresI RS LeadsTo_Basis); -Goal "[| invariant (Init,Acts) INV; \ -\ LeadsTo(Init,Acts) A A'; id: Acts; \ +Goal "[| invariant prg INV; \ +\ LeadsTo prg A A'; id: (Acts prg); \ \ INV Int B <= A; INV Int A' <= B' |] \ -\ ==> LeadsTo(Init,Acts) B B'"; +\ ==> LeadsTo prg B B'"; by (blast_tac (claset() addDs [invariant_includes_reachable] addIs [LeadsTo_weaken]) 1); qed "invariant_LeadsTo_weaken"; @@ -425,8 +427,8 @@ SELECT_GOAL (EVERY [TRY (rtac stableI 1), rtac constrainsI 1, - rewtac main_def, - REPEAT_FIRST (eresolve_tac [insertE, emptyE]), + full_simp_tac (simpset() addsimps [main_def]) 1, + REPEAT_FIRST (eresolve_tac [disjE]), rewrite_goals_tac cmd_defs, ALLGOALS (SELECT_GOAL Auto_tac)]); diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/SubstAx.thy --- a/src/HOL/UNITY/SubstAx.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/SubstAx.thy Wed Aug 05 10:57:25 1998 +0200 @@ -10,10 +10,9 @@ constdefs - LeadsTo :: "['a set * ('a * 'a)set set, 'a set, 'a set] => bool" - "LeadsTo == %(Init,Acts) A B. - leadsTo Acts (reachable (Init,Acts) Int A) - (reachable (Init,Acts) Int B)" - - + LeadsTo :: "['a program, 'a set, 'a set] => bool" + "LeadsTo prg A B == + leadsTo (Acts prg) + (reachable prg Int A) + (reachable prg Int B)" end diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Token.ML --- a/src/HOL/UNITY/Token.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Token.ML Wed Aug 05 10:57:25 1998 +0200 @@ -23,7 +23,7 @@ qed "not_E_eq"; (*This proof is in the "massaging" style and is much faster! *) -Goalw [stable_def] "stable Acts (Compl(E i) Un (HasTok i))"; +Goalw [stable_def] "stable acts (Compl(E i) Un (HasTok i))"; by (rtac constrains_weaken 1); by (rtac ([[TR2, TR4] MRS constrains_Un, TR5] MRS constrains_Un) 1); by (auto_tac (claset(), simpset() addsimps [not_E_eq])); @@ -77,7 +77,7 @@ (*From "A Logic for Concurrent Programming", but not used in Chapter 4. Note the use of case_tac. Reasoning about leadsTo takes practice!*) Goal "[| i \ -\ leadsTo Acts (HasTok i) ({s. (token s, i) : nodeOrder j} Un HasTok j)"; +\ leadsTo acts (HasTok i) ({s. (token s, i) : nodeOrder j} Un HasTok j)"; by (case_tac "i=j" 1); by (blast_tac (claset() addIs [subset_imp_leadsTo]) 1); by (rtac (TR7 RS leadsTo_weaken_R) 1); @@ -88,7 +88,7 @@ (*Chapter 4 variant, the one actually used below.*) Goal "[| i leadsTo Acts (HasTok i) {s. (token s, i) : nodeOrder j}"; +\ ==> leadsTo acts (HasTok i) {s. (token s, i) : nodeOrder j}"; by (rtac (TR7 RS leadsTo_weaken_R) 1); by (Clarify_tac 1); by (asm_full_simp_tac (simpset() addsimps [HasTok_def, nodeOrder_eq]) 1); @@ -101,7 +101,7 @@ (*Misra's TR9: the token reaches an arbitrary node*) -Goal "j leadsTo Acts {s. token s < N} (HasTok j)"; +Goal "j leadsTo acts {s. token s < N} (HasTok j)"; by (rtac leadsTo_weaken_R 1); by (res_inst_tac [("I", "Compl{j}"), ("f", "token"), ("B", "{}")] (wf_nodeOrder RS bounded_induct) 1); @@ -115,7 +115,7 @@ qed "leadsTo_j"; (*Misra's TR8: a hungry process eventually eats*) -Goal "j leadsTo Acts ({s. token s < N} Int H j) (E j)"; +Goal "j leadsTo acts ({s. token s < N} Int H j) (E j)"; by (rtac (leadsTo_cancel1 RS leadsTo_Un_duplicate) 1); by (rtac TR6 2); by (rtac leadsTo_weaken_R 1); diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Token.thy --- a/src/HOL/UNITY/Token.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Token.thy Wed Aug 05 10:57:25 1998 +0200 @@ -44,18 +44,18 @@ rules N_positive "0 s : reachable(Init,Acts)" - - Acts "[| act: Acts; s : reachable(Init,Acts); (s,s'): act |] - ==> s' : reachable(Init,Acts)" - -This amounts to an equivalence proof for the definition actually used, -****) - - -(** reachable: Deriving the Introduction rules **) - -Goal "s: Init ==> s : reachable(Init,Acts)"; -by (simp_tac (simpset() addsimps [reachable_def]) 1); -by (blast_tac (claset() addIs traces.intrs) 1); -qed "reachable_Init"; - +Goal "reachable prg = {s. EX evs. (s,evs): traces (Init prg) (Acts prg)}"; +by Safe_tac; +by (etac traces.induct 2); +be reachable.induct 1; +by (ALLGOALS (blast_tac (claset() addIs (reachable.intrs @ traces.intrs)))); +qed "reachable_equiv_traces"; -Goal "[| act: Acts; s : reachable(Init,Acts) |] \ -\ ==> (s,s'): act --> s' : reachable(Init,Acts)"; -by (asm_full_simp_tac (simpset() addsimps [reachable_def]) 1); -by (etac exE 1); -by (etac traces.induct 1); -by (ALLGOALS Asm_simp_tac); -by (ALLGOALS (blast_tac (claset() addIs traces.intrs))); -qed_spec_mp "reachable_Acts"; - - -val major::prems = -Goalw [reachable_def] - "[| za : reachable(Init,Acts); \ -\ !!s. s : Init ==> P s; \ -\ !!act s s'. \ -\ [| act : Acts; s : reachable(Init,Acts); P s; (s, s') : act |] \ -\ ==> P s' |] \ -\ ==> P za"; -by (cut_facts_tac [major] 1); -by Auto_tac; -by (etac traces.induct 1); -by (ALLGOALS (blast_tac (claset() addIs prems))); -qed "reachable_induct"; - -structure reachable = - struct - val Init = reachable_Init - val Acts = reachable_Acts - val intrs = [reachable_Init, reachable_Acts] - val induct = reachable_induct - end; - - - -Goal "stable Acts (reachable(Init,Acts))"; +Goal "stable (Acts prg) (reachable prg)"; by (blast_tac (claset() addIs ([stableI, constrainsI] @ reachable.intrs)) 1); qed "stable_reachable"; (*The set of all reachable states is an invariant...*) -Goal "invariant (Init,Acts) (reachable(Init,Acts))"; +Goal "invariant prg (reachable prg)"; by (simp_tac (simpset() addsimps [invariant_def]) 1); by (blast_tac (claset() addIs (stable_reachable::reachable.intrs)) 1); qed "invariant_reachable"; (*...in fact the strongest invariant!*) -Goal "invariant (Init,Acts) A ==> reachable(Init,Acts) <= A"; +Goal "invariant prg A ==> reachable prg <= A"; by (full_simp_tac (simpset() addsimps [stable_def, constrains_def, invariant_def]) 1); by (rtac subsetI 1); @@ -86,15 +37,15 @@ (*If "A" includes the initial states and is stable then "A" is invariant. Result is trivial from the definition, but it is handy.*) -Goal "[| Init<=A; stable Acts A |] ==> invariant (Init,Acts) A"; +Goal "[| (Init prg)<=A; stable (Acts prg) A |] ==> invariant prg A"; by (asm_simp_tac (simpset() addsimps [invariant_def]) 1); qed "invariantI"; (** Conjoining invariants **) -Goal "[| invariant (Init,Acts) A; invariant (Init,Acts) B |] \ -\ ==> invariant (Init,Acts) (A Int B)"; +Goal "[| invariant prg A; invariant prg B |] \ +\ ==> invariant prg (A Int B)"; by (asm_full_simp_tac (simpset() addsimps [invariant_def, stable_Int]) 1); by Auto_tac; qed "invariant_Int"; diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/Traces.thy --- a/src/HOL/UNITY/Traces.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/Traces.thy Wed Aug 05 10:57:25 1998 +0200 @@ -24,11 +24,22 @@ ==> (s', s#evs) : traces Init Acts" +record 'a program = + Init :: 'a set + Acts :: "('a * 'a)set set" + + +consts reachable :: "'a program => 'a set" + +inductive "reachable prg" + intrs + Init "s: Init prg ==> s : reachable prg" + + Acts "[| act: Acts prg; s : reachable prg; (s,s'): act |] + ==> s' : reachable prg" + constdefs - reachable :: "'a set * ('a * 'a)set set => 'a set" - "reachable == %(Init,Acts). {s. EX evs. (s,evs): traces Init Acts}" - - invariant :: "['a set * ('a * 'a)set set, 'a set] => bool" - "invariant == %(Init,Acts) A. Init <= A & stable Acts A" + invariant :: "['a program, 'a set] => bool" + "invariant prg A == (Init prg) <= A & stable (Acts prg) A" end diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/UNITY.ML --- a/src/HOL/UNITY/UNITY.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/UNITY.ML Wed Aug 05 10:57:25 1998 +0200 @@ -15,97 +15,97 @@ (*** constrains ***) val prems = goalw thy [constrains_def] - "(!!act s s'. [| act: Acts; (s,s') : act; s: A |] ==> s': A') \ -\ ==> constrains Acts A A'"; + "(!!act s s'. [| act: acts; (s,s') : act; s: A |] ==> s': A') \ +\ ==> constrains acts A A'"; by (blast_tac (claset() addIs prems) 1); qed "constrainsI"; Goalw [constrains_def] - "[| constrains Acts A A'; act: Acts; (s,s'): act; s: A |] \ + "[| constrains acts A A'; act: acts; (s,s'): act; s: A |] \ \ ==> s': A'"; by (Blast_tac 1); qed "constrainsD"; -Goalw [constrains_def] "constrains Acts {} B"; +Goalw [constrains_def] "constrains acts {} B"; by (Blast_tac 1); qed "constrains_empty"; -Goalw [constrains_def] "constrains Acts A UNIV"; +Goalw [constrains_def] "constrains acts A UNIV"; by (Blast_tac 1); qed "constrains_UNIV"; AddIffs [constrains_empty, constrains_UNIV]; Goalw [constrains_def] - "[| constrains Acts A A'; A'<=B' |] ==> constrains Acts A B'"; + "[| constrains acts A A'; A'<=B' |] ==> constrains acts A B'"; by (Blast_tac 1); qed "constrains_weaken_R"; Goalw [constrains_def] - "[| constrains Acts A A'; B<=A |] ==> constrains Acts B A'"; + "[| constrains acts A A'; B<=A |] ==> constrains acts B A'"; by (Blast_tac 1); qed "constrains_weaken_L"; Goalw [constrains_def] - "[| constrains Acts A A'; B<=A; A'<=B' |] ==> constrains Acts B B'"; + "[| constrains acts A A'; B<=A; A'<=B' |] ==> constrains acts B B'"; by (Blast_tac 1); qed "constrains_weaken"; (*Set difference: UNUSED*) Goalw [constrains_def] - "[| constrains Acts (A-B) C; constrains Acts B C |] \ -\ ==> constrains Acts A C"; + "[| constrains acts (A-B) C; constrains acts B C |] \ +\ ==> constrains acts A C"; by (Blast_tac 1); qed "constrains_Diff"; (** Union **) Goalw [constrains_def] - "[| constrains Acts A A'; constrains Acts B B' |] \ -\ ==> constrains Acts (A Un B) (A' Un B')"; + "[| constrains acts A A'; constrains acts B B' |] \ +\ ==> constrains acts (A Un B) (A' Un B')"; by (Blast_tac 1); qed "constrains_Un"; Goalw [constrains_def] - "ALL i:I. constrains Acts (A i) (A' i) \ -\ ==> constrains Acts (UN i:I. A i) (UN i:I. A' i)"; + "ALL i:I. constrains acts (A i) (A' i) \ +\ ==> constrains acts (UN i:I. A i) (UN i:I. A' i)"; by (Blast_tac 1); qed "ball_constrains_UN"; Goalw [constrains_def] - "[| ALL i. constrains Acts (A i) (A' i) |] \ -\ ==> constrains Acts (UN i. A i) (UN i. A' i)"; + "[| ALL i. constrains acts (A i) (A' i) |] \ +\ ==> constrains acts (UN i. A i) (UN i. A' i)"; by (Blast_tac 1); qed "all_constrains_UN"; (** Intersection **) Goalw [constrains_def] - "[| constrains Acts A A'; constrains Acts B B' |] \ -\ ==> constrains Acts (A Int B) (A' Int B')"; + "[| constrains acts A A'; constrains acts B B' |] \ +\ ==> constrains acts (A Int B) (A' Int B')"; by (Blast_tac 1); qed "constrains_Int"; Goalw [constrains_def] - "ALL i:I. constrains Acts (A i) (A' i) \ -\ ==> constrains Acts (INT i:I. A i) (INT i:I. A' i)"; + "ALL i:I. constrains acts (A i) (A' i) \ +\ ==> constrains acts (INT i:I. A i) (INT i:I. A' i)"; by (Blast_tac 1); qed "ball_constrains_INT"; Goalw [constrains_def] - "[| ALL i. constrains Acts (A i) (A' i) |] \ -\ ==> constrains Acts (INT i. A i) (INT i. A' i)"; + "[| ALL i. constrains acts (A i) (A' i) |] \ +\ ==> constrains acts (INT i. A i) (INT i. A' i)"; by (Blast_tac 1); qed "all_constrains_INT"; Goalw [stable_def, constrains_def] - "[| stable Acts C; constrains Acts A (C Un A') |] \ -\ ==> constrains Acts (C Un A) (C Un A')"; + "[| stable acts C; constrains acts A (C Un A') |] \ +\ ==> constrains acts (C Un A) (C Un A')"; by (Blast_tac 1); qed "stable_constrains_Un"; Goalw [stable_def, constrains_def] - "[| stable Acts C; constrains Acts (C Int A) A' |] \ -\ ==> constrains Acts (C Int A) (C Int A')"; + "[| stable acts C; constrains acts (C Int A) A' |] \ +\ ==> constrains acts (C Int A) (C Int A')"; by (Blast_tac 1); qed "stable_constrains_Int"; @@ -113,36 +113,36 @@ (*** stable ***) Goalw [stable_def] - "constrains Acts A A ==> stable Acts A"; + "constrains acts A A ==> stable acts A"; by (assume_tac 1); qed "stableI"; Goalw [stable_def] - "stable Acts A ==> constrains Acts A A"; + "stable acts A ==> constrains acts A A"; by (assume_tac 1); qed "stableD"; Goalw [stable_def] - "[| stable Acts A; stable Acts A' |] \ -\ ==> stable Acts (A Un A')"; + "[| stable acts A; stable acts A' |] \ +\ ==> stable acts (A Un A')"; by (blast_tac (claset() addIs [constrains_Un]) 1); qed "stable_Un"; Goalw [stable_def] - "[| stable Acts A; stable Acts A' |] \ -\ ==> stable Acts (A Int A')"; + "[| stable acts A; stable acts A' |] \ +\ ==> stable acts (A Int A')"; by (blast_tac (claset() addIs [constrains_Int]) 1); qed "stable_Int"; Goalw [constrains_def] - "[| constrains Acts A A'; id: Acts |] ==> A<=A'"; + "[| constrains acts A A'; id: acts |] ==> A<=A'"; by (Blast_tac 1); qed "constrains_imp_subset"; Goalw [constrains_def] - "[| id: Acts; constrains Acts A B; constrains Acts B C |] \ -\ ==> constrains Acts A C"; + "[| id: acts; constrains acts A B; constrains acts B C |] \ +\ ==> constrains acts A C"; by (Blast_tac 1); qed "constrains_trans"; @@ -151,23 +151,23 @@ Should the premise be !!m instead of ALL m ? Would make it harder to use in forward proof.*) Goalw [constrains_def] - "[| ALL m. constrains Acts {s. s x = m} (B m) |] \ -\ ==> constrains Acts {s. P(s x)} (UN m. {s. P(m)} Int B m)"; + "[| ALL m. constrains acts {s. s x = m} (B m) |] \ +\ ==> constrains acts {s. P(s x)} (UN m. {s. P(m)} Int B m)"; by (Blast_tac 1); qed "elimination"; (*As above, but for the trivial case of a one-variable state, in which the state is identified with its one variable.*) Goalw [constrains_def] - "[| ALL m. constrains Acts {m} (B m) |] \ -\ ==> constrains Acts {s. P s} (UN m. {s. P(m)} Int B m)"; + "[| ALL m. constrains acts {m} (B m) |] \ +\ ==> constrains acts {s. P s} (UN m. {s. P(m)} Int B m)"; by (Blast_tac 1); qed "elimination_sing"; Goalw [constrains_def] - "[| constrains Acts A (A' Un B); constrains Acts B B'; id: Acts |] \ -\ ==> constrains Acts A (A' Un B')"; + "[| constrains acts A (A' Un B); constrains acts B B'; id: acts |] \ +\ ==> constrains acts A (A' Un B')"; by (Blast_tac 1); qed "constrains_cancel"; @@ -176,11 +176,11 @@ (*** Theoretical Results from Section 6 ***) Goalw [constrains_def, strongest_rhs_def] - "constrains Acts A (strongest_rhs Acts A )"; + "constrains acts A (strongest_rhs acts A )"; by (Blast_tac 1); qed "constrains_strongest_rhs"; Goalw [constrains_def, strongest_rhs_def] - "constrains Acts A B ==> strongest_rhs Acts A <= B"; + "constrains acts A B ==> strongest_rhs acts A <= B"; by (Blast_tac 1); qed "strongest_rhs_is_strongest"; diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/UNITY.thy --- a/src/HOL/UNITY/UNITY.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/UNITY.thy Wed Aug 05 10:57:25 1998 +0200 @@ -13,15 +13,15 @@ constdefs constrains :: "[('a * 'a)set set, 'a set, 'a set] => bool" - "constrains Acts A B == ALL act:Acts. act^^A <= B" + "constrains acts A B == ALL act:acts. act^^A <= B" stable :: "('a * 'a)set set => 'a set => bool" - "stable Acts A == constrains Acts A A" + "stable acts A == constrains acts A A" strongest_rhs :: "[('a * 'a)set set, 'a set] => 'a set" - "strongest_rhs Acts A == Inter {B. constrains Acts A B}" + "strongest_rhs acts A == Inter {B. constrains acts A B}" unless :: "[('a * 'a)set set, 'a set, 'a set] => bool" - "unless Acts A B == constrains Acts (A-B) (A Un B)" + "unless acts A B == constrains acts (A-B) (A Un B)" end diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/WFair.ML --- a/src/HOL/UNITY/WFair.ML Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/WFair.ML Wed Aug 05 10:57:25 1998 +0200 @@ -12,20 +12,20 @@ (*** transient ***) Goalw [stable_def, constrains_def, transient_def] - "[| stable Acts A; transient Acts A |] ==> A = {}"; + "[| stable acts A; transient acts A |] ==> A = {}"; by (Blast_tac 1); qed "stable_transient_empty"; Goalw [transient_def] - "[| transient Acts A; B<=A |] ==> transient Acts B"; + "[| transient acts A; B<=A |] ==> transient acts B"; by (Clarify_tac 1); by (rtac bexI 1 THEN assume_tac 2); by (Blast_tac 1); qed "transient_strengthen"; Goalw [transient_def] - "[| act:Acts; A <= Domain act; act^^A <= Compl A |] \ -\ ==> transient Acts A"; + "[| act:acts; A <= Domain act; act^^A <= Compl A |] \ +\ ==> transient acts A"; by (Blast_tac 1); qed "transient_mem"; @@ -33,34 +33,34 @@ (*** ensures ***) Goalw [ensures_def] - "[| constrains Acts (A-B) (A Un B); transient Acts (A-B) |] \ -\ ==> ensures Acts A B"; + "[| constrains acts (A-B) (A Un B); transient acts (A-B) |] \ +\ ==> ensures acts A B"; by (Blast_tac 1); qed "ensuresI"; Goalw [ensures_def] - "ensures Acts A B \ -\ ==> constrains Acts (A-B) (A Un B) & transient Acts (A-B)"; + "ensures acts A B \ +\ ==> constrains acts (A-B) (A Un B) & transient acts (A-B)"; by (Blast_tac 1); qed "ensuresD"; (*The L-version (precondition strengthening) doesn't hold for ENSURES*) Goalw [ensures_def] - "[| ensures Acts A A'; A'<=B' |] ==> ensures Acts A B'"; + "[| ensures acts A A'; A'<=B' |] ==> ensures acts A B'"; by (blast_tac (claset() addIs [constrains_weaken, transient_strengthen]) 1); qed "ensures_weaken_R"; Goalw [ensures_def, constrains_def, transient_def] - "Acts ~= {} ==> ensures Acts A UNIV"; + "acts ~= {} ==> ensures acts A UNIV"; by (Asm_simp_tac 1); (*omitting this causes PROOF FAILED, even with Safe_tac*) by (Blast_tac 1); qed "ensures_UNIV"; Goalw [ensures_def] - "[| stable Acts C; \ -\ constrains Acts (C Int (A - A')) (A Un A'); \ -\ transient Acts (C Int (A-A')) |] \ -\ ==> ensures Acts (C Int A) (C Int A')"; + "[| stable acts C; \ +\ constrains acts (C Int (A - A')) (A Un A'); \ +\ transient acts (C Int (A-A')) |] \ +\ ==> ensures acts (C Int A) (C Int A')"; by (asm_simp_tac (simpset() addsimps [Int_Un_distrib RS sym, Diff_Int_distrib RS sym, stable_constrains_Int]) 1); @@ -73,34 +73,34 @@ bind_thm ("leadsTo_Basis", leadsto.Basis); bind_thm ("leadsTo_Trans", leadsto.Trans); -Goal "act: Acts ==> leadsTo Acts A UNIV"; +Goal "act: acts ==> leadsTo acts A UNIV"; by (blast_tac (claset() addIs [ensures_UNIV RS leadsTo_Basis]) 1); qed "leadsTo_UNIV"; Addsimps [leadsTo_UNIV]; (*Useful with cancellation, disjunction*) -Goal "leadsTo Acts A (A' Un A') ==> leadsTo Acts A A'"; +Goal "leadsTo acts A (A' Un A') ==> leadsTo acts A A'"; by (asm_full_simp_tac (simpset() addsimps Un_ac) 1); qed "leadsTo_Un_duplicate"; -Goal "leadsTo Acts A (A' Un C Un C) ==> leadsTo Acts A (A' Un C)"; +Goal "leadsTo acts A (A' Un C Un C) ==> leadsTo acts A (A' Un C)"; by (asm_full_simp_tac (simpset() addsimps Un_ac) 1); qed "leadsTo_Un_duplicate2"; (*The Union introduction rule as we should have liked to state it*) val prems = goal thy - "(!!A. A : S ==> leadsTo Acts A B) ==> leadsTo Acts (Union S) B"; + "(!!A. A : S ==> leadsTo acts A B) ==> leadsTo acts (Union S) B"; by (blast_tac (claset() addIs (leadsto.Union::prems)) 1); qed "leadsTo_Union"; val prems = goal thy - "(!!i. i : I ==> leadsTo Acts (A i) B) ==> leadsTo Acts (UN i:I. A i) B"; + "(!!i. i : I ==> leadsTo acts (A i) B) ==> leadsTo acts (UN i:I. A i) B"; by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1); by (blast_tac (claset() addIs (leadsto.Union::prems)) 1); qed "leadsTo_UN"; (*Binary union introduction rule*) -Goal "[| leadsTo Acts A C; leadsTo Acts B C |] ==> leadsTo Acts (A Un B) C"; +Goal "[| leadsTo acts A C; leadsTo acts B C |] ==> leadsTo acts (A Un B) C"; by (stac Un_eq_Union 1); by (blast_tac (claset() addIs [leadsTo_Union]) 1); qed "leadsTo_Un"; @@ -108,18 +108,18 @@ (*The INDUCTION rule as we should have liked to state it*) val major::prems = goal thy - "[| leadsTo Acts za zb; \ -\ !!A B. ensures Acts A B ==> P A B; \ -\ !!A B C. [| leadsTo Acts A B; P A B; leadsTo Acts B C; P B C |] \ + "[| leadsTo acts za zb; \ +\ !!A B. ensures acts A B ==> P A B; \ +\ !!A B C. [| leadsTo acts A B; P A B; leadsTo acts B C; P B C |] \ \ ==> P A C; \ -\ !!B S. ALL A:S. leadsTo Acts A B & P A B ==> P (Union S) B \ +\ !!B S. ALL A:S. leadsTo acts A B & P A B ==> P (Union S) B \ \ |] ==> P za zb"; by (rtac (major RS leadsto.induct) 1); by (REPEAT (blast_tac (claset() addIs prems) 1)); qed "leadsTo_induct"; -Goal "[| A<=B; id: Acts |] ==> leadsTo Acts A B"; +Goal "[| A<=B; id: acts |] ==> leadsTo acts A B"; by (rtac leadsTo_Basis 1); by (rewrite_goals_tac [ensures_def, constrains_def, transient_def]); by (Blast_tac 1); @@ -130,8 +130,8 @@ (*There's a direct proof by leadsTo_Trans and subset_imp_leadsTo, but it - needs the extra premise id:Acts*) -Goal "leadsTo Acts A A' ==> A'<=B' --> leadsTo Acts A B'"; + needs the extra premise id:acts*) +Goal "leadsTo acts A A' ==> A'<=B' --> leadsTo acts A B'"; by (etac leadsTo_induct 1); by (Clarify_tac 3); by (blast_tac (claset() addIs [leadsTo_Union]) 3); @@ -140,31 +140,31 @@ qed_spec_mp "leadsTo_weaken_R"; -Goal "[| leadsTo Acts A A'; B<=A; id: Acts |] ==> \ -\ leadsTo Acts B A'"; +Goal "[| leadsTo acts A A'; B<=A; id: acts |] ==> \ +\ leadsTo acts B A'"; by (blast_tac (claset() addIs [leadsTo_Basis, leadsTo_Trans, subset_imp_leadsTo]) 1); qed_spec_mp "leadsTo_weaken_L"; (*Distributes over binary unions*) -Goal "id: Acts ==> \ -\ leadsTo Acts (A Un B) C = (leadsTo Acts A C & leadsTo Acts B C)"; +Goal "id: acts ==> \ +\ leadsTo acts (A Un B) C = (leadsTo acts A C & leadsTo acts B C)"; by (blast_tac (claset() addIs [leadsTo_Un, leadsTo_weaken_L]) 1); qed "leadsTo_Un_distrib"; -Goal "id: Acts ==> \ -\ leadsTo Acts (UN i:I. A i) B = (ALL i : I. leadsTo Acts (A i) B)"; +Goal "id: acts ==> \ +\ leadsTo acts (UN i:I. A i) B = (ALL i : I. leadsTo acts (A i) B)"; by (blast_tac (claset() addIs [leadsTo_UN, leadsTo_weaken_L]) 1); qed "leadsTo_UN_distrib"; -Goal "id: Acts ==> \ -\ leadsTo Acts (Union S) B = (ALL A : S. leadsTo Acts A B)"; +Goal "id: acts ==> \ +\ leadsTo acts (Union S) B = (ALL A : S. leadsTo acts A B)"; by (blast_tac (claset() addIs [leadsTo_Union, leadsTo_weaken_L]) 1); qed "leadsTo_Union_distrib"; -Goal "[| leadsTo Acts A A'; id: Acts; B<=A; A'<=B' |] \ -\ ==> leadsTo Acts B B'"; +Goal "[| leadsTo acts A A'; id: acts; B<=A; A'<=B' |] \ +\ ==> leadsTo acts B B'"; (*PROOF FAILED: why?*) by (blast_tac (claset() addIs [leadsTo_Trans, leadsTo_weaken_R, leadsTo_weaken_L]) 1); @@ -172,8 +172,8 @@ (*Set difference: maybe combine with leadsTo_weaken_L??*) -Goal "[| leadsTo Acts (A-B) C; leadsTo Acts B C; id: Acts |] \ -\ ==> leadsTo Acts A C"; +Goal "[| leadsTo acts (A-B) C; leadsTo acts B C; id: acts |] \ +\ ==> leadsTo acts A C"; by (blast_tac (claset() addIs [leadsTo_Un, leadsTo_weaken]) 1); qed "leadsTo_Diff"; @@ -182,8 +182,8 @@ see ball_constrains_UN in UNITY.ML***) val prems = goal thy - "(!! i. i:I ==> leadsTo Acts (A i) (A' i)) \ -\ ==> leadsTo Acts (UN i:I. A i) (UN i:I. A' i)"; + "(!! i. i:I ==> leadsTo acts (A i) (A' i)) \ +\ ==> leadsTo acts (UN i:I. A i) (UN i:I. A' i)"; by (simp_tac (simpset() addsimps [Union_image_eq RS sym]) 1); by (blast_tac (claset() addIs [leadsTo_Union, leadsTo_weaken_R] addIs prems) 1); @@ -192,22 +192,22 @@ (*Version with no index set*) val prems = goal thy - "(!! i. leadsTo Acts (A i) (A' i)) \ -\ ==> leadsTo Acts (UN i. A i) (UN i. A' i)"; + "(!! i. leadsTo acts (A i) (A' i)) \ +\ ==> leadsTo acts (UN i. A i) (UN i. A' i)"; by (blast_tac (claset() addIs [leadsTo_UN_UN] addIs prems) 1); qed "leadsTo_UN_UN_noindex"; (*Version with no index set*) -Goal "ALL i. leadsTo Acts (A i) (A' i) \ -\ ==> leadsTo Acts (UN i. A i) (UN i. A' i)"; +Goal "ALL i. leadsTo acts (A i) (A' i) \ +\ ==> leadsTo acts (UN i. A i) (UN i. A' i)"; by (blast_tac (claset() addIs [leadsTo_UN_UN]) 1); qed "all_leadsTo_UN_UN"; (*Binary union version*) -Goal "[| leadsTo Acts A A'; leadsTo Acts B B' |] \ -\ ==> leadsTo Acts (A Un B) (A' Un B')"; +Goal "[| leadsTo acts A A'; leadsTo acts B B' |] \ +\ ==> leadsTo acts (A Un B) (A' Un B')"; by (blast_tac (claset() addIs [leadsTo_Un, leadsTo_weaken_R]) 1); qed "leadsTo_Un_Un"; @@ -215,27 +215,27 @@ (** The cancellation law **) -Goal "[| leadsTo Acts A (A' Un B); leadsTo Acts B B'; id: Acts |] \ -\ ==> leadsTo Acts A (A' Un B')"; +Goal "[| leadsTo acts A (A' Un B); leadsTo acts B B'; id: acts |] \ +\ ==> leadsTo acts A (A' Un B')"; by (blast_tac (claset() addIs [leadsTo_Un_Un, subset_imp_leadsTo, leadsTo_Trans]) 1); qed "leadsTo_cancel2"; -Goal "[| leadsTo Acts A (A' Un B); leadsTo Acts (B-A') B'; id: Acts |] \ -\ ==> leadsTo Acts A (A' Un B')"; +Goal "[| leadsTo acts A (A' Un B); leadsTo acts (B-A') B'; id: acts |] \ +\ ==> leadsTo acts A (A' Un B')"; by (rtac leadsTo_cancel2 1); by (assume_tac 2); by (ALLGOALS Asm_simp_tac); qed "leadsTo_cancel_Diff2"; -Goal "[| leadsTo Acts A (B Un A'); leadsTo Acts B B'; id: Acts |] \ -\ ==> leadsTo Acts A (B' Un A')"; +Goal "[| leadsTo acts A (B Un A'); leadsTo acts B B'; id: acts |] \ +\ ==> leadsTo acts A (B' Un A')"; by (asm_full_simp_tac (simpset() addsimps [Un_commute]) 1); by (blast_tac (claset() addSIs [leadsTo_cancel2]) 1); qed "leadsTo_cancel1"; -Goal "[| leadsTo Acts A (B Un A'); leadsTo Acts (B-A') B'; id: Acts |] \ -\ ==> leadsTo Acts A (B' Un A')"; +Goal "[| leadsTo acts A (B Un A'); leadsTo acts (B-A') B'; id: acts |] \ +\ ==> leadsTo acts A (B' Un A')"; by (rtac leadsTo_cancel1 1); by (assume_tac 2); by (ALLGOALS Asm_simp_tac); @@ -245,24 +245,24 @@ (** The impossibility law **) -Goal "leadsTo Acts A B ==> B={} --> A={}"; +Goal "leadsTo acts A B ==> B={} --> A={}"; by (etac leadsTo_induct 1); by (ALLGOALS Asm_simp_tac); by (rewrite_goals_tac [ensures_def, constrains_def, transient_def]); by (Blast_tac 1); val lemma = result() RS mp; -Goal "leadsTo Acts A {} ==> A={}"; +Goal "leadsTo acts A {} ==> A={}"; by (blast_tac (claset() addSIs [lemma]) 1); qed "leadsTo_empty"; (** PSP: Progress-Safety-Progress **) -(*Special case of PSP: Misra's "stable conjunction". Doesn't need id:Acts. *) +(*Special case of PSP: Misra's "stable conjunction". Doesn't need id:acts. *) Goalw [stable_def] - "[| leadsTo Acts A A'; stable Acts B |] \ -\ ==> leadsTo Acts (A Int B) (A' Int B)"; + "[| leadsTo acts A A'; stable acts B |] \ +\ ==> leadsTo acts (A Int B) (A' Int B)"; by (etac leadsTo_induct 1); by (simp_tac (simpset() addsimps [Int_Union_Union]) 3); by (blast_tac (claset() addIs [leadsTo_Union]) 3); @@ -274,15 +274,15 @@ by (blast_tac (claset() addIs [transient_strengthen, constrains_Int]) 1); qed "PSP_stable"; -Goal "[| leadsTo Acts A A'; stable Acts B |] \ -\ ==> leadsTo Acts (B Int A) (B Int A')"; +Goal "[| leadsTo acts A A'; stable acts B |] \ +\ ==> leadsTo acts (B Int A) (B Int A')"; by (asm_simp_tac (simpset() addsimps (PSP_stable::Int_ac)) 1); qed "PSP_stable2"; Goalw [ensures_def] - "[| ensures Acts A A'; constrains Acts B B' |] \ -\ ==> ensures Acts (A Int B) ((A' Int B) Un (B' - B))"; + "[| ensures acts A A'; constrains acts B B' |] \ +\ ==> ensures acts (A Int B) ((A' Int B) Un (B' - B))"; by Safe_tac; by (blast_tac (claset() addIs [constrainsI] addDs [constrainsD]) 1); by (etac transient_strengthen 1); @@ -290,8 +290,8 @@ qed "PSP_ensures"; -Goal "[| leadsTo Acts A A'; constrains Acts B B'; id: Acts |] \ -\ ==> leadsTo Acts (A Int B) ((A' Int B) Un (B' - B))"; +Goal "[| leadsTo acts A A'; constrains acts B B'; id: acts |] \ +\ ==> leadsTo acts (A Int B) ((A' Int B) Un (B' - B))"; by (etac leadsTo_induct 1); by (simp_tac (simpset() addsimps [Int_Union_Union]) 3); by (blast_tac (claset() addIs [leadsTo_Union]) 3); @@ -304,15 +304,15 @@ by (blast_tac (claset() addIs [leadsTo_Basis, PSP_ensures]) 1); qed "PSP"; -Goal "[| leadsTo Acts A A'; constrains Acts B B'; id: Acts |] \ -\ ==> leadsTo Acts (B Int A) ((B Int A') Un (B' - B))"; +Goal "[| leadsTo acts A A'; constrains acts B B'; id: acts |] \ +\ ==> leadsTo acts (B Int A) ((B Int A') Un (B' - B))"; by (asm_simp_tac (simpset() addsimps (PSP::Int_ac)) 1); qed "PSP2"; Goalw [unless_def] - "[| leadsTo Acts A A'; unless Acts B B'; id: Acts |] \ -\ ==> leadsTo Acts (A Int B) ((A' Int B) Un B')"; + "[| leadsTo acts A A'; unless acts B B'; id: acts |] \ +\ ==> leadsTo acts (A Int B) ((A' Int B) Un B')"; by (dtac PSP 1); by (assume_tac 1); by (asm_full_simp_tac (simpset() addsimps [Un_Diff_Diff, Int_Diff_Un]) 2); @@ -326,12 +326,12 @@ (*** Proving the induction rules ***) Goal "[| wf r; \ -\ ALL m. leadsTo Acts (A Int f-``{m}) \ +\ ALL m. leadsTo acts (A Int f-``{m}) \ \ ((A Int f-``(r^-1 ^^ {m})) Un B); \ -\ id: Acts |] \ -\ ==> leadsTo Acts (A Int f-``{m}) B"; +\ id: acts |] \ +\ ==> leadsTo acts (A Int f-``{m}) B"; by (eres_inst_tac [("a","m")] wf_induct 1); -by (subgoal_tac "leadsTo Acts (A Int (f -`` (r^-1 ^^ {x}))) B" 1); +by (subgoal_tac "leadsTo acts (A Int (f -`` (r^-1 ^^ {x}))) B" 1); by (stac vimage_eq_UN 2); by (asm_simp_tac (HOL_ss addsimps (UN_simps RL [sym])) 2); by (blast_tac (claset() addIs [leadsTo_UN]) 2); @@ -341,10 +341,10 @@ (** Meta or object quantifier ????? **) Goal "[| wf r; \ -\ ALL m. leadsTo Acts (A Int f-``{m}) \ +\ ALL m. leadsTo acts (A Int f-``{m}) \ \ ((A Int f-``(r^-1 ^^ {m})) Un B); \ -\ id: Acts |] \ -\ ==> leadsTo Acts A B"; +\ id: acts |] \ +\ ==> leadsTo acts A B"; by (res_inst_tac [("t", "A")] subst 1); by (rtac leadsTo_UN 2); by (etac lemma 2); @@ -354,10 +354,10 @@ Goal "[| wf r; \ -\ ALL m:I. leadsTo Acts (A Int f-``{m}) \ +\ ALL m:I. leadsTo acts (A Int f-``{m}) \ \ ((A Int f-``(r^-1 ^^ {m})) Un B); \ -\ id: Acts |] \ -\ ==> leadsTo Acts A ((A - (f-``I)) Un B)"; +\ id: acts |] \ +\ ==> leadsTo acts A ((A - (f-``I)) Un B)"; by (etac leadsTo_wf_induct 1); by Safe_tac; by (case_tac "m:I" 1); @@ -366,30 +366,30 @@ qed "bounded_induct"; -(*Alternative proof is via the lemma leadsTo Acts (A Int f-``(lessThan m)) B*) -Goal "[| ALL m. leadsTo Acts (A Int f-``{m}) \ +(*Alternative proof is via the lemma leadsTo acts (A Int f-``(lessThan m)) B*) +Goal "[| ALL m. leadsTo acts (A Int f-``{m}) \ \ ((A Int f-``(lessThan m)) Un B); \ -\ id: Acts |] \ -\ ==> leadsTo Acts A B"; +\ id: acts |] \ +\ ==> leadsTo acts A B"; by (rtac (wf_less_than RS leadsTo_wf_induct) 1); by (assume_tac 2); by (Asm_simp_tac 1); qed "lessThan_induct"; -Goal "[| ALL m:(greaterThan l). leadsTo Acts (A Int f-``{m}) \ +Goal "[| ALL m:(greaterThan l). leadsTo acts (A Int f-``{m}) \ \ ((A Int f-``(lessThan m)) Un B); \ -\ id: Acts |] \ -\ ==> leadsTo Acts A ((A Int (f-``(atMost l))) Un B)"; +\ id: acts |] \ +\ ==> leadsTo acts A ((A Int (f-``(atMost l))) Un B)"; by (simp_tac (HOL_ss addsimps [Diff_eq RS sym, vimage_Compl, Compl_greaterThan RS sym]) 1); by (rtac (wf_less_than RS bounded_induct) 1); by (assume_tac 2); by (Asm_simp_tac 1); qed "lessThan_bounded_induct"; -Goal "[| ALL m:(lessThan l). leadsTo Acts (A Int f-``{m}) \ +Goal "[| ALL m:(lessThan l). leadsTo acts (A Int f-``{m}) \ \ ((A Int f-``(greaterThan m)) Un B); \ -\ id: Acts |] \ -\ ==> leadsTo Acts A ((A Int (f-``(atLeast l))) Un B)"; +\ id: acts |] \ +\ ==> leadsTo acts A ((A Int (f-``(atLeast l))) Un B)"; by (res_inst_tac [("f","f"),("f1", "%k. l - k")] (wf_less_than RS wf_inv_image RS leadsTo_wf_induct) 1); by (assume_tac 2); @@ -405,22 +405,22 @@ (*** wlt ****) (*Misra's property W3*) -Goalw [wlt_def] "leadsTo Acts (wlt Acts B) B"; +Goalw [wlt_def] "leadsTo acts (wlt acts B) B"; by (blast_tac (claset() addSIs [leadsTo_Union]) 1); qed "wlt_leadsTo"; -Goalw [wlt_def] "leadsTo Acts A B ==> A <= wlt Acts B"; +Goalw [wlt_def] "leadsTo acts A B ==> A <= wlt acts B"; by (blast_tac (claset() addSIs [leadsTo_Union]) 1); qed "leadsTo_subset"; (*Misra's property W2*) -Goal "id: Acts ==> leadsTo Acts A B = (A <= wlt Acts B)"; +Goal "id: acts ==> leadsTo acts A B = (A <= wlt acts B)"; by (blast_tac (claset() addSIs [leadsTo_subset, wlt_leadsTo RS leadsTo_weaken_L]) 1); qed "leadsTo_eq_subset_wlt"; (*Misra's property W4*) -Goal "id: Acts ==> B <= wlt Acts B"; +Goal "id: acts ==> B <= wlt acts B"; by (asm_simp_tac (simpset() addsimps [leadsTo_eq_subset_wlt RS sym, subset_imp_leadsTo]) 1); qed "wlt_increasing"; @@ -429,17 +429,17 @@ (*Used in the Trans case below*) Goalw [constrains_def] "[| B <= A2; \ -\ constrains Acts (A1 - B) (A1 Un B); \ -\ constrains Acts (A2 - C) (A2 Un C) |] \ -\ ==> constrains Acts (A1 Un A2 - C) (A1 Un A2 Un C)"; +\ constrains acts (A1 - B) (A1 Un B); \ +\ constrains acts (A2 - C) (A2 Un C) |] \ +\ ==> constrains acts (A1 Un A2 - C) (A1 Un A2 Un C)"; by (Clarify_tac 1); by (blast_tac (claset() addSDs [bspec]) 1); val lemma1 = result(); (*Lemma (1,2,3) of Misra's draft book, Chapter 4, "Progress"*) -Goal "[| leadsTo Acts A A'; id: Acts |] ==> \ -\ EX B. A<=B & leadsTo Acts B A' & constrains Acts (B-A') (B Un A')"; +Goal "[| leadsTo acts A A'; id: acts |] ==> \ +\ EX B. A<=B & leadsTo acts B A' & constrains acts (B-A') (B Un A')"; by (etac leadsTo_induct 1); (*Basis*) by (blast_tac (claset() addIs [leadsTo_Basis] @@ -458,10 +458,10 @@ (*Misra's property W5*) -Goal "id: Acts ==> constrains Acts (wlt Acts B - B) (wlt Acts B)"; +Goal "id: acts ==> constrains acts (wlt acts B - B) (wlt acts B)"; by (forward_tac [wlt_leadsTo RS leadsTo_123] 1); by (Clarify_tac 1); -by (subgoal_tac "Ba = wlt Acts B" 1); +by (subgoal_tac "Ba = wlt acts B" 1); by (blast_tac (claset() addDs [leadsTo_eq_subset_wlt]) 2); by (Clarify_tac 1); by (asm_full_simp_tac (simpset() addsimps [wlt_increasing, Un_absorb2]) 1); @@ -470,20 +470,20 @@ (*** Completion: Binary and General Finite versions ***) -Goal "[| leadsTo Acts A A'; stable Acts A'; \ -\ leadsTo Acts B B'; stable Acts B'; id: Acts |] \ -\ ==> leadsTo Acts (A Int B) (A' Int B')"; -by (subgoal_tac "stable Acts (wlt Acts B')" 1); +Goal "[| leadsTo acts A A'; stable acts A'; \ +\ leadsTo acts B B'; stable acts B'; id: acts |] \ +\ ==> leadsTo acts (A Int B) (A' Int B')"; +by (subgoal_tac "stable acts (wlt acts B')" 1); by (asm_full_simp_tac (simpset() addsimps [stable_def]) 2); by (EVERY [etac (constrains_Un RS constrains_weaken) 2, etac wlt_constrains_wlt 2, fast_tac (claset() addEs [wlt_increasing RSN (2,rev_subsetD)]) 3, Blast_tac 2]); -by (subgoal_tac "leadsTo Acts (A Int wlt Acts B') (A' Int wlt Acts B')" 1); +by (subgoal_tac "leadsTo acts (A Int wlt acts B') (A' Int wlt acts B')" 1); by (blast_tac (claset() addIs [PSP_stable]) 2); -by (subgoal_tac "leadsTo Acts (A' Int wlt Acts B') (A' Int B')" 1); +by (subgoal_tac "leadsTo acts (A' Int wlt acts B') (A' Int B')" 1); by (blast_tac (claset() addIs [wlt_leadsTo, PSP_stable2]) 2); -by (subgoal_tac "leadsTo Acts (A Int B) (A Int wlt Acts B')" 1); +by (subgoal_tac "leadsTo acts (A Int B) (A Int wlt acts B')" 1); by (blast_tac (claset() addIs [leadsTo_subset RS subsetD, subset_imp_leadsTo]) 2); (*Blast_tac gives PROOF FAILED*) @@ -491,10 +491,10 @@ qed "stable_completion"; -Goal "[| finite I; id: Acts |] \ -\ ==> (ALL i:I. leadsTo Acts (A i) (A' i)) --> \ -\ (ALL i:I. stable Acts (A' i)) --> \ -\ leadsTo Acts (INT i:I. A i) (INT i:I. A' i)"; +Goal "[| finite I; id: acts |] \ +\ ==> (ALL i:I. leadsTo acts (A i) (A' i)) --> \ +\ (ALL i:I. stable acts (A' i)) --> \ +\ leadsTo acts (INT i:I. A i) (INT i:I. A' i)"; by (etac finite_induct 1); by (Asm_simp_tac 1); by (asm_simp_tac @@ -503,21 +503,21 @@ qed_spec_mp "finite_stable_completion"; -Goal "[| W = wlt Acts (B' Un C); \ -\ leadsTo Acts A (A' Un C); constrains Acts A' (A' Un C); \ -\ leadsTo Acts B (B' Un C); constrains Acts B' (B' Un C); \ -\ id: Acts |] \ -\ ==> leadsTo Acts (A Int B) ((A' Int B') Un C)"; -by (subgoal_tac "constrains Acts (W-C) (W Un B' Un C)" 1); +Goal "[| W = wlt acts (B' Un C); \ +\ leadsTo acts A (A' Un C); constrains acts A' (A' Un C); \ +\ leadsTo acts B (B' Un C); constrains acts B' (B' Un C); \ +\ id: acts |] \ +\ ==> leadsTo acts (A Int B) ((A' Int B') Un C)"; +by (subgoal_tac "constrains acts (W-C) (W Un B' Un C)" 1); by (blast_tac (claset() addIs [[asm_rl, wlt_constrains_wlt] MRS constrains_Un RS constrains_weaken]) 2); -by (subgoal_tac "constrains Acts (W-C) W" 1); +by (subgoal_tac "constrains acts (W-C) W" 1); by (asm_full_simp_tac (simpset() addsimps [wlt_increasing, Un_assoc, Un_absorb2]) 2); -by (subgoal_tac "leadsTo Acts (A Int W - C) (A' Int W Un C)" 1); +by (subgoal_tac "leadsTo acts (A Int W - C) (A' Int W Un C)" 1); by (simp_tac (simpset() addsimps [Int_Diff]) 2); by (blast_tac (claset() addIs [wlt_leadsTo, PSP RS leadsTo_weaken_R]) 2); -by (subgoal_tac "leadsTo Acts (A' Int W Un C) (A' Int B' Un C)" 1); +by (subgoal_tac "leadsTo acts (A' Int W Un C) (A' Int B' Un C)" 1); by (blast_tac (claset() addIs [wlt_leadsTo, leadsTo_Un_Un, PSP2 RS leadsTo_weaken_R, subset_refl RS subset_imp_leadsTo, @@ -532,10 +532,10 @@ bind_thm("completion", refl RS result()); -Goal "[| finite I; id: Acts |] \ -\ ==> (ALL i:I. leadsTo Acts (A i) (A' i Un C)) --> \ -\ (ALL i:I. constrains Acts (A' i) (A' i Un C)) --> \ -\ leadsTo Acts (INT i:I. A i) ((INT i:I. A' i) Un C)"; +Goal "[| finite I; id: acts |] \ +\ ==> (ALL i:I. leadsTo acts (A i) (A' i Un C)) --> \ +\ (ALL i:I. constrains acts (A' i) (A' i Un C)) --> \ +\ leadsTo acts (INT i:I. A i) ((INT i:I. A' i) Un C)"; by (etac finite_induct 1); by (ALLGOALS Asm_simp_tac); by (Clarify_tac 1); diff -r 1b0f14d11142 -r 82a5ca6290aa src/HOL/UNITY/WFair.thy --- a/src/HOL/UNITY/WFair.thy Wed Aug 05 10:56:58 1998 +0200 +++ b/src/HOL/UNITY/WFair.thy Wed Aug 05 10:57:25 1998 +0200 @@ -15,37 +15,37 @@ (*This definition specifies weak fairness. The rest of the theory is generic to all forms of fairness.*) transient :: "[('a * 'a)set set, 'a set] => bool" - "transient Acts A == EX act:Acts. A <= Domain act & act^^A <= Compl A" + "transient acts A == EX act:acts. A <= Domain act & act^^A <= Compl A" ensures :: "[('a * 'a)set set, 'a set, 'a set] => bool" - "ensures Acts A B == constrains Acts (A-B) (A Un B) & transient Acts (A-B)" - (*(unless Acts A B) would be equivalent*) + "ensures acts A B == constrains acts (A-B) (A Un B) & transient acts (A-B)" + (*(unless acts A B) would be equivalent*) consts leadsTo :: "[('a * 'a)set set, 'a set, 'a set] => bool" leadsto :: "[('a * 'a)set set] => ('a set * 'a set) set" translations - "leadsTo Acts A B" == "(A,B) : leadsto Acts" + "leadsTo acts A B" == "(A,B) : leadsto acts" -inductive "leadsto Acts" +inductive "leadsto acts" intrs - Basis "ensures Acts A B ==> leadsTo Acts A B" + Basis "ensures acts A B ==> leadsTo acts A B" - Trans "[| leadsTo Acts A B; leadsTo Acts B C |] - ==> leadsTo Acts A C" + Trans "[| leadsTo acts A B; leadsTo acts B C |] + ==> leadsTo acts A C" (*Encoding using powerset of the desired axiom - (!!A. A : S ==> leadsTo Acts A B) ==> leadsTo Acts (Union S) B + (!!A. A : S ==> leadsTo acts A B) ==> leadsTo acts (Union S) B *) - Union "(UN A:S. {(A,B)}) : Pow (leadsto Acts) - ==> leadsTo Acts (Union S) B" + Union "(UN A:S. {(A,B)}) : Pow (leadsto acts) + ==> leadsTo acts (Union S) B" monos "[Pow_mono]" -(*wlt Acts B is the largest set that leads to B*) +(*wlt acts B is the largest set that leads to B*) constdefs wlt :: "[('a * 'a)set set, 'a set] => 'a set" - "wlt Acts B == Union {A. leadsTo Acts A B}" + "wlt acts B == Union {A. leadsTo acts A B}" end