session stringclasses 1
value | dependency listlengths 0 0 | context stringlengths 31 38.1k | proof stringlengths 31 38.1k | proof_state stringlengths 38 24.6M | statement stringlengths 22 5.17k | name stringlengths 13 17 | theory_name stringclasses 518
values | num_steps int64 1 963 |
|---|---|---|---|---|---|---|---|---|
[] | lemma not_reachable_visits_same:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> trace_addr tr i = Some n
\<Longrightarrow> (n, m) \<notin> rtrancl (reachable_step' gf)
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> j > i
\<Longrightarrow> {... | lemma not_reachable_visits_same:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> trace_addr tr i = Some n
\<Longrightarrow> (n, m) \<notin> rtrancl (reachable_step' gf)
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> j > i
\<Longrightarrow> {... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; trace_addr tr i = Some n; (n, m) \<notin> (reachable_step' gf)\<^sup>*; wf_graph_function gf ilen olen; i < j\<rbrakk> \<Longrightarrow> {k. k < j \<and> trace_addr tr k = Some m} = {k. k < i \<and> trace_addr tr k = Some m} ... | lemma not_reachable_visits_same:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> trace_addr tr i = Some n
\<Longrightarrow> (n, m) \<notin> rtrancl (reachable_step' gf)
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> j > i
\<Longrightarrow> {... | unnamed_thy_30342 | GraphProof | 4 | |
[] | lemma not_reachable_visits_same_symm:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> trace_addr tr i = Some n
\<Longrightarrow> trace_addr tr j = Some n
\<Longrightarrow> (n, m) \<notin> rtrancl (reachable_step' gf)
\<Longrightarrow> wf_graph_function gf ilen olen... | lemma not_reachable_visits_same_symm:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> trace_addr tr i = Some n
\<Longrightarrow> trace_addr tr j = Some n
\<Longrightarrow> (n, m) \<notin> rtrancl (reachable_step' gf)
\<Longrightarrow> wf_graph_function gf ilen olen... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; trace_addr tr i = Some n; trace_addr tr j = Some n; (n, m) \<notin> (reachable_step' gf)\<^sup>*; wf_graph_function gf ilen olen\<rbrakk> \<Longrightarrow> {k. k < j \<and> trace_addr tr k = Some m} = {k. k < i \<and> trace_a... | lemma not_reachable_visits_same_symm:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> trace_addr tr i = Some n
\<Longrightarrow> trace_addr tr j = Some n
\<Longrightarrow> (n, m) \<notin> rtrancl (reachable_step' gf)
\<Longrightarrow> wf_graph_function gf ilen olen... | unnamed_thy_30343 | GraphProof | 2 | |
[] | lemma restrs_eventually_at_visit:
"restrs_eventually_condition tr (restrs_list rs)
\<Longrightarrow> trace_addr tr i = Some nn
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> distinct (map fst rs)
\<Longrightarrow> wf_graph_function gf ilen olen
... | lemma restrs_eventually_at_visit:
"restrs_eventually_condition tr (restrs_list rs)
\<Longrightarrow> trace_addr tr i = Some nn
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> distinct (map fst rs)
\<Longrightarrow> wf_graph_function gf ilen olen
... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>restrs_eventually_condition tr (restrs_list rs); trace_addr tr i = Some nn; tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; distinct (map fst rs); wf_graph_function gf ilen olen\<rbrakk> \<Longrightarrow> restrs_condition tr (restrs_list (restrs_visit rs nn gf)) i proof (p... | lemma restrs_eventually_at_visit:
"restrs_eventually_condition tr (restrs_list rs)
\<Longrightarrow> trace_addr tr i = Some nn
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> distinct (map fst rs)
\<Longrightarrow> wf_graph_function gf ilen olen
... | unnamed_thy_30344 | GraphProof | 10 | |
[] | lemma fold_double_trace_imp:
"fold double_trace_imp hyps hyp trs
= ((\<forall>h \<in> set hyps. h trs) \<longrightarrow> hyp trs)" apply (induct hyps arbitrary: hyp, simp_all) apply (auto simp add: double_trace_imp_def) done | lemma fold_double_trace_imp:
"fold double_trace_imp hyps hyp trs
= ((\<forall>h \<in> set hyps. h trs) \<longrightarrow> hyp trs)" apply (induct hyps arbitrary: hyp, simp_all) apply (auto simp add: double_trace_imp_def) done | proof (prove)
goal (1 subgoal):
1. fold double_trace_imp hyps hyp trs = ((\<forall>h\<in>set hyps. h trs) \<longrightarrow> hyp trs) proof (prove)
goal (1 subgoal):
1. \<And>x1 hyps hyp. (\<And>hyp. fold double_trace_imp hyps hyp trs = ((\<forall>h\<in>set hyps. h trs) \<longrightarrow> hyp trs)) \<Longrightarrow> ((... | lemma fold_double_trace_imp:
"fold double_trace_imp hyps hyp trs
= ((\<forall>h \<in> set hyps. h trs) \<longrightarrow> hyp trs)" | unnamed_thy_30345 | GraphProof | 3 | |
[] | lemma exec_trace_addr_Suc:
"tr \<in> exec_trace Gamma f \<Longrightarrow> trace_addr tr n = Some (NextNode m) \<Longrightarrow> tr (Suc n) \<noteq> None" apply (drule_tac i=n in exec_trace_step_cases) apply (auto dest!: trace_addr_SomeD) done | lemma exec_trace_addr_Suc:
"tr \<in> exec_trace Gamma f \<Longrightarrow> trace_addr tr n = Some (NextNode m) \<Longrightarrow> tr (Suc n) \<noteq> None" apply (drule_tac i=n in exec_trace_step_cases) apply (auto dest!: trace_addr_SomeD) done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma f; trace_addr tr n = Some (NextNode m)\<rbrakk> \<Longrightarrow> tr (Suc n) \<noteq> None proof (prove)
goal (1 subgoal):
1. \<lbrakk>trace_addr tr n = Some (NextNode m); tr n = None \<and> tr (Suc n) = None \<or> (\<exists>state. tr n = Some [sta... | lemma exec_trace_addr_Suc:
"tr \<in> exec_trace Gamma f \<Longrightarrow> trace_addr tr n = Some (NextNode m) \<Longrightarrow> tr (Suc n) \<noteq> None" | unnamed_thy_30346 | GraphProof | 3 | |
[] | lemma num_visits_equals_j_first:
"card {i. i < m \<and> trace_addr tr i = Some n} = j
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> \<exists>m'. trace_addr tr m' = Some n \<and> card {i. i < m' \<and> trace_addr tr i = Some n} = j - 1" apply (frule_tac P="\<lambda>m. card {i. i < m \<and> trace_addr tr i =... | lemma num_visits_equals_j_first:
"card {i. i < m \<and> trace_addr tr i = Some n} = j
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> \<exists>m'. trace_addr tr m' = Some n \<and> card {i. i < m' \<and> trace_addr tr i = Some n} = j - 1" apply (frule_tac P="\<lambda>m. card {i. i < m \<and> trace_addr tr i =... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>card {i. i < m \<and> trace_addr tr i = Some n} = j; j \<noteq> 0\<rbrakk> \<Longrightarrow> \<exists>m'. trace_addr tr m' = Some n \<and> card {i. i < m' \<and> trace_addr tr i = Some n} = j - 1 proof (prove)
goal (2 subgoals):
1. \<lbrakk>card {i. i < m \<and> trace_addr ... | lemma num_visits_equals_j_first:
"card {i. i < m \<and> trace_addr tr i = Some n} = j
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> \<exists>m'. trace_addr tr m' = Some n \<and> card {i. i < m' \<and> trace_addr tr i = Some n} = j - 1" | unnamed_thy_30347 | GraphProof | 8 | |
[] | lemma ex_least_nat:
"\<exists>n. P n \<Longrightarrow> \<exists>n :: nat. P n \<and> (\<forall>i < n. \<not> P i)" apply clarsimp apply (case_tac "n = 0") apply fastforce apply (cut_tac P="\<lambda>i. i \<noteq> 0 \<and> P i" in ex_least_nat_le, auto) done | lemma ex_least_nat:
"\<exists>n. P n \<Longrightarrow> \<exists>n :: nat. P n \<and> (\<forall>i < n. \<not> P i)" apply clarsimp apply (case_tac "n = 0") apply fastforce apply (cut_tac P="\<lambda>i. i \<noteq> 0 \<and> P i" in ex_least_nat_le, auto) done | proof (prove)
goal (1 subgoal):
1. \<exists>n. P n \<Longrightarrow> \<exists>n. P n \<and> (\<forall>i<n. \<not> P i) proof (prove)
goal (1 subgoal):
1. \<And>n. P n \<Longrightarrow> \<exists>n. P n \<and> (\<forall>i<n. \<not> P i) proof (prove)
goal (2 subgoals):
1. \<And>n. \<lbrakk>P n; n = 0\<rbrakk> \<Longri... | lemma ex_least_nat:
"\<exists>n. P n \<Longrightarrow> \<exists>n :: nat. P n \<and> (\<forall>i < n. \<not> P i)" | unnamed_thy_30348 | GraphProof | 5 | |
[] | theorem restr_trace_refine_Restr1:
"j \<noteq> 0
\<Longrightarrow> distinct (map fst rs1)
\<Longrightarrow> wf_graph_function f1 ilen olen \<Longrightarrow> Gamma1 fn1 = Some f1
\<Longrightarrow> i \<noteq> 0 \<longrightarrow>
pc' n (restrs_list ((n, [i - 1]) # (restrs_visit rs1 (NextNode n) f1)))... | theorem restr_trace_refine_Restr1:
"j \<noteq> 0
\<Longrightarrow> distinct (map fst rs1)
\<Longrightarrow> wf_graph_function f1 ilen olen \<Longrightarrow> Gamma1 fn1 = Some f1
\<Longrightarrow> i \<noteq> 0 \<longrightarrow>
pc' n (restrs_list ((n, [i - 1]) # (restrs_visit rs1 (NextNode n) f1)))... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>j \<noteq> 0; distinct (map fst rs1); wf_graph_function f1 ilen olen; Gamma1 fn1 = Some f1; i \<noteq> 0 \<longrightarrow> pc' n (restrs_list ((n, [i - 1]) # restrs_visit rs1 (NextNode n) f1)) tr; \<not> pc' n (restrs_list ((n, [j - 1]) # restrs_visit rs1 (NextNode n) f1)) t... | theorem restr_trace_refine_Restr1:
"j \<noteq> 0
\<Longrightarrow> distinct (map fst rs1)
\<Longrightarrow> wf_graph_function f1 ilen olen \<Longrightarrow> Gamma1 fn1 = Some f1
\<Longrightarrow> i \<noteq> 0 \<longrightarrow>
pc' n (restrs_list ((n, [i - 1]) # (restrs_visit rs1 (NextNode n) f1)))... | unnamed_thy_30350 | GraphProof | 26 | |
[] | theorem restr_trace_refine_Restr2:
"j \<noteq> 0
\<Longrightarrow> distinct (map fst rs2)
\<Longrightarrow> wf_graph_function f2 ilen olen \<Longrightarrow> Gamma2 fn2 = Some f2
\<Longrightarrow> i \<noteq> 0 \<longrightarrow> pc' n (restrs_list ((n, [i - 1]) # (restrs_visit rs2 (NextNode n) f2))) tr'
... | theorem restr_trace_refine_Restr2:
"j \<noteq> 0
\<Longrightarrow> distinct (map fst rs2)
\<Longrightarrow> wf_graph_function f2 ilen olen \<Longrightarrow> Gamma2 fn2 = Some f2
\<Longrightarrow> i \<noteq> 0 \<longrightarrow> pc' n (restrs_list ((n, [i - 1]) # (restrs_visit rs2 (NextNode n) f2))) tr'
... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>j \<noteq> 0; distinct (map fst rs2); wf_graph_function f2 ilen olen; Gamma2 fn2 = Some f2; i \<noteq> 0 \<longrightarrow> pc' n (restrs_list ((n, [i - 1]) # restrs_visit rs2 (NextNode n) f2)) tr'; \<not> pc' n (restrs_list ((n, [j - 1]) # restrs_visit rs2 (NextNode n) f2)) ... | theorem restr_trace_refine_Restr2:
"j \<noteq> 0
\<Longrightarrow> distinct (map fst rs2)
\<Longrightarrow> wf_graph_function f2 ilen olen \<Longrightarrow> Gamma2 fn2 = Some f2
\<Longrightarrow> i \<noteq> 0 \<longrightarrow> pc' n (restrs_list ((n, [i - 1]) # (restrs_visit rs2 (NextNode n) f2))) tr'
... | unnamed_thy_30351 | GraphProof | 25 | |
[] | lemma pc_Ret_Err_trace_end:
"er \<in> {Ret, Err} \<Longrightarrow> pc er restrs tr
\<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> \<exists>st g. trace_end tr = Some [(er, st, g)]" apply (clarsimp simp: pc_def visit_eqs trace_end_def dest!: trace_addr_SomeD) apply (frule_tac i=i in exec_trace_... | lemma pc_Ret_Err_trace_end:
"er \<in> {Ret, Err} \<Longrightarrow> pc er restrs tr
\<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> \<exists>st g. trace_end tr = Some [(er, st, g)]" apply (clarsimp simp: pc_def visit_eqs trace_end_def dest!: trace_addr_SomeD) apply (frule_tac i=i in exec_trace_... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>er \<in> {Ret, Err}; pc er restrs tr; tr \<in> exec_trace Gamma f\<rbrakk> \<Longrightarrow> \<exists>st g. trace_end tr = Some [(er, st, g)] proof (prove)
goal (1 subgoal):
1. \<And>i st g. \<lbrakk>er = Ret \<or> er = Err; tr \<in> exec_trace Gamma f; restrs_condition tr ... | lemma pc_Ret_Err_trace_end:
"er \<in> {Ret, Err} \<Longrightarrow> pc er restrs tr
\<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> \<exists>st g. trace_end tr = Some [(er, st, g)]" | unnamed_thy_30352 | GraphProof | 9 | |
[] | lemma exec_trace_end_SomeD:
"trace_end tr = Some v \<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> \<exists>n. tr n = Some v \<and> tr (Suc n) = None
\<and> (\<exists>nn st g. v = [(nn, st, g)] \<and> nn \<in> {Ret, Err})" apply (frule exec_trace_nat_trace) apply (drule(1) trace_end_SomeD)... | lemma exec_trace_end_SomeD:
"trace_end tr = Some v \<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> \<exists>n. tr n = Some v \<and> tr (Suc n) = None
\<and> (\<exists>nn st g. v = [(nn, st, g)] \<and> nn \<in> {Ret, Err})" apply (frule exec_trace_nat_trace) apply (drule(1) trace_end_SomeD)... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>trace_end tr = Some v; tr \<in> exec_trace Gamma f\<rbrakk> \<Longrightarrow> \<exists>n. tr n = Some v \<and> tr (Suc n) = None \<and> (\<exists>nn st g. v = [(nn, st, g)] \<and> nn \<in> {Ret, Err}) proof (prove)
goal (1 subgoal):
1. \<lbrakk>trace_end tr = Some v; tr \<i... | lemma exec_trace_end_SomeD:
"trace_end tr = Some v \<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> \<exists>n. tr n = Some v \<and> tr (Suc n) = None
\<and> (\<exists>nn st g. v = [(nn, st, g)] \<and> nn \<in> {Ret, Err})" | unnamed_thy_30353 | GraphProof | 6 | |
[] | lemma reachable_from_Ret:
"((Ret, nn) \<notin> reachable_step' gf)" by (simp add: reachable_step_def) | lemma reachable_from_Ret:
"((Ret, nn) \<notin> reachable_step' gf)" by (simp add: reachable_step_def) | proof (prove)
goal (1 subgoal):
1. (Ret, nn) \<notin> reachable_step' gf | lemma reachable_from_Ret:
"((Ret, nn) \<notin> reachable_step' gf)" | unnamed_thy_30354 | GraphProof | 1 | |
[] | lemma trace_end_visit_Ret:
"tr n = Some [(Ret, st, g)] \<Longrightarrow> tr (Suc n) = None
\<Longrightarrow> tr \<in> exec_trace Gamma gf
\<Longrightarrow> restrs_eventually_condition tr rs
\<Longrightarrow> visit tr Ret rs = Some st" apply (rule visit_known, assumption) apply (clarsimp simp: restrs_event... | lemma trace_end_visit_Ret:
"tr n = Some [(Ret, st, g)] \<Longrightarrow> tr (Suc n) = None
\<Longrightarrow> tr \<in> exec_trace Gamma gf
\<Longrightarrow> restrs_eventually_condition tr rs
\<Longrightarrow> visit tr Ret rs = Some st" apply (rule visit_known, assumption) apply (clarsimp simp: restrs_event... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr n = Some [(Ret, st, g)]; tr (Suc n) = None; tr \<in> exec_trace Gamma gf; restrs_eventually_condition tr rs\<rbrakk> \<Longrightarrow> visit tr Ret rs = Some st proof (prove)
goal (2 subgoals):
1. \<lbrakk>tr n = Some [(Ret, st, g)]; tr (Suc n) = None; tr \<in> exec_trac... | lemma trace_end_visit_Ret:
"tr n = Some [(Ret, st, g)] \<Longrightarrow> tr (Suc n) = None
\<Longrightarrow> tr \<in> exec_trace Gamma gf
\<Longrightarrow> restrs_eventually_condition tr rs
\<Longrightarrow> visit tr Ret rs = Some st" | unnamed_thy_30355 | GraphProof | 11 | |
[] | theorem restr_trace_refine_Leaf:
"wf_graph_function f1 ilen1 olen1 \<Longrightarrow> Gamma1 fn1 = Some f1
\<Longrightarrow> wf_graph_function f2 ilen2 olen2 \<Longrightarrow> Gamma2 fn2 = Some f2
\<Longrightarrow> pc Ret rs1 tr \<Longrightarrow> prec \<longrightarrow> pc Ret rs2 tr'
\<Longrightarrow> outp... | theorem restr_trace_refine_Leaf:
"wf_graph_function f1 ilen1 olen1 \<Longrightarrow> Gamma1 fn1 = Some f1
\<Longrightarrow> wf_graph_function f2 ilen2 olen2 \<Longrightarrow> Gamma2 fn2 = Some f2
\<Longrightarrow> pc Ret rs1 tr \<Longrightarrow> prec \<longrightarrow> pc Ret rs2 tr'
\<Longrightarrow> outp... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>wf_graph_function f1 ilen1 olen1; Gamma1 fn1 = Some f1; wf_graph_function f2 ilen2 olen2; Gamma2 fn2 = Some f2; pc Ret rs1 tr; prec \<longrightarrow> pc Ret rs2 tr'; output_rel orel (f1, f2) (rs1, rs2) (tr, tr')\<rbrakk> \<Longrightarrow> restr_trace_refine prec Gamma1 fn1 G... | theorem restr_trace_refine_Leaf:
"wf_graph_function f1 ilen1 olen1 \<Longrightarrow> Gamma1 fn1 = Some f1
\<Longrightarrow> wf_graph_function f2 ilen2 olen2 \<Longrightarrow> Gamma2 fn2 = Some f2
\<Longrightarrow> pc Ret rs1 tr \<Longrightarrow> prec \<longrightarrow> pc Ret rs2 tr'
\<Longrightarrow> outp... | unnamed_thy_30356 | GraphProof | 13 | |
[] | lemma first_reached_propD:
"first_reached_prop addrs propn trs
\<Longrightarrow> \<exists>addr \<in> set addrs. propn addr trs
\<and> (\<forall>propn. first_reached_prop addrs propn trs = propn addr trs)" by (induct addrs, simp_all split: if_split_asm) | lemma first_reached_propD:
"first_reached_prop addrs propn trs
\<Longrightarrow> \<exists>addr \<in> set addrs. propn addr trs
\<and> (\<forall>propn. first_reached_prop addrs propn trs = propn addr trs)" by (induct addrs, simp_all split: if_split_asm) | proof (prove)
goal (1 subgoal):
1. first_reached_prop addrs propn trs \<Longrightarrow> \<exists>addr\<in>set addrs. propn addr trs \<and> (\<forall>propn. first_reached_prop addrs propn trs = propn addr trs) | lemma first_reached_propD:
"first_reached_prop addrs propn trs
\<Longrightarrow> \<exists>addr \<in> set addrs. propn addr trs
\<and> (\<forall>propn. first_reached_prop addrs propn trs = propn addr trs)" | unnamed_thy_30357 | GraphProof | 1 | |
[] | lemma double_pc_reds:
"double_pc (False, nn, restrs) trs = pc nn restrs (fst trs)"
"double_pc (True, nn, restrs) trs = pc nn restrs (snd trs)" by (simp_all add: double_pc_def pc_def split_def) | lemma double_pc_reds:
"double_pc (False, nn, restrs) trs = pc nn restrs (fst trs)"
"double_pc (True, nn, restrs) trs = pc nn restrs (snd trs)" by (simp_all add: double_pc_def pc_def split_def) | proof (prove)
goal (1 subgoal):
1. double_pc (False, nn, restrs) trs = pc nn restrs (fst trs) &&& double_pc (True, nn, restrs) trs = pc nn restrs (snd trs) | lemma double_pc_reds:
"double_pc (False, nn, restrs) trs = pc nn restrs (fst trs)"
"double_pc (True, nn, restrs) trs = pc nn restrs (snd trs)" | unnamed_thy_30358 | GraphProof | 1 | |
[] | lemma merge_opt_simps[simp]:
"merge_opt (Some x) v = Some x"
"merge_opt None v = v" by (simp_all add: merge_opt_def) | lemma merge_opt_simps[simp]:
"merge_opt (Some x) v = Some x"
"merge_opt None v = v" by (simp_all add: merge_opt_def) | proof (prove)
goal (1 subgoal):
1. merge_opt (Some x) v = Some x &&& merge_opt None v = v | lemma merge_opt_simps[simp]:
"merge_opt (Some x) v = Some x"
"merge_opt None v = v" | unnamed_thy_30359 | GraphProof | 1 | |
[] | lemma fold_merge_opt_Nones_eq:
"(\<forall>v \<in> set xs. v = None) \<Longrightarrow> fold merge_opt xs v = v" by (induct xs, simp_all) | lemma fold_merge_opt_Nones_eq:
"(\<forall>v \<in> set xs. v = None) \<Longrightarrow> fold merge_opt xs v = v" by (induct xs, simp_all) | proof (prove)
goal (1 subgoal):
1. \<forall>v\<in>set xs. v = None \<Longrightarrow> fold merge_opt xs v = v | lemma fold_merge_opt_Nones_eq:
"(\<forall>v \<in> set xs. v = None) \<Longrightarrow> fold merge_opt xs v = v" | unnamed_thy_30360 | GraphProof | 1 | |
[] | lemma set_zip_rev:
"length xs = length ys \<Longrightarrow> set (zip xs ys) = set (zip (rev xs) (rev ys))" by (simp add: zip_rev) | lemma set_zip_rev:
"length xs = length ys \<Longrightarrow> set (zip xs ys) = set (zip (rev xs) (rev ys))" by (simp add: zip_rev) | proof (prove)
goal (1 subgoal):
1. length xs = length ys \<Longrightarrow> set (zip xs ys) = set (zip (rev xs) (rev ys)) | lemma set_zip_rev:
"length xs = length ys \<Longrightarrow> set (zip xs ys) = set (zip (rev xs) (rev ys))" | unnamed_thy_30361 | GraphProof | 1 | |
[] | lemma exec_trace_non_Call:
"\<lbrakk> tr \<in> exec_trace Gamma fn; Gamma fn = Some f;
trace_bottom_addr tr i = Some (NextNode n);
function_graph f n = Some node;
case node of Call _ _ _ _ \<Rightarrow> False | _ \<Rightarrow> True
\<rbrakk> \<Longrightarrow> trace_addr tr i = Some (NextNo... | lemma exec_trace_non_Call:
"\<lbrakk> tr \<in> exec_trace Gamma fn; Gamma fn = Some f;
trace_bottom_addr tr i = Some (NextNode n);
function_graph f n = Some node;
case node of Call _ _ _ _ \<Rightarrow> False | _ \<Rightarrow> True
\<rbrakk> \<Longrightarrow> trace_addr tr i = Some (NextNo... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some f; trace_bottom_addr tr i = Some (NextNode n); function_graph f n = Some node; case node of node.Call x xa xb xc \<Rightarrow> False | _ \<Rightarrow> True\<rbrakk> \<Longrightarrow> trace_addr tr i = Some (NextNode n) proof (pro... | lemma exec_trace_non_Call:
"\<lbrakk> tr \<in> exec_trace Gamma fn; Gamma fn = Some f;
trace_bottom_addr tr i = Some (NextNode n);
function_graph f n = Some node;
case node of Call _ _ _ _ \<Rightarrow> False | _ \<Rightarrow> True
\<rbrakk> \<Longrightarrow> trace_addr tr i = Some (NextNo... | unnamed_thy_30362 | GraphProof | 7 | |
[] | lemma visit_immediate_pred:
"\<lbrakk> tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen;
trace_addr tr i = Some nn; nn \<noteq> NextNode (entry_point f);
converse (reachable_step' f) `` {nn} \<subseteq> S \<rbrakk>
\<Longrightarrow> \<exists>i' nn'. i = Suc i' \<and> ... | lemma visit_immediate_pred:
"\<lbrakk> tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen;
trace_addr tr i = Some nn; nn \<noteq> NextNode (entry_point f);
converse (reachable_step' f) `` {nn} \<subseteq> S \<rbrakk>
\<Longrightarrow> \<exists>i' nn'. i = Suc i' \<and> ... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen; trace_addr tr i = Some nn; nn \<noteq> NextNode (entry_point f); (reachable_step' f)\<inverse> `` {nn} \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>i' nn'. i = Suc i' \<and> nn' \<in> S \<an... | lemma visit_immediate_pred:
"\<lbrakk> tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen;
trace_addr tr i = Some nn; nn \<noteq> NextNode (entry_point f);
converse (reachable_step' f) `` {nn} \<subseteq> S \<rbrakk>
\<Longrightarrow> \<exists>i' nn'. i = Suc i' \<and> ... | unnamed_thy_30363 | GraphProof | 9 | |
[] | lemma pred_restrs:
"\<lbrakk> tr \<in> exec_trace Gamma f; trace_bottom_addr tr i = Some nn \<rbrakk>
\<Longrightarrow> restrs_condition tr restrs (Suc i)
= restrs_condition tr (if trace_addr tr i = None then restrs
else pred_restrs nn restrs) i" apply (clarsimp simp: restrs_condition_def Coll... | lemma pred_restrs:
"\<lbrakk> tr \<in> exec_trace Gamma f; trace_bottom_addr tr i = Some nn \<rbrakk>
\<Longrightarrow> restrs_condition tr restrs (Suc i)
= restrs_condition tr (if trace_addr tr i = None then restrs
else pred_restrs nn restrs) i" apply (clarsimp simp: restrs_condition_def Coll... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma f; trace_bottom_addr tr i = Some nn\<rbrakk> \<Longrightarrow> restrs_condition tr restrs (Suc i) = restrs_condition tr (if trace_addr tr i = None then restrs else pred_restrs nn restrs) i proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_t... | lemma pred_restrs:
"\<lbrakk> tr \<in> exec_trace Gamma f; trace_bottom_addr tr i = Some nn \<rbrakk>
\<Longrightarrow> restrs_condition tr restrs (Suc i)
= restrs_condition tr (if trace_addr tr i = None then restrs
else pred_restrs nn restrs) i" | unnamed_thy_30364 | GraphProof | 3 | |
[] | lemma visit_merge:
assumes tr: "tr \<in> exec_trace Gamma fn" "Gamma fn = Some f"
and wf: "wf_graph_function f ilen olen"
and ns: "nn \<noteq> NextNode (entry_point f)"
"\<forall>n \<in> set ns. graph n = Some (Basic nn [])"
"converse (reachable_step graph) `` {nn} \<subseteq> NextNode ` s... | lemma visit_merge:
assumes tr: "tr \<in> exec_trace Gamma fn" "Gamma fn = Some f"
and wf: "wf_graph_function f ilen olen"
and ns: "nn \<noteq> NextNode (entry_point f)"
"\<forall>n \<in> set ns. graph n = Some (Basic nn [])"
"converse (reachable_step graph) `` {nn} \<subseteq> NextNode ` s... | proof (prove)
goal (1 subgoal):
1. visit tr nn restrs = fold merge_opt (map (\<lambda>n. visit tr (NextNode n) (pred_restrs' n restrs)) ns) None proof (state)
goal (1 subgoal):
1. visit tr nn restrs = fold merge_opt (map (\<lambda>n. visit tr (NextNode n) (pred_restrs' n restrs)) ns) None proof (state)
this:
nn \<not... | lemma visit_merge:
assumes tr: "tr \<in> exec_trace Gamma fn" "Gamma fn = Some f"
and wf: "wf_graph_function f ilen olen"
and ns: "nn \<noteq> NextNode (entry_point f)"
"\<forall>n \<in> set ns. graph n = Some (Basic nn [])"
"converse (reachable_step graph) `` {nn} \<subseteq> NextNode ` s... | unnamed_thy_30365 | GraphProof | 45 | |
[] | lemma visit_merge_restrs:
assumes tr: "tr \<in> exec_trace Gamma fn" "Gamma fn = Some gf"
and geq: "function_graph gf = graph"
assumes indep: "opt \<notin> set (opt2 # opts)"
assumes reach: "(nn, NextNode addr) \<notin> rtrancl (reachable_step graph)"
and wf: "wf_graph_function gf ilen olen"
fixes... | lemma visit_merge_restrs:
assumes tr: "tr \<in> exec_trace Gamma fn" "Gamma fn = Some gf"
and geq: "function_graph gf = graph"
assumes indep: "opt \<notin> set (opt2 # opts)"
assumes reach: "(nn, NextNode addr) \<notin> rtrancl (reachable_step graph)"
and wf: "wf_graph_function gf ilen olen"
fixes... | proof (prove)
goal (1 subgoal):
1. visit tr nn rs3 = merge_opt (visit tr nn rs1) (visit tr nn rs2) proof (state)
goal (1 subgoal):
1. visit tr nn rs3 = merge_opt (visit tr nn rs1) (visit tr nn rs2) proof (state)
goal (1 subgoal):
1. visit tr nn rs3 = merge_opt (visit tr nn rs1) (visit tr nn rs2) proof (prove)
goal (... | lemma visit_merge_restrs:
assumes tr: "tr \<in> exec_trace Gamma fn" "Gamma fn = Some gf"
and geq: "function_graph gf = graph"
assumes indep: "opt \<notin> set (opt2 # opts)"
assumes reach: "(nn, NextNode addr) \<notin> rtrancl (reachable_step graph)"
and wf: "wf_graph_function gf ilen olen"
fixes... | unnamed_thy_30366 | GraphProof | 30 | |
[] | theorem visit_explode_restr:
assumes gf1: "tr \<in> exec_trace Gamma fn"
"Gamma fn = Some gf"
"function_graph gf = graph"
and gf2: "(nn, NextNode addr) \<notin> (reachable_step graph)\<^sup>*"
"wf_graph_function gf ilen olen"
and rs: "restrs_list rs addr = set xs"
"filter (\<lambda>(addr', xs). ad... | theorem visit_explode_restr:
assumes gf1: "tr \<in> exec_trace Gamma fn"
"Gamma fn = Some gf"
"function_graph gf = graph"
and gf2: "(nn, NextNode addr) \<notin> (reachable_step graph)\<^sup>*"
"wf_graph_function gf ilen olen"
and rs: "restrs_list rs addr = set xs"
"filter (\<lambda>(addr', xs). ad... | proof (prove)
goal (1 subgoal):
1. visit tr nn (restrs_list rs) = fold (\<lambda>x. merge_opt (visit tr nn (restrs_list ((addr, [x]) # rs')))) xs None proof (state)
goal (1 subgoal):
1. visit tr nn (restrs_list rs) = fold (\<lambda>x. merge_opt (visit tr nn (restrs_list ((addr, [x]) # rs')))) xs None proof (chain)
pi... | theorem visit_explode_restr:
assumes gf1: "tr \<in> exec_trace Gamma fn"
"Gamma fn = Some gf"
"function_graph gf = graph"
and gf2: "(nn, NextNode addr) \<notin> (reachable_step graph)\<^sup>*"
"wf_graph_function gf ilen olen"
and rs: "restrs_list rs addr = set xs"
"filter (\<lambda>(addr', xs). ad... | unnamed_thy_30367 | GraphProof | 32 | |
[] | lemma visit_impossible:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> 0 \<notin> restrs n
\<Longrightarrow> (NextNode n, nn) \<notin> rtrancl (reachable_step graph)
\... | lemma visit_impossible:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> 0 \<notin> restrs n
\<Longrightarrow> (NextNode n, nn) \<notin> rtrancl (reachable_step graph)
\... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; function_graph gf = graph; wf_graph_function gf ilen olen; 0 \<notin> restrs n; (NextNode n, nn) \<notin> (reachable_step graph)\<^sup>*\<rbrakk> \<Longrightarrow> visit tr nn restrs = None proof (prove)
goal (1 subgoal):
1.... | lemma visit_impossible:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> 0 \<notin> restrs n
\<Longrightarrow> (NextNode n, nn) \<notin> rtrancl (reachable_step graph)
\... | unnamed_thy_30368 | GraphProof | 8 | |
[] | lemma visit_inconsistent:
"restrs i = {} \<Longrightarrow> visit tr nn restrs = None" by (auto simp add: visit_def restrs_condition_def) | lemma visit_inconsistent:
"restrs i = {} \<Longrightarrow> visit tr nn restrs = None" by (auto simp add: visit_def restrs_condition_def) | proof (prove)
goal (1 subgoal):
1. restrs i = {} \<Longrightarrow> visit tr nn restrs = None | lemma visit_inconsistent:
"restrs i = {} \<Longrightarrow> visit tr nn restrs = None" | unnamed_thy_30369 | GraphProof | 1 | |
[] | lemma visit_immediate_pred_step:
"tr i = Some [(nn, st, fn')]
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> converse (reachable_step (function_graph gf)) `` {nn} \<subseteq> {NextNode n}
\<Longrigh... | lemma visit_immediate_pred_step:
"tr i = Some [(nn, st, fn')]
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> converse (reachable_step (function_graph gf)) `` {nn} \<subseteq> {NextNode n}
\<Longrigh... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr i = Some [(nn, st, fn')]; tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; wf_graph_function gf ilen olen; (reachable_step' gf)\<inverse> `` {nn} \<subseteq> {NextNode n}; nn \<noteq> NextNode (entry_point gf); case function_graph gf n of None \<Rightarrow> False | Some ... | lemma visit_immediate_pred_step:
"tr i = Some [(nn, st, fn')]
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> converse (reachable_step (function_graph gf)) `` {nn} \<subseteq> {NextNode n}
\<Longrigh... | unnamed_thy_30370 | GraphProof | 10 | |
[] | lemma visit_Basic:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> graph n = Some (Basic nn upds)
\<Longrightarrow> converse (reachable_step graph) `` {nn} \<subseteq> {Nex... | lemma visit_Basic:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> graph n = Some (Basic nn upds)
\<Longrightarrow> converse (reachable_step graph) `` {nn} \<subseteq> {Nex... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; function_graph gf = graph; wf_graph_function gf ilen olen; graph n = Some (node.Basic nn upds); (reachable_step graph)\<inverse> `` {nn} \<subseteq> {NextNode n}; nn \<noteq> NextNode (entry_point gf)\<rbrakk> \<Longrightarro... | lemma visit_Basic:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> graph n = Some (Basic nn upds)
\<Longrightarrow> converse (reachable_step graph) `` {nn} \<subseteq> {Nex... | unnamed_thy_30371 | GraphProof | 19 | |
[] | lemma visit_Cond:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> graph n = Some (Cond l r cond)
\<Longrightarrow> converse (reachable_step graph) `` {nn} \<subseteq> {Next... | lemma visit_Cond:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> graph n = Some (Cond l r cond)
\<Longrightarrow> converse (reachable_step graph) `` {nn} \<subseteq> {Next... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some gf; function_graph gf = graph; wf_graph_function gf ilen olen; graph n = Some (node.Cond l r cond); (reachable_step graph)\<inverse> `` {nn} \<subseteq> {NextNode n}; nn \<noteq> NextNode (entry_point gf); \<forall>x. NextNode x ... | lemma visit_Cond:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some gf
\<Longrightarrow> function_graph gf = graph
\<Longrightarrow> wf_graph_function gf ilen olen
\<Longrightarrow> graph n = Some (Cond l r cond)
\<Longrightarrow> converse (reachable_step graph) `` {nn} \<subseteq> {Next... | unnamed_thy_30372 | GraphProof | 35 | |
[] | lemma exec_trace_pc_Call:
"pc' n restrs tr
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> function_graph f n = Some (node.Call nn fname inps outps)
\<Longrightarrow> Gamma fname = Some g
\<Longrightarrow> (\<exists>x. restrs_condition tr restrs x \... | lemma exec_trace_pc_Call:
"pc' n restrs tr
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> function_graph f n = Some (node.Call nn fname inps outps)
\<Longrightarrow> Gamma fname = Some g
\<Longrightarrow> (\<exists>x. restrs_condition tr restrs x \... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>pc' n restrs tr; tr \<in> exec_trace Gamma fn; Gamma fn = Some f; function_graph f n = Some (node.Call nn fname inps outps); Gamma fname = Some g\<rbrakk> \<Longrightarrow> \<exists>x. restrs_condition tr restrs x \<and> trace_addr tr x = Some (NextNode n) \<and> trace_drop_... | lemma exec_trace_pc_Call:
"pc' n restrs tr
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> function_graph f n = Some (node.Call nn fname inps outps)
\<Longrightarrow> Gamma fname = Some g
\<Longrightarrow> (\<exists>x. restrs_condition tr restrs x \... | unnamed_thy_30373 | GraphProof | 8 | |
[] | lemma exec_trace_step:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> tr i = Some stack
\<Longrightarrow> continuing stack
\<Longrightarrow> \<exists>stack'. tr (Suc i) = Some stack' \<and> (stack, stack') \<in> exec_graph_step Gamma" apply (frule_tac i=i in exec_trace_step_cases) apply auto done | lemma exec_trace_step:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> tr i = Some stack
\<Longrightarrow> continuing stack
\<Longrightarrow> \<exists>stack'. tr (Suc i) = Some stack' \<and> (stack, stack') \<in> exec_graph_step Gamma" apply (frule_tac i=i in exec_trace_step_cases) apply auto done | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma f; tr i = Some stack; continuing stack\<rbrakk> \<Longrightarrow> \<exists>stack'. tr (Suc i) = Some stack' \<and> (stack, stack') \<in> exec_graph_step Gamma proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma f; tr i = Some stac... | lemma exec_trace_step:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> tr i = Some stack
\<Longrightarrow> continuing stack
\<Longrightarrow> \<exists>stack'. tr (Suc i) = Some stack' \<and> (stack, stack') \<in> exec_graph_step Gamma" | unnamed_thy_30374 | GraphProof | 3 | |
[] | lemma visit_extended_pred:
"\<lbrakk> trace_addr tr i = Some addr; restrs_condition tr restrs i;
tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen;
converse (reachable_step' f) `` {addr} \<subseteq> S;
addr \<noteq> NextNode (entry_point f) \<rbrakk>
\<Longrigh... | lemma visit_extended_pred:
"\<lbrakk> trace_addr tr i = Some addr; restrs_condition tr restrs i;
tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen;
converse (reachable_step' f) `` {addr} \<subseteq> S;
addr \<noteq> NextNode (entry_point f) \<rbrakk>
\<Longrigh... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>trace_addr tr i = Some addr; restrs_condition tr restrs i; tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen; (reachable_step' f)\<inverse> `` {addr} \<subseteq> S; addr \<noteq> NextNode (entry_point f)\<rbrakk> \<Longrightarrow> \<exists>j nn'.... | lemma visit_extended_pred:
"\<lbrakk> trace_addr tr i = Some addr; restrs_condition tr restrs i;
tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen;
converse (reachable_step' f) `` {addr} \<subseteq> S;
addr \<noteq> NextNode (entry_point f) \<rbrakk>
\<Longrigh... | unnamed_thy_30376 | GraphProof | 8 | |
[] | lemma if_x_None_eq_Some:
"((if P then x else None) = Some y) = (P \<and> x = Some y)" by simp | lemma if_x_None_eq_Some:
"((if P then x else None) = Some y) = (P \<and> x = Some y)" by simp | proof (prove)
goal (1 subgoal):
1. ((if P then x else None) = Some y) = (P \<and> x = Some y) | lemma if_x_None_eq_Some:
"((if P then x else None) = Some y) = (P \<and> x = Some y)" | unnamed_thy_30377 | GraphProof | 1 | |
[] | lemma subtract_le_nat:
"((a :: nat) \<le> a - b) = (a = 0 \<or> b = 0)" by arith | lemma subtract_le_nat:
"((a :: nat) \<le> a - b) = (a = 0 \<or> b = 0)" by arith | proof (prove)
goal (1 subgoal):
1. (a \<le> a - b) = (a = 0 \<or> b = 0) | lemma subtract_le_nat:
"((a :: nat) \<le> a - b) = (a = 0 \<or> b = 0)" | unnamed_thy_30378 | GraphProof | 1 | |
[] | lemma bottom_addr_only:
"trace_addr tr i = None \<Longrightarrow> trace_bottom_addr tr i = Some nn
\<Longrightarrow> \<exists>x x' xs. tr i = Some (x # x' # xs) \<and> nn = fst (last (x' # xs))" apply (clarsimp simp: trace_addr_def trace_bottom_addr_def
split: option.split_asm list.split_asm) app... | lemma bottom_addr_only:
"trace_addr tr i = None \<Longrightarrow> trace_bottom_addr tr i = Some nn
\<Longrightarrow> \<exists>x x' xs. tr i = Some (x # x' # xs) \<and> nn = fst (last (x' # xs))" apply (clarsimp simp: trace_addr_def trace_bottom_addr_def
split: option.split_asm list.split_asm) app... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>trace_addr tr i = None; trace_bottom_addr tr i = Some nn\<rbrakk> \<Longrightarrow> \<exists>x x' xs. tr i = Some (x # x' # xs) \<and> nn = fst (last (x' # xs)) proof (prove)
goal (1 subgoal):
1. \<And>a aa b ab ac ba x22a. \<lbrakk>nn = fst (last x22a); tr i = Some ((a, aa... | lemma bottom_addr_only:
"trace_addr tr i = None \<Longrightarrow> trace_bottom_addr tr i = Some nn
\<Longrightarrow> \<exists>x x' xs. tr i = Some (x # x' # xs) \<and> nn = fst (last (x' # xs))" | unnamed_thy_30379 | GraphProof | 3 | |
[] | lemma extended_pred_trace_drop_n:
"trace_addr tr i = Some (NextNode n)
\<Longrightarrow> trace_addr tr j = Some nn
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> wf_graph_function f ilen olen
\<Longrightarrow> function_graph f n = Some (Call nn fna... | lemma extended_pred_trace_drop_n:
"trace_addr tr i = Some (NextNode n)
\<Longrightarrow> trace_addr tr j = Some nn
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> wf_graph_function f ilen olen
\<Longrightarrow> function_graph f n = Some (Call nn fna... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>trace_addr tr i = Some (NextNode n); trace_addr tr j = Some nn; tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen; function_graph f n = Some (node.Call nn fname inputs outputs); Gamma fname = Some f'; i < j; nn \<noteq> Err; trace_addr tr ` {Suc ... | lemma extended_pred_trace_drop_n:
"trace_addr tr i = Some (NextNode n)
\<Longrightarrow> trace_addr tr j = Some nn
\<Longrightarrow> tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> wf_graph_function f ilen olen
\<Longrightarrow> function_graph f n = Some (Call nn fna... | unnamed_thy_30380 | GraphProof | 32 | |
[] | lemma restrs_condition_no_change:
"restrs_condition tr restrs i
\<Longrightarrow> j \<ge> i
\<Longrightarrow> (\<forall>k \<in> {i ..< j}. trace_addr tr k = None)
\<Longrightarrow> restrs_condition tr restrs j" apply (clarsimp simp: restrs_condition_def) apply (rule_tac P="\<lambda>S. card S \<in> SS" for... | lemma restrs_condition_no_change:
"restrs_condition tr restrs i
\<Longrightarrow> j \<ge> i
\<Longrightarrow> (\<forall>k \<in> {i ..< j}. trace_addr tr k = None)
\<Longrightarrow> restrs_condition tr restrs j" apply (clarsimp simp: restrs_condition_def) apply (rule_tac P="\<lambda>S. card S \<in> SS" for... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>restrs_condition tr restrs i; i \<le> j; \<forall>k\<in>{i..<j}. trace_addr tr k = None\<rbrakk> \<Longrightarrow> restrs_condition tr restrs j proof (prove)
goal (1 subgoal):
1. \<And>m. \<lbrakk>\<forall>m. card {ia. ia < i \<and> trace_addr tr ia = Some (NextNode m)} \<i... | lemma restrs_condition_no_change:
"restrs_condition tr restrs i
\<Longrightarrow> j \<ge> i
\<Longrightarrow> (\<forall>k \<in> {i ..< j}. trace_addr tr k = None)
\<Longrightarrow> restrs_condition tr restrs j" | unnamed_thy_30381 | GraphProof | 5 | |
[] | lemma trace_end_exec_SomeI:
"tr \<in> exec_trace Gamma fn
\<Longrightarrow> tr i = Some stk
\<Longrightarrow> tr (Suc i) = None
\<Longrightarrow> trace_end tr = Some stk" apply (clarsimp simp: trace_end_def exI[where x="Suc i"]) apply (drule(1) exec_trace_None_dom_subset) apply (subst Max_eqI[where x=i], ... | lemma trace_end_exec_SomeI:
"tr \<in> exec_trace Gamma fn
\<Longrightarrow> tr i = Some stk
\<Longrightarrow> tr (Suc i) = None
\<Longrightarrow> trace_end tr = Some stk" apply (clarsimp simp: trace_end_def exI[where x="Suc i"]) apply (drule(1) exec_trace_None_dom_subset) apply (subst Max_eqI[where x=i], ... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; tr i = Some stk; tr (Suc i) = None\<rbrakk> \<Longrightarrow> trace_end tr = Some stk proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; tr i = Some stk; tr (Suc i) = None\<rbrakk> \<Longrightarrow> tr (Max (dom tr)) = So... | lemma trace_end_exec_SomeI:
"tr \<in> exec_trace Gamma fn
\<Longrightarrow> tr i = Some stk
\<Longrightarrow> tr (Suc i) = None
\<Longrightarrow> trace_end tr = Some stk" | unnamed_thy_30382 | GraphProof | 4 | |
[] | lemma function_call_trace_eq:
assumes tr: "tr \<in> exec_trace Gamma fname"
"Gamma fname = Some f"
"wf_graph_function f ilen olen"
and i: "trace_addr tr i = Some (NextNode n)"
"restrs_condition tr restrs i"
and cut: "\<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subse... | lemma function_call_trace_eq:
assumes tr: "tr \<in> exec_trace Gamma fname"
"Gamma fname = Some f"
"wf_graph_function f ilen olen"
and i: "trace_addr tr i = Some (NextNode n)"
"restrs_condition tr restrs i"
and cut: "\<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subse... | proof (prove)
goal (1 subgoal):
1. function_call_trace n restrs tr = Some (trace_drop_n (Suc i) 1 tr) proof (state)
goal (1 subgoal):
1. function_call_trace n restrs tr = Some (trace_drop_n (Suc i) 1 tr) proof (prove)
goal (1 subgoal):
1. \<forall>j<i. trace_addr tr j = Some (NextNode n) \<longrightarrow> \<not> res... | lemma function_call_trace_eq:
assumes tr: "tr \<in> exec_trace Gamma fname"
"Gamma fname = Some f"
"wf_graph_function f ilen olen"
and i: "trace_addr tr i = Some (NextNode n)"
"restrs_condition tr restrs i"
and cut: "\<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subse... | unnamed_thy_30383 | GraphProof | 11 | |
[] | lemma exec_trace_Err_propagate:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> tr i = Some ((Err, st, fname) # xs)
\<Longrightarrow> j \<le> length xs \<Longrightarrow> tr (i + j) = Some (upd_stack Err id (drop j ((Err, st, fname) # xs)))" apply (induct j arbitrary: xs) apply simp apply atomize apply clarsimp... | lemma exec_trace_Err_propagate:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> tr i = Some ((Err, st, fname) # xs)
\<Longrightarrow> j \<le> length xs \<Longrightarrow> tr (i + j) = Some (upd_stack Err id (drop j ((Err, st, fname) # xs)))" apply (induct j arbitrary: xs) apply simp apply atomize apply clarsimp... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma f; tr i = Some ((Err, st, fname) # xs); j \<le> length xs\<rbrakk> \<Longrightarrow> tr (i + j) = Some (upd_stack Err id (drop j ((Err, st, fname) # xs))) proof (prove)
goal (2 subgoals):
1. \<And>xs. \<lbrakk>tr \<in> exec_trace Gamma f; tr i = So... | lemma exec_trace_Err_propagate:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> tr i = Some ((Err, st, fname) # xs)
\<Longrightarrow> j \<le> length xs \<Longrightarrow> tr (i + j) = Some (upd_stack Err id (drop j ((Err, st, fname) # xs)))" | unnamed_thy_30384 | GraphProof | 9 | |
[] | lemma trace_end_trace_drop_n_Err:
"option_map (fst o hd) (trace_end (trace_drop_n i j tr)) = Some Err
\<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> trace_drop_n i j tr \<in> exec_trace Gamma f'
\<Longrightarrow> option_map (fst o hd) (trace_end tr) = Some Err" apply clarsimp apply (drule... | lemma trace_end_trace_drop_n_Err:
"option_map (fst o hd) (trace_end (trace_drop_n i j tr)) = Some Err
\<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> trace_drop_n i j tr \<in> exec_trace Gamma f'
\<Longrightarrow> option_map (fst o hd) (trace_end tr) = Some Err" apply clarsimp apply (drule... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>map_option (fst \<circ> hd) (trace_end (trace_drop_n i j tr)) = Some Err; tr \<in> exec_trace Gamma f; trace_drop_n i j tr \<in> exec_trace Gamma f'\<rbrakk> \<Longrightarrow> map_option (fst \<circ> hd) (trace_end tr) = Some Err proof (prove)
goal (1 subgoal):
1. \<And>z. ... | lemma trace_end_trace_drop_n_Err:
"option_map (fst o hd) (trace_end (trace_drop_n i j tr)) = Some Err
\<Longrightarrow> tr \<in> exec_trace Gamma f
\<Longrightarrow> trace_drop_n i j tr \<in> exec_trace Gamma f'
\<Longrightarrow> option_map (fst o hd) (trace_end tr) = Some Err" | unnamed_thy_30385 | GraphProof | 12 | |
[] | lemma trace_end_Nil:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> trace_end tr \<noteq> Some []" by (auto dest: exec_trace_end_SomeD simp: exec_trace_Nil) | lemma trace_end_Nil:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> trace_end tr \<noteq> Some []" by (auto dest: exec_trace_end_SomeD simp: exec_trace_Nil) | proof (prove)
goal (1 subgoal):
1. tr \<in> exec_trace Gamma f \<Longrightarrow> trace_end tr \<noteq> Some [] | lemma trace_end_Nil:
"tr \<in> exec_trace Gamma f
\<Longrightarrow> trace_end tr \<noteq> Some []" | unnamed_thy_30386 | GraphProof | 1 | |
[] | lemma visit_Call_loop_lemma:
"(nn, NextNode n) \<notin> rtrancl (reachable_step' f \<inter> S)
\<Longrightarrow> function_graph f n = Some (Call nn fname inputs outputs)
\<Longrightarrow> converse (reachable_step' f) `` {nn} \<subseteq> {NextNode n}
\<Longrightarrow> (nn, nn) \<notin> trancl (reachable_st... | lemma visit_Call_loop_lemma:
"(nn, NextNode n) \<notin> rtrancl (reachable_step' f \<inter> S)
\<Longrightarrow> function_graph f n = Some (Call nn fname inputs outputs)
\<Longrightarrow> converse (reachable_step' f) `` {nn} \<subseteq> {NextNode n}
\<Longrightarrow> (nn, nn) \<notin> trancl (reachable_st... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>(nn, NextNode n) \<notin> (reachable_step' f \<inter> S)\<^sup>*; function_graph f n = Some (node.Call nn fname inputs outputs); (reachable_step' f)\<inverse> `` {nn} \<subseteq> {NextNode n}\<rbrakk> \<Longrightarrow> (nn, nn) \<notin> (reachable_step' f \<inter> S)\<^sup>+... | lemma visit_Call_loop_lemma:
"(nn, NextNode n) \<notin> rtrancl (reachable_step' f \<inter> S)
\<Longrightarrow> function_graph f n = Some (Call nn fname inputs outputs)
\<Longrightarrow> converse (reachable_step' f) `` {nn} \<subseteq> {NextNode n}
\<Longrightarrow> (nn, nn) \<notin> trancl (reachable_st... | unnamed_thy_30387 | GraphProof | 5 | |
[] | lemma pred_restrs_list:
"pred_restrs nn (restrs_list xs)
= restrs_list (map (\<lambda>(i, ns). (i, if nn = NextNode i
then map (\<lambda>x. x - 1) (filter ((\<noteq>) 0) ns) else ns)) xs)" apply (clarsimp simp: pred_restrs_def split: next_node.split) apply (rule sym) apply (induct xs; simp, rule ext) appl... | lemma pred_restrs_list:
"pred_restrs nn (restrs_list xs)
= restrs_list (map (\<lambda>(i, ns). (i, if nn = NextNode i
then map (\<lambda>x. x - 1) (filter ((\<noteq>) 0) ns) else ns)) xs)" apply (clarsimp simp: pred_restrs_def split: next_node.split) apply (rule sym) apply (induct xs; simp, rule ext) appl... | proof (prove)
goal (1 subgoal):
1. pred_restrs nn (restrs_list xs) = restrs_list (map (\<lambda>(i, ns). (i, if nn = NextNode i then map (\<lambda>x. x - 1) (filter ((\<noteq>) 0) ns) else ns)) xs) proof (prove)
goal (1 subgoal):
1. \<And>x1. nn = NextNode x1 \<Longrightarrow> (restrs_list xs)(x1 := {x. Suc x \<in> r... | lemma pred_restrs_list:
"pred_restrs nn (restrs_list xs)
= restrs_list (map (\<lambda>(i, ns). (i, if nn = NextNode i
then map (\<lambda>x. x - 1) (filter ((\<noteq>) 0) ns) else ns)) xs)" | unnamed_thy_30388 | GraphProof | 7 | |
[] | lemma pred_restrs_cut:
"(\<exists>y. restrs x \<subseteq> {y}) \<Longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y})" apply (clarsimp simp: pred_restrs_def split: next_node.split) apply blast done | lemma pred_restrs_cut:
"(\<exists>y. restrs x \<subseteq> {y}) \<Longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y})" apply (clarsimp simp: pred_restrs_def split: next_node.split) apply blast done | proof (prove)
goal (1 subgoal):
1. \<exists>y. restrs x \<subseteq> {y} \<Longrightarrow> \<exists>y. pred_restrs nn restrs x \<subseteq> {y} proof (prove)
goal (1 subgoal):
1. \<And>y. \<lbrakk>restrs x \<subseteq> {y}; nn = NextNode x\<rbrakk> \<Longrightarrow> \<exists>y. {xa. Suc xa \<in> restrs x} \<subseteq> {y... | lemma pred_restrs_cut:
"(\<exists>y. restrs x \<subseteq> {y}) \<Longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y})" | unnamed_thy_30389 | GraphProof | 3 | |
[] | lemma pred_restrs_cut2:
"\<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subseteq> {y})
\<Longrightarrow> \<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y})" by (metis pred_restrs_cut) | lemma pred_restrs_cut2:
"\<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subseteq> {y})
\<Longrightarrow> \<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y})" by (metis pred_restrs_cut) | proof (prove)
goal (1 subgoal):
1. \<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subseteq> {y}) \<Longrightarrow> \<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y}) | lemma pred_restrs_cut2:
"\<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. restrs x \<subseteq> {y})
\<Longrightarrow> \<forall>x. NextNode x \<in> set cuts \<longrightarrow> (\<exists>y. pred_restrs nn restrs x \<subseteq> {y})" | unnamed_thy_30390 | GraphProof | 1 | |
[] | lemma visit_Call:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> wf_graph_function f ilen olen
\<Longrightarrow> function_graph f n = Some (Call nn fname inputs outputs)
\<Longrightarrow> Gamma fname = Some f'
\<Longrightarrow> length inputs = length (function_inpu... | lemma visit_Call:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> wf_graph_function f ilen olen
\<Longrightarrow> function_graph f n = Some (Call nn fname inputs outputs)
\<Longrightarrow> Gamma fname = Some f'
\<Longrightarrow> length inputs = length (function_inpu... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>tr \<in> exec_trace Gamma fn; Gamma fn = Some f; wf_graph_function f ilen olen; function_graph f n = Some (node.Call nn fname inputs outputs); Gamma fname = Some f'; length inputs = length (function_inputs f'); (reachable_step' f)\<inverse> `` {nn} \<subseteq> {NextNode n}; ... | lemma visit_Call:
"tr \<in> exec_trace Gamma fn \<Longrightarrow> Gamma fn = Some f
\<Longrightarrow> wf_graph_function f ilen olen
\<Longrightarrow> function_graph f n = Some (Call nn fname inputs outputs)
\<Longrightarrow> Gamma fname = Some f'
\<Longrightarrow> length inputs = length (function_inpu... | unnamed_thy_30391 | GraphProof | 55 | |
[] | lemma restr_trace_refine_Call_single:
"\<not> fst ccall \<and> (\<exists>nn outps. get_function_call gfs ccall = Some (nn, cfname, cinps, outps))
\<Longrightarrow> Gamma1 cfname = Some cf \<Longrightarrow> wf_graph_function cf cilen colen
\<Longrightarrow> fst acall \<and> (\<exists>nn outps. get_function_cal... | lemma restr_trace_refine_Call_single:
"\<not> fst ccall \<and> (\<exists>nn outps. get_function_call gfs ccall = Some (nn, cfname, cinps, outps))
\<Longrightarrow> Gamma1 cfname = Some cf \<Longrightarrow> wf_graph_function cf cilen colen
\<Longrightarrow> fst acall \<and> (\<exists>nn outps. get_function_cal... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<not> fst ccall \<and> (\<exists>nn outps. get_function_call gfs ccall = Some (nn, cfname, cinps, outps)); Gamma1 cfname = Some cf; wf_graph_function cf cilen colen; fst acall \<and> (\<exists>nn outps. get_function_call gfs acall = Some (nn, afname, ainps, outps)); Gamma2 ... | lemma restr_trace_refine_Call_single:
"\<not> fst ccall \<and> (\<exists>nn outps. get_function_call gfs ccall = Some (nn, cfname, cinps, outps))
\<Longrightarrow> Gamma1 cfname = Some cf \<Longrightarrow> wf_graph_function cf cilen colen
\<Longrightarrow> fst acall \<and> (\<exists>nn outps. get_function_cal... | unnamed_thy_30393 | GraphProof | 22 | |
[] | lemma not_finite_two:
"\<not> finite S \<Longrightarrow> \<exists>x y. x \<in> S \<and> y \<in> S \<and> x \<noteq> y" apply (case_tac "\<exists>x. x \<in> S") apply (rule ccontr, clarsimp) apply (erule notE, rule_tac B="{x}" in finite_subset) apply auto done | lemma not_finite_two:
"\<not> finite S \<Longrightarrow> \<exists>x y. x \<in> S \<and> y \<in> S \<and> x \<noteq> y" apply (case_tac "\<exists>x. x \<in> S") apply (rule ccontr, clarsimp) apply (erule notE, rule_tac B="{x}" in finite_subset) apply auto done | proof (prove)
goal (1 subgoal):
1. infinite S \<Longrightarrow> \<exists>x y. x \<in> S \<and> y \<in> S \<and> x \<noteq> y proof (prove)
goal (2 subgoals):
1. \<lbrakk>infinite S; \<exists>x. x \<in> S\<rbrakk> \<Longrightarrow> \<exists>x y. x \<in> S \<and> y \<in> S \<and> x \<noteq> y
2. \<lbrakk>infinite S; \... | lemma not_finite_two:
"\<not> finite S \<Longrightarrow> \<exists>x y. x \<in> S \<and> y \<in> S \<and> x \<noteq> y" | unnamed_thy_30394 | GraphProof | 5 | |
[] | lemma infinite_subset:
"\<not> finite S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<not> finite T" by (metis finite_subset) | lemma infinite_subset:
"\<not> finite S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<not> finite T" by (metis finite_subset) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>infinite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> infinite T | lemma infinite_subset:
"\<not> finite S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> \<not> finite T" | unnamed_thy_30395 | GraphProof | 1 | |
[] | lemma restr_trace_refine_Split_orig:
fixes rs1 rs2 hyps Gamma1 f1 Gamma2 f2
defines "H trs \<equiv>
exec_double_trace Gamma1 f1 Gamma2 f2 (fst trs) (snd trs)
\<and> restrs_eventually_condition (fst trs) (restrs_list rs1)
\<and> restrs_eventually_condition (snd trs) (restrs_list rs2)"
ass... | lemma restr_trace_refine_Split_orig:
fixes rs1 rs2 hyps Gamma1 f1 Gamma2 f2
defines "H trs \<equiv>
exec_double_trace Gamma1 f1 Gamma2 f2 (fst trs) (snd trs)
\<and> restrs_eventually_condition (fst trs) (restrs_list rs1)
\<and> restrs_eventually_condition (snd trs) (restrs_list rs2)"
ass... | proof (prove)
goal (1 subgoal):
1. restr_trace_refine prec Gamma1 f1 Gamma2 f2 (restrs_list rs1) (restrs_list rs2) orel tr tr' proof (state)
goal (1 subgoal):
1. restr_trace_refine prec Gamma1 f1 Gamma2 f2 (restrs_list rs1) (restrs_list rs2) orel tr tr' proof (prove)
goal (1 subgoal):
1. double_pc (ccall n) (tr, tr'... | lemma restr_trace_refine_Split_orig:
fixes rs1 rs2 hyps Gamma1 f1 Gamma2 f2
defines "H trs \<equiv>
exec_double_trace Gamma1 f1 Gamma2 f2 (fst trs) (snd trs)
\<and> restrs_eventually_condition (fst trs) (restrs_list rs1)
\<and> restrs_eventually_condition (snd trs) (restrs_list rs2)"
ass... | unnamed_thy_30396 | GraphProof | 86 | |
[] | lemma restrs_condition_unique:
"restrs_condition tr (restrs_list ((n, [x]) # rs)) k
\<Longrightarrow> restrs_condition tr (restrs_list ((n, [y]) # rs)) k
\<Longrightarrow> x = y" by (clarsimp simp: restrs_condition_def restrs_list_Cons
split: if_split_asm) | lemma restrs_condition_unique:
"restrs_condition tr (restrs_list ((n, [x]) # rs)) k
\<Longrightarrow> restrs_condition tr (restrs_list ((n, [y]) # rs)) k
\<Longrightarrow> x = y" by (clarsimp simp: restrs_condition_def restrs_list_Cons
split: if_split_asm) | proof (prove)
goal (1 subgoal):
1. \<lbrakk>restrs_condition tr (restrs_list ((n, [x]) # rs)) k; restrs_condition tr (restrs_list ((n, [y]) # rs)) k\<rbrakk> \<Longrightarrow> x = y | lemma restrs_condition_unique:
"restrs_condition tr (restrs_list ((n, [x]) # rs)) k
\<Longrightarrow> restrs_condition tr (restrs_list ((n, [y]) # rs)) k
\<Longrightarrow> x = y" | unnamed_thy_30397 | GraphProof | 1 | |
[] | lemma split_visit_rel':
"split_visit_rel rel (\<lambda>i. (False, NextNode (fst cseq), split_restr cseq rs1 i))
(\<lambda>i. (True, NextNode (fst aseq), split_restr aseq rs2 i)) j trs
= split_visit_rel' rel cseq aseq (rs1, rs2) trs j" apply (simp add: split_visit_rel_def split_visit_rel'_def
... | lemma split_visit_rel':
"split_visit_rel rel (\<lambda>i. (False, NextNode (fst cseq), split_restr cseq rs1 i))
(\<lambda>i. (True, NextNode (fst aseq), split_restr aseq rs2 i)) j trs
= split_visit_rel' rel cseq aseq (rs1, rs2) trs j" apply (simp add: split_visit_rel_def split_visit_rel'_def
... | proof (prove)
goal (1 subgoal):
1. split_visit_rel rel (\<lambda>i. (False, NextNode (fst cseq), restrs_list ((fst cseq, [fst (snd cseq) + i * snd (snd cseq)]) # rs1))) (\<lambda>i. (True, NextNode (fst aseq), restrs_list ((fst aseq, [fst (snd aseq) + i * snd (snd aseq)]) # rs2))) j trs = split_visit_rel' rel cseq ase... | lemma split_visit_rel':
"split_visit_rel rel (\<lambda>i. (False, NextNode (fst cseq), split_restr cseq rs1 i))
(\<lambda>i. (True, NextNode (fst aseq), split_restr aseq rs2 i)) j trs
= split_visit_rel' rel cseq aseq (rs1, rs2) trs j" | unnamed_thy_30398 | GraphProof | 4 | |
[] | theorem restr_trace_refine_Split:
assumes Suc: "\<forall>i. split_pc conc_seq rs1 tr (Suc i) \<longrightarrow> split_pc conc_seq rs1 tr i"
and init: "\<forall>i. i < k \<longrightarrow> split_pc conc_seq rs1 tr i
\<longrightarrow> split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr') i"
and induct: "... | theorem restr_trace_refine_Split:
assumes Suc: "\<forall>i. split_pc conc_seq rs1 tr (Suc i) \<longrightarrow> split_pc conc_seq rs1 tr i"
and init: "\<forall>i. i < k \<longrightarrow> split_pc conc_seq rs1 tr i
\<longrightarrow> split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr') i"
and induct: "... | proof (prove)
goal (1 subgoal):
1. restr_trace_refine prec Gamma1 f1 Gamma2 f2 (restrs_list rs1) (restrs_list rs2) orel tr tr' proof (prove)
goal (2 subgoals):
1. \<forall>tr i j k. restrs_condition tr (restrs_list ((fst conc_seq, [fst (snd conc_seq) + i * snd (snd conc_seq)]) # rs1)) k \<longrightarrow> restrs_condi... | theorem restr_trace_refine_Split:
assumes Suc: "\<forall>i. split_pc conc_seq rs1 tr (Suc i) \<longrightarrow> split_pc conc_seq rs1 tr i"
and init: "\<forall>i. i < k \<longrightarrow> split_pc conc_seq rs1 tr i
\<longrightarrow> split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr') i"
and induct: "... | unnamed_thy_30399 | GraphProof | 6 | |
[] | theorem restr_trace_refine_Split':
"let cpc = split_pc conc_seq rs1 tr;
rel = split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr')
in (\<forall>i. cpc (Suc i) --> cpc i)
--> (\<forall>i. i < k --> cpc i --> rel i)
--> (\<forall>i. cpc (i + k) --> (\<forall>j < k. rel (i + j)) --> rel (i + k)... | theorem restr_trace_refine_Split':
"let cpc = split_pc conc_seq rs1 tr;
rel = split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr')
in (\<forall>i. cpc (Suc i) --> cpc i)
--> (\<forall>i. i < k --> cpc i --> rel i)
--> (\<forall>i. cpc (i + k) --> (\<forall>j < k. rel (i + j)) --> rel (i + k)... | proof (prove)
goal (1 subgoal):
1. let cpc = \<lambda>i. pc' (fst conc_seq) (restrs_list ((fst conc_seq, [fst (snd conc_seq) + i * snd (snd conc_seq)]) # rs1)) tr; rel = split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr') in (\<forall>i. cpc (Suc i) \<longrightarrow> cpc i) \<longrightarrow> (\<forall>i<k. cpc ... | theorem restr_trace_refine_Split':
"let cpc = split_pc conc_seq rs1 tr;
rel = split_visit_rel' rel conc_seq abs_seq (rs1, rs2) (tr, tr')
in (\<forall>i. cpc (Suc i) --> cpc i)
--> (\<forall>i. i < k --> cpc i --> rel i)
--> (\<forall>i. cpc (i + k) --> (\<forall>j < k. rel (i + j)) --> rel (i + k)... | unnamed_thy_30400 | GraphProof | 4 | |
[] | lemma restr_trace_refine_Restr1_offset:
"induct_var (NextNode n) iv
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 (restrs_list ((n, [iv + 1 ..< iv + j]) # rs1)) rs2 orel tr tr'
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> distinct (map fst rs1)
\<Longrightarrow> wf_graph_function... | lemma restr_trace_refine_Restr1_offset:
"induct_var (NextNode n) iv
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 (restrs_list ((n, [iv + 1 ..< iv + j]) # rs1)) rs2 orel tr tr'
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> distinct (map fst rs1)
\<Longrightarrow> wf_graph_function... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>induct_var (NextNode n) iv; restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 (restrs_list ((n, [iv + 1..<iv + j]) # rs1)) rs2 orel tr tr'; j \<noteq> 0; distinct (map fst rs1); wf_graph_function f1 ilen olen; Gamma1 fn1 = Some f1; pc' n (restrs_list ((n, [iv]) # restrs_visit rs... | lemma restr_trace_refine_Restr1_offset:
"induct_var (NextNode n) iv
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 (restrs_list ((n, [iv + 1 ..< iv + j]) # rs1)) rs2 orel tr tr'
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> distinct (map fst rs1)
\<Longrightarrow> wf_graph_function... | unnamed_thy_30401 | GraphProof | 1 | |
[] | lemma restr_trace_refine_Restr2_offset:
"induct_var (NextNode n) iv
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 (restrs_list ((n, [iv + 1 ..< iv + j]) # rs2)) orel tr tr'
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> distinct (map fst rs2)
\<Longrightarrow> wf_graph_function... | lemma restr_trace_refine_Restr2_offset:
"induct_var (NextNode n) iv
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 (restrs_list ((n, [iv + 1 ..< iv + j]) # rs2)) orel tr tr'
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> distinct (map fst rs2)
\<Longrightarrow> wf_graph_function... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>induct_var (NextNode n) iv; restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 (restrs_list ((n, [iv + 1..<iv + j]) # rs2)) orel tr tr'; j \<noteq> 0; distinct (map fst rs2); wf_graph_function f2 ilen olen; Gamma2 fn2 = Some f2; pc' n (restrs_list ((n, [iv]) # restrs_visit rs... | lemma restr_trace_refine_Restr2_offset:
"induct_var (NextNode n) iv
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 (restrs_list ((n, [iv + 1 ..< iv + j]) # rs2)) orel tr tr'
\<Longrightarrow> j \<noteq> 0
\<Longrightarrow> distinct (map fst rs2)
\<Longrightarrow> wf_graph_function... | unnamed_thy_30402 | GraphProof | 1 | |
[] | lemma restr_trace_refine_PCCases1:
"pc nn rs1 tr
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> \<not> pc nn rs1 tr
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> restr_trace_refine pr... | lemma restr_trace_refine_PCCases1:
"pc nn rs1 tr
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> \<not> pc nn rs1 tr
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> restr_trace_refine pr... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>pc nn rs1 tr \<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'; \<not> pc nn rs1 tr \<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'\<rbrakk> \<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 ... | lemma restr_trace_refine_PCCases1:
"pc nn rs1 tr
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> \<not> pc nn rs1 tr
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> restr_trace_refine pr... | unnamed_thy_30403 | GraphProof | 1 | |
[] | lemma restr_trace_refine_PCCases2:
"pc nn rs2 tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> \<not> pc nn rs2 tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> restr_trace_refine ... | lemma restr_trace_refine_PCCases2:
"pc nn rs2 tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> \<not> pc nn rs2 tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> restr_trace_refine ... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>pc nn rs2 tr' \<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'; \<not> pc nn rs2 tr' \<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'\<rbrakk> \<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn... | lemma restr_trace_refine_PCCases2:
"pc nn rs2 tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> \<not> pc nn rs2 tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'
\<Longrightarrow> restr_trace_refine ... | unnamed_thy_30404 | GraphProof | 1 | |
[] | lemma restr_trace_refine_Err:
"(\<not> pc Err restrs tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr')
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'" apply (clarsimp simp: restr_trace_refine_def) apply (erule impCE) apply simp ap... | lemma restr_trace_refine_Err:
"(\<not> pc Err restrs tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr')
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'" apply (clarsimp simp: restr_trace_refine_def) apply (erule impCE) apply simp ap... | proof (prove)
goal (1 subgoal):
1. \<not> pc Err restrs tr' \<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr' \<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr' proof (prove)
goal (1 subgoal):
1. \<And>gf gf' xs ys. \<lbrakk>\<not> pc Err restrs tr' \... | lemma restr_trace_refine_Err:
"(\<not> pc Err restrs tr'
\<longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr')
\<Longrightarrow> restr_trace_refine prec Gamma1 fn1 Gamma2 fn2 rs1 rs2 orel tr tr'" | unnamed_thy_30405 | GraphProof | 9 | |
[] | lemma stepwise_graph_refine_Cond:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (l # nns, l' # nns') rel orel otr (Suc N)
\<Longrightarrow> stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (r # nns, r' # nns') rel orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some g... | lemma stepwise_graph_refine_Cond:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (l # nns, l' # nns') rel orel otr (Suc N)
\<Longrightarrow> stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (r # nns, r' # nns') rel orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some g... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (l # nns, l' # nns') rel orel otr (Suc N); stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (r # nns, r' # nns') rel orel otr (Suc N); Gamma f1 = Some gf1; Gamma f2 = Some gf2; function_graph gf1 n = Some (node.Cond l r cond... | lemma stepwise_graph_refine_Cond:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (l # nns, l' # nns') rel orel otr (Suc N)
\<Longrightarrow> stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (r # nns, r' # nns') rel orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some g... | unnamed_thy_30407 | GraphProof | 9 | |
[] | lemma stepwise_graph_refine_Call_same:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (nn # nns, nn' # nns') rel orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Call nn fname inputs outputs)
\<Longrightarrow> f... | lemma stepwise_graph_refine_Call_same:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (nn # nns, nn' # nns') rel orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Call nn fname inputs outputs)
\<Longrightarrow> f... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (nn # nns, nn' # nns') rel orel otr (Suc N); Gamma f1 = Some gf1; Gamma f2 = Some gf2; function_graph gf1 n = Some (node.Call nn fname inputs outputs); function_graph gf2 n' = Some (node.Call nn' fname inputs' outputs'); Gamm... | lemma stepwise_graph_refine_Call_same:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (nn # nns, nn' # nns') rel orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Call nn fname inputs outputs)
\<Longrightarrow> f... | unnamed_thy_30408 | GraphProof | 36 | |
[] | lemma stepwise_graph_refine_flip:
"stepwise_graph_refine Gamma (fns', fns) (nns', nns)
(\<lambda>x. rel (snd x, fst x)) (\<lambda>x. orel (snd x, fst x)) (option_map (\<lambda>x. (snd x, fst x)) otr) N
= stepwise_graph_refine Gamma (fns, fns') (nns, nns') rel orel otr N" apply (intro iffI stepwise_graph_r... | lemma stepwise_graph_refine_flip:
"stepwise_graph_refine Gamma (fns', fns) (nns', nns)
(\<lambda>x. rel (snd x, fst x)) (\<lambda>x. orel (snd x, fst x)) (option_map (\<lambda>x. (snd x, fst x)) otr) N
= stepwise_graph_refine Gamma (fns, fns') (nns, nns') rel orel otr N" apply (intro iffI stepwise_graph_r... | proof (prove)
goal (1 subgoal):
1. stepwise_graph_refine Gamma (fns', fns) (nns', nns) (\<lambda>x. rel (snd x, fst x)) (\<lambda>x. orel (snd x, fst x)) (map_option (\<lambda>x. (snd x, fst x)) otr) N = stepwise_graph_refine Gamma (fns, fns') (nns, nns') rel orel otr N proof (prove)
goal (2 subgoals):
1. \<And>tr tr... | lemma stepwise_graph_refine_flip:
"stepwise_graph_refine Gamma (fns', fns) (nns', nns)
(\<lambda>x. rel (snd x, fst x)) (\<lambda>x. orel (snd x, fst x)) (option_map (\<lambda>x. (snd x, fst x)) otr) N
= stepwise_graph_refine Gamma (fns, fns') (nns, nns') rel orel otr N" | unnamed_thy_30410 | GraphProof | 8 | |
[] | lemma stepwise_graph_refine_nop_right:
"stepwise_graph_refine Gamma (fns, f2 # fns') (nns, nn' # nns') rel orel otr N
\<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf2 n' = Some (Basic nn' [])
\<Longrightarrow> stepwise_graph_refine Gamma (fns, f2 # fns')
(nns, NextNode n'... | lemma stepwise_graph_refine_nop_right:
"stepwise_graph_refine Gamma (fns, f2 # fns') (nns, nn' # nns') rel orel otr N
\<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf2 n' = Some (Basic nn' [])
\<Longrightarrow> stepwise_graph_refine Gamma (fns, f2 # fns')
(nns, NextNode n'... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (fns, f2 # fns') (nns, nn' # nns') rel orel otr N; Gamma f2 = Some gf2; function_graph gf2 n' = Some (node.Basic nn' [])\<rbrakk> \<Longrightarrow> stepwise_graph_refine Gamma (fns, f2 # fns') (nns, NextNode n' # nns') rel orel otr N proof (prove)... | lemma stepwise_graph_refine_nop_right:
"stepwise_graph_refine Gamma (fns, f2 # fns') (nns, nn' # nns') rel orel otr N
\<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf2 n' = Some (Basic nn' [])
\<Longrightarrow> stepwise_graph_refine Gamma (fns, f2 # fns')
(nns, NextNode n'... | unnamed_thy_30411 | GraphProof | 7 | |
[] | lemma stepwise_graph_refine_inline_left:
"stepwise_graph_refine Gamma (fname # f1 # fns, f2 # fns')
(NextNode (entry_point gf3) # NextNode n # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1... | lemma stepwise_graph_refine_inline_left:
"stepwise_graph_refine Gamma (fname # f1 # fns, f2 # fns')
(NextNode (entry_point gf3) # NextNode n # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (fname # f1 # fns, f2 # fns') (NextNode (entry_point gf3) # NextNode n # nns, nn' # nns') rel' orel otr (Suc N); Gamma f1 = Some gf1; Gamma f2 = Some gf2; function_graph gf1 n = Some (node.Call nn fname inputs outputs); function_graph gf2 n' = Som... | lemma stepwise_graph_refine_inline_left:
"stepwise_graph_refine Gamma (fname # f1 # fns, f2 # fns')
(NextNode (entry_point gf3) # NextNode n # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1... | unnamed_thy_30412 | GraphProof | 7 | |
[] | lemma stepwise_graph_refine_end_inline_left:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns')
(nn # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Call nn fname inputs outputs)
... | lemma stepwise_graph_refine_end_inline_left:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns')
(nn # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Call nn fname inputs outputs)
... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (nn # nns, nn' # nns') rel' orel otr (Suc N); Gamma f1 = Some gf1; Gamma f2 = Some gf2; function_graph gf1 n = Some (node.Call nn fname inputs outputs); function_graph gf2 n' = Some (node.Basic nn' upds); Gamma fname = Some g... | lemma stepwise_graph_refine_end_inline_left:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns')
(nn # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Call nn fname inputs outputs)
... | unnamed_thy_30413 | GraphProof | 7 | |
[] | lemma stepwise_graph_refine_inline_right:
"stepwise_graph_refine Gamma (f1 # fns, fname # f2 # fns')
(nn # nns, NextNode (entry_point gf3) # NextNode n' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf... | lemma stepwise_graph_refine_inline_right:
"stepwise_graph_refine Gamma (f1 # fns, fname # f2 # fns')
(nn # nns, NextNode (entry_point gf3) # NextNode n' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (f1 # fns, fname # f2 # fns') (nn # nns, NextNode (entry_point gf3) # NextNode n' # nns') rel' orel otr (Suc N); Gamma f1 = Some gf1; Gamma f2 = Some gf2; function_graph gf1 n = Some (node.Basic nn upds); function_graph gf2 n' = Some (node.Call nn... | lemma stepwise_graph_refine_inline_right:
"stepwise_graph_refine Gamma (f1 # fns, fname # f2 # fns')
(nn # nns, NextNode (entry_point gf3) # NextNode n' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf... | unnamed_thy_30414 | GraphProof | 7 | |
[] | lemma stepwise_graph_refine_end_inline_right:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns')
(nn # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Basic nn upds)
\<Longrigh... | lemma stepwise_graph_refine_end_inline_right:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns')
(nn # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Basic nn upds)
\<Longrigh... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>stepwise_graph_refine Gamma (f1 # fns, f2 # fns') (nn # nns, nn' # nns') rel' orel otr (Suc N); Gamma f1 = Some gf1; Gamma f2 = Some gf2; function_graph gf1 n = Some (node.Basic nn upds); function_graph gf2 n' = Some (node.Call nn' fname inputs outputs); Gamma fname = Some g... | lemma stepwise_graph_refine_end_inline_right:
"stepwise_graph_refine Gamma (f1 # fns, f2 # fns')
(nn # nns, nn' # nns')
rel' orel otr (Suc N)
\<Longrightarrow> Gamma f1 = Some gf1 \<Longrightarrow> Gamma f2 = Some gf2
\<Longrightarrow> function_graph gf1 n = Some (Basic nn upds)
\<Longrigh... | unnamed_thy_30415 | GraphProof | 7 | |
[] | lemma stepwise_graph_refine_induct:
assumes Suc: "\<And>n otr. stepwise_graph_refine Gamma fns nns rel orel otr (Suc n)
\<Longrightarrow> stepwise_graph_refine Gamma fns nns rel orel otr n"
shows "stepwise_graph_refine Gamma fns nns rel orel otr n" proof - have induct: "\<And>n m otr. n \<ge> m
\<Lo... | lemma stepwise_graph_refine_induct:
assumes Suc: "\<And>n otr. stepwise_graph_refine Gamma fns nns rel orel otr (Suc n)
\<Longrightarrow> stepwise_graph_refine Gamma fns nns rel orel otr n"
shows "stepwise_graph_refine Gamma fns nns rel orel otr n" proof - have induct: "\<And>n m otr. n \<ge> m
\<Lo... | proof (prove)
goal (1 subgoal):
1. stepwise_graph_refine Gamma fns nns rel orel otr n proof (state)
goal (1 subgoal):
1. stepwise_graph_refine Gamma fns nns rel orel otr n proof (prove)
goal (1 subgoal):
1. \<And>n m otr. \<lbrakk>m \<le> n; stepwise_graph_refine Gamma fns nns rel orel otr n\<rbrakk> \<Longrightarro... | lemma stepwise_graph_refine_induct:
assumes Suc: "\<And>n otr. stepwise_graph_refine Gamma fns nns rel orel otr (Suc n)
\<Longrightarrow> stepwise_graph_refine Gamma fns nns rel orel otr n"
shows "stepwise_graph_refine Gamma fns nns rel orel otr n" | unnamed_thy_30416 | GraphProof | 27 | |
[] | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" by (simp add: global_acc_valid_def) | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" by (simp add: global_acc_valid_def) | proof (prove)
goal (1 subgoal):
1. global_acc_valid t_hrs_' t_hrs_'_update | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" | unnamed_thy_30418 | global_asm_stmt_gref | 1 | |
[] | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<forall>x\<in>set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s' \<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs)) \<and> (\<forall>v gs. g' (s v gs) = g' gs) \<and> (\<forall>v gs. g (s' v gs) = g gs)); (\<forall>v v' gs. th_s ... | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | unnamed_thy_30419 | global_asm_stmt_gref | 9 | |
[] | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" apply (rule globals_swap_ex_swap) apply (simp only: globals_list_def global_data_defs list.sim... | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" apply (rule globals_swap_ex_swap) apply (simp only: globals_list_def global_data_defs list.sim... | proof (prove)
goal (1 subgoal):
1. phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs \<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs) proof (prove)
goal (2 subgoals):
1. \<forall>x\<in>set global_data_list. is_global_data x \<longrightarrow> (case x of Glo... | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" | unnamed_thy_30421 | global_asm_stmt_gref | 4 | |
[] | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: ghost'state_update_globals_swap) | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: ghost'state_update_globals_swap) | proof (prove)
goal (1 subgoal):
1. t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs) | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" | unnamed_thy_30422 | global_asm_stmt_gref | 1 | |
[] | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: phantom_machine_state_'_update_globals_swap) | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: phantom_machine_state_'_update_globals_swap) | proof (prove)
goal (1 subgoal):
1. t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs) | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" | unnamed_thy_30423 | global_asm_stmt_gref | 1 | |
[] | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" by (metis globals_swap_twice_helper globals_list_distinct
globals_list_valid global_acc_valid) | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" by (metis globals_swap_twice_helper globals_list_distinct
globals_list_valid global_acc_valid) | proof (prove)
goal (1 subgoal):
1. gswap (gswap gs) = gs | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" | unnamed_thy_30424 | global_asm_stmt_gref | 1 | |
[] | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" by (cases gs, auto simp add: hrs_mem_update_def hrs_mem_def) | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" by (cases gs, auto simp add: hrs_mem_update_def hrs_mem_def) | proof (prove)
goal (1 subgoal):
1. f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs) \<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" | unnamed_thy_30425 | global_asm_stmt_gref | 1 | |
[] | lemma globals_list_valid:
"globals_list_valid symbol_table t_hrs_' t_hrs_'_update globals_list" apply (rule globals_list_valid_optimisation[OF _ _ globals_list_ok]) apply (simp_all add: globals_list_def globals_list_valid_def
global_data_defs
del: distinct_prop.simps split del... | lemma globals_list_valid:
"globals_list_valid symbol_table t_hrs_' t_hrs_'_update globals_list" apply (rule globals_list_valid_optimisation[OF _ _ globals_list_ok]) apply (simp_all add: globals_list_def globals_list_valid_def
global_data_defs
del: distinct_prop.simps split del... | proof (prove)
goal (1 subgoal):
1. globals_list_valid symbol_table t_hrs_' t_hrs_'_update global_data_list proof (prove)
goal (2 subgoals):
1. distinct_prop global_data_swappable (filter is_global_data global_data_list)
2. \<forall>g\<in>set global_data_list. global_data_valid t_hrs_' t_hrs_'_update g proof (prove)
... | lemma globals_list_valid:
"globals_list_valid symbol_table t_hrs_' t_hrs_'_update globals_list" | unnamed_thy_30427 | inf_loop_gref | 6 | |
[] | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" by (simp add: global_acc_valid_def) | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" by (simp add: global_acc_valid_def) | proof (prove)
goal (1 subgoal):
1. global_acc_valid t_hrs_' t_hrs_'_update | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" | unnamed_thy_30428 | inf_loop_gref | 1 | |
[] | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<forall>x\<in>set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s' \<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs)) \<and> (\<forall>v gs. g' (s v gs) = g' gs) \<and> (\<forall>v gs. g (s' v gs) = g gs)); (\<forall>v v' gs. th_s ... | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | unnamed_thy_30429 | inf_loop_gref | 9 | |
[] | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" apply (rule globals_swap_ex_swap) apply (simp only: globals_list_def global_data_defs list.sim... | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" apply (rule globals_swap_ex_swap) apply (simp only: globals_list_def global_data_defs list.sim... | proof (prove)
goal (1 subgoal):
1. phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs \<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs) proof (prove)
goal (2 subgoals):
1. \<forall>x\<in>set global_data_list. is_global_data x \<longrightarrow> (case x of Glo... | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" | unnamed_thy_30431 | inf_loop_gref | 4 | |
[] | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: ghost'state_update_globals_swap) | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: ghost'state_update_globals_swap) | proof (prove)
goal (1 subgoal):
1. t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs) | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" | unnamed_thy_30432 | inf_loop_gref | 1 | |
[] | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: phantom_machine_state_'_update_globals_swap) | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: phantom_machine_state_'_update_globals_swap) | proof (prove)
goal (1 subgoal):
1. t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs) | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" | unnamed_thy_30433 | inf_loop_gref | 1 | |
[] | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" by (metis globals_swap_twice_helper globals_list_distinct
globals_list_valid global_acc_valid) | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" by (metis globals_swap_twice_helper globals_list_distinct
globals_list_valid global_acc_valid) | proof (prove)
goal (1 subgoal):
1. gswap (gswap gs) = gs | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" | unnamed_thy_30434 | inf_loop_gref | 1 | |
[] | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" by (cases gs, auto simp add: hrs_mem_update_def hrs_mem_def) | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" by (cases gs, auto simp add: hrs_mem_update_def hrs_mem_def) | proof (prove)
goal (1 subgoal):
1. f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs) \<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" | unnamed_thy_30435 | inf_loop_gref | 1 | |
[] | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" by (simp add: global_acc_valid_def) | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" by (simp add: global_acc_valid_def) | proof (prove)
goal (1 subgoal):
1. global_acc_valid t_hrs_' t_hrs_'_update | lemma global_acc_valid:
"global_acc_valid t_hrs_' t_hrs_'_update" | unnamed_thy_30438 | global_array_swap_gref | 1 | |
[] | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | proof (prove)
goal (1 subgoal):
1. \<lbrakk>\<forall>x\<in>set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s' \<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs)) \<and> (\<forall>v gs. g' (s v gs) = g' gs) \<and> (\<forall>v gs. g (s' v gs) = g gs)); (\<forall>v v' gs. th_s ... | lemma globals_swap_ex_swap:
"\<forall>x \<in> set gxs. is_global_data x \<longrightarrow> (case x of GlobalData nm sz tg g' s'
\<Rightarrow> (\<forall>v v' gs. s' v (s v' gs) = s v' (s' v gs))
\<and> (\<forall>v gs. g' (s v gs) = g' gs)
\<and> (\<forall>v gs. g (s' v gs) = g gs))
\... | unnamed_thy_30439 | global_array_swap_gref | 9 | |
[] | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" apply (rule globals_swap_ex_swap) apply (simp only: globals_list_def global_data_defs list.sim... | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" apply (rule globals_swap_ex_swap) apply (simp only: globals_list_def global_data_defs list.sim... | proof (prove)
goal (1 subgoal):
1. phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs \<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs) proof (prove)
goal (2 subgoals):
1. \<forall>x\<in>set global_data_list. is_global_data x \<longrightarrow> (case x of Glo... | lemma phantom_machine_state_'_update_globals_swap[simp]:
"phantom_machine_state_' (gswap gs) = phantom_machine_state_' gs
\<and> gswap (phantom_machine_state_'_update f gs) = phantom_machine_state_'_update f (gswap gs)" | unnamed_thy_30441 | global_array_swap_gref | 4 | |
[] | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: ghost'state_update_globals_swap) | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: ghost'state_update_globals_swap) | proof (prove)
goal (1 subgoal):
1. t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs) | lemma t_hrs_ghost'state_update_globals_swap[simp]:
"t_hrs_' (gswap (ghost'state_'_update f gs)) = t_hrs_' (gswap gs)" | unnamed_thy_30442 | global_array_swap_gref | 1 | |
[] | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: phantom_machine_state_'_update_globals_swap) | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" by (simp add: phantom_machine_state_'_update_globals_swap) | proof (prove)
goal (1 subgoal):
1. t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs) | lemma t_hrs_phantom_machine_state_'_update_globals_swap[simp]:
"t_hrs_' (gswap (phantom_machine_state_'_update f gs)) = t_hrs_' (gswap gs)" | unnamed_thy_30443 | global_array_swap_gref | 1 | |
[] | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" by (metis globals_swap_twice_helper globals_list_distinct
globals_list_valid global_acc_valid) | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" by (metis globals_swap_twice_helper globals_list_distinct
globals_list_valid global_acc_valid) | proof (prove)
goal (1 subgoal):
1. gswap (gswap gs) = gs | lemma globals_swap_twice[simp]:
"gswap (gswap gs) = gs" | unnamed_thy_30444 | global_array_swap_gref | 1 | |
[] | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" by (cases gs, auto simp add: hrs_mem_update_def hrs_mem_def) | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" by (cases gs, auto simp add: hrs_mem_update_def hrs_mem_def) | proof (prove)
goal (1 subgoal):
1. f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs) \<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs | lemma t_hrs_'_update_hmu_triv[simp]:
"f (hrs_mem (t_hrs_' gs)) = hrs_mem (t_hrs_' gs)
\<Longrightarrow> t_hrs_'_update (hrs_mem_update f) gs = gs" | unnamed_thy_30445 | global_array_swap_gref | 1 |
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