instruct stringclasses 1
value | input stringlengths 25 4.64k | output stringlengths 17 413 |
|---|---|---|
Given a lemma, please prove it. | lemma dither_halts:
shows "{dither_halt_inv} dither {dither_halt_inv}" | apply(rule Hoare_haltI)
using dither_halts_aux
apply(auto simp add: tape_of_nat_def)
by (metis (lifting, mono_tags) holds_for.simps is_final_eq) |
Given a lemma, please prove it. | lemma less_oSuc_eq_le: "(x < oSuc y) = (x \<le> y)" | apply (rule iffI)
apply (erule contrapos_pp, simp add: linorder_not_less linorder_not_le)
apply (erule oSuc_leI)
apply (erule order_le_less_trans[OF _ less_oSuc]) done |
Given a lemma, please prove it. | lemma ThrowRule: "P \<subseteq> A \<Longrightarrow> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> (AnnExpr P) Throw Q, A" | apply (rule Conseq, simp)
apply (rule exI, rule exI, rule exI)
apply (rule conjI, rule Throw, auto) done |
Given a lemma, please prove it. | lemma checkNet2_correct: "checkNet2 c s t \<le> checkNet c s t" | apply (rule refine_IdD) unfolding checkNet_def checkNet2_def graph_of_def rev_graph_of_def reachable_spec_def reaching_spec_def
apply (refine_rcg)
apply refine_dref_type
apply auto done |
Given a lemma, please prove it. | lemma (in normal_set) mem_next: "(LEAST z. z \<in> A \<and> x < z) \<in> A" | apply (rule LeastI2_ex)
apply (fold Bex_def, rule unbounded)
apply (erule conjunct1) done |
Given a lemma, please prove it. | lemma skl1_inv5_trans [iff]:
"{skl1_inv5} trans skl1 {> skl1_inv5}" | apply (auto simp add: PO_hoare_defs intro!: skl1_inv5I)
apply (drule can_signal_trans, assumption)
apply (auto simp add: skl1_nostep_defs)
apply (auto simp add: skl1_defs ik_dy_def dest: skl1_inv5D) done |
Given a lemma, please prove it. | lemma WT_fv: "P,E \<turnstile> e :: T \<Longrightarrow> fv e \<subseteq> dom E"
and "P,E \<turnstile> es [::] Ts \<Longrightarrow> fvs es \<subseteq> dom E" | apply(induct rule:WT_WTs.inducts)
apply(simp_all del: fun_upd_apply)
apply fast+ done |
Given a lemma, please prove it. | lemma edka_partial_refine[refine]: "edka_partial \<le> \<Down>Id fofu" | unfolding edka_partial_def fofu_def
apply (refine_rcg bind_refine')
apply (refine_dref_type)
apply (vc_solve simp: find_shortest_augmenting_spec_def) done |
Given a lemma, please prove it. | lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}" | apply (simp add: In1_def)
apply (simp add: Scons_def) done |
Given a lemma, please prove it. | lemma is_ms_array_list_prec[safe_constraint_rules]: "precise (is_ms_array_list ms)" | unfolding is_ms_array_list_def[abs_def]
apply(rule preciseI)
apply(simp split: prod.splits)
using preciseD snga_prec
by fastforce |
Given a lemma, please prove it. | lemma IV: "Mutex \<in> Always IV" | apply (rule AlwaysI, force)
apply (unfold Mutex_def, safety) done |
Given a lemma, please prove it. | theorem HuntSands33: "(p,G,c,H):HS \<Longrightarrow> secure p G c H" | apply (simp add: secure_def, clarsimp)
apply (drule Theorem4, clarsimp)
apply (rule Prop4A)
apply (rule VDM_Sound_emptyCtxt)
apply fast done |
Given a lemma, please prove it. | lemma wf_subcls1_E: "wf ((subcls1 E)\<inverse>)" | apply (rule finite_acyclic_wf_converse)
apply (simp add: subcls1 del: insert_iff)
apply (rule acyclic_subcls1_E) done |
Given a lemma, please prove it. | lemma Highest_escapes_above: "system \<in> Highest i leadsTo (\<Inter>k. {s. i\<notin>above k s})" | apply (rule leadsTo_weaken_R)
apply (rule_tac [2] Lowest_above_subset)
apply (rule Highest_leadsTo_Lowest) done |
Given a lemma, please prove it. | lemma nd_fun_distr[nd_fun_ka]: "(x + y) \<cdot> z = x \<cdot> z + y \<cdot> z"
and nd_fun_distl[nd_fun_ka]: "x \<cdot> (y + z) = x \<cdot> y + x \<cdot> z" for x::"'a nd_fun" | unfolding plus_nd_fun_def times_nd_fun_def
by (simp_all add: kcomp_distr kcomp_distl) |
Given a lemma, please prove it. | lemma star_test_folk: "test p \<Longrightarrow> p \<cdot> x = x \<cdot> p \<longrightarrow> p \<cdot> y = y \<cdot> p \<longrightarrow> (p \<cdot> x + !p \<cdot> y)\<^sup>\<star> \<cdot> p = p \<cdot> (p \<cdot> x)\<^sup>\<star>" | using star_n_folk
by auto |
Given a lemma, please prove it. | lemma runErrorT_bindET [simp]:
"runErrorT\<cdot>(bindET\<cdot>m\<cdot>k) = bind\<cdot>(runErrorT\<cdot>m)\<cdot>
(\<Lambda> n. case n of Err\<cdot>e \<Rightarrow> return\<cdot>(Err\<cdot>e) | Ok\<cdot>x \<Rightarrow> runErrorT\<cdot>(k\<cdot>x))" | unfolding bindET_def
by simp |
Given a lemma, please prove it. | lemma mult_zero_right [simp]: "a \<cdot> 0 = (0::'a)" | proof - have "a \<cdot> 0 = a \<cdot> (0 - (0::'a))"
by simp also have "\<dots> = a \<cdot> 0 - a \<cdot> 0"
by (rule diff_mult_distrib1) simp_all also have "\<dots> = 0"
by simp finally show ?thesis .
qed |
Given a lemma, please prove it. | lemma phl_test_while:
assumes "test p" and "test b"
and "\<lbrace>p \<cdot> b\<rbrace> x \<lbrace>p\<rbrace>"
shows "\<lbrace>p\<rbrace> (b \<cdot> x)\<^sup>\<star> \<cdot> !b \<lbrace>p \<cdot> !b\<rbrace>" | by (metis assms phl_n_while test_double_comp_var) |
Given a lemma, please prove it. | lemma slength_0_conv[simp]:
"(slength\<cdot>xs = 0) \<longleftrightarrow> (xs = [::])"
"(slength\<cdot>xs = MkI\<cdot>0) \<longleftrightarrow> (xs = [::])"
"eq\<cdot>0\<cdot>(slength\<cdot>xs) = sNone\<cdot>xs"
"eq\<cdot>(slength\<cdot>xs)\<cdot>0 = sNone\<cdot>xs" | by (induct xs) (auto simp: one_Integer_def elim: slengthE) |
Given a lemma, please prove it. | lemma compare_EQ_iff_eq_TT:
"compare\<cdot>x\<cdot>y = EQ \<longleftrightarrow> eq\<cdot>x\<cdot>y = TT" | by (cases "compare\<cdot>x\<cdot>y") (simp add: is_EQ_def eq_conv_compare)+ |
Given a lemma, please prove it. | lemma carrier_Fheap':"carrier (Theap \<Gamma> e\<cdot>a) = edom (Aheap \<Gamma> e\<cdot>a)" | unfolding Theap_simp carrier_ccTTree
by simp |
Given a lemma, please prove it. | lemma runStateT_bindU [simp]:
"runStateT\<cdot>(bindU\<cdot>m\<cdot>k)\<cdot>s = runStateT\<cdot>m\<cdot>s \<bind> (\<Lambda> (x, s'). runStateT\<cdot>(k\<cdot>x)\<cdot>s')" | unfolding bindU_stateT_def
by simp |
Given a lemma, please prove it. | lemma conjugate_eq_aux:
"is_conjugation f g \<Longrightarrow> f (x \<cdot> -g y) \<le> f x \<cdot> -y" | by (metis f_subdist galois_aux le_inf_iff modular_1_aux) |
Given a lemma, please prove it. | lemma wagner_2_var: "x \<cdot> (y \<cdot> x)\<^sup>\<omega> \<le> (x \<cdot> y)\<^sup>\<omega>" | proof - have "x \<cdot> y \<cdot> x \<le> x \<cdot> y \<cdot> x"
by auto thus "x \<cdot> (y \<cdot> x)\<^sup>\<omega> \<le> (x \<cdot> y)\<^sup>\<omega>"
by (simp add: mult_assoc omega_simulation)
qed |
Given a lemma, please prove it. | lemma "R \<cdot> (S \<union> T) = R \<cdot> S \<union> R \<cdot> T" | apply (auto simp: s_prod_im) nitpick oops |
Given a lemma, please prove it. | lemma meet_interchange: "(w \<cdot> x) ; (y \<cdot> z) \<le> w ; y \<cdot> x ; z" | by (metis inf_le1 inf_le2 le_infI mult_isol_var) |
Given a lemma, please prove it. | lemma n_dist_var2: "n (n x \<cdot> n y \<oplus> t x \<cdot> n z) = n x \<cdot> t y \<oplus> t x \<cdot> t z" | by (metis (no_types) n_add_op_def n_dist_var1 n_t_closed) |
Given a lemma, please prove it. | lemma "matches\<cdot>[::]\<cdot>[:10::Integer, 20, 30:] = [:0, 1, 2, 3:]" | unfolding matches.unfold endswith.unfold
by simp |
Given a lemma, please prove it. | lemma
fixes t :: "['a::Eq_sym]"
shows "eq\<cdot>(take\<cdot>(length\<cdot>t)\<cdot>s)\<cdot>t \<sqsubseteq> isPrefixOf\<cdot>t\<cdot>s" | by (subst eq_sym) (rule eq_take_length_isPrefixOf) |
Given a lemma, please prove it. | lemma unstream_concatMapS:
"unstream\<cdot>(concatMapS\<cdot>f\<cdot>a) = concatMapL\<cdot>(unstream oo f)\<cdot>(unstream\<cdot>a)" | by (cases a, simp, simp add: unfold_concatMapStep) |
Given a lemma, please prove it. | lemma sup_distl: "x \<cdot> (y \<squnion> z) = (x \<cdot> y) \<squnion> (x \<cdot> z)" | using weak_Sup_distl[where Y="{y, z}"]
by (fastforce intro!: Sup_eqI) |
Given a lemma, please prove it. | lemma eq_tr_simps [simp]:
shows "eq\<cdot>TT\<cdot>TT = TT" and "eq\<cdot>TT\<cdot>FF = FF"
and "eq\<cdot>FF\<cdot>TT = FF" and "eq\<cdot>FF\<cdot>FF = TT" | unfolding TT_def FF_def eq_lift_def
by simp_all |
Given a lemma, please prove it. | lemma conjugation_prop1: "is_conjugation f g \<Longrightarrow> f y \<cdot> z \<le> f (y \<cdot> g z)" | by (metis le_infE modular_1 order_refl) |
Given a lemma, please prove it. | lemma fac_work_final_body3_eq: "fac_work_final = wrapA\<cdot>(fix\<cdot>fac_acc_body3)" | unfolding fac_work_final_def
by (rule worker_wrapper_fusion_new[OF wrapA_unwrapA_id unwrapA_strict]) (simp add: fac_acc_body3_body2 fac_acc_body2_body1_eq fac_acc_body1_fac_body_final_eq) |
Given a lemma, please prove it. | lemma conv_self_conjugate: "x\<^sup>\<smile> \<cdot> y = 0 \<longleftrightarrow> x \<cdot> y\<^sup>\<smile> = 0" | by (metis conv_invol conv_times conv_zero) |
Given a lemma, please prove it. | lemma S4: "x\<cdot>(y\<cdot>z) \<cong> (x\<cdot>y)\<cdot>z" | by (metis C\<^sub>0s C\<^sub>1s C\<^sub>1's C\<^sub>2s) |
Given a lemma, please prove it. | lemma distr: "(x \<squnion> y) \<cdot> z = x \<cdot> z \<squnion> y \<cdot> z" | by (metis local.residuated_multl local.residuated_sup) |
Given a lemma, please prove it. | lemma assoc_p_subid: "(x \<cdot> y) \<cdot> (z \<cdot> 1\<^sub>\<pi>) = x \<cdot> (y \<cdot> (z \<cdot> 1\<^sub>\<pi>))" | by (metis c_0 local.c6 local.cl5 local.inf.absorb_iff1) |
Given a lemma, please prove it. | lemma test_kat_prop: "test p \<Longrightarrow> test q \<Longrightarrow> p \<cdot> x = p \<cdot> x \<cdot> q \<longleftrightarrow> x \<cdot> !q = !p \<cdot> x \<cdot> !q" | by (metis local.annir local.test1_var local.test_comp_mult2 local.test_eq4 mult_assoc) |
Given a lemma, please prove it. | lemma kat_3': "d x \<cdot> y \<cdot> ad z = 0 \<Longrightarrow> d x \<cdot> y = d x \<cdot> y \<cdot> d z" | by (metis dm.dom_one local.am5 local.mult_1_left a_zero_def am_d_def) |
Given a lemma, please prove it. | lemma sfilter_sappend[simp]: "sfilter\<cdot>p\<cdot>(xs :@ ys) = sfilter\<cdot>p\<cdot>xs :@ sfilter\<cdot>p\<cdot>ys" | by (cases "ys"; clarsimp) (induct xs; fastforce simp: If2_def[symmetric] split: If2_splits) |
Given a lemma, please prove it. | lemma d_restrict_iff_1: "(d x \<cdot> y \<le> z) \<longleftrightarrow> (d x \<cdot> y \<le> d x \<cdot> z)" | by (metis dom_subid_aux2 domain_1'' domain_invol dsg1 mult_isol_var order_trans) |
Given a lemma, please prove it. | lemma stake_all_triv[simp]: "stake\<cdot>(slength\<cdot>xs)\<cdot>xs = xs" | by (cases "xs = \<bottom>") (auto simp: stake_all) |
Given a lemma, please prove it. | lemma sset_smap[simp]:
assumes "\<And>x. x \<in> sset xs \<Longrightarrow> f\<cdot>x \<noteq> \<bottom>"
shows "sset (smap\<cdot>f\<cdot>xs) = { f\<cdot>x | x. x \<in> sset xs }" | using assms
by (induct xs) auto |
Given a lemma, please prove it. | lemma "\<not> fix\<cdot>(unwrap oo body oo wrap) \<sqsubseteq> fix\<cdot>body'" | proof - have l: "fix\<cdot>(unwrap oo body oo wrap) = B\<cdot>A"
by (subst fix_eq) simp have r: "fix\<cdot>body' = \<bottom>"
by (simp add: fix_strict) from l r show ?thesis
by simp
qed |
Given a lemma, please prove it. | lemma iteration_unfoldr_distl [simp]: " y + y \<cdot> x\<^sup>\<infinity> \<cdot> x = y \<cdot> x\<^sup>\<infinity>" | by (metis distrib_left mult.assoc mult_oner iteration_unfoldr) |
Given a lemma, please prove it. | lemma conv_contrav': "1 \<lhd> x\<cdot>y = (1 \<lhd> y)\<cdot>(1 \<lhd> x)" | by (metis local.conv_contrav local.conv_invol local.jonsson3c) |
Given a lemma, please prove it. | lemma alw_mp:
assumes "alw \<phi> xs" and "alw (\<phi> impl \<psi>) xs"
shows "alw \<psi> xs" | proof- { assume "alw \<phi> xs \<and> alw (\<phi> impl \<psi>) xs" hence ?thesis
by coinduct auto } thus ?thesis
using assms
by auto
qed |
Given a lemma, please prove it. | lemma SPEC_rule_conj_leofI1:
assumes "m \<le> SPEC \<Phi>"
assumes "m \<le>\<^sub>n SPEC \<Psi>"
shows "m \<le> SPEC (\<lambda>s. \<Phi> s \<and> \<Psi> s)" | using assms
by (auto simp: pw_le_iff pw_leof_iff) |
Given a lemma, please prove it. | theorem goedel_first_strong_ex:
"\<omega>consistent \<Longrightarrow> \<exists> \<phi>. \<phi> \<in> fmla \<and> \<not> prv \<phi> \<and> \<not> prv (neg \<phi>) \<and> isTrue \<phi>" | using goedel_first_strong
by (intro exI[of _ \<phi>G]) blast |
Given a lemma, please prove it. | lemma af_\<mu>LTL:
"\<phi> \<in> \<mu>LTL \<Longrightarrow> w \<Turnstile>\<^sub>n \<phi> \<longleftrightarrow> (\<exists>i. af \<phi> (prefix i w) \<sim> True\<^sub>n)" | using valid_prefix_implies_ltl \<mu>LTL_implies_valid_prefix
by blast |
Given a lemma, please prove it. | lemma sas_plus_problem_has_serial_solution_iff_i':
assumes "is_valid_problem_sas_plus \<Psi>"
and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> (\<phi> (prob_with_noop \<Psi>)) t"
shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi>
(rem_noops
(map (\<lambda>op. \<phi>\<^su... | using assms noops_valid
by(force intro!: noops_sound sas_plus_problem_has_serial_solution_iff_i) |
Given a lemma, please prove it. | lemma
neg[simp]: "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> neg \<phi> \<in> fmla"
and
Fvars_neg[simp]: "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> Fvars (neg \<phi>) = Fvars \<phi>"
and
subst_neg[simp]:
"\<And>\<phi> t x. \<phi> \<in> fmla \<Longrightarrow> t \<in> trm \<Longrightarrow> x \<... | unfolding neg_def
by auto |
Given a lemma, please prove it. | lemma fbb_ifbb_galois: "Inf_pres \<phi> \<Longrightarrow> (\<phi> = bb\<^sub>\<F> f) = (f = bb\<^sup>-\<^sub>\<F> \<phi>)" | unfolding fbb_def iffb_def
by (metis (no_types, lifting) comp_apply ffb_iffb_galois ifbb_ifbd_demorgan iffb_ifbd_dual kop_galois) |
Given a lemma, please prove it. | lemma \<F>\<G>_semantics:
"\<F>\<G> \<phi> w = {\<psi> \<in> subformulas\<^sub>\<nu> \<phi>. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>)}" | by (induction \<phi>) force+ |
Given a lemma, please prove it. | lemma prv_eql_subst_trm3:
"x \<in> var \<Longrightarrow> \<phi> \<in> fmla \<Longrightarrow> t1 \<in> trm \<Longrightarrow> t2 \<in> trm \<Longrightarrow>
prv (eql t1 t2) \<Longrightarrow> prv (subst \<phi> t1 x) \<Longrightarrow> prv (subst \<phi> t2 x)" | using prv_eql_subst_trm2
by (meson subst prv_imp_mp) |
Given a lemma, please prove it. | lemma RETURN_ref_SPECD:
assumes "RETURN c \<le> \<Down>R (SPEC \<Phi>)"
obtains a where "(c,a)\<in>R" "\<Phi> a" | using assms
by (auto simp: pw_le_iff refine_pw_simps) |
Given a lemma, please prove it. | lemma nprv_ballE:
"nprv F (ball x t \<phi>) \<Longrightarrow> nprv F (LLq t1 t) \<Longrightarrow> nprv (insert (subst \<phi> t1 x) F) \<psi> \<Longrightarrow>
F \<subseteq> fmla \<Longrightarrow> finite F \<Longrightarrow> \<phi> \<in> fmla \<Longrightarrow> t \<in> atrm \<Longrightarrow> t1 \<in> atrm \<Longrightarro... | by (meson atrm_trm local.subst nprv_ballE_aux nprv_cut rev_subsetD) |
Given a lemma, please prove it. | lemma HBL1_iff: "\<phi> \<in> fmla \<Longrightarrow> Fvars \<phi> = {} \<Longrightarrow> bprv (PP \<langle>\<phi>\<rangle>) \<longleftrightarrow> prv \<phi>" | using HBL1 HBL1_rev unfolding PP_def
by auto |
Given a lemma, please prove it. | lemma GF_advice_restricted_subformulas:
"restricted_subformulas (\<phi>[X]\<^sub>\<nu>) = {}" | by (induction \<phi>) (simp_all add: GF_advice_restricted_subformulas_inner) |
Given a lemma, please prove it. | lemma GF_advice_subst:
"\<phi>[X]\<^sub>\<nu> = subst \<phi> (\<lambda>\<psi>. Some (\<psi>[X]\<^sub>\<nu>))" | by (induction \<phi>) auto |
Given a lemma, please prove it. | lemma \<phi>J[simp]:
assumes "\<psi> \<in> fmla" and "Fvars \<psi> = {inp}"
shows "\<phi>J \<psi> \<in> fmla" | unfolding \<phi>J_def
using assms
by (intro instInpP_fmla) auto |
Given a lemma, please prove it. | lemma af_letter_lang_congruent:
"\<phi> \<sim>\<^sub>L \<psi> \<Longrightarrow> af_letter \<phi> \<nu> \<sim>\<^sub>L af_letter \<psi> \<nu>" | unfolding ltl_lang_equiv_def
using af_letter_build
by blast |
Given a lemma, please prove it. | theorem Tarski_semantic:
assumes 0: "T \<in> fmla" "Fvars T = {xx}"
and 1: "\<And>\<phi>. \<phi> \<in> fmla \<Longrightarrow> Fvars \<phi> = {} \<Longrightarrow> isTrue (eqv (subst T \<langle>\<phi>\<rangle> xx) \<phi>)"
shows "\<not> consistent" | using assms prv_eq_isTrue[of "eqv (subst T \<langle>_\<rangle> xx) _"]
by (intro Tarski_proof_theoretic[OF 0]) auto |
Given a lemma, please prove it. | lemma "iADDI(\<phi>) \<Longrightarrow> supremum_closed (fp \<phi>)" | by (metis (full_types) Ra_restr_ex iADDI_def supremum_def) |
Given a lemma, please prove it. | lemma ltl_implies_satisfiable_prefix:
"w \<Turnstile>\<^sub>n \<phi> \<Longrightarrow> \<not> (af \<phi> (prefix i w) \<sim> False\<^sub>n)" | using ltl_implies_satisfiable_prefix eq_implies_lang
by blast |
Given a lemma, please prove it. | lemma nprv_bexiI:
"nprv F (subst \<phi> t1 x) \<Longrightarrow> nprv F (LLq t1 t) \<Longrightarrow>
F \<subseteq> fmla \<Longrightarrow> finite F \<Longrightarrow> \<phi> \<in> fmla \<Longrightarrow> t \<in> atrm \<Longrightarrow> t1 \<in> atrm \<Longrightarrow> x \<in> var \<Longrightarrow>
x \<notin> FvarsT t \<Lon... | unfolding bexi_def
by (nprover2 r1: nprv_exiI[of _ _ t1] r2: nprv_cnjI) |
Given a lemma, please prove it. | lemma convert_multiway_remove_neg: "safe_formula (remove_neg \<phi>) \<Longrightarrow> convert_multiway (remove_neg \<phi>) = remove_neg (convert_multiway \<phi>)" | by (cases \<phi>) (auto elim: case_NegE) |
Given a lemma, please prove it. | lemma prv_imp_triv:
assumes phi: "\<phi> \<in> fmla" and psi: "\<psi> \<in> fmla"
shows "prv \<psi> \<Longrightarrow> prv (imp \<phi> \<psi>)" | by (meson prv_imp_imp_triv prv_imp_mp imp phi psi) |
Given a lemma, please prove it. | lemma consistent_not_prv_not_prv_PPf:
assumes c: consistent
and 0[simp]: "\<phi> \<in> fmla" "Fvars \<phi> = {}" "\<not> prv \<phi>" "n \<in> num"
shows "\<not> bprv (PPf n \<langle>\<phi>\<rangle>)" | using not_prv_prv_neg_PPf[OF 0] c[THEN dwf_dwfd.consistent_B_consistent] unfolding B.consistent_def3
by auto |
Given a lemma, please prove it. | lemma wf_rexp_of: "wf_formula n \<phi> \<Longrightarrow> wf n (rexp_of n \<phi>)" | by (induct \<phi> arbitrary: n) (auto intro: wf_rexp_ENC simp: finite_FOV max_idx_vars) |
Given a lemma, please prove it. | lemma nfv_simps[simp]:
"nfv (Let p \<phi> \<psi>) = nfv \<psi>"
"nfv (Neg \<phi>) = nfv \<phi>"
"nfv (Or \<phi> \<psi>) = max (nfv \<phi>) (nfv \<psi>)"
"nfv (And \<phi> \<psi>) = max (nfv \<phi>) (nfv \<psi>)"
"nfv (Prev I \<phi>) = nfv \<phi>"
"nfv (Next I \<phi>) = nfv \<phi>"
"nfv (Since \<phi> I \<ps... | unfolding nfv_def Regex.nfv_regex_def
by (simp_all add: image_Un Max_Un[symmetric]) |
Given a lemma, please prove it. | lemma prv_rawpsubst:
assumes "\<phi> \<in> fmla" and "snd ` (set txs) \<subseteq> var" and "fst ` (set txs) \<subseteq> trm"
and "prv \<phi>"
shows "prv (rawpsubst \<phi> txs)" | using assms
apply (induct txs arbitrary: \<phi>) subgoal
by auto subgoal for tx txs \<phi>
by (cases tx) (auto intro: prv_subst) . |
Given a lemma, please prove it. | lemma create_name_gba_aimpl_refine:
"create_name_gba_aimpl \<phi> \<le> \<Down>Id (create_name_gba \<phi>)" | unfolding create_name_gba_aimpl_def create_name_gba_alt
apply (refine_rcg create_graph_aimpl_refine cr_rename_gba_refine)
by auto |
Given a lemma, please prove it. | lemma PI8: "Int_1a \<phi> \<Longrightarrow> Int_2 \<phi> \<Longrightarrow> Int_4 \<phi> \<Longrightarrow> Int_8 \<phi>" | using ADDI_b_def IDEM_def Int_8_def MONO_ADDIb MONO_MULTa dEXP_def join_def
by auto |
Given a lemma, please prove it. | lemma PI9: "Int_1a \<phi> \<Longrightarrow> Int_4 \<phi> \<Longrightarrow> Int_9 \<phi>" | using IDEM_def Int_9_def MONO_def MONO_MULTa
by simp |
Given a lemma, please prove it. | lemma ltl_prop_equiv_subst_S:
"S \<Turnstile>\<^sub>P subst \<phi> m = ((S - dom m) \<union> {\<chi> | \<chi> \<chi>'. \<chi> \<in> dom m \<and> m \<chi> = Some \<chi>' \<and> S \<Turnstile>\<^sub>P \<chi>'}) \<Turnstile>\<^sub>P \<phi>" | by (induction \<phi>) (auto split: option.split) |
Given a lemma, please prove it. | lemma af_lifted_range:
"range (\<up>af (Abs \<phi>)) \<subseteq> {Abs \<psi> | \<psi>. nested_prop_atoms \<psi> \<subseteq> nested_prop_atoms \<phi>}" | using af_lifted_semantics af_nested_prop_atoms
by blast |
Given a lemma, please prove it. | lemma CI5: "Cl_5 \<phi> \<Longrightarrow> Int_5 \<phi>\<^sup>d" | unfolding dNOR_def NOR_def dual_def conn
by simp |
Given a lemma, please prove it. | lemma flip_subst_fmla: "atom y \<sharp> \<phi> \<Longrightarrow> (x \<leftrightarrow> y) \<bullet> \<phi> = \<phi>(x::=Var y)" | apply(nominal_induct \<phi> avoiding: x y rule: fmla.strong_induct)
apply (auto simp: flip_subst_trm)
using fresh_at_base(2)
by blast |
Given a lemma, please prove it. | lemma NN_unique:
assumes "\<phi> \<in> fmla" "Fvars \<phi> = {}"
shows "bprv (all yy (all yy'
(imp (cnj (NN \<langle>\<phi>\<rangle> (Var yy)) (NN \<langle>\<phi>\<rangle> (Var yy')))
(eql (Var yy) (Var yy')))))" | using assms N_unique unfolding Let_def NN_def
by meson |
Given a lemma, please prove it. | lemma progress_remove_neg[simp]: "progress \<sigma> P (remove_neg \<phi>) j = progress \<sigma> P \<phi> j" | by (cases \<phi>) simp_all |
Given a lemma, please prove it. | lemma nf_ACI_juncts:
"\<lbrakk>\<psi> \<in> disjuncts \<phi>; nf_ACI \<phi>\<rbrakk> \<Longrightarrow> nf_ACI \<psi>"
"\<lbrakk>\<psi> \<in> conjuncts \<phi>; nf_ACI \<phi>\<rbrakk> \<Longrightarrow> nf_ACI \<psi>" | by (induct \<phi>) auto |
Given a lemma, please prove it. | lemma lformula_nFAND: "lformula (nFAND \<phi>s) = (\<forall>\<phi> \<in> set \<phi>s. lformula \<phi>)" | by (induct \<phi>s rule: nFAND.induct) auto |
Given a lemma, please prove it. | lemma af_const_congruent:
"\<phi> \<sim>\<^sub>C \<psi> \<Longrightarrow> af \<phi> w \<sim>\<^sub>C af \<psi> w" | by (induction w arbitrary: \<phi> \<psi>) (insert af_letter_const_congruent, fastforce+) |
Given a lemma, please prove it. | lemma satisfies_bounded_fold_rderiv:
"\<lbrakk>wf idx \<phi>; #\<^sub>V \<AA> = idx; size x = idx\<rbrakk> \<Longrightarrow>
\<AA> \<Turnstile>\<^sub>b fold rderiv (replicate k x) \<phi> \<longleftrightarrow> (SNOC x^^k) \<AA> \<Turnstile>\<^sub>b \<phi>" | by (induct k arbitrary: \<AA> \<phi>) (auto simp: satisfies_bounded_rderiv wf_rderiv nvars_SNOCs) |
Given a lemma, please prove it. | lemma fold_graph_OR: "finite \<Phi> \<Longrightarrow> fold_graph Or_ltlp False_ltlp \<Phi> (OR \<Phi>)" | unfolding OR_def
by (rule someI2_ex[OF finite_imp_fold_graph]) |
Given a lemma, please prove it. | lemma fold_deriv_FNot:
"fold (deriv d0) xs (FNot \<phi>) = FNot (fold (deriv d0) xs \<phi>)" | by (induct xs arbitrary: \<phi>) auto |
Given a lemma, please prove it. | lemma val_imp_close_world: "valuation M \<Turnstile> \<phi> \<Longrightarrow> M \<^sup>c\<TTurnstile>\<^sub>= \<phi>" | unfolding entailment_def
using valuation_aux_1
by blast |
Given a lemma, please prove it. | lemma satPB_gPB[simp]:
assumes "Ik.wtE \<xi>" and "GE.satPB \<xi> (gC ` \<Phi>)"
shows "Ik.satPB \<xi> \<Phi>" | using Ik.wt_\<Phi> assms unfolding GE.satPB_def Ik.satPB_def
by (auto simp add: Ik.wtPB_def) |
Given a lemma, please prove it. | lemma FIXvarFIX: "(PhiWhile b) = (\<lambda> \<Phi> . (\<lambda> (s,t,\<beta>) . (b,\<Phi>,\<beta>,s,t):var))" | apply rule
apply (rule FIXvarFIX') done |
Given a lemma, please prove it. | lemma FI3: "Int_3 \<phi> \<Longrightarrow> Fr_3(\<F>\<^sub>I \<phi>)" | using NOR_def Fr_int_def IC3 unfolding conn
by auto |
Given a lemma, please prove it. | lemma af_to_af_opt:
"Unf (af \<phi> w) = af\<^sub>\<UU> (Unf \<phi>) w"
"Unf\<^sub>G (af\<^sub>G \<phi> w) = af\<^sub>G\<^sub>\<UU> (Unf\<^sub>G \<phi>) w" | by (induction w arbitrary: \<phi>) (simp_all add: af_letter_alt_def) |
Given a lemma, please prove it. | lemma alw_impl_ev_alw:
assumes "alw (\<phi> impl ev \<psi>) xs"
shows "alw (ev \<phi> impl ev \<psi>) xs" | using assms
by coinduct (auto dest: ev_alw_impl) |
Given a lemma, please prove it. | lemma prv_\<phi>G_eqv_all_not_PPf:
"prv (eqv \<phi>G (all yy (neg (PPf (Var yy) \<langle>\<phi>G\<rangle>))))" | using bprv_prv[OF _ _ bprv_\<phi>G_eqv_all_not_PPf, simplified] . |
Given a lemma, please prove it. | lemma ASSUME_leof_rule[refine_vcg]:
assumes "\<Phi> \<Longrightarrow> \<Psi> ()"
shows "ASSUME \<Phi> \<le>\<^sub>n SPEC \<Psi>" | using assms
by (auto simp: ASSUME_leof_iff) |
Given a lemma, please prove it. | lemma ev_introR[intro]:
assumes "\<phi> \<pi>l" shows "ev \<phi> \<pi>l" | using assms unfolding ev_def
by (auto intro: until_introR) |
Given a lemma, please prove it. | lemma minit_safe_minit: "mmonitorable \<phi> \<Longrightarrow> minit_safe \<phi> = minit \<phi>" | unfolding minit_safe_def monitorable_formula_code
by simp |
Given a lemma, please prove it. | lemma FG_advice_b2:
"\<lbrakk>Y \<subseteq> \<G> \<phi> w; w \<Turnstile>\<^sub>n \<phi>[Y]\<^sub>\<mu>\<rbrakk> \<Longrightarrow> w \<Turnstile>\<^sub>n \<phi>" | by (metis FG_advice_b2_helper \<G>_elim subset_eq) |
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