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lemma InvariantsNoDecisionsWhenConflictNorUnitAfterExhaustivePropagate:
assumes
"exhaustiveUnitPropagate_dom state"
"InvariantConsistent (getM state)"
"InvariantUniq (getM state)"
"InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)" and
"InvariantWatchListsUniq (getWatchList state)" and
"InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"
"InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)" and
"InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
"InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"
"InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"
"InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"
"InvariantUniqQ (getQ state)"
"InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"
"InvariantNoDecisionsWhenUnit (getF state) (getM state) (currentLevel (getM state))"
shows
"let state' = exhaustiveUnitPropagate state in
InvariantNoDecisionsWhenConflict (getF state') (getM state') (currentLevel (getM state')) \<and>
InvariantNoDecisionsWhenUnit (getF state') (getM state') (currentLevel (getM state'))" |
lemma prefix_trans: "!!xs::'a list. [| xs <= ys; ys <= zs |] ==> xs <= zs" |
lemma simulation_silents1:
assumes bisim: "s1 \<approx> s2" and moves: "s1 -\<tau>1\<rightarrow>* s1'"
shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> s1' \<approx> s2'" |
lemma trancl_union_outside:
assumes "(v,w) \<in> (E\<union>U)\<^sup>+"
and "(v,w) \<notin> E\<^sup>+"
shows "\<exists>x y. (v,x) \<in> (E\<union>U)\<^sup>* \<and> (x,y) \<in> U \<and> (y,w) \<in> (E\<union>U)\<^sup>*" |
lemma cIsInvar_isChair_isPC: "cIsInvar isChair_isPC" |
lemma Rcd_type1: \<comment> \<open>A.13(3)\<close>
assumes H: "\<Gamma> \<turnstile> t : T"
shows "t = Rcd fs \<Longrightarrow> \<Gamma> \<turnstile> T <: RcdT fTs \<Longrightarrow>
\<forall>(l, U) \<in> set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor> \<and> \<Gamma> \<turnstile> u : U" |
lemma normalized_last_not_None:
\<comment> \<open>\ sometimes the inductive definition might work better\<close>
"normalized xs \<longleftrightarrow> xs = [] \<or> last xs \<noteq> None" |
lemma list_emb_eq_length_induct [consumes 2, case_names Nil Cons]:
assumes "length xs = length ys"
and "list_emb P xs ys"
and "Q [] []"
and "\<And>x y xs ys. \<lbrakk>P x y; list_emb P xs ys; Q xs ys\<rbrakk> \<Longrightarrow> Q (x#xs) (y#ys)"
shows "Q xs ys" |
lemma delete_edge_set_edges:
"set (edgesL (delete_edge v v' G)) = {(a,b). (a,b) \<in> set (edgesL G) \<and> (a,b) \<noteq> (v,v')}" |
lemma assumes "\<tau> \<Turnstile> \<delta> (V :: ('\<AA>,Void\<^sub>b\<^sub>a\<^sub>s\<^sub>e) Bag)"
shows "\<tau> \<Turnstile> V \<cong> Void\<^sub>n\<^sub>u\<^sub>l\<^sub>l \<or> \<tau> \<Turnstile> V \<cong> Void\<^sub>e\<^sub>m\<^sub>p\<^sub>t\<^sub>y" |
lemma member_of_subclseq_declC:
"G\<turnstile>m member_of C \<Longrightarrow> G\<turnstile>C \<preceq>\<^sub>C declclass m" |
lemma mapping_of_bind_pmf:
assumes "finite (set_pmf p)"
shows "mapping_of_pmf (bind_pmf p f) =
fold_combine_plus (\<lambda>x. Mapping.map_values (\<lambda>_. (*) (pmf p x))
(mapping_of_pmf (f x))) (set_pmf p)" |
lemma (in prob_space) indep_vars_iff_distr_eq_PiM':
fixes I :: "'i set" and X :: "'i \<Rightarrow> 'a \<Rightarrow> 'b"
assumes "I \<noteq> {}"
assumes rv: "\<And>i. i \<in> I \<Longrightarrow> random_variable (M' i) (X i)"
shows "indep_vars M' X I \<longleftrightarrow>
distr M (\<Pi>\<^sub>M i\<in>I. M' i) (\<lambda>x. \<lambda>i\<in>I. X i x) = (\<Pi>\<^sub>M i\<in>I. distr M (M' i) (X i))" |
lemma Gcd_factorial_greatest:
assumes "\<And>y. y \<in> A \<Longrightarrow> x dvd y"
shows "x dvd Gcd_factorial A" |
lemma invertibleRule:
assumes rules: "R' \<subseteq> upRules \<and> R = Ax \<union> R'"
and UC: "uniqueConclusion R'"
and IN: "(Ps,C) \<in> R*"
and der: "(C,n) \<in> derivable R*"
shows "\<forall> p \<in> set Ps. \<exists> m\<le>n. (p,m) \<in> derivable R*" |
lemma [code]: "iarray_to_vec v - iarray_to_vec u = (Matrix_To_IArray.iarray_to_vec (Rep_iarray_type v - Rep_iarray_type u))" |
lemma Nats_diff_nonpos_Ints:
"x \<in> \<nat> \<Longrightarrow> y \<in> \<int>\<^sub>\<le>\<^sub>0 \<Longrightarrow> x - y \<in> \<nat>" |
lemma \<L>_complement_filter_reg:
"\<L> (complement_reg (filter_ta_reg \<F> \<A>) \<F>) = \<T>\<^sub>G (fset \<F>) - \<L> \<A>" |
lemma dom_override_the2 [simp]:
"\<lbrakk> dom G1 \<inter> dom G2 = {}; x \<in> (dom G1) \<rbrakk> \<Longrightarrow> ((G1 ++ G2) x) = (G1 x)" |
lemma islimpt_greaterThanLessThan2:
fixes a b::"'a::{linorder_topology, dense_order}"
assumes "a < b"
shows "b islimpt {a<..<b}" |
lemma append_rows_index:
assumes "dim_col A = dim_col B"
assumes "i < dim_row A + dim_row B"
assumes "j < dim_col A"
shows "(A @\<^sub>r B) $$ (i,j) = (if i < dim_row A then A $$ (i,j) else B $$ (i-dim_row A,j))" |
lemma makeCFCorrect: "abstFn makeCF = makeAF" |
lemma length_under_max[simp]: "length xs < max (length xs + 3) fft" |
lemma (in CLF) T_thm_1_glb: "glb P cl = glb {x. (f x, x) \<in> r & x \<in> A} cl" |
lemma items_le_Predict: "items_le k (Predict k I) = Predict k (items_le k I)" |
lemma upd_nested_empty[simp]: "upd_nested m d f {} = m" |
lemma cmp_in_hom':
assumes "C.arr \<mu>" and "C.arr \<nu>" and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>"
shows "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : map\<^sub>0 (src\<^sub>C \<nu>) \<rightarrow>\<^sub>D map\<^sub>0 (trg\<^sub>C \<mu>)\<guillemotright>"
and "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>) \<Rightarrow>\<^sub>D F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)\<guillemotright>" |
lemma fds_nth_fds': "f 0 = 0 \<Longrightarrow> fds_nth (fds f) = f" |
lemma inverse_matrix:
fixes A::"'a::{field}^'n::{mod_type}^'n::{mod_type}"
shows "inverse_matrix A = (if invertible A then Some (P_Gauss_Jordan A) else None)" |
lemma succss_closed_dfs':
"invariant ys \<Longrightarrow> list_all is_node xs \<Longrightarrow>
succss (set_of ys) \<subseteq> set xs \<union> set_of ys \<Longrightarrow>
succss (set_of (dfs ys xs)) \<subseteq> set_of (dfs ys xs)" |
theorem asymmetric_master_theorem_ltr:
assumes
"w \<Turnstile>\<^sub>n \<phi>"
obtains Y and i where
"Y \<subseteq> subformulas\<^sub>\<nu> \<phi>"
and
"suffix i w \<Turnstile>\<^sub>n af \<phi> (prefix i w)[Y]\<^sub>\<mu>"
and
"\<forall>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi>\<^sub>1 R\<^sub>n \<psi>\<^sub>2 \<in> Y \<longrightarrow> suffix i w \<Turnstile>\<^sub>n G\<^sub>n (\<psi>\<^sub>2[Y]\<^sub>\<mu>)"
and
"\<forall>\<psi>\<^sub>1 \<psi>\<^sub>2. \<psi>\<^sub>1 W\<^sub>n \<psi>\<^sub>2 \<in> Y \<longrightarrow> suffix i w \<Turnstile>\<^sub>n G\<^sub>n (\<psi>\<^sub>1[Y]\<^sub>\<mu> or\<^sub>n \<psi>\<^sub>2[Y]\<^sub>\<mu>)" |
lemma preserves_comp_2:
assumes "A.seq f' f"
shows "\<tau> (f' \<cdot>\<^sub>A f) = \<tau> f' \<cdot>\<^sub>B F f" |
lemma dim_image_le:
assumes lf: "linear s1 s2 f"
shows "vs2.dim (f ` S) \<le> vs1.dim (S)" |
lemma ftv_ty_eqvt[eqvt]:
fixes pi::"tvar prm"
and T::"ty"
shows "pi\<bullet>(ftv T) = ftv (pi\<bullet>T)" |
theorem wls_skel_vsubst:
assumes "wls s X"
shows "skel (X #[y1 // y2]_ys) = skel X" |
lemma ex_moebius:
assumes "z1 \<noteq> z2" and "z1 \<noteq> z3" and "z2 \<noteq> z3"
"w1 \<noteq> w2" and "w1 \<noteq> w3" and "w2 \<noteq> w3"
shows "\<exists> M. ((moebius_pt M z1 = w1) \<and> (moebius_pt M z2 = w2) \<and> (moebius_pt M z3 = w3))" |
lemma VUnion_VPow[simp]: "\<Union>\<^sub>\<circ>(VPow A) = A" |
lemma ESem_fresh_cong:
assumes "\<rho> f|` (fv e) = \<rho>' f|` (fv e)"
shows "\<lbrakk> e \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> e \<rbrakk>\<^bsub>\<rho>'\<^esub>" |
lemma (in vsv) vsv_vrrestriction[intro, simp]: "vsv (r \<restriction>\<^sup>r\<^sub>\<circ> A)" |
lemma eq_Fn: "eq a v1 v2 \<Longrightarrow> eq (pred\<cdot>a) (v1 \<down>Fn v) (v2 \<down>Fn v)" |
lemma Complexs_0 [simp]: "0 \<in> \<complex>" and Complexs_1 [simp]: "1 \<in> \<complex>" |
lemma PO_m1_refines_init_a0_ri [iff]:
"init m1 \<subseteq> R_a0iim1_ri``(init a0i)" |
lemma lemma_3:
fixes PROB :: "'a problem" and s :: "'a state"
assumes "finite PROB" "(s \<in> valid_states PROB)" "(as \<in> valid_plans PROB)"
"(length as > (2 ^ (card (prob_dom PROB)) - 1))"
shows "(\<exists>as'.
(exec_plan s as = exec_plan s as')
\<and> (length as' < length as)
\<and> (subseq as' as)
)" |
lemma dp_map1: "is_direct_product x y \<Longrightarrow> is_map x" |
lemma foundedSubs: "founded subs P (tree subs Gamma) \<Longrightarrow> sigma \<in> subs Gamma \<Longrightarrow> founded subs P (tree subs sigma)" |
lemma mem_upd_commute:
"\<lbrakk> x \<noteq> y \<rbrakk> \<Longrightarrow> mem (x := v\<^sub>1, y := v\<^sub>2) = mem (y := v\<^sub>2, x := v\<^sub>1)" |
lemma strict_order_extension_if_consistent:
assumes "strongly_consistent r"
obtains r_ext where "strict_extends r_ext r" "total_preorder r_ext" |
lemma proj_elims_label:
assumes "k \<noteq> l"
shows "\<not>list_ex (has_LabelN l) (proj k S)" |
lemma IL_p1: "ex_coms, p1, lconst {} \<turnstile> \<lbrace>IL\<rbrace> \<lbrace>s12\<rbrace> \<xi>12\<triangleleft>(\<lambda>s. s)" |
lemma [def_pat_rules]: "Network.rg_succ2$c \<equiv> UNPROTECT rg_succ2" |
lemma inj_elem_index: "inj elem_index" |
lemma MN_eq_app_right:
"finite (UNIV // eq_app_right L) \<Longrightarrow> MyhillNerode L (eq_app_right L)" |
lemma stutter_invariantD [dest]:
assumes "stutter_invariant \<phi>" and "\<sigma> \<approx> \<tau>"
shows "(\<sigma> \<Turnstile>\<^sub>p \<phi>) = (\<tau> \<Turnstile>\<^sub>p \<phi>)" |
lemma take_apply_apply:
assumes "f \<in> carrier (SA n)"
assumes "a \<in> carrier (Q\<^sub>p\<^bsup>n\<^esup>)"
assumes "b \<in> carrier (Q\<^sub>p\<^bsup>k\<^esup>)"
shows "take_apply (n+k) n f (a@b) = f a" |
lemma norm_inverse: "norm (inverse a) = inverse (norm a)"
for a :: "'a::{real_normed_div_algebra,division_ring}" |
lemma children_rank_less:
assumes "tree_invar t"
shows "\<forall>t' \<in> set (children t). rank t' < rank t" |
lemma keyset_keysfor [iff]: "keyset (keysfor G)" |
lemma redT_updW_not_Suspend_Some:
"\<lbrakk> redT_updW t ws wa ws'; ws' t = \<lfloor>w'\<rfloor>; ws t = \<lfloor>w\<rfloor>; \<And>w. wa \<noteq> Suspend w \<rbrakk>
\<Longrightarrow> w' = w \<or> (\<exists>w'' w'''. w = InWS w'' \<and> w' = PostWS w''')" |
lemma timpl_closure'_mono:
assumes "TI \<subseteq> TI'"
shows "timpl_closure' TI \<subseteq> timpl_closure' TI'" |
lemma
flow_in_compact_right_existence:
assumes "\<And>t. 0 \<le> t \<Longrightarrow> t \<in> existence_ivl0 x \<Longrightarrow> flow0 x t \<in> K"
assumes "compact K" "K \<subseteq> X"
assumes "x \<in> X" "t \<ge> 0"
shows "t \<in> existence_ivl0 x" |
theorem ln_Gamma_complex_asymptotics_explicit:
fixes m :: nat and \<alpha> :: real
assumes "m > 0" and "\<alpha> \<in> {0<..<pi}"
obtains C :: real and R :: "complex \<Rightarrow> complex"
where "\<forall>s::complex. s \<notin> \<real>\<^sub>\<le>\<^sub>0 \<longrightarrow>
ln_Gamma s = (s - 1/2) * ln s - s + ln (2 * pi) / 2 +
(\<Sum>k=1..<m. bernoulli (k+1) / (k * (k+1) * s ^ k)) - R s"
and "\<forall>s. s \<noteq> 0 \<and> \<bar>Arg s\<bar> \<le> \<alpha> \<longrightarrow> norm (R s) \<le> C / norm s ^ m" |
lemma map_semantics_ltlc_aux:
assumes "inj_on f APs"
assumes "\<Union>(range w) \<subseteq> APs"
assumes "atoms_ltlc \<phi> \<subseteq> APs"
shows "w \<Turnstile>\<^sub>c \<phi> \<longleftrightarrow> (\<lambda>i. f ` w i) \<Turnstile>\<^sub>c map_ltlc f \<phi>" |
lemma finite_ta_productive:
"finite {p. \<exists>q q' C. p = q \<and> q' |\<in>| ta_der \<A> C\<langle>Var q\<rangle> \<and> q' |\<in>| P}" |
lemma phiDefNodes_aux_single_pred:
assumes "predecessors g n = [m]"
shows "phiDefNodes_aux g v (removeAll n un) m = phiDefNodes_aux g v un m" |
lemma list_prod_const:
"(\<And>x. x \<in> set xs \<Longrightarrow> f x \<le> c) \<Longrightarrow> (\<Prod>x\<leftarrow>xs. f x) \<le> c ^ length xs" for f :: "'a \<Rightarrow> nat" |
lemma Ln_of_nat_over_of_nat:
assumes "m > 0" "n > 0"
shows "Ln (of_nat m / of_nat n) = of_real (ln (of_nat m) - ln (of_nat n))" |
lemma vlrestriction_small[simp]: "small {\<langle>a, b\<rangle> | a b. a \<in>\<^sub>\<circ> A \<and> \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r}" |
lemma tfilter_transfer [transfer_rule]:
"(B ===> (A ===> (=)) ===> tllist_all2 A B ===> tllist_all2 A B) tfilter tfilter" |
lemma runErrorT_catchET [simp]:
"runErrorT\<cdot>(catchET\<cdot>m\<cdot>h) =
bind\<cdot>(runErrorT\<cdot>m)\<cdot>(\<Lambda> n. case n of
Err\<cdot>e \<Rightarrow> runErrorT\<cdot>(h\<cdot>e) | Ok\<cdot>x \<Rightarrow> return\<cdot>(Ok\<cdot>x))" |
lemma while_right_isotone:
"y \<le> z \<Longrightarrow> x \<star> y \<le> x \<star> z" |
lemma has_integral_nonneg:
fixes f :: "'n::euclidean_space \<Rightarrow> real"
assumes "(f has_integral i) S"
and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
shows "0 \<le> i" |
lemma pos2: "0 < (2::nat)" |
lemma matchC_ZO_def2:
"matchC_ZO theta d c =
(\<forall> s t d' t'.
s \<approx> t \<and> (d,t) \<rightarrow>c (d',t')
\<longrightarrow>
(s \<approx> t' \<and> (d',c) \<in> theta)
\<or>
(\<exists> c' s'. (c,s) \<rightarrow>c (c',s') \<and> s' \<approx> t' \<and> (d',c') \<in> theta)
\<or>
(\<exists> s'. (c,s) \<rightarrow>t s' \<and> s' \<approx> t' \<and> discr d'))" |
lemma part_circlepath_rotate_right:
"part_circlepath c r (a + x) (b + x) = (\<lambda>z. c + cis x * (z - c)) \<circ> part_circlepath c r a b" |
lemma sp_equiv_list_fv_list:
assumes "sp_equiv_list (\<sigma> \<odot>e ts) (\<tau> \<odot>e ts)"
shows "sp_equiv_list (map \<sigma> (fv_fo_terms_list ts)) (map \<tau> (fv_fo_terms_list ts))" |
lemma sup_apx_left_isotone:
assumes "x \<sqsubseteq> y"
shows "x \<squnion> z \<sqsubseteq> y \<squnion> z" |
lemma arr_compE\<^sub>E\<^sub>C [elim]:
assumes "arr (t \<cdot> u)"
and "\<lbrakk>arr t; arr u; trg t = src u\<rbrakk> \<Longrightarrow> T"
shows T |
lemma wf_strict_prefix: "wfP strict_prefix" |
lemma planning_by_cnf_dimacs_sound:
"\<A> \<Turnstile> map_formula var_to_dimacs (\<Phi>\<^sub>\<forall> (\<phi> (prob_with_noop abs_prob)) t) \<Longrightarrow>
valid_plan
(decode_abs_plan
(rem_noops
(map (\<lambda>op. \<phi>\<^sub>O\<inverse> (prob_with_noop abs_prob) op)
(concat (\<Phi>\<inverse> (\<phi> (prob_with_noop abs_prob)) (\<A> o var_to_dimacs) t)))))" |
lemma eqExcPID_N_trans:
assumes "eqExcPID_N s s1" and "eqExcPID_N s1 s2" shows "eqExcPID_N s s2" |
lemma par_by_merge_mono_2:
assumes "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2"
shows "(P \<parallel>\<^bsub>M\<^esub> Q\<^sub>1) \<sqsubseteq> (P \<parallel>\<^bsub>M\<^esub> Q\<^sub>2)" |
lemma parts_cut_eq [simp]: "X\<in> parts H ==> parts (insert X H) = parts H" |
lemma mergesort_by_rel_merge_simps[simp] :
"mergesort_by_rel_merge R (x#xs) (y#ys) =
(if R x y then x # mergesort_by_rel_merge R xs (y#ys) else y # mergesort_by_rel_merge R (x#xs) ys)"
"mergesort_by_rel_merge R xs [] = xs"
"mergesort_by_rel_merge R [] ys = ys" |
lemma TIP:
assumes 1: "\<phi> \<in> fmla" "Fvars \<phi> = {}" and 2: "isTrue (PP \<langle>\<phi>\<rangle>)"
shows "prv \<phi>" |
lemma coeff_simp:
assumes "f \<in> carrier P"
shows "coeff (UP R) f = f " |
lemma mon_loc_map_loc[simp]: "mon_loc fg (map LOC w) = mon_ww fg w" |
lemma butlast_subset:
"xs \<noteq> [] \<Longrightarrow> set xs \<subseteq> A \<Longrightarrow> set (butlast xs) \<subseteq> A" |
lemma smcf_cn_comp_ArrMap_app[smc_cn_cs_simps]:
assumes "\<GG> : \<BB> \<^sub>S\<^sub>M\<^sub>C\<mapsto>\<mapsto>\<^bsub>\<alpha>\<^esub> \<CC>" and "\<FF> : \<AA> \<^sub>S\<^sub>M\<^sub>C\<mapsto>\<mapsto>\<^bsub>\<alpha>\<^esub> \<BB>" and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>"
shows "(\<GG> \<^sub>S\<^sub>M\<^sub>C\<^sub>F\<circ> \<FF>)\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = \<GG>\<lparr>ArrMap\<rparr>\<lparr>\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr>\<rparr>" |
lemma E\<^sub>i\<^sub>i\<^sub>i: "E(x\<cdot>y) \<^bold>\<leftarrow> (dom x \<cong> cod y \<^bold>\<and> E(cod y))" |
lemma CallRedsObj:
"P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle> \<Longrightarrow>
P,E \<turnstile> \<langle>Call e Copt M es,s\<rangle> \<rightarrow>* \<langle>Call e' Copt M es,s'\<rangle>" |
lemma blinfun_to_matrix_matpow: "blinfun_to_matrix (X ^^ i) = matpow (blinfun_to_matrix X) i" |
lemma classes_above_isfields:
"\<lbrakk> classes_above P C \<inter> classes_changed P P' = {} \<rbrakk>
\<Longrightarrow>
isfields P C = isfields P' C" |
lemma geodesic_quasi_isometric_image:
fixes f::"'a::metric_space \<Rightarrow> 'b::Gromov_hyperbolic_space_geodesic"
assumes "lambda C-quasi_isometry_on UNIV f"
"geodesic_segment_between G x y"
shows "hausdorff_distance (f`G) {f x--f y} \<le> 92 * lambda^2 * (C + deltaG(TYPE('b)))" |
lemma SFAssInitReds:
assumes e\<^sub>2_steps: "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2,l\<^sub>2,sh\<^sub>2),False\<rangle>"
and cF: "P \<turnstile> C has F,Static:t in D"
and nDone: "\<nexists>sfs. sh\<^sub>2 D = Some (sfs, Done)"
and INIT_steps: "P \<turnstile> \<langle>INIT D ([D],False) \<leftarrow> unit,(h\<^sub>2,l\<^sub>2,sh\<^sub>2),False\<rangle> \<rightarrow>* \<langle>Val v',(h',l',sh'),b'\<rangle>"
and sh'D: "sh' D = Some(sfs,i)"
and sfs': "sfs' = sfs(F\<mapsto>v)" and sh'': "sh'' = sh'(D\<mapsto>(sfs',i))"
shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D}:=e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>unit,(h',l',sh''),False\<rangle>"
(*<*)(is "(?x, ?z) \<in> (red P)\<^sup>*") |
lemma simE:
fixes P :: pi
and Rel :: "(pi \<times> pi) set"
and Q :: pi
and a :: subject
and x :: name
and Q' :: pi
assumes "P \<leadsto>[Rel] Q"
shows "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q' \<Longrightarrow> x \<sharp> P \<Longrightarrow> \<exists>P'. P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P' \<and> (derivative P' Q' a x Rel)"
and "Q \<longmapsto> \<alpha> \<prec> Q' \<Longrightarrow> \<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel" |
lemma inv_on_left_moving_tl[simp]:
"\<lbrakk>abc_lm_v am n = 0; inv_locate_b (as, am) (n, aaa, []) ires\<rbrakk>
\<Longrightarrow> inv_on_left_moving (as, abc_lm_s am n 0) (s, tl aaa, [hd aaa]) ires" |
lemma num_val_iff[simp]: "e\<noteq>Infty \<Longrightarrow> Num (val e) = e" |
lemma addfunsetI :
"\<lbrakk> supp f \<subseteq> A; range f \<subseteq> M; \<forall>x\<in>A. \<forall>y\<in>A. f (x+y) = f x + f y \<rbrakk>
\<Longrightarrow> f \<in> addfunset A M" |
lemma norm_stack_Cons[simp]: "norm_stack (d # \<Sigma>) = Subx.norm_unboxed d # norm_stack \<Sigma>" |
lemma injective_vimage_restrict:
assumes J: "J \<subseteq> I"
and sets: "A \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" "B \<subseteq> (\<Pi>\<^sub>E i\<in>J. S i)" and ne: "(\<Pi>\<^sub>E i\<in>I. S i) \<noteq> {}"
and eq: "(\<lambda>x. restrict x J) -` A \<inter> (\<Pi>\<^sub>E i\<in>I. S i) = (\<lambda>x. restrict x J) -` B \<inter> (\<Pi>\<^sub>E i\<in>I. S i)"
shows "A = B" |
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