hcju/isabelleps huggingface dataset:
induct: fixes P3 :: "cell \<Rightarrow> interface \<Rightarrow> interface \<Rightarrow> bool"   and P1 :: "sdiagram \<Rightarrow> interface \<Rightarrow> interface \<Rightarrow> bool"   and P2 :: "cell \<times> interface \<Rightarrow> interface \<Rightarrow> interface \<Rightarrow> bool"   and x1 :: "sdiagram"   and x2 :: "interface"   and x3 :: "interface"   and x4 :: "cell \<times> interface"   and x5 :: "interface"   and x6 :: "interface"   and x7 :: "cell"   and x8 :: "interface"   and x9 :: "interface" assumes "\<And>P. P3 (Filler P) P P"   and "\<And>P c Q. prov_triple (asn P, c, asn Q) \<Longrightarrow> P3 (Basic P c Q) P Q"   and "\<And>D P Q x. \<lbrakk>prov_sdia D P Q; P1 D P Q\<rbrakk> \<Longrightarrow> P3 (Exists_sdia x D) (Exists_int x P) (Exists_int x Q)"   and "\<And>D P Q E. \<lbrakk>prov_sdia D P Q; P1 D P Q; prov_sdia E P Q; P1 E P Q\<rbrakk> \<Longrightarrow> P3 (Choose_sdia P D E Q) P Q"   and "\<And>D P. \<lbrakk>prov_sdia D P P; P1 D P P\<rbrakk> \<Longrightarrow> P3 (Loop_sdia P D P) P P"   and "\<And>\<gamma> P Q F. \<lbrakk>prov_cell \<gamma> P Q; P3 \<gamma> P Q; wr_cell \<gamma> \<inter> rd_int F = {}\<rbrakk> \<Longrightarrow> P2 (\<gamma>, F) (P \<otimes> F) (Q \<otimes> F)"   and "\<And>\<Pi>. \<lbrakk>chain_all (\<lambda>(P, \<rho>, Q). prov_row \<rho> P Q \<and> P2 \<rho> P Q) \<Pi>; 0 < chainlen \<Pi>\<rbrakk> \<Longrightarrow> P1 (SDiagram (comlist \<Pi>)) (pre \<Pi>) (post \<Pi>)" shows "(prov_sdia x1 x2 x3 \<longrightarrow> P1 x1 x2 x3) \<and> (prov_row x4 x5 x6 \<longrightarrow> P2 x4 x5 x6) \<and> (prov_cell x7 x8 x9 \<longrightarrow> P3 x7 x8 x9)"
tmp: shows "(fst \<sigma>2', snd \<sigma>2' @ snd \<sigma>3') \<in> seq ars"
absorb_iff2: fixes a :: "'e"   and b :: "'e" shows "(a \<le> b) = (max a b = b)"
var_eq_symp: fixes x :: "('e, 'f, 'g) var0"   and y :: "('e, 'f, 'g) var0" assumes "x \<sim> y" shows "y \<sim> x"
simps_1: shows "make_vert_equiv 0 = [[1]]"
ExitBlockFault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and f :: "'f"   and abrs :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile>nrms,css \<Down> Fault f" shows "\<Gamma>\<turnstile>[],(nrms, abrs) # css \<Down> Fault f"
cd_path_shift0: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>" shows "IFC_def.is_cdi prob i \<pi> k = IFC_def.is_cdi prob (i - k) (\<pi> \<guillemotleft> k) 0"
set_conv_list: fixes g :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sum g (set xs) = sum_list (map g (remdups xs))"
not_undeclared_declared: fixes G :: "prog"   and membr_id :: "memberid"   and C :: "qtname" assumes "\<not> G\<turnstile> membr_id undeclared_in C" shows "\<exists>m. G\<turnstile> m declared_in C \<and> membr_id = memberid m"
arc_snd_in_verts: fixes G :: "'b pair_pre_digraph"   and e :: "'b \<times> 'b" assumes "pair_wf_digraph G"   and "e \<in> parcs G" shows "snd e \<in> pverts G"
fix_transform_pair_sum: fixes F :: "('a discr \<rightarrow> 'b) \<times> ('c discr \<rightarrow> 'b) \<rightarrow> ('a discr \<rightarrow> 'b) \<times> ('c discr \<rightarrow> 'b)" shows "fix\<cdot>F = sum_to_tup\<cdot>(fix\<cdot>(tup_to_sum oo F oo sum_to_tup))"
lossless_def: shows "lossless = (\<lambda>(\<A>\<^sub>1, \<A>\<^sub>2). (\<forall>pk. lossless_gpv \<I>_full (\<A>\<^sub>1 pk)) \<and> (\<forall>cipher \<sigma>. lossless_gpv \<I>_full (\<A>\<^sub>2 cipher \<sigma>)))"
Deq_enabledE: fixes ic :: "state \<Rightarrow> 'a"   and q :: "state \<Rightarrow> 'a list"   and oc :: "state \<Rightarrow> 'a" shows "\<turnstile> Enabled (<Deq ic q oc>_(ic, q, oc)) \<longrightarrow> q \<noteq> #[]"
return_bind: fixes x :: "'a list"   and f :: "'a list \<Rightarrow> 'm" shows "bind (return x) f = f x"
continuous_on_u: fixes T :: "real set" shows "continuous_on T u"
rroots2_def: fixes p :: "real poly" shows "rroots2 p = (let a = poly.coeff p 2; b = poly.coeff p 1; c = poly.coeff p 0; b2a = b / (2 * a); bac = b2a\<^sup>2 - c / a in if bac = 0 then [- b2a] else if bac < 0 then [] else let e = sqrt bac in [- b2a + e, - b2a - e])"
simps_1: fixes h :: "'b \<Rightarrow> int" shows "sortedTree h Tip = True"
zero_pdivides: fixes p :: "'a list" shows "[] pdivides p = (p = [])"
fields_table_def: fixes G :: "prog"   and C :: "qtname"   and P :: "fspec \<Rightarrow> Decl.field \<Rightarrow> bool" shows "fields_table G C P = map_option type \<circ> table_of (filter (\<lambda>(x, y). P x y) (DeclConcepts.fields G C))"
Key_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs" shows "K \<notin> range shrK"
better_Cut_substc: fixes a :: "coname"   and c :: "coname"   and P :: "trm"   and x :: "name"   and y :: "name"   and M :: "trm"   and N :: "trm" assumes "a \<sharp> (c, P)"   and "x \<sharp> [y].P" shows "Cut <a>.M x.N{c:=y.P} = (if N = Ax x c then Cut <a>.M{c:=y.P} y.P else Cut <a>.M{c:=y.P} x.N{c:=y.P})"
index_f: shows "bij_betw MN.index_f (UNIV // eq_app_right MN.language) (hfset (ord_of MN.min_states))"
affine_hull_Int: fixes S :: "'b set"   and T :: "'b set" assumes "\<not> affine_dependent (S \<union> T)" shows "affine hull (S \<inter> T) = affine hull S \<inter> affine hull T"
RelS_Eg1Eg2_secure_refinement_\<R>: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set"   and P :: "bexp set" shows "secure_refinement (A.\<R> \<Gamma> \<S> P) RelS_Eg1Eg2 \<I>simple"
rel_gpv''_pos_distr: fixes A :: "'a \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'b \<Rightarrow> 'h \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'i \<Rightarrow> bool"   and A' :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and C' :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and R' :: "'i \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_gpv'' A C R OO rel_gpv'' A' C' R' \<le> rel_gpv'' (A OO A') (C OO C') (R OO R')"
better_LAnd2_intro: fixes a :: "coname"   and b1 :: "coname"   and M1 :: "trm"   and b2 :: "coname"   and M2 :: "trm"   and y :: "name"   and x :: "name"   and N :: "trm" assumes "a \<sharp> ([b1].M1, [b2].M2)"   and "y \<sharp> [x].N" shows "Cut <a>.AndR <b1>.M1 <b2>.M2 a y.AndL2 x.N y \<longrightarrow>\<^sub>l Cut <b2>.M2 x.N"
demand_def: fixes f :: "C \<rightarrow> 'a" shows "demand f = (if f\<cdot>C\<^sup>\<infinity> \<noteq> \<bottom> then C\<^bsup>LEAST n. f\<cdot>C\<^bsup>n\<^esup> \<noteq> \<bottom>\<^esup> else C\<^sup>\<infinity>)"
prv_imp_dsjEE: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> \<psi>)"   and "prv (imp \<chi> \<psi>)" shows "prv (imp (dsj \<phi> \<chi>) \<psi>)"
abs_eq: fixes x :: "64 word" shows "int_of_uint64 (Abs_uint64 x) = uint x"
progress: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and l :: "sfields"   and b :: "bool" assumes "wwf_J_prog P"   and "P \<turnstile> h \<surd>"   and "P,h \<turnstile>\<^sub>s sh \<surd>"   and "P,E,h,sh \<turnstile> e : T"   and "\<D> e \<lfloor>dom l\<rfloor>"   and "P,sh \<turnstile>\<^sub>b (e,b) \<surd>"   and "\<not> final e" shows "\<exists>e' s' b'. P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"
seq_UINF_distr': fixes F :: "'e \<Rightarrow> ('b, 'd) urel"   and Q :: "('d, 'c) urel" shows "(\<Sqinter> P \<bullet> F P) ;; Q = (\<Sqinter> P \<bullet> F P ;; Q)"
min_set_min_union_1: fixes A :: "'b fset set"   and B :: "'b fset set" shows "min_set A \<union>\<^sub>m B = A \<union>\<^sub>m B"
trivial_def: fixes x :: "'a set" shows "trivial x = (x \<subseteq> {the_elem x})"
Fract_of_nat_eq: fixes k :: "nat" shows "Fract (int k) 1 = rat_of_nat k"
fact_mono_nat: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "fact m \<le> fact n"
pathfinish_linepath: fixes a :: "'b"   and b :: "'b" shows "pathfinish (linepath a b) = b"
assms: shows "(t, s) \<in> timpl_closure'_step TI"
th3_1: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "B\<^sub>\<real> a b b"
axioms_3: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and remove_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_shadow_root_locs :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "l_remove_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs set_shadow_root set_shadow_root_locs get_disconnected_nodes get_disconnected_nodes_locs remove_shadow_root remove_shadow_root_locs type_wf known_ptr" shows "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"
set_def: fixes a :: "'b array"   and x :: "'b list" shows "Array.set a x = arrays_update (\<lambda>h. h(TYPEREP('b) := (h TYPEREP('b))(addr_of_array a := map to_nat x)))"
length_trace_minus_one_lt_length_trace: fixes I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" shows "length (trace_parallel_plan_strips I \<pi>) - 1 < length (trace_parallel_plan_strips I \<pi>)"
minus_finfun_apply: fixes f :: "'a \<Rightarrow>f 'b"   and g :: "'a \<Rightarrow>f 'b" shows "($) (f - g) = ($) f - ($) g"
Entry_Exit_edge: shows "\<exists>a. lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a \<and> src a = NewEntry \<and> trg a = NewExit \<and> knd a = (\<lambda>s. False)\<^sub>\<surd>"
role_comp_inv_eq: fixes y :: "role_t"   and x :: "role_t" shows "(y = role_comp x) = (x = role_comp y)"
simps_4: fixes fv :: "'b \<Rightarrow> 'c set"   and safe :: "safety \<Rightarrow> 'b \<Rightarrow> bool"   and g :: "safety"   and r :: "'b regex"   and s :: "'b regex" shows "Regex.safe_regex fv safe Futu g (Times r s) = ((g = Lax \<or> fv_regex fv r \<subseteq> fv_regex fv s) \<and> Regex.safe_regex fv safe Futu g s \<and> Regex.safe_regex fv safe Futu Lax r)"
loc_def: fixes w :: "'c el_step list" shows "loc w \<equiv> filter (\<lambda>e. case e of LOC a \<Rightarrow> True | ENV a \<Rightarrow> False) w"
mult_rel_class_of: fixes U :: "'a set"   and x :: "'b"   and V :: "'a set"   and y :: "'b"   and W :: "'a set" assumes "(U, x) \<in> Sigma I \<FF>"   and "(V, y) \<in> Sigma I \<FF>"   and "W \<in> I"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "mult_rel \<lfloor>U, x\<rfloor> \<lfloor>V, y\<rfloor> = \<lfloor>W, \<cdot>\<^bsub>W\<^esub> (\<rho> U W x) (\<rho> V W y)\<rfloor>"
filter_map: fixes P :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "List.filter P (list.map f xs) = list.map f (List.filter (P \<circ> f) xs)"
cpx_vec_length_inner_prod: fixes v :: "complex vec" shows "complex_of_real (\<parallel>v\<parallel>\<^sup>2) = \<langle>v|v\<rangle>"
rel_option_iff: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c option"   and y :: "'d option" shows "rel_option R x y = (case (x, y) of (None, None) \<Rightarrow> True | (None, Some a) \<Rightarrow> False | (Some x, None) \<Rightarrow> False | (Some x, Some y) \<Rightarrow> R x y)"
circuit_dep: fixes X :: "'a set" assumes "circuit X" shows "\<not> indep X"
return_option_def: fixes x :: "'a" shows "local.return_option x = OptionT (return (Some x))"
surjective: fixes r :: "('b, 'c) set_arrow_scheme" shows "r = \<lparr>set_dom = set_dom r, set_func = set_func r, set_cod = set_cod r, \<dots> = set_arrow.more r\<rparr>"
"1_1": shows "fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as) = fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) (exec_plan s as')"
one_cong: shows "\<one>\<^bsub>R\<^esub> = 1 mod int p"
A_3: shows "Z' = (Z \<inter> {u. u \<turnstile> g})\<^bsub>r \<rightarrow> 0\<^esub> \<inter> {u. u \<turnstile> inv_of A l'}"
image_subset_iff: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'a set" shows "(f ` A \<subseteq> B) = (\<forall>x\<in>A. f x \<in> B)"
dghm_tdghm_comp_is_tdghm': fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<HH> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and \<FF>' :: "V"   and \<GG>' :: "V" assumes "\<HH> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "\<FF>' = \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>"   and "\<GG>' = \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>" shows "\<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<NN> : \<FF>' \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>' : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
eq_2: shows "(\<Prod>i\<in>UNIV - {k}. (if i = k then a i + b i else c i) $ p i) = (\<Prod>i\<in>UNIV - {k}. (if i = k then b i else c i) $ p i)"
weak_reduction_coupled_simulation_versus_simulation_and_contrasimulation: fixes Rel :: "('a \<times> 'a) set"   and Cal :: "'a processCalculus" shows "weak_reduction_coupled_simulation Rel Cal = (weak_reduction_simulation Rel Cal \<and> weak_reduction_contrasimulation Rel Cal)"
s: fixes w :: "'a" assumes "w \<in> s" shows "eval_fps F w = f w"
insert_uconnectedV_per': fixes x :: "nat"   and y :: "nat"   and F :: "nat uprod set" assumes "x \<in> V"   and "y \<in> V"   and "F \<subseteq> E" shows "Restr (uconnected (insert (Upair x y) F)) V = per_union (Restr (uconnected F) V) x y"
red_int_inv: fixes n :: "nat"   and e :: "exp"   and P :: "bool" assumes "ENat n \<longrightarrow> e" shows "P"
intros: fixes state :: "State" assumes "\<lbrakk>\<not> getConflictFlag state; getQ state \<noteq> []\<rbrakk> \<Longrightarrow> exhaustiveUnitPropagate_dom (applyUnitPropagate state)" shows "exhaustiveUnitPropagate_dom state"
commute: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "abel_semigroup f" shows "f a b = f b a"
basis_reduction_step_cost_def: fixes upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "basis_reduction_step_cost upw i state = (if i = 0 then ((True, Suc i, inc_state state), 0) else let (state', cost_add) = basis_reduction_add_rows_cost upw i state; di = d_state state' i; dsi = d_state state' (Suc i); dim1 = d_state state' (i - 1); (num, denom) = quotient_of \<alpha>; cond = di * di * denom \<le> num * dim1 * dsi; local_cost = 5 in if cond then ((True, Suc i, inc_state state'), local_cost + cost_add) else case basis_reduction_swap_cost i state' of (res, cost_swap) \<Rightarrow> (res, local_cost + cost_swap + cost_add))"
dom: shows "\<D>\<^sub>\<circ> vreal_inverse = \<real>\<^sub>\<circ>"
LD_1'_def: fixes r :: "('b \<times> 'b) set"   and \<beta> :: "'b"   and \<alpha> :: "'b"   and \<sigma>1 :: "'b list"   and \<sigma>2 :: "'b list"   and \<sigma>3 :: "'b list" shows "LD_1' r \<beta> \<alpha> \<sigma>1 \<sigma>2 \<sigma>3 = (set \<sigma>1 \<subseteq> r \<down>s {\<beta>} \<and> length \<sigma>2 \<le> 1 \<and> set \<sigma>2 \<subseteq> {\<alpha>} \<and> set \<sigma>3 \<subseteq> r \<down>s {\<alpha>, \<beta>})"
set_rw: shows "[0..<Suc (Suc k)] = [0..<Suc k] @ [Suc k]"
fds_converges_def: fixes f :: "'a fds"   and s :: "'a" shows "fds_converges f s = summable (\<lambda>n. fds_nth f n / nat_power n s)"
weaken_rely: fixes i :: "'a::refinement_lattice"   and j :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice" assumes "(i::'a::refinement_lattice) \<sqsubseteq> (j::'a::refinement_lattice)" shows "(c::'a::refinement_lattice) // (j::'a::refinement_lattice) \<sqsubseteq> c // (i::'a::refinement_lattice)"
2: shows "\<forall>P Q \<phi>. R P Q \<and> P \<turnstile> \<phi> \<longrightarrow> (\<exists>Q'. Q \<Rightarrow> Q' \<and> R P Q' \<and> Q' \<turnstile> \<phi>)"
trivial_measure_def: fixes X :: "'b set" shows "trivial_measure X = sigma X {{}, X}"
sup_bot_right: fixes sup :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and less_eq :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and bot :: "'m"   and x :: "'m" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "sup x bot = x"
LLL_inv_initial_state: shows "LLL_invariant True 0 fs_init"
CHAOS_DF_refine_F: fixes A :: "'a set" shows "CHAOS A \<sqsubseteq>\<^sub>F DF A"
top_char: fixes x :: "'a" shows "top = x \<squnion> - x"
assms_3: shows "\<turnstile>\<^sub>\<tau> a : \<tau>"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "pred_mset P = (\<lambda>x. Multiset.Ball x P)"
asIMOD_igSwapIPresIGWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSwapIPresIGWls (asIMOD SEM)"
simps_7: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<r>\<^bold>[t\<^bold>]\<rbrace> = D.\<rho> \<lbrace>t\<rbrace>"
generate_check_def: fixes a :: "nat list"   and b :: "nat list" shows "generate_check a b = concat (map (\<lambda>ys. map (\<lambda>xs. (xs, ys)) (c1_gen_check b (fst ys) a)) (c2_gen_check a b b))"
simps_2: fixes i :: "instr"   and "is" :: "instr list" shows "store_sops (i # is) = (case i of Write volatile a sop x xa xb xc \<Rightarrow> insert sop (store_sops is) | RMW a t sop cond ret A L R W \<Rightarrow> insert sop (store_sops is) | _ \<Rightarrow> store_sops is)"
total_sync_order: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist" shows "total_onP (sactions P E) (sync_order P E)"
locally_finite_in_def: fixes X :: "'a topology"   and \<A> :: "'a set set" shows "locally_finite_in X \<A> = (\<Union> \<A> \<subseteq> topspace X \<and> (\<forall>x\<in>topspace X. \<exists>V. openin X V \<and> x \<in> V \<and> finite {U \<in> \<A>. U \<inter> V \<noteq> {}}))"
z: fixes y :: "complex" assumes "y \<in> closure S" shows "(cmod \<circ> f) y \<le> (cmod \<circ> f) z"
generate_valid_stateful_policy_IFSACS_2_filter_compliant_stateful_ACS: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph"   and edgesList :: "('b \<times> 'b) list"   and \<T> :: "'b stateful_policy" assumes "valid_reqs M"   and "wf_graph G"   and "all_security_requirements_fulfilled M G"   and "set edgesList \<subseteq> edges G"   and "\<T> = generate_valid_stateful_policy_IFSACS_2 G M edgesList" shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)"
is_pred_succ_no_node: fixes a :: "nat \<Rightarrow> nat list"   and u :: "nat" assumes "is_adj_map a"   and "u \<notin> V" shows "a u = []"
simps_12: fixes v :: "'a acom"   and va :: "'a acom"   and vb :: "bexp"   and vc :: "'a acom"   and vd :: "'a acom"   and ve :: "'a" shows "(v;; va \<le> IF vb THEN vc ELSE vd {ve}) = False"
diff_ge_0_iff_ge: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (minus a b) = less_eq b a"
list_assn_pure_conv: fixes R :: "('b \<times> 'a) set" shows "list_assn (pure R) = pure (\<langle>R\<rangle>list_rel)"
vectorspace_def: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme" shows "vectorspace K V \<equiv> module K V \<and> field K"
exec_induct: fixes Pa :: "instr list"   and x1a :: "int"   and x1b :: "char list \<Rightarrow> int"   and x1c :: "int list"   and x2a :: "int"   and x2b :: "char list \<Rightarrow> int"   and x2c :: "int list"   and P :: "int \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> int list \<Rightarrow> int \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> int list \<Rightarrow> bool" assumes "Pa \<turnstile> (x1a, x1b, x1c) \<rightarrow>* (x2a, x2b, x2c)"   and "\<And>a aa b. P a aa b a aa b"   and "\<And>a aa b ab ac ba ad ae bb. \<lbrakk>Pa \<turnstile> (a, aa, b) \<rightarrow> (ab, ac, ba); Pa \<turnstile> (ab, ac, ba) \<rightarrow>* (ad, ae, bb); P ab ac ba ad ae bb\<rbrakk> \<Longrightarrow> P a aa b ad ae bb" shows "P x1a x1b x1c x2a x2b x2c"
diff_fun_space_times: fixes charts :: "('c, 'd) chart set"   and k :: "enat"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "c_manifold charts k"   and "f \<in> c_manifold.diff_fun_space charts k"   and "g \<in> c_manifold.diff_fun_space charts k" shows "f * g \<in> c_manifold.diff_fun_space charts k"
min_add_distrib_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "plus x (ord.min less_eq y z) = ord.min less_eq (plus x y) (plus x z)"
pr1_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.dom (hfsetcat.pr1 a b) = hfsetcat.prod a b"
q_avail: shows "i1.orig.quorum I\<^sub>1"
reindex_cong: fixes l :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and h :: "'c \<Rightarrow> 'e" assumes "inj_on l B"   and "A = l ` B"   and "\<And>x. x \<in> B \<Longrightarrow> g (l x) = h x" shows "prod g A = prod h B"
semigroup_mult_def: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.semigroup_mult times \<equiv> \<forall>a b c. times (times a b) c = times a (times b c)"
from_bool_neq_0: fixes b :: "bool" shows "(from_bool b \<noteq> 0) = b"
founded_def: fixes subs :: "'b \<Rightarrow> 'b set"   and Pred :: "'b \<Rightarrow> bool" shows "founded subs Pred = (\<lambda>A. \<forall>(n, delta)\<in>A. terminal subs delta \<longrightarrow> Pred delta)"
some_sym_eq_trivial: fixes x :: "'c" shows "(SOME y. x = y) = x"
map_indets_inj_on_PolysI: fixes f :: "'b \<Rightarrow> 'c"   and X :: "'b set" assumes "inj_on f X" shows "inj_on (map_indets f) P[X]"
lm092: fixes bids :: "'a \<times> 'b set \<Rightarrow> 'c"   and N :: "'a set"   and G :: "'b set" shows "summedBidVectorRel bids N G = graph (N \<times> (Pow G - {{}})) (summedBidSecond bids)"
step: fixes y :: "hf"   and z :: "hf" assumes "y \<^bold>\<in> ya__" shows "(x @+ y = x @+ z) = (y = z)"
reachable_oseq_seqp_sos': fixes s :: "(nat \<Rightarrow> 'e) \<times> ('e, 'f, 'g, 'h) seqp"   and OA :: "((nat \<Rightarrow> 'e) \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and I :: "'f seq_action \<Rightarrow> bool"   and i :: "nat"   and A :: "('e \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp" assumes "s \<in> reachable OA I"   and "initiali i (init OA) (init A)"   and "automaton.trans OA = oseqp_sos \<Gamma> i"   and "automaton.trans A = seqp_sos \<Gamma>" shows "\<exists>\<xi>. fst s i = \<xi> \<and> (\<xi>, snd s) \<in> reachable A I"