hcju/isabelleps huggingface dataset:
lifting_intersection_def: fixes Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and Q :: "'c set"   and Inf_G_q :: "'c \<Rightarrow> 'b inference set"   and entails_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Red_I_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b inference set"   and Red_F_q :: "'c \<Rightarrow> 'b set \<Rightarrow> 'b set"   and Bot_F :: "'a set"   and \<G>_F_q :: "'c \<Rightarrow> 'a \<Rightarrow> 'b set"   and \<G>_I_q :: "'c \<Rightarrow> 'a inference \<Rightarrow> 'b inference set option"   and Prec_F_g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "lifting_intersection Inf_F Bot_G Q Inf_G_q entails_q Red_I_q Red_F_q Bot_F \<G>_F_q \<G>_I_q Prec_F_g \<equiv> inference_system_family Q \<and> consequence_relation_family Bot_G Q entails_q \<and> lifting_intersection_axioms Inf_F Bot_G Q Inf_G_q entails_q Red_I_q Red_F_q Bot_F \<G>_F_q \<G>_I_q Prec_F_g"
intro: fixes x :: "'d \<Longrightarrow> 'e" assumes "wb_lens x"   and "mwb_lens x" shows "vwb_lens x"
simps_87: fixes x101 :: "Term.var"   and x102 :: "expr"   and x31 :: "ty"   and x32 :: "expr" shows "x101:=x102 \<noteq> Cast x31 x32"
Zminf_pOp_closed: fixes x :: "ant"   and y :: "ant" assumes "x \<in> Z\<^sub>-\<^sub>\<infinity>"   and "y \<in> Z\<^sub>-\<^sub>\<infinity>" shows "x + y \<in> Z\<^sub>-\<^sub>\<infinity>"
seqI: fixes g1 :: "'a"   and f1 :: "'a"   and g2 :: "'a"   and f2 :: "'a" assumes "seq g1 f1 \<and> seq g2 f2" shows "CC.seq (g1, g2) (f1, f2)"
cases: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> 'i" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "'i" where "x = (s1, s2)"
move_def: fixes ts :: "traffic"   and ts' :: "traffic"   and v :: "view" shows "move ts ts' v = \<lparr>ext =  shift ext v pos ts' (own v) - pos ts (own v), lan = lan v, own = own v\<rparr>"
len: shows "\<^bold>|u\<^bold>| + \<^bold>|v\<^bold>| \<le> \<^bold>|w\<^bold>|"
intros: fixes uu :: "'b"   and xs :: "'b list" shows "tail_side (uu # xs)"
map_sel_2: fixes a :: "'b literal"   and f :: "'b \<Rightarrow> 'c" assumes "is_neg a" shows "atm_of (map_literal f a) = f (atm_of a)"
interval_imp_mem: fixes cl :: "'b potype"   and x :: "'b"   and a :: "'b"   and b :: "'b" assumes "PO cl"   and "x \<in> interval (order cl) a b" shows "(a, x) \<in> order cl"
typeof_lit_typeof: fixes v :: "'addr val"   and T :: "ty"   and h :: "'heap" assumes "typeof v = \<lfloor>T\<rfloor>" shows "typeof\<^bsub>h\<^esub> v = \<lfloor>T\<rfloor>"
altc_sample_state_def: fixes altc_sample :: "'b \<Rightarrow> ('d \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'d \<Rightarrow> ('s, 'm) stateT" shows "local.altc_sample_state altc_sample p f = StateT (\<lambda>s. altc_sample p (\<lambda>x. run_state (f x) s))"
simps_3: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and c1 :: "('c, 'd, 'e) cmd"   and c2 :: "('c, 'd, 'e) cmd" assumes "PL cval" shows "proper (c1 ;; c2) = (proper c1 \<and> proper c2)"
Axiom_of_Pairing: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> Vset \<alpha>"   and "b \<in>\<^sub>\<circ> Vset \<alpha>" shows "ZFC_in_HOL.set {a, b} \<in>\<^sub>\<circ> Vset \<alpha>"
ot\<omega>j: fixes j :: "nat"   and j0 :: "nat" assumes "j \<le> j0" shows "ordertype (BB j0 j ` \<K> j0 j) (lenlex less_than) = \<omega> \<up> ord_of_nat j"
fds_nth_const_Suc_0: fixes c :: "'b" shows "fds_nth (fds_const c) (Suc 0) = c"
suffix_def: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "\<sigma> |\<^sub>s i \<equiv> \<lambda>j. \<sigma> (j + i)"
fun_dual6: fixes f :: "'b set \<Rightarrow> 'c set"   and g :: "'b set \<Rightarrow> 'c set" shows "(f = (`) \<partial> \<circ> g \<circ> (`) \<partial>) = ((`) \<partial> \<circ> f \<circ> (`) \<partial> = g)"
notnotD: fixes P :: "bool" assumes "\<not> \<not> P" shows "P"
bind_spmf_mono: fixes B :: "('c \<Rightarrow> 'd spmf) \<Rightarrow> 'e spmf"   and C :: "'e \<Rightarrow> ('c \<Rightarrow> 'd spmf) \<Rightarrow> 'f spmf" assumes "monotone spmf.le_fun (ord_spmf (=)) B"   and "\<And>y. monotone spmf.le_fun (ord_spmf (=)) (C y)" shows "monotone spmf.le_fun (ord_spmf (=)) (\<lambda>f. B f \<bind> (\<lambda>y. C y f))"
add_mono_thms_linordered_field_5: fixes i :: "'f"   and j :: "'f"   and k :: "'f"   and l :: "'f" assumes "i < j \<and> k < l" shows "i + k < j + l"
simps_3: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "next_free (\<phi> implies\<^sub>p \<psi>) = (next_free \<phi> \<and> next_free \<psi>)"
A'_def: shows "A' = filter (\<lambda>v. dim_vec v = d) A"
"2_2": shows "(return_time p2 has_derivative dR2) (at (flow0 x0 h))"
lDef: shows "lDef \<equiv> lift_Def Def Entry Exit H L"
BT: shows "B\<^sup>T \<in> carrier_mat n m"
ivl_disj_un_singleton_3: fixes l :: "'b"   and u :: "'b" assumes "l < u" shows "{l} \<union> {l<..<u} = {l..<u}"
intro: fixes G :: "('a, 'b) graph_rec_scheme" assumes "g_V0 G \<subseteq> g_V G"   and "g_E G \<subseteq> g_V G \<times> g_V G" shows "graph G"
simps_2: fixes f :: "'t \<Rightarrow>\<^sub>0 'c"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and u :: "'t" shows "find_adds (f # fs) u = (if f \<noteq> 0 \<and> lt f adds\<^sub>t u then Some f else find_adds fs u)"
subdivision_pair_induct: fixes G :: "'a pair_pre_digraph"   and x :: "'a pair_pre_digraph"   and P :: "'a pair_pre_digraph \<Rightarrow> bool" assumes "subdivision_pair G x"   and "pair_bidirected_digraph G \<Longrightarrow> P G"   and "\<And>e w H. \<lbrakk>e \<in> parcs H; w \<notin> pverts H; subdivision_pair G H; P H\<rbrakk> \<Longrightarrow> P (subdivide H e w)" shows "P x"
Collect_mem_eq: fixes A :: "'b set" shows "rel_ext (\<lambda>x. x \<in> A) = A"
Cignorabimus_def: fixes \<alpha> :: "game"   and V :: "variable set" shows "Cignorabimus \<alpha> V = {M. \<forall>I \<omega> \<omega>' X. Vagree \<omega> \<omega>' (- M) \<longrightarrow> \<omega> \<in> game_sem I \<alpha> (restrictto X V) \<longrightarrow> \<omega>' \<in> game_sem I \<alpha> (restrictto X V)}"
K_block_design_axioms: shows "K_block_design \<V> \<B> \<K>"
Int_closed_segment: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b \<in> closed_segment a c \<or> \<not> collinear {a, b, c}" shows "closed_segment a b \<inter> closed_segment b c = {b}"
INT_D: fixes b :: "'k"   and B :: "'l \<Rightarrow> 'k set"   and A :: "'l set"   and a :: "'l" assumes "b \<in> \<Inter> (B ` A)"   and "a \<in> A" shows "b \<in> B a"
g_isEmpty_def: fixes m :: "'s" shows "g_isEmpty m \<equiv> g_size_abort 1 m = 0"
union_def: fixes A :: "ZF"   and B :: "ZF" shows "HOLZF.union A B \<equiv> HOLZF.Sum (Upair A B)"
Fundamental_property_wcd_aux: fixes prog :: "cmd" shows "BackwardSlice_wf sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (WWeakControlDependence.PDG_BS_w prog) reds (labels_nodes prog)"
assms: fixes bl :: "'a set" assumes "bl \<in># \<B>" shows "incomplete_block bl"
commute: fixes a :: "'g"   and b :: "'g" shows "sup a b = sup b a"
remdups_adj_replicate: fixes n :: "nat"   and x :: "'a" shows "remdups_adj (replicate n x) = (if n = 0 then [] else [x])"
Abs_SI_cases: fixes x :: "SI" obtains y :: "unit" where "x = Abs_SI y" and "y \<in> UNIV"
less_eq_view_ext_def: fixes V :: "'a view_scheme"   and V' :: "'a view_scheme" shows "V \<le> V' \<equiv> ext V \<le> ext V' \<and> lan V \<le> lan V' \<and> own V = own V' \<and> more V \<le> more V'"
nLt_le_conv: fixes x :: "'a"   and y :: "'a" shows "(comp x y \<noteq> Lt) = le y x"
length_bvxor: fixes a :: "bit list"   and b :: "bit list" assumes "length a = length b" shows "length (bvxor a b) = length a"
map_upds_def: fixes m :: "'a \<Rightarrow> 'b option"   and xs :: "'a list"   and ys :: "'b list" shows "m(xs [\<mapsto>] ys) = m ++ map_of (rev (zip xs ys))"
exhaust: fixes y :: "'b tree" obtains x1 :: "'b" and x2 :: "'b tree" and x3 :: "'b tree" where "y = Node x1 x2 x3"
finalStateCharacterizationLemma: fixes state :: "State"   and decisionVars :: "nat set"   and F0 :: "Literal list list" assumes "\<not> applicableDecide state decisionVars"   and "\<not> applicableBacktrack state F0" shows "\<not> formulaFalse F0 (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state)) \<or> formulaFalse F0 (elements (getM state)) \<and> decisions (getM state) = []"
prems_2: shows "(l # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
is_reachable_sim: fixes a :: "'a" assumes "a \<in> a.reachable" shows "\<exists>b. (a, b) \<in> R \<and> b \<in> b.reachable"
lookup_pair_tc_def: shows "lookup_pair_tc \<equiv> comparator.lookup_pair comparator_of"
replacement_preserves_congruences: fixes I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and u :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list"   and v :: "'b trm"   and t :: "'b trm"   and p :: "indices list"   and s :: "'b trm" assumes "congruence I"   and "I (u \<lhd> \<sigma>) (v \<lhd> \<sigma>)"   and "subterm t p u"   and "replace_subterm t p v s" shows "I (t \<lhd> \<sigma>) (s \<lhd> \<sigma>)"
hn_val_unfold: fixes R :: "('b \<times> 'a) set"   and a :: "'a"   and b :: "'b" shows "hn_val R a b = \<up> ((b, a) \<in> R)"
lsimulation_this: shows "lsimulation (R1 O R2) A C"
endofunctor_axioms: shows "endofunctor (\<bullet>) trg"
block_sizes: fixes bl :: "('a \<times> 'b) set" assumes "bl \<in># f B" shows "int (card bl) \<in> \<K>'"
matrix_mul_rid: fixes A :: "(('c, 'd) vec, 'e) vec" shows "A ** mat (1::'c) = A"
naturality: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.\<nu> (\<F>C.cod f) \<cdot> \<F>C.map f = (\<F>\<^sub>SC.map \<circ> \<F>C.D) f \<cdot> \<F>C.\<nu> (\<F>C.dom f)"
get_elements_by_tag_name_impl: shows "get_elements_by_tag_name = local.a_get_elements_by_tag_name"
cases: fixes a1 :: "(char list \<times> pat) list"   and a2 :: "(char list \<times> trm) list"   and a3 :: "trm list" assumes "\<turnstile> a1 [\<rhd>] a2 \<Rightarrow> a3" obtains fs :: "(char list \<times> trm) list" where "a1 = []" and "a2 = fsb__" and "a3 = []"   | fs :: "(char list \<times> trm) list" and l :: "char list" and t :: "trm" and p :: "pat" and ts :: "trm list" and fps :: "(char list \<times> pat) list" and us :: "trm list" where "a1 = (l, p) \<Colon> fps" and "a2 = fs" and "a3 = ts @ us" and "fs\<langle>l\<rangle>\<^sub>? = \<lfloor>t\<rfloor>" and "\<turnstile> p \<rhd> t \<Rightarrow> ts" and "\<turnstile> fps [\<rhd>] fs \<Rightarrow> us"
LT15: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) \<longrightarrow> G \<leadsto> H"
return_time_gt: fixes P :: "'a set"   and x :: "'a"   and t :: "real" assumes "returns_to P x"   and "closed P"   and "\<And>s. \<lbrakk>0 < s; s \<le> t\<rbrakk> \<Longrightarrow> flow0 x s \<notin> P" shows "t < return_time P x"
simps_2: fixes E :: "char list \<Rightarrow> 'a"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "char list"   and ts :: "fterm list" shows "eval\<^sub>l E F G (Neg p ts) = (\<not> G p (eval\<^sub>t\<^sub>s E F ts))"
measurable_fst': fixes f :: "'c \<Rightarrow> 'd \<times> 'e"   and M :: "'c measure"   and N :: "'d measure"   and P :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M N \<Otimes>\<^sub>M P" shows "(\<lambda>x. fst (f x)) \<in> M \<rightarrow>\<^sub>M N"
case_1: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f1 :: "'m"   and f2 :: "'m"   and f3 :: "'n \<Rightarrow> 'm" assumes "countable_bipartite_web \<Gamma>" shows "(case bi.SOURCE of bi.SOURCE \<Rightarrow> f1 | bi.SINK \<Rightarrow> f2 | bi.Inner x \<Rightarrow> f3 x) = f1"
lm005: fixes f :: "'c \<Rightarrow> 'd" shows "Graph f = graph UNIV f"
simps_5: fixes \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "ltlr_to_ltln (\<phi> and\<^sub>r \<psi>) = ltlr_to_ltln \<phi> and\<^sub>n ltlr_to_ltln \<psi>"
subfield_iff_2: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "ring R"   and "subfield K R" shows "field (R\<lparr>carrier := K\<rparr>)"
is_field_def: fixes m :: "qtname \<times> memberdecl" shows "is_field m = (\<exists>declC f. m = (declC, fdecl f))"
subset_mono: fixes N :: "('c, 'd) message set"   and M :: "('c, 'd) message set" assumes "N \<subseteq> M" shows "I ` insert_messages N \<subseteq> I ` insert_messages M"
assms_2: shows "(c, s) \<rightarrow>*c (c', s')"
is_none_code_2: fixes x :: "'c" shows "Option.is_none (Some x) = False"
assms_1: shows "w \<in> set_val_locs character_data_ptr"
cdf_is_right_cont: fixes a :: "real" shows "continuous (at_right a) C"
iso_homology_group_prod_deformation: fixes X :: "'a topology"   and f :: "'a \<Rightarrow> 'a"   and S :: "'a set"   and p :: "int" assumes "homotopic_with (\<lambda>x. True) X X id f"   and "f ` topspace X \<subseteq> S" shows "homology_group p (subtopology X S) \<cong> homology_group p X \<times>\<times> relative_homology_group (p + 1) X S"
path_component_of_maximal: fixes X :: "'c topology"   and s :: "'c set"   and x :: "'c" assumes "path_connectedin X s"   and "x \<in> s" shows "s \<subseteq> path_component_of_set X x"
assms_3: fixes i :: "nat"   and j :: "nat" assumes "i < nr"   and "j < nc" shows "r (A $$ (i, j)) (B $$ (i, j))"
simps_2: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" shows "cl_inv (Node l R k v r) = (cl_inv l \<and> cl_inv r \<and> cl l = B \<and> cl r = B)"
linorder_lattice_inf_greatest: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y"   and "x \<le> z" shows "x \<le> inf y z"
cos_le_one: fixes x :: "real" shows "cos x \<le> 1"
crel_vs_elim: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and v :: "'g"   and v\<^sub>T :: "(heap, 'h) state"   and M :: "heap" assumes "state_dp_consistency.crel_vs R v v\<^sub>T"   and "state_dp_consistency.cmem M"   and "P M" obtains v' :: "'h" and M' :: "heap" where "run_state v\<^sub>T M = (v', M')" and "R v v'" and "state_dp_consistency.cmem M'" and "P M'"
x_stays_before_y_if_y_not_moved_to_front: fixes q :: "'b"   and xs :: "'b list"   and x :: "'b"   and y :: "'b"   and n :: "nat" assumes "q \<in> set xs"   and "distinct xs"   and "x \<in> set xs"   and "y \<in> set xs"   and "y \<noteq> q"   and "x < y in xs" shows "x < y in mtf2 n q xs"
order_iff_strict: fixes a :: "'a"   and b :: "'a" shows "(b \<succeq> a) = (b \<succ> a \<or> a = b)"
Limsup_add_ereal_right: fixes F :: "'b filter"   and c :: "ereal"   and g :: "'b \<Rightarrow> ereal" assumes "F \<noteq> bot"   and "\<bar>c\<bar> \<noteq> \<infinity>" shows "Limsup F (\<lambda>n. g n + c) = Limsup F g + c"
square_free_int_rat: fixes f :: "int poly" assumes "square_free f" shows "square_free (of_int_poly f)"
rep_eq: fixes x :: "'c"   and xa :: "'c set_dlist" shows "list_of_dlist (DList_Set.insert x xa) = equal_base.list_insert ceq' x (list_of_dlist xa)"
F_injI: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> {0..<n} \<rightarrow> {0..<m}"   and "\<forall>i. i \<notin> {0..<n} \<longrightarrow> f i = i"   and "inj_on f {0..<n}" shows "f \<in> local.F_inj"
ennreal_minus_eq_0: fixes a :: "ennreal"   and b :: "ennreal" assumes "a - b = 0" shows "a \<le> b"
shiftl_word_eq: fixes w :: "'a word"   and n :: "nat" shows "w << n = push_bit n w"
UN_E: fixes b :: "'c"   and B :: "'d \<Rightarrow> 'c set"   and A :: "'d set" assumes "s2p (\<mu> (\<P> B A)) b" obtains x :: "'d" where "s2p A x" and "s2p (B x) b"
same_upto_trans: fixes j :: "nat"   and A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat" assumes "local.same_upto j A B"   and "local.same_upto j B C" shows "local.same_upto j A C"
simps: fixes a1 :: "nat \<times> nat \<times> thread"   and a2 :: "nat \<times> nat \<times> thread" shows "bstep_sng a1 a2 = ((\<exists>c n. a1 = (c, n, INIT) \<and> a2 = (c, (n + 1) mod (k * N), WAIT (n mod N))) \<or> (\<exists>c tk n. a1 = (c, n, WAIT tk) \<and> a2 = (c, n, WAIT tk) \<and> c \<noteq> tk) \<or> (\<exists>c n. a1 = (c, n, WAIT c) \<and> a2 = (c, n, HOLD c)) \<or> (\<exists>c n tk. a1 = (c, n, HOLD tk) \<and> a2 = (c, n, REL tk)) \<or> (\<exists>c n tk. a1 = (c, n, REL tk) \<and> a2 = ((tk + 1) mod N, n, INIT)))"
image_map_of: fixes xs :: "('b \<times> 'c) list" assumes "distinct (map fst xs)" shows "map_of xs ` set (map fst xs) = Some ` set (map snd xs)"
set_update_memI: fixes n :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "n < length xs" shows "x \<in> set (xs[n := x])"
coalesce_eq_OK2_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and zs :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [+f] ys) = OK zs" shows "ys [\<sqsubseteq>\<^bsub>r\<^esub>] zs"
divide_less_0_1_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (divide one a) zero = less a zero"
supports_finite: fixes S :: "'a set"   and x :: "'b" assumes "S supports x"   and "finite S" shows "finite (supp x)"
types_exist_lfilled: fixes k :: "nat"   and lholed :: "Lholed"   and es :: "e list"   and lfilled :: "e list"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" assumes "Lfilled k lholed es lfilled"   and "\<S>\<bullet>\<C> \<turnstile> lfilled : ts _> ts'" shows "\<exists>t1s t2s \<C>' arb_label. \<S>\<bullet>\<C>\<lparr>label := arb_label @ label \<C>\<rparr> \<turnstile> es : t1s _> t2s"
setVsList: fixes g1 :: "'d \<Rightarrow> 'c list list"   and G :: "'d"   and g2 :: "'c list \<Rightarrow> 'f \<Rightarrow> 'e list"   and N :: "'f"   and f2 :: "'c set \<Rightarrow> 'f \<Rightarrow> 'e set" assumes "\<forall>l\<in>set (g1 G). set (g2 l N) = f2 (set l) N" shows "set (map (\<lambda>l. set (g2 l N)) (g1 G)) = {f2 P N |P. P \<in> set (map set (g1 G))}"
r_neg2: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> (\<ominus>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> y) = y"
assms: shows "snd (execute_instr_sub1 i s) = False \<and> s' = snd (fst (execute_instr_sub1 i s)) \<and> get_S (cpu_reg_val PSR s) = 0"
I1: shows "callee_invariant_on oracle1 I1 \<I>"
hyps_2: fixes T\<^sub>1 :: "ty"   and T\<^sub>2 :: "ty" assumes "S = T\<^sub>1 \<rightarrow> T\<^sub>2"   and "val t" shows "\<exists>x ta S. t = (\<lambda>x:S. ta)"
self_sync: shows "A \<Turnstile>\<^sub>A (otherwith S {i} (orecvmsg R), other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, uu_, \<sigma>'). U (\<sigma> i) (\<sigma>' i))"
config_rand_dist_perm: fixes init :: "'b list"   and A :: "('b list \<Rightarrow> 'c pmf) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> ((nat \<times> nat list) \<times> 'c) pmf)"   and qs :: "'b list" assumes "distinct init" shows "\<forall>x\<in>set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). dist_perm (fst x) init"
signed_list_lconjaction_fst: fixes ss :: "'a list"   and x :: "'a \<times> bool" shows "fst (signed_list_lconjaction ss x) = lconjby (sum_list ss) (fst x)"
simps_4: fixes v :: "String.literal"   and va :: "typ"   and n :: "nat" shows "lift (Ct v va) n = Ct v va"
right_unit: fixes a :: "('a set \<times> 'b) set" assumes "a \<in> stx.carrier_direct_lim" shows "stx.add_rel a (stx.class_of X (zero_str\<^sub>X X)) = a"
AVar: fixes a :: "val"   and i :: "val" assumes "v = Lit a.[Lit i]" shows "thesis"
r_sigma: fixes i :: "nat"   and x :: "nat" shows "eval r_sigma [i, x] = \<phi> (the (eval r_limr [i])) x"
HSem_ignores_fresh_restr': fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "Vars.var set"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" assumes "fv \<Gamma> \<subseteq> S"   and "\<And>x \<rho>. x \<in> domA \<Gamma> \<Longrightarrow> \<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> the (map_of \<Gamma> x) \<rbrakk>\<^bsub>\<rho> f|` fv (the (map_of \<Gamma> x))\<^esub>" shows "(\<lbrace>\<Gamma>\<rbrace>\<rho>) f|` S = \<lbrace>\<Gamma>\<rbrace>\<rho> f|` S"
zero_iff_2: shows "(y = 0) = (y' = 0)"
update_matchF_base_def: fixes n :: "nat"   and I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and rels :: "'b option list set list"   and nt :: "'a" shows "update_matchF_base n I mr mrs rels nt = (let X = mrtabulate mrs (l\<epsilon>_strict n rels) in ([(nt, rels, if left I = 0 then lookup X mr else empty_table)], X))"
prv_dsj_impL: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "prv (imp \<phi> (dsj \<phi> \<chi>))"
foldri_cons_id: fixes l :: "'c list" shows "foldri l (\<lambda>_. True) (#) [] = l"
path_component_maximal: fixes x :: "'b"   and T :: "'b set"   and S :: "'b set" assumes "x \<in> T"   and "path_connected T"   and "T \<subseteq> S" shows "T \<subseteq> path_component_set S x"
ac_Zp_mult: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> nonzero Zp"   and "y \<in> nonzero Zp" shows "ac_Zp (x \<otimes> y) = ac_Zp x \<otimes> ac_Zp y"
secureI_secureFillI_Aux: fixes n :: "nat"   and k :: "nat"   and m :: "nat"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and J :: "OBJ"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and ss :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and tt :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<beta> :: "(Location \<times> Location) set"   and C :: "CtxtProg"   and I :: "OBJ"   and X :: "Var set" assumes "n \<le> k"   and "m \<le> k"   and " s , J \<rightarrow>\<^sub>n  t "   and " ss , J \<rightarrow>\<^sub>m  tt "   and "s \<equiv>\<^sub>\<beta> ss"   and "J = Fill C I"   and "LOW X C"   and "LOW X Ctxt_Body"   and "body = Fill Ctxt_Body I"   and "secure I" shows "\<exists>\<gamma>. t \<equiv>\<^sub>\<gamma> tt \<and> Pbij_extends \<gamma> \<beta>"
jpTraces_init_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envObs :: "'b \<Rightarrow> 'e \<Rightarrow> 'g"   and jviewInit :: "'b \<Rightarrow> 'g \<Rightarrow> 'f"   and jviewIncr :: "'b \<Rightarrow> 'g \<Rightarrow> 'f \<Rightarrow> 'f"   and s :: "'e"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'h list"   and envTrans :: "'h \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and jp :: "'b \<Rightarrow> ('g, 'd, 'i) Protocol" assumes "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"   and "tInit s \<in> IncrEnvironment.jpTraces envInit envAction envTrans envObs jp" shows "s \<in> set envInit"
induct: fixes P :: "'a prog"   and x1 :: "String.literal"   and x2 :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and Pa :: "String.literal \<Rightarrow> ((String.literal \<times> String.literal) \<times> ty \<times> fmod) list \<Rightarrow> bool" assumes "P \<turnstile> x1 has_fields x2"   and "\<And>C D fs ms FDTs FDTs'. \<lbrakk>class P C = \<lfloor>(D, fs, ms)\<rfloor>; C \<noteq> Object; P \<turnstile> D has_fields FDTs; Pa D FDTs; FDTs' = map (\<lambda>(F, Tm). ((F, C), Tm)) fs @ FDTs\<rbrakk> \<Longrightarrow> Pa C FDTs'"   and "\<And>D fs ms FDTs. \<lbrakk>class P Object = \<lfloor>(D, fs, ms)\<rfloor>; FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs\<rbrakk> \<Longrightarrow> Pa Object FDTs" shows "Pa x1 x2"
exec_wfp_no_error: fixes p :: "prog"   and \<sigma> :: "char list \<Rightarrow> nat option" assumes "wfp p"   and "\<sigma> \<in> states p" shows "\<not> body p \<rhd> \<sigma> \<leadsto> None"