hcju/isabelleps huggingface dataset:
finite_range_stream_space: fixes f :: "'c \<Rightarrow> 'a"   and i :: "nat"   and x :: "'c" assumes "finite (range f)" shows "(\<lambda>w. w !! i) -` open_exclude_set (f x) (range f) \<in> sets (stream_space borel)"
ltln_Until_to_StrongRelease: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> M\<^sub>n (\<phi> or\<^sub>n \<psi>)"
qGoodBinp_iff_goodBinp_asBinp: fixes qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option" shows "goodBinp (asBinp qbinp) = qGoodBinp qbinp"
uc_interr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<parallel> y \<cdot> (z \<parallel> U) = x \<cdot> (z \<parallel> U) \<parallel> (y \<cdot> (z \<parallel> U))"
row_space_is_preserved: fixes P :: "(('b, 'c) vec, 'c) vec"   and A :: "(('b, 'd) vec, 'c) vec" assumes "invertible P" shows "row_space (P ** A) = row_space A"
mult_numeral_1: fixes a :: "'a" shows "Numeral1 * a = a"
even_iff_mod_2_eq_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a = (modulo a (numeral.numeral one plus (num.Bit0 num.One)) = zero)"
leq_trans: fixes x :: "'d spmf"   and y :: "'d spmf"   and z :: "'d spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y z" shows "ord_spmf (=) x z"
gridgen_distinct: fixes p :: "(nat \<times> int) list"   and ds :: "nat set"   and l :: "nat" shows "distinct (gridgen p ds l)"
foldr_replicate: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and x :: "'d" shows "foldr f (replicate n x) = f x ^^ n"
HS_B_sup: fixes G :: "Var \<Rightarrow> L"   and b :: "BExpr"   and p :: "L"   and q :: "L" assumes "(G, b, p) \<in> HS_B"   and "LEQ p q" shows "(G, b, q) \<in> HS_B"
restrict_map_def: fixes m :: "'b \<Rightarrow> 'c option"   and A :: "'b set" shows "m |` A = (\<lambda>x. if x \<in> A then m x else None)"
assms: shows "\<forall>PROB s as. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)"
dist_at: fixes vs :: "'a list"   and a :: "'a list"   and r :: "'a"   and b :: "'a list"   and c :: "'a list"   and d :: "'a list" assumes "distinct vs"   and "vs = a @ r # b"   and "vs = c @ r # d" shows "a = c \<and> b = d"
distinct_11: fixes x3 :: "qtname"   and x4 :: "ty" shows "ClassT x3 \<noteq> ArrayT x4"
mult_cong: fixes x :: "int"   and y :: "int" shows "x mod int p \<otimes> (y mod int p) = x * y mod int p"
favorites_no_agent: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and i :: "'c" assumes "pref_profile_wf agents alts R"   and "i \<notin> agents" shows "favorites R i = {}"
eq_iff: fixes a :: "'c"   and b :: "'c" shows "(a = b) = (a \<le> b \<and> b \<le> a)"
adj_cblinfun_compose: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'a"   and B :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "(A o\<^sub>C\<^sub>L B)* = B* o\<^sub>C\<^sub>L A*"
intro: fixes bind :: "'e \<Rightarrow> ('d \<Rightarrow> 'e) \<Rightarrow> 'e"   and fail :: "'e" assumes "\<And>f. bind fail f = fail" shows "monad_fail_axioms bind fail"
dropD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m Drop]"   and "matches \<gamma> m p"   and "s = Undecided" shows "t = Decision FinalDeny"
ih_3: assumes "x \<sharp> P" shows "x \<sharp> P'"
iFROM_add: fixes n :: "nat"   and k :: "nat" shows "[n\<dots>] \<oplus> k = [n + k\<dots>]"
assms: shows "bound_intersect_2d prec Xs x = Some (m, M)"
in_lists_conv_set: fixes xs :: "'a list"   and A :: "'a set" shows "(xs \<in> lists A) = (\<forall>x\<in>set xs. x \<in> A)"
im_zero: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom R smult M smult' T" shows "T (0::'b) = (0::'c)"
degree_sum_list_le: fixes ps :: "'a poly list"   and T :: "nat" assumes "\<And>p. p \<in> set ps \<Longrightarrow> degree p \<le> T" shows "degree (sum_list ps) \<le> T"
x: shows "Derives1 (u @ v) (fst d) (snd d) x \<and> Derivation x D w"
f_1: shows "f = (\<lambda>x. ennreal (spmf p x))"
lapp_inv2T: fixes r :: "'b llist"   and s :: "'b llist"   and A :: "'b set" assumes "r @@ s \<in> A\<^sup>\<omega>" shows "r \<in> A\<^sup>\<star> \<and> s \<in> A\<^sup>\<omega> \<or> r \<in> A\<^sup>\<omega>"
canonical_retraction_uniform: fixes A :: "'a set set"   and B :: "'a set set"   and C :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "chamber C"   and "C \<in> A \<inter> B" shows "fun_eq_on (canonical_retraction A C) (the_apartment_iso B A) (\<Union> B)"
1: shows "(tab\<^sub>0 r)\<^sup>* \<star> tab\<^sub>1 s \<cong> (Maps.REP R\<^sub>0)\<^sup>* \<star> Maps.REP S\<^sub>1"
sum_diff1_nat: fixes f :: "'a \<Rightarrow> nat"   and A :: "'a set"   and a :: "'a" shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
big_iff_small: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int" shows "cs \<Rightarrow> t = cs \<rightarrow>* (SKIP, t)"
ind_eq_fin: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'a list"   and v :: "'a list" assumes "traces ind"   and "\<forall>a\<in>set u. \<forall>b\<in>set v. ind a b" shows "(traces.eq_swap ind)\<^sup>*\<^sup>* (u @ v) (v @ u)"
simps_1: fixes Q :: "hf set" shows "nextl Q [] = epsclo Q"
freeliftid_trivial_relator_freeword_R: fixes rs :: "'g list" assumes "rs \<in> R" shows "freeword_funlift id (relator_freeword rs) = (0::'g)"
mset_tree_empty: fixes t :: "('b \<times> 'c) tree" shows "(mset_tree t = {#}) = (t = \<langle>\<rangle>)"
map_upd_triv: fixes t :: "'c \<Rightarrow> 'b option"   and k :: "'c"   and x :: "'b" assumes "t k = Some x" shows "t(k \<mapsto> x) = t"
finite_Union: fixes X :: "'c set set"   and M :: "'c measure" assumes "finite X"   and "X \<subseteq> fmeasurable M" shows "\<Union> X \<in> fmeasurable M"
fstI: fixes x :: "'e \<times> 'f"   and y :: "'e"   and z :: "'f" assumes "x = (y, z)" shows "fst x = y"
list_gcd: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs" shows "gcd_list xs dvd x"
orderE: fixes b :: "'a Word.word"   and a :: "'a Word.word" assumes "b \<le>s a" obtains "a = signed.max a b"
measurable_return_val: fixes t :: "pdf_type" shows "return_val \<in> stock_measure t \<rightarrow>\<^sub>M subprob_algebra (stock_measure t)"
absorb_iff2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
pareto_efficient: fixes agents :: "'a set"   and alts :: "'b set"   and scf :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "pareto_efficient_scf agents alts scf"   and "pref_profile_wf agents alts R" shows "scf R \<inter> pareto_losers R = {}"
intro: fixes f :: "'a fds" assumes "\<And>n. 0 < n \<Longrightarrow> fds_nth f n \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "nonneg_dirichlet_series f"
non_empty_cycle_root_start_end_points_plus: fixes r :: "'a"   and x :: "'a" assumes "non_empty_cycle_root r x" shows "connected_root (x ; 1) x"
maxsimp_in_opp_half_apartment: fixes C :: "'a set" assumes "SimplicialComplex.maxsimp folding_f.Y C" shows "C \<in> folding_g.\<C> - f \<turnstile> folding_g.\<C>"
mult_less_0_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a b) zero = (less zero a \<and> less b zero \<or> less a zero \<and> less zero b)"
normalizability_I': fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> A\<^sup>*"   and "(b, c) \<in> A\<^sup>!" shows "(a, c) \<in> A\<^sup>!"
Sup_pres_ladj: fixes f :: "'c \<Rightarrow> 'd" assumes "Sup_pres f" shows "\<exists>g. f \<stileturn> g"
\<epsilon>_def: shows "\<epsilon> \<equiv> \<Sum>\<^sup>+ x\<in>B \<Gamma>. d_IN g x - d_OUT g x"
extremes_equiv: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c"   and c :: "'c" assumes "extreme X r b"   and "extreme X r c" shows "sympartp r b c"
simps_9: fixes f :: "'b \<Rightarrow> 'a"   and x2 :: "'b" shows "map_option f \<lfloor>x2\<rfloor> = \<lfloor>f x2\<rfloor>"
VVp_to_V: fixes v :: "'a"   and p :: "Player" assumes "v \<in> G'.VV p" shows "v \<in> V\<^bsub>G'\<^esub>"
decode_plan_step_element_then_1: fixes k :: "nat"   and t :: "nat"   and op :: "'b strips_operator"   and \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool" assumes "k < t"   and "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)" shows "op \<in> set (\<Pi>\<^sub>\<O>)"
x: shows "x = median (map median_le_5 (chop 5 xs))"
infinite_set_has_order_preserving_inj: fixes E :: "'a set" assumes "E \<noteq> {}"   and "\<forall>x. \<exists>y\<in>E. x < y" shows "\<exists>f. range f \<subseteq> E \<and> inj f \<and> (\<forall>m. f m < f (Suc m))"
kauff_mat_tensor_distrib: shows "\<forall>xs ys. kauff_mat (basic xs \<otimes> basic ys) = kauff_mat (basic xs) \<otimes> kauff_mat (basic ys)"
CONS_neq_NIL: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set"   and R :: "bool" assumes "CONS M N = NIL" shows "R"
unsat_subs_unsat: fixes c1 :: "('d, 'e) conf"   and c2 :: "('d, 'e) conf" assumes "\<not> Conf.sat c1"   and "c2 \<sqsubseteq> c1" shows "\<not> Conf.sat c2"
ANR_Times: fixes S :: "'a set"   and T :: "'b set" assumes "ANR S"   and "ANR T" shows "ANR (S \<times> T)"
IH: fixes acc :: "nat" shows "totient_naive k acc n = card {x \<in> {0<..k}. coprime x n} + acc"
weakPsiCongReflexive: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<doteq> P"
AE_const: fixes P :: "bool"   and M :: "'b pmf" shows "(AE x in measure_pmf M. P) = P"
collapse_2: fixes tm :: "('a, 'b) tm" assumes "is_App tm" shows "App (fun tm) (arg tm) = tm"
assms_3: fixes a :: "'al" assumes "a \<in> A" shows "Bex B ((\<le>\<^sub>o\<^sub>w) a)"
lfp_const: fixes t :: "'d" shows "(\<nu> x \<bullet> t) = t"
list_exhaust3: fixes xs :: "'a list" obtains "xs = []"   | x :: "'a" where "xs = [x__]"   | x :: "'a" and y :: "'a" and ys :: "'a list" where "xs = x # y # ys"
surj_compose: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'e"   and C :: "'e set" assumes "f ` A = B"   and "g ` B = C" shows "compose A g f ` A = C"
lens_override_put_right_out: fixes A :: "'e \<Longrightarrow> 'f"   and X :: "'g \<Longrightarrow> 'f"   and S\<^sub>1 :: "'f"   and S\<^sub>2 :: "'f"   and v :: "'g" assumes "vwb_lens A"   and "X \<bowtie> A" shows "S\<^sub>1 \<triangleleft>\<^bsub>A\<^esub> put\<^bsub>X\<^esub> S\<^sub>2 v = S\<^sub>1 \<triangleleft>\<^bsub>A\<^esub> S\<^sub>2"
\<A>'_1: shows "\<A>' \<in> set (tr\<^sub>p\<^sub>c \<A> [])"
partition: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> - y = x"
k_1: fixes G1 :: "'b poly"   and G2 :: "'b poly" shows "??.Subresultant.F G1 G2 (Suc (??.Subresultant.k G1 G2)) = 0"
mwb_lens_weak: fixes x :: "'c \<Longrightarrow> 'd" assumes "mwb_lens x" shows "weak_lens x"
disjunct_symbols: shows "\<NN> \<inter> \<TT> = {}"
omega_complete_def: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "omega_complete A r \<equiv> {range f |f. monotone (\<le>) r f}-complete A r"
lookup_update: fixes k :: "'e"   and v :: "'d"   and m :: "('e, 'd) mapping" shows "Mapping.lookup (Mapping.update k v m) k = Some v"
that: assumes "\<And>c. c \<in> set cs \<Longrightarrow> P c w" shows "thesis"
con_rews_2: fixes a :: "'a" shows "a :# \<bottom> = \<bottom>"
FSwImorph_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h, 'i) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.FSwImorph Delta h hA (FixSyn.termMOD Delta) MOD = FixSyn.termFSwImorph Delta h hA MOD"
do_opapp_cases: fixes vs :: "v list"   and env' :: "v sem_env"   and exp' :: "exp0" assumes "do_opapp vs = Some (env', exp')" obtains env :: "v sem_env" and n :: "char list" and v0 :: "v" where "vs = [Closure env__ n__ exp', v0__]" and "env' = update_v (\<lambda>_. nsBind n__ v0__ (sem_env.v env__)) env__"   | env :: "v sem_env" and funs :: "(char list \<times> char list \<times> exp0) list" and name :: "char list" and n :: "char list" and v0 :: "v" where "vs = [Recclosure env funs name, v0]" and "allDistinct (map (\<lambda>(f, uu_, uu_). f) funs)" and "find_recfun name funs = Some (n, exp')" and "env' = update_v (\<lambda>_. nsBind n v0 (build_rec_env funs env (sem_env.v env))) env"
flr: shows "\<lfloor>l'\<rfloor> \<le> \<lfloor>r'\<rfloor>"
checked_invariant_nsc: fixes init_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'c"   and assert_nsc :: "'a \<Rightarrow> 'c \<Rightarrow> 'a list + 'c"   and check_nsc :: "'c \<Rightarrow> 'a list + 'c"   and solution_nsc :: "'c \<Rightarrow> (nat, 'b) mapping"   and checkpoint_nsc :: "'c \<Rightarrow> 'd"   and backtrack_nsc :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and invariant_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and checked_nsc :: "('a \<times> 'b ns_constraint) list \<Rightarrow> 'a set \<Rightarrow> 'c \<Rightarrow> bool"   and nsc :: "('a \<times> 'b ns_constraint) list"   and J :: "'a set"   and s :: "'c" assumes "Incremental_NS_Constraint_Ops init_nsc assert_nsc check_nsc solution_nsc checkpoint_nsc backtrack_nsc invariant_nsc checked_nsc"   and "checked_nsc nsc J s" shows "invariant_nsc nsc J s"
h11: shows "i (t + 3 + k + d) \<noteq> []"
simps_5: fixes v :: "32 word" shows "is_Addr (Intg v) = False"
monotone_rel_prod_iff: fixes f :: "'a \<times> 'b \<Rightarrow> 'c" shows "monotone (rel_prod orda ordb) ordc f = ((\<forall>a. monotone ordb ordc (\<lambda>b. f (a, b))) \<and> (\<forall>b. monotone orda ordc (\<lambda>a. f (a, b))))"
gr_implies_not_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
linord_of_list_sorted_on: fixes xs :: "'a list" assumes "distinct xs" shows "sorted_on (linord_of_list (rev xs)) xs"
T_Let: fixes \<Gamma> :: "binding list"   and t\<^sub>1 :: "trm"   and T\<^sub>1 :: "POPLmarkRecord.type"   and p :: "pat"   and \<Delta> :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t\<^sub>1 : T\<^sub>1"   and "\<turnstile> p : T\<^sub>1 \<Rightarrow> \<Delta>"   and "\<Delta> @ \<Gamma> \<turnstile> t\<^sub>2 : T\<^sub>2" shows "\<Gamma> \<turnstile> (LET p = t\<^sub>1 IN t\<^sub>2) : \<down>\<^sub>\<tau> \<parallel>\<Delta>\<parallel> 0 T\<^sub>2"
gauss_jordan_transform: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and nc' :: "nat"   and A' :: "'b mat"   and B' :: "'b mat"   and b :: "'c" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc'"   and "gauss_jordan A B = (A', B')" shows "\<exists>P\<in>Units (ring_mat TYPE('b) nr b). A' = P * A \<and> B' = P * B"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "(a, b) \<in> rel D"   and "(b, c) \<in> rel D" shows "(a, c) \<in> rel D"
cases: fixes x :: "tm \<times> tm" obtains xa :: "name" and p :: "tm" and q :: "tm" and v :: "name" and y :: "name" and x' :: "name" where "atom xa \<sharp> (p, q, v, y, x')" and "atom x' \<sharp> (p, q, v, y)" and "atom y \<sharp> (p, q, v)" and "atom v \<sharp> (p, q)" and "x = (p, q)"
set_tag_name_get_tag_name: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "char list"   and h' :: "(_) heap" assumes "l_set_tag_name_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs set_tag_name set_tag_name_locs"   and "h \<turnstile> l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_tag_name element_ptr tag \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name element_ptr \<rightarrow>\<^sub>r tag"
any_node_exits_path: fixes G :: "nat graph_rec"   and v :: "nat" assumes "cfg_doms G"   and "v \<in> set (g_V G)" shows "\<exists>p. path_entry (g_E G) p v"
hd_repeat_cases: fixes P :: "'a \<Rightarrow> bool"   and m :: "nat"   and a :: "'a"   and r :: "'a list" shows "P (hd (a \<up> m @ r)) = ((m = 0 \<longrightarrow> P (hd r)) \<and> (\<forall>nat. m = Suc nat \<longrightarrow> P a))"
init_simps_8: fixes a :: "ATypeId" shows "init (AClassT a) = nullV"
lexord_linear: fixes r :: "('b \<times> 'b) set"   and x :: "'b list"   and y :: "'b list" assumes "\<forall>a b. (a, b) \<in> r \<or> a = b \<or> (b, a) \<in> r" shows "(x, y) \<in> lexord r \<or> x = y \<or> (y, x) \<in> lexord r"
qSwapAll_preserves_alphaAll: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" shows "(qGood X \<longrightarrow> (\<forall>X' zs z1 z2. X #= X' \<longrightarrow> X #[[z1 \<and> z2]]_zs #= X' #[[z1 \<and> z2]]_zs)) \<and> (qGoodAbs A \<longrightarrow> (\<forall>A' zs z1 z2. A $= A' \<longrightarrow> A $[[z1 \<and> z2]]_zs $= A' $[[z1 \<and> z2]]_zs))"
associated_polynomials_imp_same_length: fixes K :: "'a set"   and p :: "'a list"   and q :: "'a list" assumes "subring K R"   and "p \<in> carrier (K [X])"   and "q \<in> carrier (K [X])"   and "p \<sim>\<^bsub>K [X]\<^esub> q" shows "length p = length q"
\<Z>_\<beta>: fixes \<beta> :: "V" assumes "\<beta> = \<beta>" shows "\<Z> \<beta>"
jw_Fun_inv: fixes v :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty"   and c :: "Syntax.term" assumes "{$$} \<turnstile>\<^sub>W v : Fun \<tau>\<^sub>1 \<tau>\<^sub>2"   and "value v" obtains e :: "Syntax.term" and x :: "Syntax.var" where "v = Lam x e \<or> v = Rec x e" and "atom x \<sharp> c"
dres_order_simps_8: shows "dSUCCEED \<noteq> dFAIL"
IH: fixes ops :: "('a \<times> 'a option) list" assumes "insert_ops ops"   and "ops = xs @ [(oid, Some ref)] @ ys"   and "ref \<notin> set (interp_ins xs)" shows "oid \<notin> set (interp_ins ops)"
sum_eq: fixes pa :: "(nat \<times> int) list"   and \<alpha>' :: "(nat \<times> int) list \<Rightarrow> real" assumes "pa \<in> sparsegrid dm lm" shows "(\<Sum>p'\<in>lgrid (base {d} pa) {d} lm. if p' \<in> parents d (base {d} pa) pa then \<alpha>' p' * l2_\<phi> (pa ! d) (p' ! d) else 0) = (\<Sum>p'\<in>parents d (base {d} pa) pa. \<alpha>' p' * l2_\<phi> (pa ! d) (p' ! d))"
strict_trans1: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "c < b" shows "c < a"
zero_spaceN_add: fixes x :: "'c"   and N :: "'c quasinorm"   and y :: "'c" assumes "x \<in> zero_space\<^sub>N N"   and "y \<in> zero_space\<^sub>N N" shows "x + y \<in> zero_space\<^sub>N N"
correctCompositionDiffLevels_def: fixes S :: "CSet" shows "correctCompositionDiffLevels S \<equiv> \<forall>C\<in>subcomp S. \<forall>i. S \<in> AbstrLevel i \<longrightarrow> C \<notin> AbstrLevel i"
T4: shows "serial (\<lambda>w v. w r v \<and> (\<exists>x. \<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v)) \<longrightarrow> universal (\<lambda>w v. w r v \<longrightarrow> (\<exists>x. existsAt x v \<and> (\<forall>xa. (\<P> xa \<^bold>\<rightarrow> xa x) v)))"
in_sorted_domain_iff_lookup: fixes k :: "'b"   and ko :: "'b nat_term_order"   and xs :: "('b, 'c) oalist_ntm" shows "(k \<in> set (OAlist_sorted_domain_ntm ko xs)) = (OAlist_lookup_ntm xs k \<noteq> (0::'c))"
H_1: shows "C # Cs @ [D] = As @ [A, B] @ Bs"
degree_def': fixes K :: "'a set"   and p :: "'a list" assumes "polynomial K p" shows "degree p = (LEAST n. \<forall>i>n. local.coeff p i = \<zero>)"
rs_def: shows "rs = map (\<lambda>x. (x, p x)) (filter ((<) z) zs)"
ithI: fixes f :: "'a \<Rightarrow> 'b"   and B :: "('a \<Rightarrow> 'b) set"   and i :: "'a"   and x :: "'b" assumes "f \<in> B"   and "f i = x" shows "x \<in> ith B i"
SN_of_FST_of_App: fixes t :: "lam"   and s :: "lam" assumes "SN (App t s)" shows "SN (fst_app (App t s))"
integrable_pmf: fixes X :: "'e set"   and M :: "'e pmf" shows "integrable (count_space X) (pmf M)"
diff_of_aezfun_is_aezfun: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> aezfun_set"   and "g \<in> aezfun_set" shows "f - g \<in> aezfun_set"
prikey_crypt: fixes K :: "enc_key"   and K' :: "nat"   and s :: "(agent \<times> msg) set" assumes "(Spy, Crypt K (AgrKey (PriK K'))) \<in> s"   and "insert (Spy, AgrKey (PubK {})) (insert (Spy, AgrKey (PubK {Rev_PriK})) ({Spy} \<times> ((\<lambda>x. EncKey (SigK x)) ` bad_sigk \<union> (\<lambda>x. AgrKey (PriK (Sec_PriK x))) ` bad_sec_prik \<union> (\<lambda>x. AgrKey (PriK (Tok_PriK x))) ` bad_tok_prik \<union> range (\<lambda>x. EncKey (VerK x)) \<union> range (\<lambda>x. AgrKey (PubK {Sec_PriK x, Rev_PriK})) \<union> range (\<lambda>x. AgrKey (PubK {Tok_PriK x})) \<union> range (\<lambda>(n, m). AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK})) \<union> range (\<lambda>(n, m). Hash (AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK}))) \<union> range (\<lambda>x. \<langle>x, AgrKey (PubK {Tok_PriK x})\<rangle>) \<union> {\<langle>n, AgrKey (PriK (Tok_PriK n))\<rangle> |n. n \<in> bad_tok_prik} \<union> {\<langle>n, AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK})\<rangle> |n m. (n, m) \<in> bad_id} \<union> {\<langle>n, Hash (AgrKey (PubK {Tok_PriK n, Sec_PriK m, Rev_PriK}))\<rangle> |n m. (n, m) \<in> bad_id}))) \<Turnstile> s" shows "(Spy, AgrKey (PriK K')) \<in> s"
verit_sum_simplify: fixes a :: "'b" shows "a + (0::'b) = a"
llcp_eq_0_iff: fixes xs :: "'c llist"   and ys :: "'c llist" shows "(llcp xs ys = 0) = (lnull xs \<or> lnull ys \<or> lhd xs \<noteq> lhd ys)"
elims_2: fixes x :: "('c, 'd, 'varSort, 'var, 'e) qAbs" assumes "qGoodAbs x" obtains xs :: "'varSort" and xa :: "'var" and X :: "('c, 'd, 'varSort, 'var, 'e) qTerm" where "x = qAbs xs xa X" and "qGood X"
sum_list_le_sum_list': fixes xs :: "'a list"   and p :: "nat" shows "sum_list xs \<le> sum_list' p xs"
assoc: fixes c :: "'a"   and b :: "'a"   and a :: "'a" shows "c * (b * a) = c * b * a"
trace_callee_None: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('d \<times> 'e) spmf"   and xs :: "('f \<times> 'd) list"   and x :: "'f" shows "trace_callee callee (return_pmf None) xs x = return_pmf None"