hcju/isabelleps huggingface dataset:
that: assumes "set (awalk_verts u p) \<subseteq> verts G"   and "set p \<subseteq> arcs G"   and "cas u p v"   and "awhd u p = u"   and "awlast u p = v" shows "thesis"
return_op_sem: fixes x :: "val" assumes "x \<in> space (stock_measure t)" shows "return_val (op_sem oper x) = return (stock_measure t') (op_sem oper x)"
differenceSumVsCardinalityReal: fixes Z :: "'a set"   and X :: "'a set"   and Y :: "'a set" assumes "Z \<subseteq> X \<union> Y"   and "finite Z" shows "real (sum (X <| Y) X) - real (sum (X <| Y) Z) = real (card X) - real (card (Z \<inter> X))"
nabla_nu: fixes x :: "'a" shows "\<nu> (\<nabla> x) = \<nabla> x"
partial_order_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "partial_order_on A r \<equiv> preorder_on A r \<and> antisym r"
induce_subgraph_arcs: fixes G :: "('d, 'c) pre_digraph"   and vs :: "'d set" shows "arcs (G \<restriction> vs) = {e \<in> arcs G. tail G e \<in> vs \<and> head G e \<in> vs}"
f'a_nonzero: shows "f' \<bullet> a \<noteq> \<zero>"
ParTR_MtransC: fixes c2 :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state"   and c1 :: "('test, 'atom) com" assumes "(c2, s) \<rightarrow>*t s'" shows "(c1 | c2, s) \<rightarrow>*c (c1, s')"
a: shows "\<forall>x. x \<in> Q \<longrightarrow> (x, f x) \<in> r \<and> f x \<in> Q"
sep_conj_sep_true': fixes P :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "P h" shows "((\<lambda>s. True) \<and>* P) h"
fun_rel_id: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('c \<times> 'c) set" assumes "R1 = Id"   and "R2 = Id" shows "R1 \<rightarrow> R2 = Id"
sub_emb: fixes s :: "('c, 'd) tm"   and t :: "('c, 'd) tm" assumes "sub s t" shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
enum_ex_set_def: fixes P :: "'a set \<Rightarrow> bool" shows "enum_ex P = (\<exists>A\<in>set enum. P A)"
line_integral_exists: shows "\<forall>b\<in>basis. \<forall>(k, \<gamma>)\<in>one_chain. line_integral_exists F {b} \<gamma>"
sas_plus_equivalent_to_strips_ii: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "STRIPS_Semantics.is_parallel_solution_for_problem (\<phi> \<Psi> ) \<pi>" shows "list_all (list_all (\<lambda>op. ListMem op (\<Psi>\<^sub>\<O>\<^sub>+))) (\<phi>\<^sub>P\<inverse> \<Psi> \<pi>)"
monotone_lset: shows "monotone lprefix (\<subseteq>) lset"
assert_bound_nolhs_tableau_valuated: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s" shows "\<nabla> (assert_bound a s)"
akra_bazzi_integrable: fixes a :: "real"   and b :: "real" assumes "x\<^sub>0 \<le> a"   and "a \<le> b" shows "integrable (\<lambda>x. g x / x powr (p + 1)) a b"
inj_on_distinct_setlistmapim: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and Xs :: "'b set list" assumes "inj_on f A"   and "\<forall>X\<in>set Xs. order.greater_eq A X"   and "distinct Xs" shows "distinct (f \<Turnstile> Xs)"
in_set_permutationss: fixes xss :: "'b list list"   and yss :: "'b list list" assumes "distincts xss" shows "(yss \<in> set (permutationss xss)) = (distincts yss \<and> map set xss = map set yss)"
rt_fresh_as_refl: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" shows "rt \<approx>\<^bsub>dip\<^esub> rt"
redT_updWs_preserve_wset_thread_ok: fixes ws :: "'a \<Rightarrow> 'b wait_set_status option"   and ts :: "'a \<Rightarrow> ('c \<times> 'd \<Rightarrow>f nat) option"   and t :: "'a"   and was :: "('a, 'b) wait_set_action list"   and ws' :: "'a \<Rightarrow> 'b wait_set_status option"   and xln :: "'c \<times> 'd \<Rightarrow>f nat" assumes "wset_thread_ok ws ts"   and "redT_updWs t ws was ws'"   and "ts t = \<lfloor>xln\<rfloor>" shows "wset_thread_ok ws' ts"
bdd_above_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq A = (\<exists>M. \<forall>x\<in>A. less_eq x M)"
elims: fixes x :: "three"   and xa :: "three"   and y :: "three" assumes "f1 x xa = y" obtains xb :: "three" where "x = xb__" and "xa = Zero" and "y = xb__"   | "x = Zero" and "xa = One" and "y = One"   | "x = One" and "xa = One" and "y = Two"   | "x = Two" and "xa = One" and "y = Two"   | xb :: "three" where "x = xb" and "xa = Two" and "y = Two"
out_port_cases: fixes p :: "('form, 'var) out_port"   and n :: "('form, 'rule) graph_node" assumes "p |\<in>| outPorts n" obtains a :: "'form" where "n = Assumption a" and "p = Reg a"   | r :: "'rule" and h :: "'form" and c :: "('form, 'var) antecedent" where "n = Rule r__" and "p = Hyp h c"   | r :: "'rule" and f :: "'form" where "n = Rule r" and "p = Reg f"   | "n = Helper" and "p = Reg anyP"
prems_2: shows "is \<lhd> input_sizes (Pool m1 m2)"
a_get_tag_name_locs_def: fixes element_ptr :: "(_) element_ptr" shows "local.a_get_tag_name_locs element_ptr \<equiv> {preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr tag_name)}"
assms: shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r"
a: shows "snd (rep_nat_term u) < snd (rep_nat_term v)"
well_scoped_inst: fixes v :: "'vertex"   and p :: "'inPort"   and var :: "'var"   and v' :: "'vertex" assumes "valid_in_port (v, p)"   and "var \<in> local_vars (nodeOf v) p"   and "v' |\<in>| vertices"   and "freshenLC (vidx v) var \<in> subst_lconsts (inst v')" shows "v' \<in> scope (v, p)"
cases: fixes a :: "trm" assumes "nump a" obtains "a = zer"   | t :: "trm" where "a = suc t" and "nump t"
Root_CTRL_def: shows "Root_CTRL = Abs_seqauto (Root_CTRL_States, Root_CTRL_Init, Root_CTRL_Labels, Root_CTRL_Delta)"
E: fixes i :: "nat" shows "E i \<in> sets (M i)"
rec_5: shows "sorted_wrt (\<succ>\<^sub>t) ks"
inject_1: fixes x2 :: "qtname"   and y2 :: "qtname" shows "(IfaceT x2 = IfaceT y2) = (x2 = y2)"
cont: fixes x :: "real" assumes "a \<le> x"   and "x \<le> b" shows "isCont f x"
cases: fixes a1 :: "'t"   and a2 :: "(status \<times> status \<times> 'x) \<times> 'm"   and a3 :: "('l, 't, status \<times> status \<times> 'x, 'm, 'w, 'o action action) thread_action"   and a4 :: "(status \<times> status \<times> 'x) \<times> 'm" assumes "if.init_fin a1 a2 a3 a4" obtains t :: "'t" and x :: "status \<times> 'x" and m :: "'m" and ta :: "('l, 't, status \<times> 'x, 'm, 'w, 'o action) thread_action" and x' :: "status \<times> 'x" and m' :: "'m" where "a1 = t__" and "a2 = ((Running, x__), m__)" and "a3 = convert_TA_initial (convert_obs_initial ta)" and "a4 = ((Running, x'), m')" and "if.r_syntax t__ x__ m__ ta x' m'"   | t :: "'t" and x :: "status \<times> 'x" and m :: "'m" where "a1 = tb__" and "a2 = ((PreStart, xa__), ma__)" and "a3 = \<lbrace>InitialThreadAction\<rbrace>" and "a4 = ((Running, xa__), ma__)"   | x :: "status \<times> 'x" and t :: "'t" and m :: "'m" where "a1 = t" and "a2 = ((Running, x), m)" and "a3 = \<lbrace>ThreadFinishAction\<rbrace>" and "a4 = ((Finished, x), m)" and "init_fin_final x"
poly_y_x_poly_lift: fixes p :: "'c poly" shows "poly_y_x (poly_lift p) = [:p:]"
HPhase1or2ReadElse_HInv5: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadElse s s' p d r"   and "HInv5_inner s q" shows "HInv5_inner s' q"
Dynkin_system_Dynkin: fixes M :: "'c set set"   and \<Omega> :: "'c set" assumes "M \<subseteq> Pow \<Omega>" shows "Dynkin_system \<Omega> (Dynkin \<Omega> M)"
finite_insert: fixes a :: "'a"   and A :: "'a set" shows "finite (a \<triangleright> A) = finite A"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "P []"   and "\<And>x xs. \<lbrakk>P (filter (\<lambda>y. \<not> x \<le> y) xs); P (filter ((\<le>) x) xs)\<rbrakk> \<Longrightarrow> P (x # xs)" shows "P a0"
Rep_Abs_eq_Pg: fixes g :: "game" shows "Rep_Pg (Abs_Pg (eq_game_rel `` {g})) = eq_game_rel `` {g}"
add_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a + c \<le> b + c"
assms_2: fixes x :: "real" assumes "x \<in> {a..b}" shows "f' x \<le> u"
complex_In_mult_cnj_zero: fixes z :: "complex" shows "is_real (z * cnj z)"
a_b_3: shows "q \<approx>[\<R>] mix_pmf \<alpha> p r"
aboveS_Above_trans: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and C :: "'b set" assumes "trans r"   and "antisym r"   and "a \<in> aboveS r b"   and "b \<in> Above r C" shows "a \<in> AboveS r C"
wens_Int_eq_lemma: fixes T :: "'b set"   and B :: "'b set"   and F :: "'b program"   and act :: "('b \<times> 'b) set" assumes "T - B \<subseteq> awp F T"   and "act \<in> Acts F" shows "T \<inter> wens F act B \<subseteq> wens F act (T \<inter> B)"
weak_nominal_ts_axioms: shows "weak_nominal_ts (\<turnstile>) (\<rightarrow>) \<tau>"
pdom_res_upd_in: fixes k :: "'a"   and A :: "'a set"   and f :: "('a, 'b) pfun"   and v :: "'b" assumes "k \<in> A" shows "A \<lhd>\<^sub>p f(k \<mapsto> v)\<^sub>p = (A \<lhd>\<^sub>p f)(k \<mapsto> v)\<^sub>p"
bind_UNION: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "Set.bind A f = \<Union> (f ` A)"
nres_more_simps_5: fixes x :: "'b"   and X :: "'b set" shows "(RETURN x = RES X) = ({x} = X)"
cdot_square_norm: fixes x :: "'b" shows "\<langle>x, x\<rangle> = complex_of_real ((norm x)\<^sup>2)"
subgroup_imp_finite_comm_group: fixes H :: "'a set" assumes "subgroup H (G\<lparr>carrier := H'\<rparr>)" shows "finite_comm_group (G\<lparr>carrier := H', carrier := H\<rparr>)"
ocircline_set_I: fixes H :: "ocircline"   and z :: "complex_homo" assumes "on_ocircline H z" shows "z \<in> ocircline_set H"
tail_invar_def: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "tail_invar bq = (queue_invar bq \<and> rank_invar bq)"
elims: fixes x :: "dynamic"   and y :: "nat option" assumes "unbox_num x = y" obtains n :: "nat" where "x = DNum n" and "y = Some n"   | "x = DNil" and "y = None"   | v :: "bool" where "x = DBool v" and "y = None"
RBTi1_axioms: shows "RBTi1 TYPE('a) bal_i"
simps_7: fixes x2 :: "'c" shows "\<stileturn> \<noteq> \<lbrakk>x2\<rbrakk>"
psi_l_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "grover_state.psi_l n f l = grover_state.alpha_l n f l \<cdot>\<^sub>v grover_state.\<alpha> n f + grover_state.beta_l n f l \<cdot>\<^sub>v grover_state.\<beta> n f"
check_eqv_def: fixes idx :: "idx"   and r :: "((nat, nat) atomic, WS1S_Prelim.order) aformula"   and s :: "((nat, nat) atomic, WS1S_Prelim.order) aformula" shows "WS1S_Alt.check_eqv idx r s = ((ws1s_wf idx r \<and> ws1s_lformula r) \<and> (ws1s_wf idx s \<and> ws1s_lformula s) \<and> (case WS1S_Alt.closure idx (norm (RESTRICT r), norm (RESTRICT s)) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False))"
outSetCorelatedEmpty1: fixes x :: "chanID" assumes "OUTfromV x = {}" shows "outSetCorelated x = {}"
cl_inv'R: fixes l :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b"   and r :: "('a, 'b) rbt" assumes "cl_inv' (Node l R k v r)"   and "cl l = B"   and "cl r = B" shows "cl_inv (Node l R k v r)"
term_ok_Abs_fvI: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and s :: "Core.term"   and \<tau> :: "typ"   and x :: "variable" assumes "term_ok \<Theta> s"   and "typ_ok \<Theta> \<tau>" shows "term_ok \<Theta> (Abs_fv x \<tau> s)"
coinduct_strong: fixes R :: "'a msllist \<Rightarrow> 'a msllist \<Rightarrow> bool"   and msllist :: "'a msllist"   and msllist' :: "'a msllist" assumes "R msllist msllist'"   and "\<And>msllist msllist'. R msllist msllist' \<Longrightarrow> (msllist = MSLNil) = (msllist' = MSLNil) \<and> (msllist \<noteq> MSLNil \<longrightarrow> msllist' \<noteq> MSLNil \<longrightarrow> un_MSLCons1 msllist = un_MSLCons1 msllist' \<and> (R (un_MSLCons2 msllist) (un_MSLCons2 msllist') \<or> un_MSLCons2 msllist = un_MSLCons2 msllist'))" shows "msllist = msllist'"
INF_lower2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and i :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "i \<in> A"   and "less_eq (f i) u" shows "less_eq (Inf (f ` A)) u"
hol_pal_lem0: fixes d :: "'a"   and a :: "'a"   and k :: "real"   and b :: "'a" assumes "d \<bullet> a \<le> k"   and "k \<le> d \<bullet> b" obtains c :: "'a" where "c \<in> closed_segment a b" and "d \<bullet> c = k" and "\<And>z. z \<in> closed_segment a c \<Longrightarrow> d \<bullet> z \<le> k" and "\<And>z. z \<in> closed_segment c b \<Longrightarrow> k \<le> d \<bullet> z"
adv: fixes \<A> :: "nat \<Rightarrow> ('a + 'b, 'c + 'd) distinguisher" assumes "\<And>\<eta>. \<I>_middle \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<A> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<A> \<eta>) (bound_inner \<eta>)"   and "\<And>\<eta>. lossless_inner \<Longrightarrow> plossless_gpv (\<I>_middle \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<A> \<eta>)" shows "Negligible.negligible (\<lambda>\<eta>. advantage (\<A> \<eta>) (sim_inner \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal \<eta>) (middle \<eta>))"
chine_hcomp_ide_arr: fixes f :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" assumes "ide f"   and "arr \<mu>"   and "src f = trg \<mu>" shows "chine_hcomp f \<mu> = \<langle>\<p>\<^sub>1[Leg0 (Dom f), Leg1 (Dom \<mu>)] \<lbrakk>Leg0 (Cod f), Leg1 (Cod \<mu>)\<rbrakk> Chn \<mu> \<cdot> \<p>\<^sub>0[Leg0 (Dom f), Leg1 (Dom \<mu>)]\<rangle>"
Nats_downward_closed: fixes x :: "nat star"   and y :: "nat star" assumes "x \<in> \<nat>"   and "y \<le> x" shows "y \<in> \<nat>"
mem_idx_sound: fixes x :: "'b"   and as :: "'b list" shows "(x \<in> set as) = (\<exists>i. mem_idx x as = Some i)"
evaluation_functor_axioms: shows "FreeMonoidalCategory.evaluation_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D V"
poly_mapping_plus_induct: fixes P :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool"   and p :: "'b \<Rightarrow>\<^sub>0 'c" assumes "P 0"   and "\<And>p c t. \<lbrakk>c \<noteq> (0::'c); t \<notin> keys p; P p\<rbrakk> \<Longrightarrow> P (monomial c t + p)" shows "P p"
get_phantom_def: fixes X :: "'c" shows "get_phantom X = SPEC (\<lambda>R. R = X)"
transitionD_1: fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s" assumes "A \<turnstile> l \<longrightarrow>\<^bsup>g,\<mu>,X\<^esup> l'" shows "(l, g, \<mu>) \<in> PTA.trans_of A"
perm_cont_simp: fixes \<pi> :: "perm"   and x :: "'a"   and y :: "'a" shows "(\<pi> \<bullet> x \<sqsubseteq> \<pi> \<bullet> y) = (x \<sqsubseteq> y)"
surjective: fixes r :: "('a, 'b) state_scheme" shows "r = \<lparr>counter = counter r, discovered = discovered r, finished = finished r, pending = pending r, stack = stack r, tree_edges = tree_edges r, back_edges = back_edges r, cross_edges = cross_edges r, \<dots> = state.more r\<rparr>"
nd: shows "\<not> [:- a, 1::'a:] dvd q"
bisum_outside: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'a"   and lst :: "'b list" shows "bisum (\<lambda>x y. f1 x - f2 x y + f3 y) lst = sum_list (map f1 lst) + f1 (0::'b) - bisum f2 lst + sum_list (map f3 lst) + f3 (0::'b)"
leD: fixes y :: "'b multiset"   and x :: "'b multiset" assumes "y \<subseteq># x" shows "\<not> x \<subset># y"
refine: fixes node_eq_impl :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and node_rel :: "('b \<times> 'c) set"   and node_hash_impl :: "nat \<Rightarrow> 'b \<Rightarrow> nat"   and node_def_hash_size :: "nat" assumes "(node_eq_impl, (=)) \<in> node_rel \<rightarrow> node_rel \<rightarrow> bool_rel"   and "is_bounded_hashcode node_rel node_eq_impl node_hash_impl"   and "is_valid_def_hm_size TYPE('b) node_def_hash_size" shows "(find_path1_impl node_eq_impl node_hash_impl node_def_hash_size, find_path1_gen) \<in> \<langle>node_rel\<rangle>slg_rel \<rightarrow> node_rel \<rightarrow> (node_rel \<rightarrow> bool_rel) \<rightarrow> \<langle>\<langle>\<langle>node_rel\<rangle>list_rel \<times>\<^sub>r node_rel\<rangle>option_rel\<rangle>nres_rel"
adjoin: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "H0 \<subseteq> H"   and "a \<in> carrier G - H"   and "P (G\<lparr>carrier := H\<rparr>)" shows "P (G\<lparr>carrier := adjoin G H a\<rparr>)"
Min_less_iff: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less (linorder.Min less_eq A) x = (\<exists>a\<in>A. less a x)"
that: fixes p' :: "('a, 'b, 'c, 'd) seqp" assumes "p = {l}groupcast(fips, fmsg) . p'"   and "l' \<in> labels \<Gamma> qq" shows "thesis"
inf_compl_bot_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "inf x (inf y (uminus x)) = bot"
defs_in_allDefs: fixes v :: "'var \<times> 'node \<times> Def"   and g :: "'g"   and n :: "'node" assumes "v \<in> defs' g n" shows "v \<in> braun_ssa.allDefs g n"
Max_to_image2: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool" shows "Max {f t |t. P t} = Max (f ` {t. P t})"
shadow_root_delete_get_4: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and node_ptr :: "(_) node_ptr" assumes "delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr h = Some h'" shows "get node_ptr h' = get node_ptr h"
bal_insert: fixes t :: "'b tree234"   and a :: "'b" assumes "bal t" shows "bal (insert a t)"
rep_eq: fixes x :: "'a perm" shows "(\<langle>$\<rangle>) (inverse x) = inv ((\<langle>$\<rangle>) x)"
is_gb_graph: shows "gb_graph (create_gba_from_nodes \<phi> qs)"
prob_space_distr: fixes f :: "'c \<times> 'd \<times> 'e \<Rightarrow> 'b"   and M' :: "'b measure" assumes "P.random_variable M' f" shows "prob_space (distr (density (S \<Otimes>\<^sub>M T \<Otimes>\<^sub>M P) (\<lambda>x. ennreal (Pxyz x))) M' f)"
simps_2: fixes x :: "udom"   and k :: "udom \<rightarrow> udom\<cdot>maybe" shows "bindU\<cdot>(Just\<cdot>x)\<cdot>k = k\<cdot>x"
finite_incidence_system_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" shows "finite_incidence_system point_set block_collection \<equiv> incidence_system point_set block_collection \<and> finite_incidence_system_axioms point_set"
constants_are_ground: fixes x :: "'b trm" assumes "is_a_constant x" shows "ground_term x"
TERMINAL_not_BRANCH: fixes a :: "'h"   and c :: "'f"   and rs :: "'i \<Rightarrow> ('f + 'g + 'h, 'i) Old_Datatype.node set" shows "TERMINAL a \<noteq> BRANCH c rs"
assms_2: fixes x :: "'a" shows "f' (f x) = x"
append_pref: fixes l' :: "'b list"   and l'' :: "'b list"   and l :: "'b list" assumes "l' = l''" shows "l @ l' = l @ l''"
2: fixes w :: "complex" assumes "w \<in> ball z r" shows "((\<lambda>u. 1 / (2 * complex_of_real pi * \<i>) * g u / (u - w) ^ 1) has_contour_integral g w) (circlepath z r)"
members_of_only_vertical_div_line_integrable': fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "only_vertical_division one_chain two_chain_typeI"   and "(k, \<gamma>) \<in> one_chain"   and "(k, \<gamma>) \<in> one_chain"   and "finite two_chain_typeI" shows "line_integral_exists F {i} \<gamma>"
id: fixes x :: "int poly" shows "of_int_poly x = map_poly of_rat (of_int_poly x)"
total_onPD: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "total_onP A r"   and "x \<in> A"   and "y \<in> A" shows "r x y \<or> x = y \<or> r y x"
inner_simps_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<bullet> (y + z) = x \<bullet> y + x \<bullet> z"
primroot_shortest_root: fixes u :: "'b list" assumes "u \<noteq> \<epsilon>" shows "\<^bold>|\<rho> u\<^bold>| = (LEAST d. \<exists>r. u \<in> r* \<and> \<^bold>|r\<^bold>| = d)"
mbuf2t_take_add_induct': fixes f :: "'b option list set \<Rightarrow> 'b option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and xs :: "'b option list set list"   and ys :: "'b option list set list"   and buf :: "'b option list set list \<times> 'b option list set list"   and nts :: "nat list"   and z' :: "'a"   and buf' :: "'b option list set list \<times> 'b option list set list"   and nts' :: "nat list"   and \<sigma> :: "(char list \<times> 'b list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'b list set"   and \<phi> :: "'b MFOTL.formula"   and \<psi> :: "'b MFOTL.formula"   and j' :: "nat"   and U :: "nat \<Rightarrow> 'a \<Rightarrow> bool" assumes "mbuf2t_take f z (mbuf2_add xs ys buf) nts = (z', buf', nts')"   and "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)..<j'] nts"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) [Monitor.progress \<sigma> \<phi> j..<Monitor.progress \<sigma> \<phi> j'] xs"   and "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) [Monitor.progress \<sigma> \<psi> j..<Monitor.progress \<sigma> \<psi> j'] ys"   and "j \<le> j'"   and "U (min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)) z"   and "\<And>k X Y z. \<lbrakk>min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j) \<le> k; Suc k \<le> Monitor.progress \<sigma> \<phi> j'; Suc k \<le> Monitor.progress \<sigma> \<psi> j'; qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) k \<phi>) X; qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) k \<psi>) Y; U k z\<rbrakk> \<Longrightarrow> U (Suc k) (f X Y (\<tau> \<sigma> k) z)" shows "U (min (Monitor.progress \<sigma> \<phi> j') (Monitor.progress \<sigma> \<psi> j')) z'"
return_bind_spmf: fixes x :: "'h"   and f :: "'h \<Rightarrow> 'g spmf" shows "return_spmf x \<bind> f = f x"
increasing_tendsto: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in F. less_eq (f n) l"   and "\<And>x. less x l \<Longrightarrow> \<forall>\<^sub>F n in F. less x (f n)" shows "filterlim f (topological_space.nhds open l) F"
lemOutsideNotOnCone: fixes f :: "'a Point"   and cone :: "'a Cone" assumes "outsideCone f cone" shows "\<not> onCone f cone"
assms_1: shows "a \<in>\<^sub>\<circ> Vset \<alpha>"
igAbsCongUSTR_def: fixes MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'q, 'r) model_scheme" shows "igAbsCongUSTR MOD \<equiv> \<forall>xs x x' y X X'. igFresh MOD xs y X \<and> igFresh MOD xs y X' \<and> igSubst MOD xs (igVar MOD xs y) x X = igSubst MOD xs (igVar MOD xs y) x' X' \<longrightarrow> igAbs MOD xs x X = igAbs MOD xs x' X'"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "j \<le> n" shows "x i \<noteq> x j"