hcju/isabelleps huggingface dataset:
impWitnessPath_def: fixes c :: "('loc, 't, 'b) configuration_scheme"   and loc1 :: "'loc"   and loc2 :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and t :: "'t" shows "impWitnessPath c loc1 loc2 xs t = (path loc1 loc2 xs \<and> distinct xs \<and> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc1) \<and> t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) xs)) \<and> (\<forall>k<length xs. \<exists>t. t \<in>\<^sub>A frontier (c_imp c (case xs ! k of (s, l, t) \<Rightarrow> t)) \<and> t = results_in t' (sum_weights (map (\<lambda>(s, l, t). l) (take (k + 1) xs))))))"
mapl_FGcontra_def: fixes l1 :: "'p \<Rightarrow> 'ac" shows "mapl_FGcontra l1 = map_FGcontra l1 id id id id id id id id id id"
is_Tensor_def: fixes "term" :: "'b local.term" shows "is_Tensor term = (\<exists>x31 x32. term = x31 \<^bold>\<otimes> x32)"
drop_bit_of_1: fixes n :: "nat" shows "drop_bit n (1::'a) = of_bool (n = 0)"
tendsto_inverse_ereal: fixes f :: "'b \<Rightarrow> ereal"   and c :: "ereal"   and F :: "'b filter" assumes "(f \<longlongrightarrow> c) F"   and "\<forall>\<^sub>F x in F. 0 \<le> f x" shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse c) F"
sas_plus_equivalent_to_strips_i_a: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and I :: "'c \<Rightarrow> 'd option"   and G :: "'c \<Rightarrow> 'd option"   and \<pi> :: "('c \<times> 'd) strips_operator list list" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom I \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom I. the (I v) \<in> \<R>\<^sub>+ \<Psi> v"   and "dom G \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom G. the (G v) \<in> \<R>\<^sub>+ \<Psi> v"   and "\<forall>ops'\<in>set \<pi>. \<forall>op'\<in>set ops'. op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "\<phi>\<^sub>S \<Psi> G \<subseteq>\<^sub>m execute_parallel_plan (\<phi>\<^sub>S \<Psi> I) \<pi>" shows "G \<subseteq>\<^sub>m execute_parallel_plan_sas_plus I (\<phi>\<^sub>P\<inverse> \<Psi> \<pi>)"
extensional_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "f \<in> extensional A"   and "A \<subseteq> B" shows "f \<in> extensional B"
Vinv2_def: shows "Vinv2 = {s. \<forall>r. no_defection s (votes s r) r}"
convergent_def: fixes "open" :: "'e set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'e" assumes "class.topological_space open" shows "topological_space.convergent open X = (\<exists>L. filterlim X (topological_space.nhds open L) sequentially)"
I_minWeightBasis_def: shows "I_minWeightBasis \<equiv> \<lambda>(T, E). indep T \<and> T \<subseteq> carrier \<and> E \<subseteq> carrier \<and> (\<forall>x\<in>T. \<forall>y\<in>E. weight x \<le> weight y) \<and> (\<forall>e\<in>carrier - E - T. \<not> indep (insert e T)) \<and> no_smallest_element_skipped E T"
A3: shows "\<forall>a b c. norm_metric.smC a b c c \<longrightarrow> a = b"
predicate1D: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "P \<le> Q"   and "P x" shows "Q x"
append_Nil2: fixes xs :: "'e list" shows "xs @ [] = xs"
eval_upshift: fixes P :: "'v" shows "\<not> eval (upshift P) 0"
exec_instr_Pop: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" shows "exec_instr Pop P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh = (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, ics) # frs, sh)"
intro: fixes G :: "('d, 'f, 'e, 'g, 'h) galcon_scheme" assumes "\<And>x. x \<in> carrier \<X>\<^bsub>G\<^esub> \<Longrightarrow> \<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x) \<sqsubseteq>\<^bsub>\<X>\<^bsub>G\<^esub>\<^esub> x" shows "retract_axioms G"
assms: shows "falsifies\<^sub>c\<^sub>s ds Cs"
set_tag_name_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and tag :: "char list" assumes "type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok set_tag_name element_ptr tag"
t: fixes t :: "real" assumes "t \<in> A" shows "t = real (K t) / 2 ^ N t * dist x0 y0 \<and> K t \<le> 2 ^ N t"
VectorSpaceHom_composite_left: fixes smult'' :: "'f \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'v \<Rightarrow> 'a" assumes "VectorSpaceHom (\<cdot>) V smult'' S" shows "VectorSpaceHom (\<cdot>) V smult'' (S \<circ> T)"
path_conc_conv: fixes E :: "('b \<times> 'b) set"   and u :: "'b"   and la :: "'b list"   and lb :: "'b list"   and w :: "'b" shows "path E u (la @ lb) w = (\<exists>v. path E u la v \<and> path E v lb w)"
get_root_node_same_no_parent: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_root_node ptr \<rightarrow>\<^sub>r cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child" shows "h \<turnstile> Shadow_DOM.get_parent child \<rightarrow>\<^sub>r None"
atLeastatMost_psubset_iff: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset"   and d :: "'a multiset" shows "(subset_mset.atLeastAtMost a b \<subset> subset_mset.atLeastAtMost c d) = ((\<not> a \<subseteq># b \<or> c \<subseteq># a \<and> b \<subseteq># d \<and> (c \<subset># a \<or> b \<subset># d)) \<and> c \<subseteq># d)"
eq: shows "totient n = (\<Prod>p\<in>P. totient (p ^ r p)) * (\<Prod>p\<in>P'. totient (p ^ r p))"
reachable_successors: fixes w :: "'d stream"   and A :: "('d, 'c) nba"   and V :: "(nat \<times> 'c) set"   and p :: "nat \<times> 'c" shows "gsuccessors A w V p \<subseteq> greachable A w V p"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b"   and list :: "'c list" shows "(case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (if list = [] then f1 else f2 (hd list) (tl list))"
sall_TT: fixes p :: "'a \<rightarrow> tr"   and xs :: "[:'a:]" shows "(sall\<cdot>p\<cdot>xs = TT) = (xs \<noteq> \<bottom> \<and> (\<forall>x\<in>sset xs. p\<cdot>x = TT))"
sums_mult2: fixes a :: "'a"   and c :: "'a" assumes "f sums a" shows "(\<lambda>n. f n * c) sums (a * c)"
map_gpv_conv_bind: fixes f :: "'d \<Rightarrow> 'a"   and x :: "('d, 'b, 'c) gpv" shows "map_gpv f (\<lambda>x. x) x = bind_gpv x (\<lambda>x. Done (f x))"
lub_upper: fixes A :: "'a set"   and x :: "'a" assumes "Complete_Partial_Order.chain leq A"   and "x \<in> A" shows "leq x (lub A)"
formulaTrueIffAllClausesAreTrue: fixes formula :: "Literal list list"   and valuation :: "Literal list" shows "formulaTrue formula valuation = (\<forall>clause. clause el formula \<longrightarrow> clauseTrue clause valuation)"
split_box_conj: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and sigma :: "behavior" shows "(sigma \<Turnstile> \<box>(F \<and> G)) = ((sigma \<Turnstile> \<box>F) \<and> (sigma \<Turnstile> \<box>G))"
intros: fixes n :: "('form, 'rule) graph_node"   and i :: "nat"   and s :: "'subst"   and \<Gamma> :: "'form fset"   and c :: "'form" shows "no_fresh_check n i s (\<Gamma> \<turnstile> c)"
mem_diff: fixes x :: "'a"   and A :: "'a set"   and B :: "'a set" shows "(x \<in> A - B) = (x \<in> A \<and> x \<notin> B)"
Sum_any_when_equal: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.Sum_any plus zero (\<lambda>a. zero.when zero (f a) (a = b)) = f b"
l_comb_def: fixes R :: "('i, 'j) Ring_scheme"   and M :: "('h, 'i, 'k) Module_scheme"   and n :: "nat"   and s :: "nat \<Rightarrow> 'i"   and m :: "nat \<Rightarrow> 'h" shows "l_comb R M n s m = \<Sigma>\<^sub>e M (\<lambda>j. s j \<cdot>\<^sub>s\<^bsub>M\<^esub> m j) n"
components_are_iso: fixes a :: "'a" assumes "S.ide a" shows "S.iso (S.src a)"
\<K>_rw: fixes k :: "nat" shows "(\<Sum>w | length w = k \<and> set w \<subseteq> L. 1 / b ^ cw_len_concat w) = (\<Sum>m = 0..<Suc (k * max_len). real (card ({w. length w = k \<and> set w \<subseteq> L} \<inter> cw_len_concat -` {m})) * (1 / b ^ m))"
pseudo_mod_smult_right: fixes x :: "'b"   and q :: "'b poly"   and p :: "'b poly" assumes "x \<noteq> (0::'b)"   and "q \<noteq> 0" shows "pseudo_mod p (Polynomial.smult x q) = Polynomial.smult (x ^ (Suc (length (coeffs p)) - length (coeffs q))) (pseudo_mod p q)"
qFresh_qSwap_exchange1: fixes zs :: "'a"   and z2 :: "'b"   and z1 :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm" shows "qFresh zs z2 (X #[[z1 \<and> z2]]_zs) = qFresh zs z1 X"
hyps: fixes A :: "(('a, 'cols) vec, 'rows) vec" assumes "n < nrows A" shows "matrix_to_iarray (Hermite_reduce_above A n i j res) = Hermite_reduce_above_iarrays (matrix_to_iarray A) n (mod_type_class.to_nat i) (mod_type_class.to_nat j) res"
totient_conv_moebius_mu: fixes n :: "nat" shows "int (totient n) = dirichlet_prod moebius_mu int n"
intro: fixes U :: "'a set"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ls :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and top :: "'a" assumes "semilattice_inf_ow U inf le ls"   and "order_top_ow U le ls top" shows "bounded_semilattice_inf_top_ow U inf le ls top"
not_in_iff_zmset: fixes x :: "'c"   and M :: "'c zmultiset" shows "(x \<notin>#\<^sub>z M) = (zcount M x = 0)"
ty_dom_supp_1: fixes \<Gamma> :: "binding list" shows "supp (ty_dom \<Gamma>) = ty_dom \<Gamma>"
u_other_ivt: fixes t :: "real" assumes "other.s t \<le> u t" shows "collision_react {0..t}"
OUTfromVCorrect1_data11: shows "OUTfromVCorrect1 data11"
hextI: fixes h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "\<forall>a C S. h a = \<lfloor>(C, S)\<rfloor> \<longrightarrow> (\<exists>S'. h' a = \<lfloor>(C, S')\<rfloor>)" shows "h \<unlhd> h'"
simps_1: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and \<zeta> :: "('b, 'a) hd" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym (Hd \<zeta>) = (LEAST w. \<exists>f\<in>ground_heads.ground_heads ground_heads_var \<zeta>. w = wt_sym f + the_enat (enat \<delta> * arity_sym f))"
character_def: fixes G :: "('c, 'd) monoid_scheme"   and \<chi> :: "'c \<Rightarrow> complex" shows "character G \<chi> \<equiv> finite_comm_group G \<and> character_axioms G \<chi>"
StartBallot_HInv3_q: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and p :: "Proc"   and d :: "Disk" assumes "StartBallot s s' q"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
full_inclusion_functor_axioms: shows "full_inclusion_functor C Ide"
Inf_between_mono: fixes N :: "'f set"   and N' :: "'f set"   and M :: "'f set"   and M' :: "'f set" assumes "N \<subseteq> N'"   and "M \<subseteq> M'" shows "Inf_between N M \<subseteq> Inf_between N' M'"
R\<^sub>C_of_def: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and P :: "((('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set" shows "R\<^sub>C_of \<R>\<^sub>A \<R> P \<equiv> {(x, y). \<exists>x\<^sub>A y\<^sub>A. (x\<^sub>A, x) \<in> \<R> \<and> (y\<^sub>A, y) \<in> \<R> \<and> (x\<^sub>A, y\<^sub>A) \<in> \<R>\<^sub>A \<and> snd (fst x) = snd (fst y) \<and> conc.low_mds_eq (snd (fst x)) (snd x) (snd y) \<and> (x, y) \<in> P}"
in_obs_valid_2: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set" assumes "n' \<in> obs n S" shows "valid_node n'"
last_index_less_size_conv: fixes xs :: "'a list"   and x :: "'a" shows "(last_index xs x < length xs) = (x \<in> set xs)"
K3_imp_K2: fixes A :: "agent"   and authTicket :: "msg"   and authK :: "nat"   and T2 :: "nat"   and B :: "agent"   and evs :: "event list" assumes "Says A Tgs \<lbrace>authTicket, Crypt authK \<lbrace>Agent A, Number T2\<rbrace>, Agent B\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "evs \<in> kerbIV" shows "\<exists>Ta. Says Kas A (Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta, authTicket\<rbrace>) \<in> set evs"
indep_setsI: fixes I :: "'d set"   and F :: "'d \<Rightarrow> 'b set set"   and i :: "'i" assumes "\<And>i. i \<in> I \<Longrightarrow> F i \<subseteq> D'.events i"   and "\<And>A J. \<lbrakk>J \<noteq> {}; J \<subseteq> I; finite J; \<forall>j\<in>J. A j \<in> F j\<rbrakk> \<Longrightarrow> D'.prob i (\<Inter> (A ` J)) = (\<Prod>j\<in>J. D'.prob i (A j))" shows "D'.indep_sets i F I"
assms: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
totatives_def: fixes n :: "nat" shows "totatives n = {k \<in> {0<..n}. coprime k n}"
quot_Var_EQ_diff: fixes i :: "name"   and x :: "name" assumes "i \<noteq> x" shows "{\<guillemotleft>Var i\<guillemotright> EQ \<guillemotleft>Var x\<guillemotright>} \<turnstile> Fls"
mojmir_accept_alt_def: shows "accept\<^sub>M = finite {x. \<not> token_succeeds x}"
simps_2: fixes x :: "'v"   and y :: "'v" shows "capacity' (Inner x, Inner y) = capacity \<Delta> (x, y)"
simps_4: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" shows "(P,h \<turnstile> v :\<le> NT) = (P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>NT\<rfloor>)"
leg1_in_hom_2: shows "\<guillemotleft>p\<^sub>1 : p\<^sub>1 \<Rightarrow> p\<^sub>1\<guillemotright>"
SDG_proc_entry_exit_cdep: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and r :: "'ret"   and p :: "'pname"   and fs :: "(('var \<Rightarrow> 'val option) \<Rightarrow> 'val option) list"   and n :: "'node SDG_node"   and a' :: "'edge"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "n = CFG_node (targetnode a)"   and "a' \<in> get_return_edges a"   and "n' = CFG_node (sourcenode a')" shows "n \<longrightarrow>\<^bsub>cd\<^esub> n'"
earlyLateOutput: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'" shows "P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"
i_Equiv_Exec_Stream_expand_shrink_equiv_state_set: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "nat \<Rightarrow> 'h"   and i :: "nat" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "\<And>input0 m. \<lbrakk>set input0 \<subseteq> A; m \<in> A\<rbrakk> \<Longrightarrow> Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input0 \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input0 \<odot> k2) c2)"   and "range input \<subseteq> A" shows "equiv_states (localState1 ((i_Exec_Comp_Stream trans_fun1 ((input_fun1 \<circ> input) \<odot> k1) c1 \<div>\<^sub>l k1) i)) (localState2 ((i_Exec_Comp_Stream trans_fun2 ((input_fun2 \<circ> input) \<odot> k2) c2 \<div>\<^sub>l k2) i))"
rep_eq: fixes x :: "'g \<Rightarrow> 'f"   and xa :: "('g, 'e) uexpr" shows "\<lbrakk>uop x xa\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b))"
nth_default_map_eq: fixes f :: "'b \<Rightarrow> 'a"   and dflt' :: "'b"   and dflt :: "'a"   and xs :: "'b list"   and n :: "nat" assumes "f dflt' = dflt" shows "nth_default dflt (map f xs) n = f (nth_default dflt' xs n)"
Dom_cod: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "local.Dom (cod f) = local.Cod f"
stream_smap_nats: fixes s :: "'b stream" shows "s = smap ((!!) s) nats"
covib: shows "ov\<inverse> O b \<subseteq> b \<union> m \<union> ov \<union> f\<inverse> \<union> d\<inverse>"
IH: assumes "\<not> list_all P A" shows "\<exists>B x C. A = B @ x # C \<and> \<not> P x \<and> list_all P B"
sec_id_pubk_more_def: fixes n :: "nat"   and m :: "nat" shows "sec_id_pubk_more n m \<equiv> insert (Spy, \<langle>n, ID n (Sec_PubKey m)\<rangle>) (sec_id_pubk_less n)"
hm_delete_def: fixes k :: "'a"   and hm :: "('a, 'b) hashmap" shows "hm_delete k hm = RBT_HM (HashMap_Impl.delete k (impl_of_RBT_HM hm))"
pred_inject_2: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" shows "pred_option P (Some a) = P a"
conv_dist_inf: fixes x :: "'a"   and y :: "'a" shows "id (x \<sqinter> y) = id x \<sqinter> id y"
exhaust: fixes y :: "Unary_PCF.type" obtains "y = \<B>"   | x21 :: "Unary_PCF.type" and x22 :: "Unary_PCF.type" where "y = x21 \<rightarrow> x22"
two_cubes_trace_horizontal_boundaries: shows "two_chain_horizontal_boundary two_chain \<subseteq> \<gamma>"
index_smult_vec_2: fixes a :: "'b"   and v :: "'b vec" shows "dim_vec (a \<cdot>\<^sub>v v) = dim_vec v"
spec: shows "SecurityInvariant SINVAR_BLPtrusted.sinvar SINVAR_BLPtrusted.default_node_properties SINVAR_BLPtrusted.receiver_violation"
circuit_nonempty: fixes X :: "'a set" assumes "\<E>.circuit X" shows "X \<noteq> {}"
fbd_ifbd_inv2: fixes \<phi> :: "'c set \<Rightarrow> 'd set" assumes "Sup_pres \<phi>" shows "(bd\<^sub>\<F> \<circ> bd\<^sup>-\<^sub>\<F>) \<phi> = \<phi>"
filterlim_at_split: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'b filter"   and x :: "'a" shows "filterlim f F (at x) = (filterlim f F (at_left x) \<and> filterlim f F (at_right x))"
measure_lborel_box: fixes l :: "'a"   and u :: "'a" assumes "\<And>b. b \<in> Basis \<Longrightarrow> l \<bullet> b \<le> u \<bullet> b" shows "content (box l u) = prod ((\<bullet>) (u - l)) Basis"
sel_2: fixes x1 :: "'a extended"   and x2 :: "'a"   and x3 :: "'a extended"   and x4 :: "'b" shows "I (InsertMessage x1 x2 x3 x4) = x2"
FOREACHoci_by_LIST_FOREACH: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi> :: "'d set \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d set"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma>0 :: "'c" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,\<Phi>\<^esup> S c f \<sigma>0 = ASSERT (finite S) \<bind> (\<lambda>_. LIST_FOREACH \<Phi> (it_to_sorted_list R S) c f \<sigma>0)"
cr_envT_stateT_simps: fixes s :: "'d"   and M :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "cr_envT_stateT s M (EnvT f) (StateT g) = M (f s) (g s)"
hn_ctxt_option: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a option"   and y :: "'b option"   and z :: "assn" assumes "option_assn A x y = z" shows "hn_ctxt (option_assn A) x y = z"
minus_divide_right: fixes a :: "'a"   and b :: "'a" shows "- (a / b) = a / - b"
opp_opp: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c"   and p :: "'c point" assumes "ell_field R"   and "ell_field.on_curve R a b p" shows "ell_field.opp R (ell_field.opp R p) = p"
single_plus_obtain: fixes A :: "'a"   and \<Gamma> :: "'a multiset" assumes "A \<in># \<Gamma>" shows "\<exists>\<Delta>. \<Gamma> = \<Delta> \<oplus> A"
assms_3: shows "domA \<Delta> \<inter> upds S = {}"
shift_streams: fixes w :: "'b list"   and A :: "'b set"   and s :: "'b stream" assumes "w \<in> lists A"   and "s \<in> streams A" shows "w @- s \<in> streams A"
wf_atom_mono: fixes tys :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and tys' :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and a :: "'a atom" assumes "tys \<subseteq>\<^sub>m tys'"   and "wf_atom tys a" shows "wf_atom tys' a"
simple_distributed_joint2: fixes X :: "'a \<Rightarrow> 'e"   and Y :: "'a \<Rightarrow> 'f"   and Z :: "'a \<Rightarrow> 'g"   and Px :: "'e \<times> 'f \<times> 'g \<Rightarrow> real" assumes "simple_distributed M (\<lambda>x. (X x, Y x, Z x)) Px" shows "distributed M (count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M) \<Otimes>\<^sub>M count_space (Z ` space M)) (\<lambda>x. (X x, Y x, Z x)) (\<lambda>x. ennreal (if x \<in> (\<lambda>x. (X x, Y x, Z x)) ` space M then Px x else 0))"
dSUCCEED_abort_def: fixes msg :: "String.literal"   and dm :: "'a dres"   and m :: "'a dres" shows "dSUCCEED_abort msg dm m = (case m of dSUCCEEDi \<Rightarrow> Code.abort msg (\<lambda>_. dm) | _ \<Rightarrow> m)"
multi_inequalityTr1: fixes n :: "nat"   and x :: "nat \<Rightarrow> ant"   and m :: "nat" assumes "\<forall>j\<le>n. x j \<noteq> - \<infinity>" shows "\<exists>L. \<forall>N>L. \<forall>l\<le>n. an m < x l + an N"
member_le_sum_list: fixes x :: "'c multiset"   and xs :: "'c multiset list" assumes "x \<in> set xs" shows "x \<subseteq># monoid_add.sum_list (+) {#} xs"
gen_sa_impl_rel_eext_def: fixes Rm :: "('g \<times> 'j) set"   and Rl :: "('f \<times> ('h \<Rightarrow> 'i)) set" shows "\<langle>Rm, Rl\<rangle>gen_sa_impl_rel_eext = {(\<lparr>sai_L = Li, \<dots> = mi\<rparr>, \<lparr>sa_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
embed_measure_RealPairVal: shows "stock_measure (PRODUCT REAL REAL) = embed_measure lborel RealPairVal"
CompDiag_Can_Inv: fixes t :: "'c local.term" assumes "Can t"   and "Diag t" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> Inv t = Cod t"
\<phi>_in_hom: fixes y :: "'b"   and f :: "'a"   and x :: "'a" assumes "Adj.D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<guillemotleft>\<phi>' y f : y \<rightarrow>\<^sub>D G' x\<guillemotright>"
homeomorphic_maps_imp_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphic_maps X Y f g" shows "homeomorphic_map X Y f"
ucycles_def: fixes G :: "nat set \<times> nat set set" shows "ucycles G \<equiv> {p. 3 \<le> uwalk_length p \<and> p \<in> uwalks G \<and> distinct (tl p) \<and> hd p = last p}"
image_image: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"