hcju/isabelleps huggingface dataset:
connected_if_composable: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "sources \<nu> = targets \<mu>"
not_False: shows "(\<lambda>x. P x \<and> Q x) \<noteq> (\<lambda>x. False)"
SC_counterexample: fixes C :: "'b list"   and D :: "'b list"   and \<Gamma> :: "'b formula list"   and A :: "'b list"   and \<Delta> :: "'b formula list"   and B :: "'b list" assumes "(C, D) \<in> sc \<Gamma> A \<Delta> B" shows "(\<lambda>a. a \<in> set C) \<not>\<Turnstile> image_mset Atom (mset A) + mset \<Gamma> \<Rightarrow> image_mset Atom (mset B) + mset \<Delta>"
increasing_LIMSEQ: fixes f :: "nat \<Rightarrow> real"   and l :: "real" assumes "\<And>n. f n \<le> f (Suc n)"   and "\<And>n. f n \<le> l"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e" shows "f \<longlonglongrightarrow> l"
pwD2: fixes S :: "'c nres"   and S' :: "'c nres"   and x :: "'c" assumes "S \<le> S'"   and "inres S x" shows "inres S' x"
Field_card_of: fixes A :: "'b set" shows "Field |A| = A"
that: fixes n :: "nat" shows "\<not> bit a n \<or> \<not> bit b n"
slength_sfoldl: shows "slength = sfoldl\<cdot>(\<Lambda> i uu_. i + 1)\<cdot>0"
simps_2: fixes sp :: "('b, 'a) sp\<^sub>\<nu>"   and s :: "'b stream" shows "stl (run\<^sub>\<nu> sp s) = run\<^sub>\<nu> (snd (fst (run\<^sub>\<mu> (out sp) s))) (snd (run\<^sub>\<mu> (out sp) s))"
excised_chain_exists: fixes X :: "'c topology"   and U :: "'c set"   and T :: "'c set"   and S :: "'c set"   and p :: "nat"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "X closure_of U \<subseteq> X interior_of T"   and "T \<subseteq> S"   and "singular_chain p (subtopology X S) c" obtains n :: "nat" and d :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" and e :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" where "singular_chain p (subtopology X (S - U)) d" and "singular_chain p (subtopology X T) e" and "(singular_subdivision p ^^ n) c = d + e"
purged_relation_def: fixes kstep :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and output_f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and current :: "'a \<Rightarrow> 'd"   and cswitch :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and kprecondition :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and control :: "'a \<Rightarrow> 'd \<Rightarrow> 'b list list \<Rightarrow> 'b option \<times> 'b list list \<times> 'a"   and kinvolved :: "'b \<Rightarrow> 'd set"   and ifp :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and vpeq :: "'d \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and u :: "'d"   and execs1 :: "'d \<Rightarrow> 'b list list"   and execs2 :: "'d \<Rightarrow> 'b list list" assumes "Separation_Kernel kstep output_f current cswitch kprecondition control kinvolved ifp vpeq" shows "Separation_Kernel.purged_relation ifp u execs1 execs2 \<equiv> \<forall>d. ifp\<^sup>*\<^sup>* d u \<longrightarrow> execs1 d = execs2 d"
add_mono_thms_linordered_semiring_1: fixes i :: "'k"   and j :: "'k"   and k :: "'k"   and l :: "'k" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
rep_eq: fixes x :: "'a ccsubspace"   and xa :: "'a ccsubspace" shows "(x \<le> xa) = (space_as_set x \<subseteq> space_as_set xa)"
has_as_product_def: fixes J :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and D :: "'d \<Rightarrow> 'a"   and a :: "'a" shows "has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J (\<cdot>) D a \<pi>"
graph_axioms: shows "graph gbm_\<alpha> gbm_invar"
fold_conv_fold_keys': fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'c set_rbt" shows "RBT_Set2.fold f t = fold f (RBT_Impl.keys (mapping_rbt.impl_of t))"
extend_ensures: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "Extend h" shows "(extend h F \<in> extend_set h A ensures extend_set h B) = (F \<in> A ensures B)"
term_pp_rel_def: fixes r :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and a :: "'t \<times> 'a"   and b :: "'t \<times> 'a" shows "term_pp_rel r a b = r (snd b \<oplus> fst a) (snd a \<oplus> fst b)"
empty_path_nodes: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n [] n'" shows "n = n'"
eqd12: shows "stm1.d2 = stm2.d1"
merge_length2_nonempty: fixes as :: "nat list list"   and bs :: "nat list list"   and us :: "nat list list"   and vs :: "nat list list" assumes "merge as bs us vs"   and "bs \<in> lists (- {[]})" shows "vs \<in> lists (- {[]})"
chine_simps_2: shows "C.dom \<mu>.chine = \<mu>.apex"
pr_passwd_parts: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and K :: "rat"   and m :: "nat" assumes "(evs, S, A, U) \<in> protocol"   and "Crypt (sesK K) (Passwd m) \<in> parts (A \<union> spies evs)" shows "(\<exists>n run. Says n run 5 (User m) (Card n) (Crypt (sesK K) (Passwd m)) \<in> set evs) \<or> (\<exists>run. Says m run 5 Spy (Card m) (Crypt (sesK K) (Passwd m)) \<in> set evs)"
proper_implies_not_part: fixes x :: "i"   and y :: "i" assumes "PP x y" shows "\<not> P y x"
derives_word_is_prefix: fixes w :: "symbol list"   and a :: "symbol list"   and b :: "symbol list" assumes "is_word w"   and "derives (w @ a) b" shows "is_prefix w b"
restrict_modes_length: fixes mdss :: "(Mode \<Rightarrow> 'Var set) list"   and X :: "'Var set" shows "length (restrict_modes mdss X) = length mdss"
simps: fixes V :: "'d list set"   and M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and v' :: "('d \<times> 'e) list"   and io :: "('d \<times> 'e) list" shows "minimal_sequence_to_failure_extending V M1 M2 v' io = (v' \<in> L\<^sub>i\<^sub>n M1 V \<and> sequence_to_failure M1 M2 (v' @ io) \<and> (\<nexists>io'. \<exists>w'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (w' @ io') \<and> length io' < length io))"
wiringI: fixes \<I> :: "('h, 'i) \<I>"   and \<I>' :: "('j, 'k) \<I>"   and cnv :: "('h, 'i, 'j, 'k) converter"   and f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> map_converter id id f g 1\<^sub>C"   and "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "wiring \<I> \<I>' cnv (f, g)"
ev_sdropD: fixes P :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev P xs" obtains i :: "nat" where "P (sdrop i xs)"
hT: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "\<And>Q. unitary Q \<Longrightarrow> unitary (t Q)"   and "unitary P" shows "unitary (T t P)"
f_expand_assoc: fixes xs :: "'b message_af list"   and a :: "nat"   and b :: "nat" shows "xs \<odot> a \<odot> b = xs \<odot> (a * b)"
poly_eval_plus: fixes a :: "'d \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (p + q) = poly_eval a p + poly_eval a q"
intros_25: fixes i :: "nat"   and \<C> :: "t_context"   and t :: "t" assumes "i < length (local \<C>)"   and "local \<C> ! i = t" shows "\<C> \<turnstile> [Set_local i] : [t] _> []"
remove_abs2: shows "\<exists>S0'. set_iterator_genord iti S0' (\<lambda>x y. R (\<alpha> x) (\<alpha> y)) \<and> inj_on \<alpha> S0' \<and> \<alpha> ` S0' = S0 \<and> (\<forall>x\<in>S0'. invar x)"
le_iff_add: fixes a :: "'d multiset"   and b :: "'d multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "\<lbrace>t \<^bold>\<cdot> u\<rbrace> = \<lbrace>t\<rbrace> \<cdot> \<lbrace>u\<rbrace>"
assms_2: shows "\<exists>n'\<ge>n. \<parallel>uid\<parallel>\<^bsub>t n'\<^esub>"
sublist_def: fixes l :: "nat"   and r :: "nat"   and xs :: "'b list" shows "sublist l r xs = drop l (take r xs)"
C_merge_all: fixes xss :: "'c list list"   and m :: "nat"   and k :: "nat" assumes "\<forall>xs\<in>set xss. length xs = m"   and "length xss = 2 ^ k" shows "C_merge_all xss \<le> m * k * 2 ^ k"
vsv_vconverse: fixes r :: "V" assumes "v11 r" shows "vsv (r\<inverse>\<^sub>\<circ>)"
bdd_below_Icc: fixes a :: "'b multiset"   and b :: "'b multiset" shows "subset_mset.bdd_below (subset_mset.atLeastAtMost a b)"
martingale_def: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure"   and X :: "'d \<Rightarrow> 'c \<Rightarrow> real" shows "martingale M F X = (Filtration.filtration M F \<and> (\<forall>t. integrable M (X t)) \<and> borel_adapt_stoch_proc F X \<and> (\<forall>t s. t \<le> s \<longrightarrow> AEeq M (real_cond_exp M (F t) (X s)) (X t)))"
max_list_mem: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "Missing_Lemmas.max_list xs \<in> set xs"
LIMSEQ_le: fixes X :: "nat \<Rightarrow> 'a"   and x :: "'a"   and Y :: "nat \<Rightarrow> 'a"   and y :: "'a" assumes "X \<longlonglongrightarrow> x"   and "Y \<longlonglongrightarrow> y"   and "\<exists>N. \<forall>n\<ge>N. X n \<le> Y n" shows "x \<le> y"
div_int_poly_def: fixes p :: "int poly"   and q :: "int poly" shows "div_int_poly p q = (case div_mod_int_poly p q of None \<Rightarrow> None | Some (d, m) \<Rightarrow> if m = 0 then Some d else None)"
openin_Int_Inter: fixes \<F> :: "'c set set"   and T :: "'c topology"   and U :: "'c set" assumes "finite \<F>"   and "openin T U"   and "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (U \<inter> \<Inter> \<F>)"
finite_lists_distinct_length_eq: fixes A :: "'g set"   and n :: "nat" assumes "finite A" shows "finite {xs. length xs = n \<and> distinct xs \<and> set xs \<subseteq> A}"
eq: shows "(\<Sum>s\<in>keys p - {t}. lookup (monomial (lookup p s) (except s {x})) (except t {x})) = (0::'b)"
simps: fixes x :: "'a extN" shows "x\<^sup>T = x"
fic_subst1: fixes M :: "trm"   and a :: "coname"   and b :: "coname"   and P :: "trm"   and x :: "name" assumes "fic M a"   and "a \<noteq> b"   and "a \<sharp> P" shows "fic (M{b:=x.P}) a"
poly_pos_greater: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x>a. 0 < poly p x) = (poly_inf p = 1 \<and> (\<forall>x>a. poly p x \<noteq> 0))"
leg1_simps_5: shows "local.dom s\<^sub>1 = s\<^sub>1"
group_l_invI: assumes "\<And>x. x \<in> carrier H' \<Longrightarrow> \<exists>y\<in>carrier H'. y \<otimes>\<^bsub>H'\<^esub> x = \<one>\<^bsub>H'\<^esub>" shows "Group.group H'"
yoneda_lemma_for_hom_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a"   and a' :: "'a" shows "yoneda_lemma_for_hom C S \<phi> a a' \<equiv> (category C \<and> dual_category C \<and> replete_set_category S) \<and> (yoneda_functor_fixed_object C S \<phi> a \<and> yoneda_functor_fixed_object C S \<phi> a') \<and> yoneda_lemma C S \<phi> (yoneda_functor.Y C S \<phi> a') a \<and> yoneda_lemma_for_hom_axioms C a'"
Sim: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> Rel" shows "\<Psi> \<rhd> P \<leadsto>[Rel] Q"
compact_eq_seq_compact_metric: fixes S :: "'c set" shows "compact S = seq_compact S"
carrier_subgroup_generated_by_singleton: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "carrier (subgroup_generated (reduced_homology_group (int p) (nsphere p)) {x}) = range (([^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub>) x)"
wp_SKIP: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp SKIP Q = Q"
iso_same_card: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "G \<cong> H" shows "card (carrier G) = card (carrier H)"
Fin_uminus_le_iff: fixes y :: "'a"   and x :: "'a extended" shows "(Fin (- y) \<le> - x) = (x \<le> Fin y)"
match_3: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and h :: "'a \<times> 'a" assumes "VoV.seq g f"   and "VoV.seq h g" shows "VoV.seq (VoV.comp h g) f"
pref_prod_pref': fixes u :: "'b list"   and z :: "'b list"   and w :: "'b list" assumes "u \<le>p z \<cdot> u \<cdot> w" shows "u \<le>p z \<cdot> u"
def_scheme: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<beta>" shows "f X Y = (\<lambda>\<tau>. if (\<upsilon> X) \<tau> = true \<tau> \<and> (\<upsilon> Y) \<tau> = true \<tau> then g (X \<tau>) (Y \<tau>) else invalid \<tau>)"
if_eval\<^sub>w: fixes b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" shows "\<langle>Stmt.If b t e, mds, mem\<rangle> \<leadsto> \<langle>if ev\<^sub>B mem b then t else e, mds, mem\<rangle>"
trans: fixes a :: "'d word"   and b :: "'d word"   and c :: "'d word" assumes "a \<le>s b"   and "b \<le>s c" shows "a \<le>s c"
assms_2: shows "h \<turnstile> setter \<rightarrow>\<^sub>h h'"
primes_omega_def: fixes n :: "nat" shows "primes_omega n = card (prime_factors n)"
permutation_compose: fixes p :: "'b \<Rightarrow> 'b"   and q :: "'b \<Rightarrow> 'b" assumes "permutation p"   and "permutation q" shows "permutation (p \<circ> q)"
pref_profile_wf_axioms: shows "pref_profile_wf agents alts R1"
psi'_l_dim: fixes l :: "nat" shows "psi'_l l \<in> carrier_vec N"
eq_diff_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(a = minus c b) = (plus a b = c)"
map_zip_map2: fixes f :: "'h \<times> 'i \<Rightarrow> 'g"   and xs :: "'h list"   and g :: "'j \<Rightarrow> 'i"   and ys :: "'j list" shows "map f (zip xs (map g ys)) = map2 (\<lambda>x y. f (x, g y)) xs ys"
wlp_arraycpy_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and a :: "char list"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> (x[] ::= a) Q s = Q (s(x := s a))"
abs_ereal_less0: fixes x :: "ereal" assumes "x < 0" shows "\<bar>x\<bar> = - x"
simps: fixes xs :: "'a list" shows "merge_sort xs = (if length xs \<le> 1 then xs else merge_lists (merge_sort (take (length xs div 2) xs)) (merge_sort (drop (length xs div 2) xs)))"
vpairs_iff: fixes x :: "V"   and r :: "V" shows "(x \<in>\<^sub>\<circ> vpairs r) = (x \<in>\<^sub>\<circ> r \<and> (\<exists>a b. x = \<langle>a, b\<rangle>))"
m_st_def: fixes m :: "'b \<Rightarrow> 'a"   and st :: "'b Abs_State.st" shows "m_st m st = (\<Sum>x\<in>set (Abs_State.dom st). m (fun st x))"
restrict_cnf_atom_def: fixes a :: "'b"   and v :: "bool"   and C :: "'b literal set set" shows "restrict_cnf_atom a v C \<equiv> {c - {make_lit (\<not> v) a} |c. c \<in> C \<and> make_lit v a \<notin> c}"
prj'_chine_composite_3: shows "cods.Prj\<^sub>0\<^sub>0 \<cdot> \<mu>\<nu>\<pi>.chine = \<pi>.chine \<cdot> doms.Prj\<^sub>0\<^sub>0"
ortho: fixes w :: "'a vec"   and x :: "'a vec" assumes "w \<in> W"   and "x \<in> X" shows "w \<bullet> x = (0::'a)"
Sup_fold_sup: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and A :: "'c set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A" shows "Sup A = Finite_Set.fold sup bot A"
dom_map_of_conv_image_fst: fixes xys :: "('a \<times> 'b) list" shows "dom (map_of xys) = fst ` list.set xys"
distinct_11: shows "buffer\<^sub>C \<noteq> high_var\<^sub>C"
diagseq_add: fixes k :: "nat"   and n :: "nat" shows "diagseq (k + n) = (seqseq k \<circ> fold_reduce k n) (k + n)"
LeftDerivationLadder_grow: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<alpha> :: "symbol list"   and i :: "nat"   and r :: "symbol \<times> symbol list"   and ix :: "nat"   and E :: "(nat \<times> symbol \<times> symbol list) list"   and j :: "nat"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder a D L \<alpha>"   and "ladder_last_j L = i"   and "LeftDerivationIntro \<alpha> i r ix E j \<gamma>" shows "LeftDerivationLadder a (D @ [(i, r)] @ E) (L @ [mk_deriv_intro ix (Suc (length D + length E)) j]) \<gamma>"
sup_ge2: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y \<le> y"
assms_3: fixes x :: "'a"   and a :: "'b" assumes "P x"   and "Q x a" shows "R a"
char_power: fixes a :: "'a"   and k :: "nat" assumes "a \<in> carrier G" shows "\<chi> (a [^] k) = \<chi> a ^ k"
greatest_closed: fixes L :: "('c, 'd) gorder_scheme"   and l :: "'c"   and A :: "'c set" assumes "greatest L l A" shows "l \<in> carrier L"
awalk_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "awalk u (e # es) w = (e \<in> parcs G \<and> u = fst e \<and> awalk (snd e) es w)"
INT_greatest: fixes A :: "'c set"   and C :: "'d set"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<Longrightarrow> C \<subseteq> B x" shows "C \<subseteq> \<Inter> (B ` A)"
none: assumes "do_if v e1 e2 = None" shows "thesis"
assms_4: assumes "\<not> \<unrhd>\<^sub>u\<^sub>b (lt dir) c (UB dir s x)"   and "\<not> \<lhd>\<^sub>l\<^sub>b (lt dir) c (LB dir s x)"   and "x \<in> lvars (\<T> s)" shows "P (update\<B>\<I> (UBI_upd dir) i x c s)"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_all_front_tickfree_traces2: fixes s :: "'a event list" assumes "front_tickFree s" shows "s \<in> T (CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV)"
elims_2: fixes atom_ordering :: "('a \<times> 'a) set"   and x :: "'a Literal" assumes "propositional_atoms atom_ordering"   and "propositional_atoms.negative x" obtains A :: "'a" where "x = Neg A"
not_in_set_insert: fixes x :: "'d"   and xs :: "'d list" assumes "x \<notin> set xs" shows "List.insert x xs = x # xs"
space_empty_eq_bot: fixes a :: "'d measure" shows "(space a = {}) = (a = bot)"
simps_3: fixes uu :: "poscheckvariant"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" shows "fv\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uu: t \<doteq> t'\<rangle> = fv t \<union> fv t'"
cond: shows "\<alpha> * \<parallel>gs1.gso i\<parallel>\<^sup>2 < \<parallel>gs1.gso (i - 1)\<parallel>\<^sup>2"
saturated_def: fixes N :: "'g set" shows "ground.saturated N = (ground.Inf_from N \<subseteq> Red_I_G N)"
signed_list_lconjaction_fst: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and ss :: "'b list"   and x :: "'b \<times> bool" assumes "class.group_add minus plus zero uminus" shows "fst (foldr (signed_funaction (\<lambda>x y. minus (plus x y) x)) ss x) = minus (plus (monoid_add.sum_list plus zero ss) (fst x)) (monoid_add.sum_list plus zero ss)"
Im2_quat_of_real: fixes a :: "real" shows "Im2 (quat_of_real a) = 0"
assms_2: shows "\<forall>x y. f x \<le> y \<longrightarrow> (\<exists>z\<ge>x. y = f z)"
degree_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and p :: "'a list" assumes "class.semiring_0 plus zero times" shows "semiring_0.degree zero p = length (semiring_0.pnormalize zero p) - 1"
size_del: fixes x :: "'b"   and t :: "'b tree"   and t' :: "'b tree" assumes "I_RBTid2.del x t = Some t'" shows "size t' = size t - 1"
dom_locals_avar_vvar_mono: fixes s :: "state"   and G :: "prog"   and i :: "val"   and a :: "val"   and s' :: "state"   and val :: "val" shows "dom (locals (snd s)) \<subseteq> dom (locals (snd (snd (fst (avar G i a s')) val s)))"
Throw: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>Throw,Normal s\<rangle> \<Rightarrow> Abrupt s"
countable_reachable: fixes s :: "'a" shows "countable ((SIGMA x:UNIV. set_pmf (J x))\<^sup>* `` {s})"
children_h3: shows "h3 \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children_h3"
simps_5: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" shows "step ((Cn n f gs, xs, ls) # fs, rv) = (if length ls = length gs then if rv \<up> then ((f, ls, []) # (Cn n f gs, xs, ls) # fs, None) else (fs, rv) else if rv \<up> then if length ls < length gs then ((gs ! length ls, xs, []) # (Cn n f gs, xs, ls) # fs, None) else (fs, rv) else ((Cn n f gs, xs, ls @ [the rv]) # fs, None))"
cBall_cempty: fixes P :: "'d \<Rightarrow> bool" shows "cBall cempty P = True"
cs_0: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a" assumes "IFC prob" shows "IFC_def.cs prob \<pi> 0 = [\<pi> 0]"
rnn: shows "real_non_neg_vec (\<chi>i. complex_of_real (v $h i))"
ring_tOp_commute: fixes R :: "('d, 'e) Ring_scheme"   and x :: "'d"   and y :: "'d" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y = y \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
C1_differentiable_on_components: fixes f :: "real \<Rightarrow> 'b"   and s :: "real set" assumes "\<And>i. i \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s" shows "f C1_differentiable_on s"
abs: fixes \<Gamma> :: "(name, value) fmap"   and cs :: "(Term_Class.term \<times> sterm) list" shows "\<Gamma> \<turnstile>\<^sub>v Sabs cs \<down> Vabs cs \<Gamma>"
Heap_execute: fixes f :: "'b Heap" shows "Heap.Heap (execute f) = f"
h_minus_aux_le_h': fixes \<Delta> :: "('a, 'b) network_scheme"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and h' :: "'a \<times> 'a \<Rightarrow> ennreal"   and j :: "nat"   and e :: "'a \<times> 'a" assumes "flow_network \<Delta> g"   and "\<And>e. h' e \<noteq> \<top>" shows "flow_network.h_minus_aux \<Delta> h' j e \<le> h' e"
f_Exec_Stream_Acc_Output_Init_eq_f_Exec_Stream_Acc_Output_Cons_output: fixes output_fun :: "'e \<Rightarrow> 'd message_af"   and c :: "'e"   and k :: "nat"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list" assumes "output_fun c = NoMsg" shows "f_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun xs c = NoMsg # f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c"
aff_dim_affine_independent: fixes B :: "'a set" assumes "\<not> affine_dependent B" shows "int (card B) = aff_dim B + 1"
app_iso_simps_2: fixes hom :: "('g, 'h, 'f, 'e) digraph_isomorphism"   and G :: "('g, 'h) pre_digraph" shows "arcs (app_iso hom G) = iso_arcs hom ` arcs G"
do_opappE: fixes vs :: "v list"   and res :: "v sem_env \<times> exp0" assumes "do_opapp vs = Some res" obtains env :: "v sem_env" and n :: "char list" and e :: "exp0" and v2 :: "v" where "vs = [Closure env__ n__ e, v2__]" and "res = (update_v (\<lambda>_. nsBind n__ v2__ (sem_env.v env__)) env__, e)"   | env :: "v sem_env" and funs :: "(char list \<times> char list \<times> exp0) list" and n :: "char list" and v2 :: "v" where "vs = [Recclosure env funs n, v2]" and "Some res = (if allDistinct (map (\<lambda>(f, x, e). f) funs) then case find_recfun n funs of None \<Rightarrow> None | Some (n, e) \<Rightarrow> Some (update_v (\<lambda>_. nsBind n v2 (build_rec_env funs env (sem_env.v env))) env, e) else None)"