hcju/isabelleps huggingface dataset:
Class_closed: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "a \<in> rel.Class b"   and "b \<in> Sigma I \<FF>" shows "a \<in> Sigma I \<FF>"
simplex_index_1: fixes cs :: "('b \<times> constraint) list"   and I :: "'b list" assumes "simplex_index cs = Inl I" shows "set I \<subseteq> fst ` set cs \<and> (\<nexists>v. (set I, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set cs) \<and> (distinct_indices cs \<longrightarrow> (\<forall>J\<subset>set I. \<exists>v. (J, v) \<Turnstile>\<^sub>i\<^sub>c\<^sub>s set cs))"
simps_4: fixes T :: "typ"   and t :: "Core.term"   and U :: "typ"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option" shows "raw_match_term (Abs T t) (Abs U u) subs = Option.bind (raw_match' T U subs) (raw_match_term t u)"
remap_rta: fixes f :: "'a \<Rightarrow> 'd" shows "ranked_tree_automaton (ta_remap f TA) A"
nodes_inv2: fixes n :: "nat"   and sf1 :: "('c, 'd) Digit"   and pr2 :: "('c, 'd) Digit"   and ts :: "('c, 'd) Node list" assumes "is_leveln_digit n sf1"   and "is_measured_digit sf1"   and "is_leveln_digit n pr2"   and "is_measured_digit pr2"   and "\<forall>x\<in>set ts. is_leveln_node n x \<and> is_measured_node x" shows "\<forall>x\<in>set (nodes (digitToNlist sf1 @ ts @ digitToNlist pr2)). is_leveln_node (Suc n) x \<and> is_measured_node x"
comp_null_1: fixes f :: "(bool, bool list) concrete_category.arr" shows "J.null \<cdot>\<^sub>J f = J.null"
P_cp: fixes x :: "'a state \<times> 'a state \<Rightarrow> Set(\<langle>\<langle>'b\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)"   and \<tau> :: "'a state \<times> 'a state" shows "P x \<tau> = P (\<lambda>_. x \<tau>) \<tau>"
n: shows "\<not> is_zero_row_upt_k ma k A"
fin_make_untimed_inf_truncate_Nonempty_all0a: fixes z :: "nat \<Rightarrow> 'a list"   and a :: "'a" assumes "z 0 = [a]" shows "\<forall>i. fin_make_untimed (inf_truncate z i) \<noteq> []"
cases: fixes a1 :: "char list"   and a2 :: "char list"   and 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" assumes "P \<turnstile> a1 \<prec>\<^sub>R a2" obtains C :: "char list" and Bs :: "base list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" and D :: "char list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(Bs, rest)\<rfloor>" and "Repeats D \<in> set Bs"
STOP_def: shows "STOP \<equiv> Abs_process ({(s, X). s = []}, {})"
pickInp_asInp: fixes qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" assumes "qGoodInp qinp" shows "(pickInp (asInp qinp), qinp) \<in> alphaInp"
SubstTerm_imp_subst_dbtm: fixes v :: "hf"   and e :: "(name, hf) finfun"   and u :: "dbtm"   and x :: "hf"   and x' :: "hf" assumes "SubstTerm v \<lbrakk>quot_dbtm u\<rbrakk>e x x'" shows "\<exists>t. x = \<lbrakk>quot_dbtm t\<rbrakk>e \<and> x' = \<lbrakk>quot_dbtm (subst_dbtm u (decode_Var v) t)\<rbrakk>e"
integral_continuous_on_param: fixes U :: "'b set"   and a :: "'c"   and b :: "'c"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). f x t)" shows "continuous_on U (\<lambda>x. integral (cbox a b) (f x))"
psi_scale_3: fixes r :: "real"   and a :: "real \<times> real"   and b :: "real \<times> real" shows "psi 0 (r *\<^sub>R a) (r *\<^sub>R b) = (if 0 < r then psi 0 a b else if r < 0 then psi b a 0 else True)"
qiterl_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "prod.qiterl x y = prod.lfp (prod.qiter_fun x y)"
seqE: fixes g :: "'j"   and f :: "'j" assumes "J.seq g f" obtains "J.arr f" and "J.arr g" and "J.dom g = J.cod f"
in_multiset_in_set: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in># mset xs) = (x \<in> set xs)"
whilefalse: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = False" shows "SRabbr (while\<^bsub>\<iota>\<^esub> b do c od) m [] None m"
onorm_scaleR_left: fixes r :: "'d \<Rightarrow> real"   and f :: "'e" assumes "bounded_linear r" shows "onorm (\<lambda>x. r x *\<^sub>R f) = onorm r * norm f"
power_commutes: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.monoid_mult one times" shows "times (power.power one times a n) a = times a (power.power one times a n)"
H2_CSP1_commute: shows "H2 \<circ> CSP1 = CSP1 \<circ> H2"
hom_one: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
nonneg_interior: fixes x :: "'a" assumes "x \<in> S" shows "0 \<le> f x"
order_trans_rules_26: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a = b"   and "b \<le> c" shows "a \<le> c"
less_disjoint_disjointed: fixes m :: "nat"   and n :: "nat"   and A :: "nat \<Rightarrow> 'b set" assumes "m < n" shows "disjointed A m \<inter> disjointed A n = {}"
pHom_rHom: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y" shows "f \<in> rHom R A"
log_dec: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and X :: "msg"   and K :: "key"   and Y :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "s' = insert (Spy, X) s \<and> (Spy, Crypt K X) \<in> s \<and> (Spy, Key (InvK K)) \<in> s" shows "key_sets Y (crypts {Y. Log Y = X}) \<subseteq> key_sets Y (crypts (Log -` spied s))"
RID\<^sub>LI_1: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr"   and r :: "'b" assumes "s = (\<sigma>, \<tau>, e)"   and "r \<in> RID\<^sub>S \<sigma>" shows "r \<in> RID\<^sub>L s"
pi_machin: shows "pi = 16 * arctan (1 / 5) - 4 * arctan (1 / 239)"
simps_1: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and p :: "'c spmf" shows "foldl_spmf f p [] = p"
assms_5: shows "prv (imp \<chi> \<psi>)"
Lim_within: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and a :: "'c"   and S :: "'c set" shows "(f \<longlongrightarrow> l) (at a within S) = (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
unit_wfactors: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c" assumes "Group.monoid G"   and "a \<in> Units G" shows "wfactors G [] a"
assms_1: shows "rb_aux_inv (bs, ss, p # ps)"
disj_forward: fixes P' :: "bool"   and Q' :: "bool"   and P :: "bool"   and Q :: "bool" assumes "P' \<or> Q'"   and "P' \<Longrightarrow> P"   and "Q' \<Longrightarrow> Q" shows "P \<or> Q"
Spair_induced_automorph: fixes s :: "'a permutation" assumes "s \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "s = (case Spair s of (f, g) \<Rightarrow> Abs_induced_automorph f g)"
sints64: shows "sints 64 = {i. - (2 ^ 63) \<le> i \<and> i < 2 ^ 63}"
assms: shows "((\<mu> U\<^sub>r \<eta> \<in> old (fst n_ns) \<longrightarrow> \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns) \<or> \<mu> \<in> old (fst n_ns) \<union> new (fst n_ns) \<and> \<mu> U\<^sub>r \<eta> \<in> next (fst n_ns)) \<and> (\<forall>nd\<in>snd n_ns. \<mu> U\<^sub>r \<eta> \<in> old nd \<and> name nd \<in> incoming (fst n_ns) \<longrightarrow> \<eta> \<in> old nd \<or> \<mu> \<in> old nd \<and> \<mu> U\<^sub>r \<eta> \<in> old (fst n_ns) \<union> new (fst n_ns))) \<and> (\<forall>n\<in>snd n_ns. \<forall>nd\<in>snd n_ns. \<mu> U\<^sub>r \<eta> \<in> old n \<and> name n \<in> incoming nd \<longrightarrow> \<eta> \<in> old n \<or> \<mu> \<in> old n \<and> \<mu> U\<^sub>r \<eta> \<in> old nd) \<and> ((\<forall>nm\<in>incoming (fst n_ns). nm < name (fst n_ns)) \<and> 0 < name (fst n_ns) \<and> (\<forall>q\<in>snd n_ns. name q < name (fst n_ns) \<and> (\<forall>nm\<in>incoming q. nm < name (fst n_ns)))) \<and> expand_name_ident (snd n_ns)"
WCR_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "WCR_on r A"   and "a \<in> A" obtains "(b, c) \<in> r\<^sup>\<down>"   | "(a, b) \<notin> r"   | "(a, c) \<notin> r"
get_owner_document_ptr_in_heap: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_get_owner_document get_owner_document"   and "h \<turnstile> ok get_owner_document ptr" shows "ptr |\<in>| object_ptr_kinds h"
OclExcluding_valid_args_valid: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> Bag('e)"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<upsilon> X->excluding\<^sub>B\<^sub>a\<^sub>g(x)) = (\<tau> \<Turnstile> \<delta> X \<and> \<tau> \<Turnstile> \<upsilon> x)"
simps_2: fixes s :: "'c \<Rightarrow> bool option"   and ops :: "'c strips_operator list"   and opss :: "'c strips_operator list list" shows "execute_parallel_plan s (ops # opss) = (if are_all_operators_applicable s ops \<and> STRIPS_Semantics.are_all_operator_effects_consistent ops then execute_parallel_plan (execute_parallel_operator s ops) opss else s)"
exists_RestrictedP: fixes s :: "name"   and f :: "tm"   and k :: "tm"   and H :: "fm set" assumes "atom s \<sharp> (f, k)" shows "H \<turnstile> SyntaxN.Ex s (RestrictedP f k (Var s))"
less_4: fixes D :: "'a set" assumes "D \<in> \<D>" shows "\<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)"
relto_trancl_conv: fixes S :: "('a \<times> 'a) set"   and R :: "('a \<times> 'a) set" shows "(S\<^sup>* O R O S\<^sup>*)\<^sup>+ = (R \<union> S)\<^sup>* O R O (R \<union> S)\<^sup>*"
Ln_times_of_real: fixes r :: "real"   and z :: "complex" assumes "0 < r"   and "z \<noteq> 0" shows "Ln (complex_of_real r * z) = Ln (complex_of_real r) + Ln z"
emeasure_return: fixes A :: "'a set"   and M :: "'a measure"   and x :: "'a" assumes "A \<in> sets M" shows "emeasure (return M x) A = indicator A x"
residual_conj_eq: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "residual (conjugate f) = (\<lambda>x. - f (- x))"
tm_wf_t_wcode_prepare: shows "tm_wf0 t_wcode_prepare"
proves_eq_transitive: 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 t :: "Core.term"   and u :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> s"   and "term_ok \<Theta> t"   and "term_ok \<Theta> u"   and "typ_of s = typ_of t"   and "typ_of t = typ_of u"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq s t \<longmapsto> mk_eq t u \<longmapsto> mk_eq s u"
e_nth_append_big: fixes xs :: "nat"   and n :: "nat"   and ys :: "nat" assumes "e_length xs \<le> n" shows "e_nth (e_append xs ys) n = e_nth ys (n - e_length xs)"
induct: fixes x1 :: "cld list"   and x2 :: "ctx"   and x3 :: "cl"   and x4 :: "nat"   and P :: "cld list \<Rightarrow> ctx \<Rightarrow> cl \<Rightarrow> nat \<Rightarrow> bool" assumes "path_length x1 x2 x3 x4"   and "\<And>Pa ctx. P Pa ctx cl_object 0"   and "\<And>Pa ctx fqn ctx' cld cl nn. \<lbrakk>find_cld Pa ctx fqn (Some (ctx', cld)); superclass_name cld cl; path_length Pa ctx' cl nn; P Pa ctx' cl nn\<rbrakk> \<Longrightarrow> P Pa ctx (cl_fqn fqn) (nn + 1)" shows "P x1 x2 x3 x4"
mset_heap_empty_iff: fixes ts :: "'b tree list" shows "(mset_heap ts = {#}) = (ts = [])"
f01: shows "f ` {0..1} \<subseteq> {0..1}"
Re_complex_div_le_0: fixes a :: "complex"   and b :: "complex" shows "(Re (a / b) \<le> 0) = (Re (a * cnj b) \<le> 0)"
csqrt_unique: fixes w :: "complex"   and z :: "complex" assumes "w\<^sup>2 = z"   and "0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w" shows "csqrt z = w"
gstep_succE: fixes gc' :: "('c, 'ls, 'gs) global_config"   and gc :: "('c, 'ls, 'gs) global_config" assumes "gc' \<in> gstep_succ gc" obtains lc :: "('c, 'ls) local_config" and lcs :: "('c, 'ls) local_config multiset" and c' :: "'c" and a :: "'a" and ls' :: "'ls" and gs' :: "'gs" where "global_config.processes gc = {#lc#} + lcs" and "cstep (command lc) a c'" and "en (local_config.state lc, global_config.state gc) a" and "ex (local_config.state lc, global_config.state gc) a = (ls', gs')" and "gc' = \<lparr>global_config.processes = {#\<lparr>command = c', state = ls'\<rparr>#} + lcs, state = gs'\<rparr>"
a: shows "a \<in> U \<rightarrow> {\<zero>\<^bsub>K\<^esub>}"
edom_AnalBinds: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "edom (ABinds \<Gamma>\<cdot>ae) \<subseteq> fv \<Gamma>"
factorize_rat_poly_generic_def: fixes alg :: "int_poly_factorization_algorithm"   and f :: "rat poly" shows "factorize_rat_poly_generic alg f = (case rat_to_normalized_int_poly f of (c, g) \<Rightarrow> case factorize_int_poly_generic alg g of (d, fs) \<Rightarrow> (c * rat_of_int d, map (\<lambda>(fi, i). (of_int_poly fi, i)) fs))"
pareto_losers_lift_pref_profile: shows "pareto_losers R' = pareto_losers R \<union> (alts' - alts)"
simps_10: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm" shows "PApp x31 x32 \<noteq> PUnit"
left_commute: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry"   and c :: "'c DBMEntry" shows "linordered_monoid.min b (linordered_monoid.min a c) = linordered_monoid.min a (linordered_monoid.min b c)"
\<F>\<G>_suffix: fixes \<phi> :: "'a ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'a set" shows "\<F>\<G> \<phi> (suffix i w) = \<F>\<G> \<phi> w"
invar: shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"
gtr_find_thread: fixes c :: "'c multiset"   and e :: "'d"   and c' :: "'e multiset"   and gtrs :: "(('c \<times> 'c multiset) \<times> 'd \<times> 'e \<times> 'e multiset) set" assumes "(c, e, c') \<in> gtr gtrs" obtains s :: "'c" and ce :: "'c multiset" and s' :: "'e" and ce' :: "'e multiset" where "c = add_mset s ce" and "c' = add_mset s' ce'" and "((s, ce), e, s', ce') \<in> gtrs"
set_shadow_root_get_child_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) object_ptr" shows "\<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_child_nodes_locs ptr'. r h h')"
T1: shows "f' (x, y) o\<^sub>L embed2_blinfun o\<^sub>L T = 1\<^sub>L"
widen_is_class_type_of: fixes T :: "ty"   and C :: "String.literal"   and P :: "'b prog"   and T' :: "ty" assumes "class_type_of' T = \<lfloor>C\<rfloor>"   and "P \<turnstile> T' \<le> T"   and "T' \<noteq> NT" obtains C' :: "String.literal" where "class_type_of' T' = \<lfloor>C'\<rfloor>" and "P \<turnstile> C' \<preceq>\<^sup>* C"
takeWhileNot_path_entry: fixes G :: "nat graph_rec"   and E :: "(nat \<times> nat) set"   and p :: "nat list"   and x :: "nat"   and v :: "nat"   and c :: "nat list" assumes "cfg_doms G"   and "path_entry E p x"   and "v \<in> set p"   and "takeWhile ((\<noteq>) v) (rev p) = c" shows "path_entry E (rev c) v"
t_decomp: shows "t = tm_max prec I a t1 t2"
bij_betw_inv_into: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "bij_betw (inv_into A f) B A"
UnCI: fixes c :: "'f"   and B :: "'f set"   and A :: "'f set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
iso_is_arr: fixes f :: "'d" assumes "D.iso f" shows "D.arr f"
eoutKnowCorrect_L1s: fixes C :: "specID"   and m :: "Secrets" assumes "eoutKnowCorrect C (sKS m)"   and "eout C (sE m)" shows "m \<in> specSecrets C \<or> know C (sKS m)"
bit_or_int_iff: fixes k :: "int"   and l :: "int"   and n :: "nat" shows "bit (k OR l) n = (bit k n \<or> bit l n)"
bis_converse: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and B1' :: "'h set"   and B2' :: "'i set"   and s1' :: "'h \<Rightarrow> ('f, 'h, 'i) F1"   and s2' :: "'i \<Rightarrow> ('g, 'h, 'i) F2"   and R1 :: "('d \<times> 'h) set"   and R2 :: "('e \<times> 'i) set" assumes "bis B1 B2 s1 s2 B1' B2' s1' s2' R1 R2" shows "bis B1' B2' s1' s2' B1 B2 s1 s2 (R1\<inverse>) (R2\<inverse>)"
R_lin_independent_ConsI: fixes ms :: "'m list"   and m :: "'m" assumes "R_lin_independent ms"   and "\<And>r rs. \<lbrakk>set (r # rs) \<subseteq> R; (r # rs) \<bullet>\<cdot> (m # ms) = (0::'m)\<rbrakk> \<Longrightarrow> r = (0::'r)" shows "R_lin_independent (m # ms)"
ring_simprules_18: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = y"
carrier_subgroup_generated_by_singleton: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e" assumes "Group.group G"   and "x \<in> carrier G" shows "carrier (subgroup_generated G {x}) = range (([^]\<^bsub>G\<^esub>) x)"
validBoundOutputTransition: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto> a<\<nu>x> \<prec> P'"   and "valid P" shows "valid P'"
sconf_def: fixes E :: "String.literal \<Rightarrow> ty option"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)" shows "E \<turnstile> s \<surd> \<equiv> let (h, l) = s in hconf h \<and> P,h \<turnstile> l (:\<le>) E \<and> preallocated h"
divide_less_1_iff_ennreal: fixes a :: "ennreal"   and b :: "ennreal" shows "(a / b < 1) = (0 < b \<and> a < b \<or> b = 0 \<and> a = 0 \<or> b = \<top>)"
exhaust: fixes y :: "arctic" obtains "y = MinInfty"   | x2 :: "int" where "y = Num_arc x2"
simps_8: fixes \<alpha> :: "('b + 'e, 'c, 'd) hp"   and \<sigma> :: "'e \<Rightarrow> ('b, 'd) trm" shows "PsubstFO \<alpha>** \<sigma> = PsubstFO \<alpha> \<sigma>**"
between_equality: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet B A C" shows "A = B"
arrE: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.arr (fst f) \<and> B.arr (snd f) \<and> trg (fst f) = src (snd f)"
columnvector_def: fixes v :: "('b, 'd) vec" shows "columnvector v = (\<chi>i j. v $ i)"
prems: shows "suspendable (\<xi> M\<^sub>n \<xi>')"
size_le_1: fixes f :: "'b fset" shows "(size f \<le> 1) = (f = {||} \<or> (\<exists>e. f = {|e|}))"
nres_monad3: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres"   and g :: "'b \<Rightarrow> 'a nres" shows "M \<bind> f \<bind> g = M \<bind> (\<lambda>x. f x \<bind> g)"
mult_row_code: fixes A :: "(('b, 'a) vec, 'c) vec"   and a :: "'c"   and q :: "'b"   and i :: "'c" shows "($) (mult_row_row A a q i) = (\<lambda>j. if i = a then q * A $ a $ j else A $ i $ j)"
comm_semiring_1_hom_axioms: shows "comm_semiring_1_hom hom"
Domainp_iff: fixes T :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'b" shows "Domainp T x = (\<exists>y. T x y)"
fold_congs_9: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "Trap set"   and f :: "Trap set \<Rightarrow> Trap set"   and f' :: "Trap set \<Rightarrow> Trap set" assumes "r = r'"   and "traps r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "traps_update f r = traps_update f' r'"
cases: fixes a1 :: "ty"   and a2 :: "ty" assumes "a1 <: a2" obtains A :: "ty" where "a1 = A" and "a2 = A"   | f1 :: "funty" and f2 :: "funty" where "a1 = TFun f1" and "a2 = TFun f2" and "f1 <:: f2"
prems_2: shows "\<forall>x. d1 \<le> x \<and> x < c \<longrightarrow> (\<forall>q\<in>set (pders p). poly q x \<noteq> 0)"
lincomb_closed: fixes S :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "S \<subseteq> carrier_vec nc" shows "AB.NC.lincomb a S \<in> carrier_vec nc"
adjust_def: fixes t :: "('b \<times> nat) tree" shows "adjust t = (case t of \<langle>l, (x, lv), r\<rangle> \<Rightarrow> if lv - 1 \<le> lvl l \<and> lv - 1 \<le> lvl r then t else if lvl r < lv - 1 \<and> sngl l then skew \<langle>l, (x, lv - 1), r\<rangle> else if lvl r < lv - 1 then case l of \<langle>t1, (a, lva), \<langle>t2, (b, lvb), t3\<rangle>\<rangle> \<Rightarrow> \<langle>\<langle>t1, (a, lva), t2\<rangle>, (b, lvb + 1), \<langle>t3, (x, lv - 1), r\<rangle>\<rangle> else if lvl r < lv then split \<langle>l, (x, lv - 1), r\<rangle> else case r of \<langle>t1, (b, lvb), t4\<rangle> \<Rightarrow> case t1 of \<langle>t2, (a, lva), t3\<rangle> \<Rightarrow> \<langle>\<langle>l, (x, lv - 1), t2\<rangle>, (a, lva + 1), split \<langle>t3, (b, if sngl t1 then lva else lva + 1), t4\<rangle>\<rangle>)"
lprod_3_4: fixes yr :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set"   and x :: "'a"   and z :: "'a" assumes "(yr, y) \<in> R" shows "([x, yr, z], [x, y, z]) \<in> lprod R"
hypsubst: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" assumes "A = B"   and "x \<in> B" obtains "x \<in> A"
simps_1: fixes uu :: "'b" shows "chainlen \<lbrace> uu \<rbrace> = 0"
cbelow_on_second_in: fixes A :: "'c set"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and rel :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "cbelow_on A ord z rel a b" shows "b \<in> A"
compare_1_rat: fixes y :: "int poly \<times> rat \<times> rat"   and x :: "rat" assumes "invariant_1 y" shows "compare_1_rat y x = compare (real_of_rat x) (real_of_1 y)"
simps_2: fixes v :: "value" shows "value_eq (Some v) None = invalid"
Skip_Sim: fixes S :: "'e \<Rightarrow> 'd" shows "Skip \<preceq>S Skip"
swapidseq_ext_zero_imp_id: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'a" assumes "swapidseq_ext S 0 p" shows "p = id"
validNF_def: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> ('e \<times> 'd) \<times> bool"   and Q :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace>! \<equiv> \<lbrace>P\<rbrace> f \<lbrace>Q\<rbrace> \<and> no_fail P f"
le_divide_eq_1_pos: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less_eq one (divide b a) = less_eq a b"
shows_prec_arctic_delta_def: fixes d :: "nat"   and ari :: "'b arctic_delta" shows "shows_prec d ari = Show_Arctic.shows_arctic_delta ari"
set_tag_name_impl: shows "set_tag_name = a_set_tag_name"
antisym: fixes b :: "'b zmultiset"   and a :: "'b zmultiset" assumes "b \<subseteq>#\<^sub>z a"   and "a \<subseteq>#\<^sub>z b" shows "a = b"
RL: assumes "unat f < unat b" shows "UCAST('a \<rightarrow> 'b) f < b"
normal_invE_1: fixes N :: "'a set" assumes "N \<lhd> G" shows "subgroup N G"
max_less: shows "2 ^ 31 < 9223372039002259455"
BSPTaxonomyDifferentCorrections_axioms: shows "BSPTaxonomyDifferentCorrections ES1 \<parallel> ES2 \<V>"
reachable_mono: fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set"   and X :: "'a set"   and X' :: "'a set" assumes "R \<subseteq> R'"   and "X \<subseteq> X'" shows "R\<^sup>* `` X \<subseteq> R'\<^sup>* `` X'"
disc_transfer_1: fixes R1 :: "'a \<Rightarrow> 'c \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (tllist_all2 R1 R2) (=) is_TNil is_TNil"