hcju/isabelleps huggingface dataset:
obtain_idx: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and x :: "'a extended" assumes "integrate_insert_commute M a s"   and "x \<in> set (ext_ids s)" shows "\<exists>i. idx s x = Inr i"
assms_3: shows "B = \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>"
bij: shows "bij_betw p {0..<Suc n} {0..<Suc n}"
Joints_in_lv_rel_set_relD: fixes xs :: "(real \<times> real pdevs) list"   and X :: "'b set" assumes "(Joints xs, X) \<in> \<langle>lv_rel\<rangle>set_rel" shows "X = Affine (eucl_of_list_aform xs)"
lprefix_lappend: fixes xs :: "'b llist"   and ys :: "'b llist" shows "lprefix xs (xs $ ys)"
select_convs_3: fixes CatDom :: "('f, 'd, 'h) Category_scheme"   and CatCod :: "('g, 'e, 'i) Category_scheme"   and MapM :: "'d \<Rightarrow> 'e"   and more :: "'j" shows "(##) \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = MapM"
bounded_box: fixes a :: "'c"   and b :: "'c" shows "bounded (box a b)"
dg_prod_is_arrD_7: fixes a :: "V"   and b :: "V"   and f :: "V"   and i :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b"   and "i \<in>\<^sub>\<circ> I" shows "f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>"
pell_valuation_nth_solution: fixes D :: "nat"   and n :: "nat" assumes "pell D" shows "pell.pell_valuation D (case pell.nth_solution D n of (x, y) \<Rightarrow> (int x, int y)) = pell.pell_valuation D (case pell.fund_sol D of (x, y) \<Rightarrow> (int x, int y)) ^ n"
wf_trm_subst: fixes \<delta> :: "'b \<Rightarrow> ('fun, 'b) Term.term"   and t :: "('fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)" shows "wf\<^sub>t\<^sub>r\<^sub>m t = wf\<^sub>t\<^sub>r\<^sub>m (t \<cdot> \<delta>)"
take_bit_of_1: fixes n :: "nat" shows "take_bit n (1::'a) = of_bool (0 < n)"
protocol_def: fixes input\<^sub>1 :: "bool \<times> bool"   and \<sigma> :: "bool" shows "protocol input\<^sub>1 \<sigma> = (let (b\<^sub>\<sigma>, b\<^sub>\<sigma>') = input\<^sub>1 in I \<bind> (\<lambda>(\<alpha>, \<tau>). local.etp.S \<alpha> \<bind> (\<lambda>x\<^sub>\<sigma>. local.etp.S \<alpha> \<bind> (\<lambda>y\<^sub>\<sigma>'. let y\<^sub>\<sigma> = F \<alpha> x\<^sub>\<sigma>; x\<^sub>\<sigma> = F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>; x\<^sub>\<sigma>' = F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>'; \<beta>\<^sub>\<sigma> = B \<alpha> x\<^sub>\<sigma> \<oplus> b\<^sub>\<sigma>; \<beta>\<^sub>\<sigma>' = B \<alpha> x\<^sub>\<sigma>' \<oplus> b\<^sub>\<sigma>' in return_spmf ((), if \<sigma> then B \<alpha> x\<^sub>\<sigma>' \<oplus> \<beta>\<^sub>\<sigma>' else B \<alpha> x\<^sub>\<sigma> \<oplus> \<beta>\<^sub>\<sigma>)))))"
higher_differentiable_on_real_SucI: fixes S :: "real set"   and n :: "nat"   and f :: "real \<Rightarrow> 'a" assumes "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>x. nth_derivative n f x 1) differentiable at x"   and "continuous_on S (\<lambda>x. nth_derivative (Suc n) f x 1)"   and "higher_differentiable_on S f n"   and "open S" shows "higher_differentiable_on S f (Suc n)"
his_determ_base: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and n :: "nat"   and nid' :: "'a"   and t :: "nat \<Rightarrow> cnf"   and nid :: "'a" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "(n, nid') \<in> Blockchain.his active cmp bc mining t n nid" shows "nid' = nid"
intros: assumes "\<And>x y. (x, y) \<in> \<^bold>E\<^bsub>\<Delta>''\<^esub> \<Longrightarrow> (y, x) \<notin> \<^bold>E\<^bsub>\<Delta>''\<^esub>"   and "(source \<Delta>'', sink \<Delta>'') \<notin> \<^bold>E\<^bsub>\<Delta>''\<^esub>" shows "\<Delta>''.wf_residual_network"
HSem_eq: fixes \<Gamma> :: "('var \<times> 'exp) list"   and \<rho> :: "'var \<Rightarrow> 'value" shows "\<lbrace> \<Gamma> \<rbrace>\<rho> = \<rho> ++\<^bsub>domA \<Gamma>\<^esub> \<^bold>\<lbrakk> \<Gamma> \<^bold>\<rbrakk>\<^bsub>\<lbrace> \<Gamma> \<rbrace>\<rho>\<^esub>"
m_maxTr: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" shows "\<forall>l\<le>n. f l \<le> m_max n f"
consec_assoc1: fixes j :: "nat_int"   and k :: "nat_int"   and i :: "nat_int" shows "consec j k \<and> consec i (j \<squnion> k) \<longrightarrow> consec (i \<squnion> j) k"
remove: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<in> A" shows "mset_set A = add_mset x (mset_set (A - {x}))"
NF_NT: fixes s :: "'a event list"   and P :: "'a process" assumes "(s, {}) \<notin> F P" shows "s \<notin> T P"
apply_cltn2_injective: fixes p :: "proj2"   and C :: "cltn2"   and q :: "proj2" assumes "apply_cltn2 p C = apply_cltn2 q C" shows "p = q"
minimum_elem_def: shows "minimum_elem = (\<lambda>D X a. a \<in> X \<and> (\<forall>x\<in>X. a \<preceq>\<^bsub>D\<^esub> x))"
distinct_288: fixes x15 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Auth x15 \<noteq> Unroll x14"
abs_norm_cancel: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and a :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a\<bar> = norm a"
Comp_eq_iso_class_memb: fixes h :: "'a"   and G :: "'a set"   and F :: "'a set" assumes "h \<in> Maps.Comp G F" shows "Maps.Comp G F = \<lbrakk>h\<rbrakk>"
sum_over_sqrt: fixes SQ :: "real"   and f :: "nat \<Rightarrow> real"   and n :: "nat" assumes "0 \<le> SQ" shows "(\<Sum>i = 0..<n + 1. f i * SQ ^ (i div 2) * real (i mod 2) * sqrt SQ + f i * SQ ^ (i div 2) * (1 - real (i mod 2))) = (\<Sum>i = 0..<n + 1. f i * sqrt SQ ^ i)"
simps_1: fixes xs :: "[:'a:]" shows "prefix\<cdot>xs\<cdot>\<bottom> = \<bottom>"
tND1: fixes p :: "('b set set, 'c) Combinators list"   and x :: "('b set set, 'c) Combinators"   and y :: "('b set set, 'c) Combinators"   and a :: "'b set set"   and b :: "'b set set"   and c :: "'b set set"   and d :: "'b set set" assumes "allNetsDistinct p"   and "x \<in> set p"   and "y \<in> set p"   and "a = srcNet x"   and "b = destNet x"   and "c = srcNet y"   and "d = destNet y"   and "a \<noteq> c"   and "singleCombinators [x]"   and "x \<noteq> DenyAll"   and "singleCombinators [y]"   and "y \<noteq> DenyAll" shows "twoNetsDistinct a b c d"
coprime_exp_mod: fixes lu :: "int" assumes "comm_monoid_mult_class.coprime lu p"   and "prime p"   and "n \<noteq> 0" shows "lu mod p ^ n \<noteq> 0"
normF_xl_lim: fixes F :: "'a set" assumes "finite F" shows "(\<lambda>m. normF F (\<lambda>a. x m a - l a)) \<longlonglongrightarrow> 0"
domain_eq: shows "Domainp pcr_so = (\<lambda>s. s \<noteq> [] \<and> is_upper (hd s))"
eq0: shows "comp k k' = comp (fst (f (k, v))) k''"
aperiodic: shows "KN.aperiodic (C \<times> Some ` C)"
star_separate: fixes x :: "'a"   and y :: "'a" assumes "x \<sqinter> y = bot"   and "y \<sqinter> y = bot" shows "top = top \<squnion> y \<sqinter> top"
O_simps_3: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "\<gamma> trn" shows "O (trn # tr) = g trn # O tr"
vec_of_poly_def: fixes p :: "'b poly" shows "vec_of_poly p \<equiv> let m = degree p in vec (Suc m) (\<lambda>i. coeff p (m - i))"
rev_predicate1D: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and Q :: "'b \<Rightarrow> bool" assumes "P x"   and "P \<le> Q" shows "Q x"
th: shows "\<lparr>Add (Mul (head p) (Pw (Bound 0) (Reflected_Multivariate_Polynomial.degree p))) (behead p)\<rparr>\<^sub>p\<^bsup>bs\<^esup> = \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup>"
has_laplace_uminus: fixes f :: "real \<Rightarrow> complex"   and F :: "complex"   and S :: "complex" assumes "(f has_laplace F) S" shows "((\<lambda>x. - f x) has_laplace - F) S"
geK: fixes n :: "nat" assumes "N + K < n" shows "B (Suc n) \<le> C * Max {B i |i. i \<le> n}"
project_act_I: fixes z :: "'c"   and z' :: "'c"   and act :: "('c \<times> 'c) set" assumes "(z, z') \<in> act" shows "(f z, f z') \<in> project_act h act"
calculus_axioms: shows "calculus Bot_F Inf_F (\<lambda>N1 N2. entails_q q (wf_lift.\<G>_Fset N1) (wf_lift.\<G>_Fset N2)) wf_lift.Red_I_\<G> local.wf_lift.Red_F_\<G>"
goedel_rosser_first_strong: assumes "consistent" shows "\<not> prv \<phi>R \<and> \<not> prv (neg \<phi>R) \<and> isTrue \<phi>R"
disjnt_Diff1: fixes X :: "'a set"   and Y' :: "'a set"   and Y :: "'a set"   and X' :: "'a set" assumes "X \<subseteq> Y'" shows "disjnt (X - Y) (X' - Y')"
inner_sum_left: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a" shows "inner (sum f A) y = (\<Sum>x\<in>A. inner (f x) y)"
fbox_kstarI: fixes P :: "'b \<Rightarrow> bool"   and I :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'b set" assumes "P \<le> I"   and "I \<le> Q"   and "I \<le> |F] I" shows "P \<le> |F\<^sup>*] Q"
tendsto_at_iff_tendsto_nhds: fixes g :: "'a \<Rightarrow> 'b"   and l :: "'a" shows "g \<midarrow>l\<rightarrow> g l = (g \<longlongrightarrow> g l) (nhds l)"
map'_1: fixes x :: "String.literal"   and t :: "tm"   and e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and v :: "'a" assumes "\<not> occurs_in x t" shows "semantics_term e f t = semantics_term (e(x := v)) f t"
Delta_Root_CTRL: shows "Delta Root_CTRL = Root_CTRL_Delta"
remdups_adj_Cons: fixes x :: "'b"   and xs :: "'b list" shows "remdups_adj (x # xs) = (case remdups_adj xs of [] \<Rightarrow> [x] | y # xs \<Rightarrow> if x = y then y # xs else x # y # xs)"
the_map_option: fixes x :: "'c option"   and f :: "'c \<Rightarrow> 'd" assumes "\<not> Option.is_none x" shows "the (map_option f x) = f (the x)"
error_if_None: fixes c :: "bool"   and e :: "error"   and y :: "abopt" shows "(error_if c e y = None) = (\<not> c \<and> y = None)"
i_intersect: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "ideal (I \<inter> J) R"
assms: shows "eval_ubx opubx \<Sigma> = Some z"
cINF_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 f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'c"   and u :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_below less_eq (f ` A)"   and "x \<in> A"   and "less_eq (f x) u" shows "less_eq (Inf (f ` A)) u"
lcf_deg_0: fixes p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int" assumes "deg INTEG p = 0"   and "p \<in> carrier (UP INTEG)"   and "q \<in> carrier (UP INTEG)" shows "p \<otimes>\<^bsub>UP INTEG\<^esub> q = p (deg INTEG p) \<odot>\<^bsub>UP INTEG\<^esub> q"
nabla_subdist: fixes x :: "'a"   and y :: "'a" shows "\<nabla>x \<le> \<nabla>(x + y)"
clock_jviewInit_def: shows "clock_jviewInit \<equiv> \<lambda>a. Pair 0"
dependent_live_vars_Exit_empty: fixes V :: "'var"   and as' :: "'edge list"   and as :: "'edge list" assumes "(V, as', as) \<in> dependent_live_vars (_Exit_)" shows "False"
eq: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S" shows "h (g x y) = g (h x) (h y)"
obtain_set_qFresh: fixes V :: "'var set"   and XS :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" assumes "finite V \<or> |V| <o |UNIV|"   and "finite XS \<or> |XS| <o |UNIV|"   and "Ball XS qGood" shows "\<exists>W. infinite W \<and> W \<inter> V = {} \<and> (\<forall>x\<in>W. \<forall>X\<in>XS. qAFresh xs x X \<and> qFresh xs x X)"
bounded_def: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.metric_space dist uniformity open" shows "metric_space.bounded dist S = (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
gfp_while_lattice: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f"   and "finite UNIV" shows "gfp f = while (\<lambda>A. f A \<noteq> A) f top"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'b"   and generat :: "('d, 'e, 'f) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. generat = Pure x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. generat = IO x21 x22 \<longrightarrow> P (f2 x21 x22)))"
sel_2: fixes x1 :: "'a"   and x2 :: "'a tree"   and x3 :: "'a tree" shows "left (Node x1 x2 x3) = x2"
connectedin_topspace: fixes X :: "'b topology" shows "connectedin X (topspace X) = connected_space X"
red_rtrancl_plus_higher: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* p q"   and "\<And>u v. \<lbrakk>u \<in> keys p; v \<in> keys r\<rbrakk> \<Longrightarrow> u \<prec> v" shows "(punit.red F)\<^sup>*\<^sup>* (p + r) (q + r)"
ground_fm_Fvars: fixes A :: "fm" shows "ground_fm A = (Fvars A = {})"
1: shows "inj_on g (op_set_filter (Not \<circ> op_set_isEmpty \<circ> g) S)"
NatTransCompDefDom: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and f :: "'i"   and X :: "'g"   and Y :: "'g" assumes "NatTrans \<eta>"   and "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y" shows "NTDom \<eta> ## f \<approx>>\<^bsub>NTCatCod \<eta>\<^esub> \<eta> $$ Y"
Fftor: shows "Ftor F : Op C \<longrightarrow> SET"
assms_3: shows "\<forall>r\<in>R. r <o p"
invar_NodeR: fixes l :: "('b \<times> nat) tree"   and x :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree"   and r' :: "('b \<times> nat) tree" assumes "invar \<langle>l, (x, n), r\<rangle>"   and "n = lvl r + 1"   and "invar r'"   and "lvl r' = lvl r" shows "invar \<langle>l, (x, n), r'\<rangle>"
SFAssRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e,s,b\<rangle> \<rightarrow> \<langle>C\<bullet>\<^sub>sF{D} := e',s',b'\<rangle>"
eval_monom_smallo: fixes basis :: "(real \<Rightarrow> real) list"   and es :: "real list"   and e :: "real"   and c :: "real" assumes "basis \<noteq> []"   and "basis_wf basis"   and "length es = length basis"   and "hd es < e" shows "eval_monom (c, es) basis \<in> o(\<lambda>x. hd basis x powr e)"
linear_frechet_derivative: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" assumes "f differentiable net" shows "linear (frechet_derivative f net)"
Monotone_def: fixes \<phi> :: "(((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool) \<Rightarrow> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" shows "Monotone \<phi> = (\<forall>\<Phi> \<Psi>. (\<forall>s t \<beta>. \<Phi> (s, t, \<beta>) \<longrightarrow> \<Psi> (s, t, \<beta>)) \<longrightarrow> (\<forall>s t \<beta>. \<phi> \<Phi> (s, t, \<beta>) \<longrightarrow> \<phi> \<Psi> (s, t, \<beta>)))"
vconcat_vcons: fixes xs :: "V"   and xss :: "V" assumes "vfsequence xs"   and "vfsequence xss"   and "\<forall>xs\<in>elts (\<R>\<^sub>\<circ> xss). vfsequence xs" shows "vconcat (xss #\<^sub>\<circ> xs) = vconcat xss @\<^sub>\<circ> xs"
d_Suc: fixes i :: "nat" shows "fs_init.gs.d (Suc i) = fs_init.gs.\<mu>' i i"
enum_all_set_def: fixes P :: "'a set \<Rightarrow> bool" shows "enum_all P = (\<forall>A\<in>set enum. P A)"
echelon_form_condition2: fixes A :: "(('a, 'b) vec, 'c) vec"   and j :: "'c" assumes "echelon_form A" shows "\<forall>i. i < j \<and> \<not> is_zero_row i A \<and> \<not> is_zero_row j A \<longrightarrow> (LEAST n. A $ i $ n \<noteq> (0::'a)) < (LEAST n. A $ j $ n \<noteq> (0::'a))"
emv_subst_imp_is_Var: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term" assumes "emv s t"   and "s = t \<cdot> \<sigma>" shows "\<forall>x\<in>vars_term t. is_Var (\<sigma> x)"
strict_trans1: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry"   and c :: "'c DBMEntry" assumes "a \<preceq> b"   and "b \<prec> c" shows "a \<prec> c"
renames_CAs0: shows "is_renaming_list \<rho>s"
reach_nil: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a" shows "reach tr p [] p"
mC_ZOC_mono: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd"   and s :: "'state"   and t :: "'state"   and I0 :: "nat set"   and P :: "nat set set"   and F :: "nat set \<Rightarrow> nat set"   and theta' :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set" assumes "mC_ZOC theta c d s t I0 P F"   and "theta \<subseteq> theta'" shows "mC_ZOC theta' c d s t I0 P F"
simps_1: shows "rexp_ewp 0\<^sub>r = False"
full_True: shows "intT = (\<lambda>\<sigma> D. True)"
proper_cont: fixes c :: "('test, 'atom, 'choice) cmd"   and i :: "nat"   and s :: "'state" assumes "proper c"   and "i < brn c" shows "proper (local.cont c s i)"
dg_prod_Cod_app_component_app: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"   and "i \<in>\<^sub>\<circ> I" shows "dg_prod I \<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr>\<lparr>i\<rparr> = \<AA> i\<lparr>Cod\<rparr>\<lparr>f\<lparr>i\<rparr>\<rparr>"
L1_int_ineq: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" shows "\<bar>integral\<^sup>L M f\<bar> \<le> Norm (\<LL> 1 M) f"
mulex_on_self_add_singleton_right: fixes a :: "'b"   and A :: "'b set"   and M :: "'b multiset"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "a \<in> A"   and "M \<in> multisets A" shows "mulex_on P A M (add_mset a M)"
fstar_unfoldl_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "Inf_pres f" shows "x \<sqinter> f (fstar f x) = fstar f x"
elims_3: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and xb :: "lock_action" assumes "\<not> lock_action_ok x xa xb" obtains l :: "('b \<times> nat) option" and t :: "'b" where "x = la__" and "xa = tb__" and "xb = Lock" and "\<not> may_lock la__ tb__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = lb__" and "xa = tc__" and "xb = Unlock" and "\<not> has_lock lb__ tc__"   | l :: "('b \<times> nat) option" and t :: "'b" where "x = l" and "xa = t" and "xb = UnlockFail" and "\<not> \<not> has_lock l t"
pointermap_sane_def: fixes m :: "('c, 'd) pointermap_scheme" shows "pointermap_sane m \<equiv> distinct (entries m) \<and> (\<forall>n\<in>{..<length (entries m)}. getentry m (entries m ! n) = Some n) \<and> (\<forall>p i. getentry m p = Some i \<longrightarrow> entries m ! i = p \<and> i < length (entries m))"
minus_an_in_aug_minf: fixes n :: "nat" shows "- an n \<in> Z\<^sub>-\<^sub>\<infinity>"
simps_2: fixes x :: "'k"   and xs :: "'k list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
qTerm_rawInduct: fixes phi :: "('c, 'd, 'a, 'b, 'e) qTerm \<Rightarrow> bool"   and phiAbs :: "('c, 'd, 'a, 'b, 'e) qAbs \<Rightarrow> bool"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" assumes "\<And>xs x. phi (qVar xs x)"   and "\<And>delta inp binp. \<lbrakk>liftAll phi inp; liftAll phiAbs binp\<rbrakk> \<Longrightarrow> phi (qOp delta inp binp)"   and "\<And>xs x X. phi X \<Longrightarrow> phiAbs (qAbs xs x X)" shows "phi X \<and> phiAbs A"
weakBisimCasePres: fixes CsP :: "('c \<times> ('a, 'b, 'c) psi) list"   and CsQ :: "('c \<times> ('a, 'b, 'c) psi) list"   and \<Psi> :: "'b" assumes "\<And>\<phi> P. (\<phi>, P) mem CsP \<Longrightarrow> \<exists>Q. (\<phi>, Q) mem CsQ \<and> guarded Q \<and> (\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q)"   and "\<And>\<phi> Q. (\<phi>, Q) mem CsQ \<Longrightarrow> \<exists>P. (\<phi>, P) mem CsP \<and> guarded P \<and> (\<forall>\<Psi>. \<Psi> \<rhd> P \<doteq> Q)" shows "\<Psi> \<rhd> Cases CsP \<approx> Cases CsQ"
antisym_conv1: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less"   and "\<not> less x y" shows "less_eq x y = (x = y)"
inext_predicate_change_exists_aux: fixes c :: "nat"   and I :: "nat set"   and a :: "nat"   and b :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "c = card (I \<down>\<ge> a \<down>< b)"   and "a < b"   and "a \<in> I"   and "b \<in> I"   and "\<not> P a"   and "P b" shows "\<exists>n\<in>I \<down>\<ge> a \<down>< b. \<not> P n \<and> P (inext n I)"
not_numeral_le_zero: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq (numeral.numeral one plus n) zero"
cases: fixes a :: "msg"   and H :: "msg set" assumes "a \<in> synth H" obtains X :: "msg" where "a = X__" and "X__ \<in> H"   | agt :: "agent" where "a = Agent agt"   | n :: "nat" where "a = Number n"   | X :: "msg" where "a = Hash Xa__" and "Xa__ \<in> synth H"   | X :: "msg" and Y :: "msg" where "a = \<lbrace>Xb__, Y\<rbrace>" and "Xb__ \<in> synth H" and "Y \<in> synth H"   | X :: "msg" and K :: "nat" where "a = Crypt K X" and "X \<in> synth H" and "Key K \<in> H"
tendsto_le: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a"   and g :: "'b \<Rightarrow> 'a"   and y :: "'a" assumes "class.linorder_topology less_eq less open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open x) F"   and "filterlim g (topological_space.nhds open y) F"   and "\<forall>\<^sub>F x in F. less_eq (g x) (f x)" shows "less_eq y x"
type_less_x_String: fixes \<tau> :: "'a basic_type" assumes "\<tau> < String" obtains "\<tau> = OclVoid"
substrTT_assoc_Some: fixes fs :: "(char list \<times> POPLmarkRecord.type) list"   and a :: "char list"   and T :: "POPLmarkRecord.type"   and k :: "nat"   and U :: "POPLmarkRecord.type" assumes "fs\<langle>a\<rangle>\<^sub>? = \<lfloor>T\<rfloor>" shows "fs[k \<mapsto>\<^sub>\<tau> U]\<^sub>r\<^sub>\<tau>\<langle>a\<rangle>\<^sub>? = \<lfloor>T[k \<mapsto>\<^sub>\<tau> U]\<^sub>\<tau>\<rfloor>"
fps_to_fls_nth: fixes f :: "'b fps"   and n :: "int" shows "fls_nth (fps_to_fls f) n = (if n < 0 then 0::'b else f $ nat n)"
split_asm: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and option :: "'g option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"