hcju/isabelleps huggingface dataset:
invariant_def: fixes root :: "('a att_scheme \<times> 'b, 'a att_scheme, 'c) env"   and path :: "'c list" shows "local.invariant root path = (\<exists>att dir. access root path user\<^sub>1 {} = Some (Env att dir) \<and> dir \<noteq> Map.empty \<and> user\<^sub>1 \<noteq> owner att \<and> access root path user\<^sub>1 {Writable} = None)"
bind_return: fixes x :: "'m1" shows "bind1 x return1 = x"
word_and_full_mask_simp: fixes x :: "'b word" shows "x AND mask LENGTH('b) = x"
div_mult_le: fixes n :: "nat"   and m :: "nat" shows "n div m * m \<le> n"
cosh_dist_general: fixes p :: "proj2"   and q :: "proj2"   and a :: "proj2"   and b :: "proj2" assumes "are_endpoints_in_S p q a b" shows "cosh_dist a b = (sqrt (cross_ratio p q a b) + 1 / sqrt (cross_ratio p q a b)) / 2"
rtrancl_on_into_rtrancl_on: fixes a :: "'c"   and b :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set"   and c :: "'c" assumes "(a, b) \<in> rtrancl_on F r"   and "(b, c) \<in> r"   and "c \<in> F" shows "(a, c) \<in> rtrancl_on F r"
prems: assumes "d < length b" shows "child p' dir d' ! d = b ! d"
colx: 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 X :: "'a"   and Y :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Tarski_neutral_dimensionless.Col Bet X Y A"   and "Tarski_neutral_dimensionless.Col Bet X Y B"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Col Bet X Y C"
interior_translation: fixes a :: "'c"   and S :: "'c set" shows "interior ((+) a ` S) = (+) a ` interior S"
Idem_comp: fixes r :: "('b \<times> 'b trm) list"   and s :: "('b \<times> 'b trm) list"   and t :: "'b trm"   and u :: "'b trm" assumes "Idem r"   and "Unifier s (t \<lhd> r) (u \<lhd> r)"   and "\<And>q. Unifier q (t \<lhd> r) (u \<lhd> r) \<Longrightarrow> s \<lozenge> q \<doteq> q" shows "Idem (r \<lozenge> s)"
cart2_append1_between_strict: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" assumes "z_non_zero p"   and "z_non_zero q"   and "z_non_zero r"   and "B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r"   and "q \<noteq> p"   and "q \<noteq> r" shows "\<exists>k>0. k < 1 \<and> cart2_append1 q = k *\<^sub>R cart2_append1 r + (1 - k) *\<^sub>R cart2_append1 p"
leg_in_hom_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "partial_magma.in_hom C (Leg1 S) (span_in_category.apex C S) (partial_magma.cod C (Leg1 S))"
feasibleI: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> bounded_by b (t P)"   and "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> nneg (t P)" shows "feasible t"
unless_def: fixes P :: "bool"   and m :: "'d \<Rightarrow> (unit \<times> 'd) \<times> bool" shows "unless P m \<equiv> when1 (\<not> P) m"
fun_upd_restrict: fixes m :: "'b \<Rightarrow> 'c option"   and D :: "'b set"   and x :: "'b"   and y :: "'c option" shows "(m |` D)(x := y) = (m |` (D - {x}))(x := y)"
weak_ring_morphism_inv_into: fixes f :: "'a \<Rightarrow> 'd"   and I :: "'a set"   and a :: "'a" assumes "weak_ring_morphism f I R"   and "a \<in> carrier R" obtains i :: "'a" where "i \<in> I" and "inv_into (carrier R) f (f a) = i \<oplus> a"
genPoly_incr_facesquander_lb: fixes g' :: "graph"   and n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph" assumes "g' \<in> set (generatePolygon n v f g)"   and "Invariants.inv g"   and "f \<in> set (nonFinals g)"   and "v \<in> \<V> f"   and "3 \<le> n" shows "faceSquanderLowerBound g + \<d> n \<le> faceSquanderLowerBound g'"
dg_prod_is_arrD_1: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_prod I \<AA>\<^esub> b" shows "vsv f"
right_unique_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "right_unique R = (\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z)"
abs_dlist_def: shows "dlist \<equiv> quot_type.abs dlist_eq Abs_dlist"
app_iso_inv: fixes G :: "('c, 'd) pre_digraph"   and hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "pre_digraph.digraph_isomorphism G hom" shows "app_iso (pre_digraph.inv_iso G hom) (app_iso hom G) = G"
simps_32: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Grow_memory ts = (if memory \<C> \<noteq> None then type_update ts [TSome T_i32] (Type [T_i32]) else Bot)"
Assms_1: shows "obsv_state.obs s' r_w p = Some w"
divides_aux_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and qr :: "'a \<times> 'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "unique_euclidean_semiring_numeral.divides_aux zero qr = (snd qr = zero)"
fi_state: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)"   and "i < m" shows "fi_state state = fs ! i"
matchC_C_sym_trans: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and theta :: "(('d, 'c, 'a) cmd \<times> ('d, 'c, 'a) cmd) set"   and c1 :: "('d, 'c, 'a) cmd"   and c :: "('d, 'c, 'a) cmd"   and c2 :: "('d, 'c, 'a) cmd" assumes "PL_Indis cval indis"   and "PL_Indis.matchC_C aval tval cval indis theta c1 c"   and "PL_Indis.matchC_C aval tval cval indis theta c c2"   and "sym theta"   and "trans theta" shows "PL_Indis.matchC_C aval tval cval indis theta c1 c2"
known_ptr_node_or_document: fixes ptr :: "(_) object_ptr" assumes "known_ptr ptr" shows "is_node_ptr_kind ptr \<or> is_document_ptr_kind ptr"
that_2: fixes y :: "'b" shows "B y \<longrightarrow> B' y"
fresh_star_list_cons_2: fixes Xs :: "name set"   and x :: "'a"   and xs :: "'a list" shows "Xs \<sharp>* (x # xs) = (Xs \<sharp>* x \<and> Xs \<sharp>* xs)"
set_mset_single: fixes b :: "'a" shows "set_mset (\<LM> b  \<RM>) = {b}"
fminus_fsubset: fixes A :: "'a fset"   and B :: "'a fset" shows "A |-| B |\<subseteq>| A"
WTrtNew: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option" assumes "is_class P C" shows "P,E,h,sh \<turnstile> new C : Class C"
plus_of_int: fixes x :: "real"   and n :: "int" shows "cis (x + real_of_int n * (2 * pi)) = cis x"
rep_eq: fixes x :: "'a blinop"   and xa :: "'a blinop" shows "Rep_blinop (x * xa) = Rep_blinop x o\<^sub>L Rep_blinop xa"
lookup_nodes_of_phis'_subst: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "Mapping.lookup (uninst_code.nodes_of_phis' g next (uninst_code.substitution_code phis g next) nodes_of_phis) (uninst_code.substitution_code phis g next) = Some ((case Mapping.lookup nodes_of_phis (uninst_code.substitution_code phis g next) of None \<Rightarrow> {} | Some x \<Rightarrow> Set.remove next x) \<union> (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> Set.remove next x))"
assms_4: shows "h \<turnstile> get_elements_by_class_name ptr name \<rightarrow>\<^sub>r results"
determines_span: assumes "ide f" shows "span_in_category (\<odot>) \<lparr>Leg0 = \<lbrakk>\<lbrakk>tab\<^sub>0 f\<rbrakk>\<rbrakk>, Leg1 = \<lbrakk>\<lbrakk>tab\<^sub>1 f\<rbrakk>\<rbrakk>\<rparr>"
g_def: shows "g = b_least2 (\<lambda>u z. sgn1 (sf (z + 1) - u))"
smult_eq_0_iff: fixes a :: "'b"   and p :: "'b poly" shows "(smult a p = 0) = (a = (0::'b) \<or> p = 0)"
induct: fixes x :: "'b llist"   and A :: "'b set"   and P :: "'b llist \<Rightarrow> bool" assumes "x \<in> A\<^sup>\<star>"   and "P LNil"   and "\<And>l a. \<lbrakk>l \<in> A\<^sup>\<star>; P l; a \<in> A\<rbrakk> \<Longrightarrow> P (a ## l)" shows "P x"
Inv_Inv: fixes t :: "'a local.term" assumes "Can t" shows "Inv (Inv t) = t"
IS': fixes \<nu>2 :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "(\<nu>, \<nu>2) \<in> (prog_sem I ($\<alpha> vid1))\<^sup>*" shows "\<nu>2 \<in> fml_sem I (Pc pid1 \<rightarrow> [[$\<alpha> vid1]]Pc pid1)"
psi_def: shows "\<psi> = inv_into (carrier G) \<phi>"
not_a2_dvd_a1: shows "\<not> a2 dvd a1"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>x y ys. P ys \<Longrightarrow> P (x # y # ys)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
set_2: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat" shows "set_of (MKT x21 x22 x23 x24) = Set.insert x21 (set_of x22 \<union> set_of x23)"
zero_interchange: shows "is_zero_row_upt_k (mod_type_class.from_nat i) k (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n))"
ref_WHILEI_invarI: fixes I :: "'c \<Rightarrow> bool"   and s :: "'c"   and M :: "'d nres"   and R :: "('d \<times> 'c) set"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres" assumes "I s \<Longrightarrow> M \<le> \<Down> R (WHILE\<^bsup>I\<^esup> b f s)" shows "M \<le> \<Down> R (WHILE\<^bsup>I\<^esup> b f s)"
sums_0: fixes f :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n = (0::'a)" shows "f sums (0::'a)"
ccts: shows "thread_oks (redT_updT TS TA) TAS"
lookup_minus_fun: fixes s :: "'a \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (s - t) = lookup s - lookup t"
subst_floatarith_Var_fold_const_fa: fixes s :: "nat \<Rightarrow> nat"   and fa :: "floatarith" shows "subst_floatarith (\<lambda>x. Var (s x)) (fold_const_fa fa) = fold_const_fa (subst_floatarith (\<lambda>x. Var (s x)) fa)"
slice2_cdep_slice2: fixes nx :: "'node SDG_node"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "nx \<in> sum_SDG_slice2 n"   and "n s\<longrightarrow>\<^bsub>cd\<^esub> n'" shows "nx \<in> sum_SDG_slice2 n'"
trans_lex_prod: fixes R1 :: "('g \<times> 'g) set"   and R2 :: "('h \<times> 'h) set" assumes "trans R1"   and "trans R2" shows "trans (R1 <*lex*> R2)"
remap_polys_l_dom_err_def: shows "remap_polys_l_dom_err = SPEC (\<lambda>_. True)"
class_add_eff_sup_state_opt: fixes i :: "instr"   and P :: "jvm_prog"   and pc :: "nat"   and xt :: "ex_table"   and \<tau> :: "ty list \<times> ty err list"   and \<tau>s :: "(ty list \<times> ty err list) option list"   and wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and mxs :: "nat"   and T :: "ty"   and cdec :: "jvm_method class" assumes "\<forall>(pc', \<tau>')\<in>set (eff i P pc xt \<lfloor>\<tau>\<rfloor>). P \<turnstile> \<tau>' \<le>' \<tau>s ! pc'"   and "wf_prog wf_md P"   and "\<not> is_class P C"   and "app\<^sub>i (i, P, pc, mxs, T, \<tau>)" shows "\<forall>(pc', \<tau>')\<in>set (eff i (class_add P (C, cdec)) pc xt \<lfloor>\<tau>\<rfloor>). class_add P (C, cdec) \<turnstile> \<tau>' \<le>' \<tau>s ! pc'"
alphaAbs_trans: fixes A :: "('b, 'c, 'varSort, 'var, 'd) qAbs"   and A' :: "('b, 'c, 'varSort, 'var, 'd) qAbs"   and A'' :: "('b, 'c, 'varSort, 'var, 'd) qAbs" assumes "qGoodAbs A \<or> qGoodAbs A' \<or> qGoodAbs A''"   and "A $= A'"   and "A' $= A''" shows "A $= A''"
ltl_implies_satisfiable_prefix: fixes eq :: "'a ltln \<Rightarrow> 'a ltln \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln"   and i :: "nat" assumes "ltl_equivalence eq"   and "w \<Turnstile>\<^sub>n \<phi>" shows "\<not> eq (af \<phi> (prefix i w)) false\<^sub>n"
quasi_inverse_hcomp': fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and f' :: "'b"   and g :: "'b"   and g' :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.equivalence_map V H \<a> \<i> src trg f"   and "bicategory.equivalence_map V H \<a> \<i> src trg f'"   and "bicategory.equivalence_map V H \<a> \<i> src trg (H f f')"   and "bicategory.quasi_inverses V H \<a> \<i> src trg f g"   and "bicategory.quasi_inverses V H \<a> \<i> src trg f' g'" shows "bicategory.quasi_inverses V H \<a> \<i> src trg (H f f') (H g' g)"
intros_3: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<approx>2 ys" shows "ys \<approx>2 xs"
head_def: fixes S :: "'node list" shows "head S \<equiv> if S = [] then nil else hd S"
partn_lst_triv1: fixes \<alpha> :: "V list"   and i :: "nat"   and n :: "nat"   and B :: "'a set"   and r :: "('a \<times> 'a) set" assumes "\<alpha> ! i \<le> 1"   and "i < length \<alpha>"   and "1 < n"   and "B \<noteq> {}"   and "wf r" shows "Partitions.partn_lst r B \<alpha> n"
set_less_eq_aux_antisym: fixes A :: "'a set"   and B :: "'a set" assumes "A \<sqsubseteq>' B"   and "B \<sqsubseteq>' A" shows "A = B"
simps_3: fixes x2 :: "'b" shows "ext0.Val x2 \<noteq> ext0.Bot"
inject_2: fixes x21 :: "('a, 'b) tm"   and x22 :: "('a, 'b) tm"   and y21 :: "('a, 'b) tm"   and y22 :: "('a, 'b) tm" shows "(App x21 x22 = App y21 y22) = (x21 = y21 \<and> x22 = y22)"
elims_2: fixes x :: "('b \<times> 'b) list"   and xa :: "'b"   and xb :: "'b" assumes "in_trancl x xa xb" obtains TI :: "('b \<times> 'b) list" and a :: "'b" and b :: "'b" where "x = TI" and "xa = a" and "xb = b" and "if (a, b) \<in> set TI then True else list_ex (\<lambda>(c, d). c = a \<and> in_trancl (removeAll (c, d) TI) d b) TI"
poly_to_row_scalar_mult: fixes p :: "'c \<Rightarrow>\<^sub>0 'd"   and ts :: "'c list"   and c :: "'d" assumes "keys p \<subseteq> set ts" shows "row_to_poly ts (c \<cdot>\<^sub>v poly_to_row ts p) = c \<cdot> p"
rev_append: fixes xs :: "'b list"   and ys :: "'b list" shows "rev (xs @ ys) = rev ys @ rev xs"
chinese_remainder_aux_poly: fixes A :: "'c set"   and m :: "'c \<Rightarrow> 'd poly" assumes "finite A"   and "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)" shows "\<exists>b. \<forall>i\<in>A. [b i = 1] (mod m i) \<and> [b i = 0] (mod prod m (A - {i}))"
zero_in_ring_n_pd_zero_K: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "distinct_pds K n P" shows "\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub> = \<zero>"
p''_def: fixes a :: "real"   and t :: "real" shows "movement.p'' a t = a"
SeqE: fixes c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int"   and ct :: "com \<times> (char list \<Rightarrow> int)" assumes "(c1;; c2, s) \<rightarrow> ct" obtains "c1 = SKIP" and "ct = (c2, s)"   | c\<^sub>1' :: "com" and s' :: "char list \<Rightarrow> int" where "ct = (c\<^sub>1';; c2, s')" and "(c1, s) \<rightarrow> (c\<^sub>1', s')"
a_2: shows "f \<in> \<Theta>(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p' / u powr (p + 1))))"
ring_hom_trans: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and g :: "'b \<Rightarrow> 'e"   and T :: "('e, 'f) ring_scheme" assumes "f \<in> ring_hom R S"   and "g \<in> ring_hom S T" shows "g \<circ> f \<in> ring_hom R T"
discr_eff_indis: fixes c :: "('test, 'atom, 'choice) cmd"   and i :: "nat"   and s :: "'state" assumes "discr c"   and "i < brn c" shows "s \<approx> eff c s i"
it_to_it_linord_correct: fixes it :: "('a list \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a list \<Rightarrow> 'a list) \<Rightarrow> 'a list \<Rightarrow> 'a list"   and S :: "'a set" assumes "set_iterator_linord it S" shows "set_iterator_linord (it_to_it it) S"
res_def: fixes key_alg :: "'c spmf"   and enc_alg :: "'c \<Rightarrow> 'a \<Rightarrow> 'b spmf"   and dec_alg :: "'c \<Rightarrow> 'b \<Rightarrow> 'a option" shows "cipher.res key_alg enc_alg dec_alg = 1\<^sub>C |\<^sub>= cipher.enc enc_alg |\<^sub>= cipher.dec dec_alg \<rhd> 1\<^sub>C |\<^sub>= parallel_wiring \<rhd> parallel_resource1_wiring \<rhd> key.res key_alg \<parallel> CIPHER.AUTH.res"
outputFreshTrans: fixes a :: "name"   and b :: "name"   and P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi" assumes "a{b}.P \<longmapsto> \<alpha> \<prec> P'"   and "a \<sharp> \<alpha> \<or> b \<sharp> \<alpha>" shows "False"
f_props_2: shows "\<forall>cube\<in>one_chain1. chain_reparam_weak_path (rec_join [cube]) (f cube)"
fs_cont: fixes k :: "nat" assumes "k \<le> N" shows "continuous_on {real_of_int a..} (fs k)"
monoD: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and f :: "'b::type \<Rightarrow> 'c::order"   and x :: "'b::type"   and y :: "'b::type" assumes "class.order (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "order.mono (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (f::'b::type \<Rightarrow> 'c::order)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) (y::'b::type)" shows "(f::'b::type \<Rightarrow> 'c::order) (x::'b::type) \<le> f (y::'b::type)"
cinsert_absorb: fixes a :: "'f"   and A :: "'f cset" assumes "cin a A" shows "cinsert a A = A"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_tag_name get_tag_name_locs" shows "l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs"
homeomorphic_map_interior_of: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "homeomorphic_map X Y f"   and "S \<subseteq> topspace X" shows "Y interior_of f ` S = f ` (X interior_of S)"
graph_homomorphism_semantics: fixes A :: "('d, 'e) labeled_graph"   and B :: "('d, 'f) labeled_graph"   and f :: "('e \<times> 'f) set"   and a :: "'e"   and b :: "'e"   and e :: "'d allegorical_term"   and a' :: "'f"   and b' :: "'f" assumes "graph_homomorphism A B f"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>"   and "(a, a') \<in> f"   and "(b, b') \<in> f" shows "(a', b') \<in> :B:\<lbrakk>e\<rbrakk>"
f_minus: fixes v :: "'c \<Rightarrow> 'a"   and w :: "'c \<Rightarrow> 'a" assumes "v \<in> carrier (func_space A)"   and "w \<in> carrier (func_space A)" shows "lincomb (v \<ominus>\<^bsub>func_space A\<^esub> w) A = lincomb v A \<ominus>\<^bsub>M\<^esub> lincomb w A"
enum_all_bit1_def: fixes P :: "'a bit1 \<Rightarrow> bool" shows "enum_class.enum_all P = (\<forall>b\<in>set enum_class.enum. P b)"
sup_sharp_homTr: fixes N :: "('i, 'b, 'j) Module_scheme"   and L :: "('k, 'b, 'l) Module_scheme"   and u :: "'a \<Rightarrow> 'i"   and f :: "'i \<Rightarrow> 'k" assumes "R module N"   and "R module L"   and "u \<in> mHom R M N"   and "f \<in> mHom R N L" shows "sup_sharp R M N L u f \<in> mHom R M L"
intros_3: fixes n :: "nat"   and l :: "'a tree23"   and m :: "'a tree23"   and r :: "'a tree23"   and p :: "int \<times> 'a"   and q :: "int \<times> 'a" assumes "full n l"   and "full n m"   and "full n r" shows "full (Suc n) (Branch3 l p m q r)"
labelAtIn_def: fixes v :: "nat"   and p :: "(form, nat) antecedent" shows "task.labelAtIn v p = subst (inst v) (predicate.freshen (id v) (task.labelsIn (nodeOf v) p))"
nipd: shows "\<forall>l\<in>{j..k}. \<pi> l \<noteq> ipd (\<pi> j)"
assms_1: shows "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) ((A_zs @ A_x # A_xs) @ A_ws @ A_y # A_ys) ((zs @ x # xs) @ ws @ y # ys)"
simps_1: fixes x2 :: "'d"   and y2 :: "'d" shows "(\<lfloor>x2\<rfloor> = \<lfloor>y2\<rfloor>) = (x2 = y2)"
concurrent_set_ConsI: fixes a :: "'a"   and xs :: "'a list"   and x :: "'a" assumes "concurrent_set a xs"   and "a \<parallel> x" shows "concurrent_set a (x # xs)"
assoc: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "max (max a b) c = max a (max b c)"
hyps_1: fixes s :: "'a" assumes "\<Gamma>\<turnstile>merge_guards c1 \<down> Normal s" shows "\<Gamma>\<turnstile>c1 \<down> Normal s"
simps_25: fixes g1 :: "'b gexp"   and g2 :: "'b gexp"   and v :: "'b"   and va :: "value list" shows "GExp_Lexorder.less_gexp_aux (Nor g1 g2) (In v va) = False"
constructive_security_obsf_trivial: fixes \<I> :: "('m, 'o) \<I>"   and \<I>_common :: "('n, 'p) \<I>"   and res :: "('m + 'n, 'o + 'p) resource"   and \<A> :: "('m + 'n, 'o + 'p) distinguisher_obsf" assumes "\<I> \<oplus>\<^sub>\<I> \<I>_common \<turnstile>res res \<surd>" shows "constructive_security_obsf res res 1\<^sub>C \<I> \<I> \<I>_common \<A> 0"
mantissa_eq_zero_iff: fixes x :: "Float.float" shows "(mantissa x = 0) = (x = 0)"
relevant_vars_Entry: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and H :: "'c set"   and L :: "'c set"   and High :: "'b"   and Low :: "'b"   and V :: "'c"   and S :: "'b SDG_node set" assumes "NonInterferenceInterGraph sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses H L High Low"   and "V \<in> SDG.relevant_vars sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S (CFG_node Entry)"   and "High \<notin> SDG.SDG_to_CFG_set (SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S)" shows "V \<in> L"
antecedent: shows "\<not> P x y \<and> O x y"
zip_eq_def: fixes l :: "tm list"   and l' :: "tm list" shows "zip_eq l l' \<equiv> map2 (\<lambda>t t'. Atom (Rl STR ''='' [t, t'])) l l'"
inv_comp_right_2: fixes g :: "'a"   and f :: "'a" assumes "B.iso (g \<cdot>\<^sub>B f)"   and "B.iso f" shows "B.iso g"
induct: fixes P :: "('a, 'b, 'c) sum3 \<Rightarrow> bool"   and sum3 :: "('a, 'b, 'c) sum3" assumes "\<And>x. P (In1 x)"   and "\<And>x. P (In2 x)"   and "\<And>x. P (In3 x)" shows "P sum3"
invx: shows "inv x = g \<otimes> (inv hx \<otimes> inv g)"
nemp: shows "take p w \<cdot> take q w \<noteq> \<epsilon>"
unwinding_theorem_BSD: fixes ur :: "('s \<times> 's) set" assumes "modified_view.lrf ur"   and "modified_view.osc ur" shows "BSD \<V>' Tr\<^bsub>(induceES SES)\<^esub>"