hcju/isabelleps huggingface dataset:
GSMP_disjoint_term: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and l :: "'e"   and \<A> :: "('e strand_label \<times> ('a, 'b) strand_step) list"   and l' :: "'e"   and Sec :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2"   and "labeled_typed_model.GSMP_disjoint arity public Ana \<Gamma> (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) Sec" shows "t \<notin> labeled_typed_model.GSMP arity Ana \<Gamma> (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>)) \<or> t \<notin> labeled_typed_model.GSMP arity Ana \<Gamma> (trms\<^sub>l\<^sub>s\<^sub>t (proj l' \<A>)) \<or> t \<in> Sec \<or> intruder_model.intruder_synth arity public {} t"
h_OUT: fixes x :: "'v" shows "d_OUT h x = (if x \<in> X then f x else 0)"
gcd'_greatest: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and gcd' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and l :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.GCD_ring times one plus zero minus uminus"   and "GCD_ring.is_gcd times gcd'"   and "dvd.dvd times l a"   and "dvd.dvd times l b" shows "dvd.dvd times l (gcd' a b)"
inj_lift_map: fixes i :: "nat" shows "inj (lift_map i)"
CxPy: shows "C \<subseteq> {i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y}"
mm_equiv_intro: fixes \<R> :: "(((('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) \<times> (('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)) set"   and lc\<^sub>1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc\<^sub>2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "strong_low_bisim_mm \<R>"   and "(lc\<^sub>1, lc\<^sub>2) \<in> \<R>" shows "lc\<^sub>1 \<approx> lc\<^sub>2"
zero_le_odd_power: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and a :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "odd n" shows "less_eq zero (power.power one times a n) = less_eq zero a"
notin_fset: fixes x :: "'b"   and S :: "'b fset" shows "(x |\<notin>| S) = (x \<notin> \<langle>S\<rangle>\<^sub>f)"
effect_success: fixes c :: "'b Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'b" assumes "effect c h h' r" shows "success c h"
assms_3: shows "AE x in M. real_cond_exp M Invariants f x = 0"
leg_simps_3: shows "C.arr HHH\<mu>\<nu>\<pi>\<rho>.dom.leg1"
\<pi>'_def: shows "\<pi>' \<equiv> path \<sigma>'"
natural_transformation_axioms_def: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor"   and G :: "('b, 'c, 'e, 'f) functor"   and u :: "'b \<Rightarrow> 'f" shows "natural_transformation_axioms AA BB F G u \<equiv> (Functor F : AA \<longrightarrow> BB \<and> Functor G : AA \<longrightarrow> BB \<and> u \<in> Ob\<^bsub>AA\<^esub> \<rightarrow> Ar\<^bsub>BB\<^esub>) \<and> u \<in> extensional (Ob\<^bsub>AA\<^esub>) \<and> (\<forall>A\<in>Ob\<^bsub>AA\<^esub>. u A \<in> Hom\<^bsub>BB\<^esub> F \<^bsub>\<o>\<^esub> A G \<^bsub>\<o>\<^esub> A) \<and> (\<forall>A\<in>Ob\<^bsub>AA\<^esub>. \<forall>B\<in>Ob\<^bsub>AA\<^esub>. \<forall>f\<in>Hom\<^bsub>AA\<^esub> A B. G \<^bsub>\<a>\<^esub> f \<bullet>\<^bsub>BB\<^esub> u A = u B \<bullet>\<^bsub>BB\<^esub> F \<^bsub>\<a>\<^esub> f)"
Units_inv_comm: fixes x :: "'a mat"   and y :: "'a mat" assumes "x \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> y = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub>"   and "x \<in> Units (ring_mat TYPE('a) n b)"   and "y \<in> Units (ring_mat TYPE('a) n b)" shows "y \<otimes>\<^bsub>ring_mat TYPE('a) n b\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n b\<^esub>"
transfer: shows "rel_fun (pcr_ttree (=)) (rel_fun (pcr_ttree (=)) (pcr_ttree (=))) (\<union>) (\<oplus>\<oplus>)"
distributed_transform_integrable: fixes M :: "'d measure"   and N :: "'e measure"   and X :: "'d \<Rightarrow> 'e"   and Px :: "'e \<Rightarrow> real"   and P :: "'f measure"   and Y :: "'d \<Rightarrow> 'f"   and Py :: "'f \<Rightarrow> real"   and T :: "'e \<Rightarrow> 'f"   and f :: "'f \<Rightarrow> real" assumes "distributed M N X (\<lambda>x. ennreal (Px x))"   and "\<And>x. x \<in> space N \<Longrightarrow> 0 \<le> Px x"   and "distributed M P Y (\<lambda>x. ennreal (Py x))"   and "\<And>x. x \<in> space P \<Longrightarrow> 0 \<le> Py x"   and "Y = (\<lambda>x. T (X x))"   and "T \<in> N \<rightarrow>\<^sub>M P"   and "f \<in> borel_measurable P" shows "integrable P (\<lambda>x. Py x * f x) = integrable N (\<lambda>x. Px x * f (T x))"
enabled_iff_sconst: fixes K :: "'c \<Rightarrow> 'c pmf"   and x :: "'c"   and \<omega> :: "'c stream" assumes "set_pmf (K x) = {x}" shows "MC_syntax.enabled K x \<omega> = (\<omega> = sconst x)"
simps_5: fixes P' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and e' :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and C :: "acom"   and Q' :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc ({P'/Q/e'} CONSEQ C) Q' = (vc C Q \<and> (\<exists>k>0. \<forall>l s. P' l s \<longrightarrow> time C s \<le> k * e' s \<and> (\<forall>t. \<exists>l'. pre C Q l' s \<and> (Q l' t \<longrightarrow> Q' l t))))"
pos_emb_step_at_nested: fixes p :: "dir list"   and d1 :: "dir"   and q :: "dir list"   and d2 :: "dir"   and t :: "('a, 'b) tm" shows "position_of (emb_step_at (p @ [d1] @ q) d2 t) (p @ [d1]) = position_of t (p @ [d1])"
closed_INT: fixes "open" :: "'a set \<Rightarrow> bool"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'a set" assumes "class.topological_space open"   and "\<forall>x\<in>A. topological_space.closed open (B x)" shows "topological_space.closed open (\<Inter> (B ` A))"
N_def: shows "N = (\<lambda>xs. alw (ev (HLD {t})) xs \<and> alw (\<lambda>xs. \<not> (HLD {t} xs \<and> nxt (HLD {t'}) xs)) xs)"
return_bind: fixes x :: "'d"   and f :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) \<times> bool" shows "return x >>= f = f x"
DIS: fixes S :: "'a formula set"   and F :: "'a formula"   and G :: "'a formula"   and H :: "'a formula" assumes "\<And>s. \<lbrakk>s \<subseteq> S; finite s\<rbrakk> \<Longrightarrow> s \<in> C"   and "Dis F G H"   and "F \<in> S" shows "G \<triangleright> S \<in> C \<union> {S. \<forall>s\<subseteq>S. finite s \<longrightarrow> s \<in> C} \<or> H \<triangleright> S \<in> C \<union> {S. \<forall>s\<subseteq>S. finite s \<longrightarrow> s \<in> C}"
inv_RBT_Impl_diag_3: fixes t :: "('c, 'd) rbt" assumes "inv1 t"   and "inv2 t"   and "color_of t = color.B" shows "color_of (RBT_Impl_diag t) = color.B"
translation_galois: fixes T :: "'b set"   and a :: "'b"   and S :: "'b set" shows "(T = (+) a ` S) = (S = (+) (- a) ` T)"
uint32_sdiv_def: fixes x :: "uint32"   and y :: "uint32" shows "uint32_sdiv x y = (if y = 0 then undefined (div) x 0 else Abs_uint32 (Rep_uint32 x sdiv Rep_uint32 y))"
simps_7: fixes t1 :: "PrimScheme"   and t2 :: "PrimScheme" shows "sch_to_pr (Rec_op t1 t2) = UnaryRecOp (sch_to_pr t1) (sch_to_pr t2)"
openin_set_rel_interior: fixes S :: "'a set" shows "openin (top_of_set S) (rel_interior S)"
l6_16_1: fixes P :: "'p"   and Q :: "'p"   and S :: "'p"   and X :: "'p" assumes "P \<noteq> Q"   and "Col S P Q"   and "Col X P Q" shows "Col X P S"
lc_def: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.lc term_of_pair ord_term p = lookup p (ordered_term.lt term_of_pair ord_term p)"
j_def: shows "j = (LEAST j. u \<le>p concat (take (Suc j) ws))"
wt_wt': fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and e :: "expr"   and T :: "ty" assumes "P,E,h \<turnstile> e : T" shows "P,E,h \<turnstile> e :' T"
assms_1: shows "\<Gamma> \<turnstile>\<^sub>W e : \<tau>"
assms: shows "t \<in> trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (b \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>)"
can_lock_preserved: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x" assumes "s \<in> wf_state"   and "s -t'\<triangleright>ta'\<rightarrow> s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s\<rangle> \<wrong>" shows "t \<turnstile> \<langle>x, shr s'\<rangle> \<wrong>"
ord_strict_p_irreflexive: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "\<not> p \<prec>\<^sub>p p"
unit_div_mult_swap: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c one" shows "times a (divide b c) = divide (times a b) c"
hcomp_simps_4: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.hseq \<nu> \<mu>" shows "Span.cod (\<nu> \<circ> \<mu>) = Span.cod \<nu> \<circ> Span.cod \<mu>"
explicationInterference: fixes \<phi> :: "c \<Rightarrow> w \<Rightarrow> bool" shows "\<forall>c w. (\<exists>x. InterferesWith x \<phi> c w) = (\<^bold>\<not>\<^bold>\<diamond>\<^sub>a\<phi>) c w"
inres_leof_mono: fixes m :: "'e nres"   and m' :: "'e nres"   and x :: "'e" assumes "m \<le>\<^sub>n m'"   and "nofail m"   and "inres m x" shows "inres m' x"
cases: fixes x :: "nat \<times> nat list" obtains m :: "nat" and xs :: "nat list" where "x = (m, xs)"
ln_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "ln_aform_err p X = (let ivl = ivl_of_aform_err p X; l = lower ivl in if 0 < l then min_range_mono p floatarith.Ln inverse l (upper ivl) X else None)"
prems: fixes x :: "'a" assumes "x \<in> C"   and "enat (Suc n) \<le> esize x" shows "thesis"
\<epsilon>o_def: fixes x :: "'a" shows "Adj.\<epsilon>o x = \<psi> x (G x)"
inv_rem: shows "queue_invar (remove1 (getMinTree q) q)"
fun_ran_lens_def: fixes X :: "'b \<Longrightarrow> 'd"   and Y :: "('a \<Rightarrow> 'd) \<Longrightarrow> 'c" shows "fun_ran_lens X Y = \<lparr>lens_get = \<lambda>s. get\<^bsub>X\<^esub> \<circ> get\<^bsub>Y\<^esub> s, lens_put = \<lambda>s v. put\<^bsub>Y\<^esub> s (\<lambda>x. put\<^bsub>X\<^esub> (get\<^bsub>Y\<^esub> s x) (v x))\<rparr>"
inner_node_Exit_edge: fixes n :: "'node" assumes "inner_node n" obtains a :: "'edge" where "valid_edge a" and "intra_kind (kind a)" and "inner_node (sourcenode a)" and "targetnode a = (_Exit_)"
sexp_In1I: fixes M :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp" shows "In1 M \<in> sexp"
mult_aux: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" assumes "completely_multiplicative_function f"   and "1 < a"   and "1 < b" shows "f (a * b) = f a * f b"
l_get_dom_component_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
simps_2: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and more :: "'c" shows "\<one>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, \<dots> = more\<rparr>\<^esub> = one"
card_order_on_ordIso: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "card_order_on A r"   and "card_order_on A r'" shows "r =o r'"
x_in: fixes x :: "'c" assumes "x \<in> set vs" shows "x \<in> carrier M"
ncoplanar_perm_22: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Coplanar A B C D" shows "\<not> Coplanar D C A B"
hchop_weaken1: fixes \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. \<phi> ts v \<longrightarrow> (\<exists>va u. v=va\<parallel>u \<and> \<phi> ts va \<and> True)"
is_zariski_open_def: fixes U :: "'a set set" shows "is_zariski_open U \<equiv> generated_topology Spec {U. \<exists>\<aa>. ideal \<aa> R (+) (\<cdot>) \<zero> \<one> \<and> U = Spec\<setminus>\<V> \<aa>} U"
T_apply: fixes v :: "'v" assumes "v \<in> V" shows "T v = ((1::'f) / ordG) \<sharp>\<cdot> (\<Sum>g\<in>G. CP g v)"
finite_total_preorder_on_iff': fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "election agents alts" shows "finite_total_preorder_on alts R = total_preorder_on alts R"
assms: shows "SN_on {(y, z). (z, y) \<in> r} {x}"
monad_resumption_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and pause :: "'c \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_resumption return bind pause" shows "monad_resumption (return_env return) (bind_env bind) (pause_env pause)"
init_pnet_p_SubnetS: fixes s1 :: "'s net_state"   and s2 :: "'s net_state"   and p :: "net_tree" assumes "SubnetS s1 s2 \<in> init (pnet np p)" obtains p1 :: "net_tree" and p2 :: "net_tree" where "p = p1 \<parallel> p2" and "s1 \<in> init (pnet np p1)" and "s2 \<in> init (pnet np p2)"
append_butlast_last_id: fixes xs :: "'b list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
rep_eq: fixes x :: "('b, 'a) sq_matrix"   and xa :: "'a"   and xb :: "('b, 'a) vec" shows "to_fun (upd_row x xa xb) = (\<lambda>i j. if i = xa then xb $ j else to_fun x i j)"
region_cover: fixes v :: "'a \<Rightarrow> real" assumes "\<forall>x\<in>X. 0 \<le> v x" shows "\<exists>R. R \<in> \<R> \<and> v \<in> R"
scopeExtParRight: fixes x :: "name"   and P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "x \<sharp> P"   and "Id \<subseteq> Rel"   and "eqvt Rel"   and "\<And>R S y. y \<sharp> R \<Longrightarrow> (R \<parallel> <\<nu>y>S, <\<nu>y>(R \<parallel> S)) \<in> Rel"   and "\<And>R S y z. y \<sharp> R \<Longrightarrow> (<\<nu>z>(R \<parallel> <\<nu>y>S), <\<nu>y><\<nu>z>(R \<parallel> S)) \<in> Rel" shows "P \<parallel> <\<nu>x>Q \<leadsto>[Rel] <\<nu>x>(P \<parallel> Q)"
pwd_spy_i_unused: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "\<exists>A. PriKey A \<notin> used (fst (pwd_spy_i n))"
monoid_axioms: shows "monoid (\<^bold>*) \<^bold>1"
state_rank_initial_state: fixes n :: "nat" obtains i :: "nat" where "state_rank q\<^sub>0 n = Some i"
\<Delta>_simps_1: shows "seq (\<omega> \<star> r\<^sub>0) \<rho>"
permutation_validS: fixes fs :: "formula list"   and gs :: "formula list" shows "fs <~~> gs \<longrightarrow> validS fs = validS gs"
f_nonneg: fixes x :: "nat \<times> nat" shows "0 \<le> (case x of (i, j) \<Rightarrow> norm (a i) * norm (b j))"
fic_subst2: fixes M :: "trm"   and a :: "coname"   and c :: "coname"   and P :: "trm"   and x :: "name" assumes "fic M a"   and "c \<noteq> a"   and "a \<sharp> P"   and "M \<noteq> Ax x a" shows "fic (M{x:=<c>.P}) a"
arr_iff_in_hom: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "arr f = \<guillemotleft>f : local.dom f \<rightarrow> cod f\<guillemotright>"
dom_glb_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(d x \<le> d y \<cdot> d z) = (d x \<le> d y \<and> d x \<le> d z)"
\<tau>Red1_conv: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and uf :: "bool"   and P :: "'a J1_prog"   and t :: "'b"   and h :: "'c"   and ex :: "'a expr1 \<times> 'a val list"   and exs :: "('a expr1 \<times> 'a val list) list"   and ex' :: "'a expr1 \<times> 'a val list"   and exs' :: "('a expr1 \<times> 'a val list) list" shows "J1_heap_base.\<tau>Red1g addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write uf P t h (ex, exs) (ex', exs') = (J1_heap_base.Red1 addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write uf P t ex exs h \<lbrace>\<rbrace> ex' exs' h \<and> J1_heap_base.\<tau>Move1 typeof_addr P h (ex, exs))"
simulation_silent2: fixes s1 :: "'s1"   and s2 :: "'s2"   and s2' :: "'s2" assumes "s1 \<approx> s2"   and "s2 -\<tau>2\<rightarrow> s2'" shows "\<exists>s1'. s1 -\<tau>1\<rightarrow>* s1' \<and> s1' \<approx> s2'"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "S.VVV.arr f = (S.arr (fst f) \<and> S.VV.arr (snd f) \<and> S.src (fst f) = S.trg (fst (snd f)))"
adm_upward: fixes P :: "'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> P y" shows "adm P"
Times_subset_cancel2: fixes x :: "'d"   and C :: "'d set"   and A :: "'e set"   and B :: "'e set" assumes "x \<in> C" shows "(A \<times> C \<subseteq> B \<times> C) = (A \<subseteq> B)"
Hadjicostas_integral_def: fixes z :: "complex" shows "Hadjicostas_integral z = CLBINT u=0..1. complex_of_real (- ln u) powr z / complex_of_real (1 - u) * complex_of_real (- ln u + u - 1)"
aux2: assumes "\<forall>\<A>. (\<forall>G\<in>M. \<A> \<Turnstile> G) \<longrightarrow> \<A> \<Turnstile> \<phi>" shows "valuation M \<Turnstile> \<phi>"
trms\<^sub>s\<^sub>t_append: fixes A :: "('d, 'e) strand_step list"   and B :: "('d, 'e) strand_step list" shows "trms\<^sub>s\<^sub>t (A @ B) = trms\<^sub>s\<^sub>t A \<union> trms\<^sub>s\<^sub>t B"
least: fixes t' :: "real" assumes "flow0 x t' \<in> P"   and "t0 \<le> t'"   and "t' \<le> t1" shows "rt \<le> t'"
llist_of_def: shows "llist_of \<equiv> rec_list LNil (\<lambda>x xs. LCons x)"
vsv_appE: fixes A :: "V"   and a :: "V"   and b :: "V" assumes "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (r \<restriction>\<^sup>l\<^sub>\<circ> A)" obtains "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A"
length_concat_sequences: fixes key :: "'b \<Rightarrow> 'c"   and xs :: "'b list" shows "length (concat (sequences key xs)) = length xs"
assms: shows "(\<sigma>, s) \<in> init (opnet onp \<langle>i; R\<rangle>)"
ass_function_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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.ass_function normalize f = ((\<forall>a. normalize a = normalize (f a)) \<and> pairwise (\<lambda>a b. normalize a \<noteq> normalize b) (range f))"
cut_less_Min_not_empty: fixes I :: "'b set"   and t :: "'b" assumes "I \<noteq> {}"   and "iMin I < t" shows "I \<down>< t \<noteq> {}"
region_set_def: fixes R :: "('d \<Rightarrow> 'e) set"   and x :: "'d"   and c :: "'e" shows "region_set R x c = {v(x := c) |v. v \<in> R}"
I_minWeightBasisD_2: fixes T :: "'a set"   and E :: "'a set"   and e :: "'a" assumes "I_minWeightBasis (T, E)"   and "e \<in> carrier - E - T" shows "\<not> indep (insert e T)"
has_density_emeasure_space: fixes M :: "'c measure"   and N :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" assumes "has_density M N f" shows "emeasure M (space M) = integral\<^sup>N N f"
non_volatile_reads_acquired_or_read_only: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list" assumes "non_volatile_owned_or_read_only True \<S> \<O> sb"   and "sharing_consistent \<S> \<O> sb" shows "outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb \<subseteq> \<O> \<union> all_acquired sb \<union> read_only \<S>"
elims_3: fixes b :: "nat list"   and x :: "nat list"   and xa :: "nat list"   and xb :: "nat" assumes "\<not> cond1 b x xa xb" obtains ys :: "nat list" and xc :: "nat" and xs :: "nat list" and s :: "nat" where "x = ys" and "xa = xc # xs" and "xb = s" and "\<not> (s \<le> b \<bullet> ys \<and> xc \<le> maxne0_impl ys b)"
orthoU: fixes b :: "'a"   and c :: "'a" assumes "b \<in> B"   and "c \<in> B" shows "(U *\<^sub>V b) \<bullet>\<^sub>C (U *\<^sub>V c) = b \<bullet>\<^sub>C c"
nat_pow_mult: fixes x :: "int \<times> int"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> n \<otimes>\<^bsub>R\<^esub> x [^]\<^bsub>R\<^esub> m = x [^]\<^bsub>R\<^esub> (n + m)"
sp_arraycpy_eq: fixes P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and y :: "char list"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "local.sp P (x[] ::= y) t = (\<exists>vx. let s = t(x := vx) in t x = s y \<and> P s)"
clop_extensive_var: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "clop f" shows "x \<le> f x"
measurable_remove_var: fixes V :: "nat set"   and t :: "pdf_type"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" shows "remove_var \<in> state_measure (shift_var_set V) (case_nat t \<Gamma>) \<rightarrow>\<^sub>M state_measure V \<Gamma>"
refusals_def: fixes P :: "'a process"   and xs :: "'a list" shows "refusals P xs \<equiv> failures P `` {xs}"
npepow_def: fixes e :: "pexpr"   and n :: "nat" shows "npepow e n = (if n = 0 then PExpr1 (PCnst 1) else if n = 1 then e else case e of PExpr1 (PCnst c) \<Rightarrow> PExpr1 (PCnst (c ^ n)) | PExpr1 _ \<Rightarrow> PExpr2 (PPow e n) | PExpr2 pexpr2 \<Rightarrow> PExpr2 (PPow e n))"