hcju/isabelleps huggingface dataset:
hyps: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "ts_inv_ok ts I" shows "ts_inv_ok (redT_updTs' ts TAS) (upd_invs I P TAS)"
not_dbm_le_le_impl: fixes a :: "'d"   and b :: "'d" assumes "\<not> Le a \<prec> Le b" shows "b \<le> a"
inducts: fixes P :: "'b \<times> 'c \<Rightarrow> bool"   and prod :: "'b \<times> 'c" assumes "\<And>a b. P (a, b)" shows "P prod"
assms_1: fixes P :: "complex mat" assumes "is_quantum_predicate P" shows "is_quantum_predicate (WS P)"
nn_integral_spmf_neq_top: fixes p :: "'c spmf" shows "\<integral>\<^sup>+ x. ennreal (spmf p x) \<partial>count_space UNIV \<noteq> \<top>"
un_subset2: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> j \<le> i \<squnion> j"
possible_steps_def: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" shows "possible_steps e s r l i = (\<lambda>((origin, dest), t). (dest, t)) |`| ffilter (\<lambda>((origin, dest), t). origin = s \<and> Label t = l \<and> length i = Arity t \<and> apply_guards (Guards t) (join_ir i r)) e"
prems_1: shows "P,E,h \<turnstile> es [:]\<^bsub>NT\<^esub> T'' # Ts''"
apply_cltn2_line_alt_def: fixes l :: "proj2_line"   and A :: "cltn2" shows "apply_cltn2_line l A = proj2_line_abs (cltn2_rep (cltn2_inverse A) *v proj2_line_rep l)"
lossless_init: fixes init :: "'l \<Rightarrow> 'm \<Rightarrow> ('n \<times> 'o) spmf"   and response :: "'n \<Rightarrow> 'm \<Rightarrow> 'p \<Rightarrow> 'q spmf"   and check :: "'l \<Rightarrow> 'o \<Rightarrow> 'p \<Rightarrow> 'q \<Rightarrow> bool"   and Rel :: "('l \<times> 'm) set"   and S_raw :: "'l \<Rightarrow> 'p \<Rightarrow> ('o \<times> 'q) spmf"   and \<A>ss :: "'l \<Rightarrow> 'o \<times> 'p \<times> 'q \<Rightarrow> 'o \<times> 'p \<times> 'q \<Rightarrow> 'm spmf"   and challenge_space :: "'p set"   and valid_pub :: "'l set"   and G :: "('l \<times> 'm) spmf"   and h :: "'l"   and w :: "'m" assumes "\<Sigma>_protocols_to_commitments init response check Rel S_raw \<A>ss challenge_space valid_pub G" shows "lossless_spmf (init h w)"
simps_19: fixes v :: "'b"   and va :: "bexp"   and vb :: "'b acom"   and vc :: "'b"   and vd :: "char list"   and ve :: "aexp"   and vf :: "'b" shows "({v} WHILE va DO vb {vc} \<sqsubseteq> vd ::= ve {vf}) = False"
contract_prime: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "carrier (Vr K v) \<subseteq> carrier (Vr K v')" shows "prime_ideal (Vr K v) (carrier (Vr K v) \<inter> vp K v')"
wp_under_wlpI: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>P. unitary P \<Longrightarrow> wp prog P \<tturnstile> wlp prog P" shows "wp_under_wlp prog"
gen_ideal_monoid: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and x :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "x \<in> R" shows "Group_Theory.monoid (comm_ring.gen_ideal R multiplication x) addition zero"
converse_product_overlap: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y"   and "O z (x \<otimes> y)" shows "O z y"
nres_of_def: fixes r :: "'a dres" shows "nres_of r \<equiv> case r of dSUCCEEDi \<Rightarrow> SUCCEED | dFAILi \<Rightarrow> FAIL | dRETURN x \<Rightarrow> RETURN x"
init: fixes \<xi> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and l :: "'e" assumes "(\<xi>, p) \<in> init A"   and "l \<in> labels \<Gamma> p" shows "P (\<xi>, l)"
prems_2: assumes "n < length (x # xs)" shows "\<not> P ((x # xs) ! n)"
zhmset_of_inject: fixes M :: "hmultiset"   and N :: "hmultiset" shows "(zhmset_of M = zhmset_of N) = (M = N)"
eq: shows "sort_oalist ko xs' = (k, v) # xs''"
v_AbsN: fixes e :: "db" shows "val (DBAbsN e)"
analz_insert_Crypt: fixes K :: "nat"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<notin> analz H" shows "analz (insert (Crypt K X) H) = insert (Crypt K X) (analz H)"
B_def: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real" shows "B V f \<equiv> {0} \<union> {\<bar>f x\<bar> / \<parallel>x\<parallel> |x. x \<noteq> (0::'b) \<and> x \<in> V}"
rcos_const_imp_mem: fixes i :: "'a" assumes "i \<in> carrier R"   and "I +> i = I" shows "i \<in> I"
lenp: shows "\<^bold>|take p w\<^bold>| = p"
rat_to_int_factor_explicit: fixes p :: "int poly"   and g :: "rat poly"   and h :: "rat poly"   and r :: "rat"   and rg :: "int poly" assumes "of_int_poly p = g * h"   and "rat_to_normalized_int_poly g = (r, rg)" shows "\<exists>r. p = rg * Polynomial.smult (content p) r"
LTL_FG_almost_all_suffixes: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> F G \<phi> = (\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile> \<phi>)"
tInit_is_tMap: fixes s :: "'b"   and f :: "'c \<Rightarrow> 'b"   and t :: "'c Trace" shows "(tInit s = tMap f t) = (\<exists>s'. t = tInit s' \<and> f s' = s)"
assms_4: shows "M' = p @ [(bLiteral, False)]"
is_fieldD: fixes m :: "qtname \<times> memberdecl" assumes "is_field m" shows "\<exists>declC f. m = (declC, fdecl f)"
simps: fixes a :: "fmla" shows "(a \<in> boolean_axioms) = ((\<exists>A. a = A IMP A) \<or> (\<exists>A B. a = A IMP A OR B) \<or> (\<exists>A. a = A OR A IMP A) \<or> (\<exists>A B C. a = A OR B OR C IMP (A OR B) OR C) \<or> (\<exists>C A B. a = C OR A IMP (C IMP B) IMP A OR B))"
mult_sound: fixes P :: "'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (\<lambda>s. P s * Q s)"
h_def: shows "h = (\<lambda>x. if x \<in> {0..x0} then x powr (Re z - 1) else exp (- x / 2))"
vec_to_iarray_exists: fixes A :: "('e, 'd) vec" shows "(\<exists>b. A $ b \<noteq> (0::'e)) = IArray.exists (\<lambda>b. vec_to_iarray A !! b \<noteq> (0::'e)) (IArray [0..<IArray.length (vec_to_iarray A)])"
Chernoff_b: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f" shows "Chernoff_on A f = (\<forall>B C. B \<subseteq> A \<and> C \<subseteq> A \<longrightarrow> f (B \<union> C) \<subseteq> f B \<union> f C)"
prodag_zero: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "\<zero>\<^bsub>a\<Pi>\<^bsub>I\<^esub> A\<^esub> = prod_zero I A"
an_nat_pos: fixes n :: "nat" shows "0 \<le> an n"
eq_unit: fixes a :: "'a"   and b :: "'a" assumes "class.unitary TYPE('a)" shows "a = b"
lincomb_as_lincomb_list_distinct: fixes ws :: "'a vec list"   and n :: "nat"   and f :: "'a vec \<Rightarrow> 'a" assumes "set ws \<subseteq> carrier_vec n"   and "distinct ws" shows "module.lincomb (module_vec TYPE('a) n) f (set ws) = module.lincomb_list (module_vec TYPE('a) n) (\<lambda>i. f (ws ! i)) ws"
prems_2: shows "(a # t') @ u =\<^sub>F v @ w"
mset_single_iff_right: fixes x :: "'a"   and xs :: "'a list" shows "({#x#} = mset xs) = (xs = [x])"
setdist_pos_le: fixes s :: "'a set"   and t :: "'a set" shows "0 \<le> setdist s t"
confluence_var: fixes y :: "'a"   and x :: "'a" shows "(y \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>) = (y\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star> \<cdot> y\<^sup>\<star>)"
induced_automorphism_morphism_order2: shows "\
 \<circ> \
 = id"
simps_20: fixes C' :: "char list"   and C :: "char list"   and e :: "'a exp"   and Cs :: "char list list"   and e' :: "'a exp" shows "not_init C' (RI (C,e) ; Cs \<leftarrow> e') = (C' \<notin> set (C # Cs) \<and> not_init C' e \<and> not_init C' e')"
BetS_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "BetS A B C \<equiv> Bet A B C \<and> A \<noteq> B \<and> B \<noteq> C"
findLastAssertedLiteral_def: fixes state :: "State" shows "findLastAssertedLiteral state = state\<lparr>getCl := getLastAssertedLiteral (oppositeLiteralList (getC state)) (elements (getM state))\<rparr>"
tail_events_def: fixes A :: "'c \<Rightarrow> 'a set set" shows "tail_events A = (\<Inter>n. sigma_sets (space M) (\<Union> (A ` {n..})))"
Diff_subset: fixes A :: "'c set"   and B :: "'c set" shows "A - B \<subseteq> A"
simps: fixes pat :: "[:'r:]"   and usvs :: "[:'r:] \<times> [:'r:]" shows "rep2\<cdot>pat\<cdot>usvs = Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs)"
rep_eq: fixes x :: "nat"   and xa :: "float interval" shows "bounds_of_interval (exp_float_interval x xa) = (case bounds_of_interval xa of (lx, ux) \<Rightarrow> (lb_exp x lx, ub_exp x ux))"
sublist_append_rightI: fixes xs :: "'c list"   and ss :: "'c list" shows "sublist xs (xs @ ss)"
span_diff: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set"   and y :: "'b \<Rightarrow>\<^sub>0 'c" assumes "x \<in> phull S"   and "y \<in> phull S" shows "x - y \<in> phull S"
comp_mapping_complement: fixes x :: "'a"   and y :: "'a" assumes "id x \<sqinter> x \<le> top \<and> x \<sqinter> top = top" shows "x \<sqinter> - y = - (x \<sqinter> y)"
simps_1: fixes x :: "name"   and T :: "ty"   and t :: "btrm" shows "(\<lambda>[PVar x T]. t) = Bind T x t"
Spy_see_private_Key: fixes evs :: "event list"   and b :: "bool"   and A :: "agent" assumes "evs \<in> set_cr" shows "(Key (invKey (publicKey b A)) \<in> parts (knows Spy evs)) = (A \<in> bad)"
parts_insert_Key: fixes K :: "nat"   and H :: "msg set" shows "parts (insert (Key K) H) = insert (Key K) (parts H)"
add_succ_spec_def: fixes dst :: "'a"   and succ :: "'a set"   and v :: "'b"   and PRED :: "'a \<Rightarrow> 'b option"   and N :: "'a set" shows "add_succ_spec dst succ v PRED N \<equiv> assert (N \<subseteq> dom PRED) \<bind> (\<lambda>_. SPEC (\<lambda>(f, PRED', N'). case f of True \<Rightarrow> dst \<in> succ - dom PRED \<and> PRED \<subseteq>\<^sub>m PRED' \<and> PRED' \<subseteq>\<^sub>m map_mmupd PRED (succ - dom PRED) v \<and> dst \<in> dom PRED' | False \<Rightarrow> dst \<notin> succ - dom PRED \<and> PRED' = map_mmupd PRED (succ - dom PRED) v \<and> N' = N \<union> (succ - dom PRED)))"
min_absorb2: fixes y :: "'a"   and x :: "'a" assumes "y \<in> U"   and "x \<in> U"   and "y \<le>\<^sub>o\<^sub>w x" shows "local.min x y = y"
scalar_prod_left_zero: fixes v :: "'c vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "0\<^sub>v n \<bullet> v = (0::'c)"
hd_append2: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<noteq> \<epsilon>" shows "hd (xs \<cdot> ys) = hd xs"
spair_spp_def: fixes p :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)"   and q :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" shows "spair_spp p q = (let t1 = punit.lt (snd p); t2 = punit.lt (snd q); l = lcs t1 t2 in (ord_term_lin.max ((l - t1) \<oplus> fst p) ((l - t2) \<oplus> fst q), local.punit.monom_mult ((1::'b) / punit.lc (snd p)) (l - t1) (snd p) - local.punit.monom_mult ((1::'b) / punit.lc (snd q)) (l - t2) (snd q)))"
ag_pOp_closed: fixes A :: "('h, 'i) aGroup_scheme"   and x :: "'h"   and y :: "'h" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y \<in> carrier A"
intros_23: fixes i :: "nat"   and \<C> :: "t_context"   and t1s :: "t list"   and t2s :: "t list" assumes "i < length (types_t \<C>)"   and "types_t \<C> ! i = t1s _> t2s"   and "table \<C> \<noteq> None" shows "\<C> \<turnstile> [Call_indirect i] : t1s @ [T_i32] _> t2s"
arrow_from_functor_axioms: shows "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F (Go x) x (\<epsilon>o x)"
start_of_Suc1: fixes ly :: "nat list"   and ap :: "abc_inst list"   and as :: "nat"   and n :: "nat" assumes "ly = layout_of ap"   and "abc_fetch as ap = Some (Inc n)" shows "start_of ly (Suc as) = start_of ly as + 2 * n + 9"
rel_frontier_def: fixes S :: "'b set" shows "rel_frontier S = closure S - rel_interior S"
r: fixes x :: "'s" assumes "x \<in> X \<inter> Y" shows "measure_pmf.prob (K x) Y \<le> r"
class_add_sup_ty_opt: fixes P :: "jvm_prog"   and l1 :: "ty err"   and l2 :: "ty err"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P \<turnstile> l1 \<le>\<^sub>\<top> l2"   and "\<not> is_class P C" shows "class_add P (C, cdec) \<turnstile> l1 \<le>\<^sub>\<top> l2"
COMB_competitive: shows "\<forall>s0\<in>{x. distinct x \<and> x \<noteq> []}. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. T\<^sub>p_on_rand' (COMB []) (fst (COMB []) s0 \<bind> (\<lambda>is. return_pmf (s0, is))) qs \<le> real 8 / real 5 * real (T\<^sub>p_opt s0 qs) + b"
assms_1: shows "g \<turnstile> nodes_of_uses \<approx>\<^sub>\<phi> uninst_code.ssa.useNodes_of uses g"
unit_circle_fix_moebius_pt_inversion: fixes M :: "moebius"   and z :: "complex_homo" assumes "unit_circle_fix M" shows "moebius_pt M (inversion z) = inversion (moebius_pt M z)"
fresh_smallstep_I: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term"   and x :: "Syntax.var" assumes "\<lless>\<pi>, e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   and "atom x \<sharp> e" shows "atom x \<sharp> e'"
d: shows "ff = rec_nat [] (\<lambda>i t. let e = SOME e. infinite {t' \<in> Tr'. t @ [e] < t'} in t @ [e])"
residue_of_one_1: fixes k :: "nat" assumes "0 < k" shows "\<one> k = \<one>\<^bsub>residue_ring (p ^ k)\<^esub>"
fps_XD'_def: fixes a :: "'a fps" shows "fps_XD' a = (\<lambda>b. a * fps_deriv b)"
induct: fixes x1 :: "sterm"   and x2 :: "sterm"   and P :: "sterm \<Rightarrow> sterm \<Rightarrow> bool" assumes "x1 => x2"   and "\<And>x. P (Fvar x) (Fvar x)"   and "\<And>f' f L T. \<lbrakk>dom f' = dom f; finite L; \<forall>l\<in>dom f. \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t. (the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup> => t \<and> P (the (f l)\<^bsup>[Fvar s,Fvar p]\<^esup>) t) \<and> the (f' l) = \<sigma>[s,p] t); \<forall>l\<in>dom f. body (the (f l))\<rbrakk> \<Longrightarrow> P (Obj f T) (Obj f' T)"   and "\<And>t t' L u u' l. \<lbrakk>t => t'; P t t'; lc t; finite L; \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. (u\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> P (u\<^bsup>[Fvar s,Fvar p]\<^esup>) t'') \<and> u' = \<sigma>[s,p] t''); body u\<rbrakk> \<Longrightarrow> P (Upd t l u) (Upd t' l u')"   and "\<And>f T f' L t t' l. \<lbrakk>Obj f T => Obj f' T; P (Obj f T) (Obj f' T); finite L; \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. (t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> P (t\<^bsup>[Fvar s,Fvar p]\<^esup>) t'') \<and> t' = \<sigma>[s,p] t''); l \<in> dom f; lc (Obj f T); body t\<rbrakk> \<Longrightarrow> P (Upd (Obj f T) l t) (Obj (f'(l \<mapsto> t')) T)"   and "\<And>t t' u u' l. \<lbrakk>t => t'; P t t'; u => u'; P u u'; lc t; lc u\<rbrakk> \<Longrightarrow> P (Call t l u) (Call t' l u')"   and "\<And>f T f' l p p'. \<lbrakk>Obj f T => Obj f' T; P (Obj f T) (Obj f' T); l \<in> dom f; p => p'; P p p'; lc (Obj f T); lc p\<rbrakk> \<Longrightarrow> P (Call (Obj f T) l p) (the (f' l)\<^bsup>[Obj f' T,p']\<^esup>)" shows "P x1 x2"
add_all_deducible_clauses_finite: fixes R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and S :: "'at Literal set set"   and N :: "nat" assumes "derived_clauses_are_finite R"   and "all_fulfill finite S" shows "all_fulfill finite (add_all_deducible_clauses R (inferred_clause_sets R S N))"
map\<^sub>0_def: fixes a :: "'a" shows "F.map\<^sub>0 a \<equiv> src\<^sub>D (FoH.map a)"
injective_quotient_map: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c topology"   and X' :: "'d topology" assumes "inj_on f (topspace X)" shows "quotient_map X X' f = (continuous_map X X' f \<and> open_map X X' f \<and> closed_map X X' f \<and> f ` topspace X = topspace X')"
nta_bisim_mono: fixes bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool"   and ta :: "('a, 'b, 'c) new_thread_action"   and ta' :: "('a, 'd, 'e) new_thread_action"   and bisim' :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> bool" assumes "nta_bisim bisim ta ta'"   and "\<And>t s1 s2. bisim t s1 s2 \<Longrightarrow> bisim' t s1 s2" shows "nta_bisim bisim' ta ta'"
forall_intro: 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 \<Gamma> :: "Core.term set"   and B :: "Core.term"   and x :: "variable"   and \<tau> :: "typ" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "(x, \<tau>) \<notin> FV \<Gamma>"   and "wf_type (sig \<Theta>) \<tau>" shows "\<Theta>,\<Gamma> \<turnstile> mk_all x \<tau> B"
compl_inter_eq_pp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- - x \<sqinter> y = - - x \<sqinter> z"   and "- x \<sqinter> y = - x \<sqinter> z" shows "y = z"
pdom_pId_on: fixes A :: "'a set" shows "pdom (pId_on A) = A"
nonempty_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "\<gamma> K \<phi> \<noteq> {}"
ssubst_sopen_commute: fixes t' :: "sterm"   and x :: "char list"   and s :: "sterm"   and p :: "sterm"   and n :: "nat"   and t :: "sterm" assumes "lc t'"   and "x \<notin> FV s"   and "x \<notin> FV p" shows "[x \<rightarrow> t'] {n \<rightarrow> [s,p]} t = {n \<rightarrow> [s,p]} [x \<rightarrow> t'] t"
Suc_2p_unat_mask: fixes n :: "nat"   and k :: "nat" assumes "n < LENGTH('b)" shows "Suc (2 ^ n * k + unat (mask n)) = 2 ^ n * (k + 1)"
le_nhds_metric_le: fixes F :: "'c filter"   and a :: "'c" shows "(F \<le> nhds a) = (\<forall>e>0. \<forall>\<^sub>F x in F. dist x a \<le> e)"
D_det: fixes P :: "'b process"   and Q :: "'b process" shows "D (P [+] Q) = D P \<union> D Q"
symeval_def: fixes P :: "(char list \<Rightarrow> int option) \<times> 'b \<Rightarrow> bool"   and e :: "aexp"   and v :: "int" shows "symeval P e v \<equiv> \<forall>s n. P (s, n) \<longrightarrow> paval' e s = Some v"
cons_nomatch: fixes name :: "'a"   and name' :: "'a"   and rs :: "('a \<times> 'b) list"   and rhs :: "'b"   and rhs' :: "'b" assumes "name \<noteq> name'"   and "srewrite_step rs name rhs" shows "srewrite_step ((name', rhs') # rs) name rhs"
con: shows "continuous_on (ball \<xi> r) (\<lambda>x. exp (h x) / g x)"
hom_dvd: fixes p :: "'a poly poly"   and q :: "'a poly poly" assumes "p dvd q" shows "poly_y_x p dvd poly_y_x q"
fps_X_fps_compose: fixes a :: "'a fps" shows "fps_X oo a = Abs_fps (\<lambda>n. if n = 0 then 0::'a else fps_nth a n)"
param_tl: fixes R :: "('a \<times> 'b) set" shows "(tl, tl) \<in> \<langle>R\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
i_take_take_eq2: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "n \<le> m" shows "f \<Down> n \<down> m = f \<Down> n"
assms_3: shows "proj2_abs a \<noteq> proj2_abs r"
ssubst_v_tm: fixes t :: "dbtm" assumes "supp t \<subseteq> atom ` V'"   and "supp t \<subseteq> atom ` Vs" shows "ssubst (vquot_dbtm V' t) V' F' = ssubst (vquot_dbtm Vs t) Vs F"
K_argmin: fixes i :: "int"   and j :: "int" assumes "i < j" shows "K i j = argmin (min_wpl3 (i + 1) j) [i + 1..j]"
independent_and_consistent_imp_uniqueness_solution: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "vec.dim (solution_set A 0) = 0"   and "consistent A b" shows "\<exists>!x. is_solution x A b"
st_mat_inverse: fixes x :: "'a st_mat" shows "Abs_st_mat (st_mat x) = x"
red_external_wt_hconf_hext: fixes wf_md :: "'addr J_prog \<Rightarrow> String.literal \<Rightarrow> (String.literal list \<times> 'addr expr) mdecl \<Rightarrow> bool"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and va :: "'addr extCallRet"   and h' :: "'heap"   and h'' :: "'heap"   and U :: "ty" assumes "wf_prog wf_md P"   and "P,t \<turnstile> \<langle>a\<bullet>M(vs),h\<rangle> -ta\<rightarrow>ext \<langle>va,h'\<rangle>"   and "h'' \<unlhd> h"   and "P,h'' \<turnstile> a\<bullet>M(vs) : U"   and "P,h'' \<turnstile> t \<surd>t"   and "hconf h''" shows "\<exists>ta' va' h'''. P,t \<turnstile> \<langle>a\<bullet>M(vs),h''\<rangle> -ta'\<rightarrow>ext \<langle>va',h'''\<rangle> \<and> collect_locks \<lbrace>ta\<rbrace>\<^bsub>l\<^esub> = collect_locks \<lbrace>ta'\<rbrace>\<^bsub>l\<^esub> \<and> collect_cond_actions \<lbrace>ta\<rbrace>\<^bsub>c\<^esub> = collect_cond_actions \<lbrace>ta'\<rbrace>\<^bsub>c\<^esub> \<and> collect_interrupts \<lbrace>ta\<rbrace>\<^bsub>i\<^esub> = collect_interrupts \<lbrace>ta'\<rbrace>\<^bsub>i\<^esub>"
pos_less_divideC_eq: fixes c :: "complex"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a < b /\<^sub>C c) = (c *\<^sub>C a < b)"
zero_not_per: fixes w :: "'b list" shows "\<not> periodN w 0"
1: shows "(v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T = v * v\<^sup>T \<squnion> v * v\<^sup>T * e \<squnion> e\<^sup>T * v * v\<^sup>T \<squnion> e\<^sup>T * e"
vimage_comp: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f"   and x :: "'f set" shows "f -` g -` x = (g \<circ> f) -` x"
a_redu_subst1: fixes M :: "trm"   and M' :: "trm"   and y :: "name"   and c :: "coname"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>a M'" shows "M{y:=<c>.P} \<longrightarrow>\<^sub>a* M'{y:=<c>.P}"
filterlim_at_right_to_0: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter"   and a :: "real" shows "filterlim f F (at_right a) = (LIM x at_right 0. f (x + a) :> F)"
\<oo>_\<dd>s: fixes w :: "'b list"   and L :: "'b language" shows "\<oo> (\<dd>s w L) = in_language L w"
map\<^sub>0_def: fixes a :: "'b" shows "I.map\<^sub>0 a \<equiv> src\<^sub>B (local.map a)"
execn_sequence_appD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and xs :: "('e, 'd, 'f) com list"   and ys :: "('e, 'd, 'f) com list"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>sequence Seq (xs @ ys),Normal s\<rangle> =n\<Rightarrow> t" shows "\<exists>s'. \<Gamma>\<turnstile> \<langle>sequence Seq xs,Normal s\<rangle> =n\<Rightarrow> s' \<and> \<Gamma>\<turnstile> \<langle>sequence Seq ys,s'\<rangle> =n\<Rightarrow> t"
assms_1: shows "\<Gamma> \<turnstile>\<^sub>c e : t"
preserves_section_retraction: fixes e :: "'b"   and m :: "'b" assumes "Adj.D.ide (e \<cdot>\<^sub>D m)" shows "Adj.C.ide (F e \<cdot>\<^sub>C F m)"
bounded: shows "\<exists>K. \<forall>xa. norm ((if x \<in> S then f' x else (\<lambda>x. 0::'b)) xa) \<le> norm xa * K"
wellformed_inc: fixes k :: "nat"   and t :: "term"   and n :: "nat" assumes "wellformed' k t"   and "k \<le> n" shows "wellformed' n t"
Min_gr_iff: fixes A :: "'c Word.word set"   and x :: "'c Word.word" assumes "finite A"   and "A \<noteq> {}" shows "(x <s signed.Min A) = (\<forall>a\<in>A. x <s a)"
ideal_prod_inter: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "I \<cdot> J \<subseteq> I \<inter> J"
cases: fixes x :: "'a list" obtains "x = []"   | xa :: "'a" and ys :: "'a list" where "x = xa # ys"
submonoid_axioms: shows "submonoid H G (\<cdot>) \<one>"
llist_of_eq_LCons_conv: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b llist" shows "(llist_of xs = LCons y ys) = (\<exists>xs'. xs = y # xs' \<and> ys = llist_of xs')"
1: shows "\<lbrace>t\<rbrace> = local.inv \<lbrace>Cod t\<^bold>\<down>\<rbrace> \<cdot> \<lbrace>\<^bold>\<lfloor>t\<^bold>\<rfloor>\<rbrace> \<cdot> \<lbrace>Dom t\<^bold>\<down>\<rbrace>"
dom_simp: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.dom f = C.VxV.dom f"
nb\<^sub>4: fixes opa :: "('a, 'b) sas_plus_operator"   and v :: "'a"   and a :: "'b" assumes "opa \<in> set (map (strips_op_to_sasp \<Psi>) ops')"   and "(v, a) \<in> set (sas_plus_operator.precondition_of opa)" shows "\<exists>op'\<in>set ops'. opa = \<phi>\<^sub>O\<inverse> \<Psi> op' \<and> (v, a) \<in> set (strips_operator.precondition_of op')"
schur_upper_triangular_2: fixes A :: "'b mat"   and n :: "nat"   and es :: "'b list" assumes "A \<in> carrier_mat n n"   and "char_poly A = (\<Prod>a\<leftarrow>es. [:- a, 1::'b:])" shows "upper_triangular (schur_upper_triangular A es)"