hcju/isabelleps huggingface dataset:
recurrent_subset_meas_1: fixes A :: "'a set" assumes "A \<in> sets M" shows "recurrent_subset A \<in> sets M"
rec: shows "scalar_prod (w1 @ [x1]) (w2 @ [x2]) = x1 * x2 + scalar_prod w1 w2"
commute: fixes a :: "'h"   and b :: "'h" shows "inf a b = inf b a"
span_scale: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and c :: "'a" assumes "Modules.module scale"   and "x \<in> module.span scale S" shows "scale c x \<in> module.span scale S"
fdomain_fconst_on: fixes A :: "V"   and c :: "V" shows "\<D>\<^sub>\<bullet> (fconst_on A c) = A"
assms: shows "\<forall>e f g. list_all (eval e f g) ps \<longrightarrow> eval e f g p"
is_nth_power_upto_unitI: fixes x :: "'a"   and n :: "nat"   and y :: "'a" assumes "normalize (x ^ n) = normalize y" shows "is_nth_power_upto_unit n y"
le_infI1: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and x :: "'d"   and b :: "'d" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a x" shows "less_eq (inf a b) x"
simps_2: fixes x :: "'c"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and y :: "'c" shows "x # xs \<Squnion>\<^bsub>f\<^esub> y = xs \<Squnion>\<^bsub>f\<^esub> (x \<squnion>\<^bsub>f\<^esub> y)"
arrE: fixes f :: "'a" assumes "R.arr f" shows "right a f"
fn_to_set_def: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" shows "fn_to_set f = {x. \<exists>y. f x y = 0}"
embed_spmf_def: shows "local.embed_spmf = embed_pmf (\<lambda>x. case x of None \<Rightarrow> 1 - enn2real (\<integral>\<^sup>+ x. ennreal (f x) \<partial>count_space UNIV) | Some x' \<Rightarrow> max 0 (f x'))"
hyp_port_it_paths: fixes "is" :: "nat list"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and f :: "'form" assumes "is \<in> it_paths t"   and "f \<in> hyps_along t is" shows "hyp_port_path_for t is f \<in> it_paths t"
intro: fixes R :: "('e, 'f) ring_scheme" assumes "ring R"   and "Group.comm_monoid R" shows "cring R"
seq_compact_eq_countably_compact: fixes U :: "'c set" shows "seq_compact U = countably_compact U"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree"   and b :: "'a" shows "freq (Node w t\<^sub>1 t\<^sub>2) b = freq t\<^sub>1 b + freq t\<^sub>2 b"
sel_rews_6: fixes a :: "'r"   and slist :: "[:'r:]" assumes "a \<noteq> \<bottom>" shows "stail\<cdot>(a :# slist) = slist"
0: shows "(0::'a) \<in> (+) (- c) ` S"
valid_tuple_mono: fixes tas :: "nat \<times> 'a option list" assumes "valid_tuple tuple_since tas" shows "valid_tuple tuple_since' tas"
int_of_uint16_def: shows "int_of_uint16 \<equiv> map_fun Rep_uint16 id uint"
bisim1BinOpThrow: fixes P :: "'b prog"   and h :: "'heap"   and e1 :: "'addr expr1"   and bop :: "bop"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v1 :: "'addr val"   and v2 :: "'addr val" shows "P,e1 \<guillemotleft>bop\<guillemotright> e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v1, v2], xs, length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
add_vars'_fv: fixes t :: "Core.term" shows "set (add_vars' t []) = fv t"
Gamma_hurwitz_zeta_aux_integral_real: fixes x :: "real"   and a :: "real"   and n :: "nat" assumes "0 < x"   and "0 < a" shows "Gamma x / (real n + a) powr x = \<integral>s\<in>{0<..}. (s powr (x - 1) / exp ((real n + a) * s))\<partial>lebesgue"
queue_invar_def: fixes q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.queue_invar q \<equiv> \<forall>t\<in>set q. BinomialHeapStruc.tree_invar t"
pfp_wn_pfp: fixes P :: "'a acom \<Rightarrow> bool"   and f :: "'a acom \<Rightarrow> 'a acom"   and x :: "'a acom"   and p :: "'a acom" assumes "\<And>x1 x2. \<lbrakk>P x1; P x2; x1 \<le> x2\<rbrakk> \<Longrightarrow> f x1 \<le> f x2"   and "P x"   and "\<And>x. P x \<Longrightarrow> P (f x)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<nabla> x2)"   and "\<And>x1 x2. \<lbrakk>P x1; P x2\<rbrakk> \<Longrightarrow> P (x1 \<triangle> x2)"   and "pfp_wn f x = Some p" shows "P p \<and> f p \<le> p"
begin_halts: fixes x :: "nat" assumes "0 < x" shows "\<exists>stp. is_final (steps0 (1, [], Oc \<up> x) tcopy_begin stp)"
order_iff_strict: fixes b :: "'a"   and a :: "'a" shows "cless_eq b a = (cless b a \<or> a = b)"
holomorphic_onI: fixes s :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "\<And>x. x \<in> s \<Longrightarrow> f field_differentiable at x within s" shows "f holomorphic_on s"
assms: shows "A \<union>\<^sub>\<circ> B \<subseteq>\<^sub>\<circ> C"
simps_2: fixes v :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list"   and vc :: "('b \<times> 'b rose_tree\<^sub>h list) blindable\<^sub>h" shows "zipper_children\<^sub>m (v, Tree\<^sub>m (Blinded vc)) = []"
gr_zeroI: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "n = zero \<Longrightarrow> False" shows "less zero n"
is_first_drop_length: fixes k :: "nat"   and a :: "'a list"   and u :: "'a list"   and v :: "'a list"   and X :: "'a"   and w :: "'a list" assumes "k \<le> length a"   and "length u < k"   and "v = X # w"   and "take k a = take k (u @ v)" shows "is_first X (drop (length u) a)"
x_below: shows "x \<le> (x\<^sup>\<star> \<cdot> y)\<^sup>\<star> \<cdot> x\<^sup>\<star>"
ordIso_ordLeq_trans: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set"   and r'' :: "('h \<times> 'h) set" assumes "r =o r'"   and "r' \<le>o r''" shows "r \<le>o r''"
exhaust: fixes y :: "ev_wait_stage_t" obtains "y = EV_PREP"   | "y = EV_WAIT"   | "y = EV_FINISH"
anormal_decr: fixes a :: "'a" assumes "\<not> depends\<^sub>0 a"   and "anormal a" shows "anormal (decr a)"
hcomp_conc_assoc: fixes P :: "conc_interface"   and Q :: "conc_interface"   and R :: "conc_interface" shows "P \<otimes>\<^sub>c (Q \<otimes>\<^sub>c R) \<simeq> (P \<otimes>\<^sub>c Q) \<otimes>\<^sub>c R"
nn_integral_mono: fixes M :: "'d measure"   and u :: "'d \<Rightarrow> ennreal"   and v :: "'d \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x \<le> v x" shows "integral\<^sup>N M u \<le> integral\<^sup>N M v"
dimacs_lit_to_var_def: fixes l :: "int" shows "dimacs_lit_to_var l \<equiv> nat \<bar>l\<bar>"
div_dvd_iff_mult: 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 b :: "'a"   and a :: "'a"   and c :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "b \<noteq> zero"   and "dvd.dvd times b a" shows "dvd.dvd times (divide a b) c = dvd.dvd times a (times c b)"
hfun_def: fixes B :: "hf"   and C :: "hf"   and F :: "hf" shows "hfun B C F \<equiv> F \<le> B * C \<and> hfunction F \<and> hdomain F = B \<and> hrange F \<le> C"
finite_OUT: fixes A :: "'a set" shows "(\<Sum>\<^sup>+ y\<in>A. f (x, y)) \<noteq> \<top>"
wf12: shows "mat nr ncc (mat_mult nr m1 m2)"
sets_K: fixes x :: "real \<times> 'a" assumes "x \<in> space S" shows "events x = sets S"
bijective_conv_mapping: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.bijective x = matrix_stone_relation_algebra.mapping (x\<^sup>t)"
add: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e"   and y :: "'e" assumes "Modules.additive f" shows "f (x + y) = f x + f y"
less: shows "\<bar>real_of_int (sgn ai) * b\<bar> < 1"
simple_match_and_SomeD: fixes m1 :: "'a simple_match"   and m2 :: "'a simple_match"   and m :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme" assumes "simple_match_and m1 m2 = Some m" shows "simple_matches m p = (simple_matches m1 p \<and> simple_matches m2 p)"
G_def: shows "G = sample_uniform (order \<G>) \<bind> (\<lambda>w1. sample_uniform (order \<G>) \<bind> (\<lambda>w2. return_spmf (\<^bold>g [^] w1 \<otimes> g' [^] w2, w1, w2)))"
hypreal_sqrt_approx_zero: fixes x :: "real star" assumes "0 < x" shows "((*f* sqrt) x \<approx> 0) = (x \<approx> 0)"
backflows_subseteq: fixes X :: "('b \<times> 'b) set"   and Y :: "('b \<times> 'b) set" shows "(X \<subseteq> Y) = (backflows X \<subseteq> backflows Y)"
f_iso: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "x \<le> y \<longrightarrow> f x \<le> f y"
runit_inv: fixes a :: "'a" assumes "ide a" shows "inverse_arrows \<r>[a] \<r>\<^sup>-\<^sup>1[a]"
that: fixes I :: "nat \<Rightarrow> nat"   and k :: "nat" assumes "bij_betw I {0..<n} {0..<n}"   and "k \<le> n"   and "\<forall>i<k. P (I i)"   and "\<forall>i. k \<le> i \<and> i < n \<longrightarrow> \<not> P (I i)" shows "thesis"
swapEnvDom_def: fixes xs :: "'d"   and x :: "'e"   and y :: "'e"   and rho :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" shows "swapEnvDom xs x y rho \<equiv> \<lambda>zs z. rho zs (z @zs[x \<and> y]_xs)"
simps_2: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and smult :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and monom :: "'d \<Rightarrow> nat \<Rightarrow> 'c"   and coeff :: "'c \<Rightarrow> nat \<Rightarrow> 'd"   and more :: "'e" shows "coeff \<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, smult = smult, monom = monom, coeff = coeff, \<dots> = more\<rparr> = coeff"
measurable_extract_int: shows "extract_int \<in> stock_measure INTEG \<rightarrow>\<^sub>M count_space UNIV"
perfect_sec_P2_def: fixes funct :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'h) spmf"   and P2_real_view :: "'e \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> 'j \<Rightarrow> ('g \<times> 'k) spmf"   and x :: "'e"   and y :: "'f"   and z :: "'i"   and S :: "('j \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> ('f \<times> 'l) spmf) \<times> ('j \<Rightarrow> 'f \<Rightarrow> 'i \<Rightarrow> 'h \<Rightarrow> 'l \<Rightarrow> 'k spmf)"   and \<A> :: "'j" shows "malicious_base.perfect_sec_P2 funct P2_real_view x y z S \<A> = (malicious_base.ideal_view_2 funct x y z S \<A> = P2_real_view x y z \<A>)"
hd_append: fixes xs :: "'g list"   and ys :: "'g list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
div_mult_self2: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times b c)) b = plus c (divide a b)"
Suc_0: fixes f :: "nat \<Rightarrow> 'b" assumes "multiplicative_function f" shows "f (Suc 0) = (1::'b)"
map_ident: fixes t :: "'c rule" shows "map_rule (\<lambda>x. x) t = t"
Compl_Icc: fixes a :: "'c"   and b :: "'c" shows "- {a..b} = {..<a} \<union> {b<..}"
submoduleE_6: fixes H :: "('c \<Rightarrow> 'a) set"   and x :: "'c \<Rightarrow> 'a" assumes "Module.submodule H R (func_space A)"   and "x \<in> H" shows "\<ominus>\<^bsub>func_space A\<^esub> x \<in> H"
simps: fixes xs :: "'c"   and x :: "'d"   and X :: "('a, 'b, 'c, 'd, 'e) qTerm" shows "qSkelAbs (qAbs xs x X) = Branch (\<lambda>i. Some (qSkel X)) Map.empty"
commute: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "AC f" shows "f x y = f y x"
expectation_gpv_top_pfinite: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "pfinite_gpv \<I> gpv" shows "expectation_gpv \<top> \<I> (\<lambda>_. \<top>) gpv = \<top>"
trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a \<le>p b"   and "b \<le>p c" shows "a \<le>p c"
committed_reads_see_committed_writes_weak_def: fixes ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('a, 'b) justifying_execution" shows "committed_reads_see_committed_writes_weak ws J = (\<forall>n. \<forall>r'\<in>read_actions (justifying_exec (J (Suc n))) \<inter> committed (J (Suc n)). let r = action_translation (J (Suc n)) r'; committed_n = action_translation (J n) ` committed (J n) in r \<in> committed_n \<or> ws r \<in> committed_n)"
P2: shows "\<lfloor>\<lambda>w. \<exists>x. ((actualizedAt) x \<^bold>\<and> Necessary x \<^bold>\<and> Abstract x) w\<rfloor>"
prems_2: shows "\<forall>r\<in>set (r # rs1). \<not> (get_action r = Return \<and> matches \<gamma> (get_match r) p)"
rWELL: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "wo_rel2 r s" shows "Well_order r"
asym: fixes x :: "'a"   and y :: "'a" assumes "x \<sqsubset> y"   and "y \<sqsubset> x"   and "x \<in> Y"   and "y \<in> Y" shows "False"
len_takeWhile_comp1: fixes n1 :: "'b"   and xs :: "'b list"   and n2 :: "'b"   and x :: "'b" assumes "n1 \<in> set xs"   and "n2 \<in> set xs"   and "n1 \<noteq> n2" shows "length (takeWhile ((\<noteq>) n1) (rev (x # xs))) \<noteq> length (takeWhile ((\<noteq>) n2) (rev (x # xs)))"
perm_sing_imp: fixes ys :: "'a list"   and xs :: "'a list"   and y :: "'a" assumes "ys <~~> xs"   and "xs = [y]" shows "ys = [y]"
CFG_semantics_wf_axioms: fixes prog :: "cmd" shows "CFG_semantics_wf sourcenode targetnode kind (valid_edge prog) (_Entry_) reds (labels_nodes prog)"
not_neutral_contains_not_neutral: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and g :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "comm_monoid_set f z"   and "comm_monoid_set.F f z g A \<noteq> z" obtains a :: "'b" where "a \<in> A" and "g a \<noteq> z"
fmmap_empty: fixes f :: "'c \<Rightarrow> 'b" shows "fmmap f {$$} = {$$}"
fresh_star_at_base: fixes S :: "atom set"   and x :: "'a" shows "S \<sharp>* x = (atom x \<notin> S)"
assms_3: shows "H1 \<lhd> G\<lparr>carrier := H\<rparr>"
add_ac_1: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a + b + c = a + (b + c)"
spr_sim_tObsC_abs: fixes t :: "('a, 'es, 'as) BEState Trace" shows "tObsC_abs t = sprCRel (spr_sim t)"
S1: assumes "sf x \<le> sf y" shows "x \<le> y"
NO_to_NOU: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and ODE :: "('c + 'b, 'd) ODE"   and S :: "('d + 'd) set" assumes "OadmitFO \<sigma> ODE S" shows "OUadmitFO \<sigma> ODE S"
Int_space_eq2: fixes x :: "'a set"   and M :: "'a measure" assumes "x \<in> null_sets M" shows "x \<inter> space M = x"
s_AuthI_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, Auth e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'\<ggreater>" obtains e'a :: "Syntax.term" where "e' = Auth e'a" and "\<lless>\<pi>, e\<ggreater> I\<rightarrow> \<lless>\<pi>', e'a\<ggreater>"   | "\<pi>' = \<pi>" and "e' = e" and "value e"
mod_mult_self1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (times c b)) b = modulo a b"
containsNotTerminal: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "contains f iA n" shows "\<not> terminal subs (f n)"
bell10_is_state: shows "state 2 |\<beta>\<^sub>1\<^sub>0\<rangle>"
lem4: fixes gpv :: "('e, 'f, 'g) gpv" shows "inline run_converter gpv id_converter = inline id_oracle gpv id_converter"
crel_vs1_alt_def: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool" shows "crel_vs1 R = state_dp_consistency.crel_vs R OO local.rel_state (=)"
tA: shows "HMA_M (from_hma\<^sub>m A) A"
eq: shows "the_gpv gpv = TRY the_gpv gpv ELSE the_gpv gpv'"
insertE: fixes a :: "'a"   and b :: "'a"   and A :: "'a set" assumes "a \<in> {b} \<union> A" obtains "a = b"   | "a \<in> A"
q_def: shows "q = [:- of_int a, of_int b:]"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "0 \<le> f x"
assoc: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "a + b + c = a + (b + c)"
gap_in_block: fixes i :: "nat"   and t :: "nat" shows "gap i t < e_length (block i t)"
B: shows "x # xs \<simeq> {ys, zs, P}"
A_Said_YM3_lemma: fixes evs :: "event list"   and K :: "nat"   and NB :: "nat"   and B :: "agent"   and A :: "agent" assumes "evs \<in> yahalom"   and "Key K \<notin> analz (knows Spy evs)"   and "Crypt K (Nonce NB) \<in> parts (knows Spy evs)"   and "Crypt (shrK B) \<lbrace>Agent A, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "B \<notin> bad" shows "\<exists>X. Says A B \<lbrace>X, Crypt K (Nonce NB)\<rbrace> \<in> set evs"
IF2wit_def: shows "IF2wit = ctor2 wit_F2"
bind_runit: fixes x :: "'d option" shows "Option.bind x Some = x"
Sup_subset_mono: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
set_bset_bsingleton: fixes x :: "'e" shows "set_bset (bsingleton x) = {x}"
if_false: fixes mem :: "'Var \<Rightarrow> 'Val"   and b :: "'BExp"   and t :: "('Var, 'AExp, 'BExp) Stmt"   and e :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<not> ev\<^sub>B mem b" shows "eval\<^sub>w_simplep (Stmt.If b t e) mem e mem"
real_of_ereal_mult: fixes a :: "ereal"   and b :: "ereal" shows "real_of_ereal (a * b) = real_of_ereal a * real_of_ereal b"
LIM_zero_iff: fixes f :: "'e \<Rightarrow> 'f"   and l :: "'f"   and F :: "'e filter" shows "((\<lambda>x. f x - l) \<longlongrightarrow> (0::'f)) F = (f \<longlongrightarrow> l) F"
PS\<^sub>R: shows "sparse (dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) PS\<^sub>R"
q64_infinite_def: shows "q64 = (\<lambda>k. k\<^sup>2 mod 64) ` {0..}"
simps_1: fixes f :: "'a \<Rightarrow> bool"   and v0 :: "'a" shows "if_rval f (Rval v0) = f v0"
diff_Suc_eq_diff_pred: fixes m :: "nat"   and n :: "nat" shows "m - Suc n = m - 1 - n"
While: fixes P :: "'d set"   and I :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and b :: "'d set"   and c :: "('d, 'e, 'f) com"   and A :: "'d set"   and Q :: "'d set"   and V :: "('d \<times> 'd) set" assumes "P \<subseteq> I"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(I \<inter> b) c I,A"   and "I \<inter> - b \<subseteq> Q" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnno b I V c Q,A"
subindex_permutation: fixes ss :: "nat list"   and ls :: "'b list" assumes "distinct ss"   and "set ss \<subseteq> {..<length ls}" obtains f :: "nat \<Rightarrow> nat" where "f permutes {..<length ls}" and "permute_list f ls = map ((!) ls) (filter (\<lambda>i. i \<notin> set ss) [0..<length ls]) @ map ((!) ls) ss"
non_exe_mode_equal: fixes s :: "'b sparc_state"   and t :: "'b sparc_state" assumes "exe_mode_val s = False \<and> get_trap_set s = {} \<and> Some t = NEXT s" shows "t = s"
edka_def: shows "edka \<equiv> let f = \<lambda>_. 0::'capacity in while\<^sub>T\<^bsup>fofu_invar\<^esup> (\<lambda>(f, brk). \<not> brk) (\<lambda>(f, uu_). find_shortest_augmenting_spec f \<bind> (\<lambda>p. case p of None \<Rightarrow> return (f, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (NPreflow.isAugmentingPath c s t f p) \<bind> (\<lambda>_. assert (Graph.isShortestPath (cf_of f) s p t) \<bind> (\<lambda>_. let f = NFlow.augment_with_path c f p in assert (NFlow c s t f) \<bind> (\<lambda>_. return (f, False))))))) (f, False) \<bind> (\<lambda>(f, uu_). assert (NFlow c s t f) \<bind> (\<lambda>_. return f))"
DBM_reset'_sound: fixes n :: "nat"   and v :: "'e \<Rightarrow> nat"   and cs :: "'e list"   and u :: "'e \<Rightarrow> 'f"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and d :: "'f" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n"   and "u \<in> [reset' M n cs v d]\<^bsub>v,n\<^esub>" shows "\<exists>ts. set_clocks cs ts u \<in> [M]\<^bsub>v,n\<^esub>"
assign: fixes E :: "'d \<Rightarrow> ('e \<Rightarrow> 'c) \<Rightarrow> 'c"   and e :: "'d"   and m :: "'e \<Rightarrow> 'c"   and v :: "'c"   and x :: "'e"   and \<iota> :: "nat"   and BMap :: "'c \<Rightarrow> bool" assumes "E e m = v" shows "((x:=\<^bsub>\<iota>\<^esub> e, m), [], None, m(x := v)) \<in> MWLs_semantics.MWLsSteps_det E BMap"
splus_def: fixes t :: "'a"   and v :: "'a" shows "local.punit.splus t v = fst (t + local.punit.pp_of_term v, local.punit.component_of_term v)"
\<pi>_upper_bound'': fixes x :: "real" assumes "2 \<le> x" shows "\<pi> x < 4 * (x / ln x)"
refl: fixes x :: "'a" assumes "x \<in> Collect derivable" shows "x \<sqsubseteq> x"
fps_divide_compose: fixes c :: "'a fps"   and b :: "'a fps"   and a :: "'a fps" assumes "fps_nth c 0 = (0::'a)"   and "fps_nth b 0 \<noteq> (0::'a)" shows "a / b oo c = (a oo c) / (b oo c)"
distinctPermCancel: fixes p :: "(name \<times> name) list"   and T :: "'e" assumes "distinctPerm p" shows "p \<bullet> p \<bullet> T = T"
add_mono: fixes a :: "'b zmultiset"   and b :: "'b zmultiset"   and c :: "'b zmultiset"   and d :: "'b zmultiset" assumes "a \<subseteq>#\<^sub>z b"   and "c \<subseteq>#\<^sub>z d" shows "a + c \<subseteq>#\<^sub>z b + d"
signed_take_bit_Suc_minus_bit0: fixes n :: "nat"   and k :: "num" shows "signed_take_bit (Suc n) (- numeral (num.Bit0 k)) = signed_take_bit n (- numeral k) * 2"
hyp: shows "z \<cdot> x \<le> y \<cdot> z"
WT_restrict_converter: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and A :: "'i set" assumes "\<I>, \<I>' \<turnstile>\<^sub>C cnv \<surd>" shows "\<I>, \<I>' \<turnstile>\<^sub>C restrict_converter A \<I>' cnv \<surd>"
ide_def: fixes a :: "(bool, 'arr list) concrete_category.arr" shows "ide a \<equiv> a \<cdot> a \<noteq> null \<and> (\<forall>f. (f \<cdot> a \<noteq> null \<longrightarrow> f \<cdot> a = f) \<and> (a \<cdot> f \<noteq> null \<longrightarrow> a \<cdot> f = f))"
simps_8: fixes r :: "real"   and x :: "octo" shows "Im7 (r *\<^sub>R x) = r * Im7 x"
inited_gext': fixes s :: "st"   and s' :: "st"   and C :: "qtname" assumes "s\<le>|s'"   and "inited C (globs s)" shows "inited C (globs s')"
map_nth: fixes xs :: "'g list" shows "map ((!) xs) [0..<length xs] = xs"
rsp: shows "eq_onp (\<lambda>cs. \<forall>c\<in>set cs. \<not> digit7 c) [] []"
thm_2_3_6_indets: fixes fs :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) list"   and b :: "nat" assumes "extended_ord.is_GB_cofactor_bound (set fs) b" shows "extended_ord.punit.is_Groebner_basis (set (extended_ord.punit.Macaulay_list (extended_ord.deg_shifts (\<Union> (indets ` set fs)) b fs)))"
Cd_single: fixes x :: "'a"   and X :: "'a set"   and d :: "'b" assumes "x \<in> X"   and "inj_on Xd X"   and "x \<in> Field (Pd_singletons_for_ds X {Xd x} d)" shows "x \<in> Singleton_for_d.Cd d X"