hcju/isabelleps huggingface dataset:
strongly_strategyproof_profile_def: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and i :: "'agent"   and Ri' :: "'alt \<Rightarrow> 'alt \<Rightarrow> bool" shows "strongly_strategyproof_profile R i Ri' = SD (R i) (sds (R(i := Ri'))) (sds R)"
prj_tuple_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "commutative_square f g h k" shows "prj1 f g \<cdot> ECP.tuple h f g k = h"
strict_iff_order: fixes b :: "'x \<Rightarrow>\<^sub>0 nat"   and a :: "'x \<Rightarrow>\<^sub>0 nat" shows "(b \<prec> a) = (b \<preceq> a \<and> a \<noteq> b)"
density_cong: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and f' :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "f' \<in> borel_measurable M"   and "AE x in M. f x = f' x" shows "density M f = density M f'"
fls_subdegree_minus_sym: fixes g :: "'a fls"   and f :: "'a fls" shows "fls_subdegree (g - f) = fls_subdegree (f - g)"
AR_imp_locally_path_connected: fixes S :: "'a set" assumes "AR S" shows "locally path_connected S"
galois_char: fixes l :: "'a \<Rightarrow> 'a"   and u :: "'a \<Rightarrow> 'a" shows "galois l u = ((\<forall>x. x \<le> u (l x)) \<and> (\<forall>x. l (u x) \<le> x) \<and> isotone l \<and> isotone u)"
bell_11_index_1: shows "|\<beta>\<^sub>1\<^sub>1\<rangle> $$ (0, 0) = 0"
unit_factor_poly_def: fixes p :: "'b poly" shows "unit_factor_class.unit_factor p = [:unit_factor_class.unit_factor (lead_coeff p):]"
lfp_induct_set: fixes a :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> lfp f"   and "mono f"   and "\<And>x. x \<in> f (lfp f \<inter> {x. P x}) \<Longrightarrow> P x" shows "P a"
cin_def: shows "cin \<equiv> (id ---> rcset ---> id) (\<in>)"
abs_bound_def: shows "abs_bound \<equiv> Ivl 0 (max \<bar>lower bound\<bar> \<bar>upper bound\<bar>)"
removing_offending_flows_makes_invariant_hold: fixes G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" shows "\<forall>F\<in>set_offending_flows G nP. sinvar (delete_edges G F) nP"
order: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a l\<rightarrow> b = (1::'a))"
invar: fixes s :: "('a, 'b) assoc_list" shows "lm.invar s"
nat_geq_1_eq_neqz: fixes x :: "nat" shows "(1 \<le> x) = (x \<noteq> 0)"
IInf_def: shows "IInf = Inf \<circ> Rep_upset"
powser_limit_0: fixes s :: "real"   and a :: "nat \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'a" assumes "0 < s"   and "\<And>x. norm x < s \<Longrightarrow> (\<lambda>n. a n * x ^ n) sums f x" shows "f \<midarrow>0::'a\<rightarrow> a 0"
set_cases: fixes e :: "'d"   and a :: "'d Monomorphic_Monad.id" assumes "e \<in> set_id a" obtains "a = return_id e"
eout_know_s: fixes C :: "specID"   and m :: "Secrets" shows "eoutKnowCorrect C (sKS m) = (eout C (sE m) = (m \<in> specSecrets C \<or> know C (sKS m)))"
A5: shows "\<forall>SQ SR. SQ \<in>S Q \<and> SR \<in>S R \<longrightarrow> (SQ, SR) \<in> SRel\<^sup>+"
gen_igba_impl_rel_eext_def_internal: fixes Rm :: "('g \<times> 'j) set"   and Rl :: "('f \<times> ('h \<Rightarrow> 'i \<Rightarrow> bool)) set" shows "gen_igba_impl_rel_eext Rm Rl \<equiv> {(\<lparr>igbai_L = Li, \<dots> = mi\<rparr>, \<lparr>igba_L = L, \<dots> = m\<rparr>) |Li mi L m. (Li, L) \<in> Rl \<and> (mi, m) \<in> Rm}"
plus_int_def: shows "(+) \<equiv> (Rep_Integ ---> Rep_Integ ---> Abs_Integ) (\<lambda>(x, y) (u, v). (x + u, y + v))"
l_get_mode_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_mode :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, shadow_root_mode) prog"   and get_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_get_mode type_wf get_mode get_mode_locs \<equiv> ((\<forall>shadow_root_ptr h h'. reads (get_mode_locs shadow_root_ptr) (get_mode shadow_root_ptr) h h') \<and> (\<forall>h shadow_root_ptr. type_wf h \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h \<longrightarrow> h \<turnstile> ok get_mode shadow_root_ptr)) \<and> (\<forall>h shadow_root_ptr. h \<turnstile> ok get_mode shadow_root_ptr \<longrightarrow> shadow_root_ptr |\<in>| shadow_root_ptr_kinds h) \<and> (\<forall>shadow_root_ptr h. pure (get_mode shadow_root_ptr) h)"
partial: fixes \<sigma> :: "nat \<Rightarrow> state"   and \<sigma>' :: "nat \<Rightarrow> state"   and dests :: "nat \<Rightarrow> nat option"   and sip :: "nat"   and rip :: "nat"   and rsn :: "nat" assumes "\<forall>j. quality_increases (\<sigma> j) (\<sigma>' j)"   and "\<forall>rip\<in>dom dests. rip \<in> kD (rt (\<sigma> sip)) \<and> the (dests rip) - 1 \<le> nsqn (rt (\<sigma> sip)) rip"   and "dests rip = Some rsn" shows "rip \<in> kD (rt (\<sigma>' sip)) \<and> rsn - 1 \<le> nsqn (rt (\<sigma>' sip)) rip"
ex_Melem_conv: fixes A :: "'a multiset" shows "(\<exists>x. x \<in># A) = (A \<noteq> {#})"
S_def: shows "S = {s. partial_order_on A s \<and> r \<subseteq> s}"
dg_Cod_in_Vset: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<CC>\<lparr>Cod\<rparr> \<in>\<^sub>\<circ> Vset \<beta>"
diff_Suc_Suc: fixes m :: "nat"   and n :: "nat" shows "Suc (m::nat) - Suc (n::nat) = m - n"
disc_opposite_ocircline: fixes H :: "ocircline" shows "disc (opposite_ocircline H) = disc_compl H"
Diff_not_in: fixes a :: "'a"   and A :: "'a set" shows "a \<notin> A - {a}"
closure_eq: fixes S :: "'a set" shows "(closure S = S) = closed S"
subset_Intl: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A \<subseteq> C" shows "A \<inter> B \<subseteq> C"
ltrm_decomp: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)"   and "0 < deg R f" obtains g :: "nat \<Rightarrow> 'c" where "g \<in> carrier (UP R) \<and> f = g \<oplus>\<^bsub>UP R\<^esub> monom (UP R) (f (deg R f)) (deg R f) \<and> deg R g < deg R f"
setcompr_eq_image: fixes f :: "'f \<Rightarrow> 'e"   and P :: "'f \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
array_get_alloc: fixes v :: "'c"   and h :: "heap" shows "Array.get (snd (Ref.alloc v h)) = Array.get h"
vbrelation_vlrestriction: fixes r :: "V"   and A :: "V" shows "vbrelation (r \<restriction>\<^sup>l\<^sub>\<circ> A)"
composites_are_arrows_1: shows "hseq f (g \<star> f)"
simps_3: fixes r :: "'a ifex list" shows "lowest_tops (Trueif # r) = lowest_tops r"
poly_as_sum_of_monoms': fixes p :: "'b poly"   and n :: "nat" assumes "degree p \<le> n" shows "(\<Sum>i\<le>n. monom (coeff p i) i) = p"
in_pred_image_iff: fixes X :: "nat set"   and x :: "nat" assumes "0 \<notin> X" shows "(x \<in> (\<lambda>x. x - Suc 0) ` X) = (Suc x \<in> X)"
no_incoming_to_zero: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and aaa :: "nat"   and ba :: "transition"   and s :: "nat"   and d :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" assumes "\<forall>((from, to), t)|\<in>|e. 0 < to"   and "(aaa, ba) |\<in>| possible_steps e s d l i" shows "aaa \<noteq> 0"
A_def: shows "A = {(x, m) |x m. ereal m < a}"
det_row_minus: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and a :: "('c, 'd) vec"   and b :: "('c, 'd) vec" shows "Square_Matrix.det (upd_row M i (a - b)) = Square_Matrix.det (upd_row M i a) - Square_Matrix.det (upd_row M i b)"
four_two: fixes \<delta>S :: "real"   and p :: "nat"   and q :: "nat"   and i :: "nat" assumes "\<beta> \<le> rmin"   and "\<mu> \<le> \<delta>S"   and "\<gamma>1 \<delta>S \<le> \<delta>S"   and "\<gamma>2 \<delta>S \<le> \<delta>"   and "\<gamma>3 \<delta>S \<le> \<delta>" shows "okClocks p q i"
extender_subset: fixes S :: "'b formula set"   and k :: "nat" shows "S \<subseteq> extender k S"
focus_coeffs_subset_Polys': fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and Y :: "'a set"   and X :: "'a set" assumes "p \<in> P[Y]" shows "range (lookup (focus X p)) \<subseteq> P[Y - X]"
falling_fact_pochhammer': fixes a :: "int"   and k :: "nat" shows "(\<Prod>i = 0..<k. a - int i) = pochhammer (a - int k + 1) k"
of_int_fact: fixes n :: "nat" shows "of_int (fact n) = fact n"
R: fixes S' :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> U <: S"   and "TVarB U \<Colon> \<Gamma> \<turnstile> s : S'"   and "TVarB U \<Colon> \<Gamma> \<turnstile> S' <: U'" shows "P"
inducts: fixes x :: "nat set"   and P :: "nat set \<Rightarrow> bool" assumes "i_set_contp x"   and "\<And>n. P [n\<dots>]"   and "\<And>n. P [\<dots>n]"   and "\<And>n d. P [n\<dots>,d]" shows "P x"
dim_factorization_lattice: fixes x :: "nat"   and u :: "'a poly"   and d :: "nat"   and m :: "'a" assumes "x < degree u + d" shows "dim_vec (factorization_lattice u d m ! x) = degree u + d"
pr_var_mwb_lens: fixes x :: "'b \<Longrightarrow> 'c" assumes "mwb_lens x" shows "mwb_lens (&x)"
not_less: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<sqsubset> y) = (y \<sqsubseteq> x)"
real_divr: fixes x :: "real"   and y :: "real"   and prec :: "nat" shows "x / y \<le> real_divr prec x y"
code: fixes x :: "complex"   and y :: "complex" shows "x * y = Complex (Re x * Re y - Im x * Im y) (Re x * Im y + Im x * Re y)"
m_rep_def: fixes p :: "int extended \<times> int extended" shows "m_rep p = (if is_empty_rep p then 3 else let (l, h) = p in (case l of -\<infinity> \<Rightarrow> 0 | _ \<Rightarrow> 1) + (case h of \<infinity> \<Rightarrow> 0 | _ \<Rightarrow> 1))"
Union_image_insert: fixes f :: "'c \<Rightarrow> 'b set"   and a :: "'c"   and B :: "'c set" shows "\<Union> (f ` (a \<triangleright> B)) = f a \<union> \<Union> (f ` B)"
ext_cont_cancel_cbox: fixes x :: "'c"   and a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> cbox a b" shows "ext_cont f a b x = f x"
K_record_comp: fixes c :: "'d"   and f :: "'c \<Rightarrow> 'e" shows "(\<lambda>x. c) \<circ> f = (\<lambda>x. c)"
measurable_restrict_subset: fixes J :: "'c set"   and L :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" assumes "J \<subseteq> L" shows "(\<lambda>f. restrict f J) \<in> Pi\<^sub>M L M \<rightarrow>\<^sub>M Pi\<^sub>M J M"
bot_lifted_pair_def: shows "bot \<equiv> Abs_lifted_pair (\<lambda>f. (bot, Filters.filter.Abs_filter UNIV))"
thetaSeqW_Wretr: shows "thetaSeqW \<subseteq> Wretr (thetaSeqW \<union> Wbis)"
assms: shows "(R \<upharpoonleft> P \<otimes> Q) x y"
lowerPowers_def: shows "lowerPowers \<equiv> map_fun id (map_fun id (map_fun lookup Abs_poly_mapping)) (\<lambda>b i p x. if x \<in> {b..<b + i} then 0::'b else p (dec_above b i x))"
standard_sound: fixes P :: "'a \<Rightarrow> bool" shows "sound \<guillemotleft> P \<guillemotright>"
assms: shows "vec_of_basis_enum a = vec_of_basis_enum b"
cs_done_implies_both_snapshotted_1: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat"   and cid :: "nat"   and p :: "'d"   and q :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final"   and "snd (cs (distributed_system.s channel trans send recv init t i) cid) = Done"   and "i < length t"   and "channel cid = Some (p, q)" shows "ps (distributed_system.s channel trans send recv init t i) p \<noteq> None"
collect_interrupts_prefix: fixes ias :: "'b interrupt_action list"   and ias' :: "'b interrupt_action list" shows "collect_interrupts ias \<subseteq> collect_interrupts (ias @ ias')"
inf_fixed_only_moebius_similarity: fixes M :: "moebius" assumes "moebius_pt M \<infinity>\<^sub>h = \<infinity>\<^sub>h" shows "\<exists>a b. a \<noteq> 0 \<and> M = moebius_similarity a b"
fps_shift_times_fps_X_power': fixes n :: "nat"   and f :: "'b fps" shows "fps_shift n (f * fps_X ^ n) = f"
coprime_imp_gcd_eq_1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a b" shows "gcd a b = one"
mI_principalTr: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and I :: "'a set"   and x :: "'a" assumes "0 < n"   and "distinct_pds K n P"   and "ideal (O\<^bsub>K P n\<^esub>) I"   and "I \<noteq> {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}"   and "I \<noteq> carrier (O\<^bsub>K P n\<^esub>)"   and "x \<in> I" shows "\<forall>j\<le>n. (\<nu>\<^bsub>K P j\<^esub>) (mIg\<^bsub> K n P I\<^esub>) \<le> (\<nu>\<^bsub>K P j\<^esub>) x"
HEG: shows "mat_of_rows n (map of_int_hom.vec_hom fs') = E * mat_of_rows n (map of_int_hom.vec_hom fs)"
assms_5: fixes x :: "'a" assumes "x \<in> B - A" shows "0 \<le> g x"
powr_0: fixes z :: "'c" shows "(0::'c) powr z = (0::'c)"
properfactor_prod_r: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Group.monoid G"   and "properfactor G a b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "c \<in> carrier G" shows "properfactor G a (b \<otimes>\<^bsub>G\<^esub> c)"
rel_def: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a" shows "x \<sim> y \<equiv> \<exists>s1. s1 \<in> S \<and> s1 \<cdot> (snd y \<cdot> fst x - snd x \<cdot> fst y) = \<zero>"
K'_def: fixes x :: "'a"   and n' :: "nat"   and \<omega>' :: "nat \<Rightarrow> 'a" shows "K' x n' \<omega>' = K (case n' of 0 \<Rightarrow> x | Suc x \<Rightarrow> \<omega>' x)"
fdisjnt_union_right: fixes M :: "'b fset"   and A :: "'b fset"   and B :: "'b fset" assumes "fdisjnt M A"   and "fdisjnt M B" shows "fdisjnt M (A |\<union>| B)"
nlistToDigit_list: fixes xs :: "('c, 'd) Node list" assumes "1 \<le> length xs \<and> length xs \<le> 4" shows "digitToList (nlistToDigit xs) = nlistToList xs"
aeq: shows "aS\<^bsub>T\<^esub> = iS\<^bsub>CI T\<^esub>"
split_szip: fixes P :: "('c \<times> 'd) stream \<Rightarrow> prop" shows "(\<And>x. PROP P x) \<equiv> (\<And>y z. PROP P (y ||| z))"
inf_sup_ord_2: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) y"
ordsetmap: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and a :: "'b"   and b :: "'b" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "a \<in> P"   and "b \<in> P"   and "less_eq a b" shows "less_eq' (f a) (f b)"
assms: shows "\<^bold>|z\<^bold>| + \<^bold>|x\<^bold>| \<le> \<^bold>|x \<^sup>@ a\<^bold>|"
lin_mul: fixes t :: "Reflection_Examples.num"   and i :: "nat"   and bs :: "nat list" shows "Inum (lin_mul t i) bs = Inum (num.Mul i t) bs"
lcf_ltrm: fixes p :: "nat \<Rightarrow> 'a" shows "monom P (p (deg R p)) (deg R p) = monom P (p (deg R p)) (deg R p)"
nxt_wrong_hyp: fixes b :: "nat"   and b' :: "nat"   and i :: "nat" assumes "nxt b \<down>= b'"   and "s b \<down>= i" shows "\<exists>y<e_length b'. \<phi> i y \<down>\<noteq> e_nth b' y"
simps_1: fixes s :: "state"   and x :: "val" shows "(s:x \<ge> 0) = (x \<noteq> Null)"
can_Ide_self: fixes t :: "'a E.term" assumes "E.Ide t" shows "B.can t t = EVAL t"
countable_Un_Int_2: fixes I :: "'b set"   and N :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> N i \<in> sets M" shows "\<Inter> (N ` I) \<in> sets M"
w: shows "w \<in> lang (seq [Times (Atom y) (Atom x), Star (Times (Atom y) (Atom x)), Atom y, Atom y])"
S5Basic_4: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<^bold>\<diamond>\<phi> in v]"
sandwich_def: fixes A :: "'o"   and a :: "'a" shows "\<sigma>(A,a) \<equiv> \<lambda>B\<in>Ob. \<lparr>set_dom = Hom A B, set_func = \<lambda>f\<in>Hom A B. set_func (F \<^bsub>\<a>\<^esub> f) a, set_cod = F \<^bsub>\<o>\<^esub> B\<rparr>"
left_unit: fixes a :: "'b" assumes "a \<in> B" shows "addB zeroB a = a"
game0: shows "ind_cpa.ind_cpa \<A> = game0"
basis_card: fixes B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set" assumes "basis B\<^sub>1"   and "basis B\<^sub>2" shows "card B\<^sub>1 = card B\<^sub>2"
affine_independent_iff_card: fixes s :: "'b set" shows "(\<not> affine_dependent s) = (finite s \<and> aff_dim s = int (card s) - 1)"
prime_factor_int: fixes k :: "int" assumes "\<bar>k\<bar> \<noteq> 1" obtains p :: "int" where "prime p" and "p dvd k"
is_hashmap_def: fixes m :: "'c \<Rightarrow> 'd option"   and ht :: "('c, 'd) hashtable" shows "is_hashmap m ht = (\<exists>\<^sub>Al. is_hashmap' m l ht)"
rep_def: fixes r :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c" shows "pmdl.rep r = (\<Sum>v\<in>keys r. lookup r v \<odot> v)"
SCallRedsThrowParams: fixes es :: "char list exp list"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and vs\<^sub>1 :: "val list"   and a :: "char list exp"   and es\<^sub>2 :: "char list exp list"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s\<^sub>0,b\<^sub>0\<rangle> [\<rightarrow>]* \<langle>map Val vs\<^sub>1 @ throw a # es\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>2,b\<^sub>2\<rangle>"
hyper_isin_def: fixes a :: "'f"   and A :: "'f set option" shows "a \<in>\<in> A \<equiv> case A of None \<Rightarrow> True | \<lfloor>A\<rfloor> \<Rightarrow> a \<in> A"
that: fixes r :: "real" assumes "0 < r"   and "ball \<xi> r \<subseteq> S"   and "inj_on f (ball \<xi> r)" shows "thesis"
modifies_mono: fixes vs\<^sub>1 :: "char list set"   and vs\<^sub>2 :: "char list set"   and a :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "vs\<^sub>1 \<subseteq> vs\<^sub>2"   and "modifies vs\<^sub>1 a b" shows "modifies vs\<^sub>2 a b"
fixp_def: fixes f :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b" shows "ab.fixp f = prod_lub luba lubb (ab.iterates f)"
measurable_piecewise_restrict: fixes C :: "'d set set"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and N :: "'e measure" assumes "countable C"   and "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> \<Omega> \<inter> space M \<in> sets M"   and "space M \<subseteq> \<Union> C"   and "\<And>\<Omega>. \<Omega> \<in> C \<Longrightarrow> f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
bij_betw_imp_inj_on: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set" assumes "bij_betw f A B" shows "inj_on f A"
M1: shows "h \<Turnstile> R1 \<and>\<^sub>A R2"
prob_space_uniform_count_measure: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "prob_space (uniform_count_measure A)"
at_within_open: fixes "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and S :: "'e set" assumes "class.topological_space open"   and "a \<in> S"   and "open S" shows "topological_space.at_within open a S = topological_space.at_within open a UNIV"
simply_connected_eq_homotopic_circlemaps2a: fixes h :: "complex \<Rightarrow> 'a"   and S :: "'a set" assumes "continuous_on (sphere 0 1) h"   and "h ` sphere 0 1 \<subseteq> S"   and "\<And>f g. \<lbrakk>continuous_on (sphere 0 1) f; f ` sphere 0 1 \<subseteq> S; continuous_on (sphere 0 1) g; g ` sphere 0 1 \<subseteq> S\<rbrakk> \<Longrightarrow> homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g" shows "\<exists>a. homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S h (\<lambda>x. a)"
finite_backflows_state: fixes \<T> :: "'a stateful_policy" assumes "wf_stateful_policy \<T>" shows "finite (backflows (flows_state \<T>))"
f'_bounded: fixes t :: "real"   and x :: "'a" assumes "t \<in> T"   and "x \<in> X'" shows "onorm (f' (t, x)) \<le> B'"
rtranclpD: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "R\<^sup>*\<^sup>* a b" shows "a = b \<or> a \<noteq> b \<and> R\<^sup>+\<^sup>+ a b"
COMPARE_intros_1: fixes x :: "real"   and y :: "real" assumes "x < y" shows "COMPARE x y \<equiv> LT"
Pi_I': fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>x. x \<in> A \<longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
red_supset_transitive: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and C :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "A \<sqsupset>p B"   and "B \<sqsupset>p C" shows "A \<sqsupset>p C"
tl_listsN: fixes A :: "'b list set"   and S :: "'b set"   and n :: "nat" assumes "A \<subseteq> listsN (n + 1) S" shows "tl ` A \<subseteq> listsN n S"
span_eq_dim: fixes S :: "'c set"   and T :: "'c set" assumes "vs2.span S = vs2.span T" shows "vs2.dim S = vs2.dim T"
eventually_at_topological: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" assumes "class.topological_space open" shows "eventually P (topological_space.at_within open a s) = (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
finite: fixes m :: "('a, 'b) HashMap.hashmap" assumes "hm.invar m" shows "finite (dom (hm.\<alpha> m))"
induct: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c list"   and x2 :: "'d list"   and P :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "for_all2 r x1 x2"   and "P [] []"   and "\<And>x y xs ys. \<lbrakk>r x y; for_all2 r xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
ALG'_def: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list"   and init :: "'c list"   and i :: "nat"   and x :: "'c" shows "ALG' A qs init i x = E (map_pmf (\<lambda>xa. real (ALG x qs i xa)) (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take i qs)))"
ext_inject: fixes dwa_cond :: "'a \<Rightarrow> bool"   and dwa_step :: "'a \<Rightarrow> 'a"   and dwa_initial :: "'a"   and dwa_invar :: "'a set"   and more :: "'b"   and dwa_cond' :: "'a \<Rightarrow> bool"   and dwa_step' :: "'a \<Rightarrow> 'a"   and dwa_initial' :: "'a"   and dwa_invar' :: "'a set"   and more' :: "'b" shows "(\<lparr>dwa_cond = dwa_cond, dwa_step = dwa_step, dwa_initial = dwa_initial, dwa_invar = dwa_invar, \<dots> = more\<rparr> = \<lparr>dwa_cond = dwa_cond', dwa_step = dwa_step', dwa_initial = dwa_initial', dwa_invar = dwa_invar', \<dots> = more'\<rparr>) = (dwa_cond = dwa_cond' \<and> dwa_step = dwa_step' \<and> dwa_initial = dwa_initial' \<and> dwa_invar = dwa_invar' \<and> more = more')"
2: fixes x :: "'a" assumes "x \<noteq> a" shows "g x \<le> f x"
insert_commute: fixes x :: "'f"   and y :: "'f"   and A :: "'f set" shows "insert x (insert y A) = insert y (insert x A)"
P2_sec_OT_14: fixes M :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "R2_14 M C = funct_OT_14 M C \<bind> (\<lambda>(out1, out2). S2_14 C out2)"
DBInit: shows "\<turnstile> Init DBInit \<longrightarrow> Init BInit inp qc out"
UNIV: fixes S :: "'b set set" shows "generate_topology S UNIV"