hcju/isabelleps huggingface dataset:
distinct_6: fixes x31 :: "'a convnet"   and x32 :: "'a convnet"   and x21 :: "'a"   and x22 :: "'a convnet" shows "Pool x31 x32 \<noteq> Conv x21 x22"
add: fixes b1 :: "'a \<Rightarrow> real"   and b2 :: "'a \<Rightarrow> real" assumes "\<forall>x. x \<in> src.diff_fun_space \<longrightarrow> X x \<in> UNIV"   and "b1 \<in> src.diff_fun_space"   and "b2 \<in> src.diff_fun_space" shows "X (b1 + b2) = X b1 + X b2"
take_subseq_def: fixes s :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> nat" shows "take_subseq s f = (\<lambda>k. s (f k))"
d_IN_Sup: fixes Y :: "('k \<times> 'k \<Rightarrow> ennreal) set"   and y :: "'k" assumes "Complete_Partial_Order.chain (\<le>) Y"   and "Y \<noteq> {}"   and "countable (support_flow (\<Squnion> Y))" shows "d_IN (\<Squnion> Y) y = (\<Squnion>f\<in>Y. d_IN f y)"
checker'_def: fixes p :: "nat"   and s\<^sub>e :: "float interval option"   and v\<^sub>e :: "float interval option"   and a\<^sub>e :: "float interval option"   and s\<^sub>o :: "float interval option"   and v\<^sub>o :: "float interval option"   and a\<^sub>o :: "float interval option" shows "checker' p s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o = approx_form p (Conj (Conj (Less (Var (Suc (Suc 0))) (Var (Suc (Suc (Suc 0))))) (Conj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Var (Suc (Suc (Suc (Suc (Suc 0))))))) (Conj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Var (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Conj (Less (Var 0) (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))) (Less (Var (Suc 0)) (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))))))) (Disj (Less (Add (Var (Suc (Suc 0))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc 0)))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var 0))))) (Var (Suc (Suc (Suc 0))))) (Conj (LessEqual (Var (Suc (Suc (Suc 0)))) (Add (Var (Suc (Suc 0))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc 0)))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var 0)))))) (Conj (Less (Add (Var (Suc (Suc 0))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc 0)))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var 0))))) (Add (Var (Suc (Suc (Suc 0)))) (Mult (Minus (Power (Var (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) 2)) (Inverse (Mult (Var (Suc (Suc (Suc (Suc 0))))) (Var (Suc 0))))))) (Disj (LessEqual (Var (Suc 0)) (Var 0)) (Disj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc 0)))))) (Var (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Disj (LessEqual (Var (Suc (Suc (Suc (Suc (Suc (Suc (Suc 0)))))))) (Add (Var (Suc (Suc (Suc (Suc (Suc 0)))))) (Minus (Mult (Mult (Var 0) (Inverse (Var (Suc 0)))) (Var (Suc (Suc (Suc (Suc (Suc (Suc 0))))))))))) (Less (Mult (Power (Add (Var (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) (Minus (Var (Suc (Suc (Suc (Suc (Suc 0)))))))) 2) (Inverse (Var (Suc (Suc (Suc (Suc 0))))))) (Mult (Add (Var (Suc (Suc (Suc 0)))) (Minus (Var (Suc (Suc 0))))) (Add (Var (Suc 0)) (Minus (Var 0)))))))))))) [a\<^sub>e, a\<^sub>o, s\<^sub>e, s\<^sub>o, Interval' (Float 2 0) (Float 2 0), v\<^sub>e, v\<^sub>o, Interval' (Float 0 1) (Float 0 1)] (replicate 8 0)"
hyps: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "I < n" shows "det (swap_row_to_front A I) = (- (1::'a)) ^ I * det A"
poly_has_sgnx_values_1: fixes p :: "real poly"   and a :: "real" assumes "p \<noteq> 0" shows "(poly p has_sgnx 1) (at_left a) \<or> (poly p has_sgnx - 1) (at_left a)"
split_option_ex: fixes P :: "'b option \<Rightarrow> bool" shows "(\<exists>x. P x) = (P None \<or> (\<exists>x. P (Some x)))"
x_2: fixes y :: "'a" assumes "y \<in> set new_pre" shows "\<not> G.hitting_paths y"
assms_7: shows "\<not> (\<exists>(a', b', c')\<in>set d. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
prj_chine_assoc_1: shows "doms.Prj\<^sub>1 \<cdot> doms.chine_assoc = doms.Prj\<^sub>1\<^sub>1"
i_f_join_i_shrink_iMODb: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat"   and d :: "nat" assumes "0 < k" shows "f \<longmapsto> k \<Join>\<^sub> [ n * k + (k - 1), mod k, d ] = f \<div> k \<Join>\<^sub> [n\<dots>,d]"
zero_not_in_Suc_image: fixes A :: "nat set" shows "0 \<notin> Suc ` A"
isecure_def: shows "isecure \<equiv> NI_indirect_sources \<and> NI_unrelated"
algebraicE: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set"   and x :: "'c" assumes "domain R"   and "subring K R"   and "x \<in> carrier R"   and "((\<lambda>K x. \<not> ring.transcendental R K x) over K) x" obtains p :: "'c list" where "p \<in> carrier (K [X]\<^bsub>R\<^esub>)" and "p \<noteq> []" and "ring.eval R p x = \<zero>\<^bsub>R\<^esub>"
is_none_def: fixes x :: "'d option" shows "Option.is_none x = (x = None)"
simps: fixes f :: "pheap \<Rightarrow> bool"   and h :: "pheap" shows "aseval (Assn f) h = f h"
set_2: fixes x21 :: "'d"   and x22 :: "'d list" shows "set (x21 # x22) = insert x21 (set x22)"
nth_image: fixes l :: "nat"   and xs :: "'c list" assumes "l \<le> length xs" shows "(!) xs ` {0..<l} = set (take l xs)"
eq_TT_2: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and xs :: "'a"   and ys :: "'a" assumes "class.Eq_def eq" shows "(eq\<cdot>xs\<cdot>ys = TT) = (xs \<noteq> \<bottom> \<and> ys \<noteq> \<bottom> \<and> xs = ys)"
is_ft_dghm_axioms: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>f\<^sub>a\<^sub>i\<^sub>t\<^sub>h\<^sub>f\<^sub>u\<^sub>l\<^bsub>\<alpha>\<^esub> \<CC>"
filterlim_at_withinI: fixes f :: "'c \<Rightarrow> 'a"   and c :: "'a"   and F :: "'c filter"   and A :: "'a set" assumes "(f \<longlongrightarrow> c) F"   and "\<forall>\<^sub>F x in F. f x \<in> A - {c}" shows "filterlim f (at c within A) F"
strict_coboundedI1: fixes a :: "'c"   and c :: "'c"   and b :: "'c" assumes "a < c" shows "min a b < c"
getMinTree_cons: fixes y :: "('b, 'a) BinomialHeapStruc.BinomialTree"   and x :: "('b, 'a) BinomialHeapStruc.BinomialTree"   and xs :: "('b, 'a) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.prio (BinomialHeapStruc.getMinTree (y # x # xs)) \<le> BinomialHeapStruc.prio (BinomialHeapStruc.getMinTree (x # xs))"
ref_def: fixes v :: "'a" shows "ref v = Heap_Monad.heap (Ref.alloc v)"
inner_sum_Basis: fixes i :: "'a" assumes "i \<in> Basis" shows "One \<bullet> i = 1"
supset_lbound_lcoset_shift: fixes X :: "'a set"   and Y :: "'a set"   and B :: "'a set"   and a :: "'a" assumes "dual_order.lbound_of X Y B" shows "dual_order.lbound_of (a +o X) (a +o Y) (a +o B)"
span_redundant: fixes x :: "'c"   and S :: "'c set" assumes "x \<in> vs2.span S" shows "vs2.span (insert x S) = vs2.span S"
MarkGuardsD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and P :: "'e set"   and f :: "'f"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P mark_guards f c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c Q,A"
Closed': fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "(\<Psi>', S, T) \<in> Rel'" shows "(p \<bullet> \<Psi>', p \<bullet> S, p \<bullet> T) \<in> Rel'"
ipresIGSubstAll_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'k, 'l, 'a, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresIGSubstAll Delta h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresSubstAll Delta h hA MOD"
strict_implies_order: fixes b :: "'e multiset"   and a :: "'e multiset" assumes "b \<subset># a" shows "b \<subseteq># a"
cod_char: fixes f :: "'c \<times> 'c" shows "C.VV.cod f = (if C.VV.arr f then C.VxV.cod f else C.VxV.null)"
E: shows "(if i < index key x n mi ma then (\<le>) else (<)) (offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i) (offs_next (offs ((enum xs index key n mi ma)[index key x n mi ma := Suc (enum xs index key n mi ma ! index key x n mi ma)]) k) (Suc (length xs + k)) (x # xs) index key mi ma i)"
sorted_list_of_multiset_singleton: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_multiset less_eq {#x#} = [x]"
det: shows "Re (mat_det (A, B, C, D)) \<le> 0"
map_def: shows "local.map \<equiv> \<lambda>j. if j = J.Zero then C.dom f0 else if j = J.One then C.cod f0 else if j = J.j0 then f0 else if j = J.j1 then f1 else C.null"
is_process_Rep: fixes P :: "'c process" shows "is_process (Rep_process P)"
subsumption_and_max_literal: fixes atom_ordering :: "('a \<times> 'a) set"   and C1 :: "'a Literal set"   and C2 :: "'a Literal set"   and L1 :: "'a Literal"   and L2 :: "'a Literal"   and A1 :: "'a"   and A2 :: "'a" assumes "propositional_atoms atom_ordering"   and "subsumes C1 C2"   and "propositional_atoms.strictly_maximal_literal atom_ordering C1 L1"   and "propositional_atoms.strictly_maximal_literal atom_ordering C2 L2"   and "A1 = atom L1"   and "A2 = atom L2" shows "A1 = A2 \<or> (A1, A2) \<in> atom_ordering"
fromQ_inject: fixes x :: "'e['g, 'f]"   and y :: "'e['g, 'f]" shows "(fromQ x = fromQ y) = (x = y)"
image_ring_is_ring: fixes f :: "'a \<Rightarrow> 'd"   and I :: "'a set" assumes "weak_ring_morphism f I R" shows "ring (image_ring f R)"
mk_parts_connection_TCP_def: fixes sport :: "16 word"   and dport :: "16 word" shows "mk_parts_connection_TCP sport dport = \<lparr>pc_iiface = ''1'', pc_oiface = ''1'', pc_proto = TCP, pc_sport = sport, pc_dport = dport\<rparr>"
floor_eq: fixes b :: "real" assumes "\<bar>b\<bar> \<le> 1 / 2" shows "\<lfloor>log 2 (1 + (real_of_int r + b) / 2 powr real_of_int k)\<rfloor> = (if r = 0 \<and> b < 0 then - 1 else 0)"
NSLIMSEQ_sin_pi: shows "(\<lambda>n. real n * sin (pi / real n)) \<longlonglongrightarrow>\<^sub>N\<^sub>S pi"
coneToFun_def: fixes \<chi> :: "'j \<Rightarrow> 's" shows "coneToFun \<chi> \<equiv> \<lambda>j. if J.arr j then S.img (\<chi> j) else S.unity"
fps_compose_power: fixes c :: "'a fps"   and a :: "'a fps"   and n :: "nat" assumes "fps_nth c 0 = (0::'a)" shows "(a oo c) ^ n = a ^ n oo c"
crdt_ops_unique_last: fixes xs :: "('c \<times> 'd) list"   and oid :: "'c"   and oper :: "'d"   and deps :: "'d \<Rightarrow> 'c set" assumes "crdt_ops (xs @ [(oid, oper)]) deps" shows "oid \<notin> set (map fst xs)"
USdB_Strong_d_Bisimulation: fixes d :: "'d" shows "Strong_d_Bisimulation d \<approx>\<^bsub>d\<^esub>"
split_scaleR_pos_le: fixes a :: "real"   and b :: "'a" assumes "0 \<le> a \<and> (0::'a) \<le> b \<or> a \<le> 0 \<and> b \<le> (0::'a)" shows "(0::'a) \<le> a *\<^sub>R b"
bind'_cong: fixes f :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and g :: "'b \<Rightarrow> ('a \<times> 'b) option"   and M :: "'b \<Rightarrow> ('c \<times> 'b) option" assumes "f \<sigma> = g \<sigma>" shows "(f ;- M) \<sigma> = (g ;- M) \<sigma>"
scale_left_distrib: fixes a :: "'a"   and b :: "'a"   and x :: "'c" shows "(a + b) *b x = a *b x + b *b x"
span_empty: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "module.span scale {} = {0::'b}"
fun_upd_smaller_oexp: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a" assumes "f \<in> FINFUNC"   and "x \<in> Field s"   and "y \<in> Field r"   and "(y, f x) \<in> r" shows "(f(x := y), f) \<in> oexp"
eq: fixes x :: "real" assumes "a \<le> b"   and "a < ereal x"   and "ereal x < b"   and "x \<notin> X" shows "f x = g x"
bind_lunit: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a option" shows "Option.bind (pure x) f = f x"
infinite_iff_countable_subset: fixes S :: "'d set" shows "infinite S = (\<exists>f. inj f \<and> range f \<subseteq> S)"
inf_top_right: fixes x :: "'b \<times> 'c" shows "inf_prod x top_prod = x"
int_pow_closed: fixes G :: "('d, 'e) monoid_scheme"   and x :: "'d"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> i \<in> carrier G"
hmset_of_enat_less: fixes m :: "enat"   and n :: "enat" shows "(hmset_of_enat m < hmset_of_enat n) = (m < n)"
inorder_insert: fixes t :: "('b ivl \<times> 'b) tree"   and x :: "'b ivl" assumes "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (Interval_Tree.insert x t) = ins_list x (Tree2.inorder t)"
Mapping_eq_lookup: fixes m :: "('a, 'b) mapping"   and m' :: "('a, 'b) mapping" shows "(m = m') = (Mapping.lookup m = Mapping.lookup m')"
normalized_nnf_match_alist_and': fixes as :: "'b negation_type list" shows "normalized_nnf_match (alist_and' as)"
local: fixes \<sigma> :: "nat \<Rightarrow> state"   and m :: "msg" assumes "P \<sigma> m" shows "P (\<sigma>(i := \<sigma> i\<lparr>msg := m\<rparr>)) m"
intros: fixes xss :: "'b list list list" assumes "\<And>x xa. \<lbrakk>x \<in> set xss; xa \<in> set x\<rbrakk> \<Longrightarrow> xa \<noteq> []" shows "map_head'_side xss"
htai_union_def: fixes H1 :: "(nat, nat) hashedTa"   and H2 :: "(nat, nat) hashedTa" shows "htai_union H1 H2 \<equiv> hta_reindex (hta_union H1 H2)"
funs_term_Fun_subterm': fixes f :: "'a"   and T :: "('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term" assumes "Fun f T \<sqsubseteq> t" shows "f \<in> funs_term t"
gbinomial_r_part_sum: fixes m :: "nat" shows "sum ((gchoose) ((2::'a) * of_nat m + (1::'a))) {..m} = (2::'a) ^ (2 * m)"
ta_5: shows "fv (ta \<cdot> \<delta>a) = {}"
proj7_def: shows "\<pi>\<^sub>7 \<equiv> \<lambda>(uu_, uu_, uu_, uu_, uu_, pre). pre"
locally_compactE: fixes S :: "'b set" assumes "locally compact S" obtains u :: "'b \<Rightarrow> 'b set" and v :: "'b \<Rightarrow> 'b set" where "\<And>x. x \<in> S \<Longrightarrow> x \<in> u x \<and> u x \<subseteq> v x \<and> v x \<subseteq> S \<and> openin (top_of_set S) (u x) \<and> compact (v x)"
cases: fixes x :: "('a \<Rightarrow> ((bool list \<times> bool list) \<times> 'c, 'a, bool list) gpv) \<times> ('a \<times> bool list \<Rightarrow> 'c \<Rightarrow> (bool, 'a, bool list) gpv)" obtains \<A>1 :: "'a \<Rightarrow> ((bool list \<times> bool list) \<times> 'c, 'a, bool list) gpv" and \<A>2 :: "'a \<times> bool list \<Rightarrow> 'c \<Rightarrow> (bool, 'a, bool list) gpv" where "x = (\<A>1, \<A>2)"
fimage_Suc_inj: fixes A :: "nat fset"   and B :: "nat fset" shows "(Suc |`| A = Suc |`| B) = (A = B)"
Un_Image: fixes R :: "('d \<times> 'c) set"   and S :: "('d \<times> 'c) set"   and A :: "'d set" shows "(R \<union> S) `` A = R `` A \<union> S `` A"
keys_add': fixes p :: "'a"   and f :: "'a \<Rightarrow>\<^sub>0 'b"   and g :: "'a \<Rightarrow>\<^sub>0 'b" assumes "p \<in> keys (f + g)" shows "p \<in> keys f \<union> keys g"
SFAcc: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and sh :: "sheap"   and sfs :: "sfields"   and v :: "val"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C has F,Static:t in D"   and "sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "sfs F = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D},(h, l, sh)\<rangle> \<Rightarrow> \<langle>Val v,(h, l, sh)\<rangle>"
univ_rw'': shows "UNIV = UNIV - {i} \<union> {i}"
Knaster_Tarski_bottom: fixes L :: "('a, 'b) gorder_scheme"   and f :: "'a \<Rightarrow> 'a" assumes "weak_complete_lattice L"   and "Mono\<^bsub>L\<^esub> f"   and "f \<in> carrier L \<rightarrow> carrier L" shows "\<bottom>\<^bsub>fpl L f\<^esub> .=\<^bsub>L\<^esub> LFP\<^bsub>L\<^esub> f"
distinct_11: fixes x31 :: "'d"   and x32 :: "'b"   and x41 :: "'d"   and x42 :: "'c"   and x43 :: "('a, 'b, 'c, 'd) Tree" shows "tPred x31 x32 \<noteq> tAct x41 x42 x43"
asym_less: fixes a :: "'a"   and b :: "'a" assumes "cmp a b = Less" shows "cmp b a \<noteq> Less"
update_convs_5: fixes roomk' :: "(room \<Rightarrow> key) \<Rightarrow> room \<Rightarrow> key"   and owns :: "room \<Rightarrow> guest option"   and currk :: "room \<Rightarrow> key"   and issued :: "key set"   and cards :: "guest \<Rightarrow> (key \<times> key) set"   and roomk :: "room \<Rightarrow> key"   and isin :: "room \<Rightarrow> guest set"   and safe :: "room \<Rightarrow> bool"   and more :: "'a" shows "roomk_update roomk' \<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk, isin = isin, safe = safe, \<dots> = more\<rparr> = \<lparr>owns = owns, currk = currk, issued = issued, cards = cards, roomk = roomk' roomk, isin = isin, safe = safe, \<dots> = more\<rparr>"
factorization_MP_Rel: shows "(MP_Rel ===> MF_Rel ===> (=)) factorization_m (factorization Irr_Mon)"
parent_opt: shows "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r parent_opt"
Diff_Diff_Int: fixes A :: "'g set"   and B :: "'g set" shows "A - (A - B) = A \<inter> B"
L: fixes u :: "nat \<Rightarrow> real" assumes "Gromov_converging_at_boundary u" shows "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> \<infinity> \<or> (\<lambda>x. ereal (u x)) \<longlonglongrightarrow> - \<infinity>"
diff_right_commute: fixes a :: "'b"   and c :: "'b"   and b :: "'b" shows "a - c - b = a - b - c"
chain_subset_antisym_Union: fixes R :: "('b \<times> 'b) set set" assumes "chain\<^sub>\<subseteq> R"   and "\<forall>r\<in>R. antisym r" shows "antisym (\<Union> R)"
zero_in_finite_cone: fixes Vs :: "'b vec set"   and n :: "nat" assumes "Vs \<subseteq> carrier_vec n" shows "0\<^sub>v n \<in> gram_schmidt.finite_cone n Vs"
permutations_of_set_nonempty: fixes A :: "'a set" assumes "A \<noteq> {}" shows "permutations_of_set A = (\<Union>x\<in>A. (#) x ` permutations_of_set (A - {x}))"
negligible_multc: fixes f :: "nat \<Rightarrow> real"   and c :: "real" shows "negligible (\<lambda>\<eta>. f \<eta> * c) = (negligible f \<or> c = 0)"
design_H1: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool"   and Q :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "(P \<turnstile> Q) is H1 healthy"
retract_of_imp_subset: fixes S :: "'b set"   and T :: "'b set" assumes "S retract_of T" shows "S \<subseteq> T"
b_lt_n: shows "\<forall>x\<in>set b. x < n"
asIMOD_igWlsAbsIsInBar: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsNE SEM" shows "igWlsAbsIsInBar (asIMOD SEM)"
monotone_Pair: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c" assumes "monotone ord orda f"   and "monotone ord ordb g" shows "monotone ord (rel_prod orda ordb) (\<lambda>x. (f x, g x))"
emeasure_lborel_box_finite: fixes a :: "'b"   and b :: "'b" shows "emeasure lborel (box a b) < \<infinity>"
ring_primeE_2: fixes p :: "'a list" assumes "p \<in> carrier (K [X])"   and "pprime K p" shows "prime (Ring_Divisibility.mult_of (K [X])) p"
advantage_def: fixes \<A> :: "('h, 'i) distinguisher"   and res1 :: "('h, 'i) resource"   and res2 :: "('h, 'i) resource" shows "advantage \<A> res1 res2 = \<bar>spmf (connect \<A> res1) True - spmf (connect \<A> res2) True\<bar>"
strip_while: fixes f :: "'a acom \<Rightarrow> 'a acom"   and P :: "'a acom \<Rightarrow> bool"   and C :: "'a acom"   and C' :: "'a acom" assumes "\<forall>C. strip (f C) = strip C"   and "while_option P f C = Some C'" shows "strip C' = strip C"
eq: shows "\<forall>\<^sub>F z in nhds z. (\<Sum>n. c n * z ^ n) = f z"
encRelS: shows "\<forall>S1 S2. \<lbrakk>S1\<rbrakk> = \<lbrakk>S2\<rbrakk> \<longrightarrow> (S1, S2) \<in> RelS"
6: shows "\<Union> {if xa \<in> FvarsT r - {x} \<union> f t x then FvarsT ta else {} |ta xa. (ta, xa) \<in> set txs} = \<Union> {f ta xa |ta xa. (ta, xa) \<in> set txs}"
vimage_eq_UN: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'e set" shows "f -` B = (\<Union>y\<in>B. f -` {y})"
Bang: fixes P :: "pi"   and V :: "residual" assumes "P \<parallel> !P \<longmapsto> V" shows "!P \<longmapsto> V"
OTNaux: fixes a :: "('b, 'c) Combinators"   and x :: "'b"   and y :: "'b" assumes "onlyTwoNets a"   and "\<not> member DenyAll a"   and "(x, y) \<in> sdnets a" shows "x = first_srcNet a \<and> y = first_destNet a \<or> x = first_destNet a \<and> y = first_srcNet a"
oreachable_init: fixes s :: "'c \<times> 'd"   and A :: "('c \<times> 'd, 'e) automaton"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "s \<in> init A" shows "s \<in> oreachable A S U"
O_def: shows "O \<equiv> BD_Security.filtermap \<gamma> g"
RETURN_dRETURN: fixes f :: "'a"   and f' :: "'a nres" assumes "RETURN f \<le> f'" shows "nres_of (dRETURN f) \<le> f'"
simps_2: fixes i :: "'a"   and l :: "'a factor_tree"   and r :: "'a factor_tree" shows "factor_node_info (Factor_Node i l r) = i"
degree_smult_le: fixes a :: "'d"   and p :: "'d poly" shows "degree (smult a p) \<le> degree p"
infinite_growing: fixes X :: "'a set" assumes "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. x < y" shows "infinite X"
is_ground_subst_list_def: fixes \<sigma>s :: "'s list" shows "is_ground_subst_list \<sigma>s = Ball (set \<sigma>s) is_ground_subst"
subprob_algebra_def: fixes K :: "'c measure" shows "subprob_algebra K = (SUP A\<in>sets K. vimage_algebra {M. subprob_space M \<and> sets M = sets K} (\<lambda>M. emeasure M A) borel)"
function_ring_car_memI: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "\<And>a. a \<in> S \<Longrightarrow> f a \<in> carrier R"   and "\<And>a. a \<notin> S \<Longrightarrow> f a = undefined" shows "f \<in> carrier (function_ring S R)"
a: shows "\<Gamma>' \<turnstile> \<theta> is \<theta>' over \<Gamma>"
continuous_at_Ln: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "isCont Ln z"
seq_symeval: fixes \<Gamma> :: "'a \<Rightarrow> 'a"   and P :: "'a hrel"   and Q :: "'a hrel" shows "\<Gamma> \<Turnstile> P ;; Q = (\<Gamma> \<Turnstile> P) ;; Q"
antimonoD: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "order.antimono less_eq f"   and "less_eq x y" shows "f y \<le> f x"
hyperpow_hrabs: fixes r :: "'b star"   and n :: "nat star" shows "\<bar>r\<bar> pow n = \<bar>r pow n\<bar>"
take_Cons: fixes n :: "nat"   and x :: "'g"   and xs :: "'g list" shows "take n (x # xs) = (case n of 0 \<Rightarrow> [] | Suc m \<Rightarrow> x # take m xs)"
lm08: fixes \<Omega> :: "'c list"   and N :: "integer set"   and bids :: "integer \<times> 'c set \<Rightarrow> 'd"   and random :: "integer" assumes "distinct \<Omega>"   and "set \<Omega> \<noteq> {}"   and "finite N" shows "card ((argmax \<circ> sum) (summedBidVector (pseudoAllocation (randomEl (takeAll (\<lambda>x. winningAllocationRel (N \<union> {seller}) (set \<Omega>) ((\<in>) x) bids) (allAllocationsAlg (N \<union> {seller}) \<Omega>)) random) <| ((N \<union> {seller}) \<times> finestpart (set \<Omega>))) (N \<union> {seller}) (set \<Omega>)) ((argmax \<circ> sum) bids (allAllocations (N \<union> {seller}) (set \<Omega>)))) = 1"
Rep_mod_ring_def: fixes x :: "'a mod_ring" shows "Rep x = to_int_mod_ring x"
hoare_post_imp: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b \<Rightarrow> 'b"   and Q :: "'b \<Rightarrow> bool"   and R :: "'b \<Rightarrow> bool" assumes "\<lbrace>P\<rbrace> a \<lbrace>Q\<rbrace>"   and "\<And>s. Q s \<Longrightarrow> R s" shows "\<lbrace>P\<rbrace> a \<lbrace>R\<rbrace>"