hcju/isabelleps huggingface dataset:
type_less_eq_x_OclAny: fixes \<tau> :: "'a basic_type" assumes "\<tau> \<le> OclAny" obtains "\<tau> = OclVoid"   | "\<tau> = OclAny"   | "\<tau> = Boolean"   | "\<tau> = Integer"   | "\<tau> = UnlimitedNatural"   | "\<tau> = Real"   | "\<tau> = String"   | \<E> :: "('a, String.literal) phantom" where "\<tau> = Enum \<E>"   | \<C> :: "'a" where "\<tau> = \<langle>\<C>\<rangle>\<^sub>\<T>"
gR_rel_defined: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "partial_semigroup.gR_rel times D x y" shows "D x (cancellative_partial_semigroup.rquot times D y x)"
crit_pair_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.crit_pair p q = (if local.punit.component_of_term (punit.lt p) = local.punit.component_of_term (punit.lt q) then (local.punit.monom_mult ((1::'c) / punit.lc p) (lcs (punit.lp p) (punit.lp q) - punit.lp p) (punit.tail p), local.punit.monom_mult ((1::'c) / punit.lc q) (lcs (punit.lp p) (punit.lp q) - punit.lp q) (punit.tail q)) else (0, 0))"
minus_order: fixes a :: "'a"   and b :: "'a" assumes "- a \<le> - b" shows "b \<le> a"
not_regular_langI: fixes B :: "'a list set"   and A :: "'a list set" assumes "infinite B"   and "\<And>x y. \<lbrakk>x \<in> B; y \<in> B; x \<noteq> y\<rbrakk> \<Longrightarrow> \<exists>w. (x @ w \<in> A) \<noteq> (y @ w \<in> A)" shows "\<nexists>r. lang r = A"
quasinorm_sum_maximum: fixes N1 :: "'c quasinorm"   and N :: "'c quasinorm"   and N2 :: "'c quasinorm" assumes "N1 \<subseteq>\<^sub>N N"   and "N2 \<subseteq>\<^sub>N N" shows "N1 +\<^sub>N N2 \<subseteq>\<^sub>N N"
g_1_def: fixes res :: "int \<times> int" shows "g_1 res = (THE x. P_1 res x)"
not_qf: fixes p :: "fm" assumes "qfree p" shows "qfree (not p)"
LLq_def: shows "LLq \<equiv> \<lambda>t1 t2. psubst Lq [(t1, xx), (t2, yy)]"
homology_homotopy_empty: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and p :: "int" assumes "homotopic_with (\<lambda>h. True) X Y f g" shows "hom_induced p X {} Y {} f = hom_induced p X {} Y {} g"
usc_def: fixes f :: "'b \<Rightarrow> 'c" shows "usc f = (\<forall>x. usc_at x f)"
subseqs_powset: fixes xs :: "'b list" shows "sset (subseqs xs) = Pow (set xs)"
set_intros_5: fixes yl :: "'a"   and x32 :: "('a, 'b, 'c) expr"   and x31 :: "('a, 'b, 'c) expr"   and x33 :: "('a, 'b, 'c) expr" assumes "yl \<in> RID\<^sub>E x32" shows "yl \<in> RID\<^sub>E (Ite x31 x32 x33)"
induct: fixes P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a list"   and a1 :: "'a list" assumes "\<And>xs. P xs []"   and "\<And>v va. P [] (v # va)"   and "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)" shows "P a0 a1"
assms_1: shows "(x, y) = split_i_j_x i j Z"
Rep_linear_poly: fixes x :: "linear_poly" shows "Rep_linear_poly x \<in> {c. Abstract_Linear_Poly.inv c}"
SUP_cong: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b \<Rightarrow> 'c \<times> 'd"   and D :: "'b \<Rightarrow> 'c \<times> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup_prod (C ` A) = Sup_prod (D ` B)"
deadlockedAcquire: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and l :: "'l"   and t' :: "'t" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> waiting (wset s t)"   and "0 < ln $ l"   and "has_lock (locks s $ l) t'"   and "t' \<noteq> t"   and "t' \<in> deadlocked s \<or> t' \<in> final_threads s" shows "t \<in> deadlocked s"
simps_4: fixes xs :: "'a list"   and a :: "fexpr"   and b :: "fexpr" shows "feval xs (FSub a b) = feval xs a \<ominus> feval xs b"
ExcessTable_subset: fixes g :: "graph"   and vs :: "nat list" shows "set (map fst (ExcessTable g vs)) \<subseteq> set vs"
sel_1: fixes x21 :: "'b"   and x22 :: "'b llist" shows "lhd (x21 ## x22) = x21"
cInf_eq_non_empty: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and 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 set"   and a :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq a x"   and "\<And>y. (\<And>x. x \<in> X \<Longrightarrow> less_eq y x) \<Longrightarrow> less_eq y a" shows "Inf X = a"
vimage_null_part_null_sets: fixes f :: "'a \<Rightarrow> 'c"   and N :: "'c measure"   and A :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "null_sets N \<subseteq> null_sets (distr M N f)"   and "A \<in> sets (completion N)" shows "f -` null_part N A \<inter> space M \<in> null_sets M"
a_def: shows "a = min \<bar>l\<bar> \<bar>u\<bar>"
compatAtm_def: fixes atm :: "'atom" shows "compatAtm atm \<equiv> \<forall>s t. s \<approx> t \<longrightarrow> aval atm s \<approx> aval atm t"
is_Inner_def: fixes vertex :: "'a vertex" shows "is_Inner vertex = (\<exists>x3. vertex = \<langle>x3\<rangle>)"
xt3: shows "x \<in> set (fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))"
zeta_partial_sum_le_pos_bigo: fixes s :: "real" assumes "0 < s"   and "s \<noteq> 1" shows "sum_upto (\<lambda>n. real n powr - s) \<in> O(\<lambda>x. x powr max 0 (1 - s))"
Alexander_subbase_alt: fixes U :: "'d set"   and \<B> :: "'d set set"   and X :: "'d topology" assumes "U \<subseteq> \<Union> \<B>"   and "\<And>C. \<lbrakk>C \<subseteq> \<B>; U \<subseteq> \<Union> C\<rbrakk> \<Longrightarrow> \<exists>C'. finite C' \<and> C' \<subseteq> C \<and> U \<subseteq> \<Union> C'"   and "topology (arbitrary union_of (finite intersection_of (\<lambda>x. x \<in> \<B>) relative_to U)) = X" shows "compact_space X"
Suc_le: fixes a :: "'a" assumes "mod_type_class.to_nat a + 1 < CARD('a)" shows "a < a + (1::'a)"
a: shows "limsup (\<lambda>x. ereal (real (card (A \<inter> {k..nat (int x + l)})) / real x)) \<le> limsup (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x))"
assms_2: fixes f :: "nat \<Rightarrow> 'a" assumes "f 0 \<in> A"   and "\<forall>i. (f i, f (Suc i)) \<in> r" shows "P"
linalw: fixes a :: "nat"   and b :: "nat"   and A :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and w :: "nat \<Rightarrow> 'b" assumes "a \<le> b"   and "(\<box>A) (w |\<^sub>s a)" shows "(\<box>A) (w |\<^sub>s b)"
simps_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(\<^bold>\<langle>x1\<^bold>\<rangle> = \<^bold>\<langle>y1\<^bold>\<rangle>) = (x1 = y1)"
simps_1: fixes x :: "Vars.var" shows "isVal (Var x) = False"
cod_closed: fixes f :: "'a" assumes "Arr f" shows "Arr (B.cod f)"
SUP_UNION: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "Sup (f ` \<Union> (g ` A)) = (SUP y\<in>A. Sup (f ` g y))"
callee_invariant: fixes y :: "'b"   and s' :: "'s"   and s :: "'s"   and x :: "'a" assumes "(y, s') \<in> set_spmf (callee s x)"   and "I s"   and "x \<in> outs_\<I> \<I>" shows "I s'"
JNF_6: shows "of_real_hom.mat_hom A = P * J * iP"
simps_2: fixes \<Gamma> :: "char list set"   and v :: "char list" shows "wfe \<Gamma> (Var v) = (v \<in> \<Gamma>)"
OrR: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F, G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta>"
prod_zero_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "finite A" shows "(prod f A = (0::'a)) = (\<exists>a\<in>A. f a = (0::'a))"
simps: fixes a1 :: "bool"   and a2 :: "char list list"   and a3 :: "v sem_env"   and a4 :: "'b state"   and a5 :: "dec list"   and a6 :: "'b state \<times> (v sem_env, v) result" shows "evaluate_decs a1 a2 a3 a4 a5 a6 = ((\<exists>ck mn env s. a1 = ck \<and> a2 = mn \<and> a3 = env \<and> a4 = s \<and> a5 = [] \<and> a6 = (s, Rval (make_sem_env nsEmpty nsEmpty))) \<or> (\<exists>ck mn s1 s2 env d ds e. a1 = ck \<and> a2 = mn \<and> a3 = env \<and> a4 = s1 \<and> a5 = d # ds \<and> a6 = (s2, Rerr e) \<and> evaluate_dec ck mn env s1 d (s2, Rerr e)) \<or> (\<exists>ck mn s1 s2 s3 env d ds new_env r. a1 = ck \<and> a2 = mn \<and> a3 = env \<and> a4 = s1 \<and> a5 = d # ds \<and> a6 = (s3, combine_dec_result new_env r) \<and> evaluate_dec ck mn env s1 d (s2, Rval new_env) \<and> evaluate_decs ck mn (extend_dec_env new_env env) s2 ds (s3, r)))"
valid_in_impl: fixes I :: "interp"   and \<phi> :: "fml"   and \<psi> :: "fml" shows "valid_in I (\<phi> \<rightarrow> \<psi>) = (fml_sem I \<phi> \<subseteq> fml_sem I \<psi>)"
assms_2: shows "\<exists>n'\<ge>n. \<parallel>tid\<parallel>\<^bsub>t n'\<^esub>"
RSpan_contains_spanset_single: fixes m :: "'m" assumes "m \<in> M" shows "m \<in> RSpan [m]"
finite_cone_iff_cone_list: fixes Vs :: "'a vec set"   and Vsl :: "'a vec list" assumes "Vs \<subseteq> carrier_vec n"   and "Vs = set Vsl" shows "finite_cone Vs = cone_list Vsl"
rep_eq: fixes x :: "'d set"   and xa :: "'d \<Rightarrow> 'e" shows "fmlookup (finmap_of x xa) = (\<lambda>xb. if xb \<in> x \<and> finite x then Some (xa xb) else None)"
bit_or_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (or a b) n = (bit a n \<or> bit b n)"
a2_map''_keys: shows "Mapping.keys a2_map'' = {tp - (len - 1)..tp}"
neg_le_iff_le: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d"   and a :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus b) (uminus a) = less_eq a b"
hyps_3: shows "lbl \<in>\<^sub>A weights l2 l3"
rtlE: fixes L :: "'a tree set"   and A :: "'a \<Rightarrow> nat option" assumes "L \<in> regular_languages A" obtains TA :: "(nat, 'a) tree_automaton_rec" where "L = ta_lang TA" and "ranked_tree_automaton TA A"
remove_child_heap_is_wellformed_preserved: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "type_wf h"   and "known_ptrs h"   and "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'" shows "heap_is_wellformed h'"
term_eta_trans: fixes s :: "dB"   and t :: "dB"   and u :: "dB" assumes "s \<leftrightarrow> t"   and "t \<rightarrow>\<^sub>\<eta> u" shows "s \<leftrightarrow> u"
UINF_mem_UNIV: fixes P :: "'f \<Rightarrow> ('d, 'e) uexpr" shows "(\<Sqinter> x \<in> UNIV \<bullet> P x) = (\<Sqinter> x \<bullet> P x)"
generator_cases: fixes y :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" assumes "y \<in> {g. terminates g}" obtains x :: "('b, 'a) generator" where "y = generator x"
insert_subset: fixes x :: "'b"   and A :: "'b set"   and B :: "'b set" shows "(insert x A \<subseteq> B) = (x \<in> B \<and> A \<subseteq> B)"
normal_form_id_iff: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "(normal_form F p = p) = (\<not> punit.is_red (punit.reduced_GB F) p)"
AbsNat_plus: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and RepNat :: "'a \<Rightarrow> nat"   and AbsNat :: "nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat" assumes "class.nat minus plus zero RepNat AbsNat" shows "AbsNat (i + j) = plus (AbsNat i) (AbsNat j)"
span_mul: fixes x :: "'c"   and S :: "'c set"   and c :: "real" assumes "x \<in> span S" shows "c *\<^sub>R x \<in> span S"
pairwise_orthogonal_insert: fixes S :: "'a set"   and x :: "'a" assumes "pairwise orthogonal S"   and "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y" shows "pairwise orthogonal (insert x S)"
dX_union: fixes X :: "'a set"   and Y :: "'a set"   and d :: "'b" shows "PdXXX.dX (X \<union> Y) d = PdXXX.dX X d \<union> PdXXX.dX Y d"
periodN_fac: fixes u :: "'b list"   and w :: "'b list"   and v :: "'b list"   and p :: "nat" assumes "periodN (u \<cdot> w \<cdot> v) p"   and "w \<noteq> \<epsilon>" shows "periodN w p"
filter_compliant_stateful_ACS_accu_correct_induction: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and accu :: "('b \<times> 'b) list"   and edgesList :: "('b \<times> 'b) list"   and \<T> :: "'b stateful_policy" assumes "valid_reqs (get_ACS M)"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "set accu \<union> set edgesList \<subseteq> E"   and "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<lparr>hosts = V, flows_fix = E, flows_state = set accu\<rparr>). F \<subseteq> backflows (set accu)"   and "\<forall>a\<in>set accu. a \<notin> backflows E"   and "\<T> = \<lparr>hosts = V, flows_fix = E, flows_state = set (filter_compliant_stateful_ACS_accu \<lparr>nodes = V, edges = E\<rparr> M accu edgesList)\<rparr>" shows "\<forall>F\<in>get_offending_flows (get_ACS M) (stateful_policy_to_network_graph \<T>). F \<subseteq> backflows (filternew_flows_state \<T>)"
pCons_0_0: shows "[:0::'b:] = 0"
assms_4: shows "\<not> witnessed p a ((ps, a) # branch)"
cnj_Dirichlet_L: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and s :: "complex" assumes "dcharacter n \<chi>" shows "cnj (Dirichlet_L n \<chi> s) = Dirichlet_L n (inv_character \<chi>) (cnj s)"
R1_14_def: fixes R1_12_0T :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and n :: "nat"   and msgs :: "bool \<times> bool \<times> bool \<times> bool"   and choice :: "bool \<times> bool" shows "R1_14 R1_12_0T n msgs choice = (let (m00, m01, m10, m11) = msgs; (c0, c1) = choice in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. R1_12_0T n (S0, S1) c0 \<bind> (\<lambda>a. R1_12_0T n (S2, S3) c1 \<bind> (\<lambda>b. R1_12_0T n (S4, S5) c1 \<bind> (\<lambda>c. return_spmf (msgs, (S0, S1, S2, S3, S4, S5), a, b, c)))))))))))"
converseD: fixes a :: "'b"   and b :: "'c"   and r :: "('c \<times> 'b) set" assumes "(a, b) \<in> r\<inverse>" shows "(b, a) \<in> r"
nth_append_take: fixes i :: "nat"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" assumes "i \<le> length xs" shows "(take i xs @ y # ys) ! i = y"
ring_irreducibleE_1: fixes R :: "('c, 'd) ring_scheme"   and r :: "'c" assumes "domain R"   and "r \<in> carrier R"   and "ring_irreducible\<^bsub>R\<^esub> r" shows "r \<noteq> \<zero>\<^bsub>R\<^esub>"
list_wset_relD: fixes a :: "'a list"   and b :: "'b set"   and R :: "('a \<times> 'b) set" assumes "(a, b) \<in> \<langle>R\<rangle>list_wset_rel" shows "(set a, b) \<in> \<langle>R\<rangle>set_rel"
interPrefixEmpty: fixes s :: "'c event list"   and t :: "'c event list"   and A :: "'c event set"   and t1 :: "'c event list" assumes "s setinterleaves ((t, []), A) \<and> t1 < t" shows "\<exists>s1<s. s1 setinterleaves ((t1, []), A)"
distinct_9: fixes x2 :: "('e, 'f, 'g) Tree"   and x41 :: "'g"   and x42 :: "('e, 'f, 'g) Tree" shows "tNot x2 \<noteq> tAct x41 x42"
sum_nonzero_representation_eq: fixes basis :: "'b set"   and v :: "'b" assumes "local.independent basis"   and "v \<in> local.span basis" shows "(\<Sum>b | local.representation basis v b \<noteq> (0::'a). local.representation basis v b *s b) = v"
rep_eq: fixes x :: "('c, 'd) fmap" shows "fset (fset_of_fmap x) = set_of_map (fmlookup x)"
delete_append_prefix_trie: fixes xs :: "bool list"   and ys :: "bool list"   and t :: "trie" shows "delete (xs @ ys) (prefix_trie xs t) = (if delete ys t = Lf then Lf else prefix_trie xs (delete ys t))"
iMODb_singleton_eq_conv: fixes r :: "nat"   and m :: "nat"   and c :: "nat"   and a :: "nat" shows "([ r, mod m, c ] = {a}) = (r = a \<and> (m = 0 \<or> c = 0))"
filter_set: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "Set.filter P (set xs) = set (filter P xs)"
residue_times_zero_l: fixes k :: "nat"   and y :: "int" shows "y \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> 0 = 0"
fresh_3: fixes a :: "name"   and x1 :: "name"   and x2 :: "trm" shows "a \<sharp> \<Lambda> x1 . x2 = a \<sharp> [x1].x2"
full_Empty_iff: fixes n :: "nat" shows "full n Empty = (n = 0)"
typeII_chain_line_integral_exists_boundary': shows "\<forall>(k, \<gamma>)\<in>two_chain_vertical_boundary two_chain. line_integral_exists F {j} \<gamma>"
rep_eq: fixes x :: "'b zmultiset" shows "set_antichain (frontier x) = minimal_antichain {t. 0 < zcount x t}"
isCont_tendsto_compose: fixes l :: "'b"   and g :: "'b \<Rightarrow> 'c"   and f :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "isCont g l"   and "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
vsv: fixes a :: "V"   and b :: "V"   and c :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vint_uminus"   and "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vint_uminus" shows "b = c"
termFSwSbImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'd"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'e"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) model_scheme" shows "termFSwSbImorph h hA MOD \<equiv> termFSwImorph h hA MOD \<and> ipresSubstAll h hA MOD"
src_hcomp': fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "src (\<nu> \<star> \<mu>) = src \<mu>"
perm_length: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <~~> ys" shows "length xs = length ys"
f_ex: fixes n :: "'v" shows "\<exists>v'. n \<in> V \<longrightarrow> n \<noteq> v \<longrightarrow> adjacent n v' \<and> adjacent v v'"
QR_decomposition_iarrays_def: fixes A :: "real iarray iarray" shows "QR_decomposition_iarrays A = (let Q = divide_by_norm_iarray (Gram_Schmidt_matrix_iarrays A) in (Q, transpose_iarray Q **i A))"
rep_eq: fixes x :: "uint" shows "msb x = msb (Rep_uint x)"
Standard_star_of: fixes x :: "'b" shows "star_of x \<in> Standard"
tailcall: fixes P :: "'s \<Rightarrow> real"   and u :: "('s \<Rightarrow> real) \<Rightarrow> 's \<Rightarrow> real" assumes "sound P" shows "T u P = t P (u P)"
Q_def: fixes Q :: "value \<Rightarrow> value \<Rightarrow> bool"   and name :: "name"   and ts :: "value list"   and name' :: "name"   and us :: "value list" assumes "value_struct_rel Q" shows "Q (Vconstr name ts) (Vconstr name' us) = (name = name' \<and> list_all2 Q ts us)"
word_ao_equiv: fixes w :: "'b word"   and w' :: "'b word" shows "(w = w OR w') = (w' = w AND w')"
circuit_in_min_dep_in: fixes \<E> :: "'a set"   and X :: "'a set"   and x :: "'a" assumes "\<E> \<subseteq> \<E>"   and "\<E>.circuit_in \<E> X"   and "x \<in> X" shows "\<E>.indep_in \<E> (X - {x})"
1: fixes a :: "'a set" assumes "path_components_of X \<subseteq> {a}" shows "path_components_of Y \<subseteq> {h a}"
low_proc_non0\<^sub>1_def: fixes p :: "prog" shows "low_proc_non0\<^sub>1 p \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> the (\<sigma> ''lowIn'') \<noteq> 0 \<and> body p \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma>' ''lowOut'') = the (\<sigma> ''lowIn'') + 1"
partition_main_poly_def: shows "partition_main_poly = partition_main (\<le>) fst"
full_status: fixes f :: "'b"   and n :: "nat" shows "status full_status (f, n) = [0..<n]"
sq_norm_vec_rat_def: fixes x :: "rat vec" shows "sq_norm_vec_rat x = \<parallel>x\<parallel>\<^sup>2"
node2: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, \<sigma>'). a \<noteq> \<tau> \<and> (\<forall>d. a \<noteq> i:deliver(d)) \<longrightarrow> S (\<sigma> i) (\<sigma>' i))"
lc_monomial: fixes t :: "'a"   and c :: "'c" shows "punit.lc (monomial c t) = c"
irreducible_imp_primitive: fixes f :: "'a poly" assumes "irreducible f"   and "degree f \<noteq> 0" shows "primitive f"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<bullet>"
leq_zero_imp: fixes x :: "'a" assumes "(x, r.zero) \<in> r" shows "x = r.zero"
zero_notin_Suc_image: fixes A :: "nat set" shows "0 \<notin> Suc ` A"
concrete_security_elgamal: fixes \<A> :: "('grp \<Rightarrow> ((bool list \<times> bool list) \<times> 'b, 'grp, bool list) gpv) \<times> ('grp \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, 'grp, bool list) gpv)"   and \<eta> :: "nat" assumes "lossless \<A>" shows "advantage \<eta> (oracle \<eta>, empty) \<A> \<le> advantage \<eta> (elgamal_reduction \<eta> \<A>)"
connected_component_mono: fixes S :: "'c set"   and T :: "'c set"   and x :: "'c" assumes "S \<subseteq> T" shows "connected_component_set S x \<subseteq> connected_component_set T x"
negligible_on_intervals: fixes s :: "'b set" shows "negligible s = (\<forall>a b. negligible (s \<inter> cbox a b))"
sel_8: fixes x51 :: "Core.term"   and x52 :: "Core.term" shows "un_App1 (x51 $ x52) = x51"
\<R>\<^sub>2_mem_eq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
prems_2: shows "A3 \<in> carrier_mat (Suc m) n"
empty: fixes G :: "prog"   and A :: "'b triple set" shows "G,A|\<turnstile>{}"
dom_comp: fixes g :: "'c"   and f :: "'c" assumes "Cop.seq g f" shows "Cop.dom (g \<cdot>\<^sup>o\<^sup>p f) = Cop.dom f"