hcju/isabelleps huggingface dataset:
merge_sum_alt_def: fixes ma :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and mb :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option"   and x :: "'c + 'd"   and y :: "'c + 'd" shows "merge_sum ma mb x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> map_option Inl (ma x y) | (Inl x, Inr xa) \<Rightarrow> Map.empty xa | (Inr x, Inl xa) \<Rightarrow> Map.empty xa | (Inr x, Inr y) \<Rightarrow> map_option Inr (mb x y))"
def_homo: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "\<delta> x->including\<^sub>S\<^sub>e\<^sub>t(y) = (\<delta> x and \<upsilon> y)"
sup1: fixes A :: "'b"   and k :: "'b"   and a :: "'b"   and b :: "'b" assumes "A::'b \<equiv> sup (k::'b) (a::'b)" shows "sup (A::'b) (b::'b) \<equiv> sup (k::'b) (sup (a::'b) b)"
image_ring_carrier: fixes f :: "'d \<Rightarrow> 'c"   and R :: "('d, 'e) ring_scheme" shows "carrier (image_ring f R) = f ` carrier R"
canon_ang_arg: fixes z :: "complex" shows "\<downharpoonright>arg z\<downharpoonleft> = arg z"
finite_measure_UNION: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> events"   and "disjoint_family A" shows "(\<lambda>i. prob (A i)) sums prob (\<Union> (range A))"
lens_override_idem: fixes X :: "'a \<Longrightarrow> 'b"   and S :: "'b" assumes "vwb_lens X" shows "S \<triangleleft>\<^bsub>X\<^esub> S = S"
measure_negligible_symdiff: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> lmeasurable"   and "negligible (S - T \<union> (T - S))" shows "Sigma_Algebra.measure lebesgue T = Sigma_Algebra.measure lebesgue S"
dims1_def: shows "ps_P.dims1 = nths ps2_P.dims0 ps2_P.vars1'"
MGFn_FVar: fixes G :: "prog"   and A :: "state triple set"   and n :: "nat"   and statDeclC :: "qtname"   and e :: "expr"   and accC :: "qtname"   and stat :: "bool"   and fn :: "vname" assumes "G,A\<turnstile>{=:n} In1r (Init statDeclC)\<succ> {G\<rightarrow>}"   and "G,A\<turnstile>{=:n} In1l e\<succ> {G\<rightarrow>}"   and "wf_prog G" shows "G,A\<turnstile>{=:n} In2 ({accC,statDeclC,stat}e..fn)\<succ> {G\<rightarrow>}"
Min_disc_is_scc_root: fixes v :: "'v"   and scc :: "'v set" assumes "v \<in> scc"   and "v \<in> dom (discovered s)"   and "is_scc E scc"   and "\<delta> s v = Min (\<delta> s ` (scc \<inter> dom (discovered s)))" shows "scc_root s v scc"
simps_1: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and a :: "'d node_action"   and i :: "nat"   and p1 :: "net_tree"   and p2 :: "net_tree"   and s1 :: "'c net_state"   and s2 :: "'c net_state"   and s1' :: "'c net_state"   and s2' :: "'c net_state" shows "net_ip_action np a i (p1 \<parallel> p2) (SubnetS s1 s2) (SubnetS s1' s2') = ((i \<in> net_ips s1 \<longrightarrow> (s1, a, s1') \<in> automaton.trans (pnet np p1) \<and> s2' = s2 \<and> net_ip_action np a i p1 s1 s1') \<and> (i \<in> net_ips s2 \<longrightarrow> (s2, a, s2') \<in> automaton.trans (pnet np p2) \<and> s1' = s1 \<and> net_ip_action np a i p2 s2 s2'))"
E_def: shows "E = (1 + real n) * real_of_int qmax * real n * exp (real n) / real n"
multiset_eqI: fixes A :: "'a multiset"   and B :: "'a multiset" assumes "\<And>x. count A x = count B x" shows "A = B"
snd_add: fixes x :: "'b \<times> 'a"   and y :: "'b \<times> 'a" shows "snd (x + y) = snd x + snd y"
OperationCallTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and op :: "(any_unop + boolean_unop + numeric_unop + string_unop + collection_unop) + (super_binop + boolean_binop + numeric_binop + string_binop + collection_binop) + (string_ternop + collection_ternop) + String.literal"   and params :: "'a expr list"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E OperationCall src k op params : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<pi> :: "'a OCL_Types.type list" where "\<Gamma> \<turnstile>\<^sub>E src : \<tau>'" and "\<Gamma> \<turnstile>\<^sub>L params : \<pi>" and "op_type op k \<tau>' \<pi> \<tau>"
pred_pmf_bind: fixes P :: "'c \<Rightarrow> bool"   and p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf" shows "pred_pmf P (p \<bind> f) = pred_pmf (pred_pmf P \<circ> f) p"
is_processT3_SR: fixes s :: "'a event list"   and t :: "'a event list"   and X :: "'a event set"   and P :: "'a process" assumes "(s @ t, X) \<in> F P" shows "(s, {}) \<in> F P"
assms_2: shows "N \<lhd> G\<lparr>carrier := H\<rparr>"
step: fixes S :: "'a set" assumes "S \<subseteq> V"   and "P S" shows "P (W \<union> S \<union> directly_attracted p S)"
conga_preserves_in_angle: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and I :: "'p"   and I' :: "'p" assumes "A B C CongA A' B' C'"   and "A B I CongA A' B' I'"   and "I InAngle A B C"   and "A' B' OS I' C'" shows "I' InAngle A' B' C'"
f1: shows "Sup {norm (f x) / norm x |x. x \<noteq> (0::'a)} = Inf {K. \<forall>x. x \<noteq> (0::'a) \<longrightarrow> norm (f x) \<le> norm x * K}"
filter_min_aux_subset: fixes xs :: "'a list"   and ys :: "'a list" shows "set (local.filter_min_aux xs ys) \<subseteq> set xs \<union> set ys"
card_Gram_Schmidt: fixes xs :: "('a, 'b) vec list" assumes "distinct xs" shows "card (set (Gram_Schmidt xs)) \<le> card (set xs)"
eq_\<I>_restrict_converter_cong: fixes \<I> :: "('i, 'j) \<I>"   and \<I>' :: "('k, 'l) \<I>"   and cnv :: "('i, 'j, 'k, 'l) converter"   and cnv' :: "('i, 'j, 'k, 'l) converter"   and A :: "'i set" assumes "\<I>,\<I>' \<turnstile>\<^sub>C cnv \<sim> cnv'"   and "A \<subseteq> outs_\<I> \<I>" shows "restrict_converter A \<I>' cnv = restrict_converter A \<I>' cnv'"
pol_coeff_def: fixes S :: "('e, 'f) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "pol_coeff S c = (\<forall>j\<le>fst c. snd c j \<in> carrier S)"
u1: shows "u\<^sub>1 \<equiv>\<^sub>U r1"
hom_GB_bound_is_GB_cofactor_bound: fixes X :: "'a set"   and F :: "(('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and b :: "nat" assumes "finite X"   and "F \<subseteq> P[X]"   and "extended_ord.is_hom_GB_bound (homogenize None ` extend_indets ` F) b" shows "is_GB_cofactor_bound F b"
Fun_param_in_subterms: fixes x :: "('a, 'b) Term.term"   and X :: "('a, 'b) Term.term list"   and f :: "'a" assumes "x \<in> set X" shows "x \<sqsubseteq> Fun f X"
quasisym_not_greater: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b" assumes "comparator cmp"   and "cmp a b \<noteq> Greater"   and "cmp b a \<noteq> Greater" shows "cmp a b = Equiv"
hyper_isin_def: fixes a :: "'b"   and A :: "'b set option" shows "a \<in>\<in> A \<equiv> case A of None \<Rightarrow> True | \<lfloor>A\<rfloor> \<Rightarrow> a \<in> A"
isSurjAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'sort, 'f, 'g, 'h, 'c, 'i) Recursion.model_scheme" shows "isSurjAbs hA MOD \<equiv> \<forall>us s A. gWlsAbs MOD (us, s) A \<longrightarrow> (\<exists>A'. wlsAbs (us, s) A' \<and> hA A' = A)"
vfsequence_vcons_vrange: fixes x :: "V" shows "\<R>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = vinsert x (\<R>\<^sub>\<circ> xs)"
permutes_image: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S" shows "p ` S = S"
X: shows "X = map (\<lambda>(x, y). Fraction_Field.Fract x y) xs"
scaleRat_uminus: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "scaleRat (- 1) x = uminus x"
dflt_cmp_id: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "(dflt_cmp less_eq less, dflt_cmp less_eq less) \<in> Id \<rightarrow> Id \<rightarrow> comp_res_rel"
assms_1: shows "a \<in> {real_of_float al..real_of_float au}"
UnCI: fixes c :: "'a"   and B :: "'a set"   and A :: "'a set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
excd_impl_exicd: fixes i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n" assumes "\<exists>k. i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k" shows "\<exists>k. i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
assms_4: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "a <\<^sub>\<int> x"
finite_field_factorization_main_def: fixes p :: "int"   and f_ops :: "'a arith_ops_record"   and f :: "int poly" shows "finite_field_factorization_main p f_ops f \<equiv> let (c', fs') = finite_field_factorization_i p f_ops (of_int_poly_i f_ops (poly_mod.Mp p f)) in (arith_ops_record.to_int f_ops c', map (to_int_poly_i f_ops) fs')"
ins_size_1: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Bal t'" shows "size t' = size t + 1"
permute_zero: fixes permute :: "perm \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e" assumes "class.pt permute" shows "permute 0 x = x"
sel: fixes x :: "((common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>h" shows "the_Transaction\<^sub>h (Transaction\<^sub>h x) = x"
arr_Par_arr_RelD_2: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Par \<alpha> T" shows "vsv (T\<lparr>ArrVal\<rparr>)"
assms_4: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) Tree\<^sub>\<alpha>"   and f :: "'d"   and act :: "'c" assumes "P t\<^sub>\<alpha>" shows "P (Act\<^sub>\<alpha> f act t\<^sub>\<alpha>)"
prod_mset_multiplicity: fixes M :: "'a multiset" shows "\<Prod>\<^sub># M = (\<Prod>x\<in>set_mset M. x ^ count M x)"
bisim1AAss4: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and i :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (unit, xs) \<leftrightarrow> ([], xs, Suc (length (compE2 a) + length (compE2 i) + length (compE2 e)), None)"
insort_is_Cons: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'d list"   and f :: "'d \<Rightarrow> 'c"   and a :: "'d" assumes "class.linorder less_eq less"   and "\<forall>x\<in>set xs. less_eq (f a) (f x)" shows "linorder.insort_key less_eq f a xs = a # xs"
in_VEndSet: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T" shows "T \<in> {S. VectorSpaceEnd smult V S}"
Gamma_times_zeta_has_integral_real: fixes z :: "real" assumes "1 < z" shows "((\<lambda>x. x powr (z - 1) / (exp x - 1)) has_integral Gamma z * Re (zeta (complex_of_real z))) {0<..}"
do_widening_spec_def: fixes i :: "nat" shows "do_widening_spec i = SPEC (\<lambda>b. True)"
euler_incr_slp_def: shows "euler_incr_slp = slp_of_fas (euler_incr_fas' D)"
"2_2": fixes y :: "'a" assumes "y \<in> set xs" shows "(x, y) \<in> R"
Disj_eqvt: fixes xset :: "('a, 'b, 'c) formula set['a]"   and p :: "perm" assumes "finite (supp xset)" shows "p \<bullet> Disj xset = Disj (p \<bullet> xset)"
not_less: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<sqsubset>\<^sub>a y) = (y \<sqsubseteq>\<^sub>a x)"
id_is_mor_pr_rngs: shows "morphism_presheaves_of_rings S is_open local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf (\<lambda>U. identity (local.im_sheaf U))"
y_eq: shows "s2 \<cdot> s' \<cdot> fst y = s2 \<cdot> snd y \<cdot> r'"
set_insert: fixes x :: "'k"   and A :: "'k set" assumes "x \<in> A" obtains B :: "'k set" where "A = insert x B" and "x \<notin> B"
Heap_ordE: fixes x :: "'a Heap"   and y :: "'a Heap"   and h :: "heap" assumes "Heap_ord x y" obtains "execute x h = None"   | "execute x h = execute y h"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "\<exists>y\<in>carrier (G\<lparr>carrier := S\<rparr>). y \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x = \<one>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub>"
power_int_add: fixes x :: "'a"   and m :: "int"   and n :: "int" assumes "x \<noteq> (0::'a) \<or> m + n \<noteq> 0" shows "x powi (m + n) = x powi m * x powi n"
comp_inv_arr': fixes f :: "'c \<times> 'c" assumes "C.VV.iso f" shows "C.VV.comp (C.VV.inv f) f = C.VV.dom f"
icd_path_shift: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and l :: "nat"   and k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "IFC prob"   and "l \<le> k"   and "IFC_def.is_path prob \<pi>" shows "IFC_def.is_icdi prob i \<pi> k = IFC_def.is_icdi prob (i - l) (\<pi> \<guillemotleft> l) (k - l)"
vec_vec_Tensor_def: shows "vec_vec_Tensor \<equiv> rec_list (\<lambda>ys. []) (\<lambda>x xs xsa ys. local.times x ys @ xsa ys)"
vdomain_vrange_is_vempty: shows "(\<D>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = []\<^sub>\<circ>) = (\<R>\<^sub>\<circ> (\<BB>\<lparr>Cod\<rparr>) = []\<^sub>\<circ>)"
vec_lambda_eta: fixes g :: "('a, 'b) Finite_Cartesian_Product.vec" shows "(\<chi>i. g $h i) = g"
bt_augment_v_commute_2: fixes t :: "('d, 'c) bintree"   and rs :: "('d, 'c) bintree list"   and q :: "('c, 'd) pq" shows "set |bt_augment t (bts_augment rs q)| = set |bts_augment rs (bt_augment t q)|"
Carmichael_number_odd: fixes n :: "nat" assumes "Carmichael_number n" shows "odd n"
orecvmsg_simps_6: fixes P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'd"   and d :: "nat" shows "orecvmsg P \<sigma> (deliver d) = True"
unique_NA: fixes A :: "agent"   and NA :: "msg"   and B :: "agent"   and evs :: "event list"   and C :: "agent" assumes "Crypt (shrK A) \<lbrace>NA, Agent A, Agent B\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>NA, Agent A, Agent C\<rbrace> \<in> parts (knows Spy evs)"   and "evs \<in> otway"   and "A \<notin> bad" shows "B = C"
Dg_def: shows "Dg \<equiv> \<lambda>n s. if n < p then (- 1) ^ n * (b - s) ^ (p - 1 - n) / fact (p - 1 - n) else 0"
klift_eta_inv1: fixes f :: "'c \<Rightarrow> 'd set" shows "bd\<^sup>-\<^sub>\<F> (bd\<^sub>\<F> f) = f"
jump_jump_poly: fixes q :: "real poly"   and p :: "real poly"   and a :: "real" shows "jump (\<lambda>x. poly q x / poly p x) a = jump_poly q p a"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright>"
card_of_Plus2: fixes B :: "'c set"   and A :: "'d set" shows "|B| \<le>o |A <+> B|"
filterlim_filtermap: fixes f :: "'a \<Rightarrow> 'b"   and F1 :: "'b filter"   and g :: "'c \<Rightarrow> 'a"   and F2 :: "'c filter" shows "filterlim f F1 (filtermap g F2) = (LIM x F2. f (g x) :> F1)"
Cd_single: fixes x :: "'x"   and X :: "'x set"   and d :: "'d" assumes "x \<in> X"   and "inj_on Xd X"   and "x \<in> Field (Pd d)" shows "x \<in> Cd d X"
\<Delta>\<Phi>_pass1: fixes hs :: "'b heap list" assumes "hs \<noteq> []"   and "no_Emptys hs" shows "\<Phi>_hps (pass\<^sub>1 hs) - \<Phi>_hps hs \<le> 2 * log 2 (real (size_hps hs)) - real (length hs) + 2"
F: shows "P \<turnstile> C' sees F,Static:vT' in D"
in_ideal_gen_by_prod: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and \<aa> :: "'c set"   and \<bb> :: "'c set"   and a :: "'c"   and b :: "'c" assumes "comm_ring R addition multiplication zero unit"   and "ideal \<aa> R addition multiplication zero unit"   and "ideal \<bb> R addition multiplication zero unit"   and "a \<in> R"   and "b \<in> comm_ring.ideal_gen_by_prod R addition multiplication zero \<aa> \<bb>" shows "multiplication a b \<in> comm_ring.ideal_gen_by_prod R addition multiplication zero \<aa> \<bb>"
cong: fixes solvable\<^sub>0 :: "'c \<Rightarrow> bool"   and trivial :: "'c \<Rightarrow> bool"   and subst\<^sub>0 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "ATOM_EQ.lift_eq_qe solvable\<^sub>0 trivial subst\<^sub>0 \<equiv> ATOM_EQ.lift_eq_qe solvable\<^sub>0 trivial subst\<^sub>0"
closest_pair_bf_c1: fixes ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length ps"   and "(c\<^sub>0, c\<^sub>1) = closest_pair_bf ps" shows "c\<^sub>1 \<in> set ps"
exec_step_ind_Putstatic_Called: fixes D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and D :: "char list"   and F :: "char list"   and C :: "char list"   and sh :: "sheap"   and sfs :: "sfields"   and i :: "init_state"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and Cs :: "char list list"   and frs :: "frame list" assumes "(D', b, t) = field P D F"   and "P \<turnstile> C has F,Static:t in D"   and "the (sh D') = (sfs, i)" shows "exec_step_ind (StepI (Putstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc (Called Cs) frs sh (None, h, (tl stk, loc, C\<^sub>0, M\<^sub>0, Suc pc, No_ics) # frs, sh(D' \<mapsto> (sfs(F \<mapsto> hd stk), i)))"
has_integral_integrable: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and s :: "'b set" assumes "(f has_integral i) s" shows "f integrable_on s"
cobounded2: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<inter># b \<subseteq># b"
butlast_snoc: fixes xs :: "'b list"   and x :: "'b" shows "butlast (xs \<cdot> [x]) = xs"
SUP_le_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and u :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (f ` A)) u = (\<forall>i\<in>A. less_eq (f i) u)"
asIMOD_igWls: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'a) semDom"   and s :: "'sort"   and X :: "('varSort \<Rightarrow> 'var \<Rightarrow> 'a) \<Rightarrow> 'a" shows "igWls (asIMOD SEM) s X = (\<forall>val. (sWlsVal SEM val \<or> X val = undefined) \<and> (sWlsVal SEM val \<longrightarrow> sWls SEM s (X val)))"
is_nth_power_0: fixes n :: "nat" assumes "0 < n" shows "is_nth_power n (0::'a)"
abs_eq: shows "(\<cdot>) = (\<lambda>f g. (f\<^sub>\<bullet> \<circ>\<^sub>K g\<^sub>\<bullet>)\<^sup>\<bullet>)"
fref_param0I: fixes c :: "'e"   and a :: "'f"   and R :: "('e \<times> 'f) set" assumes "(c, a) \<in> R" shows "(uncurry0 c, uncurry0 a) \<in> unit_rel \<rightarrow>\<^sub>f R"
span_li_not_depend_2: fixes S :: "'e set"   and N :: "'e set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule K N W" shows "LinearCombinations.module.lin_dep K (W.vs N) S = W.module.lin_dep S"
select_convs_1: fixes pid :: "pname"   and tid :: "tname"   and more :: "'a" shows "pid \<lparr>pid = pid, tid = tid, \<dots> = more\<rparr> = pid"
restrict_relpE: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" assumes "(R \<upharpoonleft> P \<otimes> Q) x y" obtains "R x y" and "P x" and "Q y"
ANeg_1: fixes \<phi> :: "\<o>" shows "[\<^bold>\<not>\<^bold>\<A>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<phi> in dw]"
fr_Send6: fixes n :: "nat"   and nReturn :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and nC :: "nat \<Rightarrow> Config"   and nStore :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and nGet :: "nat \<Rightarrow> nat \<Rightarrow> nat list"   and t :: "nat"   and k :: "nat" assumes "\<forall>i<n. FlexRayController (nReturn i) recv (nC i) (nStore i) (nSend i) (nGet i)"   and "DisjointSchedules n nC"   and "IdenticCycleLength n nC"   and "t mod cycleLength (nC k) mem schedule (nC k)"   and "k < n"   and "nReturn k t \<noteq> []" shows "\<exists>k<n. nSend k t \<noteq> []"
interval_tendsto_0: shows "(\<lambda>i. 0) \<longlonglongrightarrow>\<^sub>i (0::'a)"
intros_5: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com"   and t :: "char list \<Rightarrow> int"   and c\<^sub>1 :: "com" assumes "\<not> bval b s"   and "(c\<^sub>2, s) \<Rightarrow> t" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
one_star_app_congL: fixes t1 :: "lam"   and t2 :: "lam"   and s :: "lam" assumes " t1 \<longrightarrow>\<^sub>1\<^sup>* t2" shows " App t1 s \<longrightarrow>\<^sub>1\<^sup>* App t2 s"
inv_all_imp_inv_from: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat"   and k :: "nat" assumes "local.inv_all p A" shows "local.inv_from p A k"
coprime_crossproduct_nat: fixes a :: "nat"   and d :: "nat"   and b :: "nat"   and c :: "nat" assumes "coprime a d"   and "coprime b c" shows "(a * c = b * d) = (a = b \<and> c = d)"
eq_fst_iff: fixes a :: "'e"   and p :: "'e \<times> 'f" shows "(a = fst p) = (\<exists>b. p = (a, b))"
op_list_append_elem_def: fixes s :: "'a list"   and x :: "'a" shows "op_list_append_elem s x \<equiv> s @ [x]"
Restrp_transitive: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transitive A less_eq" shows "transitive UNIV (less_eq \<restriction> A)"
process_rule_2_futures: fixes ys :: "'c list"   and ys' :: "'c list"   and Y :: "'c set"   and P :: "'c process"   and xs :: "'c list" assumes "(ys @ ys', Y) \<in> futures P xs" shows "(ys, {}) \<in> futures P xs"
minus_add_uminus_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "A - B = A + - B"
False: shows "\<not> (\<forall>b\<in>set bs. fst b = 0)"
step: fixes i :: "Integer" assumes "le\<cdot>1\<cdot>i = TT"   and "P (i - 1)" shows "P i"
fres_def: fixes z :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "prod.fres z y = Sup_prod {x. less_eq_prod (times_prod' x y) z}"
split_half_not_empty: fixes xs :: "('b btree \<times> 'b) list" assumes "1 \<le> length xs" shows "\<exists>ls sub sep rs. split_half xs = (ls, (sub, sep) # rs)"
dblock: shows "{dblock s' q} \<subseteq> {dblock s q}"
smallomega_iff_smallo: fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" shows "(g \<in> \<omega>[F](f)) = (f \<in> o[F](g))"
pred_map: fixes Q :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b list" shows "list_all Q (map f x) = list_all (Q \<circ> f) x"
DERIV_deriv_iff_field_differentiable: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "(f has_field_derivative deriv f x) (at x) = (f field_differentiable at x)"
dom_m_def: fixes N :: "('a, 'b) fmap" shows "dom_m N = mset_fset (fmdom N)"
f2r_def: fixes f :: "'d \<Rightarrow> 'e set" shows "\<R> f = {(x, y). s2p (f x) y}"
\<UU>\<bb>\<xx>\<OO>\<pp>_to_\<II>\<nn>\<ll>: fixes opubx :: "'opubx"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and z :: "('dyn, 'ubx1, 'ubx2) unboxed" assumes "\<UU>\<bb>\<xx>\<OO>\<pp> opubx \<Sigma> = Some z" shows "\<II>\<nn>\<ll> (\<DD>\<ee>\<II>\<nn>\<ll> (\<BB>\<oo>\<xx> opubx)) (map Subx.norm_unboxed \<Sigma>) = Some (\<BB>\<oo>\<xx> opubx)"
subgroup_generated_is_monoid: fixes H :: "'a set" shows "Group_Theory.monoid (subgroup_generated H) (\<cdot>) \<one>"
lower_comp: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme"   and x :: "'d" assumes "galois_connection G"   and "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> (\<pi>\<^sub>*\<^bsub>G\<^esub> (\<pi>\<^sup>*\<^bsub>G\<^esub> x)) = \<pi>\<^sup>*\<^bsub>G\<^esub> x"
intros_3: fixes i :: "nat"   and l :: "color list" assumes "2 \<le> i"   and "i < length l"   and "l ! (i - 2) = R"   and "l ! (i - 1) = R"   and "l ! i = R" shows "valid_point l i"
in_hom_def: fixes f :: "'a2"   and a :: "'a2"   and b :: "'a2" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>2 b\<guillemotright> \<equiv> a \<in> C2.domains f \<and> b \<in> C2.codomains f"