hcju/isabelleps huggingface dataset:
split_rule_abs_split: fixes ts :: "('b \<times> 'a) list"   and c :: "nat"   and tsi :: "('a btnode ref option \<times> 'a) list"   and a :: "('a btnode ref option \<times> 'a) array"   and n :: "nat"   and A :: "'b \<Rightarrow> 'a btnode ref option \<Rightarrow> assn"   and p :: "'a" assumes "sorted_less (separators ts)" shows "<is_pfa c tsi (a, n) * list_assn (A \<times>\<^sub>a id_assn) ts tsi> split_fun (a, n) p <\<lambda>r. is_pfa c tsi (a, n) * list_assn (A \<times>\<^sub>a id_assn) ts tsi * \<up> (split_relation ts (abs_split ts p) r)>\<^sub>t"
assoc_end_min_less_eq_max: fixes assoc :: "String.literal"   and ends :: "String.literal \<rightharpoonup>\<^sub>f (classes1 \<times> nat \<times> enat \<times> bool \<times> bool)"   and role :: "String.literal"   and "end" :: "classes1 \<times> nat \<times> enat \<times> bool \<times> bool" assumes "assoc |\<in>| fmdom assocs"   and "fmlookup assocs assoc = Some ends"   and "role |\<in>| fmdom ends"   and "fmlookup ends role = Some end" shows "enat (assoc_end_min end) \<le> assoc_end_max end"
rep_eq: fixes x :: "('b, 'c, 'd, 'e) formula" shows "Rep_formula (FL_Formula.Not x) = Not\<^sub>\<alpha> (Rep_formula x)"
fmapU_cast_eq: fixes A :: "udom defl" shows "fmapU\<cdot>(cast\<cdot>A) = PRJ(udom\<cdot>'a) oo cast\<cdot>(TC('a)\<cdot>A) oo EMB(udom\<cdot>'a)"
matchC_C_sym_trans: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c1 :: "('test, 'atom, 'choice) cmd"   and c :: "('test, 'atom, 'choice) cmd"   and c2 :: "('test, 'atom, 'choice) cmd" assumes "matchC_C theta c1 c"   and "matchC_C theta c c2"   and "sym theta"   and "trans theta" shows "matchC_C theta c1 c2"
eq: fixes X :: "(real \<times> real \<times> real) list" shows "(\<exists>(a, b, c)\<in>set X. eq_fun a b c eq les leq neq) = (\<exists>x\<in>F ` \<Union> (roots ` set X). x)"
res: shows "list_all2 (rel_prod (=) poly_rel) Res res"
viewL_def: fixes t :: "('c, 'd) FingerTreeStruc" shows "viewL t = (case viewLn t of None \<Rightarrow> None | Some (a, t2) \<Rightarrow> Some (n_unwrap a, t2))"
rel_T_witness1: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "((('z, 'n, 'q, 't, 'w, 'aa) T \<times> (('ab, 'p, 's, 'v, 'y, 'aa) T \<Rightarrow> ('z \<times> 'ab, 'o, 'r, 'u, 'x, 'aa) T)) \<times> ((('z, 'n, 'q, 't, 'w, 'aa) T \<times> (('ab, 'p, 's, 'v, 'y, 'aa) T \<Rightarrow> ('z \<times> 'ab, 'o, 'r, 'u, 'x, 'aa) T)) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) T) \<times> ('ab, 'p, 's, 'v, 'y, 'aa) T \<times> 'z \<times> ('z \<times> 'ab) \<times> 'ab \<times> 'aa) itself"   and L1 :: "'z \<Rightarrow> 'ab \<Rightarrow> bool"   and x :: "('z, 'n, 'q, 't, 'w, 'aa) T"   and y :: "('ab, 'p, 's, 'v, 'y, 'aa) T" assumes "rel_G_neg_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G"   and "rel_T L1 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y" shows "rel_T (\<lambda>x (x', y). x' = x \<and> L1 x y) Co1 Co2 Contra1 Contra2 x (rel_T_witness L1 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' x y)"
cINF_greatest: fixes A :: "'a set"   and m :: "'b multiset"   and f :: "'a \<Rightarrow> 'b multiset" assumes "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># f x" shows "m \<subseteq># \<Sqinter> (f ` A)"
SUP_least: fixes Inf :: "'k set \<Rightarrow> 'k"   and Sup :: "'k set \<Rightarrow> 'k"   and inf :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and sup :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bot :: "'k"   and top :: "'k"   and A :: "'l set"   and f :: "'l \<Rightarrow> 'k"   and u :: "'k" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
dsem_to_ssem: fixes \<theta> :: "('b, 'c) trm"   and I :: "('b, 'd, 'c) interp"   and \<nu> :: "(real, 'c) vec \<times> (real, 'c) vec" assumes "dfree \<theta>" shows "dterm_sem I \<theta> \<nu> = sterm_sem I \<theta> (fst \<nu>)"
assms_1: shows "theta \<subseteq> ZOretrT theta"
eq_LConsD: fixes xs :: "'c llist"   and y :: "'c"   and ys :: "'c llist" assumes "xs = LCons y ys" shows "xs \<noteq> LNil \<and> lhd xs = y \<and> ltl xs = ys"
has_densityI: fixes f :: "'a \<Rightarrow> ennreal"   and N :: "'a measure"   and M :: "'a measure" assumes "f \<in> borel_measurable N"   and "M = density N f"   and "space N \<noteq> {}" shows "has_density M N f"
wfactors_unique: fixes as :: "'a list"   and a :: "'a"   and as' :: "'a list" assumes "wfactors (Ring_Divisibility.mult_of R) as a"   and "wfactors (Ring_Divisibility.mult_of R) as' a"   and "a \<in> carrier (Ring_Divisibility.mult_of R)"   and "set as \<subseteq> carrier (Ring_Divisibility.mult_of R)"   and "set as' \<subseteq> carrier (Ring_Divisibility.mult_of R)" shows "essentially_equal (Ring_Divisibility.mult_of R) as as'"
arc_linorder_matrix: fixes f :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_stone_relation_algebra.arc f = (\<exists>e. f e = top \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot))"
contf: fixes x :: "real" assumes "a < ereal x"   and "ereal x < b" shows "isCont f (g x)"
get_owner_document_is_l_get_owner_document: shows "l_get_owner_document Shadow_DOM.get_owner_document"
birkhoff_sum_1_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "qmpt M T" shows "qmpt.birkhoff_sum T f 1 x = f x"
while_rule: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and Q :: "'a \<Rightarrow> bool"   and r :: "('a \<times> 'a) set" assumes "P s"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. \<lbrakk>P s; \<not> b s\<rbrakk> \<Longrightarrow> Q s"   and "wf r"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r" shows "Q (while b c s)"
natToVertexList_last: fixes f :: "face"   and v :: "nat"   and es :: "nat list" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "incrIndexList es |es| |vertices f|" shows "last (natToVertexList v f es) = Some (last (verticesFrom f v))"
prod_filter_mono: fixes F :: "'a filter"   and F' :: "'a filter"   and G :: "'b filter"   and G' :: "'b filter" assumes "F \<le> F'"   and "G \<le> G'" shows "F \<times>\<^sub>F G \<le> F' \<times>\<^sub>F G'"
finite_HPT: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status" assumes "TS \<in> HPT ST" shows "finite TS"
Copied_8: shows "Nom a' at b in (ps, a) # branch"
of_int_0_le_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (ring_1.of_int one plus zero minus z) = (0 \<le> z)"
idiff_enat_0_right: fixes n :: "enat" shows "n - enat 0 = n"
mat_of_row_def: fixes y :: "'a vec" shows "mat_of_row y = mat 1 (dim_vec y) (\<lambda>ij. y $ snd ij)"
divide_minus1: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide x (uminus one) = uminus x"
fls_unit_base_subdegree_power_1: fixes x :: "'a"   and f :: "'a fls"   and n :: "nat" assumes "x * fls_nth f (fls_subdegree f) = (1::'a)" shows "fls_subdegree (f ^ n) = int n * fls_subdegree f"
lunit_hcomp_4: fixes f :: "'d"   and g :: "'d" assumes "D.ide f"   and "D.ide g"   and "src\<^sub>D f = trg\<^sub>D g" shows "\<l>\<^sub>D\<^sup>-\<^sup>1[f \<star>\<^sub>D g] = \<a>\<^sub>D[trg\<^sub>D f, f, g] \<cdot>\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[f] \<star>\<^sub>D g)"
incseq_le: fixes X :: "nat \<Rightarrow> 'a"   and L :: "'a"   and n :: "nat" assumes "incseq X"   and "X \<longlonglongrightarrow> L" shows "X n \<le> L"
id_eq_prop_prop_4_b: fixes v :: "i"   and F :: "\<Pi>\<^sub>3" shows "[F \<^bold>= F in v]"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "predicate \<Rightarrow> 'b list \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and atom :: "'b atom" shows "P (case atom of predAtm x xa \<Rightarrow> f1 x xa | Eq x xa \<Rightarrow> f2 x xa) = ((\<forall>x11 x12. atom = predAtm x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. atom = Eq x21 x22 \<longrightarrow> P (f2 x21 x22)))"
program_typedef_6: fixes F :: "'c program" shows "Acts F \<equiv> case Rep_Program F of (init, acts, allowed) \<Rightarrow> acts"
list_order_insert_none: fixes ops :: "('a \<times> 'a option) list"   and oid :: "'a"   and x :: "'a" assumes "insert_ops (ops @ [(oid, None)])"   and "x \<in> set (interp_ins ops)" shows "list_order (ops @ [(oid, None)]) oid x"
ring_simprules_2: fixes R :: "('a, 'b) ring_scheme" assumes "ring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
full_quicksort_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs :: "'c list" shows "full_quicksort R h xs \<equiv> if xs = [] then RETURN xs else WB_Sort.quicksort R h (0, length xs - 1, xs)"
Ord_odiff: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "Ord (odiff \<beta> \<alpha>)"
rep_eq: fixes x :: "nat \<Rightarrow> bool" shows "Rep_uint (set_bits x) = set_bits x"
d2: shows "\<forall>D. fine d2 (a, b) D \<longrightarrow> \<bar>rsum D f - k2\<bar> < \<bar>k1 - k2\<bar> / 2"
simps: fixes a1 :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and a2 :: "'f set"   and a3 :: "'f \<Rightarrow> 'g" shows "valid_abstraction a1 a2 a3 = (\<exists>A k X. a1 = A \<and> a2 = X \<and> a3 = k \<and> (\<forall>(x, m)\<in>clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>) \<and> collect_clkvt (trans_of A) \<subseteq> X \<and> finite X)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.cod f1, f2)) (\<lambda>f2. F (A1.cod f1', f2)) (\<lambda>f2. F (f1', f2))"
in_hom_char: fixes f :: "('o, 'a) arr"   and a :: "('o, 'a) arr"   and b :: "('o, 'a) arr" shows "in_hom f a b = (arr f \<and> ide a \<and> ide b \<and> Dom f = Dom a \<and> Cod f = Dom b)"
parallel_callee_parametric: fixes A :: "'h \<Rightarrow> 'l \<Rightarrow> bool"   and B :: "'i \<Rightarrow> 'm \<Rightarrow> bool"   and C :: "'j \<Rightarrow> 'n \<Rightarrow> bool"   and R :: "'k \<Rightarrow> 'o \<Rightarrow> bool"   and A' :: "'p \<Rightarrow> 'r \<Rightarrow> bool"   and B' :: "'q \<Rightarrow> 's \<Rightarrow> bool" shows "rel_fun (rel_converter A B C R) (rel_fun (rel_converter A' B' C R) (rel_converter (rel_sum A A') (rel_sum B B') C R)) (|\<^sub>\<propto>) (|\<^sub>\<propto>)"
qmpt_I: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "sigma_finite_measure M"   and "T \<in> M \<rightarrow>\<^sub>M M"   and "\<And>A. A \<in> sets M \<Longrightarrow> (T -` A \<inter> space M \<in> null_sets M) = (A \<in> null_sets M)" shows "qmpt M T"
assms: fixes x :: "'a" shows "mf x \<equiv> ASSERT (P x) \<bind> (\<lambda>_. RETURN (f x))"
abs_eq: fixes x :: "1 word \<times> 'a word \<times> 'b word" shows "IEEE.exponent (Abs_float x) = (case x of (uu_, e, uua_) \<Rightarrow> unat e)"
discrim_def: fixes X :: "msg" shows "discrim X = the_elem (\<Union>U\<in>Rep_Msg X. {freediscrim U})"
hyps_2: fixes l :: "char list \<Rightarrow> val option" assumes "P \<turnstile> h \<surd>"   and "P \<turnstile> E \<surd>"   and "\<D> e \<lfloor>dom l\<rfloor>"   and "\<not> final e" shows "\<exists>e' s'. P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"
simps_3: fixes carrier :: "'c set"   and pop :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mop :: "'c \<Rightarrow> 'c"   and zero :: "'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, pop = pop, mop = mop, zero = zero, \<dots> = more\<rparr>\<^esub> = zero"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc (x ::= a) Q = True"
F_def: shows "F \<equiv> rec_nat [M] (\<lambda>n NL. Eps (\<Phi> NL) # NL)"
kernel_to_trivial_group: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c" assumes "group_hom G H h"   and "trivial_group H" shows "kernel G H h = carrier G"
eq_sym: fixes eq :: "'b \<rightarrow> 'b \<rightarrow> tr"   and x :: "'b"   and y :: "'b" assumes "class.Eq_sym eq" shows "eq\<cdot>x\<cdot>y = eq\<cdot>y\<cdot>x"
unfold_lesub_prod: fixes x :: "'c \<times> 'd"   and r\<^sub>A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r\<^sub>B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'c \<times> 'd" shows "(x <=(r\<^sub>A,r\<^sub>B) y) = Product.le r\<^sub>A r\<^sub>B x y"
simps_18: shows "op_sem Snd = snd \<circ> extract_pair"
P1_ideal_ddh1_\<sigma>_false: fixes \<sigma> :: "bool"   and n :: "nat"   and M :: "'grp \<times> 'grp"   and z :: "'e"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)"   and D :: "'d \<times> 'grp \<Rightarrow> bool spmf" assumes "\<sigma> = False" shows "P1_ideal_model n M \<sigma> z \<A> \<bind> D = local.ddh.DDH1 n (P1_DDH_mal_adv_\<sigma>_false n M z \<A> D)"
linear_neg: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "linear f" shows "f (- x) = - f x"
pred_of_enex_def: fixes enex :: "('d \<Rightarrow> 'f set) \<times> ('f \<Rightarrow> 'd \<Rightarrow> 'e)" shows "pred_of_enex enex \<equiv> \<lambda>s s'. let (en, ex) = enex in \<exists>a\<in>en s. s' = ex a s"
hf_ext: fixes a :: "hf"   and b :: "hf" shows "(a = b) = (\<forall>x. (x \<^bold>\<in> a) = (x \<^bold>\<in> b))"
vsv_ex1_app1: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> I" shows "(b = f\<lparr>a\<rparr>) = (\<langle>a, b\<rangle> \<in>\<^sub>\<circ> f)"
1: assumes "\<And>x. 0 \<le> x \<and> x \<le> 1 \<Longrightarrow> (\<exists>n. x * (t - s) = 2 * (real_of_int n * pi)) \<longrightarrow> x = 0 \<or> x = 1" shows "\<bar>s - t\<bar> \<le> 2 * pi"
absorb_iff2: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a \<le>s b) = (signed.max a b = b)"
prime_factorization_normalize: fixes n :: "'b" shows "prime_factorization (normalize n) = prime_factorization n"
foldl_absorb0: fixes x :: "'a"   and zs :: "'a list" shows "x + foldl (+) (0::'a) zs = foldl (+) x zs"
vpairwiseD_1: fixes R :: "V \<Rightarrow> V \<Rightarrow> bool"   and S :: "V"   and x :: "V"   and y :: "V" assumes "vpairwise R S"   and "x \<in>\<^sub>\<circ> S"   and "y \<in>\<^sub>\<circ> S"   and "x \<noteq> y" shows "R x y"
is_SNF_Smith_mxn: fixes Smith_1x2 :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat"   and Smith_2x2 :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat"   and div_op :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b mat"   and m :: "nat"   and n :: "nat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "A \<in> carrier_mat m n" shows "is_SNF A (Smith_Impl.Smith_mxn Smith_1x2 Smith_2x2 div_op A)"
Congruence_def: shows "additive.Congruence = {(a, b). a \<in> R \<and> b \<in> R \<and> - a + b \<in> I}"
p_max_def: fixes a :: "real"   and v :: "real"   and s0 :: "real" shows "movement.p_max a v s0 = movement.p a v s0 (movement.t_stop a v)"
measure_measure_1: shows "M = Abs_measure (\<Omega>, A, \<mu>)"
bin_sc_FP: fixes n :: "nat" shows "bin_sc n False 0 = 0"
parZero: fixes P :: "pi" shows "P \<parallel> \<zero> \<sim>\<^sub>l P"
tranclp_eq_rtranclp: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "(\<lambda>x y. P x y \<or> x = y)\<^sup>+\<^sup>+ = P\<^sup>*\<^sup>*"
intros_2: fixes u :: "'e \<Rightarrow> 'd"   and c :: "'e"   and d :: "'d" assumes "- u c \<le> d" shows "dbm_entry_val u None (Some c) (Le d)"
beukers_integral1_swap: fixes r :: "nat"   and s :: "nat" shows "beukers_integral1 r s = beukers_integral1 s r"
linear_def: fixes s1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s2 :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c" shows "Vector_Spaces.linear s1 s2 f \<equiv> vector_space s1 \<and> vector_space s2 \<and> module_hom s1 s2 f"
\<sigma>: fixes f :: "'f"   and n :: "nat" shows "set (\<sigma> (f, n)) \<subseteq> {0..<n}"
erl1: shows "l\<^sub>1 \<equiv>\<^sub>E rl1"
r_null: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "semiring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
boundedI: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e" assumes "semilattice_order_set f less_eq less"   and "finite A"   and "A \<noteq> {}"   and "\<And>a. a \<in> A \<Longrightarrow> less_eq x a" shows "less_eq x (semilattice_set.F f A)"
Im_divide_of_real: fixes z :: "complex"   and r :: "real" shows "Im (z / cor r) = Im z / r"
subseq_Cons2': fixes x :: "'c"   and xs :: "'c list"   and ys :: "'c list" assumes "subseq (x # xs) (x # ys)" shows "subseq xs ys"
not_less: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" shows "(\<not> x \<prec> y) = (y \<preceq> x)"
Tree_wf\<^sub>p_def: shows "Tree_wf\<^sub>p \<equiv> map_prod abs_Tree\<^sub>p abs_Tree\<^sub>p ` Tree_wf"
map_fun_def: fixes f :: "'e \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'f"   and h :: "'g \<Rightarrow> 'h" shows "(f ---> g) h = g \<circ> h \<circ> f"
ba: fixes b :: "'b" assumes "b \<in> B" shows "b_to_a b \<in> A"
outputCases: fixes substTerm :: "'d \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'd"   and substAssert :: "'e \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'e"   and substCond :: "'f \<Rightarrow> name list \<Rightarrow> 'd list \<Rightarrow> 'f"   and SCompose' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and SImp' :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and SBottom' :: "'e"   and SChanEq' :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and \<Psi> :: "'e"   and M :: "'d"   and N :: "'d"   and P :: "('d, 'e, 'f) psi"   and \<alpha> :: "'d action"   and P' :: "('d, 'e, 'f) psi"   and Prop :: "'d action \<Rightarrow> ('d, 'e, 'f) psi \<Rightarrow> bool" assumes "env substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq'"   and "env.semantics substTerm substAssert substCond SCompose' SImp' SBottom' SChanEq' \<Psi> (M\<langle>N\<rangle>.P) (\<alpha> \<prec> P')"   and "\<And>K. SImp' \<Psi> (SChanEq' M K) \<Longrightarrow> Prop (K\<langle>N\<rangle>) P" shows "Prop \<alpha> P'"
gh: fixes x :: "'a" assumes "x \<in> (\<Union> \<F> - C) \<inter> \<Union> \<H>" shows "g x = h x"
less_Quantity_ext_def: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
sw: fixes Aa :: "'a mat"   and xs :: "(nat \<times> nat) list" assumes "dim_row Aa = dim_col Aa"   and "\<And>i j. (i, j) \<in> set xs \<Longrightarrow> i < dim_col Aa \<and> j < dim_col Aa \<and> i \<noteq> j" shows "det (sw Aa xs) = (- (1::'a)) ^ length xs * det Aa"
simps_2: fixes A :: "('c, 'd) ioa"   and s :: "'c"   and p :: "'d \<times> 'c" shows "is_exec_frag_of A (s, [p]) = s \<midarrow>fst p\<midarrow>A\<longrightarrow> snd p"
singleton2: fixes i :: "nat_int" shows "(\<exists>n. Rep_nat_int i = {n}) \<longrightarrow> |i| = 1"
prime_sum_card_elements_with_ord_eq_totient: shows "(\<Sum>d | d dvd totient n. \<psi> d) = totient n"
that_1: shows "((=) ===> pcr_nonzero A ===> pcr_nonzero A) (*\<^sub>R) (*\<^sub>R)"
inv_renaming_def: fixes \<sigma> :: "'s" shows "inv_renaming \<sigma> = (SOME \<tau>. \<sigma> \<odot> \<tau> = id_subst)"
leg_simps_3: shows "C.arr HfHgh.dom.leg1"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int" assumes "t_design point_set block_collection u_block_size grouping 1" shows "steiner_system point_set block_collection u_block_size grouping"
induct: fixes P :: "('a, 'b) aformula list \<Rightarrow> bool"   and a0 :: "('a, 'b) aformula list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x v va. P (v # va) \<Longrightarrow> P (x # v # va)" shows "P a0"
brec_step_def: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and Qi :: "('a, unit) HashMap.hashmap" shows "brec_step rqrm Qi \<equiv> \<lambda>(Q, W, rcm, qwit). let (q, W') = fifo.removel W in ls.iterate (rqrm_lookup rqrm q) (brec_inner_step Qi) (Q, W', rcm, qwit)"
retract_of_refl: fixes S :: "'b set" shows "S retract_of S"
exec_bind_SE_failure: fixes A :: "'e \<Rightarrow> ('d \<times> 'e) option"   and \<sigma> :: "'e"   and M :: "'d \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "A \<sigma> = None" shows "\<not> (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E A M)"
finfun_Diag_update_update: fixes f :: "'c \<Rightarrow>f 'd"   and a :: "'c"   and b :: "'d"   and g :: "'c \<Rightarrow>f 'e"   and a' :: "'c"   and c :: "'e" shows "($f(a $:= b), g(a' $:= c)$) = (if a = a' then ($f, g$)(a $:= (b, c)) else ($f, g$)(a $:= (b, g $ a))(a' $:= (f $ a', c)))"
factorization_m_lead_coeff: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "factorization_m f (c, fs)" shows "lead_coeff (Mp f) = M c"
is_gcd_some_gcdI: fixes a :: "'a"   and b :: "'a" assumes "\<exists>x. is_gcd x a b" shows "is_gcd (some_gcd a b) a b"
hom_components_nonzero: fixes q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "q \<in> hom_components p" shows "q \<noteq> 0"
case_1: fixes f1 :: "name \<Rightarrow> 'a"   and f2 :: "name \<Rightarrow> 'a"   and x1 :: "name" shows "(case InputS x1 of InputS x \<Rightarrow> f1 x | BoundOutputS x \<Rightarrow> f2 x) = f1 x1"
feasibleI: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> bounded_by b (t P)"   and "\<And>b P. \<lbrakk>bounded_by b P; nneg P\<rbrakk> \<Longrightarrow> nneg (t P)" shows "feasible t"
nb\<^sub>8_2: shows "dom (map_of (map (\<lambda>v. (v, False)) (delete_effects_of op'))) = (\<Union>(v, a)\<in>set (effect_of op). {(v, a') |a'. a' \<in> \<R>\<^sub>+ \<Psi> v \<and> a' \<noteq> a})"
ns: shows "n (i - 2) - n (i - 1) + (n (i - 1) - n i) = n (i - 2) - n i"
not_final_thread_existsE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "not_final_thread s t" obtains x :: "'x" and ln :: "'a \<Rightarrow>f nat" where "thr s t = \<lfloor>(x, ln)\<rfloor>"