hcju/isabelleps huggingface dataset:
disjunctive_diff: 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 uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "\<And>n. bit b n \<Longrightarrow> bit a n" shows "minus a b = and a (not b)"
CHECK_def: fixes f :: "unit \<Rightarrow> 'e"   and P :: "bool" shows "CHECK f P = (if P then RETURN () else let _ = f () in SUCCEED)"
get_element_by_id_impl: shows "get_element_by_id = local.a_get_element_by_id"
action_closed: fixes s :: "'s"   and cfg :: "'s cfg"   and t :: "'s" assumes "s \<in> S"   and "cfg \<in> cfg_on s"   and "t \<in> set_pmf (action cfg)" shows "t \<in> S"
cexpr_sem_dist_dens_cexpr_nonneg: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and dst :: "pdf_dist"   and e2 :: "cexpr"   and V :: "nat set"   and \<sigma> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : dist_param_type dst"   and "\<Gamma> \<turnstile>\<^sub>c e2 : dist_result_type dst"   and "free_vars e1 \<subseteq> V"   and "free_vars e2 \<subseteq> V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "ennreal (extract_real (cexpr_sem \<sigma> (dist_dens_cexpr dst e1 e2))) = dist_dens dst (cexpr_sem \<sigma> e1) (cexpr_sem \<sigma> e2) \<and> 0 \<le> extract_real (cexpr_sem \<sigma> (dist_dens_cexpr dst e1 e2))"
h'_nonneg: fixes x :: "nat"   and y :: "nat" shows "0 \<le> h' x y"
f12: shows "(r'' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r'), s'' \<otimes> (s \<otimes> s')) \<in> carrier rel"
kop_epi_iff: fixes f :: "('d \<Rightarrow> 'c set) \<Rightarrow> 'e"   and g :: "('d \<Rightarrow> 'c set) \<Rightarrow> 'e" shows "(f \<circ> op\<^sub>K = g \<circ> op\<^sub>K) = (f = g)"
fst_splitAt_rev: fixes xs :: "'a list"   and x :: "'a" assumes "distinct xs"   and "x \<in> set xs" shows "fst (splitAt x (rev xs)) = rev (snd (splitAt x xs))"
of_nat_0: shows "of_nat 0 = (0::'a)"
intro: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "tiny_digraph \<alpha> \<AA>"   and "tiny_digraph \<alpha> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
path_acyclic: fixes x :: "'a" assumes "has_start_points_path x" shows "is_acyclic x"
random_bst_reduce: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "random_bst A = pmf_of_set A \<bind> (\<lambda>x. random_bst {y \<in> A. y < x} \<bind> (\<lambda>l. random_bst {y \<in> A. x < y} \<bind> (\<lambda>r. return_pmf \<langle>l, x, r\<rangle>)))"
interaction_bound_bind_step: fixes interaction_bound' :: "('g, 'e, 'f) gpv \<Rightarrow> enat"   and f :: "'d \<Rightarrow> ('g, 'e, 'f) gpv"   and "consider" :: "'e \<Rightarrow> bool"   and p :: "('d, 'e, 'f) gpv" assumes "\<And>p. interaction_bound' (p \<bind> f) \<le> interaction_bound consider p + (\<Squnion>x\<in>results'_gpv p. interaction_bound' (f x))"   and "\<And>gpv. interaction_bound' gpv \<le> (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))" shows "(\<Squnion>generat\<in>set_spmf (the_gpv (p \<bind> f)). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input)) \<le> interaction_bound consider p + (\<Squnion>x\<in>results'_gpv p. \<Squnion>generat\<in>set_spmf (the_gpv (f x)). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))"
tObsC: shows "tObsC (t \<leadsto> s) = tObsC (t' \<leadsto> s')"
sz: shows "(sz, length) \<in> arel \<rightarrow> nat_rel"
simps_1: fixes l :: "int"   and h :: "int" shows "is_empty {l\<dots>h} = (h < l)"
mono_def: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "class.preord le" shows "preord.mono le f = (\<forall>x y. le x y \<longrightarrow> le (f x) (f y))"
yaux: shows "y' \<sharp> [(x, x')] \<bullet> t"
minus: fixes x :: "complex"   and xa :: "'a" shows "- x *\<^sub>C xa = - (x *\<^sub>C xa)"
assms: shows "eSwapAbs MOD zs z1 z2 eA = OK B"
Min_singleton: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.linorder less_eq less" shows "linorder.Min less_eq {x} = x"
assms_3: shows "h = [] \<or> (\<exists>hs. h = [x, x] @ hs)"
gval_gOr: fixes x :: "'a gexp"   and y :: "'a gexp"   and r :: "'a \<Rightarrow> value option" shows "gval (gOr x y) r = gval x r \<or>? gval y r"
that: fixes a :: "'edge" assumes "valid_edge a"   and "intra_kind (kind a)"   and "inner_node (sourcenode a)"   and "targetnode a = (_Exit_)" shows "thesis"
Qg_tassoc: fixes G :: "('c, 'd) Group_scheme"   and N :: "'c set" assumes "Group G"   and "G \<triangleright> N" shows "\<forall>X\<in>set_rcs G N. \<forall>Y\<in>set_rcs G N. \<forall>Z\<in>set_rcs G N. c_top G N X (c_top G N Y Z) = c_top G N (c_top G N X Y) Z"
eq_mono: fixes a :: "Arity"   and a' :: "Arity"   and v1 :: "Value"   and v2 :: "Value" assumes "a \<sqsubseteq> a'"   and "eq a' v1 v2" shows "eq a v1 v2"
bdd_below_Ioc: fixes a :: "'a"   and b :: "'a" shows "bdd_below {a<..b}"
fr_rot_rhs: fixes A :: "assn"   and B :: "assn"   and C :: "assn" assumes "A \<Longrightarrow>\<^sub>A B * C" shows "A \<Longrightarrow>\<^sub>A C * B"
bin_rcat_def: fixes n :: "nat" shows "bin_rcat n = horner_sum (take_bit n) (2 ^ n) \<circ> rev"
distinct_299: fixes x151 :: "floatarith"   and x152 :: "nat"   and x18 :: "float" shows "x151 ^\<^sub>e x152 \<noteq> Num x18"
prod: shows "x \<otimes>\<^bsub>R Quot I\<^esub> y = I +> a \<otimes> b"
fpxs_val_const: fixes c :: "'b" shows "fpxs_val (fpxs_const c) = 0"
assms_3: shows "src\<^sub>C f = src\<^sub>C f'"
tail_swap: fixes x :: "'b" shows "tail G ((a \<rightleftharpoons>\<^sub>F b) x) = tail G x"
partial_restr: fixes a :: "'e \<Rightarrow> 'f"   and G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set"   and J :: "'e set" assumes "a \<in> carrier (DirProds G I)"   and "J \<subseteq> I" shows "restrict a J \<in> carrier (DirProds G J)"
finite_neq_0_inv: fixes h :: "'a \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b" assumes "finite {x. h x (f x) (g x) \<noteq> (0::'b)}"   and "finite {x. f x \<noteq> (0::'c)}"   and "\<And>x y. h x (0::'c) y = y" shows "finite {x. g x \<noteq> (0::'b)}"
remove_def: shows "Shadow_DOM.remove \<equiv> l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_remove Shadow_DOM.get_child_nodes Shadow_DOM.set_child_nodes Shadow_DOM.get_parent Shadow_DOM.get_owner_document get_disconnected_nodes set_disconnected_nodes"
det_addrow: fixes l :: "nat"   and n :: "nat"   and k :: "nat"   and A :: "'b mat"   and a :: "'b" assumes "l < n"   and "k \<noteq> l"   and "A \<in> carrier_mat n n" shows "det (addrow a k l A) = det A"
sup_inf_distrib1: fixes x :: "'a upred"   and y :: "'a upred"   and z :: "'a upred" shows "(x \<or> y \<and> z) = ((x \<or> y) \<and> (x \<or> z))"
Lp_Lq_duality_1: fixes f :: "'b \<Rightarrow> real"   and p :: "ennreal"   and M :: "'b measure"   and q :: "ennreal" assumes "f \<in> space\<^sub>N (\<LL> p M)"   and "1 / p + 1 / q = 1"   and "p = \<infinity> \<Longrightarrow> sigma_finite_measure M" shows "bdd_above ((\<lambda>g. LINT x|M. f x * g x) ` {g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1})"
perm_fun_def: fixes pi :: "('f \<times> 'f) list"   and f :: "'d \<Rightarrow> 'e" shows "pi \<bullet> f = (\<lambda>x. pi \<bullet> f (rev pi \<bullet> x))"
simps_9: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlr"   and \<psi> :: "'b ltlr" shows "\<xi> \<Turnstile>\<^sub>r \<phi> R\<^sub>r \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>r \<psi> \<or> (\<exists>j<i. suffix j \<xi> \<Turnstile>\<^sub>r \<phi>))"
assms_1: shows "is_path_undir (add_edge a w b G) v p v'"
mem_wwa_Worder: fixes W :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "W \<in> WWa D g a" shows "Worder (Iod D W)"
lits_subseteq_imp_atms_subseteq: fixes C :: "'b literal multiset"   and D :: "'b literal multiset" assumes "set_mset C \<subseteq> set_mset D" shows "atms_of C \<subseteq> atms_of D"
cross7_simps_51: fixes a :: "'b"   and x :: "'b"   and y :: "'b" shows "a * (x - y) = a * x - a * y"
remove_child_wf2_is_l_remove_child_wf2: shows "l_remove_child_wf2 ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs Shadow_DOM.remove_child Shadow_DOM.heap_is_wellformed Shadow_DOM.get_child_nodes Shadow_DOM.remove"
tensor_in_hom: fixes f :: "'c local.term set"   and a :: "'c local.term set"   and b :: "'c local.term set"   and g :: "'c local.term set"   and c :: "'c local.term set"   and d :: "'c local.term set" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : c \<rightarrow> d\<guillemotright>" shows "\<guillemotleft>f \<otimes> g : a \<otimes> c \<rightarrow> b \<otimes> d\<guillemotright>"
inv_le: fixes i :: "nat"   and x :: "real" assumes "inverse (real (Suc i)) \<le> x" shows "\<not> x \<le> 0"
YFtorPreFtor: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "PreFunctor (YFtor' C)"
assms: shows "snd (quot_of_fract x) = (1::'a)"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set" assumes "Problem wtFsym wtPsym arOf resOf parOf \<Phi>"   and "MonotProblem_axioms wtFsym wtPsym arOf resOf parOf \<Phi>" shows "MonotProblem wtFsym wtPsym arOf resOf parOf \<Phi>"
good_item_simps_35: fixes xsa :: "'varSort"   and xa :: "'var"   and delta :: "'c"   and inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option" shows "Var xsa xa \<noteq> Op delta inp binpa"
d\<mu>_def: fixes gs :: "int vec list"   and i :: "nat"   and j :: "nat" shows "d\<mu> gs i j = int_of_rat (rat_of_int (d gs (Suc j)) * gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom gs) i j)"
good_ruleset_def: fixes rs :: "'a rule list" shows "good_ruleset rs \<equiv> \<forall>r\<in>set rs. (\<nexists>chain. get_action r = Call chain) \<and> get_action r \<noteq> Return \<and> (\<nexists>chain. get_action r = Goto chain) \<and> get_action r \<noteq> Unknown"
merge_None: fixes m :: "('e \<times> 'd) list"   and n :: "('e \<times> 'd) list"   and k :: "'e" shows "(map_of (AList.merge m n) k = None) = (map_of n k = None \<and> map_of m k = None)"
all_local_consts_listed': fixes n :: "('form, 'rule) graph_node"   and p :: "('form, 'var) antecedent" assumes "n \<in> sset nodes"   and "p |\<in>| inPorts n" shows "lconsts (a_conc p) \<union> \<Union> (lconsts ` fset (a_hyps p)) \<subseteq> a_fresh p"
indep_sets_X: fixes I :: "'c set"   and i :: "'c \<Rightarrow> 'i" assumes "\<And>ia. ia \<in> I \<Longrightarrow> i ia \<in> I" shows "indep_sets (\<lambda>ia. sigma_sets (space M) {X (i ia) -` A \<inter> space M |A. A \<in> E (i ia)}) I = indep_sets (\<lambda>ia. {X (i ia) -` A \<inter> space M |A. A \<in> E (i ia)}) I"
multiple_block_in_original: fixes b :: "'a set"   and n :: "nat" assumes "b \<in># multiple_blocks n" shows "b \<in># \<B>"
matrix_to_iarray_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "matrix_to_iarray (Finite_Cartesian_Product.transpose A) = transpose_iarray (matrix_to_iarray A)"
arcs_in: shows "{a, a'} \<subseteq> arcs G"
delete_def: fixes x :: "'c"   and t :: "('c \<times> 'd) tree234" shows "Tree234_Map.delete x t = tree\<^sub>d (Tree234_Map.del x t)"
strict_implies_order: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a \<prec> b" shows "a \<preceq> b"
map_sub_consts_Nil: fixes G :: "('a, 'b) form list" shows "map (sub_consts []) G = G"
simps_1: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and e :: "'a" shows "insert_sort_insert_by cmp e [] = [e]"
Abs_FingerTree_inject: fixes x :: "('a, 'b) FingerTreeStruc"   and y :: "('a, 'b) FingerTreeStruc" assumes "x \<in> {t. FingerTreeStruc.ft_invar t}"   and "y \<in> {t. FingerTreeStruc.ft_invar t}" shows "(Abs_FingerTree x = Abs_FingerTree y) = (x = y)"
R_a0iim1_ri_def: shows "R_a0iim1_ri \<equiv> {(s, t). signals s = ri_runs2sigs (runs t) \<and> corrupted s = {}}"
interchange_columns_iarray_def: fixes A :: "'b iarray iarray"   and a :: "nat"   and b :: "nat" shows "interchange_columns_iarray A a b = tabulate2 (nrows_iarray A) (ncols_iarray A) (\<lambda>i j. if j = a then A !! i !! b else if j = b then A !! i !! a else A !! i !! j)"
is_weakest_liberal_precondition_def: fixes W :: "complex mat"   and S :: "com"   and P :: "complex mat" shows "is_weakest_liberal_precondition W S P = (is_quantum_predicate W \<and> \<Turnstile>\<^sub>p {W} S {P} \<and> (\<forall>Q. is_quantum_predicate Q \<longrightarrow> \<Turnstile>\<^sub>p {Q} S {P} \<longrightarrow> Q \<le>\<^sub>L W))"
correct_def: fixes funct :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and protocol :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and m1 :: "'d"   and m2 :: "'e" shows "malicious_base.correct funct protocol m1 m2 = (protocol m1 m2 = funct m1 m2)"
mult: fixes a :: "nat"   and b :: "nat" shows "f (a * b) = f a * f b"
perm_aux_fold: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" shows "perm_aux pi x = pi \<bullet> x"
cons_2: assumes "coll 0 a (sum_list xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> coll 0 x y"   and "\<And>x. x \<in> set xs \<Longrightarrow> coll 0 x (sum_list xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> 0 < snd x"   and "a \<noteq> 0" shows "coll 0 a y"
rel_bset_def: shows "rel_bset \<equiv> map_fun id (map_fun set_bset (map_fun set_bset id)) rel_set"
induct: fixes P :: "'c Tree \<Rightarrow> 'c Tree \<Rightarrow> bool"   and a0 :: "'c Tree"   and a1 :: "'c Tree" assumes "\<And>t1. P t1 E"   and "\<And>t2. P E t2"   and "\<And>v1 l1 r1 v2 l2 r2. \<lbrakk>v2 \<le> v1 \<Longrightarrow> P l1 (T v2 l2 r2); \<not> v2 \<le> v1 \<Longrightarrow> P l2 (T v1 l1 r1)\<rbrakk> \<Longrightarrow> P (T v1 l1 r1) (T v2 l2 r2)" shows "P a0 a1"
sw_inj: fixes xs :: "'e"   and x :: "'d"   and y :: "'d"   and zs :: "'e"   and z :: "'d"   and z' :: "'d" shows "(z @zs[x \<and> y]_xs = z' @zs[x \<and> y]_xs) = (z = z')"
mult_eq: fixes x :: "nat"   and y :: "nat" shows "x \<otimes>\<^bsub>G\<^esub> y = x * y mod p"
eq_diff_eq': fixes x :: "real"   and y :: "real"   and z :: "real" shows "(x = y - z) = (y = x + z)"
sum_ennreal: fixes I :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" shows "(\<Sum>i\<in>I. ennreal (f i)) = ennreal (sum f I)"
list_cases_Cons_snoc: fixes xs :: "'b list" obtains "xs = []"   | x :: "'b" where "xs = [xa__]"   | x :: "'b" and ys :: "'b list" and y :: "'b" where "xs = x # ys @ [y]"
inf_dense: fixes 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 x :: "'c"   and y :: "'c" assumes "class.linear_bounded_lattice inf less_eq less sup bot top"   and "x \<noteq> bot"   and "y \<noteq> bot" shows "inf x y \<noteq> bot"
card_Diff_singleton: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) = card A - 1"
Q_hit_vertices: shows "set Q_hit \<subseteq> V - {new_last}"
cycle_root_non_empty: fixes x :: "'a"   and r :: "'a" assumes "x \<noteq> 0" shows "cycle_root r x = non_empty_cycle_root r x"
the_\<theta>_props_1: shows "\<guillemotleft>\<mu>.the_\<theta> : s\<^sub>0 \<star> \<mu>.chine \<Rightarrow> r\<^sub>0\<guillemotright>"
lbk_option_def: fixes p :: "real"   and M :: "bool stream measure"   and geom_proc :: "nat \<Rightarrow> bool stream \<Rightarrow> real"   and u :: "real"   and d :: "real"   and init :: "real"   and Mkt :: "(bool stream, 'b) discrete_market"   and r :: "real"   and risk_free_asset :: "'b"   and stk :: "'b"   and G :: "nat \<Rightarrow> bool stream measure"   and T :: "nat" assumes "CRR_market p M geom_proc u d init Mkt r risk_free_asset stk G" shows "CRR_market.lbk_option Mkt stk T = (\<lambda>w. (MAX i\<in>{0..T}. prices Mkt stk i w) - prices Mkt stk T w)"
intro: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "\<And>a a' b. prod (a + a') b = prod a b + prod a' b"   and "\<And>a b b'. prod a (b + b') = prod a b + prod a b'"   and "\<And>r a b. prod (r *\<^sub>C a) b = r *\<^sub>C prod a b"   and "\<And>a r b. prod a (r *\<^sub>C b) = r *\<^sub>C prod a b"   and "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K" shows "bounded_cbilinear prod"
group_commutes_pow: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x [^] n \<otimes> y = y \<otimes> x [^] n"
pderivs_Times: fixes s :: "'a list"   and r1 :: "'a rexp"   and r2 :: "'a rexp" shows "pderivs s (Times r1 r2) \<subseteq> Timess (pderivs s r1) r2 \<union> pderivs_lang (PSuf s) r2"
degree_poly_x_minus_y: fixes p :: "'a poly" shows "degree (poly_x_minus_y p) = degree p"
with_proj_simps_1: fixes G :: "'b pair_pre_digraph" shows "verts (with_proj G) = pverts G"
dghm_cn_cov_comp_ObjMap_vdomain: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> ((\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ObjMap\<rparr>) = \<AA>\<lparr>Obj\<rparr>"
sumE: fixes s :: "'g + 'h" obtains x :: "'g" where "s = Inl x"   | y :: "'h" where "s = Inr y"
M: shows "M = (\<lambda>n. tensor_P (testN n) (1\<^sub>m K))"
of_nat_less_iff: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m < n)"
DERIV_cmult: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and c :: "'a" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
cring_simprules_3: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "\<ominus>\<^bsub>R\<^esub> x \<in> carrier R"
hash_blindable_simps_1: fixes h :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "hash_blindable h (Unblinded x) = Content (h x)"
size_4: fixes x21 :: "'c"   and x22 :: "'c list" shows "length (x21 # x22) = length x22 + Suc 0"
map_value_image_filter_axioms: shows "map_value_image_filter \<alpha>1 invar1 \<alpha>2 invar2 map_value_image_filter"
REMOVE_INTERNAL_def: fixes x :: "'c" shows "REMOVE_INTERNAL x \<equiv> x"
map_pre_coG_def: shows "map_pre_coG \<equiv> \<lambda>f1 f2. BNF_Composition.id_bnf \<circ> map_coG0 f1 (map_coF f2) \<circ> BNF_Composition.id_bnf"
And_def: fixes \<phi> :: "'c MFOTL.formula"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.And \<phi> \<psi> = formula.Neg (formula.Or (formula.Neg \<phi>) (formula.Neg \<psi>))"
mult_zero_left: fixes V :: "'b set"   and x :: "'b" assumes "vectorspace V"   and "x \<in> V" shows "0 \<cdot> x = (0::'b)"
less_eq_real_int_def: fixes r :: "real_int"   and s :: "real_int" shows "r \<le> s \<equiv> left s \<le> left r \<and> right r \<le> right s"
get_root_node_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_root_node ptr" shows "ptr |\<in>| object_ptr_kinds h"
j: shows "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
point_in_vector_or_complement: fixes p :: "'a"   and v :: "'a" assumes "point p"   and "vector v"   and "regular v" shows "p \<le> v \<or> p \<le> - v"
simps_2: fixes t :: "'a tree23"   and a :: "'a"   and ts :: "'a tree23s" shows "trees (TTs t a ts) = {t} \<union> trees ts"
partition_onD1: fixes A :: "'c set"   and P :: "'c set set" assumes "partition_on A P" shows "A = \<Union> P"
assms_3: shows "x\<^sup>2 + y\<^sup>2 = p"
supplement2_Jacobi: fixes p :: "int" assumes "odd p"   and "1 < p" shows "Jacobi 2 p = (- 1) ^ (((nat p)\<^sup>2 - 1) div 8)"
valid_Tree\<^sub>\<alpha>_eqvt: fixes P :: "'state"   and t :: "('b, 'pred, 'act) Tree\<^sub>\<alpha>"   and p :: "perm" assumes "valid_Tree\<^sub>\<alpha> P t" shows "valid_Tree\<^sub>\<alpha> (p \<bullet> P) (p \<bullet> t)"
m: fixes i :: "nat" assumes "m < i" shows "a i = (0::'a)"
Inf: assumes "infinite A"   and "|A| =o |UNIV|" shows "thesis"
continuous_on_blinfun_of_matrix: fixes S :: "'c set"   and g :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> real" assumes "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk> \<Longrightarrow> continuous_on S (\<lambda>s. g s i j)" shows "continuous_on S (\<lambda>s. blinfun_of_matrix (g s))"
reachable_weakenE: fixes s :: "'c"   and A :: "('c, 'd) automaton"   and P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "s \<in> reachable A P"   and "\<And>a. P a \<Longrightarrow> Q a" shows "s \<in> reachable A Q"
safe_regex_safe_formula: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and \<phi> :: "Formula.formula" assumes "safe_regex m g r"   and "\<phi> \<in> regex.atms r" shows "safe_formula \<phi> \<or> (\<exists>\<psi>. \<phi> = formula.Neg \<psi> \<and> safe_formula \<psi>)"
uniformly_convergent_EM_remainder': fixes a :: "real"   and G :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and A :: "'a set"   and n :: "nat"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b" assumes "\<And>y. a \<le> y \<Longrightarrow> (G has_real_derivative g y) (at y within {a..})"   and "\<And>a' b y. \<lbrakk>y \<in> A; a \<le> a'; a' \<le> b\<rbrakk> \<Longrightarrow> (\<lambda>t. pbernpoly n t *\<^sub>R f y t) integrable_on {a'..b}"   and "convergent (\<lambda>y. G (real y))"   and "\<forall>\<^sub>F x in at_top. \<forall>y\<in>A. norm (f y x) \<le> g x" shows "uniformly_convergent_on A (\<lambda>x s. EM_remainder' n (f s) a (real x))"
alg_closed: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" assumes "class.alg_closed_field divide inverse times one plus zero minus uminus"   and "0 < n"   and "f n \<noteq> zero" shows "\<exists>x. comm_monoid_add.sum plus zero (\<lambda>k. times (f k) (power.power one times x k)) {..n} = zero"