hcju/isabelleps huggingface dataset:
get_scdom_component_def: shows "Shadow_DOM_SC_DOM_Components.get_scdom_component \<equiv> l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_scdom_component Shadow_DOM.get_owner_document get_disconnected_nodes Shadow_DOM.to_tree_order"
distinct_29: fixes x51 :: "expr"   and x52 :: "expr"   and x61 :: "Unary_PCF.type"   and x62 :: "expr" shows "x51 \<cdot> x52 \<noteq> \<Lambda>\<langle>x61\<rangle> x62"
member_set_of_pred: fixes x :: "'a"   and P :: "'a Predicate.pred" shows "(x \<in> set_of_pred P) = pred.eval P x"
preserves_reflects_arr: fixes f :: "'c \<times> 'c" shows "D.arr (cmp f) = C.VV.arr f"
simps_10: fixes P :: "pi" shows "summands (!P) = {}"
FSwSbImorph_igWlsBinp: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'e, 'f, 'g) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'f option"   and h :: "'e \<Rightarrow> 'h"   and hA :: "'f \<Rightarrow> 'i"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'd, 'h, 'i, 'j) model_scheme" assumes "igWlsBinp MOD delta binp"   and "FSwSbImorph h hA MOD MOD'" shows "igWlsBinp MOD' delta (lift hA binp)"
hnz_prop: fixes l :: "rat list" shows "has_no_zeros l = (\<not> (\<exists>k<length l. l ! k = 0))"
alg_mult_def: fixes p :: "'a list"   and x :: "'a" shows "alg_mult p x = (if p = [] then 0 else if x \<in> carrier R then GREATEST n. [\<one>, \<ominus> x] [^]\<^bsub>poly_ring R\<^esub> n pdivides p else 0)"
vfsequence_vfsequence_of_vlist: fixes xs :: "V list" shows "vfsequence (vfsequence_of_vlist xs)"
min_term_min: fixes v :: "'t" shows "min_term \<preceq>\<^sub>t v"
split_zmod: fixes P :: "int \<Rightarrow> bool"   and n :: "int"   and k :: "int" shows "P (n mod k) = ((k = 0 \<longrightarrow> P n) \<and> (0 < k \<longrightarrow> (\<forall>i j. 0 \<le> j \<and> j < k \<and> n = k * i + j \<longrightarrow> P j)) \<and> (k < 0 \<longrightarrow> (\<forall>i j. k < j \<and> j \<le> 0 \<and> n = k * i + j \<longrightarrow> P j)))"
rtrancl3p_refl: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" shows "rtrancl3p r a [] a"
n_S2: fixes s :: "'s" assumes "s \<in> S2" shows "n s = 1"
i_expand_i_append: fixes xs :: "'d message_af list"   and f :: "nat \<Rightarrow> 'd message_af"   and k :: "nat" shows "(xs \<frown> f) \<odot> k = xs \<odot> k \<frown> f \<odot> k"
cases: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b llist" assumes "chain' R a" obtains x :: "'b" where "a = LCons x__ LNil"   | x :: "'b" and zs :: "'b list" and xs :: "'b llist" where "a = LCons x (prepend zs xs)" and "chain R (llist_of (x # zs @ [lhd xs]))" and "chain' R xs"
ereal_le_divide_neg: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" assumes "x < 0"   and "x \<noteq> - \<infinity>" shows "(y \<le> z / x) = (z \<le> x * y)"
relation_of_Prefix0: fixes c :: "'c"   and P :: "('c, 'd) action" shows "action.relation_of (Prefix0 c P) = (R (true \<turnstile> do (\<lambda>_. c) \<and> (\<lambda>(A, A'). alpha_rp.more A' = alpha_rp.more A)) ;; action.relation_of P)"
ring_homD: fixes Phi :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "Phi \<in> ring_hom P R" shows "ring_hom_cring P R Phi"
not_strict_par: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p" assumes "A B Par C D"   and "Col A B X"   and "Col C D X" shows "Col A B C \<and> Col A B D"
assms_1: shows "\<forall>s. ksimplex p (Suc n) s \<longrightarrow> rl ` s \<subseteq> {..Suc n}"
closest_pair_time_def: fixes n :: "nat" shows "closest_pair_time n = 1 + mergesort_recurrence n + closest_pair_recurrence n"
Ex_cong: fixes H :: "fm set"   and A :: "fm"   and A' :: "fm"   and i :: "name" assumes "H \<turnstile> A IFF A'"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> SyntaxN.Ex i A IFF SyntaxN.Ex i A'"
a_rcos_self: fixes x :: "'a" assumes "x \<in> carrier G" shows "x \<in> H +> x"
cases: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c" assumes "integrable M f" obtains x :: "'c" where "has_bochner_integral M f x"
AT_EF: fixes A :: "'a attree"   and I :: "'a set"   and s :: "'a set" assumes "\<turnstile>A"   and "attack A = (I, s)" shows "Kripke {s. \<exists>i\<in>I. i \<rightarrow>\<^sub>i* s} I \<turnstile> EF s"
replicate_0: fixes x :: "'c" shows "replicate 0 x = []"
iv: shows "(- 1, \<lambda>y. (- (d / 2), 0)) \<noteq> (- 1, \<lambda>x. ((x - 1 / 2) * d, d * sqrt (1 / 4 - (x - 1 / 2) * (x - 1 / 2))))"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "C.seq g f" shows "C.cod (g \<cdot>\<^sub>C f) = C.cod g"
independent_backwards_1: fixes K :: "'a set"   and u :: "'a"   and Us :: "'a list" assumes "local.independent K (u # Us)" shows "u \<notin> Span K Us"
arrE: fixes f :: "'a" assumes "S.arr f" shows "left b f"
fbd_comp_pres: fixes f :: "'c \<Rightarrow> 'e set"   and g :: "'e \<Rightarrow> 'd set" shows "bd\<^sub>\<F> (f \<circ>\<^sub>K g) = bd\<^sub>\<F> g \<circ> bd\<^sub>\<F> f"
bigo_lesseq2: fixes f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> O(h)"   and "\<forall>x. \<bar>g x\<bar> \<le> f x" shows "g \<in> O(h)"
le_fininf_ind'': fixes a :: "'action"   and w :: "'action stream"   and b :: "'action" assumes "[a] \<preceq>\<^sub>F\<^sub>I w"   and "[b] \<preceq>\<^sub>F\<^sub>I w"   and "a \<noteq> b" shows "ind a b"
elims_3: fixes x :: "('a, 'b) rbt" assumes "\<not> cl_inv' x" obtains l :: "('a, 'b) rbt" and c :: "color" and k :: "'a" and v :: "'b" and r :: "('a, 'b) rbt" where "x = Node l c k v r" and "\<not> (cl_inv l \<and> cl_inv r)"
simps: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal" shows "local.trimming g = (current \<Gamma> g \<and> wave \<Gamma> g \<and> g \<le> f \<and> (\<forall>x. x \<in> RF\<^sup>\<circ> (TER f) \<longrightarrow> x \<notin> A \<Gamma> \<longrightarrow> KIR g x) \<and> \<E> (TER g) - A \<Gamma> = \<E> (TER f) - A \<Gamma>)"
S: shows "S = (\<lambda>s. if s ''a'' \<le> s ''z'' then s else s(''y'' := s ''a'' * s ''b'', ''z'' := s ''a''))"
simps_5: fixes t :: "'c local.term" shows "Src \<^bold>\<l>\<^bold>[t\<^bold>] = Src t"
2: shows "(- - v\<^sup>T * (w \<sqinter> - v\<^sup>T) * w\<^sup>\<star> \<squnion> - - v\<^sup>T) * w \<le> - - v\<^sup>T * (w \<sqinter> - v\<^sup>T) * w\<^sup>\<star> \<squnion> - - v\<^sup>T"
merge_single2: fixes A :: "'s" shows "merge A (\<lambda>x. return (single x)) = return A"
a3: fixes x :: "'a" assumes "x \<in> F" shows "norm (f x) \<le> norm (f x)"
that: fixes a' :: "'c node_action" assumes "((\<sigma>, \<zeta>), a', \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)" shows "thesis"
dvdm_l_ll: fixes a :: "int poly"   and b :: "int poly" assumes "pl.dvdm a b" shows "pll.dvdm a b"
wf_converse: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "wfP r\<inverse>\<inverse> = wf ({(x, y). r x y}\<inverse>)"
complex_mod_mult_cnj: fixes z :: "complex" shows "cmod (z * cnj z) = (cmod z)\<^sup>2"
index_rev: fixes xs :: "'b list"   and x :: "'b" assumes "distinct xs"   and "x \<in> set xs" shows "index (rev xs) x = length xs - index xs x - 1"
supp_9: fixes x1 :: "name"   and x2 :: "pi" shows "supp (<\<nu>x1>x2) = supp ([x1].x2)"
connected_existence_ivl: fixes x0 :: "'a" shows "connected (existence_ivl t0 x0)"
plus_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
regular_Derivs_finite: fixes r :: "'a rexp" shows "finite (range (\<lambda>w. Derivs w (lang r)))"
subg_Qsubg: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> K"   and "K \<subseteq> H" shows "G / K \<guillemotright> carrier (\<natural>H / K) "
that: fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<And>y. y \<in> closure S \<Longrightarrow> a \<bullet> x \<le> a \<bullet> y"   and "\<And>y. y \<in> rel_interior S \<Longrightarrow> a \<bullet> x < a \<bullet> y" shows "thesis"
img_point_in_Univ: fixes x :: "'s"   and a :: "'s" assumes "\<guillemotleft>x : unity \<rightarrow> a\<guillemotright>" shows "img x \<in> Univ"
simps_76: fixes x8 :: "'a OCL_Types.type"   and x7 :: "'a OCL_Types.type" shows "Sequence x8 \<noteq> Bag x7"
WT1: shows "\<I> \<turnstile>res res1 \<surd>"
assms: shows "a \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
length_gen_akiyama_tanigawa_row: fixes f :: "nat \<Rightarrow> real"   and n :: "nat"   and l :: "nat"   and u :: "nat" shows "length (gen_akiyama_tanigawa_row f n l u) = u - l"
bmod_assoc: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a" shows "\<otimes> \<rho> (\<otimes> \<rho> f g) h x = \<otimes> \<rho> f (\<otimes> \<rho> g h) x"
zdiv_le_dividend: fixes a :: "int"   and b :: "int" assumes "0 \<le> a"   and "0 < b" shows "a div b \<le> a"
\<A>\<^sub>d_3: shows "\<I> \<Turnstile>\<^sub>c \<langle>to_st \<A>\<^sub>d\<rangle>"
n_as: fixes n :: "nat"   and a :: "'a" assumes "(n, a) \<in> set n_as"   and "0 < n" shows "norm a \<le> 1"
game_def: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv" shows "local.ind_cca.game \<A> = key_gen \<bind> (\<lambda>key. coin_spmf \<bind> (\<lambda>b. exec_gpv (ind_cca'.oracle_encrypt key b \<oplus>\<^sub>O local.ind_cca.oracle_decrypt key) \<A> {} \<bind> (\<lambda>(b', L'). return_spmf (b = b'))))"
assms: shows "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2"
MapsToObj_1: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "X \<in> obj\<^bsub>C\<^esub>"
rep_eq: fixes x :: "'d \<Rightarrow> 'c option"   and xa :: "'d fset" shows "fset (fselect x xa) = select x (fset xa)"
simps_10: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "list_all2 R [] []"
simps_6: fixes var :: "nat" shows "freeIn var FalseF = True"
cmp_imp_equal: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res"   and x :: "'a"   and y :: "'a" assumes "eq_linorder_on D cmp"   and "x \<in> D"   and "y \<in> D"   and "cmp x y = EQUAL" shows "x = y"
prems: shows "i < Monitor.progress \<sigma> (formula.Until \<phi>1 I \<phi>2) (plen \<pi>)"
rtrancl_beta_Call: fixes t :: "sterm"   and t' :: "sterm"   and u :: "sterm"   and u' :: "sterm"   and l :: "Label" assumes "t ->> t'"   and "lc t"   and "u ->> u'"   and "lc u" shows "Call t l u ->> Call t' l u'"
singleton_abelian_group: fixes a :: "'b" shows "comm_group (singleton_group a)"
csfi: shows "s O f\<inverse> \<subseteq> b \<union> m \<union> ov"
case_cong: fixes option :: "'d option"   and option' :: "'d option"   and f1 :: "'e"   and g1 :: "'e"   and f2 :: "'d \<Rightarrow> 'e"   and g2 :: "'d \<Rightarrow> 'e" assumes "option = option'"   and "option' = None \<Longrightarrow> f1 = g1"   and "\<And>x2. option' = Some x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case option of None \<Rightarrow> f1 | Some x2 \<Rightarrow> f2 x2) = (case option' of None \<Rightarrow> g1 | Some x2 \<Rightarrow> g2 x2)"
assms_2: shows "x (Suc n) \<noteq> 0"
nth_0: fixes n :: "nat" shows "\<not> 0 !! n"
dim_image_eq: fixes f :: "'b \<Rightarrow> 'b \<times> 'c"   and S :: "'b set" assumes "Vector_Spaces.linear (*a) local.scale f"   and "inj_on f (vs1.span S)" shows "p.dim (f ` S) = vs1.dim S"
complex_Taylor: fixes S :: "complex set"   and n :: "nat"   and f :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and B :: "real"   and w :: "complex"   and z :: "complex" assumes "convex S"   and "\<And>i x. \<lbrakk>x \<in> S; i \<le> n\<rbrakk> \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> cmod (f (Suc n) x) \<le> B"   and "w \<in> S"   and "z \<in> S" shows "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / fact i)) \<le> B * cmod (z - w) ^ Suc n / fact n"
i_in_inf_fl: shows "\<iota> \<in> Inf_FL"
subgroupE_1: fixes H :: "nat set" assumes "subgroup H G" shows "H \<subseteq> carrier G"
remove_child_parent_child_rel_subset: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "h \<turnstile> remove_child ptr child \<rightarrow>\<^sub>h h'"   and "known_ptrs h"   and "type_wf h" shows "parent_child_rel h' \<subseteq> parent_child_rel h"
SINV_post_def: fixes par :: "Var list"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and B :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool" shows "SINV_post par T B = (\<lambda>s0 s t. \<forall>ops1 ops2 S R h k w r. (ops1, par, R, ops2) \<in> Frame \<longrightarrow> T (R, k) (h, w) \<longrightarrow> s = (w # ops2, S, h) \<longrightarrow> r = (ops1, S, k) \<longrightarrow> B s0 r t)"
concurrentD2: fixes s1 :: "'a"   and s2 :: "'a" assumes "s1 \<parallel> s2" shows "\<not> s2 \<prec> s1"
pow_mono: fixes n :: "nat"   and n' :: "nat"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'" shows "PoW t n \<le> PoW t n'"
pas_alt_comm: fixes x :: "'a"   and y :: "'a" assumes "D x y" shows "D y x \<and> x \<oplus> y = y \<oplus> x"
islimpt_Un_finite: fixes s :: "'b set"   and x :: "'b"   and t :: "'b set" assumes "finite s" shows "x islimpt s \<union> t = x islimpt t"
GroupHomSetD_Im: fixes T :: "'a \<Rightarrow> 'b"   and G :: "'a set"   and H :: "'b set" assumes "T \<in> GroupHomSet G H" shows "T ` G \<subseteq> H"
simps_2: fixes bs :: "'b list"   and n :: "nat" shows "Ipoly bs (Bound n) = bs ! n"
dvd_imp_mod_0: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semidom_modulo divide plus minus zero times one modulo"   and "dvd.dvd times a b" shows "modulo b a = zero"
cong_four_block_mat: fixes A1 :: "'a mat"   and B1 :: "'a mat"   and A2 :: "'a mat"   and B2 :: "'a mat"   and A3 :: "'a mat"   and B3 :: "'a mat"   and A4 :: "'a mat"   and B4 :: "'a mat" assumes "A1 = B1"   and "A2 = B2"   and "A3 = B3"   and "A4 = B4" shows "four_block_mat A1 A2 A3 A4 = four_block_mat B1 B2 B3 B4"
card_le_pick_inf: fixes S :: "nat set"   and i :: "nat"   and n :: "nat" assumes "infinite S"   and "i \<le> pick S n" shows "card {a \<in> S. a < i} \<le> n"
store_typing_imp_glob_length_eq: fixes s :: "s"   and \<S> :: "s_context" assumes "store_typing s \<S>" shows "length (s.globs s) = length (s_globs \<S>)"
notin_fset: fixes x :: "'a"   and S :: "'a fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
size_replicate_mset: fixes n :: "nat"   and M :: "'c" shows "size (replicate_mset n M) = n"
norm_eq: shows "norm_instr (body fd2 ! pc) = norm_instr instr"
keys_MP_oalist: fixes xs :: "('c, 'd) oalist_ntm" shows "keys (MP_oalist xs) = set (map fst (fst (list_of_oalist_ntm xs)))"
wtSB_def: fixes \<pi> :: "'b \<Rightarrow> 'fsym trm" shows "wtSB \<pi> \<equiv> \<forall>x. wt (\<pi> x)"
smult_pCons: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "smult a (pCons b p) = pCons (a * b) (smult a p)"
translation_diff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and s :: "'a set"   and t :: "'a set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (s - t) = plus a ` s - plus a ` t"
min_int_poly_nonzero: fixes x :: "'b" shows "min_int_poly x \<noteq> 0"
takeWhile_hd_tl: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "xs \<noteq> []" shows "takeWhile P xs = (if P (hd xs) then hd xs # takeWhile P (tl xs) else [])"
strong_delete_point_proper: fixes p :: "'a" assumes "\<And>bl. \<lbrakk>bl \<in># str_del_point_blocks x; p \<in> bl\<rbrakk> \<Longrightarrow> 2 \<le> card bl"   and "str_del_point_blocks x rep p < int (size (str_del_point_blocks x))" shows "proper_design (des.del_point p) (des.str_del_point_blocks p)"
induct: fixes P :: "('b \<times> nat) option \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "('b \<times> nat) option"   and a1 :: "'b"   and a2 :: "nat" assumes "\<And>L t. P L t 0"   and "\<And>L t m. P (lock_lock L t) t m \<Longrightarrow> P L t (Suc m)" shows "P a0 a1 a2"
inject: fixes x21 :: "nat"   and x22 :: "nat"   and x23 :: "'b option"   and x24 :: "'b tree"   and x25 :: "'b tree"   and y21 :: "nat"   and y22 :: "nat"   and y23 :: "'b option"   and y24 :: "'b tree"   and y25 :: "'b tree" shows "(Node x21 x22 x23 x24 x25 = Node y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
T_def: fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c" shows "MC_syntax.T K s = distr (stream_space (Pi\<^sub>M UNIV (\<lambda>x. measure_pmf (K x)))) (stream_space (count_space UNIV)) (MC_syntax.walk K s)"
inorder_delete: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t \<and> Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = del_list x (Tree2.inorder t)"
is_parser_scan_upto: fixes "end" :: "char list" shows "is_parser (scan_upto end)"
uminus_add_closed: fixes g :: "'g"   and h :: "'g" assumes "g \<in> G"   and "h \<in> G" shows "- g + h \<in> G"
fold2_o_ctor2: fixes s1 :: "('d, 'f, 'e) F1 \<Rightarrow> 'f"   and s2 :: "('d, 'f, 'e) F2 \<Rightarrow> 'e" shows "fold2 s1 s2 \<circ> ctor2 = s2 \<circ> F2map id (fold1 s1 s2) (fold2 s1 s2)"
Compl_insert: fixes x :: "'b"   and A :: "'b set" shows "- insert x A = - A - {x}"
set_less_eq_aux_finite_total: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B" shows "A \<sqsubseteq>' B \<or> A = B \<or> B \<sqsubseteq>' A"
rank_def: fixes n :: "nat"   and A :: "'b mat" shows "vec_space.rank n A = vectorspace.dim class_ring (module_vec TYPE('b) n\<lparr>carrier := LinearCombinations.module.span class_ring (module_vec TYPE('b) n) (set (cols A))\<rparr>)"
Max_ge: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and x :: "'d" assumes "class.linorder less_eq less"   and "finite A"   and "x \<in> A" shows "less_eq x (linorder.Max less_eq A)"
app_subst_list: fixes S :: "nat \<Rightarrow> typ" shows "$ S = map ($ S)"
asymp_equiv_def: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "(f \<sim>[F] g) = ((\<lambda>x. if f x = (0::'c) \<and> g x = (0::'c) then 1::'c else f x / g x) \<longlongrightarrow> (1::'c)) F"
factors_odd: shows "odd (c - b) \<and> odd (c + b)"
dim_row_mat_of_rows_list: fixes nr :: "nat"   and ls :: "'a list list" shows "dim_row (mat_of_rows_list nr ls) = length ls"
fcomp_fid_on: fixes A :: "V"   and B :: "V" shows "fid_on A \<circ>\<^sub>\<bullet> fid_on B = fid_on (A \<inter>\<^sub>\<circ> B)"
compatibleI_with_proj: fixes G :: "'c pair_pre_digraph"   and H :: "'c pair_pre_digraph" shows "compatible (with_proj G) (with_proj H)"
holomorphic_on_chain: fixes g :: "complex \<Rightarrow> complex"   and t :: "complex set"   and f :: "complex \<Rightarrow> complex"   and s :: "complex set" assumes "g holomorphic_on t"   and "f holomorphic_on s"   and "f ` s \<subseteq> t" shows "(\<lambda>x. g (f x)) holomorphic_on s"
real_of_ereal_ord_simps_4: fixes y :: "ereal"   and x :: "real" shows "(real_of_ereal y < x) = ((\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x))"
less_imp_lex: fixes xs :: "nat list"   and ys :: "nat list" assumes "xs <\<^sub>v ys" shows "xs <\<^sub>l\<^sub>e\<^sub>x ys"
assms: shows "NextPerformOp' c0 c1 p \<Delta>neg \<Delta>mint_msg \<Delta>mint_self"
epath_restrict: fixes p :: "'b uprod list"   and I :: "'b uprod set"   and E :: "'b uprod set"   and u :: "'b"   and v :: "'b" assumes "set p \<subseteq> I"   and "epath E u p v" shows "epath (E \<inter> I) u p v"
proves_eq_combination: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term"   and x :: "Core.term"   and y :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> x"   and "typ_of x = Some \<tau>"   and "term_ok \<Theta> y"   and "typ_of y = Some \<tau>"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq f g \<longmapsto> mk_eq x y \<longmapsto> mk_eq (f $ x) (g $ y)"
simps: fixes \<epsilon> :: "'a"   and n :: "'a"   and x :: "'a" shows "sqrt_approx_main_impl \<epsilon> n x = (if x * x - n < \<epsilon> then x else sqrt_approx_main_impl \<epsilon> n ((n / x + x) / (2::'a)))"
foldl_snd_zip: fixes ys :: "'a list"   and xs :: "'b list"   and f :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and b :: "'c" assumes "length ys \<le> length xs" shows "foldl (\<lambda>b (x, y). f b y) b (zip xs ys) = foldl f b ys"
bintrunc_numeral: fixes k :: "num"   and x :: "int" shows "take_bit (numeral k) x = of_bool (odd x) + 2 * take_bit (pred_numeral k) (x div 2)"
accepts_rexp2na: fixes r :: "'b rexp"   and w :: "'b list" shows "NA.accepts (rexp2na r) w = (w \<in> lang r)"
select_convs_16: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and rreqid :: "nat"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and more :: "'a" shows "sip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, rreqid = rreqid, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = sip"
strip_bot_acom: fixes c :: "com" shows "strip (\<bottom>\<^sub>c c) = c"
aezdeltafun_plus: fixes b :: "'a"   and c :: "'a"   and a :: "'b" shows "(b + c) \<delta>\<delta> a = b \<delta>\<delta> a + c \<delta>\<delta> a"
common_boundary_subdiv_exists_refl': fixes C :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "\<forall>(k, \<gamma>)\<in>C. valid_path \<gamma>"   and "boundary_chain C" shows "common_boundary_sudivision_exists C C"
MGT: shows "\<forall>Z. \<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
ipoly_poly_add: fixes p :: "int poly"   and q :: "int poly"   and x :: "'a"   and y :: "'a" assumes "of_int (lead_coeff p) \<noteq> (0::'a)"   and "of_int (lead_coeff q) \<noteq> (0::'a)"   and "ipoly p x = (0::'a)"   and "ipoly q y = (0::'a)" shows "ipoly (poly_add p q) (x + y) = (0::'a)"
valid_prefix: shows "mask (LENGTH('a) - len) AND base = 0"
quasi_isometry_on_bounded: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "lambda C -quasi_isometry_on X f"   and "bounded X" shows "bounded (f ` X)"