hcju/isabelleps huggingface dataset:
rel_option_iff: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b option"   and y :: "'c option" shows "rel_option R x y = (case (x, y) of (None, None) \<Rightarrow> True | (None, Some a) \<Rightarrow> False | (Some x, None) \<Rightarrow> False | (Some x, Some y) \<Rightarrow> R x y)"
assms: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list"   and data :: "nat \<times> 'd" shows "snd ` set (np gs bs hs data) \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs) \<and> set hs \<times> (set gs \<union> set bs) \<subseteq> snd ` set (np gs bs hs data) \<and> (\<forall>a b. a \<in> set hs \<longrightarrow> b \<in> set hs \<longrightarrow> a \<noteq> b \<longrightarrow> (a, b) \<in>\<^sub>p snd ` set (np gs bs hs data)) \<and> (\<forall>p q. (True, p, q) \<in> set (np gs bs hs data) \<longrightarrow> q \<in> set gs \<union> set bs)"
arcosh_ge_0: fixes x :: "real" assumes "1 \<le> x" shows "0 \<le> arcosh x"
dom_closed: fixes f :: "'a" assumes "left (trg \<mu>) f" shows "left (trg \<mu>) (local.dom f)"
rank_of_eq_card_basis_in: fixes \<E> :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> insert x X"   and "X_insert.basis_in \<E> B" shows "X_insert.rank_of \<E> = card B"
LIST_FOREACH'_def: fixes tsl :: "'d list nres"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and \<sigma> :: "'c" shows "LIST_FOREACH' tsl c f \<sigma> \<equiv> tsl \<bind> (\<lambda>xs. nfoldli xs c f \<sigma>)"
positively_invariant_eq_rev: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and M :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.positively_invariant f X M = auto_ll_on_open.negatively_invariant (- f) X M"
div_less: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "a < b" shows "a div b = (0::'a)"
rep_eq: fixes x :: "('b, 'd) urel"   and xa :: "('d, 'c) urel" shows "\<lbrakk>x ;; xa\<rbrakk>\<^sub>e = (\<lambda>b. b \<in> {p. \<lbrakk>x\<rbrakk>\<^sub>e p} O {q. \<lbrakk>xa\<rbrakk>\<^sub>e q})"
inf_th_14_OT_P4: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "R2_14_OT msgs C = funct_OT_14 msgs C \<bind> (\<lambda>(s1, s2). S2_14_OT C s2)"
induct: fixes G :: "prog"   and x1 :: "qtname \<times> mdecl"   and x2 :: "qtname \<times> mdecl"   and P :: "qtname \<times> mdecl \<Rightarrow> qtname \<times> mdecl \<Rightarrow> bool" assumes "G \<turnstile> x1 overrides x2"   and "\<And>new old. \<lbrakk>\<not> is_static new; \<not> is_static old; accmodi new \<noteq> Private; msig new = msig old; G\<turnstile>declclass new\<prec>\<^sub>C declclass old; G\<turnstile>Method new declared_in declclass new; G\<turnstile>Method old declared_in declclass old; G \<turnstile>Method old inheritable_in pid (declclass new) ; G\<turnstile>resTy new\<preceq>resTy old\<rbrakk> \<Longrightarrow> P new old"   and "\<And>new intr old. \<lbrakk>G \<turnstile> new overrides intr; P new intr; G \<turnstile> intr overrides old; P intr old\<rbrakk> \<Longrightarrow> P new old" shows "P x1 x2"
complementary_def: fixes H1 :: "'a set"   and H2 :: "'a set" shows "H.complementary H1 H2 = (H1 \<inter> H2 = {\<one>\<^bsub>G\<lparr>carrier := H\<rparr>\<^esub>})"
elementary_category_with_pullbacks_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj0 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and prj1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "elementary_category_with_pullbacks C prj0 prj1 \<equiv> category C \<and> elementary_category_with_pullbacks_axioms C prj0 prj1"
simps_4: fixes bs :: "real list"   and a :: "MIR.num" shows "Ifm bs (Gt a) = (0 < Inum bs a)"
semiconnex_def: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "semiconnex A less \<equiv> \<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> less x y \<or> x = y \<or> less y x"
isCont_o2: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd" assumes "isCont f a"   and "isCont g (f a)" shows "isCont (\<lambda>x. g (f x)) a"
lset_lzipD1: fixes x :: "'c"   and y :: "'d"   and xs :: "'c llist"   and ys :: "'d llist" assumes "(x, y) \<in> lset (lzip xs ys)" shows "x \<in> lset xs"
assms_6: assumes "s1\<Colon>\<preceq>(G, L)"   and "normal s1 \<Longrightarrow> G,L,snd s1\<turnstile>t\<succ>v\<Colon>\<preceq>T"   and "error_free s0 = error_free s1" shows "P"
NotBot_provable: fixes \<Gamma> :: "'b formula set" shows "\<Gamma> \<turnstile> \<^bold>\<not> \<bottom>"
inf_bot_left: fixes x :: "'k multiset" shows "{#} \<inter># x = {#}"
last_index_append: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" shows "last_index (xs @ ys) x = (if x \<in> set ys then length xs + last_index ys x else if x \<in> set xs then last_index xs x else length xs + length ys)"
filter_cong: fixes xs :: "'c list"   and ys :: "'c list"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "xs = ys"   and "\<And>x. x \<in> set ys \<Longrightarrow> P x = Q x" shows "filter P xs = filter Q ys"
hfun_sigma_def: fixes r :: "hf" shows "hfun_sigma r \<equiv> \<forall>z\<^bold>\<in>r. \<forall>z'\<^bold>\<in>r. \<exists>x y x' y'. z = \<langle>x, y\<rangle> \<and> z' = \<langle>x', y'\<rangle> \<and> (x = x' \<longrightarrow> y = y')"
rtb_rbl_ariths_3: fixes v :: "'a word"   and ys :: "bool list"   and w :: "'a word"   and xs :: "bool list" assumes "rev (to_bl v) = ys"   and "rev (to_bl w) = xs" shows "rev (to_bl (v * w)) = rbl_mult ys xs"
sorted_wrt_check_generate: fixes a :: "nat list"   and b :: "nat list" shows "sorted_wrt (<\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2) (check_generate a b)"
length_ladder_cut: fixes L :: "(nat \<times> nat \<times> nat) list"   and n :: "nat" assumes "L \<noteq> []" shows "length (ladder_cut L n) = length L"
Max_by_def: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "Max_by f S = (SOME x. x \<in> S \<and> f x = Max (f ` S))"
that: fixes f :: "complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "f a = 0"   and "f holomorphic_on ball 0 1"   and "f ` ball 0 1 \<subseteq> ball 0 1"   and "g holomorphic_on ball 0 1"   and "g ` ball 0 1 \<subseteq> ball 0 1"   and "\<And>z. z \<in> ball 0 1 \<Longrightarrow> f (g z) = z"   and "\<And>z. z \<in> ball 0 1 \<Longrightarrow> g (f z) = z" shows "thesis"
ex_map_conv: fixes ys :: "'h list"   and f :: "'g \<Rightarrow> 'h" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
d_x_zero: fixes x :: "'a" shows "d (x \<cdot> (0::'a)) = x \<cdot> (0::'a) \<parallel> 1\<^sub>\<sigma>"
reaches_def: fixes x :: "'h"   and y :: "'h" shows "x reaches y = (\<lambda>s. (\<lambda>x y. (x points_to y) s)\<^sup>*\<^sup>* x y)"
sigma_property_disjoint_lemma: fixes C :: "'a set set" assumes "M \<subseteq> C"   and "closed_cdi \<Omega> C" shows "sigma_sets \<Omega> M \<subseteq> C"
OT_14_P2_sec: fixes R2_12OT :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT122 spmf"   and n :: "nat"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "OT_14.perfect_sec_P2 R2_12OT n m1 m2"
assms_14: shows "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
2: fixes z :: "('fun, 'atom) Term.term \<times> nat" assumes "Var z \<in> set V" shows "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var z) = \<Gamma> (Var y)"
Fun_subterm_inside_params: fixes t :: "('c, 'd) Term.term"   and f :: "'c"   and X :: "('c, 'd) Term.term list" shows "(t \<sqsubset> Fun f X) = (t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set X)"
tst: shows "ts t = \<lfloor>xw\<rfloor>"
State_works: fixes h :: "nat"   and n_ops :: "nat"   and t :: "nat"   and k :: "nat" assumes "cnf_to_dimacs.valid_state_var h n_ops (State t k)" shows "cnf_to_dimacs.dimacs_to_var h n_ops (cnf_to_dimacs.var_to_dimacs h n_ops (State t k)) = State t k"
ii: assumes "\<exists>C'\<in>fst ` N. C' \<prec>\<cdot> C" shows "(C, L) \<in> Red_F_\<G> N"
r_badblock: fixes n :: "nat"   and v :: "nat" shows "eval r_badblock [n, v] \<down>= Partial_Recursive.list_encode (replicate n 0 @ [1 - v])"
max2_equals1: fixes a :: "'b \<Rightarrow> 'a"   and b :: "'b \<Rightarrow> 'a" assumes "a \<in> Field oexp"   and "b \<in> Field oexp" shows "(o.max2 a b = a) = ((b, a) \<in> oexp)"
abs_eq: fixes xa :: "'c list"   and x :: "'c \<Rightarrow> 'd" shows "Mapping.tabulate xa x = mapping.Mapping (map_of (map (\<lambda>k. (k, x k)) xa))"
in_keys_monom_mult_le: fixes v :: "'t"   and c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<in> keys (monom_mult c t p)" shows "v \<preceq>\<^sub>t t \<oplus> lt p"
vec_eq_f: shows "\<forall>i. vec_to_iarray (A $ i) = f (mod_type_class.to_nat i)"
ID_ccompare: fixes ccompare :: "('c \<Rightarrow> 'c \<Rightarrow> order) option"   and c :: "'c \<Rightarrow> 'c \<Rightarrow> order" assumes "class.ccompare ccompare"   and "ID ccompare = Some c" shows "class.linorder (le_of_comp c) (lt_of_comp c)"
degree_linear_power_neg: fixes a :: "'a"   and n :: "nat" shows "degree ([:a, - (1::'a):] ^ n) = n"
to_fun_ctrm: fixes f :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "f \<in> carrier P"   and "b \<in> carrier R" shows "to_fun (monom P (f 0) 0) b = f 0"
l_sort_form_aux: fixes xs :: "'b list" shows "l_sort_form (l_sort_aux (xs, [], []))"
scale_spmf_1: fixes p :: "'a spmf" shows "scale_spmf 1 p = p"
mult_right: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" shows "(\<lambda>x. f x * h x) \<in> L F (\<lambda>x. g x * h x)"
fun_pair_subst_set: fixes \<delta> :: "'b \<Rightarrow> ('fun, 'a) Term.term"   and M :: "(('fun, 'b) Term.term \<times> ('fun, 'b) Term.term) set" shows "pair ` M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta> = pair ` (M \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<delta>)"
preserves_dom: fixes f :: "'a" assumes "S.arr f" shows "S.dom (\<rr> f) = R (S.dom f)"
isFDERIV_isDERIV_D: fixes n :: "nat"   and xs :: "nat list"   and fas :: "floatarith list"   and vs :: "real list"   and i :: "nat"   and j :: "nat" assumes "isFDERIV n xs fas vs"   and "i < n"   and "j < n" shows "isDERIV (xs ! i) (fas ! j) vs"
divide_zero_left: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b" assumes "field R"   and "a \<in> carrier R"   and "a \<noteq> \<zero>\<^bsub>R\<^esub>" shows "\<zero>\<^bsub>R\<^esub> \<oslash>\<^bsub>R\<^esub> a = \<zero>\<^bsub>R\<^esub>"
Q5''_a_def: shows "Q5''_a \<equiv> [: p, t, lnk, lbl, mrk \<leadsto> p', t', lnk', lbl', mrk' . p \<noteq> nil \<and> (\<forall>i. \<not> g mrk lbl (lnk i) p) \<and> t = nil \<and> p' = nil \<and> t' = t \<and> lnk' = lnk \<and> lbl' = lbl \<and> mrk' = mrk :]"
dirichlet_prod'_floor_conv_sum_upto: fixes f :: "nat \<Rightarrow> real"   and x :: "real" shows "dirichlet_prod' f (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x = sum_upto (\<lambda>xa. sum_upto f (x / real xa)) x"
prems: shows "a \<in> outstanding_refs is_Write\<^sub>s\<^sub>b (x # xs)"
rcs_Unit1: fixes H :: "'a set" assumes "G \<guillemotright> H " shows "H \<bullet> \<one> = H"
inv_iso_def: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" shows "H.inv_iso hom \<equiv> \<lparr>iso_verts = the_inv_into (verts (app_iso iso_f G)) (iso_verts hom), iso_arcs = the_inv_into (arcs (app_iso iso_f G)) (iso_arcs hom), iso_head = head (app_iso iso_f G), iso_tail = tail (app_iso iso_f G)\<rparr>"
invt_invt_inImage: fixes \<sigma> :: "'tp"   and a :: "univ" assumes "\<not> unprot \<sigma>"   and "\<not> protFw \<sigma>"   and "a \<in> (eintF (Tag \<sigma>) \<circ> singl) ` {b. eintT \<sigma> b}" shows "eintT \<sigma> (invt \<sigma> a) \<and> eintF (Tag \<sigma>) [invt \<sigma> a] = a"
dom_trg: fixes \<mu> :: "'a" assumes "C.arr \<mu>" shows "C.dom (trg\<^sub>C \<mu>) = trg\<^sub>C \<mu>"
is_threshold_def: fixes c :: "(nat set \<times> nat set set) set"   and t :: "nat \<Rightarrow> real" shows "is_threshold c t \<equiv> ugraph_property c \<and> (\<forall>p. nonzero_prob_fun p \<longrightarrow> (p \<lless> t \<longrightarrow> prob_in_class p c \<longlonglongrightarrow> 0) \<and> (t \<lless> p \<longrightarrow> prob_in_class p c \<longlonglongrightarrow> 1))"
SPN_r_SPN_s_apex_eq: shows "SPN_r_SPN_s.apex = Maps.MkIde (src rs.\<rho>\<sigma>.tab)"
rel_fundefs_Some2: fixes f :: "'a \<Rightarrow> ('b, 'c, 'd, 'op) Std.instr fundef option"   and g :: "'a \<Rightarrow> ('b, 'c, 'd, 'op, 'opinl) Inca.instr fundef option"   and x :: "'a"   and y :: "('b, 'c, 'd, 'op, 'opinl) Inca.instr fundef" assumes "rel_fundefs f g"   and "g x = Some y" shows "\<exists>z. f x = Some z \<and> rel_fundef (\<lambda>x y. x = norm_instr y) z y"
n_inter_num_choose_design_inter: fixes b1 :: "'a set"   and b2 :: "'a set"   and n :: "nat" assumes "b1 \<in># \<B>"   and "b2 \<in># \<B>" shows "(b1 |\<inter>|\<^sub>n b2) = int (nat (b1 |\<inter>| b2) choose n)"
exhaust_sel: fixes envT :: "('d, 'e) envT" obtains "envT = EnvT (run_env envT)"
defined_StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_I: fixes \<tau> :: "'d state \<times> 'd state"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e"   and y :: "'d state \<times> 'd state \<Rightarrow> 'e" assumes "\<tau> \<Turnstile> \<upsilon> x"   and "\<tau> \<Turnstile> \<upsilon> y" shows "\<tau> \<Turnstile> \<delta> StrictRefEq\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t x y"
arctan_upper_13_def: shows "arctan_lower_12 \<equiv> \<lambda>x. x"
FL_valid_Act: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act, 'effect) formula" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>\<alpha>' x' P'. Act f \<alpha> x = Act f \<alpha>' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>',P'\<rangle> \<and> P' \<Turnstile> x')"
axioms_1: fixes \<U> :: "'c set"   and \<A> :: "'c set multiset"   and \<V> :: "'c set"   and \<B> :: "'c set multiset" assumes "sub_design \<U> \<A> \<V> \<B>" shows "sub_incidence_system \<U> \<A> \<V> \<B>"
asym: fixes a :: "'a word"   and b :: "'a word" assumes "a <s b"   and "b <s a" shows "False"
AND_NOT_mask_plus_AND_mask_eq: fixes w :: "'a word"   and n :: "nat" shows "(w AND NOT (mask n)) + (w AND mask n) = w"
sc_eq_box_box_2: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi>) \<^bold>\<rightarrow> (\<^bold>\<not>\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<not>\<phi>) in v]"
markedElementsAreElements: fixes m :: "'a"   and M :: "('a \<times> bool) list" assumes "m \<in> set (markedElements M)" shows "m \<in> set (elements M)"
elem_in_singleton_append: fixes x :: "'a vec"   and a :: "'a" shows "(x @\<^sub>v [a]\<^sub>v) $ dim_vec x = a"
sup_Sup: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "sup x y = Sup (insert x (\<eta> y))"
nrename_interesting1: fixes x :: "name"   and y :: "name"   and z :: "name"   and M :: "trm" assumes "distinct [x, y, z]" shows "M[x\<turnstile>n>z][z\<turnstile>n>y] = M[z\<turnstile>n>y][x\<turnstile>n>y]"
small_to_big: fixes cs :: "com \<times> (char list \<Rightarrow> int)"   and t :: "char list \<Rightarrow> int" assumes "cs \<rightarrow>* (SKIP, t)" shows "cs \<Rightarrow> t"
packet_assume_new_simple_ruleset: fixes rs :: "'a common_primitive rule list" assumes "simple_ruleset (rs::'a common_primitive rule list)" shows "simple_ruleset (packet_assume_new (rs::'a common_primitive rule list))"
induct: fixes P :: "('c \<times> ('c, 'd) RHS) list \<Rightarrow> (nat \<times> nat \<Rightarrow> 'c list) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "('c \<times> ('c, 'd) RHS) list"   and a1 :: "nat \<times> nat \<Rightarrow> 'c list"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat" assumes "\<And>G T len i j. \<lbrakk>\<And>x. \<lbrakk>x = T((i, j) := inner G T i 1 j); i + j < len\<rbrakk> \<Longrightarrow> P G x len (i + 1) j; \<And>x. \<lbrakk>x = T((i, j) := inner G T i 1 j); \<not> i + j < len; j < len\<rbrakk> \<Longrightarrow> P G x len 0 (j + 1)\<rbrakk> \<Longrightarrow> P G T len i j" shows "P a0 a1 a2 a3 a4"
hyps_2: shows "\<R> = {region X I r |I r. valid_region X k I r}"
mem_extend_set_iff: fixes z :: "'c"   and A :: "'a set" shows "(z \<in> extend_set h A) = (f z \<in> A)"
take_bit_eq_mask: 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 n :: "nat"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "take_bit n a = and a (mask n)"
mem_Times_mset_iff: fixes x :: "'b \<times> 'c"   and A :: "'b multiset"   and B :: "'c multiset" shows "(x \<in># A \<times># B) = (fst x \<in># A \<and> snd x \<in># B)"
rel_fun_def_butlast: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and T :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'g" shows "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> rel_fun S T (f x) (g y))"
vertices_def: fixes labeled_graph :: "('c, 'b) labeled_graph" shows "vertices labeled_graph = (case labeled_graph of LG x1 x2 \<Rightarrow> x2)"
ALG_P_split: fixes qs :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and q :: "'c"   and init :: "'c list"   and x :: "'c"   and y :: "'c" assumes "length qs < length Strat" shows "ALG_Pxy Strat (qs @ [q]) init x y = ALG_Pxy Strat qs init x y + ALG_P' Strat (qs @ [q]) init (length qs) x y"
v2_1: shows "is_measured_ftree s \<and> is_leveln_ftree 0 s \<and> is_leveln_node 0 nd \<and> is_measured_node nd"
Un_ac_4: fixes A :: "'e set"   and B :: "'e set"   and C :: "'e set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
cxt_inv_stop: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt" assumes "cxt_to_stmt E c = Stop" shows "c = Stop \<and> E = []"
eval_wp_Mu: fixes T :: "(bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wp (\<mu>t. T t) = lfp_trans (\<lambda>t. wp (T (Embed t)))"
nths_append: fixes l :: "'b list"   and l' :: "'b list"   and A :: "nat set" shows "nths (l @ l') A = nths l A @ nths l' {j. j + length l \<in> A}"
equiv_trans_def: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "equiv_trans t u = (le_trans t u \<and> le_trans u t)"
sumlist_carrier: fixes xs :: "'c list" assumes "set xs \<subseteq> carrier M" shows "M.sumlist xs \<in> carrier M"
distinct_13: fixes x41 :: "'b formula"   and x42 :: "'b formula" shows "\<bottom> \<noteq> x41 \<^bold>\<and> x42"
Trivial_Aexp_def: fixes e :: "exp" shows "Trivial_Aexp e = (\<Lambda> n. (\<lambda>x. up\<cdot>0) f|` fv e)"
transE: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" obtains "(x, z) \<in> r"
\<L>\<^sub>P_derives: fixes a :: "symbol list" assumes "a \<in> \<L>\<^sub>P" shows "\<exists>b. derives [\<SS>] (a @ b)"
col_trivial_3: fixes A :: "'p"   and B :: "'p" shows "Col A B A"
compl_unique: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "inf x y = bot"   and "sup x y = top" shows "uminus x = y"
authKeys_simp: fixes K :: "nat"   and A :: "agent"   and K' :: "nat"   and Peer :: "agent"   and Ta :: "nat"   and evs :: "event list" assumes "K \<in> authKeys (Says Kas A \<lbrace>Crypt (shrK A) \<lbrace>Key K', Agent Peer, Number Ta\<rbrace>, Crypt (shrK Peer) \<lbrace>Agent A, Agent Peer, Key K', Number Ta\<rbrace>\<rbrace> # evs)" shows "K = K' \<or> K \<in> authKeys evs"
AbstFormP_Eq: fixes v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and y :: "tm"   and y' :: "tm" shows "{AbstTermP v i x x', AbstTermP v i y y'} \<turnstile> AbstFormP v i (Q_Eq x y) (Q_Eq x' y')"
map\<^sub>0_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>D (\<sigma>.map\<^sub>0 a) = G.map\<^sub>0 a"
f: shows "((\<lambda>x. F (\<gamma>1 x) \<bullet> b * (vector_derivative \<gamma>1 (at x within {0..1}) \<bullet> b)) has_integral integral {\<phi> 0..\<phi> 1} (\<lambda>x. F (\<gamma>2 x) \<bullet> b * (vector_derivative \<gamma>2 (at x within {0..1}) \<bullet> b))) {0..1}"
real_mult_le_cancel_left_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b"   and "less zero c" shows "less_eq (times c a) (times c b) = less_eq a b"
collinear_affine_hull_collinear: fixes s :: "'b set" shows "collinear (affine hull s) = collinear s"
narrow_type_1: fixes \<Delta> :: "binding list"   and Q :: "POPLmarkRecord.type"   and \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and P :: "POPLmarkRecord.type" assumes "\<Delta> @ TVarB Q \<Colon> \<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> P <: Q" shows "\<Delta> @ TVarB P \<Colon> \<Gamma> \<turnstile> t : T"
asym: fixes a :: "'b list"   and b :: "'b list" assumes "strict_prefix a b"   and "strict_prefix b a" shows "False"
is_map_to_list_def: fixes Rk :: "('f \<times> 'g) set"   and Rv :: "('h \<times> 'i) set"   and Rm :: "('f \<times> 'g) set \<Rightarrow> ('h \<times> 'i) set \<Rightarrow> ('j \<times> ('g \<Rightarrow> 'i option)) set"   and tsl :: "'j \<Rightarrow> ('f \<times> 'h) list" shows "is_map_to_list Rk Rv Rm tsl \<equiv> is_map_to_sorted_list (\<lambda>_ _. True) Rk Rv Rm tsl"
p_altdef: fixes k :: "nat" shows "p k = insertion x (powsum_mpoly {..<n} k)"
ide_MkIde: fixes A :: "'e set" assumes "A \<in> {A. |A| <o \<AA>}" shows "ide (MkIde A)"
listall_cong: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "xs = ys" shows "listall P xs = listall P ys"
wf_r_cases: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and P :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set \<Rightarrow> bool" assumes "update_arg_wf r"   and "\<And>nhip pre. P (0, unk, val, Suc 0, nhip, pre)"   and "\<And>dsn hops nhip pre. 0 < dsn \<Longrightarrow> P (dsn, kno, val, hops, nhip, pre)" shows "P r"