hcju/isabelleps huggingface dataset:
project_act_extend_act_restrict: fixes C :: "'c set"   and act :: "('a \<times> 'a) set" shows "project_act h (Restrict C (extend_act h act)) = Restrict (project_set h C) act"
inline1_parametric': fixes S :: "'k \<Rightarrow> 'p \<Rightarrow> bool"   and C :: "'l \<Rightarrow> 'q \<Rightarrow> bool"   and R :: "'m \<Rightarrow> 'r \<Rightarrow> bool"   and C' :: "'n \<Rightarrow> 's \<Rightarrow> bool"   and R' :: "'o \<Rightarrow> 't \<Rightarrow> bool"   and A :: "'u \<Rightarrow> 'v \<Rightarrow> bool" shows "((S ===> C ===> rel_gpv'' (rel_prod R S) C' R') ===> rel_gpv'' A C R ===> S ===> rel_spmf (rel_sum (rel_prod A S) (rel_prod C' (rel_prod (R' ===> rel_gpv'' (rel_prod R S) C' R') (R ===> rel_gpv'' A C R))))) inline1 inline1"
finsum_vec: shows "finsum_vec TYPE(rat) n = finsum gs.V"
src_hcomp: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "D.hseq \<mu> \<nu>" shows "src\<^sub>D (\<mu> \<star>\<^sub>D \<nu>) = src\<^sub>D \<nu>"
map_upd_Some_swap: fixes tab :: "'b \<Rightarrow> 'a option"   and r :: "'b"   and w :: "'a"   and u :: "'b"   and v :: "'a"   and vn :: "'b"   and z :: "'a" assumes "(tab(r \<mapsto> w, u \<mapsto> v)) vn = Some z" shows "\<exists>z. (tab(u \<mapsto> v, r \<mapsto> w)) vn = Some z"
minus_splus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and s :: "'b"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "s adds t" shows "term_powerprod.splus pair_of_term term_of_pair (t - s) v = term_powerprod.sminus pair_of_term term_of_pair (term_powerprod.splus pair_of_term term_of_pair t v) s"
extremum: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "mbot \<preceq> a"
f8: shows "cpu_reg_val PSR s2 = cpu_reg_val PSR s1"
NatTrans: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and f :: "'f"   and X :: "'d"   and Y :: "'d" assumes "NatTransP \<eta>"   and "f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y" shows "NTDom \<eta> ## f ;;\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> $$ Y) = \<eta> $$ X ;;\<^bsub>NTCatCod \<eta>\<^esub> (NTCod \<eta> ## f)"
arrow_from_functor_axioms: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a d (J_C.MkArr \<chi>.A.map (J_C.Map d) \<chi>)"
proper_design_axioms_def: fixes block_collection :: "'b set multiset" shows "proper_design_axioms block_collection \<equiv> int (size block_collection) \<noteq> 0"
pfsubset_eq: fixes A :: "'a fset"   and B :: "'a fset" shows "(A |\<subset>| B) = (A |\<subseteq>| B \<and> A \<noteq> B)"
assms: shows "ys \<in> listset (xs1 @ xs2)"
preserves_ide: fixes a :: "'b" assumes "B.ide a" shows "C.ide (F a)"
strict_iff_order: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "ordering less_eq less" shows "less a b = (less_eq a b \<and> a \<noteq> b)"
lookup2_index: fixes l1 :: "'b list"   and l2 :: "'a list"   and f :: "'b \<Rightarrow> bool"   and e :: "'a" assumes "lookup2 l1 l2 f = Some e"   and "length l1 = length l2" shows "\<exists>i<length l2. e = l2 ! i \<and> lookup l1 f = Some (l1 ! i)"
perm_contr': fixes P :: "'b list \<Rightarrow> 'c"   and n :: "nat" assumes "\<forall>xs ys. xs <~~> ys \<longrightarrow> P xs = P ys"   and "\<forall>x xs. P (x # x # xs) = P (x # xs)" shows "\<forall>xs. length xs = n \<longrightarrow> P xs = P (remdups xs)"
intro: fixes step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and ins :: "'b list"   and c :: "'a list"   and T :: "'a"   and B :: "'a"   and A :: "'a set" assumes "bounded step (length ins)"   and "cert_ok c (length ins) T B A"   and "pres_type step (length ins) A" shows "lbvs_axioms A T B step c ins"
card_complex_roots_unity_explicit: fixes n :: "nat" shows "card {exp (2 * complex_of_real pi * \<i> * of_nat j / of_nat n) |j. j < n} = n"
Smith_normal_form_upt_k_condition1: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat"   and a :: "'c"   and b :: "'b" assumes "Smith_normal_form_upt_k A k"   and "to_nat a = to_nat b"   and "to_nat a + 1 < k"   and "to_nat b + 1 < k" shows "A $ a $ b dvd A $ (a + (1::'c)) $ (b + (1::'b))"
has_density_sets: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal" assumes "has_density M N f" shows "sets M = sets N"
of_nat_0: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero 0 = zero"
integral_minus: shows "((-) 1 has_integral 1 / 2) (cbox 0 1)"
abs_in_Basis_iff: fixes u :: "'a" shows "(\<bar>u\<bar> \<in> Basis) = (u \<in> Basis \<or> - u \<in> Basis)"
bdt_Some_Zero_iff: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat" shows "(bdt t var = Some Zero) = (\<exists>p. t = Node Tip p Tip \<and> var p = 0)"
uminus_unit_def: fixes uu :: "unit" shows "- uu = ()"
disjunctive_def: shows "Apply.disjunctive = {x. \<forall>y z. x (sup y z) = sup (x y) (x z)}"
semiring_normalization_rules_7: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "times a b = times b a"
is_column: fixes n :: "nat"   and v :: "complex mat" assumes "state n v" shows "dim_col v = 1"
special_cf_pol_coeff: shows "pol_coeff S (C\<^sub>0 S)"
Suc_k: shows "Suc k < min (nrows A) (ncols A)"
bin_rep_index: fixes n :: "nat"   and m :: "nat"   and i :: "nat" assumes "1 \<le> n"   and "m < 2 ^ n"   and "i < n"   and "0 \<le> m" shows "bin_rep n m ! i = m mod 2 ^ (n - i) div 2 ^ (n - 1 - i)"
emb_step_equiv: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" shows "(t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s) = ((\<exists>p d. emb_step_at p d t = s) \<and> t \<noteq> s)"
aff: fixes X :: "'a set" assumes "X \<in> \<F> - \<G>" shows "aff_dim X < aff_dim T"
assms: fixes i :: "nat" shows "f i \<in> borel_measurable M"
conc_fst: fixes n :: "nat"   and w :: "'e list"   and x :: "nat \<Rightarrow> 'e" assumes "n < length w" shows "(w \<frown> x) n = w ! n"
mset_set_Diff: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "B \<subseteq> A" shows "mset_set (A - B) = mset_set A - mset_set B"
strict_trans: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subset># b"   and "b \<subset># c" shows "a \<subset># c"
assms_2: shows "[a ^ (n - 1) = 1] (mod n)"
sep_disj_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.pre_sep_algebra plus zero sep_disj" shows "sep_disj x zero"
step_def: fixes A :: "'b \<times> ('c \<Rightarrow> 'b \<Rightarrow> 'b set) \<times> ('b \<Rightarrow> bool)"   and a :: "'c" shows "step A a = {(p, q). q \<in> next A a p}"
succsr_def: fixes succs :: "'c \<Rightarrow> 'd list" shows "succsr succs = {(x, y). y \<in> set (succs x)}"
measurable_Sup1: fixes m :: "'b measure"   and M :: "'b measure set"   and f :: "'b \<Rightarrow> 'c"   and N :: "'c measure" assumes "m \<in> M"   and "f \<in> m \<rightarrow>\<^sub>M N"   and "\<And>m n. \<lbrakk>m \<in> M; n \<in> M\<rbrakk> \<Longrightarrow> space m = space n" shows "f \<in> Sup M \<rightarrow>\<^sub>M N"
Inf_enat_def: fixes A :: "enat set" shows "\<Sqinter> A = (if A = {} then \<infinity> else LEAST x. x \<in> A)"
locally_refl_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset" shows "locally_refl R A \<equiv> \<forall>a. a \<in># A \<longrightarrow> (a, a) \<in> R"
assms_1: shows "v \<in> set (delete_effects_of op)"
ProblemIkPolMcalc2C_axioms: shows "ProblemIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf"
of_nat_divide_in_Ints_iff: fixes b :: "nat"   and a :: "nat" assumes "b \<noteq> 0" shows "(of_nat a / of_nat b \<in> \<int>) = (b dvd a)"
diff_size_le_size_Diff: fixes M :: "'b multiset"   and M' :: "'b multiset" shows "size M - size M' \<le> size (M - M')"
cone_hull_expl: fixes S :: "'a set" shows "cone hull S = {c *\<^sub>R x |c x. 0 \<le> c \<and> x \<in> S}"
card_order_csum: fixes r1 :: "('k \<times> 'k) set"   and r2 :: "('l \<times> 'l) set" assumes "card_order r1"   and "card_order r2" shows "card_order (r1 +c r2)"
factorize_polys_map_square_free: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and fss :: "real poly list"   and f :: "real poly" assumes "factorize_polys ps = (fs, data)"   and "fss = map real_of_rat_poly fs"   and "f \<in> set fss" shows "square_free f"
group_l_invI: assumes "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes> x = \<one>" shows "Group.group G"
\<theta>\<^sub>f: shows "\<guillemotleft>\<theta>\<^sub>f : tab\<^sub>0 f \<star> w\<^sub>f \<rightarrow> u\<^sub>f\<guillemotright>"
at_right_to_0: fixes a :: "real" shows "at_right a = filtermap (\<lambda>x. x + a) (at_right 0)"
simps_1: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "'d" shows "weight_fun.weight w w0 scf (Var x) = w0"
inj_swap0: fixes b :: "nat"   and i :: "nat" shows "inj (swap0 b i)"
lfilter_LNil: fixes P :: "'b \<Rightarrow> bool" shows "lfilter P LNil = LNil"
mset_subset_eq_exists_conv: fixes A :: "'k multiset"   and B :: "'k multiset" shows "(A \<subseteq># B) = (\<exists>C. B = A + C)"
composite_simps_6: shows "local.dom ((cod \<mu> \<star> cod \<nu>) \<star> cod \<pi>) = (local.dom (cod \<mu>) \<star> local.dom (cod \<nu>)) \<star> local.dom (cod \<pi>)"
rev_conv_fold: fixes xs :: "'d list" shows "rev xs = fold (#) xs []"
loop_init: shows "inv\<^sub>3 {} {} {} {} {u \<in> U. w u \<le> real c / 2} {u \<in> U. real c / 2 < w u}"
equal_suffix_states: shows "\<forall>k\<ge>j. S (swap_events i (j - 1) (t[j - 1 := t ! j, j := t ! (j - 1)])) k = S (t[j - 1 := t ! j, j := t ! (j - 1)]) k"
even_mult_iff: fixes a :: "'a"   and b :: "'a" shows "even (a * b) = (even a \<or> even b)"
ta1: shows "ta1 = convert_TA_initial (convert_obs_initial ta1')"
assms: shows "rt = update rt dip (dsn, dsk, flag, hops, nhip)"
perfect_sec_P2_def: fixes m1 :: "bool \<times> bool"   and m2 :: "bool" shows "OT_12_sim.perfect_sec_P2 m1 m2 \<equiv> R2_OT12 m1 m2 = funct_OT_12 m1 m2 \<bind> (\<lambda>(s1, s2). S2_OT12 m2 s2)"
norm_blinfun: fixes f :: "'e \<Rightarrow>\<^sub>L 'd"   and x :: "'e" shows "norm (blinfun_apply f x) \<le> norm f * norm x"
sum_case_Inr: fixes f :: "hf \<Rightarrow> 'b"   and g :: "hf \<Rightarrow> 'b"   and y :: "hf" shows "sum_case f g (HF.Inr y) = g y"
norm_of_real_add1: fixes x :: "real" shows "norm (of_real x + (1::'c)) = \<bar>x + 1\<bar>"
Opair: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF"   and d :: "ZF" shows "(Opair a b = Opair c d) = (a = c \<and> b = d)"
sams_lea2_suma2__conga456: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and D' :: "'p"   and E' :: "'p"   and F' :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C LeA A' B' C'"   and "D E F LeA D' E' F'"   and "SAMS A' B' C' D' E' F'"   and "A B C D E F SumA G H I"   and "A' B' C' D' E' F' SumA G H I" shows "D E F CongA D' E' F'"
replicate_Suc: fixes n :: "nat"   and x :: "'g" shows "replicate (Suc n) x = x # replicate n x"
simps_246: shows "BinXor \<noteq> BinAnd"
prems_2: shows "\<sigma>' \<in> small_nstep P \<sigma> (Suc n1)"
abs_eq: fixes xa :: "'b poly"   and x :: "'b poly" shows "xa + x = Abs_poly (\<lambda>n. poly.coeff xa n + poly.coeff x n)"
LT13: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and H :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> (F \<leadsto> G) \<longrightarrow> (G \<leadsto> H) \<longrightarrow> F \<leadsto> H"
C_restr_eq_lower: fixes r :: "C"   and f :: "C \<rightarrow> 'a"   and g :: "C \<rightarrow> 'a"   and r' :: "C" assumes "f|\<^bsub>r\<^esub> = g|\<^bsub>r\<^esub>"   and "r' \<sqsubseteq> r" shows "f|\<^bsub>r'\<^esub> = g|\<^bsub>r'\<^esub>"
intv_of_def: fixes c :: "nat"   and v :: "real" shows "intv_of c v \<equiv> if real c < v then Greater c else if \<exists>x. real x = v then Const (nat \<lfloor>v\<rfloor>) else Intv (nat \<lfloor>v\<rfloor>)"
plus_vec_def: fixes v\<^sub>1 :: "'b vec"   and v\<^sub>2 :: "'b vec" shows "v\<^sub>1 + v\<^sub>2 \<equiv> vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $ i + v\<^sub>2 $ i)"
sum_goestozero_theorem: fixes a :: "nat \<Rightarrow> real" assumes "summable (\<lambda>i. a i / real i)"   and "\<And>n. a n + ln (real n) \<le> a (Suc n) + ln (real (Suc n))" shows "a \<longlonglongrightarrow> 0"
orderE: fixes a :: "'t"   and b :: "'t" assumes "a \<preceq>\<^sub>t b" obtains "a = ord_term_lin.min a b"
rep_eq: fixes x :: "('b, 'c) uexpr"   and xa :: "'c \<Longrightarrow> 'a" shows "\<lbrakk>x \<oplus>\<^sub>p xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>x\<rbrakk>\<^sub>e (get\<^bsub>xa\<^esub> b))"
SubstTerm_subst_dbtm: fixes e :: "(name, hf) finfun"   and u :: "dbtm"   and v :: "name"   and t :: "dbtm" assumes "Term \<lbrakk>quot_dbtm u\<rbrakk>e" shows "SubstTerm (q_Var v) \<lbrakk>quot_dbtm u\<rbrakk>e \<lbrakk>quot_dbtm t\<rbrakk>e \<lbrakk>quot_dbtm (subst_dbtm u v t)\<rbrakk>e"
Cons_replicate_eq: fixes x :: "'a"   and xs :: "'a list"   and n :: "nat"   and y :: "'a" shows "(x # xs = replicate n y) = (x = y \<and> 0 < n \<and> xs = replicate (n - 1) x)"
adapt_aexp_def: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" shows "adapt_aexp e s = (\<lambda>\<sigma>\<^sub>s\<^sub>y\<^sub>m. e (\<lambda>v. \<sigma>\<^sub>s\<^sub>y\<^sub>m (symvar v s)))"
assms_2: fixes g :: "'a set" assumes "g \<in> \<G>" shows "int (card g) = \<k> - 1"
diff_def: fixes x :: "'b bit1"   and y :: "'b bit1" shows "x - y = Abs_bit1 ((Rep_bit1 x - Rep_bit1 y) mod int CARD('b bit1))"
not_less_LNil: fixes r :: "'b llist" shows "\<not> r < LNil"
Exp_Exp_Gen_inj2: fixes X :: "msg"   and X' :: "msg"   and Z :: "msg"   and Y' :: "msg" assumes "Exp (Exp Gen X) X' = Exp Z Y'" shows "Y' = X \<and> Z = Exp Gen X' \<or> Y' = X' \<and> Z = Exp Gen X"
init_fin_bisim_iff: fixes t :: "'t"   and s1 :: "status"   and x1 :: "'x1"   and m1 :: "'m1"   and s2 :: "status"   and x2 :: "'x2"   and m2 :: "'m2" shows "t \<turnstile> ((s1, x1), m1) \<approx>i ((s2, x2), m2) = (s1 = s2 \<and> t \<turnstile> (x1, m1) \<approx> (x2, m2) \<and> (s2 = Finished \<longrightarrow> final1 x1 \<and> final2 x2))"
sub_deg: fixes g :: "nat \<Rightarrow> 'a"   and f :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier P"   and "f \<in> carrier P"   and "g \<noteq> \<zero>\<^bsub>P\<^esub>" shows "deg R (Cring_Poly.compose R f g) = deg R f * deg R g"
constructive_security_obsf_refl: fixes \<eta> :: "nat"   and \<A> :: "('a + 'e, 'b + 'f) distinguisher_obsf" shows "constructive_security_obsf (real_resource \<eta>) (ideal_resource \<eta>) (sim \<eta>) (\<I>_real \<eta>) (\<I>_ideal \<eta>) (\<I>_common \<eta>) \<A> (advantage \<A> (obsf_resource (sim \<eta> |\<^sub>= 1\<^sub>C \<rhd> ideal_resource \<eta>)) (obsf_resource (real_resource \<eta>)))"
tangent_spaceI: fixes k :: "enat"   and X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "extensional0 (manifold_eucl.diff_fun_space k) X"   and "linear_on (manifold_eucl.diff_fun_space k) UNIV (*\<^sub>R) (*\<^sub>R) X"   and "\<And>f g. \<lbrakk>f \<in> manifold_eucl.diff_fun_space k; g \<in> manifold_eucl.diff_fun_space k\<rbrakk> \<Longrightarrow> X (f * g) = f p * X g + g p * X f" shows "X \<in> manifold_eucl.tangent_space k p"
lens_equiv_sym: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "X \<approx>\<^sub>L Y" shows "Y \<approx>\<^sub>L X"
val: fixes n :: "nat" assumes "dec (\<sigma> n p) = Some v" shows "v \<in> range vals"
sum_le_suminf: fixes f :: "nat \<Rightarrow> 'd"   and I :: "nat set" assumes "summable f"   and "finite I"   and "\<And>n. n \<in> - I \<Longrightarrow> (0::'d) \<le> f n" shows "sum f I \<le> suminf f"
restrict_def': fixes v :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "restrict v f c = lan v \<sqinter> f c"
right_options_neg: fixes u :: "game" shows "right_options (neg_game u) = zimage neg_game (left_options u)"
simps_4: fixes e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" shows "E (EApp e1 e2) \<rho> = {v3. \<exists>f v2 v2' v3'. VFun f \<in> E e1 \<rho> \<and> v2 \<in> E e2 \<rho> \<and> (v2', v3') \<in> fset f \<and> v2' \<sqsubseteq> v2 \<and> v3 \<sqsubseteq> v3'}"
lim: shows "passive_subset (Liminf_llist Ns) = {}"
intro: fixes step :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and final :: "'a \<Rightarrow> bool" assumes "semantics step final" shows "language step final"
finite_char_def: fixes C :: "'b set set" shows "finite_char C = (\<forall>S. (S \<in> C) = (\<forall>S'. finite S' \<longrightarrow> S' \<subseteq> S \<longrightarrow> S' \<in> C))"
i_def: shows "i \<equiv> LEAST i. \<not> suffix i w \<Turnstile>\<^sub>n \<psi>"
inter_add_left1: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<notin># N" shows "add_mset x M \<inter># N = M \<inter># N"
sum_pos: fixes I :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> (0::'b) < f i" shows "(0::'b) < sum f I"
emsapss_decode_help11_def: fixes H' :: "bit list"   and H :: "bit list" shows "emsapss_decode_help11 H' H = (if H' \<noteq> H then False else True)"
select_convs_1: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) 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 pre :: "nat set"   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 "ip \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, pre = pre, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, \<dots> = more\<rparr> = ip"
reductio_aa_3: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<^bold>\<not>\<psi> in v]"   and "[\<^bold>\<not>\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<phi> in v]"
defined: shows "apply_chart c ` a.sub.carrier \<subseteq> a.dest.carrier"
npeSum2: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<plusminus> y)^\<^bsup>R n\<^esup> = \<Sigma>\<^sub>e R (\<lambda>i.  (n choose i) \<times>\<^bsub>R\<^esub> (x^\<^bsup>R (n - i)\<^esup> \<cdot>\<^sub>r y^\<^bsup>R i\<^esup>)) n"
apply_cm_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'c, 'e) Propagate.configuration_scheme"   and loc :: "'d"   and t :: "'c"   and n :: "int" assumes "dataflow_topology summary results_in" shows "dataflow_topology.apply_cm c loc t n = (let new_pointstamps = \<lambda>loc'. if loc' = loc then update_zmultiset (c_pts c loc') t n else c_pts c loc' in c\<lparr>c_pts := new_pointstamps, c_work := \<lambda>loc'. c_work c loc' + (zmset_of (mset_set (set_antichain (frontier (new_pointstamps loc')))) - zmset_of (mset_set (set_antichain (frontier (c_pts c loc')))))\<rparr>)"
div_right_ineq: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "x < y *n z" shows "x div y < z"
f: fixes p :: "'b"   and q :: "'b" shows "((p, q) \<in> f) = (\<exists>k l u. k \<parallel> l \<and> l \<parallel> p \<and> p \<parallel> u \<and> k \<parallel> q \<and> q \<parallel> u)"
multiset_inter_single: fixes a :: "'c"   and b :: "'c" assumes "a \<noteq> b" shows "{#a#} \<inter># {#b#} = {#}"
pcr_cr_eq: shows "transaction_merkle.pcr_transaction\<^sub>h = transaction_merkle.cr_transaction\<^sub>h"
fup_Aexp_edom: fixes e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "edom (Aexp_bot_syn' a e) \<subseteq> fv e"
A_meas: fixes n :: "nat" shows "A n \<in> sets M"
assoc'_in_hom'_1: fixes \<mu> :: "'a"   and \<nu> :: "'a"   and \<tau> :: "'a" assumes "arr \<mu>"   and "arr \<nu>"   and "arr \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<guillemotleft>\<a>\<^sup>-\<^sup>1[\<mu>, \<nu>, \<tau>] : src \<tau> \<rightarrow> trg \<mu>\<guillemotright>"
weak_preservation_of_barbs_and_closures_2: fixes Rel :: "('a \<times> 'a) set"   and CWB :: "('a, 'b) calculusWithBarbs" assumes "rel_weakly_preserves_barbs Rel CWB" shows "rel_weakly_preserves_barbs (Rel\<^sup>+) CWB"
positive_rat: fixes a :: "int"   and b :: "int" shows "positive (Fract a b) = (0 < a * b)"
tensor_gate_sqr_mat: fixes m :: "nat"   and G1 :: "complex Matrix.mat"   and n :: "nat"   and G2 :: "complex Matrix.mat" assumes "gate m G1"   and "gate n G2" shows "square_mat (G1 \<Otimes> G2)"
alphaAll_refl: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(FixVars.qGood X \<longrightarrow> X #= X) \<and> (FixVars.qGoodAbs A \<longrightarrow> A $= A)"
Le_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" shows "A B Le C D \<equiv> \<exists>E. Bet C E D \<and> Cong A B C E"
preserves_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.cod (\<F>C.map f) = \<F>C.map (\<F>C.cod f)"
set_disconnected_nodes_get_shadow_root: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
S1_S2: shows "S1 \<inter> S2 = {}"
lookup_one: fixes k :: "'d" shows "lookup 1 k = ((1::'c) when k = (0::'d))"
set_isin: fixes s :: "'b trie"   and x :: "'b list" assumes "True" shows "Trie_Fun.isin s x = (x \<in> Trie_Fun.set s)"
binomial_code: fixes n :: "nat"   and k :: "nat" shows "n choose k = (if n < k then 0 else if n < 2 * k then n choose (n - k) else fold_atLeastAtMost_nat (*) (n - k + 1) n 1 div fact k)"
mult_assoc2: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> V" shows "a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
inclusion: fixes f :: "'a" assumes "\<guillemotleft>f : trg e\<^sub>0 \<rightarrow> trg e\<^sub>1\<guillemotright>" shows "arr f"
domain_eq: shows "Domainp (pcr_pdevs (=)) = (\<lambda>x. finite {i. x i \<noteq> (0::'b)})"
append1_eq_conv: fixes xs :: "'e list"   and x :: "'e"   and ys :: "'e list"   and y :: "'e" shows "(xs @ [x] = ys @ [y]) = (xs = ys \<and> x = y)"
rev_path_double: fixes xs :: "('a \<times> 'b \<times> 'a) list" shows "rev_path (rev_path xs) = xs"
to_fract_1: shows "to_fract (1::'b) = 1"
const_sound: fixes c :: "real" assumes "0 \<le> c" shows "sound (\<lambda>s. c)"
lenc_length_le: fixes xs :: "nat list" shows "length xs \<le> lenc xs"
openin_Un: fixes U :: "'b topology"   and S :: "'b set"   and T :: "'b set" assumes "openin U S"   and "openin U T" shows "openin U (S \<union> T)"
weakly_bisimilar_tau_transition_weakly_bisimilar: fixes P :: "'state"   and R :: "'state"   and Q :: "'state" assumes "P \<approx>\<cdot>\<^sub>S R"   and "P \<Rightarrow>\<^sub>S Q"   and "Q \<Rightarrow>\<^sub>S R" shows "Q \<approx>\<cdot>\<^sub>S R"
summable_zero_power': fixes f :: "nat \<Rightarrow> 'a" shows "summable (\<lambda>n. f n * (0::'a) ^ n)"
k_def: shows "k \<equiv> \<lambda>y. if fst y \<le> 1 / 2 then (k1 \<circ> (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y else (k2 \<circ> (\<lambda>x. (2 *\<^sub>R fst x - 1, snd x))) y"
wf_ruleset_rest: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and m :: "'b match_expr"   and a :: "action"   and rs :: "'b rule list" assumes "wf_ruleset \<gamma> p (Rule m a # rs)" shows "wf_ruleset \<gamma> p [Rule m a]"