hcju/isabelleps huggingface dataset:
SafeGlobVacantUptoImpliesStickyNrec_def: fixes s :: "('c, 'd) configuration stream" shows "SafeGlobVacantUptoImpliesStickyNrec s = (let c = shd s in \<forall>t q. GlobVacantUpto c q t \<longrightarrow> alw (holds (\<lambda>c. NrecVacantUpto c t)) s)"
Span_eq_combine_set: fixes K :: "'a list set"   and Us :: "'a list list" assumes "subfield K (K [X])"   and "set Us \<subseteq> carrier (K [X])" shows "UP.Span K Us = {UP.combine Ks Us |Ks. set Ks \<subseteq> K}"
alphaAbs_qAbs_iff_alphaAbs_all_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_all_qFresh xs x X xs' x' X'"
lOfWax_def: fixes \<sigma> :: "'b" shows "lOfWax \<sigma> = Fn (Tag \<sigma>) [rOfWax \<sigma>]"
triangle_strict_reverse_inequality: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and D :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet A B D"   and "Cong A C A D"   and "\<not> Tarski_neutral_dimensionless.Out Bet A B C" shows "Tarski_neutral_dimensionless.Lt Bet Cong B D B C"
induct: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and a1 :: "nat"   and a2 :: "nat"   and a3 :: "nat"   and a4 :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less"   and "\<And>m n k. P m n k 0 0"   and "\<And>m n k i. P m n k i n \<Longrightarrow> P m n k (Suc i) 0"   and "\<And>m n k i j. P m n k i j \<Longrightarrow> P m n k i (Suc j)" shows "P a0 a1 a2 a3 a4"
red2_Diag: fixes a :: "'c local.term"   and b :: "'c local.term" assumes "Diag (a \<^bold>\<otimes> b)" shows "a \<^bold>\<Down> b = a \<^bold>\<otimes> b"
no_failure: fixes i :: "example_peers \<times> nat" assumes "example.is_valid_state_id i" shows "isOK (example.state i)"
ennreal_inverse_le: fixes x :: "ennreal"   and y :: "ennreal" shows "(inverse x \<le> inverse y) = (y \<le> x)"
normal_polyI: fixes p :: "'a poly" assumes "p \<noteq> 0"   and "\<And>i. (0::'a) \<le> coeff p i"   and "\<And>i. coeff p i * coeff p (i + 2) \<le> (coeff p (i + 1))\<^sup>2"   and "\<And>i j k. \<lbrakk>i \<le> j; j \<le> k; (0::'a) < coeff p i; (0::'a) < coeff p k\<rbrakk> \<Longrightarrow> (0::'a) < coeff p j" shows "normal_poly p"
precise_pure_iff_sv: fixes R :: "('b \<times> 'a) set" shows "precise (pure R) = single_valued R"
FIRST: shows "((f, l), a, fh, lh) \<in> pr_algo_lts'"
length_nat: fixes n :: "nat" shows "length (nat_to_bv n) = length_nat n"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes> y \<in> carrier R"
path_connected_Euclidean_space: fixes n :: "nat" shows "path_connected_space (Euclidean_space n)"
axioms_1: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" assumes "sylow G p a m" shows "Group.group G"
lt_monic: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "lpp (punit.monic p) = lpp p"
dom_simp_3: shows "J.dom J.j0 = J.Zero"
before_in_irefl: fixes x :: "'c"   and xs :: "'c list" shows "x < x in xs = False"
br_dsq_def: fixes \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d" shows "br_dsq \<delta> q \<equiv> \<lambda>(Q, W). {lhs r |r. r \<in> \<delta> \<and> set (rhsq r) \<subseteq> Q - (W - {q})}"
warywary_subst_subst\<^sub>h_conv: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm" shows "wary_subst \<rho> = (\<forall>x. wary (\<rho> x) \<and> arity_var\<^sub>h x \<le> arity\<^sub>h (\<rho> x) \<and> local.ground_heads (head (\<rho> x)) \<subseteq> ground_heads_var x)"
Suc: shows "real n * x + 1 \<le> (x + 1) ^ n"
hyps_2: fixes t :: "trm" assumes "[] \<turnstile> t : T"   and "t \<in> value" shows "\<exists>a. \<turnstile> p \<rhd> t \<Rightarrow> a"
mem_range_tms_Cons_iff: fixes x :: "real"   and xs :: "real list"   and e :: "nat \<Rightarrow> real"   and X :: "taylor_model"   and XS :: "taylor_model list" shows "(x # xs all_in\<^sub>i range_tms e (X # XS)) = (x \<in>\<^sub>i range_tm e X \<and> xs all_in\<^sub>i range_tms e XS)"
surj_rename_imp_surj: fixes h :: "'d \<Rightarrow> 'c" assumes "surj (rename h)" shows "surj h"
g: fixes x :: "'d" assumes "x \<in> S" shows "g differentiable at x"
simps: fixes a :: "'c set" shows "finite a = (a = {} \<or> (\<exists>A aa. a = {aa} \<union> A \<and> finite A))"
col_mat_of_cols: fixes j :: "nat"   and vs :: "'c vec list"   and n :: "nat" assumes "j < length vs"   and "vs ! j \<in> carrier_vec n" shows "col (mat_of_cols n vs) j = vs ! j"
f_def: fixes z :: "'c" shows "f z \<equiv> fst (inv h z)"
conjugate_def: shows "conjugate \<equiv> map_fun rep_complex_homo abs_complex_homo conjugate_hcoords"
degree_smult_eq: fixes a :: "'a"   and p :: "'a poly" shows "degree (Polynomial.smult a p) = (if a = (0::'a) then 0 else degree p)"
eq_P: shows "(\<lambda>b. {x \<in> A. f x = b}) ` B - {{}} = P"
SI_insert_mod: fixes x :: "'a"   and s :: "'s" assumes "I x"   and "SI s" shows "SI (insert_mod x s)"
axioms_1: fixes G :: "('a, 'b) graph" assumes "valid_unMultigraph G" shows "valid_graph G"
lookup_not_eq_zero_eq_in_keys: fixes f :: "'e \<Rightarrow>\<^sub>0 'd"   and k :: "'e" shows "(lookup f k \<noteq> (0::'d)) = (k \<in> keys f)"
simps_1: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "F.satL \<xi> (Pos a) = F.satA \<xi> a"
iptables_goto_bigstep_Undecided_deterministic: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and t :: "state"   and t' :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t"   and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, Undecided\<rangle> \<Rightarrow> t'" shows "t' = t"
bb_Suc_n: shows "bb (Suc (card X)) = 0"
isomorphic_def: fixes a :: "'b"   and a' :: "'b" shows "B.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>B a'\<guillemotright> \<and> B.iso f)"
le_iff_sup: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (sup x y = y)"
bij_betw_iff_card: fixes A :: "'d set"   and B :: "'e set" assumes "finite A"   and "finite B" shows "(\<exists>f. bij_betw f A B) = (card A = card B)"
seqE: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'c"   and f :: "'c" assumes "category C"   and "partial_magma.arr C (C g f)" obtains "partial_magma.arr C f" and "partial_magma.arr C g" and "partial_magma.dom C g = partial_magma.cod C f"
normalize_alt_simps_1: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a :: "'a literal_expr"   and b :: "'a expr" shows "(\<Gamma> \<turnstile> Literal a \<Rrightarrow> b) = (b = Literal a)"
move_matrix_col_mult: fixes A :: "'b matrix"   and B :: "'b matrix"   and i :: "int" shows "move_matrix (A * B) 0 i = A * move_matrix B 0 i"
scale_measure_0: fixes M :: "'b measure" shows "scale_measure 0 M = null_measure M"
covering_space_lift_general: fixes C :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set"   and a :: "'c"   and z :: "'e"   and U :: "'e set"   and f :: "'e \<Rightarrow> 'd" assumes "covering_space C p S"   and "a \<in> C"   and "z \<in> U"   and "path_connected U"   and "locally path_connected U"   and "continuous_on U f"   and "f ` U \<subseteq> S"   and "f z = p a"   and "\<And>r. \<lbrakk>path r; path_image r \<subseteq> U; pathstart r = z; pathfinish r = z\<rbrakk> \<Longrightarrow> \<exists>q. path q \<and> path_image q \<subseteq> C \<and> pathstart q = a \<and> pathfinish q = a \<and> homotopic_paths S (f \<circ> r) (p \<circ> q)" obtains g :: "'e \<Rightarrow> 'c" where "continuous_on U g" and "g ` U \<subseteq> C" and "g z = a" and "\<And>y. y \<in> U \<Longrightarrow> p (g y) = f y"
conc_fun_SPEC: fixes R :: "('b \<times> 'c) set"   and \<Phi> :: "'c \<Rightarrow> bool" shows "\<Down> R (SPEC \<Phi>) = SPEC (\<lambda>y. \<exists>x. (y, x) \<in> R \<and> \<Phi> x)"
boundr_def: fixes x :: "nat list"   and y :: "nat list" shows "boundr x y = (\<forall>j<n. y ! j \<le> max_y x j)"
d: shows "DS (M1 * \<rho> * adjoint M1) \<in> carrier_mat m m"
Out1_def: fixes x :: "nat"   and y :: "nat"   and s1 :: "nat" shows "Out1 x y s1 = (let s2 = (x * y + (q - s1)) mod q in return_spmf (s1, s2))"
inf_K_enc_orb: fixes next_behaviour :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and next_stimulus :: "'c \<times> 'd \<Rightarrow> 'c"   and orbit :: "'d \<Rightarrow> 'd set"   and strong_orbit :: "'d \<Rightarrow> 'd set"   and stabiliser :: "'d \<Rightarrow> 'c set"   and fixed :: "'d \<Rightarrow> bool"   and encompassing_relation_behaviours :: "'d set \<Rightarrow> 'd set \<Rightarrow> bool"   and encompassing_relation_stimuli :: "'c set \<Rightarrow> 'c set \<Rightarrow> bool"   and induced :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and orbit_equivalent :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "topology_c2ka next_behaviour next_stimulus orbit strong_orbit stabiliser fixed encompassing_relation_behaviours encompassing_relation_stimuli induced orbit_equivalent"   and "a \<le>\<^sub>\<K> b" shows "encompassing_relation_behaviours (orbit a) (orbit b)"
assms_2: shows "A *\<^sub>m\<^sub>m M = B *\<^sub>m\<^sub>m M"
union_of_inc: fixes P :: "'b set set \<Rightarrow> bool"   and S :: "'b set"   and Q :: "'b set \<Rightarrow> bool" assumes "P {S}"   and "Q S" shows "(P union_of Q) S"
in_eq_out_arc_ends: fixes u :: "'a"   and v :: "'a" shows "((u, v) \<in> arc_to_ends G ` in_arcs G v) = ((v, u) \<in> arc_to_ends G ` out_arcs G v)"
le_supI2: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x b" shows "less_eq x (sup a b)"
complex_cnj_of_int: fixes z :: "int" shows "cnj (of_int z) = of_int z"
tuple_simps_1: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "span f g" shows "arr (hfsetcat.tuple f g)"
distinct_48: fixes x71 :: "'c"   and x72 :: "('a, 'b, 'c) Stmt"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "Await x71 x72 \<noteq> Stmt.If x51 x52 x53"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> (\<phi> and\<^sub>n \<psi>) w = \<G>\<F> \<phi> w \<union> \<G>\<F> \<psi> w"
mk_all_list_empty: fixes t :: "Core.term" shows "mk_all_list [] t = t"
simps: fixes x :: "'b list" shows "\<rho> x = The (primitive_rootP x)"
bres_fres_eq: fixes x :: "'a"   and y :: "'a" shows "proto_near_quantale.bres Sup (\<le>) (\<sqinter>) x y = proto_near_quantale.fres Sup (\<le>) (\<sqinter>) y x"
assms_2: shows "A \<inter> \<real>\<^sub>\<le>\<^sub>0 = {}"
configuration_step_eq_q\<^sub>0: fixes n :: "nat" shows "configuration q\<^sub>0 (Suc n) = {Suc n} \<union> \<Union> {configuration q' n |q'. q\<^sub>0 = \<delta> q' (w n)}"
simps_2: fixes x :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "list_diff x (y # ys) = list_diff (removeAll y x) ys"
less_eq_score_ext_def: fixes s1 :: "'a score_scheme"   and s2 :: "'a score_scheme" shows "(s1 \<le> s2) = (s1 < s2 \<or> s1 = s2)"
from_hma\<^sub>m_add: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and b :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>m a + Mod_Type_Connect.from_hma\<^sub>m b = Mod_Type_Connect.from_hma\<^sub>m (a + b)"
the_inv_into_map_Some': fixes V :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and y :: "'a" shows "(the_inv_into_map V f x = Some y) = (x \<in> f ` V \<and> y = the_inv_into V f x)"
cobounded2: fixes b :: "'e multiset"   and a :: "'e multiset" shows "b \<subseteq># a \<union># b"
elim_def: fixes \<sigma> :: "('b \<times> 'b trm) list"   and v :: "'b" shows "elim \<sigma> v \<equiv> \<forall>t. v \<notin> vars_of (t \<lhd> \<sigma>)"
enn2ereal_ennreal: fixes x :: "real" assumes "0 \<le> x" shows "enn2ereal (ennreal x) = ereal x"
del_block_def: fixes b :: "'a set" shows "del_block b \<equiv> remove1_mset b \<B>"
term_def: fixes x :: "'b" shows "TERM x \<equiv> (\<And>A. PROP A \<Longrightarrow> PROP A)"
RD_def: fixes Pi :: "(('c, bool) fmap \<times> ('c, bool) fmap) set" shows "RD Pi \<equiv> Sup {length p - 1 |p. valid_path Pi p \<and> distinct p}"
lucas_lehmer_to_real_def: shows "lucas_lehmer_to_real = (\<lambda>(a, b). real_of_int a + real_of_int b * sqrt 3)"
edges_less_eq_def: fixes a :: "'c \<times> 'd \<times> 'c"   and b :: "'c \<times> 'd \<times> 'c" shows "edges_less_eq a b \<equiv> fst (snd a) \<le> fst (snd b)"
pentagon: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "B.ide k"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h"   and "src\<^sub>B h = trg\<^sub>B k" shows "(f \<star>\<^sub>B \<a>\<^sub>B[g, h, k]) \<cdot>\<^sub>B \<a>\<^sub>B[f, g \<star>\<^sub>B h, k] \<cdot>\<^sub>B \<a>\<^sub>B[f, g, h] \<star>\<^sub>B k = \<a>\<^sub>B[f, g, h \<star>\<^sub>B k] \<cdot>\<^sub>B \<a>\<^sub>B[f \<star>\<^sub>B g, h, k]"
inf1D2: fixes A :: "'f \<Rightarrow> bool"   and B :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "inf A B x" shows "B x"
ccSUP_least: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and 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 A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and u :: "'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
fua: shows "lesvars (map (\<lambda>(x, e, v). (x, \<lambda>s. e (S s), \<Union> (SS ` v))) upds) = lesvars upds"
ring_hom_memE_2: fixes h :: "'a \<Rightarrow> 'b"   and R :: "('a, 'c) ring_scheme"   and S :: "('b, 'd) ring_scheme"   and x :: "'a"   and y :: "'a" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<otimes>\<^bsub>R\<^esub> y) = h x \<otimes>\<^bsub>S\<^esub> h y"
evenperm_swap: fixes a :: "'a"   and b :: "'a" shows "evenperm (Fun.swap a b id) = (a = b)"
iMOD_div_eq_imp_iMODb_div_eq: fixes k :: "nat"   and m :: "nat"   and r' :: "nat"   and m' :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < k"   and "k \<le> m"   and "[ r', mod m' ] = [ r, mod m ] \<oslash> k" shows "[ r', mod m', c ] = [ r, mod m, c ] \<oslash> k"
length_getpren_params: fixes m :: "nat"   and n :: "nat" shows "length (get_fstn_args m n) = n"
v1_iteratei_impl: shows "set_iteratei ls.\<alpha> ls.invar ls.iteratei"
image_block_set_constant_size: fixes B :: "'c set multiset"   and f :: "'c \<Rightarrow> 'd" shows "size B = size (blocks_image B f)"
sublinear_intros_7: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> supp (P s) \<Longrightarrow> sublinear (wp (p a))"   and "\<And>s. sum (P s) (supp (P s)) \<le> 1"   and "\<And>s a. 0 \<le> P s a"   and "\<And>s. finite (supp (P s))" shows "sublinear (wp (SetPC p P))"
zero_order_5: fixes n :: "'d multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
bij: shows "bij (\<lambda>(l, n, m). (m, l, n))"
Quotient3_rel_rep: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and a :: "'f"   and b :: "'f" assumes "Quotient3 R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)"
lub_bin: fixes below :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.po below"   and "below x y" shows "po.lub below {x, y} = y"
return_edges_only: fixes a :: "'a"   and Q' :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and p :: "'f"   and f' :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<forall>a'. valid_edge a' \<and> sourcenode a' = sourcenode a \<longrightarrow> (\<exists>Qx fx. kind a' = Qx\<hookleftarrow>\<^bsub>p\<^esub>fx)"
diff_fun_spaceD: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> diff_fun_space" shows "diff_fun k charts f"
a: shows "\<forall>x. (x \<in> A) = (a + x \<in> B)"
abs_supp_2: fixes a :: "name"   and x :: "'a" shows "supp ([a].x) = supp x - {a}"
collinear_lemma: fixes x :: "'a"   and y :: "'a" shows "collinear {0::'a, x, y} = (x = (0::'a) \<or> y = (0::'a) \<or> (\<exists>c. y = c *\<^sub>R x))"
cont: shows "\<forall>A. range A \<subseteq> M \<longrightarrow> decseq A \<longrightarrow> \<Inter> (range A) = {} \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> 0"
bit_signed_take_bit_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and m :: "nat"   and a :: "'b"   and n :: "nat" 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" shows "bit (ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not m a) n = (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n \<noteq> zero \<and> bit a (min m n))"
wlp_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "wlp \<pi> c P s"   and "\<And>s. P s \<Longrightarrow> Q s" shows "wlp \<pi> c Q s"
assms_3: shows "emb_step_at' p t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
RedBlack_subpaths_are_black_subpaths: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and rv :: "'d \<times> nat" assumes "RedBlack prb" shows "RedBlack_subpaths_from prb rv \<subseteq> subpaths_from (black prb) (fst rv)"
signum_Integer_def: shows "signum = (\<Lambda> (MkI\<cdot>x). MkI\<cdot>(sgn x))"
rel_converter_map_converter'1: shows "rel_converter A\<inverse>\<inverse> B\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> (map_converter' f g conv2) conv1 = rel_converter A\<inverse>\<inverse> (\<lambda>x. B\<inverse>\<inverse> (f x)) (\<lambda>x. C\<inverse>\<inverse> (g x)) R\<inverse>\<inverse> conv2 conv1"
assms_2: shows "0\<^sub>h \<notin> circline_set l"
rel_F_pos_distr_cond_eq: fixes tytok :: "('t \<times> 'u \<times> 'v \<times> 'w \<times> 'x \<times> 'y \<times> 'z \<times> 'aa \<times> 'ab \<times> 'ac) itself" shows "rel_F_pos_distr_cond (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) (=) tytok"
notin_image_iff: fixes y :: "'a"   and f :: "'b \<Rightarrow> 'a"   and I :: "'b set" shows "(y \<notin> f ` I) = (\<forall>i\<in>I. f i \<noteq> y)"
FAccRedsVal: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and fs :: "Objects.fields"   and F :: "char list"   and D :: "char list"   and v :: "val"   and t :: "ty" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>addr a,s',b'\<rangle>"   and "hp s' a = \<lfloor>(C, fs)\<rfloor>"   and "fs (F, D) = \<lfloor>v\<rfloor>"   and "P \<turnstile> C has F,NonStatic:t in D" shows "P \<turnstile> \<langle>e\<bullet>F{D},s,b\<rangle> \<rightarrow>* \<langle>Val v,s',b'\<rangle>"
rep_eq: shows "(*\<^sub>V) id_cblinfun = (\<lambda>x. x)"
inversion_of_complex: fixes z :: "complex" assumes "z \<noteq> 0" shows "inversion (of_complex z) = of_complex (1 / cnj z)"
sel: fixes x3 :: "'b" shows "vertex.inner (Inner x3) = x3"
f_Streams_Connected_Init_def: fixes outS :: "'d message_af list"   and inS :: "'d message_af list" shows "f_Streams_Connected_Init outS inS \<equiv> inS = outS"
validV1: shows "isViewOn \<V>1 E\<^bsub>ES1\<^esub>"
bvars\<^sub>s\<^sub>t_split: fixes S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list" shows "bvars\<^sub>s\<^sub>t (S @ S') = bvars\<^sub>s\<^sub>t S \<union> bvars\<^sub>s\<^sub>t S'"
dropUpd: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and x :: "Vars.var"   and S :: "stack_elem list" shows "(\<Gamma>, e, Upd x # S) \<Rightarrow>\<^sub>G (\<Gamma>, e, S @ [Dummy x])"
diff_eq: fixes x :: "'a"   and y :: "'a" shows "x - y = x \<cdot> - y"
bot_ideal: fixes X :: "'a set" assumes "idealp X" shows "\<bottom> \<in> X"
execute_parallel_plan_is_execute_sequential_plan_if: fixes is_valid_problem :: "'b \<Rightarrow> bool"   and \<Pi> :: "'b"   and G :: "'c \<Rightarrow> bool option"   and I :: "'c \<Rightarrow> bool option"   and \<pi> :: "'c strips_operator list list" assumes "is_valid_problem \<Pi>"   and "G \<subseteq>\<^sub>m last (trace_parallel_plan_strips I \<pi>)"   and "\<forall>k<length \<pi>. are_all_operators_applicable (trace_parallel_plan_strips I \<pi> ! k) (\<pi> ! k) \<and> are_all_operator_effects_consistent (\<pi> ! k) \<and> are_all_operators_non_interfering (\<pi> ! k)" shows "G \<subseteq>\<^sub>m execute_serial_plan I (concat \<pi>)"
in_Inf_FL_imp_to_F_in_Inf_F: fixes \<iota> :: "('f \<times> 'l) inference" assumes "\<iota> \<in> Inf_FL" shows "to_F \<iota> \<in> Inf_F"
preserves_hom: fixes f :: "'a \<times> 'a \<times> 'a"   and a :: "'a \<times> 'a \<times> 'a"   and b :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.in_hom f a b" shows "\<guillemotleft>\<a>\<^sub>B[fst f, fst (snd f), snd (snd f)] : B.HoHV a \<rightarrow>\<^sub>B B.HoVH b\<guillemotright>"
ordermap_in_ordertype: fixes a :: "'a"   and A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "a \<in> A"   and "small A" shows "ordermap A r a \<in> elts (ordertype A r)"
complex_cnj_i: shows "cnj \<i> = - \<i>"
simps_1: fixes c :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)"   and c' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list \<times> ('Var \<Rightarrow> 'Val)" shows "c \<rightarrow>\<^bsub>0\<^esub> c' = (c = c')"
function_factors_right: fixes g :: "'c \<Rightarrow> 'd"   and f :: "'b \<Rightarrow> 'd" shows "(\<forall>x. \<exists>y. g y = f x) = (\<exists>h. f = g \<circ> h)"
int_pow_closed: fixes x :: "'a"   and i :: "int" assumes "x \<in> carrier R" shows "[i] \<cdot> x \<in> carrier R"
subprob_imp_subprob_cexpr: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr" assumes "density_context V V' \<Gamma> (\<lambda>x. ennreal (extract_real (cexpr_sem x \<delta>)))" shows "subprob_cexpr V V' \<Gamma> \<delta>"