hcju/isabelleps huggingface dataset:
dvd_mult_left: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times a c"
vds_tys_length_eq'': fixes vds :: "vd list"   and tys :: "'a list" assumes "length vds = length tys" shows "tys = map ((\<lambda>(cl, var, ty). ty) \<circ> (\<lambda>(vd, ty). case vd of vd_def cl var \<Rightarrow> (cl, var, ty))) (zip vds tys)"
rep_eq: fixes x :: "String.literal"   and xa :: "String.literal" shows "(x < xa) = ord.lexordp (\<lambda>c d. of_char c < of_char d) (literal.explode x) (literal.explode xa)"
interior_diff: fixes S :: "'a set"   and T :: "'a set" shows "interior (S - T) = interior S - closure T"
assms_2: shows "f \<midarrow>pointwise\<rightarrow> F"
pmf_cond_bind_pmf: fixes A :: "'h set"   and f :: "'i \<Rightarrow> 'h pmf"   and p :: "'i pmf"   and x :: "'i" assumes "A \<inter> (\<Union>x\<in>set_pmf p. set_pmf (f x)) \<noteq> {}" shows "pmf (cond_bind_pmf p f A) x = pmf p x * measure_pmf.prob (f x) A / measure_pmf.prob (p \<bind> f) A"
div_exp: fixes x :: "'a"   and n :: "nat"   and y :: "'a" assumes "to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract" shows "to_fract (div_exp x y n) = to_fract x ^ n / to_fract y ^ (n - 1)"
existence_ivl_subset: fixes t0 :: "real"   and x0 :: "'a" shows "Y.existence_ivl t0 x0 \<subseteq> S"
Gcd_mult: fixes c :: "'a"   and A :: "'a set" shows "Gcd ((*) c ` A) = normalize (c * Gcd A)"
split_max: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and i :: "'b"   and j :: "'b" assumes "class.linorder less_eq less" shows "P (ord.max less_eq i j) = ((less_eq i j \<longrightarrow> P j) \<and> (\<not> less_eq i j \<longrightarrow> P i))"
pw_conc_inres: fixes R :: "('b \<times> 'c) set"   and S' :: "'c nres" shows "inres (\<Down> R S') = (\<lambda>s. nofail S' \<longrightarrow> (\<exists>s'. (s, s') \<in> R \<and> inres S' s'))"
del_left_mset_plus: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "mset_tree t = {#x#} + mset_tree t'"
assms: fixes \<sigma> :: "'a"   and s :: "'b" assumes "(\<sigma>, s) \<in> oreachable A S U" shows "P (\<sigma>, s)"
cases: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "a1 \<R>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> a2" obtains S :: "'procS" where "S \<in>S a1" and "\<lbrakk>S\<rbrakk> \<in>T a2"   | T1 :: "'procT" and T2 :: "'procT" where "T1 \<in>T a1" and "T2 \<in>T a2" and "(T1, T2) \<in> TRel"
Rat_cases: fixes q :: "rat" obtains a :: "int" and b :: "int" where "q = Rat.Fract a b" and "0 < b" and "coprime a b"
fag_gen_by_def: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and i :: "'e \<Rightarrow> 'e"   and z :: "'e" shows "fag_gen_by A f i z = \<lparr>carrier = addition_set f (aug_pm_set z i A), pop = \<lambda>x\<in>addition_set f (aug_pm_set z i A). restrict (BOP x f) (addition_set f (aug_pm_set z i A)), mop = restrict (ipp i) (addition_set f (aug_pm_set z i A)), zero = z\<rparr>"
elims_2: fixes x :: "('c, 'd) BinomialTree list" assumes "rank_invar x" obtains "x = []"   | t :: "('c, 'd) BinomialTree" where "x = [t__]"   | t :: "('c, 'd) BinomialTree" and t' :: "('c, 'd) BinomialTree" and bq :: "('c, 'd) BinomialTree list" where "x = t # t' # bq" and "rank t < rank t' \<and> rank_invar (t' # bq)"
Fvars_Pinp: shows "Fvars Pinp = {inp}"
seq_filter_pre_increasing: fixes P :: "'b \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<forall>n. \<exists>m>n. P (s m)" shows "seq_filter s P n < seq_filter s P (Suc n)"
dims2_def: fixes dims :: "nat list"   and vars :: "nat set" shows "partial_state.dims2 dims vars = nths dims (- vars)"
eq_onp_swap: fixes m :: "'b \<Rightarrow>\<^sub>0 'c"   and b :: "'b"   and i :: "'b" shows "eq_onp (\<lambda>f. finite {x. f x \<noteq> (0::'c)}) (\<lambda>x. lookup m (swap b i x)) (\<lambda>x. lookup m (swap b i x))"
quasi_inverts_components_3: fixes a :: "'d" assumes "D.obj a" shows "D.quasi_inverses (\<eta>\<^sub>0 a) (\<eta>'.map\<^sub>0 a)"
order_refl: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and x :: "'b::type" assumes "class.preorder (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (x::'b::type) x"
indexed_members_def: fixes s :: "'b fset" shows "indexed_members s = zip [0..<size s] (list_of_fset s)"
K_def: fixes y :: "'e" shows "K y \<equiv> \<lambda>x. y"
e_def: shows "e = (\<lambda>(i, j). d i j)"
local_order_carrier_closed: fixes e1 :: "'a event"   and i :: "nat"   and e2 :: "'a event" assumes "e1 \<sqsubset>\<^sup>i e2" shows "{e1, e2} \<subseteq> set (history i)"
dm_def: fixes r :: "('a \<times> 'a) set"   and M :: "'a multiset" shows "r \<down>m M = r \<down>s set_mset M"
hconf_allocate_mono: fixes h' :: "'heap"   and a :: "'addr"   and h :: "'heap"   and hT :: "htype" assumes "(h', a) \<in> allocate h hT"   and "hconf h"   and "is_htype P hT" shows "hconf h'"
not_cong_1243: 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 C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Cong A B C D" shows "\<not> Cong A B D C"
exhaust: fixes y :: "AST.varRef" obtains x1 :: "String.literal" and x2 :: "AST.expr option" and x3 :: "AST.varRef option" where "y = AST.VarRef x1 x2 x3"
N1_unsat: shows "N1 \<Turnstile>Q {B1}"
and_mask_less': fixes n :: "nat"   and w :: "'b word" assumes "n < LENGTH('b)" shows "w AND mask n < 2 ^ n"
unit_closed: shows "stk1.one_stalk U \<in> stk1.carrier_stalk"
mrw_values_new_actionD: fixes w :: "('a, 'b) obs_event action"   and obs :: "('a, 'b) obs_event action list"   and adal :: "'a \<times> addr_loc"   and P :: "'c prog"   and vs0 :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option" assumes "w \<in> set obs"   and "is_new_action w"   and "adal \<in> action_loc_aux P w" shows "\<exists>v b. mrw_values P vs0 obs adal = \<lfloor>(v, b)\<rfloor>"
rimp_lan_def: fixes X :: "'a list set"   and Z :: "'a list set" shows "rimp_lan X Z = {y. \<forall>x\<in>X. x @ y \<in> Z}"
inf_left_commute: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" shows "x \<inter># (y \<inter># z) = y \<inter># (x \<inter># z)"
n_def: shows "n = max (length (coeffs p)) (length (coeffs q))"
triangle_growth: fixes k :: "nat"   and A :: "'a mat" assumes "triangular_to k A"   and "triangular_column k A" shows "triangular_to (Suc k) A"
subset_inj_on: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'d set" assumes "inj_on f B"   and "A \<subseteq> B" shows "inj_on f A"
affine_extension2E: fixes fnctn_aff :: "nat \<Rightarrow> 'a \<times> 'a pdevs \<Rightarrow> 'b \<times> 'b pdevs \<Rightarrow> ('c \<times> 'c pdevs) option"   and fnctn :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and d :: "nat"   and a1 :: "'a \<times> 'a pdevs"   and a2 :: "'b \<times> 'b pdevs"   and X :: "'c \<times> 'c pdevs"   and e :: "nat \<Rightarrow> real" assumes "affine_extension2 fnctn_aff fnctn"   and "fnctn_aff d a1 a2 = Some X"   and "e \<in> funcset UNIV {- 1..1}"   and "degree_aform a1 \<le> d"   and "degree_aform a2 \<le> d" obtains e' :: "nat \<Rightarrow> real" where "e' \<in> funcset UNIV {- 1..1}" and "fnctn (aform_val e a1) (aform_val e a2) = aform_val e' X" and "\<And>n. n < d \<Longrightarrow> e' n = e n" and "aform_val e a1 = aform_val e' a1" and "aform_val e a2 = aform_val e' a2"
carrDp: shows "a ` Is p \<subseteq> carrier_vec n"
p_prec: shows "split_by_prec prec p = (pl, pr)"
fps_lr_inverse_starting0_2: fixes g :: "'a fps" shows "fps_right_inverse g (0::'a) = 0"
assms_3: shows "\<forall>x\<in>B. f x \<noteq> 0"
not_\<omega>_le_numeral: fixes n :: "num" shows "\<not> \<omega> \<le> numeral n"
zhmset_of_enat_le_iff_le: fixes m :: "enat"   and n :: "enat" shows "(zhmset_of_enat m \<le> zhmset_of_enat n) = (m \<le> n)"
has_fields_mono_lem: fixes P :: "'a prog"   and D :: "String.literal"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "P \<turnstile> D \<preceq>\<^sup>* C"   and "P \<turnstile> C has_fields FDTs" shows "\<exists>pre. P \<turnstile> D has_fields pre @ FDTs \<and> dom (map_of pre) \<inter> dom (map_of FDTs) = {}"
rep_in_hom: shows "\<guillemotleft>T0.trnr\<^sub>\<epsilon> r \<rho> : g \<star> f\<^sup>* \<Rightarrow> r\<guillemotright>"
set_iterator_genord_emp_correct: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" shows "set_iterator_genord set_iterator_emp {} R"
many_strongly_connected_implies_11_eq: fixes x :: "'a" assumes "many_strongly_connected x" shows "x\<^sup>\<star> ; x\<^sup>T ; x = x\<^sup>+"
emeasure_return_pmf: fixes x :: "'b"   and X :: "'b set" shows "emeasure (measure_pmf (return_pmf x)) X = indicator X x"
fun_subst: fixes s :: "('c, 'd) tm"   and \<rho> :: "'d \<Rightarrow> ('c, 'd) tm" assumes "is_App s" shows "subst \<rho> (fun s) = fun (subst \<rho> s)"
elims_2: fixes x :: "Core.term" assumes "is_variable x" obtains uu_ :: "variable" and uv_ :: "typ" where "x = Fv uu_ uv_"
0: shows "igSubstInpIPresIGWlsInp MOD \<and> igSubstBinpIPresIGWlsBinp MOD"
Fvars_exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "Fvars (exi x \<phi>) = Fvars \<phi> - {x}"
nat_pow_Suc: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and n :: "nat" assumes "Group.monoid G" shows "x [^]\<^bsub>G\<^esub> Suc n = x [^]\<^bsub>G\<^esub> n \<otimes>\<^bsub>G\<^esub> x"
assms_1: shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs, s\<rangle> \<Rightarrow> t"
two_comparisons_into_case_order_1: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and P :: "'b"   and Q :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if le_of_comp comp x y then if x = y then P else Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
relcomp_UNION_distrib: fixes s :: "('a \<times> 'c) set"   and r :: "'d \<Rightarrow> ('c \<times> 'b) set"   and I :: "'d set" shows "s O \<Union> (r ` I) = (\<Union>i\<in>I. s O r i)"
a: shows " t1 \<longrightarrow>\<^sub>1 t2"
extr_mono: fixes c :: "atom"   and T :: "atom set"   and G :: "chmsg set"   and H :: "chmsg set" assumes "c \<in> extr T G"   and "G \<subseteq> H" shows "c \<in> extr T H"
t: shows "\<langle>\<V> s'\<rangle> \<Turnstile>\<^sub>t \<T> s'"
runit\<^sub>F\<^sub>M\<^sub>C_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.runit\<^sub>F\<^sub>M\<^sub>C C a = free_monoidal_category.mkarr C \<^bold>\<r>\<^bold>[free_monoidal_category.rep C a\<^bold>]"
ccard_ex: fixes c :: "nat"   and n :: "nat"   and n'' :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "1 \<le> c"   and "c < card {i. n < i \<and> i \<le> n'' \<and> P i}" shows "\<exists>n'<n''. n < n' \<and> card {i. n < i \<and> i \<le> n' \<and> P i} = c"
path_component_of_subset: fixes S :: "'a set"   and T :: "'a set"   and x :: "'a"   and y :: "'a" assumes "S \<subseteq> T"   and "path_component S x y" shows "path_component T x y"
div_twopow_def: fixes x :: "int"   and n :: "nat" shows "div_twopow x n = x div 2 ^ n"
inject_3: fixes x3 :: "nat"   and y3 :: "nat" shows "(Nonce x3 = Nonce y3) = (x3 = y3)"
Un_Diff: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<union> B - C = A - C \<union> (B - C)"
simps: fixes a :: "'d"   and A :: "('e, 'f, 'd, 'g) FSM_scheme" shows "(a \<in> nodes A) = ((\<exists>p. a = p \<and> p = initial A) \<or> (\<exists>p aa. a = snd aa \<and> p \<in> nodes A \<and> snd aa \<in> succ A (fst aa) p))"
quotient_ClassE: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "A \<in> Partition"   and "\<And>a. a \<in> M \<Longrightarrow> P (Class a)" shows "P A"
Ball_def: fixes A :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
prems_1: fixes init :: "(pat \<times> exp0) list"   and rest :: "(pat \<times> exp0) list" assumes "pe # pes = init @ (p, e) # rest"   and "distinct (pat_bindings p [])"   and "list_all (\<lambda>a. case a of (p, e) \<Rightarrow> cupcake_pmatch cenv p v0 [] = No_match \<and> distinct (pat_bindings p [])) init"   and "cupcake_pmatch cenv p v0 [] = Match env'" shows "thesis"
fmpredD: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and x :: "'a"   and y :: "'b" assumes "fmpred P m"   and "fmlookup m x = Some y" shows "P x y"
simps_7: fixes V :: "char list"   and e :: "expr"   and A :: "char list set option" shows "\<D> (V:=e) A = \<D> e A"
Rel_def: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Transfer.Rel r \<equiv> r"
approximation_preproc_int_16: fixes i :: "int"   and j :: "int" shows "(i < j) = (real_of_int i < real_of_int j)"
control_independence_def: fixes upd :: "('c \<Rightarrow> 'c) \<Rightarrow> 'd control_state_scheme \<Rightarrow> 'd control_state_scheme" shows "\<sharp> upd \<equiv> \<forall>\<sigma> T b. break_status (upd T \<sigma>) = break_status \<sigma> \<and> return_status (upd T \<sigma>) = return_status \<sigma> \<and> upd T (\<sigma>\<lparr>return_status := b\<rparr>) = upd T \<sigma>\<lparr>return_status := b\<rparr> \<and> upd T (\<sigma>\<lparr>break_status := b\<rparr>) = upd T \<sigma>\<lparr>break_status := b\<rparr>"
fps_annir: fixes f :: "('c, 'd) fps" shows "f \<cdot> 0 = 0"
add_mset_remove_trivial: fixes x :: "'a"   and M :: "'a multiset" shows "remove1_mset x (add_mset x M) = M"
4: shows "rat_poly_plus (rat_poly_times B (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) B)) (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) (rat_poly_times A (A - rat_poly_times (rat_poly_times B B) B))) = A ^ 3 - B"
A_def: fixes Q :: "'b \<Rightarrow> 'd"   and \<pi> :: "'b \<Rightarrow> 'c option" shows "A Q \<pi> \<equiv> {(u, v). \<pi> u = Some v \<and> Q u = \<infinity>}"
plus_reg_lan_def: shows "(+) \<equiv> map_fun Rep_reg_lan (map_fun Rep_reg_lan Abs_reg_lan) (+)"
q: shows "q = fst (snd (bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)))"
lunit_in_vhom: fixes f :: "'a"   and b :: "'a" assumes "B.ide f"   and "trg\<^sub>B f = b" shows "\<guillemotleft>B.lunit f : b \<star>\<^sub>B f \<rightarrow>\<^sub>B f\<guillemotright>"
simple_bochner_integrable_eq_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "Bochner_Integration.simple_bochner_integrable M f" shows "Bochner_Integration.simple_bochner_integral M f = integral\<^sup>L M f"
finter_finsert_left_ifffempty: fixes a :: "'c"   and C :: "'c fset"   and B :: "'c fset" assumes "a |\<notin>| C" shows "finsert a B |\<inter>| C = B |\<inter>| C"
qSwap_preserves_qFresh_distinct: fixes ys :: "'d"   and zs :: "'d"   and y :: "'e"   and z1 :: "'e"   and z2 :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm" assumes "ys \<noteq> zs \<or> y \<notin> {z1, z2}" shows "qFresh ys y (X #[[z1 \<and> z2]]_zs) = qFresh ys y X"
span_subspace: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "B \<subseteq> span A"   and "subspace B" shows "span A = B"
eq_conv_compare: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "eq\<cdot>x\<cdot>y = is_EQ\<cdot>(compare\<cdot>x\<cdot>y)"
card_UNIV_length_enum: fixes enum :: "'c list"   and enum_all :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "CARD('c) = length enum"
bound: fixes i :: "nat" shows "AE x in M. norm (s i x) \<le> w x"
a2: shows "wf_graph \<lparr>nodes = V, edges = E\<rparr>"
one_divide_one_divide_ennreal: fixes c :: "ennreal" shows "1 / (1 / c) = c"
k'_def: shows "k' = (LEAST k. x ^ k = (1::'a) \<and> k \<noteq> 0)"
cases: fixes x :: "prop_rule" obtains "x = allI"   | "x = allE"   | "x = impI"   | "x = impE"
intra_edge1_4: shows "kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
t_3: shows "\<forall>s\<in>T \<union> subst_range \<sigma>. \<forall>u\<in>subterms s. u \<notin> subterms t"
hyps_3: fixes e' :: "char list exp" assumes "init_exp_of e1 = \<lfloor>unit\<rfloor>" shows "P \<turnstile> \<langle>init_switch e1 e',s1,b1\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
rep_eq: fixes f :: "'c \<Rightarrow> 'b"   and x :: "('c, 'a) generator" shows "generator (map_generator f x) = (case generator x of (has_next, next) \<Rightarrow> (has_next, \<lambda>s. let (a, s') = next s in (f a, s')))"
subdivision_step_def: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd" shows "subdivision_step G rev_G H rev_H \<equiv> \<lambda>(u, v, w) (uv, uw, vw). bidirected_digraph G rev_G \<and> bidirected_digraph H rev_H \<and> perm_restrict rev_H (arcs G) = perm_restrict rev_G (arcs H) \<and> compatible G H \<and> verts H = verts G \<union> {w} \<and> w \<notin> verts G \<and> arcs H = {uw, rev_H uw, vw, rev_H vw} \<union> arcs G - {uv, rev_G uv} \<and> uv \<in> arcs G \<and> distinct [uw, rev_H uw, vw, rev_H vw] \<and> arc_to_ends G uv = (u, v) \<and> arc_to_ends H uw = (u, w) \<and> arc_to_ends H vw = (v, w)"
mem_simps_7: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
vals_no_step: fixes es :: "char list exp list"   and vs :: "val list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "map_vals_of es = \<lfloor>vs\<rfloor>" shows "((es, s, b), es', s', b') \<notin> reds P"
tautology: fixes p :: "'b fm"   and M :: "('b, 'c) kripke"   and s :: "'c" assumes "tautology p" shows "M, s \<Turnstile> p"
CondHighCompositionality: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and b :: "expr"   and c1 :: "com"   and c2 :: "com" assumes "\<Gamma>,High \<turnstile> if (b) c1 else c2" shows "nonInterference \<Gamma> (if (b) c1 else c2)"
elims: fixes x :: "complex"   and xa :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex \<times> complex \<times> complex" assumes "x *\<^sub>s\<^sub>m xa = y" obtains k :: "complex" and a :: "complex" and b :: "complex" and c :: "complex" and d :: "complex" where "x = k" and "xa = (a, b, c, d)" and "y = (k * a, k * b, k * c, k * d)"
prems_4: shows "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m m"
tensor_state: fixes m :: "nat"   and u :: "complex Matrix.mat"   and n :: "nat"   and v :: "complex Matrix.mat" assumes "state m u"   and "state n v" shows "state (m + n) (u \<Otimes> v)"
stat_acc: shows "G \<turnstile> membr of statC accessible_from accC"
hom_sum: fixes c :: "'c"   and f :: "'d \<Rightarrow> 'c"   and X :: "'d set" shows "c * sum f X = (\<Sum>x\<in>X. c * f x)"
independent_some_basis: fixes V :: "'b set" shows "local.independent (some_basis V)"
simply_connected_eq_empty_inside: fixes S :: "complex set" assumes "open S" shows "simply_connected S = (connected S \<and> inside S = {})"
concE: fixes w :: "'a list"   and A :: "'a list set"   and B :: "'a list set" assumes "w \<in> A \<cdot> B" obtains u :: "'a list" and v :: "'a list" where "u \<in> A" and "v \<in> B" and "w = u @ v"
nn_integral_noteq_infinite: fixes g :: "'a \<Rightarrow> ennreal"   and M :: "'a measure" assumes "g \<in> borel_measurable M"   and "integral\<^sup>N M g \<noteq> \<infinity>" shows "AE x in M. g x \<noteq> \<infinity>"
bvxor: fixes a :: "bit list"   and b :: "bit list" shows "bvxor a b = bv_mapzip (\<oplus>\<^sub>b) a b"
not_open_singleton: fixes "open" :: "'d set \<Rightarrow> bool"   and x :: "'d" assumes "class.perfect_space open" shows "\<not> open {x}"
all_subset_image_inj: fixes f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and P :: "'a set \<Rightarrow> bool" shows "(\<forall>T\<subseteq>f ` S. P T) = (\<forall>T. T \<subseteq> S \<and> inj_on f T \<longrightarrow> P (f ` T))"
h2: shows "\<forall>i\<in>{m..<n}. f i = 0"
Gateway_L3: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "msg (Suc 0) req"   and "msg (Suc 0) a"   and "msg (Suc 0) stop"   and "ts lose"   and "ack t = [connection_ok]"   and "req (Suc t) = [send]"   and "\<forall>k\<le>Suc d. lose (t + k) = [False]" shows "ack (Suc t) = [sending_data]"
assert_commute: fixes y :: "'b"   and p :: "'b Assertion" assumes "y \<in> conjunctive" shows "y * {\<cdot> p } = {\<cdot> wp y p } * y"
assms: shows "t \<in> jkbpC\<^bsub>n\<^esub>"
range_enum_v: shows "range enum_v \<subseteq> \<^bold>V - {source \<Delta>, sink \<Delta>}"
vf_def: shows "vf = (\<lambda>w. inverse (f w))"
ideal_subset: fixes R :: "('h, 'i) Ring_scheme"   and I :: "'h set"   and h :: "'h" assumes "Ring R"   and "ideal R I"   and "h \<in> I" shows "h \<in> carrier R"
graph_homomorphism_Id: fixes a :: "('e, 'f) labeled_graph" shows "graph_homomorphism (restrict a) (restrict a) (Id_on (vertices a))"
assms: shows "eSubst MOD ys eY y eX = OK Z"
mult_scalar_zero_left: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "0 \<odot> p = 0"
Cons_3: fixes d :: "nat" assumes "d \<in> set (d__ # ds)" shows "m \<le> d"
trnl\<^sub>\<epsilon>_def: fixes u :: "'a"   and \<nu> :: "'a" shows "trnl\<^sub>\<epsilon> u \<nu> \<equiv> \<l>[u] \<cdot> (\<epsilon> \<star> u) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, u] \<cdot> (f \<star> \<nu>)"
refine: fixes G :: "('b \<Rightarrow> bool, 'b \<Rightarrow> 'b list, 'b list, 'b \<Rightarrow> bool, 'c) gen_bg_impl_scheme" shows "RETURN (code_blue_dfs_hash G) \<le> impl_blue_dfs_hash G"
InitDoneReds: fixes sh :: "sheap"   and C :: "char list"   and sfs :: "sfields"   and C' :: "char list"   and Cs :: "char list list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "sh C = \<lfloor>(sfs, Done)\<rfloor>"   and "P \<turnstile> \<langle>INIT C' (Cs,True) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>INIT C' (C # Cs,False) \<leftarrow> e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',s',b'\<rangle>"
basis_enum_eq_vec_of_basis_enumI: fixes a :: "'a"   and b :: "'a" assumes "vec_of_basis_enum a = vec_of_basis_enum b" shows "a = b"
set_s'_A: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat"   and m :: "nat" assumes "BIT_Off acts qs init" shows "set (BIT_Off.s'_A acts qs init n m) = set init"
same_laws_1: fixes R :: "'a ring"   and a :: "'a"   and b :: "'a" assumes "R \<in> C"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "a \<otimes>\<^bsub>union_ring C\<^esub> b = a \<otimes>\<^bsub>R\<^esub> b"
Inf_finite_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Sqinter> (insert a A) = a \<sqinter> \<Sqinter> A"
map_def: shows "\<psi>.map = transformation_by_components.map hom'.comp hom'.comp FG.map (\<lambda>a. hom'.inv (d\<^sub>0e\<^sub>0.\<phi> a))"
well_analyzed_split_left: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" assumes "well_analyzed (A @ B)" shows "well_analyzed A"
Min_ne_def: fixes X :: "'a set" shows "Min_ne X = FOLD_bin_ne min X"