hcju/isabelleps huggingface dataset:
union: fixes M :: "'a multiset"   and N :: "'a multiset" shows "\<Prod>\<^sub># (M + N) = \<Prod>\<^sub># M * \<Prod>\<^sub># N"
top_implies: fixes x :: "'c" shows "(if bounded_distrib_lattice_class.comp_inf.reflexive x then top else x) = x"
mod_Suc_le_divisor: fixes m :: "nat"   and n :: "nat" shows "m mod Suc n \<le> n"
add_minus_cancel: fixes x :: "'a"   and y :: "'a" assumes "x \<in> V"   and "y \<in> V" shows "x + (- x + y) = y"
fls_pow_subdegree_eqI: fixes f :: "'a fls"   and n :: "nat" assumes "fls_nth f (fls_subdegree f) ^ n \<noteq> (0::'a)" shows "fls_subdegree (f ^ n) = int n * fls_subdegree f"
simps_3: fixes x :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators"   and zs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "FWNormalisationCore.sorted (x # y # zs) l = (smaller x y l \<and> FWNormalisationCore.sorted (y # zs) l)"
oalist_inv_raw_sort_oalist: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp" shows "comparator.oalist_inv_raw comp (comparator.sort_oalist comp xs)"
x: shows "y = snd (split_n_m_x (dim_col A) (dim_row A) X)"
equiv_Worder1: fixes D :: "('c, 'd) Order_scheme"   and T :: "('e, 'f) Order_scheme" assumes "Worder D"   and "Order T"   and "ord_equiv T D" shows "Worder T"
C1_differentiable_on_eq: fixes f :: "real \<Rightarrow> 'a"   and S :: "real set" shows "(f C1_differentiable_on S) = ((\<forall>x\<in>S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x)))"
lec_rel_def: fixes le_constraint :: "'a le_constraint" shows "lec_rel le_constraint = (case le_constraint of Le_Constraint x1 x2 x3 \<Rightarrow> x1)"
iv0: shows "adjoint W * W = 1\<^sub>m n"
Sr_ring: fixes R :: "('c, 'd) Ring_scheme"   and S :: "'c set" assumes "Ring R"   and "sr R S" shows "Ring (Sr R S)"
abs_hnorm_cancel: fixes a :: "'a star" shows "\<bar>hnorm a\<bar> = hnorm a"
Sum_any_zeroI': fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x. P x \<Longrightarrow> f x = (0::'c)" shows "(\<Sum>x. f x when P x) = (0::'c)"
nat_pow_eq': shows "([^]\<^bsub>units_of R\<^esub>) = (\<lambda>x k. x ^ k mod p)"
continuous_ident: fixes x :: "'a"   and S :: "'a set" shows "continuous (at x within S) (\<lambda>x. x)"
list_of_oalist_shift_keys: fixes t :: "('a, 'b) pp"   and f :: "'d \<Rightarrow> 'd"   and xs :: "(('a, 'b) pp \<times> 'c, 'd) oalist_ntm" shows "list_of_oalist_ntm (shift_map_keys_pprod t f xs) = map_raw (\<lambda>kv. (splus_pprod t (fst kv), f (snd kv))) (list_of_oalist_ntm xs)"
ds_ord_sl_ord: fixes x :: "'a"   and y :: "'a" shows "(d x \<sqsubseteq> d y) = (d x \<cdot> d y = d x)"
ImZ_of_int: fixes n :: "int" shows "ImZ (of_int n) = 0"
count_terminals_upper_bound: fixes p :: "symbol list" shows "count_terminals p \<le> length p"
14: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z \<le> x \<sqinter> (y \<squnion> z)"
D_def: shows "D = the_adj_chamber C0 (C0 - {v})"
floor_add2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<int> \<or> y \<in> \<int>" shows "\<lfloor>x + y\<rfloor> = \<lfloor>x\<rfloor> + \<lfloor>y\<rfloor>"
apath_ends: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and u' :: "'a"   and v' :: "'a" assumes "apath u p v"   and "apath u' p v'" shows "p \<noteq> [] \<and> u \<noteq> v \<and> u = u' \<and> v = v' \<or> p = [] \<and> u = v \<and> u' = v'"
abs_id: shows "\<bar>x\<bar> = \<bar>x + (2::'a) * of_int i * y\<bar>"
cospan_\<nu>\<pi>: shows "C.cospan f.leg0 g.leg1"
abs_insert: fixes t :: "'b trie_map"   and xs :: "'b list" assumes "invar t" shows "Trie_Map.abs (Trie_Map.insert xs t) = Trie_Fun.insert xs (Trie_Map.abs t)"
apex_def: shows "src_\<mu>.cod.apex \<equiv> C.dom src_\<mu>.cod.leg0"
WCFG_Skip: shows "Skip \<turnstile> (_ 0 _) -\<Up>id\<rightarrow> (_Exit_)"
eventually_nhds_top: fixes b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "b < top" shows "eventually P (nhds top) = (\<exists>b<top. \<forall>z>b. P z)"
count_Suc: fixes a :: "'b"   and B :: "'b list"   and m :: "nat" assumes "PermutationLemmas.count a B = Suc m" shows "a \<in> set B"
mk_poly_dim_2: fixes A :: "'b mat" shows "dim_row (mk_poly A) = dim_row A"
abs_eq: fixes x :: "j \<Rightarrow> i \<Rightarrow> bool" shows "\<^bold>\<not>make\<o> x = make\<o> (\<lambda>s w. s = dj \<and> \<not> x dj w \<or> s \<noteq> dj \<and> I_NOT s (x s) w)"
Ioc_disjoint: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.linorder less_eq less" shows "(ord.greaterThanAtMost less_eq less a b \<inter> ord.greaterThanAtMost less_eq less c d = {}) = (less_eq b a \<or> less_eq d c \<or> less_eq b c \<or> less_eq d a)"
arctan_series_stream_def: shows "arctan_series_stream = arctan_series_stream_aux False 1"
image_mset_empty: fixes f :: "'b \<Rightarrow> 'a" shows "image_mset f {#} = {#}"
Abs_image: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "type_definition Rep Abs A" shows "Abs ` A = UNIV"
argmaxEquivalence: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x\<in>X. f x = g x" shows "argmax f X = argmax g X"
sw_simps1: fixes xs :: "'c"   and x :: "'b"   and y :: "'b" shows "x @xs[x \<and> y]_xs = y"
dom_cod: fixes f :: "'d" assumes "D.arr f" shows "D.dom (D.cod f) = D.cod f"
sup1I1: fixes A :: "'a \<Rightarrow> bool"   and x :: "'a"   and B :: "'a \<Rightarrow> bool" assumes "A x" shows "sup A B x"
d2: shows "column ia (fst (QR_decomposition A)) = (1 / norm (\<chi>i. Gram_Schmidt_matrix A $ i $ ia)) *\<^sub>R column ia (Gram_Schmidt_matrix A)"
gt_than_any_outside: fixes A :: "'a set"   and b :: "'a" assumes "A \<subseteq> carrier D"   and "b \<in> carrier D"   and "\<forall>x\<in>A. x \<prec> b" shows "b \<notin> A"
w_bound: fixes i :: "nat" shows "weight (f i) \<le> n"
foo2: fixes a :: "'b"   and ps :: "'b list"   and ss :: "'b list"   and p :: "'b list"   and s :: "'b list" assumes "a \<notin> set ps"   and "a \<notin> set ss"   and "set p = set s"   and "p = a # ps"   and "s = a # ss" shows "set ps = set ss"
path_component_of_equiv: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "path_component_of X x y = (x \<in> topspace X \<and> y \<in> topspace X \<and> path_component_of X x = path_component_of X y)"
set_eqD2: fixes R :: "('a, 'b) eq_object_scheme"   and A :: "'a set"   and A' :: "'a set"   and a' :: "'a" assumes "A {.=}\<^bsub>R\<^esub> A'"   and "a' \<in> A'" shows "\<exists>a\<in>A. a' .=\<^bsub>R\<^esub> a"
dual_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<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 dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "dual one = one"
LSC_Sim_resolution_la: fixes k :: "'b"   and F :: "'b literal list"   and G :: "'b literal list"   and \<Gamma> :: "'b formula multiset" assumes "disj_of_clause (removeAll (k\<^sup>+) F @ removeAll (k\<inverse>) G), \<Gamma> \<Rightarrow>\<^sub>n"   and "k\<^sup>+ \<in> set F"   and "k\<inverse> \<in> set G" shows "disj_of_clause F, disj_of_clause G, \<Gamma> \<Rightarrow>\<^sub>n"
absdiff_norm_less: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and e :: "real" assumes "(\<Sum>x\<in>S. norm (f x - g x)) < e" shows "\<bar>(\<Sum>x\<in>S. norm (f x)) - (\<Sum>x\<in>S. norm (g x))\<bar> < e"
bags_separate: fixes s :: "'c"   and t :: "'c"   and v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "s \<rightarrow>\<^bsub>T\<^esub> t"   and "v \<in> left_part s t"   and "w \<in> left_part t s"   and "v \<leadsto>xs\<leadsto> w" shows "set xs \<inter> bag s \<inter> bag t \<noteq> {}"
exe_complete_tcsigs_imp_complete_tcsigs: fixes arss :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list"   and cs :: "(String.literal \<times> String.literal) list" assumes "exe_ars_conds arss"   and "exe_complete_tcsigs cs arss" shows "complete_tcsigs (set cs) (translate_ars arss)"
has_field_derivative_sinh: fixes g :: "'a \<Rightarrow> 'a"   and Db :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(g has_field_derivative Db) (at x within s)" shows "((\<lambda>x. sinh (g x)) has_field_derivative cosh (g x) * Db) (at x within s)"
altc_sample_writer_def: fixes altc_sample :: "'b \<Rightarrow> ('c \<Rightarrow> 'm) \<Rightarrow> 'm"   and p :: "'b"   and f :: "'c \<Rightarrow> ('w, 'a, 'm) writerT" shows "local.altc_sample_writer altc_sample p f = WriterT (altc_sample p (\<lambda>p. run_writer (f p)))"
assms_2: shows "vpeq (Step.partition tid) s1 s2"
push_tyvar_def: fixes \<Gamma> :: "(ty \<times> nat) list \<times> nat" shows "push_tyvar \<Gamma> \<equiv> (fst \<Gamma>, Suc (snd \<Gamma>))"
simplex_likeI: fixes P :: "'a set"   and f :: "'a \<Rightarrow> nat set"   and A :: "nat set" assumes "finite P"   and "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f"   and "f ` P = Pow A" shows "domain.simplex_like P"
gtrp_preserve: fixes s :: "'f"   and c :: "'f multiset"   and w :: "'g el_step list"   and s' :: "'f"   and c' :: "'f multiset"   and T :: "(('f \<times> 'f multiset) \<times> 'g \<times> 'f \<times> 'f multiset) set"   and P :: "'f multiset \<Rightarrow> bool" assumes "((s, c), w, s', c') \<in> trcl (gtrp T)"   and "P (add_mset s c)"   and "\<And>s c s' c' e. \<lbrakk>P (add_mset s c); ((s, c), e, s', c') \<in> T\<rbrakk> \<Longrightarrow> P (add_mset s' c')" shows "P (add_mset s' c')"
sff'_4: assumes "f = 0" shows "a = 0 \<and> psi = []"
path_Entry_target_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as Entry" shows "n = Entry"
pred_rel: fixes P :: "'a \<Rightarrow> bool"   and x :: "'a pmf" shows "pred_pmf P x = rel_pmf (eq_onp P) x x"
deg_not_zero_imp_not_unit: fixes f :: "'a poly" assumes "0 < Polynomial.degree f" shows "\<not> is_unit f"
Q_fs_def: fixes x :: "'a" shows "Q_fs x \<equiv> if x \<in> Q then fs else []"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and grouping :: "int"   and index :: "int"   and sizes :: "int set" assumes "t_wise_balance point_set block_collection grouping index"   and "K_block_design point_set block_collection sizes"   and "tBD_axioms grouping sizes" shows "tBD point_set block_collection grouping index sizes"
Nml_Cod: fixes t :: "'a local.term" assumes "Nml t" shows "Nml (Cod t)"
prime_factorization_normalize: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.prime_factorization zero times one normalize (normalize x) = factorial_semiring.prime_factorization zero times one normalize x"
assms_1: shows "set Us \<subseteq> Span K Vs"
butterfly_apply: fixes \<psi> :: "'a"   and \<psi>' :: "'b"   and \<phi> :: "'b" shows "butterfly \<psi> \<psi>' *\<^sub>V \<phi> = (\<psi>' \<bullet>\<^sub>C \<phi>) *\<^sub>C \<psi>"
subst_elim_def: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term"   and v :: "'e" shows "subst_elim \<sigma> v \<equiv> \<forall>t. v \<notin> fv (t \<cdot> \<sigma>)"
prems_2: shows "CT \<turnstile>+ Ds <: Es"
isOK_iff: fixes m :: "'a + 'b" shows "isOK m = (\<exists>x. m = Inr x)"
ncols_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "ncols (Finite_Cartesian_Product.transpose A) = nrows A"
arc_imp_inj_on: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "inj_on g {0..1}"
R: shows "UNlift01 cl dl P F ` UNpart01 cl dl P = {UNlift01 cl dl P F (BrnFT cl dl)} \<union> UNlift01 cl dl P F ` UNpart1 cl dl P"
p_gt_2: fixes p :: "nat" assumes "mersenne_prime p" shows "2 < p"
closed_cbox: fixes a :: "'a"   and b :: "'a" shows "closed (cbox a b)"
\<G>_F_def: fixes C :: "'a literal multiset" shows "\<G>_F C = {C \<cdot> \<sigma> |\<sigma>. is_ground_subst \<sigma>}"
simps_4: fixes x :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list"   and ys :: "('c, 'd) bintree option list" shows "meld (x # xs) (None # ys) = x # meld xs ys"
sum_list_add_le_fold_eucl_truncate_up: fixes xs :: "'a list"   and z :: "'a"   and p :: "nat" shows "sum_list xs + z \<le> fold (\<lambda>x y. eucl_truncate_up p (x + y)) xs z"
IS: fixes f :: "'x \<Rightarrow> 'a"   and x :: "'x" assumes "\<And>x. \<Phi> x \<Longrightarrow> f x \<le> M x"   and "\<Phi> x"   and "f \<le> REC body" shows "body f x \<le> M x"
H: shows "(xs @ ipurge_tr I D (D y) ys, ipurge_ref I D (D y) ys X) \<in> failures P"
elims_1: fixes x :: "'a st option"   and xa :: "char list set"   and y :: "bool" assumes "top_on\<^sub>o x xa = y" obtains S :: "'a st" and X :: "char list set" where "x = Some S" and "xa = X__" and "y = top_on\<^sub>s S X__"   | X :: "char list set" where "x = None" and "xa = X" and "y"
proper_it'_def: fixes it :: "'b \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it' :: "'b \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" shows "proper_it' it it' \<equiv> \<forall>s. proper_it (it s) (it' s)"
strict_implies_order: fixes b :: "'b list"   and a :: "'b list" assumes "strict_prefix b a" shows "prefix b a"
cComm1: fixes a :: "name"   and b :: "name"   and P' :: "pi"   and P'' :: "pi"   and C :: "'a" assumes "P \<longmapsto> a<b> \<prec> P'"   and "!P \<longmapsto> a[b] \<prec> P''"   and "\<And>C. F C (!P) (a[b] \<prec> P'')" shows "F C (P \<parallel> !P) (\<tau> \<prec> P' \<parallel> P'')"
transfer_all: fixes p :: "'b star \<Rightarrow> bool"   and P :: "nat \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>X. p (star_n X) \<equiv> \<forall>\<^sub>F n in \<U>. P n (X n)" shows "\<forall>x. p x \<equiv> \<forall>\<^sub>F n in \<U>. \<forall>x. P n x"
q2: fixes cipher :: "bool list \<times> bool list"   and \<sigma> :: "'a" shows "interaction_bounded_by' (\<lambda>_. True) (\<A>2 cipher \<sigma>) q2"
filterlim_pow_at_top: fixes n :: "nat"   and f :: "'a \<Rightarrow> real"   and F :: "'a filter" assumes "0 < n"   and "filterlim f at_top F" shows "LIM x F. f x ^ n :> at_top"
set_intros_4: fixes x22 :: "'c"   and x21 :: "'d"   and x23 :: "('d, 'c) chain" shows "x22 \<in> set2_chain (\<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23)"
UP_pre_univ_prop_def: fixes R :: "('a, 'b) ring_scheme"   and S :: "('c, 'd) ring_scheme"   and h :: "'a \<Rightarrow> 'c" shows "UP_pre_univ_prop R S h \<equiv> ring_hom_cring R S h \<and> UP_cring R"
DSourcesA11_L1: shows "DSources level1 sA11 = {}"
E21: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>\<diamond>\<box>F = \<diamond>\<box>F"
add_edge_def: fixes v :: "'b"   and v' :: "'b"   and G :: "('b, 'c) graph_scheme" shows "add_edge v v' G = \<lparr>nodes = nodes G \<union> {v, v'}, edges = {(v, v')} \<union> edges G\<rparr>"
nofail_simps_2: fixes X :: "'b set" shows "nofail (RES X) = True"
weak_duality_aux: fixes i :: "'a"   and j :: "'a" assumes "i \<in> {c \<bullet> x |x. x \<in> sat_primal A b}"   and "j \<in> {b \<bullet> y |y. y \<in> sat_dual A c}" shows "i \<le> j"
Suc_Rep_inject: fixes x :: "ind"   and y :: "ind" assumes "Suc_Rep x = Suc_Rep y" shows "x = y"
map_13: fixes f :: "'c \<Rightarrow> 'b"   and x131 :: "'c ltlc"   and x132 :: "'c ltlc" shows "map_ltlc f (x131 W\<^sub>c x132) = map_ltlc f x131 W\<^sub>c map_ltlc f x132"
composite_simps_1: shows "src (src f \<star> src f) = src (src f)"
ParT_ZObis: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list" assumes "properL cl"   and "properL dl"   and "SbisL cl dl" shows "ParT cl \<approx>01 ParT dl"
theI': fixes P :: "'g \<Rightarrow> bool" assumes "\<exists>!x. P x" shows "P (THE x. P x)"
positive_eigenvalue_positive: fixes A :: "complex mat"   and n :: "nat"   and es :: "complex list"   and B :: "complex mat"   and P :: "complex mat"   and Q :: "complex mat"   and i :: "nat" assumes "A \<in> carrier_mat n n"   and "positive A"   and "char_poly A = (\<Prod>e\<leftarrow>es. [:- e, 1:])"   and "unitary_schur_decomposition A es = (B, P, Q)"   and "i < n" shows "0 \<le> B $$ (i, i)"
not_cl: shows "\<forall>u\<in>V. \<forall>v\<in>V. u \<noteq> v \<longrightarrow> \<not> close g u v"
diff_fract: fixes b :: "'a"   and d :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)"   and "d \<noteq> (0::'a)" shows "Fraction_Field.Fract a b - Fraction_Field.Fract c d = Fraction_Field.Fract (a * d - c * b) (b * d)"
i_less_suc: shows "mod_type_class.to_nat i < Suc k"
trans_DT: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<sqsubseteq>\<^sub>D\<^sub>T Q"   and "Q \<sqsubseteq>\<^sub>D\<^sub>T S" shows "P \<sqsubseteq>\<^sub>D\<^sub>T S"
ccSup_le_iff: fixes A :: "'a set"   and b :: "'a" assumes "countable A" shows "(Sup A \<le> b) = (\<forall>a\<in>A. a \<le> b)"
mat_row_first_last_append: fixes A :: "'b mat"   and m :: "nat"   and n :: "nat" assumes "dim_row A = m + n" shows "mat_row_first A m @\<^sub>r mat_row_last A n = A"
dinvarm_def: fixes \<sigma> :: "('V \<Rightarrow> 'W infty option) \<times> ('V \<Rightarrow> (('V \<times> 'W \<times> 'V) list \<times> 'W) option)" shows "dinvarm \<sigma> \<equiv> let (wl, res) = \<sigma> in (\<forall>v\<in>dom wl. the (wl v) = mpath_weight' (res v)) \<and> res_invarm res"
FGModuleHomI_fromaxioms: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and T :: "'c \<Rightarrow> 'd"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd" assumes "FGModule G smult V"   and "\<And>v v'. \<lbrakk>v \<in> V; v' \<in> V\<rbrakk> \<Longrightarrow> T (v + v') = T v + T v'"   and "supp T \<subseteq> V"   and "\<And>r m. \<lbrakk>r \<in> aezfun_setspan G; m \<in> V\<rbrakk> \<Longrightarrow> T (smult r m) = smult' r (T m)" shows "FGModuleHom G smult V smult' T"
assms_5: shows "\<T> = generate_valid_stateful_policy_IFSACS_2 G M edgesList"
map_getOneIp_distinct: fixes xs :: "'a wordinterval list" assumes "distinct xs"   and "\<forall>x1\<in>set xs. \<forall>x2\<in>set xs. x1 \<noteq> x2 \<longrightarrow> wordinterval_to_set x1 \<inter> wordinterval_to_set x2 = {}"   and "\<forall>x\<in>set xs. \<not> wordinterval_empty x" shows "distinct (map getOneIp xs)"
ext_cf_inductTl: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "pol_coeff S (Suc n, f)" shows "polyn_expr R X (n + j) (ext_cf S j (Suc n, f)) = polyn_expr R X (n + j) (ext_cf S j (n, f))"
set_of_distrib_right: fixes A1 :: "'b interval"   and A2 :: "'b interval"   and B :: "'b interval" shows "set_of ((A1 + A2) * B) \<subseteq> set_of (A1 * B + A2 * B)"
exhaust_sel: fixes heap :: "(_) heap" obtains "heap = Heap (the_heap heap)"
V_iff_non_\<phi>: fixes \<phi> :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and f :: "('b, 'c, 'd) trans \<Rightarrow> 'a"   and trn :: "('b, 'c, 'd) trans"   and tr :: "('b, 'c, 'd) trans list" shows "(BD_Security.V \<phi> f (trn # tr) = BD_Security.V \<phi> f tr) = (\<not> \<phi> trn)"
flow_continuous_on_state_space: shows "continuous_on (Sigma X existence_ivl0) (\<lambda>(x, y). flow0 x y)"
separable_def: fixes f :: "'a poly" shows "separable f = coprime f (pderiv f)"
poly_deg_poly_subst_le: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and f :: "'b \<Rightarrow> ('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "\<And>x. x \<in> indets p \<Longrightarrow> poly_deg (f x) \<le> 1" shows "poly_deg (poly_subst f p) \<le> poly_deg p"
shift_snth_less: fixes p :: "nat"   and xs :: "'b list"   and s :: "'b stream" assumes "p < length xs" shows "(xs @- s) !! p = xs ! p"
ltl_prop_equiv_false: fixes \<phi> :: "'a ltln" shows "\<phi> \<sim>\<^sub>P false\<^sub>n = (\<not> UNIV \<Turnstile>\<^sub>P \<phi>)"
equivI: fixes A :: "'e set"   and r :: "('e \<times> 'e) set" assumes "refl_on A r"   and "sym r"   and "trans r" shows "equiv A r"
mono_mndet_FD_process: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a process"   and P' :: "'a \<Rightarrow> 'a process" assumes "\<forall>x\<in>A. P x \<sqsubseteq>\<^sub>F\<^sub>D P' x" shows "mndet A P \<sqsubseteq>\<^sub>F\<^sub>D mndet A P'"
smap_eq_shift: fixes f :: "'c \<Rightarrow> 'b"   and w :: "'c stream"   and u :: "'b list"   and v :: "'b stream" shows "(smap f w = u @- v) = (\<exists>w\<^sub>1 w\<^sub>2. w = w\<^sub>1 @- w\<^sub>2 \<and> map f w\<^sub>1 = u \<and> smap f w\<^sub>2 = v)"
ind_sheaf_axioms: shows "Comm_Ring.ind_sheaf X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str X"
PostConjI: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and R :: "'e set"   and B :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c R,B" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c (Q \<inter> R),(A \<inter> B)"
VS_exprHigh: fixes e :: "Expr" shows "(e, high) \<in> VS_expr"
del_in_traverse_pairs: fixes t :: "('a, 'b) rbt"   and x :: "'a" assumes "rbt_sorted t" shows "rbt_in_traverse_pairs (del x t) = remove_elt_pairs x (rbt_in_traverse_pairs t)"
the1I2: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool" assumes "\<exists>!x. P x"   and "\<And>x. P x \<Longrightarrow> Q x" shows "Q (THE x. P x)"
A: shows "g1 \<in> \<Theta>[F](g2)"
sup_inf_distrib2: fixes y :: "'a multiset"   and z :: "'a multiset"   and x :: "'a multiset" shows "y \<inter># z \<union># x = y \<union># x \<inter># (z \<union># x)"
split_llist: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<in> lset xs" shows "\<exists>ys zs. xs = lappend ys (LCons x zs) \<and> lfinite ys"