hcju/isabelleps huggingface dataset:
acyclic_2_implies_1: fixes star :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.pd_kleene_allegory star one times sup less_eq less bot inf top conv uminus"   and "pd_kleene_allegory.acyclic_2 star one times less_eq bot inf conv uminus x" shows "pd_kleene_allegory.acyclic_1 star one times sup less_eq bot inf conv uminus x"
lossless_init: fixes h :: "'pub_input"   and w :: "'witness" shows "lossless_spmf (init h w)"
solve_start: fixes \<Psi> :: "'c TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Psi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<lbrakk> [], 0 \<turnstile> \<Psi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
assms: fixes \<alpha> :: "act"   and Q' :: "ccs" assumes "Q \<longmapsto>\<alpha> \<prec> Q'" shows "\<exists>P'. P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
head_eq_headn0: fixes p :: "'b poly" shows "head p = headn p 0"
intro: fixes block_collection :: "'b set multiset" assumes "\<And>bl. bl \<in># block_collection \<Longrightarrow> bl \<noteq> {}" shows "design_axioms block_collection"
single_subset_iff: fixes a :: "'b"   and M :: "'b multiset" shows "({#a#} \<subseteq># M) = (a \<in># M)"
assms: shows "local.ind_cpa.ibounded_by \<A> (enat q)"
Pbij_def: shows "Pbij = {\<beta>. \<forall>l1 l2 l3 l4. (l1, l2) \<in> \<beta> \<longrightarrow> (l3, l4) \<in> \<beta> \<longrightarrow> (l1 = l3) = (l2 = l4)}"
IH: assumes "\<nexists>n\<^sub>0. f n\<^sub>0 = n" shows "\<exists>n\<^sub>p. f n\<^sub>p < n \<and> (\<forall>k. f n\<^sub>p < k \<and> k \<le> n \<longrightarrow> (\<nexists>k\<^sub>0. f k\<^sub>0 = k))"
assume_SE_def: fixes P :: "'b \<Rightarrow> bool" shows "assume\<^sub>S\<^sub>E P = (\<lambda>\<sigma>. if \<exists>\<sigma>. P \<sigma> then Some ((), SOME \<sigma>. P \<sigma>) else None)"
nonempty_Basis: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "Basis \<noteq> {}"
returns: shows "eventually (returns_to {x \<in> S. s x = 0}) (at x)"
U: shows "RModule FG (\<cdot>) U \<and> U \<subseteq> V"
as_def: shows "as = map (local.dim_gen_eigenspace A ev) [0..<Suc (Suc kk)]"
equalOn_vblcaseI': fixes A :: "vbl set"   and f :: "vbl \<Rightarrow> 'a"   and g :: "vbl \<Rightarrow> 'a"   and x :: "'a" assumes "equalOn (preImage nextX A) f g" shows "equalOn A (vblcase x f) (vblcase x g)"
inv: fixes x :: "'a" assumes "x \<in> S" shows "p' (p x) = x"
not_gr0: fixes n :: "nat" shows "(\<not> 0 < (n::nat)) = (n = 0)"
h_minus_source: fixes n :: "nat"   and y :: "'v" assumes "cycles \<Delta> \<noteq> {}" shows "h_minus n (source \<Delta>, y) = 0"
finite_map_def: fixes \<alpha> :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and invar :: "'c \<Rightarrow> bool" shows "finite_map \<alpha> invar \<equiv> \<forall>m. invar m \<longrightarrow> finite (dom (\<alpha> m))"
exhaust: fixes y :: "('e, 'f) step" obtains "y = Done"   | x2 :: "'f" where "y = Skip x2"   | x31 :: "'e" and x32 :: "'f" where "y = Yield x31 x32"
measurable_eq_stream_space: fixes f :: "'a \<Rightarrow> 'b stream"   and M :: "'a measure"   and c :: "'b stream" assumes "f \<in> M \<rightarrow>\<^sub>M stream_space (count_space UNIV)" shows "Measurable.pred M (\<lambda>x. f x = c)"
CR3_RED_def: fixes t :: "lam"   and \<tau> :: "ty" shows "CR3_RED t \<tau> \<equiv> \<forall>t'.  t \<longrightarrow>\<^sub>\<beta> t' \<longrightarrow> t' \<in> RED \<tau>"
case_1: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g" shows "(case [] of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f1"
assms_1: shows "set_iterator_genord it_a S_a R_a"
connected_components_of_def: fixes X :: "'c topology" shows "connected_components_of X \<equiv> connected_component_of_set X ` topspace X"
nreach: shows "(i, j) \<notin> (graph_of_mat A)\<^sup>*"
aux_52: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry"   and z :: "'c DBMEntry" assumes "\<not> x \<prec> y"   and "dbm_add z x \<prec> dbm_add z y" shows "dbm_add z y = dbm_add z x"
consistent_swapLeaves: fixes t :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" assumes "consistent t" shows "consistent (swapLeaves t w\<^sub>a a w\<^sub>b b)"
mycasexxy_def: fixes x :: "'a"   and y :: "'a" shows "mycasexxy x y = Plus (seq [Star (Plus (Atom x) (Atom y)), Atom x, Atom x, Atom y]) (seq [Star (Plus (Atom x) (Atom y)), Atom y, Atom y, Atom x])"
plus_le_left_cancel_nowrap: fixes x :: "'b word"   and y' :: "'b word"   and y :: "'b word" assumes "x \<le> x + y'"   and "x \<le> x + y" shows "(x + y' < x + y) = (y' < y)"
abelian_free_Abelian_group: fixes S :: "'b set" shows "comm_group (free_Abelian_group S)"
mono_sinvar: fixes m :: "'b SecurityInvariant_configured"   and N :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = N, edges = E\<rparr>"   and "E' \<subseteq> E"   and "c_sinvar m \<lparr>nodes = N, edges = E\<rparr>" shows "c_sinvar m \<lparr>nodes = N, edges = E'\<rparr>"
tm_union_range_right: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t2 :: "taylor_model"   and I :: "float interval list"   and a :: "float interval list"   and prec :: "nat"   and t1 :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e t2"   and "num_params (tm_poly t2) \<le> length I"   and "develops_at_within e a I" shows "x \<in>\<^sub>i range_tm e (tm_union prec I a t1 t2)"
proper_cont: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and c :: "('c, 'd, 'a) cmd"   and i :: "nat"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" assumes "PL cval"   and "proper c"   and "i < PL.brn c" shows "proper (PL.cont aval tval cval c s i)"
lmap_eq_lmap_conv_llist_all2: fixes f :: "'f \<Rightarrow> 'e"   and xs :: "'f llist"   and g :: "'g \<Rightarrow> 'e"   and ys :: "'g llist" shows "(lmap f xs = lmap g ys) = llist_all2 (\<lambda>x y. f x = g y) xs ys"
paths_def: fixes w :: "'a \<times> 'a \<Rightarrow> enat"   and u :: "'a"   and v :: "'a" shows "WGraph.paths w u v \<equiv> {p. WGraph.path w u p v}"
tendsto_cmult_ereal: fixes c :: "ereal"   and f :: "'a \<Rightarrow> ereal"   and x :: "ereal"   and F :: "'a filter" assumes "\<bar>c\<bar> \<noteq> \<infinity>"   and "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x) \<longlongrightarrow> c * x) F"
lin_dep_cols_imp_det_0: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "module.lin_dep class_ring (module_vec TYPE('b) n) (set (cols A))" shows "det A = (0::'b)"
ap_set_def: fixes F :: "('e \<Rightarrow> 'd) set"   and X :: "'e set" shows "Applicative.ap F X = {f x |f x. f \<in> F \<and> x \<in> X}"
Sigma_Diff_distrib2: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'c set" shows "(SIGMA i:I. A i - B i) = Sigma I A - Sigma I B"
r_4: fixes x :: "'a" assumes "x \<in> S" shows "r x = x"
osum_minus_Id2: fixes r' :: "('d \<times> 'd) set"   and r :: "('e \<times> 'e) set" assumes "r' \<subseteq> Id" shows "r +o r' - Id \<subseteq> map_prod Inl Inl ` (r - Id) \<union> Inl ` Field r \<times> Inr ` Field r'"
fBall_ffilter: fixes X :: "'b fset"   and f :: "'b \<Rightarrow> bool" assumes "fBall X f" shows "ffilter f X = X"
le_PlusR: fixes s :: "'a language"   and r :: "'a language" shows "s \<le> Plus r s"
iMODb_div: fixes k :: "nat"   and m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < k"   and "m mod k = 0" shows "[ r, mod m, c ] \<oslash> k = [ r div k, mod m div k, c ]"
in_image_eucl_of_list_eucl_of_list_aform: fixes x :: "(real \<times> real pdevs) list"   and xa :: "'a" assumes "length x = DIM('a)"   and "xa \<in> Affine (eucl_of_list_aform x)" shows "xa \<in> eucl_of_list ` Joints x"
assms: shows "subst \<rho> s \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"
c_failures_tr: fixes xps :: "('a \<times> 'b) list"   and X :: "('a \<times> 'b) set"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c" assumes "(xps, X) \<in> c_failures step out s\<^sub>0" shows "xps = c_tr step out s\<^sub>0 (map fst xps)"
fix_def2: fixes F :: "'a \<rightarrow> 'a" shows "fix\<cdot>F = (\<Squnion>i. iterate i\<cdot>F\<cdot>\<bottom>)"
unidx_def: fixes i :: "nat" shows "unidx i = map fst xs ! i"
nat_into_Card: fixes \<alpha> :: "V" assumes "\<alpha> \<in> elts \<omega>" shows "Card \<alpha>"
LIMSEQ_Suc: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a" assumes "f \<longlonglongrightarrow> l" shows "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
is_longest_prefix_routing_def: shows "is_longest_prefix_routing \<equiv> sorted \<circ> map routing_rule_sort_key"
scc_ofI_awalk: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "H.awalk u p v" shows "u \<in> H.scc_of v"
single_Diff_lessThan: fixes k :: "'a" shows "{k} - {..<k} = {k}"
replicate_mset_subseteq_iff_le: fixes m :: "nat"   and x :: "'b"   and n :: "nat" shows "(replicate_mset m x \<subseteq># replicate_mset n x) = (m \<le> n)"
rel_gpv'_def: fixes A :: "'f \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_gpv' A = rel_gpv A (=)"
hyps: assumes "b < degree f" shows "P (\<Sum>i\<le>b. monom (g i) i) n"
ctr: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "sfilter P s = shd (sdrop_while (Not \<circ> P) s) ## sfilter P (stl (sdrop_while (Not \<circ> P) s))"
insert_mono: fixes C :: "'d set"   and D :: "'d set"   and a :: "'d" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
assms: shows "\<forall>a b. (a, b) \<in> R \<or> a = b \<or> (b, a) \<in> R"
icsSp_Nil: fixes n :: "'node SDG_node" assumes "valid_SDG_node n" shows "n ics-[]\<rightarrow>\<^sub>d* n"
lcp_pref_conv: fixes u :: "'b list"   and v :: "'b list" shows "(u \<and>\<^sub>p v = u) = (u \<le>p v)"
homotopic_join_subpaths3: fixes s :: "'b set"   and u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'b"   and w :: "real" assumes "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"   and "path g"   and "path_image g \<subseteq> s"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "w \<in> {0..1}" shows "homotopic_paths s (subpath v w g +++ subpath w u g) (subpath v u g)"
induced_subgraph_altdef: fixes H :: "('a, 'b) pre_digraph" shows "induced_subgraph H H__ = (subgraph H H__ \<and> (\<forall>H'. subgraph H' H__ \<longrightarrow> verts H' \<noteq> verts H \<or> arcs H' \<subseteq> arcs H))"
g_def: shows "local.g \<equiv> LLL_short_polynomial (p ^ local.ll) (Suc j') u"
int_simps_8: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> #True = P = P"
ex_mset: fixes X :: "'d multiset" shows "\<exists>xs. mset xs = X"
section_and_retraction_eq_homeomorphic_map: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "(section_map X Y f \<and> retraction_map X Y f) = homeomorphic_map X Y f"
a2: shows "(case virt_to_phys addr (mmu (add_data_cache s1 addr data bm)) (mem (add_data_cache s1 addr data bm)) of None \<Rightarrow> None | Some pair \<Rightarrow> if mmu_writable (get_acc_flag (snd pair)) 10 then Some (mem_mod_w32 10 (fst pair) bm data (add_data_cache s1 addr data bm)) else None) = None"
l_new_shadow_root_get_disconnected_nodes_def: fixes get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_disconnected_nodes get_disconnected_nodes_locs \<equiv> \<forall>h new_shadow_root_ptr h' r ptr'. h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr \<longrightarrow> h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_disconnected_nodes_locs ptr' \<longrightarrow> r h h'"
cut_less_Diff_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>< t = I - {t..}"
simps_2: fixes e :: "'c" shows "alist_and' [Pos e] = Match e"
induced_map_iterates: fixes n :: "nat"   and A :: "'a set"   and x :: "'a" shows "(induced_map A ^^ n) x = (T ^^ (\<Sum>i<n. return_time_function A ((induced_map A ^^ i) x))) x"
wp_g_ode_subset: fixes f :: "'a \<Rightarrow> 'a"   and T :: "real set"   and S :: "'a set"   and \<phi> :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and U :: "'a \<Rightarrow> real set"   and G :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "local_flow f T S \<phi>"   and "\<And>s. s \<in> S \<Longrightarrow> 0 \<in> U s \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "wp x\<acute>=\<lambda>t. f & G on U S @ 0 \<lceil>Q\<rceil> = \<lceil>\<lambda>s. s \<in> S \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> Q (\<phi> t s))\<rceil>"
bind_split_asm: fixes P :: "'a option \<Rightarrow> bool"   and m :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "P (m \<bind> f) = (\<not> (m = None \<and> \<not> P None \<or> (\<exists>x. m = \<lfloor>x\<rfloor> \<and> \<not> P (f x))))"
dbm_abstr_zone_eq: fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and cc :: "('c, 'd) cconstraint" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" shows "[abstr cc (\<lambda>i j. \<infinity>) v]\<^bsub>v,n\<^esub> = {u. u \<turnstile> cc}"
stake_Suc: fixes n :: "nat"   and s :: "'a stream" shows "stake (Suc n) s = stake n s @ [s !! n]"
f_join_snoc: fixes xs :: "'b list"   and x :: "'b"   and I :: "nat set" shows "(xs @ [x]) \<Join>\<^sub> I = xs \<Join>\<^sub> I @ (if length xs \<in> I then [x] else [])"
has_all_wt_instances_ofD': fixes N :: "('fun, 'var) Term.term set"   and M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s N"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s M"   and "has_all_wt_instances_of \<Gamma> N M"   and "t \<in> N" shows "\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
Nil_elem_T: fixes P :: "'a process" shows "[] \<in> T P"
ideal_in_kernel_axioms_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and zero' :: "'d"   and I :: "'c set" shows "ideal_in_kernel_axioms \<eta> R zero' I \<equiv> I \<subseteq> group_homomorphism.Ker \<eta> R zero'"
less_Cons: fixes x :: "nat"   and xs :: "nat list"   and y :: "nat"   and ys :: "nat list" shows "(x # xs <\<^sub>v y # ys) = (length xs = length ys \<and> (x \<le> y \<and> xs <\<^sub>v ys \<or> x < y \<and> xs \<le>\<^sub>v ys))"
nat_pow_0: fixes x :: "'a" shows "x ^ 0 = (1::'a)"
le_sup_iff: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
cmaps_fresh: fixes \<theta>_c :: "(coname \<times> name \<times> trm) list"   and a :: "coname"   and y :: "name"   and P :: "trm"   and x :: "'a" assumes "\<theta>_c cmaps a to Some (y, P)"   and "x \<sharp> \<theta>_c" shows "x \<sharp> (y, P)"
\<sigma>'_dec: shows "\<sigma>' = (fst \<sigma>1', snd \<sigma>1' @ snd \<sigma>2' @ snd \<sigma>3')"
last_lift: fixes xs :: "('b com option \<times> 'b) list"   and P :: "'b com" assumes "xs \<noteq> []"   and "fst (xs ! (length xs - Suc 0)) = None" shows "fst (map (lift P) xs ! (length (map (lift P) xs) - Suc 0)) = Some P"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "b \<le>lex a" shows "b <lex a"
tranclp_eq_rtranclp: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "(\<lambda>x y. P x y \<or> x = y)\<^sup>+\<^sup>+ = P\<^sup>*\<^sup>*"
in_degree_eq_card_arc_blocks: fixes G :: "('a, 'b) pre_digraph"   and v :: "'a" assumes "graph G" shows "in_degree G v = card (graph.arc_to_block G ` in_arcs G v)"
integral_minus_sets: fixes c :: "real"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "c \<le> a"   and "c \<le> b"   and "f integrable_on {c..max a b}" shows "integral {c..a} f - integral {c..b} f = (if a \<le> b then - integral {a..b} f else integral {b..a} f)"
oinvariantD: fixes A :: "('c \<times> 'd, 'e) automaton"   and S :: "'c \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool"   and U :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<times> 'd \<Rightarrow> bool"   and s :: "'c \<times> 'd" assumes "A \<Turnstile> (S, U \<rightarrow>) P"   and "s \<in> oreachable A S U" shows "P s"
elem_mset_set: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "(x \<in># mset_set A) = (x \<in> A)"
cs_equal_to_snapshot_def: fixes c :: "('d, 'e, 'f, 'g) configuration_scheme"   and c' :: "('h, 'i, 'f, 'j) configuration_scheme" shows "cs_equal_to_snapshot c c' \<equiv> \<forall>cid. channel cid \<noteq> None \<longrightarrow> filter ((\<noteq>) Marker) (msgs c cid) = map Msg (fst (cs c' cid))"
legal_rules_def: fixes Q :: "'a set" shows "legal_rules Q \<equiv> {q \<rightarrow> f qs |q f qs. q \<in> Q \<and> qs \<in> lists Q \<and> A f = Some (length qs)}"
trunc_degree: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P"   and "0 < deg R f" shows "deg R (Cring_Poly.truncate R f) < deg R f"
assms_1: shows "g \<turnstile> n-ns\<rightarrow>m"
scaleC_minus_left: fixes a :: "complex"   and x :: "'b" shows "- a *\<^sub>C x = - (a *\<^sub>C x)"
mod_ring_rel_def: fixes p :: "int"   and x :: "int"   and x' :: "'b mod_ring" assumes "mod_ring_locale TYPE('b) p" shows "mod_ring_locale.mod_ring_rel x x' = (x = to_int_mod_ring x')"
PDERIVS_def: fixes R :: "'b rexp set" shows "PDERIVS R = \<Union> (pderivs_lang UNIV ` R)"
simps_2: fixes k :: "nat"   and u :: "nat" shows "\<P> k (Suc u) = limit (Append (\<Z> k (Suc u)) k) (\<P> k u)"
suffix_imp_subseq: fixes xs :: "'c list"   and ys :: "'c list" assumes "suffix xs ys" shows "subseq xs ys"
mult_right_eq: fixes x :: "'a vec"   and c :: "'a vec"   and y :: "'a vec"   and A :: "'a mat" assumes "dim_vec x = dim_vec c"   and "[y\<^sub>v*A]=c" shows "(A\<^sup>T *\<^sub>v y) \<bullet> x = c \<bullet> x"
towards_inf_compl: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> - x = y \<sqinter> - y"
cSUP_least: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and M :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) M" shows "less_eq (Sup (f ` A)) M"
evals_preserves_sconf: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Ts :: "ty list" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"   and "P,E \<turnstile> es [::] Ts"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
IntI: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
non_zero_mult_invertible_non_zero: fixes v :: "(real, 'a) vec"   and M :: "((real, 'a) vec, 'a) vec" assumes "v \<noteq> 0"   and "invertible M" shows "v v* M \<noteq> 0"
collapse_1: fixes trm :: "trm" assumes "is_Var trm" shows "Var (un_Var trm) = trm"
make_affine_infinity: fixes x :: "int"   and y :: "int" shows "make_affine (x, y, \<zero>) = Infinity"
insert_DiffM: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "x, M - (x, {#}) = M"
le_sup_equiv2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<squnion> b = b)"
xt1: shows "(x, t) \<in> Sigma X existence_ivl0"
find_index_eq_none: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(find_index P xs = None) = (\<forall>x\<in>set xs. \<not> P x)"
not_arr_null: shows "\<not> VoV.arr VoV.null"
jpTraces_init_length_inv: fixes t :: "'s Trace"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "t \<in> jpTraces jp" shows "(tLength t = 0) = (\<exists>s. s \<in> set envInit \<and> t = tInit s)"
apath_sd: shows "S.apath u (sd_path e w p) v"
r_cancel: fixes a :: "'b"   and c :: "'b"   and b :: "'b" assumes "a \<otimes>\<^bsub>G i\<^esub> c = b \<otimes>\<^bsub>G i\<^esub> c"   and "a \<in> carrier (G i)"   and "b \<in> carrier (G i)"   and "c \<in> carrier (G i)" shows "a = b"
lists_digraph_map_axioms: shows "lists_digraph_map G_list xss"
assms_1: shows "(\<box>(\<lambda>s. P' s \<longrightarrow> P s)) \<sigma>"
ide_char': fixes a :: "'c" shows "C.ide a = (C.arr a \<and> (C.dom a = a \<or> C.cod a = a))"
load_tmps_read_tmps_distinct_nth_update: fixes i :: "nat"   and "is" :: "instr list"   and sb :: "'a memref list"   and p :: "'a"   and xs :: "nat \<Rightarrow> nat option"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "load_tmps is \<inter> read_tmps sb = {}" shows "load_tmps_read_tmps_distinct (ts\<^sub>s\<^sub>b[i := (p, is, xs, sb, \<D>, \<O>, \<R>)])"
I_3: fixes i :: "'b"   and \<omega> :: "'a" assumes "i \<in> I" shows "0 \<le> X i \<omega>"
term_ok_TFrees_imp_sort_ok_pre: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term"   and x :: "variable"   and S :: "String.literal set" assumes "is_std_sig \<Sigma>"   and "term_ok' \<Sigma> t"   and "(x, S) \<in> tvs t" shows "wf_sort (subclass (osig \<Sigma>)) S"
functor_ToTC: shows "functor CCC.comp (\<cdot>) T.ToTC"
resolvent_upon_and_partial_saturation: fixes P1 :: "'at Literal set"   and S :: "'at Literal set set"   and P2 :: "'at Literal set"   and A :: "'at"   and R :: "'at Literal set set"   and C :: "'at Literal set" assumes "redundant P1 S"   and "redundant P2 S"   and "partial_saturation S A (S \<union> R)"   and "C = resolvent_upon P1 P2 A" shows "redundant C (S \<union> R)"
countable_web_quotient_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "countable_web (quotient_web \<Gamma> f)"
XNv: shows "0 \<le> inner_prod v (X N *\<^sub>v v)"