hcju/isabelleps huggingface dataset:
GB_alt_2_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "G \<subseteq> dgrad_p_set d m" shows "is_Groebner_basis G = (\<forall>f\<in>pmdl G. f \<noteq> 0 \<longrightarrow> is_red G f)"
compile_env_def: fixes e :: "'fenv_std" shows "compile_env e = Sinca.Fenv.from_list (map (map_prod id compile_fundef) (Fstd_to_list e))"
scopeExtPar: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sub>e P \<parallel> <\<nu>x>Q"
uncross_positive_flip_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "uncross_positive_flip w1 w2" shows "kauff_mat w1 = kauff_mat w2"
hyps_4: shows "\<Gamma>\<^sub>1' \<sqsubseteq>\<^sub>c \<Gamma>'"
simps_6: fixes \<phi> :: "'a dfmla"   and \<psi> :: "'a dfmla" shows "wff (Until \<phi> \<psi>) = False"
SUP_emeasure_incseq: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "range A \<subseteq> sets M"   and "incseq A" shows "(SUP n. emeasure M (A n)) = emeasure M (\<Union> (range A))"
K2_def: fixes C :: "acom"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Es :: "char list set"   and Q :: "'a" shows "K2 C e Es Q \<equiv> \<forall>s1 s2. s1 = s2 on fune C Es \<longrightarrow> preT C e s1 = preT C e s2"
irreducibleD': fixes a :: "'a"   and b :: "'a" assumes "irreducible a"   and "b dvd a" shows "a dvd b \<or> is_unit b"
diff_list_def: fixes xs :: "'b list"   and ys :: "'b list" shows "diff_list xs ys = filter (\<lambda>x. x \<notin> set ys) xs"
uminus_add_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc" shows "- (A + B) = - B + - A"
index_unit_vec_2: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "unit\<^sub>v n i $ i = (1::'a)"
card_insert_le_m1: fixes n :: "nat"   and y :: "'a set"   and x :: "'a" assumes "0 < n"   and "card y \<le> n - 1" shows "card (insert x y) \<le> n"
find_consistent_signs_at_roots_def: fixes p :: "real poly"   and qs :: "real poly list" shows "find_consistent_signs_at_roots p qs = (let (M, S, \<Sigma>) = calculate_data p qs in \<Sigma>)"
hypnat_not_less0: fixes n :: "nat star" shows "\<not> n < 0"
C2: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and yvec :: "name list" assumes "(\<Psi>', S, T) \<in> Rel'"   and "yvec \<sharp>* \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>*yvec\<rparr>S, \<lparr>\<nu>*yvec\<rparr>T) \<in> Rel'"
pair_size_def: fixes p :: "'d \<times> 'e" shows "pair_size p = size (fst p) + size (snd p) + 1"
q_def: shows "q = q'(g := q0 + q' g)"
bigomega_iff_bigo: fixes g :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'c" shows "(g \<in> \<Omega>[F](f)) = (f \<in> O[F](g))"
bind_UNION: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'd set" shows "A \<bind> f = \<Union> (f ` A)"
rorc_channel_send_def: fixes s :: "(bool \<times> unit) \<times> (bool list \<Rightarrow> bool list option) \<times> (bool list \<times> bool list) cstate"   and m :: "bool list" shows "local.rorc_channel_send s m \<equiv> if fst (fst s) then return_spmf ((), (True, ()), snd s) else (rorc.rnd_oracle (rnd \<eta>))\<dagger> (snd s) m \<bind> (\<lambda>(r, s). mac \<eta> r m \<bind> (\<lambda>a. \<dagger>channel.send_oracle s (a, m) \<bind> (\<lambda>(uu_, s). return_spmf ((), (True, ()), s))))"
bij_betw_totatives_gcd_eq: fixes d :: "nat"   and n :: "nat" assumes "d dvd n"   and "0 < n" shows "bij_betw (\<lambda>k. k * d) (totatives (n div d)) {k \<in> {0<..n}. gcd k n = d}"
unfold_llist_Lazy_llist: fixes IS_LNIL :: "'b \<Rightarrow> bool"   and LHD :: "'b \<Rightarrow> 'a"   and LTL :: "'b \<Rightarrow> 'b"   and b :: "'b" shows "unfold_llist IS_LNIL LHD LTL b = Lazy_llist (\<lambda>_. if IS_LNIL b then None else Some (LHD b, unfold_llist IS_LNIL LHD LTL (LTL b)))"
unsigned_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero 0 = zero"
axioms_1: fixes return :: "'b \<Rightarrow> 'c"   and bind :: "'c \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and altc :: "'d cset \<Rightarrow> ('d \<Rightarrow> 'c) \<Rightarrow> 'c" assumes "monad_altc return bind altc" shows "monad return bind"
Abs_bit_eq_of_bool: shows "Abs_bit = of_bool"
r: shows "tree.root (mkTree (sdrop n rs) s') = shd (sdrop m steps)"
lossless_S2_14: fixes c :: "bool \<times> bool"   and out :: "bool" shows "lossless_spmf (S2_14 c out)"
hyps_23: shows "A\<^sub>Q \<sharp>* K"
inverse_fract: fixes a :: "'b"   and b :: "'b" shows "inverse (Fraction_Field.Fract a b) = Fraction_Field.Fract b a"
sorted_wrt_subset: fixes X :: "'a set"   and l :: "'a set list" assumes "X \<in> set l"   and "sorted_wrt (\<subseteq>) l" shows "hd l \<subseteq> X"
prime_elemI: fixes p :: "'a" assumes "p \<noteq> (0::'a)"   and "\<not> is_unit p"   and "\<And>a b. p dvd a * b \<Longrightarrow> p dvd a \<or> p dvd b" shows "prime_elem p"
nn_integral_const: fixes M :: "'h measure"   and c :: "ennreal" shows "\<integral>\<^sup>+ x. c \<partial>M = c * emeasure M (space M)"
dom_simp: fixes f :: "'c \<times> 'c" assumes "CopxC.arr f" shows "CopxC.dom f = (Cop.dom (fst f), C.dom (snd f))"
m_closed: shows "x \<otimes>\<^bsub>DirProds G I\<^esub> y \<in> carrier (DirProds G I)"
xivp_3: shows "X \<in> {0--t} \<rightarrow> S"
eq_foldr: fixes xs :: "'d list" shows "sum_list xs = foldr (+) xs (0::'d)"
converse_iff: fixes a :: "'a"   and b :: "'b"   and r :: "('b \<times> 'a) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
opp_fold_A_fixespointwise2: shows "fixespointwise opp_fold_A D"
simps: fixes cto :: "(String.literal \<times> typ) list"   and tao :: "(String.literal \<times> nat) list"   and sa :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "exe_wf_sig (ExeSignature cto tao sa) = (exe_wf_osig sa \<and> fst ` set (snd sa) = fst ` set tao \<and> (\<forall>type\<in>fst ` set (snd sa). \<forall>ars\<in>snd ` set (the (lookup (\<lambda>k. k = type) (snd sa))). the (lookup (\<lambda>k. k = type) tao) = length ars) \<and> Ball (snd ` set cto) (typ_ok_sig (map_of cto, map_of tao, translate_osig sa)))"
\<alpha>_\<gamma>': shows "LeftDerivation \<alpha> D' \<gamma>'"
keys_pm_of_idx_pm_subset: fixes xs :: "'c list"   and f :: "nat \<Rightarrow>\<^sub>0 'd" shows "keys (pm_of_idx_pm xs f) \<subseteq> set xs"
case_one_not_learn_V: fixes j :: "nat" assumes "\<nexists>t. local.prefixes t j \<up>"   and "2 \<le> j"   and "\<phi> j = \<psi> j" shows "\<not> learn_lim \<phi> V_constotal s"
msub_submodule: fixes M :: "('e, 'a, 'f) Module_scheme"   and M1 :: "('e, 'a, 'f) Module_scheme" assumes "R module M"   and "R module M1"   and "msubmodule R M M1" shows "submodule R M (carrier M1)"
vertex_3_int_def: shows "vertex_3 \<equiv> 3"
decreasing: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "x \<le> y" shows "f y \<le> f x"
outs_in_Use: fixes a :: "'edge"   and Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs"   and "V \<in> set outs" shows "V \<in> Use (sourcenode a)"
integration_by_parts: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'a"   and g' :: "real \<Rightarrow> 'b"   and y :: "'c" assumes "bounded_bilinear prod"   and "a \<le> b"   and "continuous_on {a..b} f"   and "continuous_on {a..b} g"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"   and "((\<lambda>x. prod (f x) (g' x)) has_integral prod (f b) (g b) - prod (f a) (g a) - y) {a..b}" shows "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
App_apps: fixes s :: "('c, 'd) tm"   and ts :: "('c, 'd) tm list"   and t :: "('c, 'd) tm" shows "App (apps s ts) t = apps s (ts @ [t])"
obs_safe_def: fixes r :: "nat"   and s :: "'a obsv_state_scheme"   and v :: "val" shows "obs_safe r s v \<equiv> \<forall>r'<r. \<exists>p. obsv_state.obs s r' p \<in> {None, Some v}"
flatf_admissible_pointwise: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b" assumes "\<And>x. P x b" shows "ccpo.admissible (flatf_lub b) (flatf_ord b) (\<lambda>g. \<forall>x. P x (g x))"
filterp_idealp_dual: shows "filterp = idealp \<circ> (`) \<partial>"
zero_less_dist_iff: fixes x :: "'b"   and y :: "'b" shows "(0 < dist x y) = (x \<noteq> y)"
leq_antisym: fixes x :: "'f option"   and y :: "'f option" assumes "option_ord x y"   and "option_ord y x" shows "x = y"
fixed_length_sublist_plus: fixes xs1 :: "'b list"   and c :: "nat"   and l :: "nat"   and xs2 :: "'b list"   and i :: "nat" assumes "length xs1 = c * l"   and "length xs2 = c * l"   and "i < c" shows "fixed_length_sublist (vec_plus xs1 xs2) l i = vec_plus (fixed_length_sublist xs1 l i) (fixed_length_sublist xs2 l i)"
prv_dsj_cases: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (dsj \<phi>1 \<phi>2)"   and "bprv (imp \<phi>1 \<chi>)"   and "bprv (imp \<phi>2 \<chi>)" shows "bprv \<chi>"
of_nat_diff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat"   and m :: "nat" assumes "class.semiring_1_cancel plus minus zero one times"   and "n \<le> m" shows "semiring_1.of_nat one plus zero (m - n) = minus (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
fps_mult_nth_eq0: fixes n :: "nat"   and f :: "'a fps"   and g :: "'a fps" assumes "n < subdegree f + subdegree g" shows "fps_nth (f * g) n = (0::'a)"
le_fun_def: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
assms: shows "(\<exists>n''\<ge>n. \<gamma> t n'' \<and> (\<forall>n'\<ge>n. n' < n'' \<longrightarrow> \<gamma>' t n')) \<or> (\<forall>n'\<ge>n. \<gamma>' t n')"
order_iff: fixes b :: "'e multiset"   and a :: "'e multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
sets_join: fixes M :: "'c measure measure" shows "sets (join M) = sets (select_sets M)"
span_superset: fixes S :: "'b set" shows "S \<subseteq> ideal S"
prems: shows "equivalent_on \<sigma> \<eta> (vars_of (t1 \<cdot> t2)) I"
assms_2: shows "module R (M\<lparr>carrier := H\<rparr>)"
pp_one: shows "- - (1::'a) = (1::'a)"
AgentX_X_PPA: shows "\<lfloor>PPA (Agent X)\<rfloor>\<^sub>X"
prod_topology_subtopology_1: fixes X :: "'d topology"   and S :: "'d set"   and Y :: "'e topology" shows "prod_topology (subtopology X S) Y = subtopology (prod_topology X Y) (S \<times> topspace Y)"
mcont_if_bot: fixes lub' :: "'c set \<Rightarrow> 'c"   and ord :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bound :: "'d llist"   and bot :: "'c"   and f :: "'d llist \<Rightarrow> 'c"   and g :: "'d llist \<Rightarrow> 'c" assumes "class.ccpo lub' ord (mk_less ord)"   and "\<And>x. \<not> x \<sqsubseteq> bound \<Longrightarrow> ord bot (f x)"   and "\<And>x. g x = (if x \<sqsubseteq> bound then bot else f x)"   and "\<And>x y. \<lbrakk>x \<sqsubseteq> y; \<not> x \<sqsubseteq> bound\<rbrakk> \<Longrightarrow> ord (f x) (f y)"   and "\<And>Y. \<lbrakk>Complete_Partial_Order.chain (\<sqsubseteq>) Y; Y \<noteq> {}; \<And>x. x \<in> Y \<Longrightarrow> \<not> x \<sqsubseteq> bound\<rbrakk> \<Longrightarrow> f (lSup Y) = lub' (f ` Y)" shows "mcont lSup (\<sqsubseteq>) lub' ord g"
sublinear_sub_distrib: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "sublinear t" shows "sub_distrib t"
prj_chine_assoc_3: shows "f_src_src.Prj\<^sub>0\<^sub>0 \<cdot> f_src_src.chine_assoc = f_src_src.Prj\<^sub>0"
elims: fixes x :: "perm"   and xa :: "('a, 'b) L_action"   and y :: "('a, 'b) L_action" assumes "x \<bullet> xa = y" obtains p :: "perm" and \<alpha> :: "'a" where "x = pa__" and "xa = L_action.Act \<alpha>" and "y = L_action.Act (pa__ \<bullet> \<alpha>)"   | p :: "perm" and f :: "'b" where "x = p" and "xa = Eff f" and "y = Eff (p \<bullet> f)"
token_run_step: fixes x :: "nat"   and n :: "nat"   and q' :: "'b"   and q :: "'b" assumes "x \<le> n"   and "token_run x n = q'"   and "q = \<delta> q' (w n)" shows "token_run x (Suc n) = q"
lim: shows "LIM x at_right a. f' x / g' x :> at_top"
hom_closed: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (reduced_homology_group (int n - 1) (nsphere (n - Suc 0)))" shows "gn x \<in> carrier (relative_homology_group (int n) (subtopology (nsphere n) {x. 0 \<le> x n}) {x. x n = 0})"
map_to_set_def: fixes m :: "'a \<Rightarrow> 'b option" shows "map_to_set m = {(k, v). m k = Some v}"
connectedI: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "Graph.walk G xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "Graph.connected G v w"
r_cancel_one': fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and a :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x = a \<otimes>\<^bsub>G\<^esub> x) = (a = \<one>\<^bsub>G\<^esub>)"
chain_Sync1: fixes Y :: "nat \<Rightarrow> 'c process"   and A :: "'c set"   and S :: "'c process" assumes "chain Y" shows "chain (\<lambda>i. Y i \<lbrakk>A\<rbrakk> S)"
one_induced_iff_principal: shows "induced_modulus 1 = (\<chi> = principal_dchar n)"
less_eq_matrix_def: fixes A :: "('a, 'b, 'c) matrix"   and B :: "('a, 'b, 'c) matrix" shows "A \<le> B \<equiv> A + B = B"
ftv_tyS: fixes S :: "tyS" shows "supp S = set (ftv S)"
card_product: shows "card ({1..m} \<times> {1..n}) = m * n"
PO_m1_refines_m1a: shows "refines R1a1 med1a1 m1a m1"
hyps_1: fixes p :: "complex poly" assumes "n = degree p" shows "degree p = proots_count p UNIV"
comp_ide_arr: fixes b :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "H.VxV.ide b"   and "H.VxV.seq b f" shows "H.VxV.comp b f = f"
Abs_finfun_inverse_finite: fixes x :: "'a \<Rightarrow> 'b" assumes "finite UNIV" shows "finfun_apply (Abs_finfun x) = x"
\<xi>_Inf_many: shows "\<exists>\<^sub>\<infinity>k. \<bar>\<xi> - (\<Sum>k = 0..k. bb k / aa k)\<bar> < 1 / prod aa {0..k} powr (2 + \<delta>)"
simps: fixes x :: "rat"   and y :: "int poly \<times> rat \<times> rat" shows "mult_rat_1 x y = (if x < 0 then uminus_2 (mult_rat_1_pos (- x) y) else if x = 0 then Rational 0 else mult_rat_1_pos x y)"
monomial_decompD_1: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "monomial_decomp ps"   and "(h, U) \<in> set ps" shows "is_monomial h"
boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b \<cdot> c" obtains "a \<le> b" and "a \<le> c"
ordLess_ordLeq_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('c \<times> 'c) set" assumes "r <o r'"   and "r' \<le>o r''" shows "r <o r''"
const_in_iff: fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "((\<lambda>_. c) \<in> o[F](f)) = ((\<lambda>_. 1::'a) \<in> o[F](f))"
berl: shows "finite_field_factorization_int p f = (c, fs)"
is_empty_def: fixes A :: "'b set" shows "Set.is_empty A = (A = {})"
ext: fixes g :: "'b"   and f :: "'b" assumes "D g f \<noteq> B.null" shows "B.seq g f"
contlub_cfun_arg: fixes Y :: "nat \<Rightarrow> 'a"   and f :: "'a \<rightarrow> 'b" assumes "chain Y" shows "f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
dg_1_is_arrD_3: fixes \<aa> :: "V"   and \<ff> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_1 \<aa> \<ff>\<^esub> b" shows "f = \<ff>"
inner_same_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"   and u :: "'c" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "inner u u = 1"
wf_trm_param: fixes f :: "('fun, 'atom, 'sets) prot_fun"   and T :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term list"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term" assumes "wf\<^sub>t\<^sub>r\<^sub>m (Fun f T)"   and "t \<in> set T" shows "wf\<^sub>t\<^sub>r\<^sub>m t"
assms_5: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s A"
flows_reverse: fixes t :: "real"   and x0a :: "'a"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "t \<in> Y.mvar.existence_ivl x0a t0 x0" shows "Y.mvar.flow x0a t (Y.mvar.flow x0a t0 x0 t) t0 = x0"
elims_2: fixes x :: "'v" assumes "terminal_vertex x" obtains v :: "'v" where "x = v" and "v |\<in>| vertices \<and> terminal_node (nodeOf v)"
dvd_sgn_mult_iff: fixes l :: "int"   and r :: "int"   and k :: "int" shows "(l dvd sgn r * k) = (l dvd k \<or> r = 0)"
tyAbs: shows "typ_of (Abs T t $ x) = Some propT"
mono_transD: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "mono_trans t"   and "sound P"   and "sound Q"   and "P \<tturnstile> Q" shows "t P \<tturnstile> t Q"
insert_absorb2: fixes x :: "'c"   and A :: "'c set" shows "Set.insert x (Set.insert x A) = Set.insert x A"
valid_vmsaux_def: shows "valid_vmsaux = (\<lambda>_ cur (t, aux) auxlist. t = cur \<and> aux = auxlist)"
pmdl_filter_syzygy_basis_subset: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "distinct bs"   and "pmdl (set gs) = pmdl (set (init_syzygy_list bs))" shows "pmdl (set (filter_syzygy_basis (length bs) gs)) \<subseteq> pmdl (syzygy_module_list bs)"
walk_in_V: fixes G :: "('a, 'b) Graph_scheme"   and xs :: "'a list" assumes "Digraph G"   and "Digraph.walk G xs" shows "set xs \<subseteq> V\<^bsub>G\<^esub>"
bits_of_int_simps_2: shows "bits_of (- 1) = [True]"
less_2: fixes u :: "'a list"   and a :: "'a" assumes "w = u @- sconst a" shows "thesis"
assms_1: shows "imp_set_iterate is_set is_it it_init it_has_next it_next"
Sigma_def: fixes A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" shows "Sigma A B \<equiv> \<Union>x\<in>A. \<Union>y\<in>B x. {(x, y)}"
right_gpv_map: fixes f :: "'m \<Rightarrow> 'h"   and g :: "'n \<Rightarrow> 'j"   and gpv :: "('m, 'n, 'l) gpv"   and h :: "'o \<Rightarrow> 'i" shows "right_gpv (map_gpv f g gpv) = map_gpv f (map_sum h g) (right_gpv gpv)"
S2_14_def: fixes C :: "bool \<times> bool"   and out :: "bool" shows "S2_14 C out = (let (c0, c1) = C in coin_spmf \<bind> (\<lambda>S0. coin_spmf \<bind> (\<lambda>S1. coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. coin_spmf \<bind> (\<lambda>a0. coin_spmf \<bind> (\<lambda>a1. coin_spmf \<bind> (\<lambda>a2. coin_spmf \<bind> (\<lambda>a3. let a0' = if \<not> c0 \<and> \<not> c1 then S0 \<oplus> S2 \<oplus> out else a0; a1' = if \<not> c0 \<and> c1 then S0 \<oplus> S3 \<oplus> out else a1; a2' = if c0 \<and> \<not> c1 then S1 \<oplus> S4 \<oplus> out else a2; a3' = if c0 \<and> c1 then S1 \<oplus> S5 \<oplus> out else a3 in S2_OT12 c0 (if c0 then S1 else S0) \<bind> (\<lambda>a. S2_OT12 c1 (if c1 then S3 else S2) \<bind> (\<lambda>b. S2_OT12 c1 (if c1 then S5 else S4) \<bind> (\<lambda>c. return_spmf ((c0, c1), (a0', a1', a2', a3'), a, b, c)))))))))))))))"
m_comm: fixes x :: "'a"   and y :: "'a" shows "x + y = y + x"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "one_hom hom"   and "semigroup_mult_hom hom" shows "monoid_mult_hom hom"
distinct_21: fixes x31 :: "('a, 'b) MWLsCom"   and x32 :: "('a, 'b) MWLsCom"   and x51 :: "nat"   and x52 :: "'a"   and x53 :: "('a, 'b) MWLsCom" shows "x31;x32 \<noteq> while\<^bsub>x51\<^esub> x52 do x53 od"
eq_neutr_iff: fixes a :: "'b"   and b :: "'b" shows "(sup a b = bot) = (a = bot \<and> b = bot)"
nonzero_neg_divide_eq_eq: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)" shows "(- (a / b) = c) = (- a = c * b)"
valid_index_lt: fixes "is" :: "nat list"   and ds :: "nat list"   and m :: "nat" assumes "is \<lhd> ds"   and "m < length ds" shows "is ! m < ds ! m"
stream_all2_combine: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a stream"   and ys :: "'b stream"   and Q :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and zs :: "'d stream"   and R :: "'a \<Rightarrow> 'd \<Rightarrow> bool" assumes "stream_all2 P xs ys"   and "stream_all2 Q ys zs"   and "\<And>x y z. P x y \<and> Q y z \<Longrightarrow> R x z" shows "stream_all2 R xs zs"
inverse_permutation_of_list_unique': fixes xs :: "('b \<times> 'b) list"   and x :: "'b"   and y :: "'b" assumes "distinct (map snd xs)"   and "(x, y) \<in> set xs" shows "inverse_permutation_of_list xs y = x"
map_spmf_of_pmf: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b pmf" shows "map_spmf f (spmf_of_pmf p) = spmf_of_pmf (map_pmf f p)"
sorted_wrt_snoc: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and y :: "'a" assumes "transp P"   and "sorted_wrt P xs"   and "P (last xs) y" shows "sorted_wrt P (xs @ [y])"
not_lindepD: fixes S :: "'a vec set"   and A :: "'a vec set"   and f :: "'a vec \<Rightarrow> 'a" assumes "lin_indpt S"   and "finite A"   and "A \<subseteq> S"   and "f \<in> A \<rightarrow> UNIV"   and "lincomb f A = 0\<^sub>v n" shows "f \<in> A \<rightarrow> {0::'a}"
UnCI: fixes c :: "'b"   and B :: "'b set"   and A :: "'b set" assumes "c \<notin> B \<Longrightarrow> c \<in> A" shows "c \<in> A \<union> B"
nsqn_update_changed_kno_val: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and dsn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt ip (dsn, kno, val, hops, nhip)" shows "nsqn (update rt ip (dsn, kno, val, hops, nhip)) ip = dsn"
iT_arith_iAll_conv_1: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool" shows "(\<box> t I \<oplus> k. P t) = (\<box> t I. P (t + k))"
algebraicI: fixes p :: "'a list"   and K :: "'a set"   and x :: "'a" assumes "p \<in> carrier (K [X])"   and "p \<noteq> []"   and "local.eval p x = \<zero>" shows "(algebraic over K) x"
assms_4: shows "T\<lparr>ArrVal\<rparr>\<lparr>x\<rparr> \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (S\<lparr>ArrVal\<rparr>)"
ma: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'a list list"   and nc :: "nat"   and m2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "mat nr n m1"   and "mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI zer (+) (*) nr m1 m2 ! j ! i = scalar_prodI zer (+) (*) (row m1 i) (col m2 j)"