hcju/isabelleps huggingface dataset:
submonoid_inverse_closed: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and u :: "'c" assumes "submonoid N M composition unit"   and "monoid.invertible N composition unit u"   and "u \<in> N" shows "monoid.inverse M composition unit u \<in> N"
c: fixes s :: "'a" assumes "P s"   and "b s" shows "P (c s)"
sound: fixes \<iota> :: "'a inference" assumes "\<iota> \<in> Inf" shows "set (prems_of \<iota>) \<Turnstile> {concl_of \<iota>}"
insertI1: fixes a :: "'b"   and B :: "'b set" shows "(a::'b) \<in> a \<triangleright> (B::'b set)"
noStuck_start: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and t :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> t"   and "t \<noteq> Stuck" shows "s \<noteq> Stuck"
Nmlize_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>Cod t\<^bold>\<rfloor> = Cod \<^bold>\<lfloor>t\<^bold>\<rfloor>"
card_mono: fixes B :: "'e set"   and A :: "'e set" assumes "finite B"   and "A \<subseteq> B" shows "card A \<le> card B"
sorted_remove1: fixes xs :: "'a list"   and a :: "'a" assumes "sorted xs" shows "sorted (remove1 a xs)"
size_split_min: fixes t :: "'c tree"   and a :: "'c"   and t' :: "'c tree" assumes "Root_Balanced_Tree.split_min t = (a, t')"   and "t \<noteq> \<langle>\<rangle>" shows "size t' = size t - 1"
hyps: fixes m1 :: "'a list list"   and m2 :: "'a list list" assumes "length m1 = nn \<and> (\<forall>a\<in>set m1. vec nr a)"   and "length m2 = nn \<and> (\<forall>a\<in>set m2. vec nr a)"   and "length m1 = nn \<and> (\<forall>a\<in>set m1. vec nr a)"   and "length m2 = nn \<and> (\<forall>a\<in>set m2. vec nr a)" shows "\<forall>x\<in>set (zip m1 m2). vec nr (vec_plusI pl (fst x) (snd x))"
p\<^sub>0_simps: shows "trg p\<^sub>0 = src h"
FactRing_def: fixes R :: "('e, 'f) ring_scheme"   and I :: "'e set" shows "R Quot I = \<lparr>carrier = a_rcosets\<^bsub>R\<^esub> I, mult = rcoset_mult R I, one = I +>\<^bsub>R\<^esub> \<one>\<^bsub>R\<^esub>, zero = I, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
IN: fixes x :: "'v"   and y :: "'v" assumes "edge \<Delta> x y" shows "edg (Edge x y) (Vertex y)"
a: shows "is_zero_row_upt_k a (Suc k) A"
Cons_4: shows "j_to_jb_index ((m, b) # n_as) i = (bi, li)"
is_open_subset: fixes X :: "'a set"   and is_open :: "'a set \<Rightarrow> bool"   and U :: "'a set"   and I :: "real set"   and C :: "real \<Rightarrow> 'a set" assumes "open_cover_of_open_subset X is_open U I C" shows "is_open U"
elims: fixes x :: "'b tree"   and y :: "'b tree" assumes "splay_max x = y" obtains "x = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | A :: "'b tree" and a :: "'b" where "x = \<langle>A__, a__, \<langle>\<rangle>\<rangle>" and "y = \<langle>A__, a__, \<langle>\<rangle>\<rangle>"   | A :: "'b tree" and a :: "'b" and B :: "'b tree" and b :: "'b" and CD :: "'b tree" where "x = \<langle>A, a, \<langle>B, b, CD\<rangle>\<rangle>" and "y = (if CD = \<langle>\<rangle> then \<langle>\<langle>A, a, B\<rangle>, b, \<langle>\<rangle>\<rangle> else case splay_max CD of \<langle>C, x, xa\<rangle> \<Rightarrow> \<langle>\<langle>\<langle>A, a, B\<rangle>, b, C\<rangle>, x, xa\<rangle>)"
scale_cancel_right: fixes a :: "complex"   and x :: "'b"   and b :: "complex" shows "(a *\<^sub>C x = b *\<^sub>C x) = (a = b \<or> x = (0::'b))"
Acc: fixes G :: "prog"   and s0 :: "st"   and va :: "Term.var"   and v :: "val"   and f :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>va=\<succ>(v, f)\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>Acc va-\<succ>v\<rightarrow> s1"
Spy_analz_priEK: fixes evs :: "event list"   and A :: "agent" assumes "evs \<in> ns_public" shows "(Key (priEK A) \<in> analz (knows Spy evs)) = (A \<in> bad)"
VVp_to_V: fixes v :: "'a"   and p :: "Player" assumes "v \<in> VV p" shows "v \<in> V"
smult_closed: fixes a :: "'a"   and x :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "x \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> x \<in> carrier P"
hcomp_eqI: fixes f :: "'a"   and g :: "'a"   and f' :: "'a"   and g' :: "'a" assumes "S.hseq f g"   and "f = f'"   and "g = g'" shows "S.hcomp f g = f' \<star> g'"
sqrt_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "sqrt_aform_err p X = (let ivl = ivl_of_aform_err p X; l = lower ivl; u = upper ivl in if 0 < l then min_range_mono p Sqrt (\<lambda>x. Half (Inverse (Sqrt x))) l u X else Some (ivl_err (real_interval (sqrt_float_interval p ivl))))"
assms_1: shows "n -as\<rightarrow>\<^sub>\<surd>* (_Exit_)"
abs_eq: fixes xa :: "'b set"   and x :: "'b set" assumes "eq_onp ord_class.filter xa xa"   and "eq_onp ord_class.filter x x" shows "(Filters.filter.Abs_filter xa \<le> Filters.filter.Abs_filter x) = (xa \<subseteq> x)"
limg: shows "(sum g \<longlongrightarrow> infsetsum' g A) (finite_subsets_at_top A)"
iff_simps_2: fixes p :: "fm" shows "Ferrack.iff p (NOT p) = F"
iarray_to_vec_def: fixes A :: "'b iarray" shows "iarray_to_vec A = (\<chi>i. A !! mod_type_class.to_nat i)"
member_notin: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> S - {y}" shows "x \<noteq> y"
hyps: shows "cis (arg (x ^ n)) = cis (arg x * real n)"
get_IFS_def: fixes M :: "'a SecurityInvariant_configured list" shows "TopoS_Composition_Theory.get_IFS M \<equiv> filter c_isIFS M"
Red_F_\<G>_q_def: fixes q :: "'q"   and N :: "'f set" shows "Red_F_\<G>_q q N = {C. \<forall>D\<in>\<G>_F_q q C. D \<in> Red_F_q q (\<G>_Fset_q q N) \<or> (\<exists>E\<in>N. Prec_F_g D E C \<and> D \<in> \<G>_F_q q E)}"
sublinear_sub_conj: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "sublinear t" shows "sub_conj t"
assms_4: shows "Inr m = create_insert s n \<sigma> new_id"
scale_left_distrib_on: fixes x :: "'b"   and a :: "'a"   and b :: "'a" assumes "x \<in> U\<^sub>M" shows "(a + b) *s x = a *s x + b *s x"
lcm_pos_int: fixes m :: "int"   and n :: "int" assumes "m \<noteq> 0"   and "n \<noteq> 0" shows "0 < lcm m n"
mssnth_exp_series_stream_aux: fixes n :: "nat"   and m :: "nat" shows "mssnth (exp_series_stream_aux (1 / fact n) (real (n + 1))) m = 1 / fact (n + m)"
append: fixes xs :: "'a list"   and ys :: "'a list" shows "sum_list (xs @ ys) = sum_list xs + sum_list ys"
IH_2: fixes stk :: "int list" shows "ccomp (WHILE b DO c) \<turnstile> (0, s2, stk) \<rightarrow>* (size (ccomp (WHILE b DO c)), s3, stk)"
distinct_269: fixes x12a :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Exp x12a \<noteq> Power x151 x152"
finite_image_iff: fixes f :: "'k \<Rightarrow> 'l"   and A :: "'k set" assumes "inj_on f A" shows "finite (f ` A) = finite A"
assms: shows "ModalSequents.mset c = \<LM> A  \<RM>"
almost_all_commutative'': fixes S :: "'b ltl set"   and P :: "'b ltl \<Rightarrow> nat \<Rightarrow> bool" assumes "finite S"   and "Only_G S"   and "\<And>x. G x \<in> S \<Longrightarrow> \<forall>\<^sub>\<infinity>i. P x i" shows "\<forall>\<^sub>\<infinity>i. \<forall>x. G x \<in> S \<longrightarrow> P x i"
widen_i_i_iI: fixes x :: "'a prog"   and xa :: "ty"   and xb :: "ty" assumes "x \<turnstile> xa \<le> xb" shows "pred.eval (widen_i_i_i x xa xb) ()"
eq_append_same1: fixes xs' :: "'d list"   and ys' :: "'d list"   and xs :: "'d list" assumes "equivclp cancel1 xs' ys'" shows "equivclp cancel1 (xs @ xs') (xs @ ys')"
measures_def: fixes fs :: "('a \<Rightarrow> nat) list" shows "measures fs = inv_image (lex less_than) (\<lambda>a. map (\<lambda>f. f a) fs)"
g_inv: fixes n :: "nat" shows "n \<le> g (g_inv n)"
B: shows "\<forall>\<^sub>F n in sequentially. ereal (Busemann_function_at xi x y - 2 * deltaG TYPE('a)) - ereal e < ereal (Busemann_function_at (to_Gromov_completion (t n)) x y)"
ifp_reflexive: shows "\<forall>p. Policy.ifp p p"
trans: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b \<le> a"   and "c \<le> b" shows "c \<le> a"
comp_cod_arr: fixes f :: "'c local.term set"   and b :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f"   and "\<F>\<^sub>SC.cod f = b" shows "\<F>\<^sub>SC.comp b f = f"
case_3: fixes f1 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f3 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x31 :: "nat"   and x32 :: "'b"   and x33 :: "'b"   and x34 :: "'c"   and x35 :: "'c"   and x36 :: "'d" shows "(case Recv x31 x32 x33 x34 x35 x36 of Trans x xa xb \<Rightarrow> f1 x xa xb | Send x xa xb xc xd xe \<Rightarrow> f2 x xa xb xc xd xe | Recv x xa xb xc xd xe \<Rightarrow> f3 x xa xb xc xd xe | Snapshot x \<Rightarrow> f4 x | RecvMarker x xa xb \<Rightarrow> f5 x xa xb) = f3 x31 x32 x33 x34 x35 x36"
wary_subst_wary: fixes \<rho> :: "'v \<Rightarrow> ('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary_subst \<rho>"   and "wary s" shows "wary (subst \<rho> s)"
stake_shift: fixes i :: "nat"   and w :: "'c list"   and s :: "'c stream" shows "stake i (w @- s) = take i w @ stake (i - length w) s"
inf_prefix_length_correct: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "i < inf_prefix_length xs ys" shows "xs ! i = ys ! i"
pcr_cr_eq: shows "pcr_ennreal = cr_ennreal"
ideriv_def: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and as :: "nat \<Rightarrow> 'a" shows "ideriv R S as = ((\<forall>i. (as i, as (Suc i)) \<in> R \<union> S) \<and> (\<exists>\<^sub>\<infinity>i. (as i, as (Suc i)) \<in> R))"
lookup_fresh: fixes z :: "name"   and \<theta> :: "(name \<times> trm) list"   and x :: "name" assumes "z \<sharp> \<theta>"   and "z \<sharp> x" shows "z \<sharp> lookup \<theta> x"
part_equiv_refl_2: fixes R :: "('a \<times> 'a) set"   and x :: "'a"   and y :: "'a" assumes "part_equiv R"   and "(x, y) \<in> R" shows "(y, y) \<in> R"
1: shows "- - v \<squnion> - w = - - (- - v \<squnion> - w) \<and> - v \<squnion> - x = - - (- v \<squnion> - x) \<and> - - v \<squnion> - - y = - - (- - v \<squnion> - - y) \<and> - v \<squnion> - - z = - - (- v \<squnion> - - z)"
supp_perm_eq: fixes x :: "'e"   and p :: "perm" assumes "supp x \<sharp>* p" shows "p \<bullet> x = x"
one_poly_eq_simps_2: shows "([:1::'c:] = 1) = True"
inverse_add: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "plus (inverse a) (inverse b) = times (times (plus a b) (inverse a)) (inverse b)"
fst_convol: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "fst \<circ> \<langle>f, g\<rangle> = f"
mult_0_right: fixes m :: "nat" shows "m * 0 = 0"
issummable: fixes A :: "real"   and d :: "nat \<Rightarrow> real"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "1 < A"   and "\<forall>n. 1 < d n"   and "\<forall>n. 0 < a n"   and "\<forall>n. 0 < b n"   and "(\<lambda>n. real_of_int (a n) powr (1 / real_of_int (2 ^ n))) \<longlonglongrightarrow> A"   and "LIM n sequentially. d n ^ 2 ^ n / real_of_int (b n) :> at_top"   and "convergent_prod d" shows "summable (\<lambda>j. real_of_int (b j) / real_of_int (a j))"
sp_cf_val: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and j :: "nat"   and a :: "'a" assumes "pol_coeff S c"   and "j \<le> fst c"   and "a \<in> carrier S" shows "snd (sp_cf S a c) j = a \<cdot>\<^sub>r\<^bsub>S\<^esub> snd c j"
algebraic_of_real_iff: fixes x :: "real" shows "algebraic (of_real x) = algebraic x"
c_cons_aux3: shows "c_cons = (\<lambda>x u. sgn2 u * (c_pair 0 x + 1) + sgn1 u * (c_pair (c_len u) (c_pair x (c_snd (u - 1))) + 1))"
conf_def: fixes P :: "'a prog"   and h :: "heap"   and v :: "val"   and T :: "ty" shows "P,h \<turnstile> v :\<le> T \<equiv> \<exists>T'. typeof\<^bsub>h\<^esub> v = \<lfloor>T'\<rfloor> \<and> P \<turnstile> T' \<le> T"
xy: shows "(x, y) \<in> Pd'_tax (Xd x)"
ys: shows "list_all2 (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<psi>)) [Monitor.progress \<sigma> P \<psi> j..<Monitor.progress \<sigma> P' \<psi> j'] ys"
il: shows "real_of_rat il \<le> root n (real_of_1 x)"
simps_5: fixes \<nu> :: "'a ltln"   and \<nu>' :: "'a ltln" shows "pure_universal (\<nu> U\<^sub>n \<nu>') = (pure_universal \<nu> \<and> pure_universal \<nu>')"
unfold_congs_2: fixes r :: "('b, 'c) node_scheme"   and r' :: "('b, 'c) node_scheme"   and v' :: "nat set"   and f :: "nat set \<Rightarrow> nat set"   and f' :: "nat set \<Rightarrow> nat set" assumes "r = r'"   and "incoming r' = v'"   and "\<And>v. v = v' \<Longrightarrow> f v = f' v" shows "incoming_update f r = incoming_update f' r'"
inject_5: fixes x1 :: "trm"   and y1 :: "trm" shows "([x1] = [y1]) = (x1 = y1)"
simps_41: fixes x61 :: "msg"   and x62 :: "msg"   and x4 :: "nat" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Key x4"
Gcd_mod_ring_def: shows "Gcd = normalization_euclidean_semiring_class.Gcd"
stables_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and step :: "nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> 'b) list"   and \<tau>s :: "'b list" shows "stables r step \<tau>s = (\<forall>p<length \<tau>s. stable r step \<tau>s p)"
build_rec_isSubobj: 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 D :: "char list"   and Cs :: "char list list"   and C :: "char list" assumes "is_subobj P (D, D # Cs)"   and "P \<turnstile> C \<prec>\<^sub>R D" shows "is_subobj P (C, C # D # Cs)"
simps_1: fixes s :: "'a" shows "iter 0 s = return_spmf s"
thetaSeq_Sretr: shows "thetaSeq \<subseteq> Sretr (thetaSeq \<union> Sbis)"
integrableD_1: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "f \<in> borel_measurable M"
treewidth_upper_bound_ex: fixes G :: "('a, 'b) Graph_scheme"   and T :: "nat Graph"   and bag :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "Graph G"   and "TreeDecomposition G T bag"   and "TreeDecomposition.width T bag \<le> n" shows "Graph.treewidth G \<le> n"
F_def: shows "F \<equiv> \<lambda>outputs xs. case xs of Resumption.resumption.Done x \<Rightarrow> {} | Resumption.resumption.Pause out c \<Rightarrow> insert out (\<Union>input. outputs (c input))"
WT_implies_WTrt: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option" assumes "P,E \<turnstile> e :: T" shows "P,E,h \<turnstile> e : T"
reflect_factorization': fixes p :: "'a poly"   and fctrs :: "'a \<times> ('a \<times> nat) list" assumes "coeff p 0 \<noteq> (0::'a)"   and "is_alt_factorization_of fctrs p" shows "is_factorization_of fctrs (reflect_poly p)"
not_rt_fresh_asI1: fixes dip :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "\<not> rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2" shows "\<not> rt1 \<approx>\<^bsub>dip\<^esub> rt2"
distinct_remdups: fixes xs :: "'c list" shows "distinct (remdups xs)"
limit_def: fixes f :: "'b set \<Rightarrow> 'b set"   and x :: "'b set" shows "limit f x = natUnion (\<lambda>n. funpower f n x)"
sum_wt_ParT_sub_notWtFT_notFinished_shift: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set"   and s :: "'state" assumes "properL cl"   and "WtFT cl \<noteq> 1"   and "n < length cl"   and "\<not> finished (cl ! n)"   and "I \<subseteq> {..<brn (cl ! n)}" shows "sum (wt (ParT cl) s) (local.shift cl n I) = 1 / real (length cl) / (1 - WtFT cl) * sum (wt (cl ! n) s) I"
P5: shows "Bet A C X \<and> Bet D X B \<longrightarrow> A B OS C D"
countableE: fixes S :: "'b set" assumes "countable S" obtains f :: "'b \<Rightarrow> nat" where "inj_on f S"
must_syncE: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and x :: "'addr option \<times> 'addr frame list"   and m :: "'heap" assumes "execd_mthr.must_sync P t x m" obtains taa :: "('addr, 'thread_id, 'heap) jvm_thread_action" and x' :: "'addr option \<times> 'addr frame list" and m' :: "'heap" and s :: "('addr, 'thread_id) locks \<times> (('thread_id \<Rightarrow> (('addr option \<times> 'addr frame list) \<times> 'addr \<Rightarrow>f nat) option) \<times> 'heap) \<times> ('thread_id \<Rightarrow> 'addr wait_set_status option) \<times> 'thread_id set" where "execd_mthr.r_syntax P t x m taa x' m'" and "exec_mthr.actions_ok s t taa" and "m = shr s"
oFix_fixed: fixes F :: "ordinal \<Rightarrow> ordinal"   and a :: "ordinal" assumes "continuous F"   and "a \<le> F a" shows "F (oFix F a) = oFix F a"
scc_root_disc_le: fixes v :: "'v"   and scc :: "'v set"   and x :: "'v" assumes "scc_root s' v scc"   and "x \<in> scc"   and "x \<in> dom (discovered s')" shows "\<delta> s' v \<le> \<delta> s' x"
False: fixes c :: "com"   and Q :: "(char list \<Rightarrow> int option) \<times> nat \<Rightarrow> bool" shows "\<turnstile>\<^sub>3\<^sub>' {\<lambda>(ps, n). False} c { Q}"
fset_finite_supp: fixes S :: "'a fset" shows "finite (supp S)"
fixp_induct: fixes P :: "(nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bdd \<Rightarrow> (nat \<times> bdd) option) \<Rightarrow> bool" assumes "option.admissible (\<lambda>ite_impl. P (curry (curry (curry ite_impl))))"   and "P (\<lambda>ite_impl i t. Map.empty)"   and "\<And>s. P s \<Longrightarrow> P (\<lambda>a aa aaa aaaa. mi.lowest_tops_impl [a, aa, aaa] aaaa \<bind> (\<lambda>aaaaa. case aaaaa of (None, uu_) \<Rightarrow> mi.case_ifexi (\<lambda>_. Some (aa, aaaa)) (\<lambda>_. Some (aaa, aaaa)) (\<lambda>_ _ _. Map.empty) a aaaa | (Some ab, uu_) \<Rightarrow> mi.restrict_top_impl a ab True aaaa \<bind> (\<lambda>aaaaa. case aaaaa of (ti, uu_) \<Rightarrow> mi.restrict_top_impl aa ab True aaaa \<bind> (\<lambda>aaaaa. case aaaaa of (tt, uu_) \<Rightarrow> mi.restrict_top_impl aaa ab True aaaa \<bind> (\<lambda>aaaaa. case aaaaa of (te, uu_) \<Rightarrow> mi.restrict_top_impl a ab False aaaa \<bind> (\<lambda>a. case a of (fi, uu_) \<Rightarrow> mi.restrict_top_impl aa ab False aaaa \<bind> (\<lambda>a. case a of (ft, uu_) \<Rightarrow> mi.restrict_top_impl aaa ab False aaaa \<bind> (\<lambda>a. case a of (fe, uu_) \<Rightarrow> s ti tt te aaaa \<bind> (\<lambda>a. case a of (tb, sa) \<Rightarrow> s fi ft fe sa \<bind> (\<lambda>a. case a of (fb, s) \<Rightarrow> ifmi' ab tb fb s))))))))))" shows "P mi.ite_impl"
field_differentiable_diff: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'a"   and g' :: "'a" assumes "(f has_field_derivative f') F"   and "(g has_field_derivative g') F" shows "((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
short_vector_impl: shows "LLL_Impl.short_vector \<alpha> fs_init = local.short_vector"
scaleR_Pair: fixes r :: "real"   and a :: "'c"   and b :: "'d" shows "r *\<^sub>R (a, b) = (r *\<^sub>R a, r *\<^sub>R b)"
11: fixes x :: "'a" shows "x \<squnion> x = x"
hereditarily_fs_implies_finite_supp: fixes t\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" assumes "hereditarily_fs t\<^sub>\<alpha>" shows "finite (supp t\<^sub>\<alpha>)"
span_superset: fixes S :: "'d set" shows "S \<subseteq> cspan S"
host_candidates: shows "h \<turnstile> filter_M (\<lambda>element_ptr. Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>shadow_root_opt. return (shadow_root_opt = Some shadow_root_ptr))) (sorted_list_of_set (fset (element_ptr_kinds h))) \<rightarrow>\<^sub>r host_candidates"
iIN_inext_nth: fixes a :: "nat"   and d :: "nat"   and n :: "nat" assumes "a \<le> d" shows "[n\<dots>,d] \<rightarrow> a = n + a"
even_mod_2_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (modulo a (numeral.numeral one plus (num.Bit0 num.One))) = dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a"
range_from_nat_into: fixes A :: "'b set" assumes "A \<noteq> {}"   and "countable A" shows "range (from_nat_into A) = A"
intros_10: fixes fop :: "binop_f"   and c1 :: "f32"   and c2 :: "f32" assumes "app_binop_f fop c1 c2 = None" shows "\<lparr>[$C ConstFloat32 c1, $C ConstFloat32 c2, $Binop_f T_f32 fop]\<rparr> \<leadsto> \<lparr>[Trap]\<rparr>"
\<delta>: fixes i :: "nat" shows "\<delta> i = n i - n (Suc i)"
decrease: fixes j :: "nat" assumes "\<not> j \<le> 1"   and "odd j" shows "Suc (j div 2) < j"
et_2: fixes v :: "'a"   and e :: "'a"   and t :: "'a" assumes "v \<sqinter> top = v"   and "e \<le> v \<sqinter> - id v"   and "t \<le> v \<sqinter> id v" shows "e \<sqinter> id t = bot"
top_eq_inf_iff: fixes x :: "'a"   and y :: "'a" shows "(1 = x \<cdot> y) = (x = 1 \<and> y = 1)"
map_upds_Nil2: fixes m :: "'c \<Rightarrow> 'd option"   and as :: "'c list" shows "m(as [\<mapsto>] []) = m"
util_def_conf: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "ordinal_utility carrier relation u"   and "x \<in> carrier"   and "y \<in> carrier" shows "(u y \<le> u x) = x \<succeq>[relation] y"
generate_one: shows "generate (Z 1) {\<one>\<^bsub>Z 1\<^esub>} = {\<one>\<^bsub>Z 1\<^esub>}"
degeneralize_nodes_card: fixes A :: "('d, 'c) dgba" shows "card (DBA.nodes (degeneralize A)) \<le> max 1 (length (dgba.accepting A)) * card (DGBA.nodes A)"
ipath_to_rtrancl: fixes E :: "('b \<times> 'b) set"   and r :: "nat \<Rightarrow> 'b"   and i1 :: "nat"   and i2 :: "nat" assumes "ipath E r"   and "i1 \<le> i2" shows "(r i1, r i2) \<in> E\<^sup>*"
getTvars_def: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and \<sigma>l :: "'e list" assumes "Signature TYPE('e) wtFsym wtPsym" shows "Signature.getTvars \<sigma>l \<equiv> map Var (Signature.getVars \<sigma>l)"
surjective: fixes r :: "('e, 'f) lasso_scheme" shows "r = \<lparr>lasso_reach = lasso_reach r, lasso_va = lasso_va r, lasso_cysfx = lasso_cysfx r, \<dots> = lasso.more r\<rparr>"
endofunctor_R: shows "endofunctor (\<cdot>\<^sub>B) B.R"
RES_leof_RES_iff: fixes X :: "'a set"   and Y :: "'a set" shows "(RES X \<le>\<^sub>n RES Y) = (X \<subseteq> Y)"
rel_endofun'_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "rel_endofun' R = R ===> R"
qtable1: shows "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) (ne + length auxlist) \<phi>) rel1"
related_match: fixes x :: "term"   and u :: "'b"   and env :: "(name, 'b) fmap"   and t :: "'a" assumes "match x u = Some env"   and "P t u" obtains env' :: "(name, 'a) fmap" where "match x t = Some env'" and "P_env env' env"
a_rcos_inv: fixes x :: "'a" assumes "x \<in> carrier R" shows "a_set_inv (I +> x) = I +> \<ominus> x"
not_cong_1243: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "\<not> Cong A B C D" shows "\<not> Cong A B D C"
vdomain_vrange_vtimes: shows "r \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<circ> r \<times>\<^sub>\<circ> \<R>\<^sub>\<circ> r"