hcju/isabelleps huggingface dataset:
simE2_1: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi"   and Q :: "pi"   and a :: "name"   and x :: "name"   and Q' :: "pi" assumes "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto><Rel> S"   and "eqvt Rel"   and "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "\<exists>P'. P \<Longrightarrow>a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel"
dickson_less_pD2: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and d :: "'d \<Rightarrow> nat"   and m :: "nat"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "gd_term.dickson_less_p pair_of_term ord_term_strict d m p q" shows "q \<in> gd_term.dgrad_p_set pair_of_term d m"
of_nat_numeral: fixes n :: "num" shows "of_nat (numeral n) = numeral n"
eval_ring_hom: fixes s :: "'a" assumes "s \<in> carrier R" shows "eval R R id s \<in> ring_hom P R"
equalityI: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
add_mono_thms_linordered_field_2: fixes i :: "'a"   and j :: "'a"   and k :: "'a"   and l :: "'a" assumes "i = j \<and> k < l" shows "i + k < j + l"
wordinterval_intersection_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_intersection r1 r2 \<equiv> wordinterval_compress (wordinterval_intersection' r1 r2)"
le_infI2: fixes b :: "'a"   and x :: "'a"   and a :: "'a" assumes "b \<le> x" shows "inf a b \<le> x"
ddlat_axioms: shows "ddlat (\<sqsubseteq>)"
lim_null_mult_right_bounded: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and B :: "real" assumes "(f \<longlongrightarrow> (0::'b)) F"   and "\<forall>\<^sub>F x in F. norm (g x) \<le> B" shows "((\<lambda>z. f z * g z) \<longlongrightarrow> (0::'b)) F"
that: fixes xs' :: "V"   and x :: "V" assumes "xs = xs' #\<^sub>\<circ> x"   and "xs' \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs'"   and "x \<in>\<^sub>\<circ> A"   and "n = vcard (xs' #\<^sub>\<circ> x)" shows "thesis"
csum_com: fixes p1 :: "('c \<times> 'c) set"   and p2 :: "('d \<times> 'd) set" shows "p1 +c p2 =o p2 +c p1"
Deduct_with_False_Disj_axioms: shows "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"
iT_Plus_neg_Plus_le_inverse: fixes k :: "nat"   and I :: "nat set" assumes "k \<le> iMin I" shows "I \<oplus>- k \<oplus> k = I"
intros: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_function M f"   and "emeasure M {y \<in> space M. f y \<noteq> (0::'d)} \<noteq> \<infinity>" shows "simple_bochner_integrable M f"
netlimit_at: fixes a :: "'a" shows "netlimit (at a) = a"
vp_gen_t: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "\<exists>t\<in>carrier (Vr K v). vp K v = Vr K v \<diamondsuit>\<^sub>p t"
prems_5: shows "s' \<in> set_pmf (action cfg)"
sup_finite_2_def: fixes x :: "finite_2"   and y :: "finite_2" shows "x \<squnion> y = (if x = a\<^sub>2 \<or> y = a\<^sub>2 then a\<^sub>2 else a\<^sub>1)"
not_less_iff_gr_or_eq: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "class.linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
invariant_1I: fixes plr :: "int poly \<times> rat \<times> rat" assumes "\<exists>!x. root_cond plr x"   and "sgn (rai_lb plr) = sgn (rai_ub plr)"   and "poly_cond (poly_real_alg_1 plr)" shows "invariant_1 plr"
disj_imp: fixes P :: "bool"   and Q :: "bool" shows "(P \<or> Q) = (\<not> P \<longrightarrow> Q)"
hcomp_char: shows "(\<star>) = (\<lambda>f g. if arr f \<and> arr g \<and> src f = trg g then f \<star>\<^sub>B g else null)"
inv_unique: fixes prod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and y :: "'a"   and x :: "'a"   and y' :: "'a" assumes "Dmonoid prod one"   and "prod y x = one"   and "prod x y' = one" shows "y = y'"
cInf_less_iff: fixes X :: "'a set"   and y :: "'a" assumes "X \<noteq> {}"   and "bdd_below X" shows "(Inf X < y) = (\<exists>x\<in>X. x < y)"
ncols_singleton: fixes j :: "nat"   and i :: "nat"   and e :: "'b" shows "ncols (singleton_matrix j i e) = (if e = (0::'b) then 0 else Suc i)"
std_apart_instance_of\<^sub>l\<^sub>s2: fixes C2 :: "fterm literal set" shows "instance_of\<^sub>l\<^sub>s C2 (C2 \<cdot>\<^sub>l\<^sub>s (\<lambda>x. \<epsilon> (''2'' @ x)))"
sys_xcpts_def: shows "sys_xcpts \<equiv> {NullPointer, ClassCast, OutOfMemory, NoClassDefFoundError, IncompatibleClassChangeError, NoSuchFieldError, NoSuchMethodError}"
simps: fixes a1 :: "(nat \<Rightarrow> nat \<Rightarrow> 'a) \<times> nat \<times> nat"   and a2 :: "nat \<Rightarrow> nat \<Rightarrow> 'a" shows "fw_graph a1 a2 = ((\<exists>fw_sum m n. a1 = (m, n, 0) \<and> a2 = fwi m n 0 n n) \<or> (\<exists>fw_sum m n k. a1 = (m, n, Suc k) \<and> a2 = fwi (fw_sum (m, n, k)) n (Suc k) n n \<and> fw_graph (m, n, k) (fw_sum (m, n, k))))"
graph_restrict_def: fixes G :: "('c, 'd) graph_rec_scheme"   and R :: "'c set" shows "graph_restrict G R \<equiv> \<lparr>g_V = g_V G, g_E = rel_restrict (g_E G) R, g_V0 = g_V0 G - R, \<dots> = graph_rec.more G\<rparr>"
vsubstEnv_psubstEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "goodEnv rho"   and "goodEnv rho'" shows "rho &[rho'] &[y1 // y]_ys = rho &[rho' &[y1 // y]_ys]"
oreachable_par_subnet_induct: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and t :: "'e net_state"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'd) \<times> 'e, 'f seq_action) automaton"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree"   and S :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> 'f node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'd) \<Rightarrow> (nat \<Rightarrow> 'd) \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'd) \<Rightarrow> 'e net_state \<Rightarrow> 'e net_state \<Rightarrow> bool" assumes "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) S U"   and "\<And>\<sigma> s t. (\<sigma>, SubnetS s t) \<in> init (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) \<Longrightarrow> P \<sigma> s t"   and "\<And>\<sigma> s t \<sigma>'. \<lbrakk>(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) S U; U \<sigma> \<sigma>'; P \<sigma> s t\<rbrakk> \<Longrightarrow> P \<sigma>' s t"   and "\<And>\<sigma> s t \<sigma>' s' t' a. \<lbrakk>(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) S U; ((\<sigma>, SubnetS s t), a, \<sigma>', SubnetS s' t') \<in> automaton.trans (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)); S \<sigma> \<sigma>' a; P \<sigma> s t\<rbrakk> \<Longrightarrow> P \<sigma>' s' t'" shows "P \<sigma> s t"
Sretr_sym_trans: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and theta :: "(('c, 'd, 'a) cmd \<times> ('c, 'd, 'a) cmd) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis"   and "sym theta \<and> trans theta" shows "trans (PL_Indis.Sretr aval tval cval indis theta)"
weakly_fair_def2: fixes enabled :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and taken :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "weakly_fair enabled taken = \<box>(\<lambda>s. \<not> (\<box>(\<lambda>s. enabled s \<and> \<not> taken s)) s)"
drop_bit_take_bit: 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 bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "drop_bit m (take_bit n a) = take_bit (n - m) (drop_bit m a)"
weakBisimParNil: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> \<zero> \<approx> P"
SUCC: fixes i :: "'a" assumes "local.aboveS i \<noteq> {}"   and "P i" shows "P (succ i)"
image_int_atLeastLessThan: fixes a :: "nat"   and b :: "nat" shows "int ` {a..<b} = {int a..<int b}"
red_preserves_sconf: fixes E :: "char list \<Rightarrow> ty option"   and e :: "expr"   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 e' :: "expr"   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 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 T :: "ty" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow> \<langle>e',s'\<rangle>"   and "P,E,hp s \<turnstile> e : T"   and "P,E \<turnstile> s \<surd>"   and "wwf_prog P" shows "P,E \<turnstile> s' \<surd>"
rec: shows "dist (x t (Suc j)) (grid_function (discrete_evolution \<psi>) x0 t (Suc j)) \<le> B * stepsize j ^ (p + 1) + (1 + stepsize j * L) * dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j)"
llength_lnths_ile: fixes xs :: "'a llist"   and A :: "nat set" shows "llength (lnths xs A) \<le> llength xs"
6: shows "(w \<sqinter> - d)\<^sup>\<star> * d * top \<le> - (d\<^sup>T * top)"
trace_core_eq_simI: fixes S :: "'a spmf \<Rightarrow> 'b spmf \<Rightarrow> bool"   and p :: "'a spmf"   and q :: "'b spmf"   and E :: "'c set"   and core1 :: "('a, 'c, 'd, 'e, 'f, 'g) core"   and core2 :: "('b, 'c, 'd, 'e, 'f, 'g) core"   and IA :: "'d set"   and IU :: "'e set" assumes "S p q"   and "\<And>p q e. \<lbrakk>S p q; e \<in> E\<rbrakk> \<Longrightarrow> weight_spmf (p \<bind> (\<lambda>s. cpoke core1 s e)) = weight_spmf (q \<bind> (\<lambda>s. cpoke core2 s e))"   and "\<And>p q e. \<lbrakk>S p q; e \<in> E\<rbrakk> \<Longrightarrow> S (mk_lossless (p \<bind> (\<lambda>s. cpoke core1 s e))) (mk_lossless (q \<bind> (\<lambda>s. cpoke core2 s e)))"   and "\<And>p q ia. \<lbrakk>S p q; ia \<in> IA\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (cfunc_adv core1 s1 ia)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_adv core2 s2 ia))"   and "\<And>p q ia s1 s2 s1' s2' oa. \<lbrakk>S p q; ia \<in> IA; s1 \<in> set_spmf p; s2 \<in> set_spmf q; (oa, s1') \<in> set_spmf (cfunc_adv core1 s1 ia); (oa, s2') \<in> set_spmf (cfunc_adv core2 s2 ia)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_adv core2 s2 ia)) oa)"   and "\<And>p q iu. \<lbrakk>S p q; iu \<in> IU\<rbrakk> \<Longrightarrow> p \<bind> (\<lambda>s1. map_spmf fst (cfunc_usr core1 s1 iu)) = q \<bind> (\<lambda>s2. map_spmf fst (cfunc_usr core2 s2 iu))"   and "\<And>p q iu s1 s2 s1' s2' ou. \<lbrakk>S p q; iu \<in> IU; s1 \<in> set_spmf p; s2 \<in> set_spmf q; (ou, s1') \<in> set_spmf (cfunc_usr core1 s1 iu); (ou, s2') \<in> set_spmf (cfunc_usr core2 s2 iu)\<rbrakk> \<Longrightarrow> S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_usr core1 s1 iu)) ou) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_usr core2 s2 iu)) ou)" shows "trace_core_eq core1 core2 E IA IU p q"
nsets_one: fixes A :: "'b set" shows "[A]\<^bsup>Suc 0\<^esup> = (\<lambda>x. {x}) ` A"
IHL: shows "\<forall>q\<in>set_kdt l \<union> set ps - set (nearest_nbors n ps p l). dist (last (nearest_nbors n ps p l)) p \<le> dist q p"
onll_def: fixes \<Gamma> :: "'j \<Rightarrow> ('h, 'i, 'j, 'k) seqp"   and P :: "('g \<times> 'k) \<times> 'l \<times> 'g \<times> 'k \<Rightarrow> bool" shows "onll \<Gamma> P \<equiv> \<lambda>((\<xi>, p), a, \<xi>', p'). \<forall>l\<in>labels \<Gamma> p. \<forall>l'\<in>labels \<Gamma> p'. P ((\<xi>, l), a, \<xi>', l')"
select_convs_3: fixes g_V :: "'b set"   and g_E :: "('b \<times> 'b) set"   and g_V0 :: "'b set"   and more :: "'c" shows "g_V0 \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_V0"
mult_left_mono_neg: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.ordered_ring plus minus zero less_eq less times uminus"   and "less_eq b a"   and "less_eq c zero" shows "less_eq (times c a) (times c b)"
iso_\<psi>: fixes f :: "'a" assumes "C.ide f" shows "D.iso (\<psi> f)"
ord_0_right_nat: fixes n :: "nat" shows "Pocklington.ord n 0 = (if n = 1 then 1 else 0)"
frontier_of_connected_component_subset: fixes S :: "'a set"   and x :: "'a" shows "frontier (connected_component_set S x) \<subseteq> frontier S"
squanderVertex_def: fixes p :: "nat"   and q :: "nat" shows "\<b> p q \<equiv> if p = 0 \<and> q = 3 then 6177 else if p = 0 \<and> q = 4 then 9696 else if p = 1 \<and> q = 2 then 6557 else if p = 1 \<and> q = 3 then 6176 else if p = 2 \<and> q = 1 then 7967 else if p = 2 \<and> q = 2 then 4116 else if p = 2 \<and> q = 3 then 12846 else if p = 3 \<and> q = 1 then 3106 else if p = 3 \<and> q = 2 then 8165 else if p = 4 \<and> q = 0 then 3466 else if p = 4 \<and> q = 1 then 3655 else if p = 5 \<and> q = 0 then 395 else if p = 5 \<and> q = 1 then 11354 else if p = 6 \<and> q = 0 then 6854 else if p = 7 \<and> q = 0 then 14493 else squanderTarget"
asymmetric_axioms: shows "asymmetric (Collect derivable) (\<sqsubset>)"
worst_chain_not_gt: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and i :: "nat" assumes "wf {(x, y). gt y x}"   and "bad p x"   and "\<And>z y x. \<lbrakk>p z y; p y x\<rbrakk> \<Longrightarrow> p z x" shows "\<not> gt (worst_chain p gt i) (worst_chain p gt (Suc i))"
empty_plan_is_valid: fixes PROB :: "'c set" shows "[] \<in> valid_plans PROB"
not_Some_eq: fixes x :: "'a option" shows "(\<forall>y. x \<noteq> pure y) = (x = None)"
extend_map_lemma_cofinite: fixes \<F> :: "'a set set"   and \<G> :: "'a set set"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" assumes "finite \<F>"   and "\<G> \<subseteq> \<F>"   and "convex T"   and "bounded T"   and "\<And>X. X \<in> \<F> \<Longrightarrow> polytope X"   and "continuous_on (\<Union> \<G>) f"   and "f ` \<Union> \<G> \<subseteq> rel_frontier T"   and "\<And>X Y. \<lbrakk>X \<in> \<F>; Y \<in> \<F>\<rbrakk> \<Longrightarrow> X \<inter> Y face_of X"   and "\<And>X. X \<in> \<F> - \<G> \<Longrightarrow> aff_dim X \<le> aff_dim T" obtains C :: "'a set" and g :: "'a \<Rightarrow> 'b" where "finite C" and "disjnt C (\<Union> \<G>)" and "card C \<le> card \<F>" and "continuous_on (\<Union> \<F> - C) g" and "g ` (\<Union> \<F> - C) \<subseteq> rel_frontier T" and "\<And>x. x \<in> \<Union> \<G> \<Longrightarrow> g x = f x"
no_smallest_element_skipped_def: fixes E :: "'a set"   and F :: "'a set" shows "no_smallest_element_skipped E F = (\<forall>e\<in>carrier - E. \<forall>g>weight e. indep (insert e (limi F g)) \<longrightarrow> e \<in> limi F g)"
simps_1: fixes \<theta> :: "(W.tvar \<times> ty) list"   and T :: "ty" shows "\<theta><Ty T> = Ty (\<theta><T>)"
simps_8: fixes f :: "'c \<Rightarrow> 'b" shows "map_node f SOURCE = SOURCE"
total_preorder_on_def: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "total_preorder_on carrier le \<equiv> Order_Predicates.preorder_on carrier le \<and> total_preorder_on_axioms carrier le"
option_transp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "transp R" shows "transp (rel_option R)"
case_prod_map_prod: fixes h :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and x :: "'g \<times> 'h" shows "(case map_prod f g x of (x, xa) \<Rightarrow> h x xa) = (case x of (l, r) \<Rightarrow> h (f l) (g r))"
swapEnv_updEnv_fresh: fixes ys :: "'varSort"   and xs :: "'varSort"   and x :: "'var"   and y1 :: "'var"   and y2 :: "'var"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "ys \<noteq> xs \<or> x \<notin> {y1, y2}"   and "good X"   and "fresh ys y1 X"   and "fresh ys y2 X" shows "rho [x \<leftarrow> X]_xs &[y1 \<and> y2]_ys = rho &[y1 \<and> y2]_ys [x \<leftarrow> X]_xs"
lNactLe_nact: fixes n' :: "nat"   and n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "n' = n \<or> n' < n \<and> \<not> (\<exists>k\<ge>n'. k < n \<and> active c (t k))" shows "dynamic_component.lNAct active c t n \<le> n'"
dim1I: fixes v :: "'c" assumes "gen_set {v}"   and "v \<noteq> \<zero>\<^bsub>V\<^esub>"   and "v \<in> carrier V" shows "dim = 1"
lr_l2r_1: fixes fs :: "'a \<Rightarrow> ValD"   and R :: "('a \<Rightarrow> ValD) set"   and xs :: "'a \<Rightarrow> ValD" assumes "fs \<in> R"   and "xs \<in> R"   and "logical_relation R" shows "(\<lambda>j. appF\<cdot>(fs j)\<cdot>(xs j)) \<in> R"
h_def: shows "h = (\<lambda>x. ennreal \<bar>g x\<bar>)"
filtermap_tan_at_right: fixes a :: "real" assumes "cos a \<noteq> 0" shows "filtermap tan (at_right a) = at_right (tan a)"
max2_def: fixes a :: "'a"   and b :: "'a" shows "max2 a b \<equiv> if (a, b) \<in> r then b else a"
in_netmap: fixes i :: "nat"   and ns :: "'d net_state" assumes "i \<in> net_ips ns" shows "netmap ns i \<noteq> None"
nnfis_mono: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> nnfis f M"   and "b \<in> nnfis g M"   and "f \<le> g" shows "a \<le> b"
Abs_inverse: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and y :: "'d" assumes "type_definition Rep Abs A"   and "y \<in> A" shows "Rep (Abs y) = y"
RHT_Cd_card: fixes d :: "'b" assumes "d \<in> ds" shows "card (Cd d X) \<le> card (Cd d (cop ds))"
get_shadow_root_ptr_simp8: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and document_ptr :: "(_) document_ptr"   and f :: "('m, 'n, 'o, 'w option) RDocument_scheme"   and h :: "(_) heap" shows "get shadow_root_ptr (put document_ptr f h) = get shadow_root_ptr h"
pgallery_chamber_system: fixes X :: "'b set set"   and Cs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.pmaxsimpchain X Cs" shows "order.greater_eq (ChamberComplex.chamber_system X) (set Cs)"
linear: assumes "b \<noteq> 0" shows "(\<forall>f\<in>set L2. eval (substInfinitesimalLinear var (- isolate_variable_sparse p var 0) (isolate_variable_sparse p var (Suc 0)) f) (xs' @ x # xs)) = (\<forall>l\<in>set L2'. evalUni (substInfinitesimalLinearUni b c l) x)"
Obj: shows "X \<in> obj\<^bsub>C\<^esub>"
gallery_append_reduce2: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "folding.gallery (xs @ ys)" shows "folding.gallery ys"
ex_unique_reduced_GB_dgrad_p_set': fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "\<exists>!G. finite G \<and> is_reduced_GB G \<and> pmdl G = pmdl F"
epi_implies_arr: fixes f :: "'a \<times> 'a" assumes "VoV.epi f" shows "VoV.arr f"
B_posA: shows "0 < emeasure (restrict_space M A) B"
Ra_def: fixes P :: "'a process"   and s :: "'a event list" shows "Ra P s = {X. (s, X) \<in> F P}"
map_fst_auxlist_take: fixes t :: "nat" assumes "t \<in> set (map fst (take (length done) auxlist))" shows "t \<le> nt"
rep_def: shows "rep = (\<lambda>U\<in>carrier (G Mod stabilizer m). SOME x. x \<in> U)"
unit_char_3: fixes a :: "'c" assumes "C.obj a" shows "F.unit a \<cdot>\<^sub>D \<i>\<^sub>D[F.map\<^sub>0 a] = (F \<i>\<^sub>C[a] \<cdot>\<^sub>D \<Phi> (a, a)) \<cdot>\<^sub>D (F.unit a \<star>\<^sub>D F.unit a)"
equals0D: fixes A :: "'k set"   and a :: "'k" assumes "A = {}" shows "a \<notin> A"
poly_neg_inf_def: fixes p :: "'a poly" shows "poly_neg_inf p \<equiv> if even (degree p) then sgn (lead_coeff p) else - sgn (lead_coeff p)"
h4: assumes "(\<forall>(d, e, f)\<in>set eq. d = 0 \<and> e = 0 \<and> f = 0) \<and> (\<exists>x. \<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)" shows "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
val_Zp_pow: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> nonzero Zp" shows "val_Zp (x [^] n) = eint (int n * ord_Zp x)"
Inf_le_iff_less: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and x :: "'b" shows "(Inf (f ` A) \<le> x) = (\<forall>y>x. \<exists>i\<in>A. f i \<le> y)"
memS_def: fixes x :: "'a"   and l :: "'a list" shows "memS x l \<equiv> x \<in> set l"
div_to_zero_def: fixes x :: "integer"   and y :: "integer" shows "div_to_zero x y = (let z = fst (Code_Numeral.divmod_abs x y) in if (x < 0) \<noteq> (y < 0) then - z else z)"
inv_2: shows "length (map of_int_hom.vec_hom fs) = m"
rtrancl_refl: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "r\<^sup>*\<^sup>* a a"
floor_eq_iff: fixes x :: "'c"   and a :: "int" shows "(\<lfloor>x\<rfloor> = a) = (of_int a \<le> x \<and> x < of_int a + (1::'c))"
subspace_orthogonal_complement: fixes W :: "'a set" shows "subspace (orthogonal_complement W)"
eqBA: shows "c \<cdot>\<^sub>m (B - A) = c \<cdot>\<^sub>m B - c \<cdot>\<^sub>m A"
gx0_ge_x1: shows "x\<^sub>1 \<le> gx0"
I_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and v :: "nat" shows "ast_problem.I problem v \<equiv> if v < length (ast_problem.astI problem) then Some (ast_problem.astI problem ! v) else None"
parity_cases: fixes a :: "'a" obtains "even a" and "a mod (2::'a) = (0::'a)"   | "odd a" and "a mod (2::'a) = (1::'a)"
dom: shows "\<D>\<^sub>\<circ> [a, b]\<^sub>\<circ> = set {0\<^sub>\<nat>, 1\<^sub>\<nat>}"
x0_hb_bound0: fixes x :: "real"   and b :: "real" assumes "x\<^sub>0 \<le> x"   and "b \<in> set bs" shows "hb / ln x powr (1 + e) < b / 2"
index_pow_mod: fixes i :: "nat"   and r :: "'b list"   and k :: "nat" assumes "i < \<^bold>|r \<^sup>@ k\<^bold>|" shows "(r \<^sup>@ k) ! i = r ! (i mod \<^bold>|r\<^bold>|)"
obj_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "obj (trg \<mu>)"
simps_50: fixes I :: "\<I>"   and r :: "formula Regex.regex" shows "safe_formula (MatchP I r) = safe_regex Past Strict r"
intro: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int"   and index :: "int" assumes "bibd point_set block_collection u_block_size index"   and "symmetric_bibd_axioms point_set block_collection" shows "symmetric_bibd point_set block_collection u_block_size index"
ts''_def_2: shows "\<S>\<bullet>\<C> \<turnstile> [$Set_local i] : ts'' _> ts'"
assms: fixes x :: "'a"   and y :: "'b" shows "mf x y \<equiv> RETURN (f x y)"
pn3: shows "6 * m + 1 \<noteq> 3"