hcju/isabelleps huggingface dataset:
find_adds_SomeD_red_single: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and f :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "find_adds fs (lt p) = Some f" shows "red_single p (tail p - monom_mult (lc p / lc f) (lp p - lp f) (tail f)) f (lp p - lp f)"
find_path_def: fixes E :: "('b \<times> 'b) set"   and U0 :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "find_path E U0 P \<equiv> ASSERT (finite U0) \<bind> (\<lambda>_. ASSERT (finite (E\<^sup>* `` U0)) \<bind> (\<lambda>_. SPEC (\<lambda>p. case p of None \<Rightarrow> \<forall>u0\<in>U0. \<forall>v\<in>E\<^sup>* `` {u0}. \<not> P v | Some (p, v) \<Rightarrow> \<exists>u0\<in>U0. path E u0 p v \<and> P v \<and> (\<forall>v\<in>set p. \<not> P v))))"
assms_2: shows "x \<noteq> 0\<^sub>h"
not_shared_not_acquired_switch: fixes a :: "nat"   and sb :: "'b memref list"   and X :: "nat set"   and Y :: "nat set" assumes "a \<notin> all_shared sb"   and "a \<notin> X"   and "a \<notin> acquired True sb X"   and "a \<notin> Y" shows "a \<notin> acquired True sb Y"
is_extensional: fixes f :: "'c local.term set" assumes "\<not> \<F>\<^sub>SC.arr f" shows "\<F>\<^sub>SC.map f = null"
simps_1: fixes f :: "'c" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<^sub>0\<rbrace> = E f"
nat_pow_nonzero: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a"   and n :: "nat" assumes "domain R"   and "x \<in> nonzero R" shows "x [^]\<^bsub>R\<^esub> n \<in> nonzero R"
assms: fixes x :: "'a"   and y :: "'b" shows "ovalidNF (P x y) (B x y) Q"
simps_2: fixes G :: "('c, 'd) ParityGame_scheme"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c"   and v0 :: "'c" assumes "ParityGame G" shows "lhd (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0) = v0"
det_0_iff_vec_prod_zero: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "(det A = (0::'b)) = (\<exists>v. v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> A *\<^sub>v v = 0\<^sub>v n)"
prop_atoms_notin_2: fixes \<phi> :: "'b ltln" shows "false\<^sub>n \<notin> prop_atoms \<phi>"
continuous_logarithm_on_contractible: fixes S :: "'a set"   and f :: "'a \<Rightarrow> complex" assumes "continuous_on S f"   and "contractible S"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" obtains g :: "'a \<Rightarrow> complex" where "continuous_on S g" and "\<And>x. x \<in> S \<Longrightarrow> f x = exp (g x)"
common_matcher_SrcDst_Inter_2: fixes X :: "'a ipt_iprange list"   and \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "(\<forall>m\<in>set X. matches (common_matcher, \<alpha>) (Match (Dst m)) a p) = (p_dst p \<in> \<Inter> (ipt_iprange_to_set ` set X))"
Diagonalize_in_Hom: fixes t :: "'a local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t\<^bold>\<rfloor> \<in> Hom \<^bold>\<lfloor>Dom t\<^bold>\<rfloor> \<^bold>\<lfloor>Cod t\<^bold>\<rfloor>"
pmc_inclusion: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "K \<div>\<^bsub>\<gamma>\<^sub>A\<^sub>G\<^sub>M\<^esub> \<phi> \<subseteq> K"
redT_acquire: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and n :: "'l"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln"   and "0 < ln $ n"   and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)" shows "s -t\<triangleright>(K$ [], [], [], [], [], convert_RA ln)\<rightarrow> s'"
vmatch_rel: fixes Q :: "value \<Rightarrow> value \<Rightarrow> bool"   and t\<^sub>1 :: "value"   and t\<^sub>2 :: "value"   and p :: "pat" assumes "value_struct_rel Q"   and "Q t\<^sub>1 t\<^sub>2" shows "rel_option (fmrel Q) (vmatch p t\<^sub>1) (vmatch p t\<^sub>2)"
has_field_derivative_rGamma_complex_no_nonpos_Int: fixes z :: "complex" assumes "z \<notin> \<int>\<^sub>\<le>\<^sub>0" shows "(rGamma has_field_derivative - rGamma z * Digamma z) (at z)"
eventually_nhds_x_imp_x: fixes "open" :: "'b set \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "class.topological_space open"   and "eventually P (topological_space.nhds open x)" shows "P x"
rtr_nomac_e1: fixes rt :: "'b routing_rule list"   and fw :: "'b simple_rule list"   and mlf :: "'b word \<Rightarrow> 48 word option"   and ifl :: "interface list"   and pi :: "'b Simple_Packet.simple_packet_ext"   and po :: "'b Simple_Packet.simple_packet_ext"   and po' :: "'b Simple_Packet.simple_packet_ext" assumes "simple_linux_router rt fw mlf ifl pi = Some po"   and "simple_linux_router_nol12 rt fw pi = Some po'" shows "\<exists>x. po = po'\<lparr>p_l2dst := x\<rparr>"
sets_embed_measure_alt: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "inj f" shows "sets (embed_measure M f) = (`) f ` sets M"
Tn_measure_preserving: fixes n :: "nat" shows "natural_extension_map ^^ n \<in> measure_preserving natural_extension_measure natural_extension_measure"
t_ins_in_def: fixes x :: "'b"   and xt :: "'b bintree" shows "t_ins_in x xt \<equiv> \<lparr>t_type.folding = False, item = x, subtrees = [xt]\<rparr>"
DERIV_isconst2: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and x :: "real" assumes "a < b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_real_derivative 0) (at x)"   and "a \<le> x"   and "x \<le> b" shows "f x = f a"
fix_F'_stable_pair_on: fixes A :: "'a set \<times> 'a set dual" assumes "A = F' A" shows "stable_pair_on (ds - {d'}) (map_prod id undual A)"
Lim_transform_within_set: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and a :: "'c"   and S :: "'c set"   and T :: "'c set" assumes "(f \<longlongrightarrow> l) (at a within S)"   and "\<forall>\<^sub>F x in at a. (x \<in> S) = (x \<in> T)" shows "(f \<longlongrightarrow> l) (at a within T)"
simps_3: fixes x1 :: "'a"   and x21 :: "'a"   and x22 :: "'a" shows "Vertex x1 \<noteq> Edge x21 x22"
vector_matrix_mult_def: fixes v :: "('b, 'd) vec"   and m :: "(('b, 'c) vec, 'd) vec" shows "v v* m \<equiv> \<chi>j. \<Sum>i\<in>UNIV. m $ i $ j * v $ i"
jumpF_polyL_smult_1: fixes c :: "real"   and q :: "real poly"   and p :: "real poly"   and x :: "real" shows "jumpF_polyL (smult c q) p x = sgn c * jumpF_polyL q p x"
Sup_pres_multl: fixes x :: "'c" shows "Sup_pres ((\<cdot>) x)"
ordLess_imp_ordLeq: fixes r :: "('b \<times> 'b) set"   and r' :: "('c \<times> 'c) set" assumes "r <o r'" shows "r \<le>o r'"
foldr: fixes f :: "'b \<Rightarrow> 'a"   and ys :: "'b list"   and y :: "'a" shows "foldr (\<lambda>x. (+) (f x)) ys (0::'a) + y = foldr (\<lambda>x. (+) (f x)) ys y"
BIGTHETA_CONST'_def: fixes c :: "'d" shows "BIGTHETA_CONST' c = \<Theta>(\<lambda>x. c)"
parallel_eoracle_def: fixes eoracle1 :: "'g \<Rightarrow> 'a \<Rightarrow> (('c \<times> 'e list) \<times> 'g) spmf"   and eoracle2 :: "'h \<Rightarrow> 'b \<Rightarrow> (('d \<times> 'f list) \<times> 'h) spmf"   and state :: "'g \<times> 'h" shows "parallel_eoracle eoracle1 eoracle2 state \<equiv> map_spmf (map_prod (case_sum (map_prod Inl (map Inl)) (map_prod Inr (map Inr))) id) \<circ> (eoracle1 \<ddagger>\<^sub>O eoracle2) state"
semilattice_neutr_axioms: shows "semilattice_neutr (\<sqinter>) \<top>"
add_sign_intros_1: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "{#} \<subset># a"   and "{#} \<subseteq># b" shows "{#} \<subset># a + b"
LFP_closed: fixes f :: "'a \<Rightarrow> 'a" shows "LFP f \<in> carrier L"
StrongEq_L_subst2: fixes P :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "cp P"   and "\<tau> \<Turnstile> x \<triangleq> y"   and "\<tau> \<Turnstile> P x" shows "\<tau> \<Turnstile> P y"
d_lt: fixes x :: "real" assumes "x < t"   and "0 < \<bar>x - t\<bar>"   and "\<bar>x - t\<bar> < d_lt" shows "\<bar>(f x - f t) / (x - t) - D_lt\<bar> < 1"
Collect_mono_iff: fixes P :: "'b::type \<Rightarrow> bool"   and Q :: "'b::type \<Rightarrow> bool" shows "(Collect (P::'b::type \<Rightarrow> bool) \<subseteq> Collect (Q::'b::type \<Rightarrow> bool)) = (\<forall>x::'b::type. P x \<longrightarrow> Q x)"
count_decrease1: fixes p :: "nat \<Rightarrow> nat"   and m :: "nat"   and k :: "nat" assumes "p partitions m"   and "sum p {..m} = k"   and "p 1 = 0" shows "(\<Sum>i\<le>m - k. p (i + 1)) = k"
1: shows "eSubst MOD ys eY y eX \<noteq> ERR"
zip_append_right: fixes xs :: "'a list"   and ys :: "'b list"   and zs :: "'b list" shows "zip xs (ys @ zs) = zip xs ys @ zip (drop (length ys) xs) zs"
dep_def: fixes nid' :: "'nid" shows "dep nid' \<equiv> pout (\<sigma>\<^bsub>nid'\<^esub>t n)"
simps_1: fixes k :: "nat"   and T :: "POPLmarkRecord.type" shows "\<down>\<^sub>\<tau> 0 k T = T"
inject_2: fixes x3 :: "num"   and y3 :: "num" shows "(num.Bit1 x3 = num.Bit1 y3) = (x3 = y3)"
LeftDerivation_append_prefix: fixes v :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and w :: "symbol list"   and u :: "symbol list" assumes "LeftDerivation v D w"   and "is_word u" shows "LeftDerivation (u @ v) (derivation_shift D 0 (length u)) (u @ w)"
D_gchain_is_d_gchain: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set" assumes "D_gchain G n f" shows "d_gchain G n f"
int_simps_3: shows "\<turnstile> (\<not> #False) = #True"
simps_2: fixes v :: "'c"   and va :: "'c list"   and p2 :: "'c list" shows "S.poly_mult (v # va) p2 = S.poly_add (map ((\<otimes>\<^bsub>S\<^esub>) (lead_coeff (v # va))) p2 @ replicate (degree (v # va)) \<zero>\<^bsub>S\<^esub>) (S.poly_mult (tl (v # va)) p2)"
nodes_trans: fixes p :: "'state"   and q :: "'state" assumes "p \<in> nodes"   and "q \<in> reachable p" shows "q \<in> nodes"
Bernoulli_inequality: fixes x :: "real"   and n :: "nat" assumes "- 1 \<le> x" shows "1 + real n * x \<le> (1 + x) ^ n"
simps: fixes x :: "'b"   and uu :: "'b heap list" shows "get_min (Hp x uu) = x"
max_word_bl: shows "to_bl (- 1) = replicate LENGTH('b) True"
homogeneous_homogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "homogeneous (homogenize x p)"
integral_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>L (restrict_space M \<Omega>) f = LINT x|M. indicat_real \<Omega> x *\<^sub>R f x"
bind_singleton_conv_image: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a" shows "Set.bind A (\<lambda>x. {f x}) = f ` A"
emptyI: fixes e :: "'b"   and D :: "'b set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b" assumes "e \<in> D" shows "({}, e) \<in> foldSetD D f e"
divides_mult: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "c \<in> carrier R"   and "a divides b" shows "c \<otimes> a divides c \<otimes> b"
simps_4: fixes p :: "real mpoly"   and L :: "real list" shows "aEval (Neq p) L = (insertion (nth_default 0 L) p \<noteq> 0)"
cINF_lower: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and x :: "'c" assumes "bdd_below (f ` A)"   and "x \<in> A" shows "Inf (f ` A) \<le> f x"
V_nonempt_PPV_nonempt: fixes V :: "('a, 'b) MWLsCom list" assumes "V \<noteq> []" shows "PPV V \<noteq> []"
subset_distinct_le: fixes A :: "'b list"   and B :: "'b list" assumes "set A \<subseteq> set B"   and "distinct A"   and "distinct B" shows "length A \<le> length B"
extremum: fixes a :: "'q multiset" shows "{#} \<subseteq># a"
T_eq_T': fixes s :: "'s" shows "T s = T' (K s)"
eq: shows "punit.reduced_GB (ideal F \<inter> P[X]) = punit.reduced_GB F"
Unbal: shows "ins (size t + dl) 0 x t = Unbal t'"
simps_1: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "(\<odot>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, smult = smult, \<dots> = more\<rparr>\<^esub>) = smult"
finite_set_choice: fixes A :: "'c set"   and P :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "finite A"   and "\<forall>x\<in>A. \<exists>y. P x y" shows "\<exists>f. \<forall>x\<in>A. P x (f x)"
code: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" shows "wit R xs = (case xs of LNil \<Rightarrow> xs | LCons x LNil \<Rightarrow> xs | LCons x (LCons y xs) \<Rightarrow> LCons x (prepend (pick R x y) (wit R (LCons y xs))))"
OrL: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and G :: "'b formula" assumes "F, \<Gamma> \<Rightarrow>\<^sub>n"   and "G, \<Gamma> \<Rightarrow>\<^sub>n" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow>\<^sub>n"
infdist_mono: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" assumes "A \<subseteq> B"   and "A \<noteq> {}" shows "infdist x B \<le> infdist x A"
lcm_div_unit2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "is_unit a" shows "lcm b (c div a) = lcm b c"
W_1: shows "RModule FG (\<cdot>) W \<and> W \<subseteq> V"
TrueInWorldNecc: fixes v :: "i"   and x :: "\<nu>"   and p :: "\<o>" shows "[(x\<^sup>P \<^bold>\<Turnstile> p) \<^bold>\<equiv> \<^bold>\<box>(x\<^sup>P \<^bold>\<Turnstile> p) in v]"
leftderives_start: shows "leftderives [\<SS>] (terminals (take r p) @ [N] @ \<gamma>)"
same_length_iff_eq_lengths: fixes xs :: "'c list"   and ys :: "'d list" shows "same_length xs ys = (length xs = length ys)"
trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g"   and "g \<in> L F h" shows "f \<in> L F h"
ltln_StrongRelease_to_Until: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n \<phi> M\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi> U\<^sub>n (\<phi> and\<^sub>n \<psi>)"
subset_imp: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "A \<noteq> {}"   and "finite B" shows "Max A \<le> Max B"
assms_2: shows "e2 \<sqsubset>\<^sup>i e3"
wt_lam: fixes x :: "nat"   and A :: "ty"   and \<Gamma> :: "(nat \<times> ty) list"   and e :: "exp"   and B :: "ty" assumes "(x, A) # \<Gamma> \<turnstile> e : B" shows "\<Gamma> \<turnstile> ELam x e : TFun (A \<rightarrow> B)"
wtC_wax: fixes \<sigma> :: "'tp" assumes "prot \<sigma> \<or> protFw \<sigma>"   and "\<not> isRes \<sigma> \<or> protCl \<sigma>" shows "GE.wtC (wax \<sigma>)"
axioms_2: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "partially_ordered_set A less_eq" shows "antisymmetric A less_eq"
distrib_left: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and c :: "'b \<times> 'c" shows "times_prod' a (sup_prod b c) = sup_prod (times_prod' a b) (times_prod' a c)"
hn_ctxt_def: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and a :: "'a"   and c :: "'b" shows "hn_ctxt P a c \<equiv> P a c"
A: shows "y = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e"
isNewArr_def: fixes t :: "New" shows "isNewArr t = (case t of new_instance C \<Rightarrow> False | new_array T l \<Rightarrow> True)"
Fail_HInv3_nL: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "Fail s s' t"   and "\<not> HInv3_L s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "\<not> HInv3_L s' p q d"
cbsub_2: shows "cbox (- b) b \<subseteq> cbox (- c) c"
simps_1: fixes t :: "lam" shows "\<box>\<lbrakk>t\<rbrakk> = t"
map\<^sub>0_in_hhom: fixes a :: "'d" assumes "D.obj a" shows "\<guillemotleft>\<eta>\<^sub>0 a : src\<^sub>D (I\<^sub>D.map a) \<rightarrow>\<^sub>D src\<^sub>D (GF.map a)\<guillemotright>"
set_nths: fixes xs :: "'c list"   and I :: "nat set" shows "set (nths xs I) = {xs ! i |i. i < length xs \<and> i \<in> I}"
simps_1: fixes prec :: "nat"   and x :: "real" shows "bound_intersect_2d prec [] x = None"
leg_simps_1: shows "Maps.arr SPN_fgh.\<mu>\<nu>.leg0"
hyps: fixes y :: "'a" assumes "|filter P xs| = |xs|"   and "y \<in> set xs" shows "P y"
a1: shows "low_equal s1 s2 \<and> (fst instr = load_store_type STB \<or> fst instr = load_store_type STH \<or> fst instr = load_store_type ST \<or> fst instr = load_store_type STA \<or> fst instr = load_store_type STD) \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (store_instr instr s1)) \<and> t2 = snd (fst (store_instr instr s2))"
axioms_1: fixes M :: "'c measure" assumes "subprob_space M" shows "finite_measure M"
4: fixes p :: "'a"   and q :: "'a" assumes "p \<in> dom f"   and "q \<in> transition A a p" shows "the (g q) \<le> the (f p)"
\<tau>rtrancl3p_step: fixes s :: "'s1"   and tl :: "'tl1"   and s' :: "'s1"   and tls :: "'tl1 list"   and s'' :: "'s1" assumes "s -1-tl\<rightarrow> s'"   and "\<not> \<tau>move1 s tl s'"   and "s' -\<tau>1-tls\<rightarrow>* s''" shows "s -\<tau>1-tl # tls\<rightarrow>* s''"
Tree\<^sub>\<alpha>_abs_rep: fixes t\<^sub>\<alpha> :: "('b, 'c, 'd) Tree\<^sub>\<alpha>" shows "abs_Tree\<^sub>\<alpha> (rep_Tree\<^sub>\<alpha> t\<^sub>\<alpha>) = t\<^sub>\<alpha>"
lookup_tl: fixes xs :: "'x"   and k :: "'a" shows "lookup (local.tl xs) k = (if \<forall>k'\<in>fst ` set (fst (list_of_oalist xs)). le (snd (list_of_oalist xs)) k k' then 0::'b else lookup xs k)"
assms: shows "\<guillemotleft>F : A \<rightarrow> B\<guillemotright>"
conj_superexpansion2: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> (p .\<or>. q) .\<and>. p) = (A \<turnstile> p)"
foldr_def: fixes f :: "'b \<times> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "('b, 'c) FingerTree"   and \<sigma> :: "'a" shows "FingerTree.foldr f t \<sigma> \<equiv> FingerTreeStruc.foldr f (Rep_FingerTree t) \<sigma>"
setE: fixes y :: "'a"   and L :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "y \<in> set L"   and "P y" shows "list_ex P L"
REL_FORCE_ID_def: fixes R :: "('a \<times> 'a) set" shows "REL_FORCE_ID R \<equiv> R = Id"
not_dvd: fixes q :: "int" assumes "ilc * lead_coeff f = p ^ n * q" shows "False"
avg_def: fixes s :: "(nat, unit) HashMap.hashmap" shows "avg s \<equiv> case avg_aux s of (c, s) \<Rightarrow> s div c"
subdegree_div_unit'': fixes q :: "'a fps"   and p :: "'a fps" assumes "fps_nth q 0 \<noteq> (0::'a)"   and "inverse (fps_nth q 0) \<noteq> (0::'a)" shows "subdegree (p / q) = subdegree p"
one_uint32_def: shows "1 \<equiv> Abs_uint32 1"
comp_arr_dom: fixes f :: "'j"   and a :: "'j" assumes "J.arr f"   and "J.dom f = a" shows "f \<cdot>\<^sub>J a = f"
wf_nFOR: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and LESS :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> bool"   and assigns :: "nat \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'd"   and nvars :: "'c \<Rightarrow> 'b"   and Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'c"   and CONS :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and SNOC :: "'e \<Rightarrow> 'c \<Rightarrow> 'c"   and Length :: "'c \<Rightarrow> nat"   and extend :: "'a \<Rightarrow> bool \<Rightarrow> 'e \<Rightarrow> 'e"   and size :: "'e \<Rightarrow> 'b"   and zero :: "'b \<Rightarrow> 'e"   and alphabet :: "'b \<Rightarrow> 'e list"   and eval :: "'d \<Rightarrow> nat \<Rightarrow> bool"   and downshift :: "'d \<Rightarrow> 'd"   and upshift :: "'d \<Rightarrow> 'd"   and add :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and cut :: "nat \<Rightarrow> 'd \<Rightarrow> 'd"   and len :: "'d \<Rightarrow> nat"   and restrict :: "'a \<Rightarrow> 'd \<Rightarrow> bool"   and Restrict :: "'a \<Rightarrow> nat \<Rightarrow> ('f, 'a) aformula"   and lformula0 :: "'f \<Rightarrow> bool"   and FV0 :: "'a \<Rightarrow> 'f \<Rightarrow> nat set"   and find0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> bool"   and wf0 :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and decr0 :: "'a \<Rightarrow> nat \<Rightarrow> 'f \<Rightarrow> 'f"   and satisfies0 :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and nullable0 :: "'f \<Rightarrow> bool"   and lderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and rderiv0 :: "'e \<Rightarrow> 'f \<Rightarrow> ('f, 'a) aformula"   and idx :: "'b"   and \<phi>s :: "('f, 'a) aformula list" assumes "Formula SUC LESS assigns nvars Extend CONS SNOC Length extend size zero alphabet eval downshift upshift add cut len restrict Restrict lformula0 FV0 find0 wf0 decr0 satisfies0 nullable0 lderiv0 rderiv0" shows "Formula_Operations.wf SUC wf0 idx (nFOR \<phi>s) = (\<forall>\<phi>\<in>set \<phi>s. Formula_Operations.wf SUC wf0 idx \<phi>)"
ToCT_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and f :: "'b \<times> 'b \<times> 'b" assumes "binary_endofunctor C T" shows "binary_endofunctor.ToCT C T f \<equiv> if partial_magma.arr (product_category.comp C (product_category.comp C C)) f then T (fst f, T (fst (snd f), snd (snd f))) else partial_magma.null C"
has_uniq_bottom: fixes P :: "'a set" assumes "has_bottom P" shows "\<exists>!z. z \<in> P \<and> (\<forall>x\<in>P. x \<succeq> z)"
CYK_eq2: fixes i :: "nat"   and j :: "nat"   and w :: "'c list"   and G :: "('d \<times> ('d, 'c) RHS) list" assumes "i + j \<le> length w"   and "1 < j" shows "CYK G w i j = {uu_. \<exists>X A B k. uu_ = X \<and> (X, Branch A B) \<in> set G \<and> A \<in> CYK G w i k \<and> B \<in> CYK G w (i + k) (j - k) \<and> 1 \<le> k \<and> k < j}"
rel_fun_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "(f, g) \<in> Id" shows "(f x, g x) \<in> Id"
closed_contains_Inf: fixes S :: "real set" assumes "S \<noteq> {}"   and "bdd_below S"   and "closed S" shows "Inf S \<in> S"
DomainI: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "r a b" shows "Domainp r a"
termi_call_steps_def: shows "termi_call_steps = {((q, t), p, s). body p \<down> s \<and> (\<exists>cs. ([body p], s) \<rightarrow>\<^sup>* (CALL q # cs, t))}"
g_def: fixes i :: "nat" shows "g i = (case i of 0 \<Rightarrow> (Seq c1 c2 # cs, css, Normal s) | Suc j \<Rightarrow> f j)"
rep_eq: fixes x :: "uint8" shows "Rep_uint (uint_of_uint8 x) = ucast (Rep_uint8 x)"
rInput: fixes \<Psi> :: "'b"   and M :: "'a"   and K :: "'a"   and xvec :: "name list"   and N :: "'a"   and Tvec :: "'a list"   and C :: "'d"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "set xvec \<subseteq> supp N"   and "length xvec = length Tvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* M"   and "xvec \<sharp>* K"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (M\<lparr>\<lambda>*xvec N\<rparr>.P) (K\<lparr>N[xvec::=Tvec]\<rparr> \<prec> P[xvec::=Tvec]) [] (\<one>)"
open_fbint_imp_infinite: fixes U :: "fbint set" assumes "open U"   and "U \<noteq> {}" shows "infinite U"
cases: fixes a :: "('s \<times> ('c \<Rightarrow> real)) pmf"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "a \<in> K st" obtains l :: "'s" and u :: "'c \<Rightarrow> real" and t :: "real" where "a = return_pmf (la__, ua__ \<oplus> t)" and "st \<in> S" and "st = (la__, ua__)" and "0 \<le> t" and "ua__ \<oplus> t \<turnstile> PTA.inv_of A la__"   | l :: "'s" and u :: "'c \<Rightarrow> real" and g :: "('c, real) cconstraint" and \<mu> :: "('c set \<times> 's) pmf" where "a = map_pmf (\<lambda>(X, l). (l, [(SOME r. set r = X)\<rightarrow>0]u)) \<mu>" and "st \<in> S" and "st = (l, u)" and "(l, g, \<mu>) \<in> PTA.trans_of A" and "u \<turnstile> g"   | "a = return_pmf st"
DISJNT: shows "\<forall>A\<in>P. \<forall>B\<in>P. A \<noteq> B \<longrightarrow> A \<inter> B = {}"