hcju/isabelleps huggingface dataset:
simps: fixes e :: "'c"   and a :: "'d"   and t :: "('c, 'd) SkewBinomialTree"   and t' :: "('c, 'd) SkewBinomialTree" shows "skewlink e a t t' = (if a \<le> prio t \<and> a \<le> prio t' then Node e a (Suc (rank t)) [t, t'] else if prio t \<le> prio t' then Node (val t) (prio t) (Suc (rank t)) (Node e a 0 [] # t' # children t) else Node (val t') (prio t') (Suc (rank t')) (Node e a 0 [] # t # children t'))"
no_output_none_nxt: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and r :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and t :: "(String.literal \<times> value list) stream" shows "alw (nxt (output_eq [])) (make_full_observation e None r p t)"
bmp: shows "0 < bitlen \<bar>mantissa x\<bar>"
set_iterator_abs_simple_rule_insert_P: assumes "set_iterator_abs \<alpha> invar it S0"   and "\<And>xc. invar xc \<Longrightarrow> f xc = f' (\<alpha> xc)"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>True; x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0\<rbrakk> \<Longrightarrow> I (insert x S) (f' x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> True; I S \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it (\<lambda>_. True) f \<sigma>0)"
sup_ge2: fixes sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and y :: "'x"   and x :: "'x" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
assms_7: shows "r = 2 ^ (2 * k - 1) * 9 - 1"
Suc_mult_cancel1: fixes k :: "nat"   and m :: "nat"   and n :: "nat" shows "(Suc k * m = Suc k * n) = (m = n)"
ord_Zp_eq_frac: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int"   and d :: "nat \<Rightarrow> int" assumes "a \<in> nonzero Zp"   and "b \<in> nonzero Zp"   and "c \<in> nonzero Zp"   and "d \<in> nonzero Zp"   and "a \<otimes> d = b \<otimes> c" shows "ord_Zp a - ord_Zp b = ord_Zp c - ord_Zp d"
supp_uminus: fixes f :: "'b \<Rightarrow> 'c" shows "Formal_Puiseux_Series.supp (\<lambda>x. - f x) = Formal_Puiseux_Series.supp f"
simple_packet_set_union_def: fixes ps :: "'b simple_match list"   and m :: "'b simple_match" shows "simple_packet_set_union ps m = m # ps"
schroeder_6_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> - z) = (z\<^sup>T * x \<le> - y\<^sup>T)"
assms: shows "pR \<noteq> 0 \<or> pI \<noteq> 0"
components_are_iso: fixes a :: "'a \<times> 'a" assumes "B.VV.ide a" shows "local.iso (cmp\<^sub>U\<^sub>P a)"
add_fract: fixes b :: "'a"   and d :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<noteq> (0::'a)"   and "d \<noteq> (0::'a)" shows "Fract a b + Fract c d = Fract (a * d + c * b) (b * d)"
SPLIT_2: shows "((sh, ch), ww, s', c') \<in> trcl (ntrs fg)"
obs_def: fixes prob :: "('b, 'c, 'd, 'a) ifc_problem"   and \<sigma> :: "'c \<Rightarrow> 'd"   and k :: "nat" shows "IFC_def.obs prob \<sigma> k = (if \<exists>i. IFC_def.is_kth_obs prob (IFC_def.path prob \<sigma>) k i then IFC_def.obsp prob \<sigma> (THE i. IFC_def.is_kth_obs prob (IFC_def.path prob \<sigma>) k i) else None)"
msubstltneg_def: fixes c :: "poly"   and t :: "tm"   and a :: "poly"   and b :: "tm" shows "msubstltneg c t a b = Lt (tm.Neg (tm.Add (tm.Mul a t) (tm.Mul c b)))"
x_1: shows "a \<bullet> i \<le> x \<bullet> i"
vmc_path_ldropn: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "vmc_path G P v0 p \<sigma>"   and "enat n < llength P" shows "vmc_path G (ldropn n P) (P $ n) p \<sigma>"
eqvtBangRel: fixes Rel :: "(pi \<times> pi) set" assumes "eqvt Rel" shows "eqvt (bangRel Rel)"
pow_mono_leq_imp_lt: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "x < 2 ^ y"
Sum2_2: fixes Q :: "pi"   and \<alpha> :: "freeRes"   and Q' :: "pi"   and P :: "pi" assumes "Q \<Longrightarrow>\<alpha> \<prec> Q'" shows "P \<oplus> Q \<Longrightarrow>\<alpha> \<prec> Q'"
cblinfun_compose_scaleC_right: fixes A :: "'c \<Rightarrow>\<^sub>C\<^sub>L 'b"   and a :: "complex"   and B :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "A o\<^sub>C\<^sub>L a *\<^sub>C B = a *\<^sub>C (A o\<^sub>C\<^sub>L B)"
cases: fixes x :: "tm" obtains y :: "name" and x :: "tm" and z :: "name" where "atom y \<sharp> (x, z)" and "atom z \<sharp> x" and "xa__ = x"
b_src: shows "b = b1 @ [fst q] @ b2"
simps_1: fixes P :: "'a \<Rightarrow> bool" shows "delete_first P [] = None"
Bij_inv_into_lemma: fixes S :: "'c set"   and h :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> h (g x y) = g (h x) (h y)"   and "h \<in> Bij S"   and "g \<in> S \<rightarrow> S \<rightarrow> S"   and "x \<in> S"   and "y \<in> S" shows "inv_into S h (g x y) = g (inv_into S h x) (inv_into S h y)"
that: fixes u :: "'t" assumes "u \<in> keys p"   and "v = t \<oplus> u" shows "thesis"
iexp_eq2: fixes x :: "real"   and n :: "nat" shows "iexp x = (\<Sum>k\<le>Suc n. (\<i> * complex_of_real x) ^ k / fact k) + \<i> ^ Suc n / fact n * (CLBINT s=0..ereal x. complex_of_real ((x - s) ^ n) * (iexp s - 1))"
subgroupE_1: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set" assumes "Group.group G"   and "subgroup H G" shows "H \<subseteq> carrier G"
mult_isol: fixes x :: "('a \<times> 'a) set"   and y :: "('a \<times> 'a) set"   and z :: "('a \<times> 'a) set" assumes "x \<subseteq> y" shows "z ; x \<subseteq> z ; y"
less_asym': fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and P :: "bool" assumes "class.preorder less_eq less"   and "less a b"   and "less b a" shows "P"
assms: shows "\<forall>\<^sub>F x in F. f x < g x"
head_notin: shows "x \<notin> fst ` succ_rel xs"
simps_1: fixes q :: "'b queue" shows "do_act Deq q = deq q"
const_type_inv: fixes c :: "'fun"   and X :: "('fun, 'var) Term.term list"   and a :: "'atom" assumes "\<Gamma> (Fun c X) = Var a" shows "arity c = 0"
lucas_corollary: fixes p :: "nat"   and n :: "nat"   and k :: "nat" assumes "prime p" shows "(n choose k) mod p = (n div p choose k div p) * (n mod p choose k mod p) mod p"
centered_def: fixes i :: "float interval" shows "centered i = i - interval_of (mid i)"
ccProd_eqvt: fixes \<pi> :: "perm"   and S :: "Vars.var set"   and S' :: "Vars.var set" shows "\<pi> \<bullet> S G\<times> S' = (\<pi> \<bullet> S) G\<times> (\<pi> \<bullet> S')"
oalist_inv_sort_oalist_raw: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and xs :: "('b \<times> 'c) list \<times> 'a" shows "oalist_raw.oalist_inv rep_key_order (oalist_raw.sort_oalist_raw rep_key_order xs)"
sep_conj_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P Q \<equiv> \<lambda>h. \<exists>x y. sep_disj x y \<and> h = plus x y \<and> P x \<and> Q y"
cases: fixes a1 :: "('a\<^sub>m, 'a\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) G\<^sub>m"   and a2 :: "('a\<^sub>m, 'a\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) G\<^sub>m" assumes "local.blinding_of_G a1 a2" obtains x :: "(('a\<^sub>m, 'a\<^sub>h) T\<^sub>m, 'a\<^sub>h T\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) F\<^sub>m" and y :: "(('a\<^sub>m, 'a\<^sub>h) T\<^sub>m, 'a\<^sub>h T\<^sub>h, 'b\<^sub>m, 'b\<^sub>h) F\<^sub>m" where "a1 = G\<^sub>m x" and "a2 = G\<^sub>m y" and "blinding_of_F (root_hash_T rha) (blinding_of_T rha boa) rhb bob x y"
ord_conv_Least: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "\<exists>n>0. x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>" shows "group.ord G x = (LEAST n. 0 < n \<and> x [^]\<^bsub>G\<^esub> n = \<one>\<^bsub>G\<^esub>)"
case_9: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'b \<Rightarrow> 'a"   and f4 :: "'b \<Rightarrow> 'a"   and f5 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f6 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f7 :: "'b ltln \<Rightarrow> 'a"   and f8 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f9 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f10 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and f11 :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> 'a"   and x91 :: "'b ltln"   and x92 :: "'b ltln" shows "(case x91 R\<^sub>n x92 of true\<^sub>n \<Rightarrow> f1 | false\<^sub>n \<Rightarrow> f2 | prop\<^sub>n(x) \<Rightarrow> f3 x | nprop\<^sub>n(x) \<Rightarrow> f4 x | x and\<^sub>n xa \<Rightarrow> f5 x xa | x or\<^sub>n xa \<Rightarrow> f6 x xa | X\<^sub>n x \<Rightarrow> f7 x | x U\<^sub>n xa \<Rightarrow> f8 x xa | x R\<^sub>n xa \<Rightarrow> f9 x xa | x W\<^sub>n xa \<Rightarrow> f10 x xa | x M\<^sub>n xa \<Rightarrow> f11 x xa) = f9 x91 x92"
progress: fixes h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty"   and l :: "String.literal \<Rightarrow> 'addr val option"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'a, 'heap) Jinja_thread_action"   and t :: "'thread_id" assumes "wwf_J_prog P"   and "hconf h"   and "P,E,h \<turnstile> e : T"   and "\<D> e \<lfloor>dom l\<rfloor>"   and "\<not> final e" shows "\<exists>e' s' taa. extTA,P,t \<turnstile> \<langle>e,(h, l)\<rangle> -taa\<rightarrow> \<langle>e',s'\<rangle>"
dup_def: fixes n :: "nat" shows "dup n = n + n"
tendsto_eval_fds_Re_going_to_at_top': fixes f :: "complex fds" assumes "conv_abscissa f \<noteq> \<infinity>" shows "(eval_fds f \<longlongrightarrow> fds_nth f 1) (Re going_to at_top)"
alw_disjoint_ccontr: fixes S :: "'a set"   and \<omega> :: "'a stream"   and R :: "'a set" assumes "alw (HLD S) \<omega>"   and "ev (alw (HLD R)) \<omega>"   and "R \<inter> S = {}" shows "False"
that: fixes Z :: "'a set" assumes "Z \<in> sets M"   and "Z \<subseteq> W"   and "emeasure M Z < \<infinity>"   and "ennreal C < emeasure M Z" shows "thesis"
z: shows "z \<in> interior (convex hull {a, b, c})"
a_a_z_0: fixes z :: "int"   and a :: "ant" assumes "0 < z"   and "a * ant z = 0" shows "a = 0"
prime_divisorE: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero"   and "\<not> dvd.dvd times a one" obtains p :: "'a" where "normalization_semidom.prime zero times one normalize p" and "dvd.dvd times p a"
2: shows "[\<^bold>\<diamond>\<^bold>\<not>(\<^bold>\<not>\<phi> \<^bold>\<or> \<^bold>\<not>\<^bold>\<diamond>\<psi>) \<^bold>\<equiv> \<^bold>\<not>(\<^bold>\<not>\<^bold>\<diamond>\<phi> \<^bold>\<or> \<^bold>\<not>\<^bold>\<diamond>\<psi>) in v]"
in_outs_\<I>_iff_responses_\<I>: fixes out :: "'a"   and \<I> :: "('a, 'b) \<I>" shows "(out \<in> outs_\<I> \<I>) = (responses_\<I> \<I> out \<noteq> {})"
simps_1: fixes r :: "rat"   and p :: "linear_poly"   and c :: "'a" shows "r *R LEQ_ns p c = (if r < 0 then GEQ_ns (r *R p) (r *R c) else LEQ_ns (r *R p) (r *R c))"
push: shows "S O R \<subseteq> R O S\<^sup>*"
subcocycle_ineq: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat"   and m :: "nat"   and x :: "'a" assumes "subcocycle u" shows "u (n + m) x \<le> u n x + u m ((T ^^ n) x)"
array_grow_rule: fixes la :: "'b list"   and s :: "nat"   and a :: "'b array"   and x :: "'b" assumes "length la \<le> s" shows "<a \<mapsto>\<^sub>a la> array_grow a s x <\<lambda>r. r \<mapsto>\<^sub>a (la @ replicate (s - length la) x)>\<^sub>t"
pcompose_const: fixes a :: "'a"   and q :: "'a poly" shows "[:a:] \<circ>\<^sub>p q = [:a:]"
bigo_powr_nonneg: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and p :: "real" assumes "f \<in> O[F](g)"   and "0 \<le> p"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "\<forall>\<^sub>F x in F. 0 \<le> g x" shows "(\<lambda>x. f x powr p) \<in> O[F](\<lambda>x. g x powr p)"
vint_assoc_law_multiplication: fixes x :: "V"   and y :: "V"   and z :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "y \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "z \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "x *\<^sub>\<int> y *\<^sub>\<int> z = x *\<^sub>\<int> (y *\<^sub>\<int> z)"
that: fixes bf' :: "'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'b" assumes "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative blinfun_apply (bf' x)) (at x within S)"   and "continuous_on S bf'"   and "\<And>x. x \<in> S \<Longrightarrow> blinfun_apply (bf' x) = f' x" shows "thesis"
csols_unique: fixes x :: "real \<Rightarrow> 'a"   and t1 :: "real"   and x0 :: "'a"   and y :: "real \<Rightarrow> 'a"   and t2 :: "real" assumes "(x, t1) \<in> csols t0 x0"   and "(y, t2) \<in> csols t0 x0" shows "\<forall>t\<in>{t0--t1} \<inter> {t0--t2}. x t = y t"
any_le_inf: fixes x :: "'c DBMEntry" shows "x \<le> \<infinity>"
cod_inv: fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.cod (\<phi>.B.inv f) = \<phi>.B.dom f"
meval: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a mformula"   and \<phi>' :: "'a MFOTL.formula" assumes "wf_mformula \<sigma> j n R \<phi> \<phi>'" shows "case meval n (\<tau> \<sigma> j) (\<Gamma> \<sigma> j) \<phi> of (xs, \<phi>\<^sub>n) \<Rightarrow> wf_mformula \<sigma> (Suc j) n R \<phi>\<^sub>n \<phi>' \<and> list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>')) [Monitor.progress \<sigma> \<phi>' j..<Monitor.progress \<sigma> \<phi>' (Suc j)] xs"
snd_apsnd: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'b \<times> 'c" shows "snd (apsnd f x) = f (snd x)"
empty_open: fixes T :: "'b set set" assumes "topology T" shows "{} open\<^bsub>T\<^esub>"
dfa_isomorphism_axioms_def: fixes M :: "'b dfa"   and N :: "'b dfa"   and h :: "hf \<Rightarrow> hf" shows "dfa_isomorphism_axioms M N h \<equiv> (bij_betw h (dfa.states M) (dfa.states N) \<and> h (dfa.init M) = dfa.init N) \<and> h ` dfa.final M = dfa.final N \<and> (\<forall>q x. q \<in> dfa.states M \<longrightarrow> h (dfa.nxt M q x) = dfa.nxt N (h q) x)"
insertion_lowerPoly: fixes p :: "'b mpoly"   and i :: "nat"   and j :: "nat"   and prfx :: "'b list"   and xs :: "'b list"   and L :: "'b list" assumes "vars p \<inter> {i..<i + j} = {}"   and "length prfx = i"   and "length xs = j" shows "insertion (nth_default (0::'b) (prfx @ L)) (lowerPoly i j p) = insertion (nth_default (0::'b) (prfx @ xs @ L)) p"
solves_ode_supset_range: fixes x :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set"   and Y :: "'a set" assumes "(x solves_ode f) T X"   and "X \<subseteq> Y" shows "(x solves_ode f) T Y"
op_set_isEmpty_def: fixes s :: "'g set" shows "op_set_isEmpty s \<equiv> s = {}"
simps_3: fixes x4 :: "nat"   and y4 :: "nat" shows "(REL x4 = REL y4) = (x4 = y4)"
map_eq_Cons_D: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and y :: "'a"   and ys :: "'a list" assumes "map f xs = y # ys" shows "\<exists>z zs. xs = z # zs \<and> f z = y \<and> map f zs = ys"
reachableI2: fixes x :: "'e"   and s :: "'g process_name \<Rightarrow> ('f, 'g, 'h, 'e) Model.local_state"   and m :: "'g" assumes "x \<in> ghost_honorary_root (s (mutator m))" shows "mut_m.reachable m x s"
preserves_dom: fixes f :: "'a" assumes "A.arr f" shows "B.dom (\<tau>' f) = G (A.dom f)"
finfun_comp2_def: fixes g :: "('c, 'b) finfun"   and f :: "'a \<Rightarrow> 'c" shows "finfun_comp2 g f = Abs_finfun (finfun_apply g \<circ> f)"
augment_with_path_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and p :: "(nat \<times> nat) list" assumes "NFlow c s t f" shows "NFlow.augment_with_path c f p \<equiv> NFlow.augment c f (NPreflow.augmentingFlow c f p)"
intro: fixes s :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and t :: "'b set"   and k :: "'b \<Rightarrow> 'a" assumes "continuous_on s h"   and "h ` s = t"   and "continuous_on t k"   and "k ` t \<subseteq> s"   and "\<And>y. y \<in> t \<Longrightarrow> h (k y) = y" shows "Retracts s h t k"
extf_singleton_nil_if_\<delta>\<^sub>h_eq_\<epsilon>\<^sub>h: fixes f :: "'s"   and gt :: "('s, 'v) tm \<Rightarrow> ('s, 'v) tm \<Rightarrow> bool"   and s :: "('s, 'v) tm" assumes "\<delta>\<^sub>h = \<epsilon>\<^sub>h" shows "extf f gt [s] []"
obs_def: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "obs \<sigma> k = (if \<exists>i. is_kth_obs (path \<sigma>) k i then obsp \<sigma> (THE i. is_kth_obs (path \<sigma>) k i) else None)"
gbinomial_int_binomial: fixes a :: "int"   and k :: "nat" shows "a gchoose k = (if 0 \<le> a then int (nat a choose k) else (- 1) ^ k * int (k + nat (- a) - 1 choose k))"
bipartite_web_of_sel_5: fixes \<Gamma> :: "('a, 'b) web_scheme"   and v :: "'a" shows "weight (bipartite_web_of \<Gamma>) (Inr v) = (if v \<in> A \<Gamma> then 0 else weight \<Gamma> v)"
prod_c: fixes CA :: "'a literal multiset" assumes "CA \<in> set CAs" shows "production N CA \<noteq> {}"
lm068: fixes x :: "'a"   and f :: "('a \<times> 'b) set"   and X :: "'a set" assumes "x \<in> Domain (f || X)" shows "(f || X) `` {x} = f `` {x}"
h2: shows "(0 < poly (qs ! a * prod_list (retrieve_polys qs I)) x) = (0 < poly (qs ! a) x \<and> 0 < poly (prod_list (retrieve_polys qs I)) x \<or> poly (qs ! a) x < 0 \<and> poly (prod_list (retrieve_polys qs I)) x < 0)"
empty_subl: fixes l :: "fterm literal" shows "l \<cdot>\<^sub>l \<epsilon> = l"
dependent_nat_choice: fixes P :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<exists>x. P 0 x"   and "\<And>x n. P n x \<Longrightarrow> \<exists>y. P (Suc n) y \<and> Q n x y" shows "\<exists>f. \<forall>n. P n (f n) \<and> Q n (f n) (f (Suc n))"
singleDSourceEmpty4isNotDSource: fixes i :: "AbstrLevelsID"   and C :: "CSet"   and S :: "CSet"   and Z :: "CSet" assumes "DAcc i C = {S}"   and "Z \<noteq> S" shows "C \<notin> DSources i Z"
DERIV_nonneg_imp_nondecreasing: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. (f has_real_derivative y) (at x) \<and> 0 \<le> y" shows "f a \<le> f b"
separating_hyperplane_set_0_inspan: fixes S :: "'a set" assumes "convex S"   and "S \<noteq> {}"   and "(0::'a) \<notin> S" obtains a :: "'a" where "a \<in> span S" and "a \<noteq> (0::'a)" and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> a \<bullet> x"
assms_2: shows "N \<subseteq> subformulas\<^sub>\<nu> \<phi>"
bound_def: fixes X :: "'c set"   and r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and b :: "'d" shows "bound X r b \<equiv> \<forall>x\<in>X. r x b"
C1: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and x :: "name" assumes "(\<Psi>', R, S) \<in> Rel"   and "x \<sharp> \<Psi>'" shows "(\<Psi>', \<lparr>\<nu>x\<rparr>R, \<lparr>\<nu>x\<rparr>S) \<in> Rel'"
OneState_HPT_Target_Source: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and S :: "'d"   and SA :: "('d, 'e, 'f) seqauto" assumes "TS \<in> HPT ST"   and "S \<in> States SA"   and "SA \<in> SAs (HA ST)"   and "States SA \<inter> Source TS = {}" shows "S \<notin> Target TS"
rep_eq: fixes x :: "'c set_dlist" shows "null x = List.null (list_of_dlist x)"
liveness_def: fixes A :: "'b set"   and P :: "'b llist set" shows "liveness A P \<equiv> \<forall>t\<in>A\<^sup>\<star>. \<exists>s\<in>A\<^sup>\<infinity>. t @@ s \<in> P"
refl_genPrefix: fixes r :: "('a \<times> 'a) set" assumes "refl r" shows "refl (genPrefix r)"
f2: fixes a :: "'a"   and aa :: "'a" shows "ad (ad a) + ad aa = ad (ad a \<cdot> aa)"
ws_def_4: shows "\<forall>i. Suc i < length ws \<longrightarrow> prio_emb_step prio (ws ! i) u = ws ! Suc i \<and> ws ! i \<rightarrow>\<^sub>e\<^sub>m\<^sub>b ws ! Suc i"
is_sup_upper: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and s :: "'a" assumes "dlat le"   and "dpo.is_sup le x y s" obtains "le x s" and "le y s"
interp_ins_tail_unfold: fixes xs :: "('b \<times> 'b option) list"   and x :: "'b \<times> 'b option" shows "interp_ins (xs @ [x]) = insert_spec (interp_ins xs) x"
mempty_left: fixes below :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and liftemb :: "'d\<^sub>\<bottom> \<rightarrow> udom\<^sub>\<bottom>"   and liftprj :: "udom\<^sub>\<bottom> \<rightarrow> 'd\<^sub>\<bottom>"   and liftdefl :: "'d itself \<Rightarrow> (udom\<^sub>\<bottom>) defl"   and emb :: "'d \<rightarrow> udom"   and prj :: "udom \<rightarrow> 'd"   and defl :: "'d itself \<Rightarrow> udom defl"   and mempty :: "'d"   and mappend :: "'d \<rightarrow> 'd \<rightarrow> 'd"   and ys :: "'d" assumes "class.monoid below liftemb liftprj liftdefl emb prj defl mempty mappend" shows "mappend\<cdot>mempty\<cdot>ys = ys"
walk_iarray_Cons: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c"   and xs :: "'c list"   and b :: "'b" shows "walk_iarray f (IArray (a # xs)) b = walk_iarray f (IArray xs) (f b a)"
is_expansion_map: fixes G :: "'a ms"   and b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and f :: "'a \<Rightarrow> 'b"   and basis' :: "(real \<Rightarrow> real) list" assumes "is_expansion G (b # basis)"   and "\<And>C. is_expansion C basis \<Longrightarrow> is_expansion (f C) basis'"   and "\<And>C. eval (f C) = eval C"   and "length basis' = expansion_level TYPE('b)" shows "is_expansion (map_ms f G) (b # basis')"
dist_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "dist x y = sqrt ((dist (fst x) (fst y))\<^sup>2 + (dist (snd x) (snd y))\<^sup>2)"
upto: shows "\<pi> =\<^bsub>m\<^esub> \<pi>\<^sub>1"
nat_pow_pow: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier (units_of R)" shows "(x [^]\<^bsub>units_of R\<^esub> n) [^]\<^bsub>units_of R\<^esub> m = x [^]\<^bsub>units_of R\<^esub> (n * m)"
post_le_one: shows "post \<le>\<^sub>L 1\<^sub>m N"
leg1_simps_3: shows "src\<^sub>C g = src\<^sub>C f"
measurable_proj_singleton: fixes i :: "'d"   and I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "i \<in> I" shows "(\<lambda>x. (x)\<^sub>F i) \<in> Pi\<^sub>F {I} M \<rightarrow>\<^sub>M M i"
klift_def: fixes f :: "'a \<Rightarrow> 'b set" shows "f\<^sup>\<dagger> = \<mu> \<circ> \<P> f"
c'_def: shows "c' \<equiv> augment_cf edges (c e)"
Upair_singleton: fixes a :: "ZF"   and c :: "ZF"   and d :: "ZF" shows "(Upair a a = Upair c d) = (a = c \<and> a = d)"
stutter_extend_ex_None_eq: fixes ex :: "'d \<Rightarrow> 'b \<Rightarrow> 'b" shows "stutter_extend_ex ex None = id"
assoc_scalar_prod: fixes v\<^sub>1 :: "'c vec"   and nr :: "nat"   and A :: "'c mat"   and nc :: "nat"   and v\<^sub>2 :: "'c vec" assumes "v\<^sub>1 \<in> carrier_vec nr"   and "A \<in> carrier_mat nr nc"   and "v\<^sub>2 \<in> carrier_vec nc" shows "vec nc (\<lambda>j. v\<^sub>1 \<bullet> col A j) \<bullet> v\<^sub>2 = v\<^sub>1 \<bullet> vec nr (\<lambda>i. row A i \<bullet> v\<^sub>2)"
uint_mod: fixes x :: "'b word"   and y :: "'b word" shows "uint (x mod y) = uint x mod uint y"
ineq_subterm_inj_cond_subst: fixes X :: "'a set"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and S :: "('b, 'a) Term.term set" assumes "X \<inter> range_vars \<theta> = {}"   and "\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t S \<longrightarrow> T = [] \<or> (\<exists>u\<in>set T. u \<notin> Var ` X)" shows "\<forall>f T. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t S \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> \<longrightarrow> T = [] \<or> (\<exists>u\<in>set T. u \<notin> Var ` X)"