hcju/isabelleps huggingface dataset:
in_listset_nth: fixes x :: "'c list"   and As :: "'c set list"   and i :: "nat" assumes "x \<in> listset As"   and "i < length As" shows "x ! i \<in> As ! i"
Ap_parcontractE: fixes p :: "comb"   and q :: "comb"   and r :: "comb" assumes "p \<bullet> q \<Rrightarrow>\<^sup>1 r" obtains "r = p \<bullet> q"   | "p = K \<bullet> r"   | x :: "comb" and y :: "comb" where "p = S \<bullet> x \<bullet> y__" and "r = x \<bullet> q \<bullet> (y__ \<bullet> q)"   | y :: "comb" and w :: "comb" where "r = y \<bullet> w" and "p \<Rrightarrow>\<^sup>1 y" and "q \<Rrightarrow>\<^sup>1 w"
simps_2: fixes p :: "real mpoly"   and i :: "nat"   and L :: "(real mpoly \<times> nat) list" shows "chop ((p, i) # L) = (if p = 0 then chop L else (p, i) # L)"
poly_deg_def: fixes p :: "('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd" shows "poly_deg p = (if keys p = {} then 0::'b else Max (deg_pm ` keys p))"
prob_space_PiM: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)" shows "prob_space (Pi\<^sub>M I M)"
states_closed: shows "\<forall>(s, \<mu>)\<in>set \<K>. \<forall>(t, uu_)\<in>set \<mu>. t \<in> set \<K>\<^sub>s"
assms: shows "\<Gamma> \<turnstile> t : \<tau>"
div_add1_eq1_mod_0_left: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m = 0" shows "(a + b) div m = a div m + b div m"
diff_def: fixes x :: "'a bit0"   and y :: "'a bit0" shows "x - y = Abs_bit0 ((Rep_bit0 x - Rep_bit0 y) mod int CARD('a bit0))"
term_subst_eq: fixes t :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'c) Term.term"   and \<tau> :: "'a \<Rightarrow> ('b, 'c) Term.term" assumes "\<And>x. x \<in> vars_term t \<Longrightarrow> \<sigma> x = \<tau> x" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
real_cond_exp_intA: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "integrable M f"   and "A \<in> sets G" shows "set_lebesgue_integral M A f = set_lebesgue_integral M A (real_cond_exp M G f)"
sc_sim: fixes \<Gamma> :: "'b formula list"   and A :: "'b list"   and \<Delta> :: "'b formula list"   and B :: "'b list" assumes "sc \<Gamma> A \<Delta> B = {}" shows "image_mset Atom (mset A) + mset \<Gamma> \<Rightarrow> image_mset Atom (mset B) + mset \<Delta>"
ac_comm: shows "\<langle>a\<rangle> \<rhd> \<langle>x\<rangle> = \<langle>x\<rangle> \<rhd> \<langle>a\<rangle>"
zero_le_hmset: fixes M :: "hmultiset" shows "0 \<le> M"
add: fixes f :: "complex \<Rightarrow> real"   and g :: "complex \<Rightarrow> real" assumes "f \<in> cx_poly"   and "g \<in> cx_poly" shows "(\<lambda>x. f x + g x) \<in> cx_poly"
inf_absorb2: fixes y :: "'b multiset"   and x :: "'b multiset" assumes "y \<subseteq># x" shows "x \<inter># y = y"
v: shows "\<phi> ` {0..1} \<subseteq> {0..1}"
minus_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier (UP INTEG)"   and "y \<in> carrier (UP INTEG)" shows "x \<ominus>\<^bsub>UP INTEG\<^esub> y \<in> carrier (UP INTEG)"
adjacent_to_adjacent_int_subset: fixes C :: "'c set"   and D :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "C \<sim> D"   and "f ` C \<sim> f ` D"   and "f ` C \<noteq> f ` D" shows "order.greater_eq (f ` (C \<inter> D)) (f ` C \<inter> f ` D)"
lset_inf_llist: fixes f :: "nat \<Rightarrow> 'a" shows "lset (inf_llist f) = range f"
eucl_ordinal_utility_def: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real" shows "eucl_ordinal_utility carrier relation u \<equiv> ordinal_utility carrier relation u"
le_numeral_extra_3: shows "(0::'a) \<le> (0::'a)"
insert_remove: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
zeta_partial_sum_le': fixes x :: "real"   and m :: "nat" assumes "0 < x"   and "0 < m" shows "(\<Sum>n = 1..m. real n powr (x - 1)) \<le> real m powr x * (1 / x + 1 / real m)"
enum_ex_ChildAct_def: fixes P :: "ChildAct \<Rightarrow> bool" shows "enum_class.enum_ex P = (P SayIKnow \<or> P SayNothing)"
col_def: fixes A :: "'c mat"   and j :: "nat" shows "col A j = vec (dim_row A) (\<lambda>i. A $$ (i, j))"
split_sel_asm: fixes P :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<not> (prod = (fst prod, snd prod) \<and> \<not> P (f (fst prod) (snd prod))))"
monic_degree_map_poly_hom: fixes p :: "'a poly" assumes "monic p" shows "degree (map_poly hom p) = degree p"
distinct_5: fixes x2 :: "nat"   and x3 :: "nat" shows "Card x2 \<noteq> User x3"
addmod32len: fixes a :: "bit list"   and b :: "bit list" shows "length (addmod32 a b) = 32"
restrict_eq: fixes A :: "'c set" shows "AList.restrict A = filter (\<lambda>(k, v). k \<in> A)"
typ_of_Abs_body_typ: fixes Ts :: "typ list"   and T :: "typ"   and t :: "Core.term"   and ty :: "typ" assumes "typ_of1 Ts (Abs T t) = Some ty" shows "\<exists>rty. ty = T \<rightarrow> rty"
fls_eq0_below_subdegree: fixes n :: "int"   and f :: "'b fls" assumes "n < fls_subdegree f" shows "fls_nth f n = (0::'b)"
bidel_arc_diff_face_axioms: shows "bidel_arc_diff_face G M a"
simps_2: fixes n :: "nat"   and m :: "nat" shows "s'_A n (Suc m) = swap (paid_A ! n ! (length (paid_A ! n) - Suc m)) (s'_A n m)"
1: shows "{s. equal_mod_timpls TI t s} \<subseteq> {s. timpls_transformable_to_pred (the_Abs ` {f \<in> funs_term t. is_Abs f} \<union> fst ` (set TI)\<^sup>+ \<union> snd ` (set TI)\<^sup>+) t s}"
ennreal_divide_zero: fixes x :: "ennreal" shows "x / 0 = (if x = 0 then 0 else \<top>)"
assms_8: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and D :: "nat"   and C :: "nat" assumes "CT;\<Gamma> \<turnstile> e0 : D"   and "P CT \<Gamma> e0 D"   and "CT \<turnstile> C \<not><: D"   and "CT \<turnstile> D \<not><: C" shows "P CT \<Gamma> (Cast C e0) C"
case_3: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and x3 :: "'c" shows "(case Fault x3 of Normal x \<Rightarrow> f1 x | Abrupt x \<Rightarrow> f2 x | Fault x \<Rightarrow> f3 x | Stuck \<Rightarrow> f4) = f3 x3"
consistency_on_irc_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "f_range_on A f"   and "consistency_on A f" shows "irc_on A f"
rank_invar_rank_skew: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_invar q" shows "SkewBinomialHeapStruc.rank_skew_invar q"
assms_2: fixes u :: "nat \<Rightarrow> 'a"   and b :: "'a" assumes "\<forall>n. u n \<in> S"   and "b \<in> T"   and "u \<longlonglongrightarrow> b" shows "convergent (\<lambda>n. f (u n))"
abs_summable_product: fixes x :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and y :: "'c \<Rightarrow> 'd" assumes "(\<lambda>i. x i * x i) abs_summable_on A"   and "(\<lambda>i. y i * y i) abs_summable_on A" shows "(\<lambda>i. x i * y i) abs_summable_on A"
setinv_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and K :: "'c set" assumes "H \<lhd> G"   and "K \<in> rcosets\<^bsub>G\<^esub> H" shows "set_inv\<^bsub>G\<^esub> K \<in> rcosets\<^bsub>G\<^esub> H"
cases: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and a :: "'b stream" assumes "alw \<phi> a" obtains xs :: "'b stream" where "a = xs" and "\<phi> xs" and "alw \<phi> (stl xs)"
A_def: shows "A \<equiv> \<lambda>m n. {x \<in> space M. dist (f x) (e n) < 1 / real (Suc m) \<and> 1 / real (Suc m) \<le> dist (f x) z}"
Rep_af_cases: fixes y :: "'d" assumes "y \<in> UNIV" obtains x :: "'d af" where "y = ??.Abstract_AF.af.Rep_af x"
IH_1: shows "(\<exists>a\<in>set (dnf \<phi>1). case a of (al, fl) \<Rightarrow> (\<forall>a\<in>set al. aEval a xs) \<and> (\<forall>f\<in>set fl. eval f xs)) = eval \<phi>1 xs"
vinsert_iff: fixes x :: "V"   and y :: "V"   and A :: "V" shows "(x \<in>\<^sub>\<circ> vinsert y A) = (x = y \<or> x \<in>\<^sub>\<circ> A)"
Nonce_notin_initState: fixes N :: "nat"   and B :: "agent" shows "Nonce N \<notin> parts (initState B)"
IH: shows "map f (sorted_list_of_set A) = sorted_list_of_set (f ` A)"
top_def: shows "\<top> = \<Squnion>UNIV"
RModule_inner_dirsum_el_decomp_Rsmult: fixes Ns :: "'m set list"   and r :: "'r"   and x :: "'m" assumes "\<forall>N\<in>set Ns. RSubmodule N"   and "add_independentS Ns"   and "r \<in> R"   and "x \<in> (\<Oplus>N\<leftarrow>Ns. N)" shows "\<Oplus>Ns\<leftarrow> (r \<cdot> x) = map ((\<cdot>) r) (\<Oplus>Ns\<leftarrow> x)"
that: fixes k :: "'a" assumes "f k < g k"   and "\<And>k'. k' < k \<Longrightarrow> f k' = g k'" shows "thesis"
Festino: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A E B"   and "A I C" shows "B Z C"
Ch_singular': fixes h :: "'c"   and X :: "'a set" shows "inj_on Xd (Ch h X)"
exp_half_le2: shows "exp (1 / 2) \<le> 2"
oalist_inv_raw_update_by_fun_pair: fixes xs :: "('a \<times> 'b) list"   and k :: "'a"   and f :: "'b \<Rightarrow> 'b" assumes "tc.oalist_inv_raw xs" shows "tc.oalist_inv_raw (update_by_fun_pair_tc k f xs)"
simps_9: shows "eval_ternary_And TernaryUnknown TernaryUnknown = TernaryUnknown"
trans_WhileTrue: fixes tst :: "'test"   and s :: "'state"   and c :: "('test, 'atom) com" assumes "tval tst s" shows "(while tst do c, s) \<rightarrow>c (c ;; while tst do c, s)"
canonical_basis_length_ell2: shows "length canonical_basis = CARD('a)"
limitin_sequentially: fixes X :: "'b topology"   and S :: "nat \<Rightarrow> 'b"   and l :: "'b" shows "limitin X S l sequentially = (l \<in> topspace X \<and> (\<forall>U. openin X U \<and> l \<in> U \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. S n \<in> U)))"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (save_retore_sub1 result new_cwp rd s1)) \<and> t2 = snd (fst (save_retore_sub1 result new_cwp rd s2))"
bdd_below_Int2: fixes B :: "'b multiset set"   and A :: "'b multiset set" assumes "subset_mset.bdd_below B" shows "subset_mset.bdd_below (A \<inter> B)"
obtain_list_from_setrel: fixes A :: "('a \<times> 'b) set"   and l :: "'a list"   and s :: "'b set" assumes "single_valued A"   and "(set l, s) \<in> \<langle>A\<rangle>set_rel" obtains m :: "'b list" where "s = set m" and "(l, m) \<in> \<langle>A\<rangle>list_rel"
mat_minus_limit: fixes A :: "complex mat"   and m :: "nat"   and X :: "nat \<Rightarrow> complex mat"   and B :: "complex mat" assumes "A \<in> carrier_mat m m"   and "limit_mat X A m" shows "limit_mat (minus_mat_seq B X) (B - A) m"
sg1: shows "tiTable_SampleT req a1 stop lose (fin_inf_append [init_state] st) (fin_inf_append [[]] buffer) ack i1 vc st"
simps: fixes \<I> :: "('b, 'd) \<I>"   and callee :: "'b \<Rightarrow> ('d \<times> 'e) spmf" shows "\<I> \<turnstile>c callee \<surd> = (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> set_spmf (callee x) \<longrightarrow> xa \<in> responses_\<I> \<I> x)"
finite_indets: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "finite (indets p)"
cop_out__osp: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and X :: "'p"   and P :: "'p"   and Y :: "'p" assumes "\<not> Coplanar A B C X"   and "Coplanar A B C P"   and "P Out X Y" shows "A B C OSP X Y"
cases: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a list" assumes "class.linorder less_eq less" obtains "xa__ = []"   | x :: "'a" and ys :: "'a list" where "xa__ = x # ys"
orientation_irreflexive_inf: fixes y :: "'a"   and x :: "'a" assumes "orientation x y" shows "irreflexive_inf x"
pderiv_cfs: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "g \<in> carrier (UP R)" shows "pderiv g n = [Suc n] \<cdot> g (Suc n)"
assms_4: shows "z \<noteq> y \<or> zs \<noteq> ys"
inv_hcomp: fixes \<nu> :: "'b"   and \<mu> :: "'b" assumes "local.iso \<nu>"   and "local.iso \<mu>"   and "src \<nu> = trg \<mu>" shows "local.inv (\<nu> \<star> \<mu>) = local.inv \<nu> \<star> local.inv \<mu>"
step_r_sound: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and \<R> :: "('e \<Rightarrow> real) set set"   and l :: "'d"   and R :: "('e \<Rightarrow> real) set"   and l' :: "'d"   and R' :: "('e \<Rightarrow> real) set"   and X :: "'e set"   and k :: "'e \<Rightarrow> nat" assumes "A,\<R> \<turnstile> \<langle>l, R\<rangle> \<leadsto> \<langle>l', R'\<rangle>"   and "\<R> = {region X I r |I r. valid_region X k I r}"   and "R' \<noteq> {}" shows "\<forall>u\<in>R. \<exists>u'\<in>R'. A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
\<C>: fixes U :: "'a set" assumes "U \<in> \<C>" shows "openin X U"
power_mono: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less_eq a b"   and "less_eq zero a" shows "less_eq (power.power one times a n) (power.power one times b n)"
inner_add_right: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (plus y z) = inner x y + inner x z"
iso_img_block_orig_exists: fixes x :: "'b set" assumes "x \<in># \<B>'" shows "\<exists>bl. bl \<in># \<B> \<and> x = \<pi> ` bl"
simps_3: fixes f1 :: "'a"   and f2 :: "'a" shows "(case left of left \<Rightarrow> f1 | right \<Rightarrow> f2) = f1"
5: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> x"
semiring_norm_83: fixes n :: "num" shows "(num.One = num.Bit0 n) = False"
FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m_refines_FORK: fixes N :: "nat"   and i :: "nat" assumes "DiningPhilosophers N" shows "DiningPhilosophers.FORK N i \<sqsubseteq> DiningPhilosophers.FORK\<^sub>n\<^sub>o\<^sub>r\<^sub>m N i 0"
homogeneous_setD: fixes A :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and a :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" assumes "homogeneous_set A"   and "a \<in> A" shows "hom_component a n \<in> A"
in_hhom_def: fixes \<mu> :: "'d"   and a :: "'d"   and b :: "'d" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright> \<equiv> D.arr \<mu> \<and> src\<^sub>D \<mu> = a \<and> trg\<^sub>D \<mu> = b"
proots_fcompose_bij_eq: fixes q1 :: "'b poly"   and q2 :: "'b poly"   and A :: "'b set"   and B :: "'b set"   and p :: "'b poly" assumes "bij_betw (\<lambda>x. poly q1 x / poly q2 x) A B"   and "p \<noteq> 0"   and "\<forall>x\<in>A. poly q2 x \<noteq> (0::'b)"   and "max (degree q1) (degree q2) \<le> 1"   and "\<forall>c. q1 \<noteq> smult c q2"   and "infinite UNIV" shows "proots_count p B = proots_count (fcompose p q1 q2) A"
dirichlet_prod_fds_nth_fds_left: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "($) (fds f) \<star> g = f \<star> g"
new_Marker_in_set_implies_nonregular_occurence: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and i :: "nat"   and p :: "'p"   and q :: "'p" assumes "c \<turnstile> ev \<mapsto> c'"   and "Marker \<notin> set (msgs c i)"   and "Marker \<in> set (msgs c' i)"   and "channel i = Some (p, q)" shows "\<not> regular_event ev"
proj2_incident_def: fixes p :: "proj2"   and l :: "proj2_line" shows "proj2_incident p l \<equiv> proj2_rep p \<bullet> proj2_line_rep l = 0"
card_cartesian_product_singleton: fixes x :: "'c"   and A :: "'d set" shows "card ({x} \<times> A) = card A"
size_list_simp_2: fixes xs :: "'c list" shows "length xs = (if xs = [] then 0 else Suc (length (tl xs)))"
fid_on_vintersection: fixes A :: "V"   and B :: "V" shows "fid_on (A \<inter>\<^sub>\<circ> B) = fid_on A \<inter>\<^sub>\<circ> fid_on B"
ArityAnalysisSafe_def: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "ArityAnalysisSafe Aexp \<equiv> SubstArityAnalysis Aexp \<and> ArityAnalysisSafe_axioms Aexp"
source_code_def: fixes M :: "'c measure"   and b :: "real"   and fi :: "'d \<Rightarrow> real"   and X :: "'c \<Rightarrow> 'd"   and enc :: "'d list \<Rightarrow> bool list"   and dec :: "bool list \<Rightarrow> 'd list option" shows "source_code M b fi X enc dec \<equiv> information_space M b \<and> source_code_axioms M b fi X enc dec"
keys_monomial_pmdl: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "is_monomial_set F"   and "p \<in> local.punit.pmdl F"   and "t \<in> keys p" obtains f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" where "f \<in> F" and "f \<noteq> 0" and "local.punit.adds_term (lpp f) t"
assms: shows "aEvalUni (EqUni (a, b, c)) x \<noteq> aEvalUni (EqUni (a, b, c)) y \<and> x < y"
Oadmit_Prod_simps: fixes \<sigma> :: "('a, 'b, 'c) subst"   and ODE1 :: "('a, 'c) ODE"   and ODE2 :: "('a, 'c) ODE"   and U :: "('c + 'c) set" shows "Oadmit \<sigma> (OProd ODE1 ODE2) U = (Oadmit \<sigma> ODE1 U \<and> Oadmit \<sigma> ODE2 U \<and> ODE_dom (Osubst ODE1 \<sigma>) \<inter> ODE_dom (Osubst ODE2 \<sigma>) = {})"
distinct_1: fixes x1 :: "agent"   and x2 :: "agent" shows "sharK x1 \<noteq> publK x2"
map_endpow_Suc: fixes n :: "nat" shows "map endpow [0..<Suc n] = id \<down> V # map ((\<circ>) T) (map endpow [0..<n])"
rbt_lookup_rbt_product: fixes leq_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less_a :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and leq_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less_b :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and rbt1 :: "('c, 'e) rbt"   and rbt2 :: "('d, 'f) rbt"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> 'g"   and a :: "'c"   and c :: "'d" assumes "class.linorder leq_a less_a"   and "class.linorder leq_b less_b"   and "ord.is_rbt less_a rbt1"   and "ord.is_rbt less_b rbt2" shows "ord.rbt_lookup (less_prod leq_a less_a less_b) (rbt_product f rbt1 rbt2) (a, c) = (case ord.rbt_lookup less_a rbt1 a of None \<Rightarrow> None | Some b \<Rightarrow> map_option (f a b c) (ord.rbt_lookup less_b rbt2 c))"
nat_pow_eone: fixes x :: "'a" assumes "x \<in> carrier R" shows "[1] \<cdot> x = x"
wf_digraph_wp_iff: fixes G :: "'b pair_pre_digraph" shows "wf_digraph (with_proj G) = pair_wf_digraph G"
exhaust_sel: fixes resource :: "('c, 'd) resource" obtains "resource = Resource (run_resource resource)"
exhaust: fixes y :: "('a, 'b) monoid_ap" obtains x1 :: "'a" and x2 :: "'b" where "y = Monoid_ap x1 x2"
set_disconnected_nodes_get_disconnected_nodes_wf_is_l_set_disconnected_nodes_get_disconnected_nodes_wf: shows "l_set_disconnected_nodes_get_disconnected_nodes_wf DocumentClass.type_wf DocumentClass.known_ptr heap_is_wellformed parent_child_rel get_child_nodes get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs"
fold_invalid_means_one_invalid: fixes G :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" assumes "gval (fold gAnd G (Bc True)) s = invalid" shows "\<exists>g\<in>set G. gval g s = invalid"
2: shows "conjunct (pred c') \<sigma>\<^sub>s\<^sub>y\<^sub>m"
fun_lub_apply: fixes Sup :: "'c set \<Rightarrow> 'b"   and Y :: "('d \<Rightarrow> 'c) set"   and x :: "'d" shows "fun_lub Sup Y x = Sup ((\<lambda>f. f x) ` Y)"
word_reverse_def: fixes w :: "'b word" shows "word_reverse w = horner_sum of_bool 2 (rev (map (bit w) [0..<LENGTH('b)]))"
simps_7: fixes x :: "name"   and P :: "('a, 'b, 'c) psi" shows "extractFrame (\<lparr>\<nu>x\<rparr>P) = \<lparr>\<nu>x\<rparr>extractFrame P"
last_append: fixes xs :: "'a list"   and ys :: "'a list" shows "last (xs @ ys) = (if ys = [] then last xs else last ys)"
measurable_compose_countable': fixes I :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'c \<Rightarrow> 'b" assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> M \<rightarrow>\<^sub>M count_space I"   and "countable I" shows "(\<lambda>x. f (g x) x) \<in> M \<rightarrow>\<^sub>M N"
cases: fixes a1 :: "lam"   and a2 :: "lam" assumes " a1 \<longrightarrow>\<^sub>\<beta>\<^sup>* a2" obtains M :: "lam" where "a1 = M" and "a2 = M"   | M1 :: "lam" and M2 :: "lam" and M3 :: "lam" where "a1 = M1" and "a2 = M3" and " M1 \<longrightarrow>\<^sub>\<beta>\<^sup>* M2" and " M2 \<longrightarrow>\<^sub>\<beta> M3"
1: assumes "\<not> k < j" shows "(\<exists>n\<ge>j + 1. n \<le> k \<and> is_some (f n)) = is_some (Quickcheck_Exhaustive.exhaustive_natural' f k (j + 1))"
cfun_fun_cong: fixes f :: "'b \<rightarrow> 'c"   and g :: "'b \<rightarrow> 'c"   and x :: "'b" assumes "f = g" shows "f\<cdot>x = g\<cdot>x"
lt_proj_poly_syz: fixes n :: "nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_pot_ord"   and "proj_poly_syz n p \<noteq> 0" shows "lt (proj_poly_syz n p) = map_component (\<lambda>k. k - n) (lt p)"
d_gso_Ints: fixes i :: "nat"   and k :: "nat" assumes "i < n"   and "k < length (map of_int_hom.vec_hom fs)" shows "(fs.gs.d k \<cdot>\<^sub>v fs.gs.gso k) $ i \<in> \<int>"
smult_pCons: fixes a :: "'d"   and b :: "'d"   and p :: "'d poly" shows "smult a (pCons b p) = pCons (a * b) (smult a p)"
swap_cs_Trans_RecvMarker: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and d :: "('p, 's, 'm) configuration"   and ev' :: "('p, 's, 'm) event"   and e :: "('p, 's, 'm) configuration"   and d' :: "('p, 's, 'm) configuration"   and e' :: "('p, 's, 'm) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
simps_1: fixes c :: "int" shows "nozerocoeff (C c) = True"
PS\<^sub>L_def: shows "PS\<^sub>L = {p \<in> ps\<^sub>L. dist (case case p of (x, y) \<Rightarrow> (real_of_int x, y) of (x, y) \<Rightarrow> (x, real_of_int y)) (case (l, snd p) of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < dist (case C\<^sub>0' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case C\<^sub>1' of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))}"
f_aggregate_one: fixes k :: "nat"   and xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "0 < k"   and "length xs = k" shows "f_aggregate xs k ag = [ag xs]"
openin_subtopology_empty: fixes U :: "'b topology"   and S :: "'b set" shows "openin (subtopology U {}) S = (S = {})"
IH: shows "P \<parallel> Q \<Longrightarrow>\<^sub>\<tau> P'' \<parallel> Q"
length_slp_of_fas_le: fixes fas :: "floatarith list" shows "length fas \<le> length (slp_of_fas fas)"
minim_UnderS_underS: fixes A :: "'a set" assumes "A \<noteq> {}"   and "A \<subseteq> Field r" shows "local.UnderS A = local.underS (minim A)"
OAlist_tc_min_key_val_def: fixes xs :: "('a, 'b) oalist_tc" shows "OAlist_tc_min_key_val xs = hd (list_of_oalist_tc xs)"
emeasure_not_in_N: fixes x1 :: "real" assumes "x1 \<notin> N" shows "emeasure (lborel_f n) {x \<in> space (lborel_f n). x(n := x1) \<in> A} = 0"
uniqueness_of_ctxt': fixes \<Gamma> :: "binding list"   and x :: "vrs"   and T :: "ty"   and S :: "ty" assumes "\<turnstile> \<Gamma> ok"   and "VarB x T \<in> set \<Gamma>"   and "VarB x S \<in> set \<Gamma>" shows "T = S"
scaleRat_scaleRat: fixes a :: "rat"   and b :: "rat"   and x :: "'a" shows "a *R b *R x = (a * b) *R x"
prod_prime_factors: fixes n :: "nat" assumes "0 < n" shows "divisor_count n = (\<Prod>p\<in>prime_factors n. divisor_count (p ^ multiplicity p n))"