hcju/isabelleps huggingface dataset:
bf_less_sets: fixes M :: "nat set"   and q :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and ij :: "nat \<times> nat" assumes "infinite M" shows "fst (bf M q ij) \<lless> snd (bf M q ij)"
Obtuse_def: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Tarski_neutral_dimensionless.Obtuse Bet Cong A B C \<equiv> \<exists>A' B' C'. Tarski_neutral_dimensionless.Per Bet Cong A' B' C' \<and> Tarski_neutral_dimensionless.LtA Bet Cong A' B' C' A B C"
well_def_wlp_nearly_healthy: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "well_def prog" shows "nearly_healthy (wlp prog)"
assms_3: shows "lex_ord_raw ko f xs ys = aux"
simps_7: fixes f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'h list \<Rightarrow> 'g \<Rightarrow> 'g"   and x21 :: "'h"   and x22 :: "'h list" shows "rec_list f1 f2 (x21 # x22) = f2 x21 x22 (rec_list f1 f2 x22)"
assms_2: shows "r \<in> range (proj_stoch_proc X n)"
clock_numbering'_1: shows "\<forall>c\<in>X. 0 < v c"
ccpo: shows "class.ccpo \<Or> (\<sqsubseteq>) lt_b"
hitting_paths_def: shows "Q'.hitting_paths \<equiv> \<lambda>x. x \<noteq> v0 \<and> ((\<exists>xs\<in>Q. x \<in> set xs) \<or> x = v1)"
strongly_connected_imp_induce_subgraph_strongly_connected: fixes G :: "('c, 'd) pre_digraph"   and H :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "subgraph H G"   and "strongly_connected H" shows "strongly_connected (G \<restriction> verts H)"
Pow_iff: fixes A :: "'a set"   and B :: "'a set" shows "(A \<in> Pow B) = (A \<subseteq> B)"
singleE: fixes x :: "'a"   and y :: "'a" assumes "pred.eval (Predicate.single x) y" obtains "y = x"
induct: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and x1 :: "Core.term set"   and x2 :: "Core.term"   and P :: "Core.term set \<Rightarrow> Core.term \<Rightarrow> bool" assumes "\<Theta>,x1 \<turnstile> x2"   and "\<And>A \<rho> \<Gamma>. \<lbrakk>wf_theory \<Theta>; A \<in> axioms \<Theta>; wf_inst \<Theta> \<rho>\<rbrakk> \<Longrightarrow> P \<Gamma> (tsubst A \<rho>)"   and "\<And>A \<Gamma>. \<lbrakk>wf_term (sig \<Theta>) A; \<turnstile>\<^sub>\<tau> A : propT; A \<in> \<Gamma>\<rbrakk> \<Longrightarrow> P \<Gamma> A"   and "\<And>\<Gamma> B x \<tau>. \<lbrakk>wf_theory \<Theta>; \<Theta>,\<Gamma> \<turnstile> B; P \<Gamma> B; (x, \<tau>) \<notin> FV \<Gamma>; wf_type (sig \<Theta>) \<tau>\<rbrakk> \<Longrightarrow> P \<Gamma> (mk_all x \<tau> B)"   and "\<And>\<Gamma> \<tau> B a. \<lbrakk>\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> B; P \<Gamma> (Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> B); \<turnstile>\<^sub>\<tau> a : \<tau>; wf_term (sig \<Theta>) a\<rbrakk> \<Longrightarrow> P \<Gamma> (subst_bv a B)"   and "\<And>\<Gamma> B A. \<lbrakk>wf_theory \<Theta>; \<Theta>,\<Gamma> \<turnstile> B; P \<Gamma> B; wf_term (sig \<Theta>) A; \<turnstile>\<^sub>\<tau> A : propT\<rbrakk> \<Longrightarrow> P (\<Gamma> - {A}) (A \<longmapsto> B)"   and "\<And>\<Gamma>\<^sub>1 A B \<Gamma>\<^sub>2. \<lbrakk>\<Theta>,\<Gamma>\<^sub>1 \<turnstile> A \<longmapsto> B; P \<Gamma>\<^sub>1 (A \<longmapsto> B); \<Theta>,\<Gamma>\<^sub>2 \<turnstile> A; P \<Gamma>\<^sub>2 A\<rbrakk> \<Longrightarrow> P (\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2) B"   and "\<And>c T \<Gamma>. \<lbrakk>wf_theory \<Theta>; const_type (sig \<Theta>) (const_of_class c) = Some (itselfT Core.aT \<rightarrow> propT); wf_type (sig \<Theta>) T; has_sort (osig (sig \<Theta>)) T (insert c full_sort)\<rbrakk> \<Longrightarrow> P \<Gamma> (mk_of_class T c)"   and "\<And>T t u \<Gamma>. \<lbrakk>wf_theory \<Theta>; wt_term (sig \<Theta>) (Abs T t); wf_term (sig \<Theta>) u; \<turnstile>\<^sub>\<tau> u : T\<rbrakk> \<Longrightarrow> P \<Gamma> (mk_eq (Abs T t $ u) (subst_bv u t))"   and "\<And>t \<tau> \<tau>' \<Gamma>. \<lbrakk>wf_theory \<Theta>; wf_term (sig \<Theta>) t; \<turnstile>\<^sub>\<tau> t : \<tau> \<rightarrow> \<tau>'\<rbrakk> \<Longrightarrow> P \<Gamma> (mk_eq (Abs \<tau> (t $ Bv 0)) t)" shows "P x1 x2"
pre_uvar_def: fixes x :: "'d \<Longrightarrow> 'e" shows "x\<^sub>< = x ;\<^sub>L mrg_prior"
erlang_ith_moment: fixes l :: "real"   and X :: "'a \<Rightarrow> real"   and k :: "nat"   and i :: "nat" assumes "0 < l"   and "distributed M lborel X (\<lambda>x. ennreal (erlang_density k l x))" shows "expectation (\<lambda>x. X x ^ i) = fact (k + i) / (fact k * l ^ i)"
assms_2: fixes n :: "nat" shows "\<parallel>y (Suc n) - y n\<parallel> \<le> c ^ n"
is_interval_def: fixes s :: "'a set" shows "is_interval s = (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i) \<longrightarrow> x \<in> s)"
trans: fixes b :: "'d list"   and a :: "'d list"   and c :: "'d list" assumes "subseq b a"   and "subseq c b" shows "subseq c a"
flatten_call: fixes init :: "'d \<Rightarrow> 'd"   and p :: "'e"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and result :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" shows "flatten (call init p return result) = [call init p return result]"
un_pred_entailment_mono: fixes P :: "'BExp set"   and P' :: "'BExp set"   and P'' :: "'BExp set"   and P''' :: "'BExp set" assumes "(P \<union> P') \<turnstile> P''"   and "P''' \<turnstile> P" shows "(P''' \<union> P') \<turnstile> P''"
to_hma_from_hma\<^sub>m: fixes v :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "HMA_Connect.to_hma\<^sub>m (HMA_Connect.from_hma\<^sub>m v) = v"
inf_eqvt: fixes p :: "perm"   and X :: "'a set" assumes "class.inf_eqvt TYPE('a)" shows "p \<bullet> Inf X = Inf (p \<bullet> X)"
S2Clerk: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkNext memCh crCh cst p \<and> $S2 rmhist p \<longrightarrow> MClkFwd memCh crCh cst p"
simps_1: fixes l2 :: "'b list" shows "inter_sorted [] l2 = []"
countable: fixes X :: "'a set" assumes "countable X" shows "small X"
mult_le_cancel_left2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (times c a) c = ((less zero c \<longrightarrow> less_eq a one) \<and> (less c zero \<longrightarrow> less_eq one a))"
simps: fixes a :: "Regions.intv" shows "Regions.isIntv a = (\<exists>uu_. a = Regions.intv.Intv uu_)"
hom_sum_list: fixes xs :: "int list" shows "of_int (sum_list xs) = sum_list (map of_int xs)"
distinct_3: fixes x11 :: "bool"   and x12 :: "nat"   and x13 :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and x14 :: "nat"   and x15 :: "nat set"   and x16 :: "nat set"   and x17 :: "nat set"   and x18 :: "nat set"   and x31 :: "'b"   and x32 :: "'b"   and x33 :: "instr list" shows "Write\<^sub>s\<^sub>b x11 x12 x13 x14 x15 x16 x17 x18 \<noteq> Prog\<^sub>s\<^sub>b x31 x32 x33"
simps_6: fixes \<pi> :: "'b prm"   and P :: "'b ptrm" shows "\<pi> \<bullet> PFst P = PFst (\<pi> \<bullet> P)"
split: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "nat \<Rightarrow> 'f"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
subset_fSuc_strict: fixes f :: "nat \<Rightarrow> 'a set"   and u :: "nat"   and v :: "nat" assumes "\<And>u. f u \<subseteq> f (Suc u)"   and "u < v" shows "f u \<subseteq> f v"
A4: shows "((\<sigma>, p'), a, \<sigma>', q) \<in> automaton.trans A"
image2_def: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b" shows "BNF_Greatest_Fixpoint.image2 A f g = {(f a, g a) |a. a \<in> A}"
expand: fixes sum :: "'d + 'e"   and sum' :: "'d + 'e" assumes "isl sum = isl sum'"   and "\<lbrakk>isl sum; isl sum'\<rbrakk> \<Longrightarrow> projl sum = projl sum'"   and "\<lbrakk>\<not> isl sum; \<not> isl sum'\<rbrakk> \<Longrightarrow> projr sum = projr sum'" shows "sum = sum'"
htuple_nonzero: fixes k :: "nat" shows "htuple k \<noteq> 0"
Klim: shows "(z \<circ> K) \<longlonglongrightarrow> y"
point_in_group: fixes G :: "'a set"   and x :: "'a" assumes "G \<in> \<G>"   and "x \<in> G" shows "x \<in> \<V>"
cases_scheme: fixes r :: "'a s_scheme" obtains inst :: "inst list" and funcs :: "cl list" and tab :: "cl option list list" and mem :: "mem list" and globs :: "global list" and more :: "'a" where "r = \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr>"
minus_max_eq_min: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "uminus (ord.max less_eq x y) = ord.min less_eq (uminus x) (uminus y)"
solve_depressed_quartic_real_def: fixes p :: "real"   and q :: "real"   and r :: "real" shows "solve_depressed_quartic_real p q r = remdups (if q = 0 then concat (map (\<lambda>z. rroots2 [:- z, 0, 1:]) (rroots2 [:r, p, 1:])) else let cubics = rroots3 [:- q\<^sup>2, 2 * p\<^sup>2 - 8 * r, 8 * p, 8:]; m = the (find ((<) 0) cubics); a = sqrt (2 * m); p2m = p / 2 + m; q2a = q / (2 * a); b1 = p2m - q2a; b2 = p2m + q2a in rroots2 [:b1, a, 1:] @ rroots2 [:b2, - a, 1:])"
linordered_field_no_ub: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "\<forall>x. \<exists>y. less x y"
lnull_imp_lfinite: fixes xs :: "'f llist" assumes "lnull xs" shows "lfinite xs"
augment_cf_def: fixes edges :: "(nat \<times> nat) set"   and cap :: "'capacity" shows "augment_cf edges cap \<equiv> \<lambda>e. if e \<in> edges then c e - cap else if prod.swap e \<in> edges then c e + cap else c e"
better_o4_intro: fixes t1 :: "lam"   and t2 :: "lam"   and s1 :: "lam"   and s2 :: "lam"   and x :: "name" assumes " t1 \<longrightarrow>\<^sub>1 t2"   and " s1 \<longrightarrow>\<^sub>1 s2" shows " App (Lam [x].t1) s1 \<longrightarrow>\<^sub>1 t2[x::=s2]"
assms_4: fixes x :: "real" assumes "x \<in> {a..b} - S" shows "norm (f x) \<le> B"
Inl_inject: fixes x :: "'d"   and y :: "'d" assumes "Inl x = Inl y" shows "x = y"
Frame_deterministic: fixes ops :: "Val list"   and par :: "Var list"   and S :: "(Var \<times> Val) list"   and os :: "Val list"   and R :: "(Var \<times> Val) list"   and opsa :: "Val list" assumes "(ops, par, S, os) \<in> Frame"   and "(ops, par, R, opsa) \<in> Frame" shows "R = S \<and> opsa = os"
slp_indexl_lookup_length: fixes xs :: "'b list"   and i :: "('a, nat) mapping" shows "slp_indexl_lookup (length xs) i = slp_index_lookup xs i"
clauseTrueAppendValuation: fixes clause :: "Literal list"   and valuation :: "Literal list"   and valuation' :: "Literal list" assumes "clauseTrue clause valuation" shows "clauseTrue clause (valuation @ valuation')"
init_result: shows "init users \<Midarrow>bs\<Rightarrow> root"
95: fixes x :: "'a" shows "- x \<squnion> - - - x = - - - x"
a1: shows "s' = snd (fst (div_write_new_val i result temp_V s)) \<and> get_S (cpu_reg_val PSR s) = 0"
distinct_15: fixes x31 :: "com"   and x32 :: "com"   and x41 :: "expr"   and x42 :: "com"   and x43 :: "com" shows "x31;; x32 \<noteq> if (x41) x42 else x43"
Lcm_fin_eq_Lcm: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'b set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "finite A" shows "semiring_gcd.Lcm_fin one zero lcm A = Lcm A"
DC_def: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Sqinter> b \<equiv> \<lambda>ab Q s. min (a ab Q s) (b ab Q s)"
r_lang_def: shows "r_lang \<equiv> map_fun id Abs_reg_lan lang"
R: fixes x :: "'b" assumes "x \<cdot> x = x" shows "x\<^sup>\<star> = (1::'b) + x"
prems: shows "abc_steps_l (as, am) ap (Suc stp) = (bs, bm)"
a: shows "\<phi> .\<longrightarrow>. (\<psi> .\<longrightarrow>. \<chi>) \<in> Cn A \<div> .\<not> \<phi>"
subresultant_prs_impl_def: fixes f :: "'c poly \<times> 'c \<Rightarrow> 'b"   and G1 :: "'c poly"   and G2 :: "'c poly" shows "subresultant_prs_impl f G1 G2 = (let pmod = pseudo_mod G1 G2; n2 = degree G2; delta_1 = degree G1 - n2; g2 = lead_coeff G2; h2 = g2 ^ delta_1 in if pmod = 0 then f (G2, h2) else let G3 = (- 1) ^ (delta_1 + 1) * pmod; g3 = lead_coeff G3; n3 = degree G3; d2 = n2 - n3; pmod = pseudo_mod G2 G3 in if pmod = 0 then f (G3, if d2 = 1 then g3 else dichotomous_Lazard g3 h2 d2) else let divisor = (- (1::'c)) ^ (d2 + 1) * g2 * h2 ^ d2; G4 = sdiv_poly pmod divisor in subresultant_prs_main_impl f G3 G4 n3 d2 h2)"
eq_mrexp3_refl: fixes r :: "'c mrexp3"   and s :: "'c mrexp3" assumes "r = s" shows "eq_mrexp3 r s"
Compl_set_less_aux_Compl: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and B :: "'a set" assumes "finite UNIV" shows "ord.set_less_aux less_eq (- A) (- B) = ord.set_less_aux less_eq B A"
carrier_quotient_ring_iff: fixes X :: "('a \<times> 'a) set" shows "(X \<in> cq.carrier_quotient_ring) = cq.valid_frac X"
sorted_upper2: shows "sorted_sublist_map R h xs''' (Suc p) hi"
proj_list_Cons_3: fixes i :: "'a"   and j :: "'a"   and d :: "'b"   and D :: "('a strand_label \<times> 'b) list" assumes "i \<noteq> j" shows "proj i ((ln j, d) # D) = proj i D"
range_eqI: fixes b :: "'g"   and f :: "'h \<Rightarrow> 'g"   and x :: "'h" assumes "b = f x" shows "b \<in> range f"
finitnpC: fixes C :: "nat set" shows "finite (PR - C)"
nonpos_Ints_diff_Nats: fixes x :: "'a"   and y :: "'a" assumes "x \<in> \<int>\<^sub>\<le>\<^sub>0"   and "y \<in> \<nat>" shows "x - y \<in> \<int>\<^sub>\<le>\<^sub>0"
root_sign_change: fixes p :: "real poly"   and x :: "real" assumes "poly p x = 0"   and "poly (pderiv p) x \<noteq> 0" obtains d :: "real" where "0 < d" and "sgn (poly p (x - d)) \<noteq> sgn (poly p (x + d))" and "sgn (poly p (x - d)) \<noteq> 0" and "0 \<noteq> sgn (poly p (x + d))" and "\<forall>d'>0. d' \<le> d \<longrightarrow> sgn (poly p (x + d')) = sgn (poly p (x + d)) \<and> sgn (poly p (x - d')) = sgn (poly p (x - d))"
A: shows "\<forall>F\<in>s1 - {F} \<union> (s2 - {\<^bold>\<not> F}). A \<Turnstile> F"
inverse_rel_def: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" shows "cinv\<^bsub>C\<^esub> f g \<equiv> f \<approx>>\<^bsub>C\<^esub> g \<and> f ;;\<^bsub>C\<^esub> g = id\<^bsub>C\<^esub> (dom\<^bsub>C\<^esub> f) \<and> g ;;\<^bsub>C\<^esub> f = id\<^bsub>C\<^esub> (cod\<^bsub>C\<^esub> f)"
locks_conv: fixes ls :: "'g"   and tsmws :: "('h \<times> 'i) \<times> 'j \<times> 'k" shows "locks (ls, tsmws) = ls"
Transaction'_inverse: fixes x :: "(common_metadata \<times> participant_metadata) \<times> view list" shows "transaction_merkle.the_Transaction (transaction_merkle.Transaction' x) = x"
lemma_conf: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and Tk :: "nat"   and K :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> bankerberos"   and "Says Server A (Crypt (shrK A) \<lbrace>Number Tk, Agent B, Key K, Crypt (shrK B) \<lbrace>Number Tk, Agent A, Key K\<rbrace>\<rbrace>) \<in> set evs"   and "Key K \<in> analz (knows Spy evs)" shows "Notes Spy \<lbrace>Number Tk, Key K\<rbrace> \<in> set evs"
get_IFS_def: fixes M :: "'a SecurityInvariant_configured list" shows "get_IFS M \<equiv> filter c_isIFS M"
eq: shows "component_of_term ` Keys (fst ` set (gs @ ab gs bs hs data')) = component_of_term ` Keys (fst ` set (gs @ bs))"
floor_mono: fixes x :: "'d"   and y :: "'d" assumes "x \<le> y" shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
natfun_bigo_1E: fixes f :: "nat \<Rightarrow> 'a"   and lb :: "real" assumes "f \<in> O(\<lambda>_. 1::'a)" obtains C :: "real" where "lb \<le> C" and "\<And>n. norm (f n) \<le> C"
length_negGorbit_list: fixes gs :: "'g list"   and T :: "'b \<Rightarrow> 'v"   and as :: "'b list" shows "length (negGorbit_list gs T as) = length gs"
h1: shows "affine hull (+) a ` S \<subseteq> (+) a ` (affine hull S)"
bounded_byI: fixes P :: "'c \<Rightarrow> real"   and b :: "real" assumes "\<And>x. P x \<le> b" shows "bounded_by b P"
F2set2_IF2set: fixes a :: "'d IF1"   and x :: "('d, 'd IF1, 'd IF2) F2" assumes "a \<in> F2set2 x" shows "IF1set a \<subseteq> IF2set (ctor2 x)"
prime_imp_nonzero: fixes x :: "'a" assumes "ASSUMPTION (prime x)" shows "x \<noteq> (0::'a)"
mu2_def: shows "mu2 = monomial ((1::'b) / lc g2) (lcs (lp g1) (lp g2) - lp g2)"
psubst_subst_compose: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "good X"   and "good Y"   and "goodEnv rho" shows "X #[Y / y]_ys #[rho] = X #[rho [y \<leftarrow> Y #[rho]]_ys]"
sum_conv: fixes x :: "'a" shows "sum_class.sum (x\<^sup>T) = sum_class.sum x"
filter_oseq_idemp: fixes p :: "'e \<Rightarrow> bool"   and s :: "nat \<Rightarrow> 'e option" shows "filter_oseq p (filter_oseq p s) = filter_oseq p s"
vpairs_vsingleton: fixes a :: "V"   and b :: "V" shows "vpairs (set {\<langle>a, b\<rangle>}) = set {\<langle>a, b\<rangle>}"
ExecRandomOK: fixes v :: "char list"   and \<sigma> :: "char list \<Rightarrow> nat option"   and u :: "nat" assumes "v \<in> dom \<sigma>" shows "Random v \<rhd> \<sigma> \<leadsto> Some (\<sigma>(v \<mapsto> u))"
E: shows "index key x n mi ma < Min (offs_set_next (offs ((enum xs index key n mi ma)[index key x n mi ma := Suc (enum xs index key n mi ma ! index key x n mi ma)]) k) (x # xs) index key mi ma i)"
whilei_def: fixes p :: "'a"   and i :: "'a"   and x :: "'a" shows "antidomain_kleene_algebra.whilei antirange_op (\<lambda>x y. y \<cdot> x) star p i x = antidomain_kleene_algebra.while antirange_op (\<lambda>x y. y \<cdot> x) star p x"
sup_commute: fixes x :: "'c multiset"   and y :: "'c multiset" shows "x \<union># y = y \<union># x"
col_mat: fixes j :: "nat"   and nc :: "nat"   and nr :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" assumes "j < nc" shows "col (mat nr nc f) j = vec nr (\<lambda>i. f (i, j))"
domain_fm: fixes n :: "nat"   and x :: "'i \<Rightarrow> 'e" shows "domain (fm n x) = Utn ` J n"
prime_ge_2_nat: fixes p :: "nat" assumes "prime p" shows "2 \<le> p"
N1: fixes N :: "nat" assumes "N1 \<le> N" shows "n + 1 \<le> u N"
compute_mantissa: fixes m :: "int"   and e :: "int" shows "mantissa (Float m e) = (if m = 0 then 0 else if even m then mantissa (normfloat (Float m e)) else m)"
term_of_pair_pair: fixes v :: "'c \<times> 'd" shows "(pprod.pp_of_term v, pprod.component_of_term v) = v"
extremum_strict: fixes a :: "'a list" shows "\<not> strict_border a []"
mod_0_le_imp_diff_div_conv: fixes n :: "nat"   and m :: "nat"   and r :: "nat" assumes "n mod m = 0"   and "0 < r"   and "r \<le> m" shows "(n - r) div m = n div m - Suc 0"
round_robin_reschedule_NoneD: fixes t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   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 "round_robin_reschedule t0 queue n0 s = None"   and "t0 \<in> set queue" shows "set (takeWhile (\<lambda>t'. t' \<noteq> t0) queue) \<inter> active_threads s = {}"
surjective: fixes r :: "('a, 'b, 'c) Axioms_scheme" shows "r = \<lparr>aAxioms = aAxioms r, aSignature = aS\<^bsub>r\<^esub>, \<dots> = Axioms.more r\<rparr>"
weakly_bisimilar_eqvt: fixes P :: "'state"   and Q :: "'state"   and p :: "perm" assumes "P \<approx>\<cdot> Q" shows "(p \<bullet> P) \<approx>\<cdot> (p \<bullet> Q)"
sign_simps_17: fixes aa :: "'a"   and ba :: "'a"   and ca :: "'a" shows "(aa + ba) * ca = aa * ca + ba * ca"
alpha_d_ext_def: fixes ok :: "bool"   and more :: "'b" shows "\<lparr>ok = ok, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons alpha_d_ext_Tuple_Iso ok more"
has_bottomI: fixes z :: "'b"   and P :: "'b set" assumes "z \<in> P"   and "\<And>x. x \<in> P \<Longrightarrow> codomain.greater_eq x z" shows "codomain.has_bottom P"
bigtheta_pow_eq_set_pow: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and n :: "nat" shows "\<Theta>[F](\<lambda>x. f x ^ n) = bigtheta_pow F \<Theta>[F](f) n"
vanishes_diff_inverse: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "\<not> Real.vanishes X"   and "Real.cauchy Y"   and "\<not> Real.vanishes Y"   and "Real.vanishes (\<lambda>n. X n - Y n)" shows "Real.vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
finite_measure_axioms_def: fixes M :: "'a measure" shows "finite_measure_axioms M \<equiv> emeasure M (space M) \<noteq> top"
r_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.r p q = (p - 1) div 2 * ((q - 1) div 2)"
implies_intro: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and A :: "Core.term" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "wf_term (sig \<Theta>) A"   and "\<turnstile>\<^sub>\<tau> A : propT" shows "\<Theta>,\<Gamma> - {A} \<turnstile> A \<longmapsto> B"
inner_Pair_0_2: fixes x :: "'a \<times> 'b"   and a :: "'a" shows "x \<bullet> (a, 0::'b) = fst x \<bullet> a"
tau_inj1: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and P :: "('f \<times> 'h, 'a, 'g) Module_scheme" assumes "zeroring R"   and "R module M"   and "R module N"   and "FM\<^bsub>R\<^esub> P M N" shows "inj_on (tau R M N P) (M \<times>\<^sub>c N)"
dropWhile_idem: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "dropWhile P (dropWhile P xs) = dropWhile P xs"
star_inductr_var: fixes z :: "'a"   and y :: "'a"   and x :: "'a" assumes "z + y \<cdot> x \<le> y" shows "z \<cdot> x\<^sup>\<star> \<le> y"
f_join_iIN: fixes xs :: "'b list"   and n :: "nat"   and d :: "nat" shows "xs \<Join>\<^sub> [n\<dots>,d] = xs \<up> n \<down> Suc d"