hcju/isabelleps huggingface dataset:
cases: fixes a1 :: "'c llist"   and a2 :: "'c llist" assumes "a1 \<sqsubseteq> a2" obtains xs :: "'c llist" where "a1 = LNil" and "a2 = xs__"   | xs :: "'c llist" and ys :: "'c llist" and x :: "'c" where "a1 = LCons x xs" and "a2 = LCons x ys" and "xs \<sqsubseteq> ys"
scaleR_norm_sgn_eq: fixes x :: "'d" shows "(\<parallel>x\<parallel>) *\<^sub>R sgn x = x"
pseudo_proj_True_snth': fixes n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "\<And>i. Suc i \<le> n \<Longrightarrow> w !! i = x !! i" shows "pseudo_proj_True n w = pseudo_proj_True n x"
comp_fun_commute_axioms: shows "comp_fun_commute insort"
sinh_ln_complex: fixes x :: "complex" assumes "x \<noteq> 0" shows "sinh (Ln x) = (x - inverse x) / 2"
stutter_equiv_trans: fixes \<rho> :: "nat \<Rightarrow> 'b"   and \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "\<rho> \<approx> \<sigma>"   and "\<sigma> \<approx> \<tau>" shows "\<rho> \<approx> \<tau>"
bin_nth_uint_imp: fixes w :: "'b word"   and n :: "nat" assumes "bit (uint w) n" shows "n < LENGTH('b)"
PrimReplace_def: fixes A :: "hf"   and R :: "hf \<Rightarrow> hf \<Rightarrow> bool" shows "PrimReplace A R = (THE z. \<forall>v. (v \<^bold>\<in> z) = (\<exists>u. u \<^bold>\<in> A \<and> R u v))"
STAR_sqrt_zero: shows "(*f* sqrt) 0 = 0"
times_MonoTran_def: shows "(*) \<equiv> map_fun Rep_MonoTran (map_fun Rep_MonoTran Abs_MonoTran) (\<circ>)"
subset_singleton_iff: fixes X :: "'b set"   and a :: "'b" shows "order.greater_eq {a} X = (X = {} \<or> X = {a})"
minus_divide_divide: fixes a :: "'a"   and b :: "'a" shows "- a / - b = a / b"
assms_7: shows "d' \<turnstile> ev \<mapsto> e'"
priority_fst: fixes xp :: "'c \<times> 'd"   and q :: "('c, 'd) pq" assumes "xp \<in> set (pq.alist_of q)" shows "PQ.priority q (fst xp) = Some (snd xp)"
inf1I: fixes A :: "'b \<Rightarrow> bool"   and x :: "'b"   and B :: "'b \<Rightarrow> bool" assumes "A x"   and "B x" shows "inf A B x"
assms_2: fixes I :: "(ZF, ZF, ZF, ZF) Interpretation" assumes "Model I T" shows "L\<lbrakk>\<phi>\<rbrakk>\<^bsub>I\<^esub> \<rightarrow> IBool True"
bij_betw_inv_into_right: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A' :: "'d set"   and a' :: "'d" assumes "bij_betw f A A'"   and "a' \<in> A'" shows "f (inv_into A f a') = a'"
rm_off_valid: assumes "\<not> c_sinvar m \<lparr>nodes = V, edges = E\<rparr>" shows "c_sinvar m \<lparr>nodes = V, edges = E - set (minimalize_offending_overapprox (c_sinvar m) Es [] \<lparr>nodes = V, edges = E\<rparr>)\<rparr>"
keys_PP_oalist: fixes xs :: "('b, 'c) oalist_tc" shows "keys_pp (PP_oalist xs) = set (OAlist_tc_sorted_domain xs)"
rank_bound: fixes A :: "((real, 'a) vec, 'b) vec" shows "rank A \<le> min CARD('b) CARD('a)"
disc_eq_case_1: fixes option :: "'a option" shows "(option = None) = (case option of None \<Rightarrow> True | Some x \<Rightarrow> bf_False x)"
steps_hext_incr: fixes es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>" shows "hp s \<unlhd> hp s'"
ST: fixes C :: "'a set" assumes "closedin (top_of_set S) C" shows "closedin (top_of_set T) (f ` C)"
Diff: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a - b \<in> fmeasurable M"
P: shows "pre @ [Broadcast (r, Expunge e i r)] prefix of k"
butlast_eval1: fixes x :: "'a" shows "butlast [x] = []"
AE_uniform_measureI: fixes A :: "'a set"   and M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "A \<in> sets M"   and "set_almost_everywhere A M P" shows "almost_everywhere (uniform_measure M A) P"
simps_3: fixes x2 :: "'g" shows "Some x2 \<noteq> None"
collapse: fixes list :: "'b list" assumes "list \<noteq> []" shows "hd list # tl list = list"
slp_of_fa_cnst_def: fixes Const :: "'a"   and Const' :: "floatarith"   and M :: "('a, nat) mapping"   and vs :: "floatarith list" shows "slp_of_fa_cnst Const Const' M vs = (Mapping.update Const (length vs) M, vs @ [case Mapping.lookup M Const of None \<Rightarrow> Const' | Some i \<Rightarrow> Var (slp_index vs i)])"
single_of_nat: fixes n :: "nat" shows "single (0::'b) (of_nat n) = of_nat n"
ENR_homeomorphic_ENR: fixes T :: "'a set"   and S :: "'b set" assumes "ENR T"   and "S homeomorphic T" shows "ENR S"
order_iff: fixes b :: "'f multiset"   and a :: "'f multiset" shows "(b \<subseteq># a) = (a = a \<union># b)"
sup_bot_left: fixes x :: "'a" shows "0 + x = x"
if_sum: fixes B :: "bool"   and E :: "real"   and T :: "real" shows "(if B then 0 else E) + (if B then T else 0) = (if B then T else E)"
nat_mult_closed: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp" shows "[n] \<cdot> x \<in> carrier Zp"
P2: shows "Bet X P' X' \<and> Cong P' X' Q A"
set_mode_is_l_set_mode: shows "l_set_mode ShadowRootClass.type_wf set_mode set_mode_locs"
assms_1: fixes i :: "nat" assumes "i \<le> k" shows "foldl1 f1 (map (h1 k i) js) = One"
strip_list_comb_const: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and name :: "name"   and xs :: "'a list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (term.list_comb app (const name) xs) = (const name, xs)"
two_t_designs_eq_points_axioms: shows "two_t_designs_eq_points \<V> \<B> \<k> \<B>' \<t> \<Lambda>\<^sub>t \<Lambda>\<^sub>t'"
simpleDefs_phiDefs_disjoint: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "defs g n \<inter> phiDefs g n = {}"
subgroup_def: fixes H :: "'b set"   and G :: "('b, 'c) monoid_scheme" shows "subgroup H G \<equiv> (H \<subseteq> carrier G \<and> (\<forall>x y. x \<in> H \<longrightarrow> y \<in> H \<longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> H)) \<and> \<one>\<^bsub>G\<^esub> \<in> H \<and> (\<forall>x. x \<in> H \<longrightarrow> inv\<^bsub>G\<^esub> x \<in> H)"
lg_safe_3: shows "lg (Suc (Suc 0)) = 1"
row_add_1: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'a mat"   and i :: "nat" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "i < nr" shows "row (A + B) i = row A i + row B i"
cong_ex_bigtheta: fixes f1 :: "real \<Rightarrow> real"   and F :: "real filter"   and f2 :: "real \<Rightarrow> real"   and g1 :: "real \<Rightarrow> real"   and g2 :: "real \<Rightarrow> real" assumes "f1 \<in> \<Theta>[F](f2)"   and "g1 \<in> \<Theta>[F](g2)" shows "(f1 \<in> L F g1) = (f2 \<in> L F g2)"
below_trans: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.po below"   and "below x y"   and "below y z" shows "below x z"
ide_def: fixes a :: "'a" shows "Left.ide a \<equiv> Left.comp a a \<noteq> Left.null \<and> (\<forall>f. (Left.comp f a \<noteq> Left.null \<longrightarrow> Left.comp f a = f) \<and> (Left.comp a f \<noteq> Left.null \<longrightarrow> Left.comp a f = f))"
rep_eq: fixes x :: "'b prefix" shows "plen x = length (Rep_prefix x)"
simps_17: fixes va :: "('b, 'c, 'd) Tree set['b]"   and v :: "'c" shows "(tConj va =\<^sub>\<alpha> tPred v) = False"
"1_4": shows "(poincare_map p1 has_derivative blinfun_apply D1) (at x0 within S)"
code: shows "1 = Octo 1 0 0 0 0 0 0 0"
rep_eq: fixes x :: "'a"   and xa :: "'b" shows "pp.mapping_of (single_pp x xa) = Poly_Mapping.single x xa"
direct_decomp_ideal_insert: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and f :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "F \<subseteq> P[X]"   and "f \<in> P[X]" shows "direct_decomp (ideal (insert f F) \<inter> P[X]) [ideal F \<inter> P[X], (*) f ` normal_form ((ideal F \<div> f) \<inter> P[X]) ` P[X]]"
RS3_NMT: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "DenyAll \<in> set p" shows "rm_MT_rules C p \<noteq> []"
rComm2: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list"   and P' :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and N :: "'a"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and A\<^sub>Q :: "name list" assumes "cP = P \<parallel> Q"   and "cRs = \<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"   and "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop"
length_selsort: fixes le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "length (linorder_list0.selsort le xs) = length xs"
order_refl: fixes x :: "'a" shows "domain_semigroup.ds_ord d (\<cdot>) x x"
vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_cases_4: fixes t :: "('d, 'c) Term.term"   and s :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,s\<rangle>) = fv t \<union> fv s"
Int_empty_right: fixes A :: "'c set" shows "A \<inter> {} = {}"
circ_elimination: fixes x :: "'a"   and y :: "'a" assumes "x * y = bot" shows "x * y\<^sup>\<star> \<le> x"
carrier_not_empty: shows "carrier (G\<lparr>carrier := H\<rparr>) \<noteq> {}"
angle_sum_triangle: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<noteq> b \<or> b \<noteq> c \<or> a \<noteq> c" shows "angle c a b + angle a b c + angle b c a = pi"
wf_code_thms_7: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and ps :: "(nat \<times> nat) list" shows "ast_problem.wf_partial_state problem ps \<equiv> distinct (map fst ps) \<and> (\<forall>(x, v)\<in>set ps. x < ast_problem.numVars problem \<and> v < ast_problem.numVals problem x)"
wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t_split_1: fixes x :: "('fun, 'var) extstrand_step"   and S :: "('fun, 'var) extstrand_step list" shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (x # S) = wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t [x] \<union> wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t S"
res: shows "LLL_Impl.basis_reduction_add_rows_loop n st i (j - 1) fs = res"
IH: fixes y :: "('b \<times> ('a \<times> 'b) list) \<times> 'b \<times> ('a \<times> 'b) list" assumes "(y, (s, [\<beta>_step] @ \<upsilon>_step), s, [\<alpha>_step] @ \<rho>_step) \<in> pex r"   and "peak ars y" shows "\<exists>\<sigma>' \<tau>'. DD ars r (fst y, snd y, \<sigma>', \<tau>')"
itrm_brackets_all: fixes x :: "nat itrm"   and vs :: "nat list"   and x' :: "nat itrm" assumes "set (opaque x) \<subseteq> set vs"   and "itrm_brackets vs x = Some x'" shows "opaque x' = []"
unwinding_theorem_BSI: fixes SES :: "('a, 'b) SES_rec"   and \<V> :: "'b V_rec"   and ur :: "('a \<times> 'a) set" assumes "Unwinding SES \<V>"   and "Unwinding.lrb SES \<V> ur"   and "Unwinding.osc SES \<V> ur" shows "BSI \<V> Tr\<^bsub>(induceES SES)\<^esub>"
onorm_triangle: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "bounded_linear f"   and "bounded_linear g" shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
times_mem_sum_mult: fixes R :: "('d, 'e) Ring_scheme"   and A :: "'d set"   and B :: "'d set"   and a :: "'d"   and b :: "'d" assumes "Ring R"   and "A \<subseteq> carrier R"   and "B \<subseteq> carrier R"   and "a \<in> A"   and "b \<in> B" shows "a \<cdot>\<^sub>r\<^bsub>R\<^esub> b \<in> sum_mult R A B"
intro: fixes m :: "int" assumes "1 < m" shows "poly_mod_2 m"
hoare_conseq_right: fixes P :: "'a set"   and R :: "('a \<times> 'a) set"   and Q' :: "'a set"   and Q :: "'a set" assumes "{P} R {> Q'}"   and "Q' \<subseteq> Q" shows "{P} R {> Q}"
subst_eq_intro: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<theta> :: "('b \<times> 'b trm) list" assumes "\<And>t. t \<lhd> \<sigma> = t \<lhd> \<theta>" shows "\<sigma> \<doteq> \<theta>"
normal_Cantor_\<omega>\<omega>: fixes \<gamma> :: "V" assumes "\<gamma> \<in> elts (\<omega> \<up> \<omega>)" shows "normal (Cantor_\<omega>\<omega> \<gamma>)"
False: shows "\<chi> \<noteq> \<chi>\<^sub>0"
cases: fixes Enc :: "'a \<Rightarrow> 'b"   and a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc" assumes "encoding.indRelRp Enc a1 a2" obtains S :: "'a" where "a1 = SourceTerm S" and "a2 = TargetTerm (Enc S)"
le: fixes n :: "nat" shows "mu urev n \<le> f n"
Int_Un_eq_1: fixes S :: "'a set"   and T :: "'a set" shows "S \<inter> T \<union> S = S"
runit_in_hom_2: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>\<r>\<^sub>D[f] : f \<star>\<^sub>D src\<^sub>D f \<Rightarrow>\<^sub>D f\<guillemotright>"
toplevel_inters_nonempty: fixes r :: "'c rexp" shows "toplevel_inters r \<noteq> {}"
below_monic_cfcomp2: fixes f :: "'a \<rightarrow> 'b"   and g :: "'c \<rightarrow> 'a" assumes "below_monic_cfun f"   and "below_monic_cfun g" shows "below_monic_cfun (f oo g)"
assms_3: shows "f \<in> dgrad_sig_set' (length fs) d"
DiffD2: fixes c :: "'c"   and A :: "'c set"   and B :: "'c set"   and P :: "bool" assumes "c \<in> A\<setminus>B"   and "c \<in> B" shows "P"
simps: fixes q :: "'b"   and \<nu> :: "'c"   and q' :: "'b" shows "embed_transition_snd (q, \<nu>, q') = {((x, q), \<nu>, y, q') |x y. True}"
prime_elem_dvd_power_iff: fixes p :: "'b"   and a :: "'b"   and n :: "nat" assumes "prime_elem p" shows "(p dvd a ^ n) = (p dvd a \<and> 0 < n)"
selected_part_def: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'b set"   and C :: "'b Literal set" assumes "propositional_atoms atom_ordering" shows "selected_part Sel C = {L \<in> C. \<exists>A\<in>Sel. L = Neg A}"
degree_aform_err_power_aform_err: fixes X :: "(real \<times> real pdevs) \<times> real"   and n :: "nat"   and p :: "nat"   and m :: "nat" assumes "degree_aform_err X \<le> n" shows "degree_aform_err (power_aform_err p X m) \<le> n"
hsize_chop: fixes t :: "('c, 'd) tm" assumes "is_App t" shows "Suc (hsize (chop t)) = hsize t"
not_in_verts_G: shows "w \<notin> verts (with_proj I')"
assms_2: shows "net_ips s = net_tree_ips n"
unat_less_helper: fixes x :: "'a word"   and n :: "nat" assumes "x < word_of_nat n" shows "unat x < n"
advantage\<^sub>n_def: fixes key_gen :: "nat \<Rightarrow> ('a \<times> 'b) spmf"   and sign :: "nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd spmf"   and verify :: "nat \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> bool"   and valid_message :: "nat \<Rightarrow> 'c \<Rightarrow> bool"   and \<A> :: "nat \<Rightarrow> ('e \<times> 'c \<times> 'd, 'e \<times> (unit + 'c), 'a + 'd) gpv"   and \<eta> :: "nat" shows "suf_cma.advantage\<^sub>n key_gen sign verify valid_message \<A> \<eta> = spmf (suf_cma.suf_cma\<^sub>n key_gen sign verify valid_message \<A> \<eta>) True"
exesort_les_code: fixes cs :: "('a \<times> 'b) list"   and c1 :: "'a set"   and c2 :: "'b set" shows "sort_leq (set cs) c1 c2 = exesort_leq cs c1 c2"
assms: shows "\<forall>ops\<in>set \<pi>. \<forall>op\<in>set ops. op \<in> set (\<Pi>\<^sub>\<O>)"
min_zpassign_simps_2: fixes x :: "'v"   and i :: "nat" shows "zhmset_of (min_passign (PCoef x i)) = 1"
compile_env_def: fixes Finca_add :: "'l \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef \<Rightarrow> 'l"   and Finca_to_list :: "'l \<Rightarrow> ('m \<times> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef) list"   and Fubx_empty :: "'r"   and Fubx_get :: "'r \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef option"   and Fubx_add :: "'r \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef \<Rightarrow> 'r"   and Fubx_to_list :: "'r \<Rightarrow> ('m \<times> ('n, 'o, 'm, 'p, 'q, 's, 't, 'u) Ubx.instr fundef) list"   and heap_empty :: "'v"   and heap_get :: "'v \<Rightarrow> 'o \<times> 'n \<Rightarrow> 'n option"   and heap_add :: "'v \<Rightarrow> 'o \<times> 'n \<Rightarrow> 'n \<Rightarrow> 'v"   and heap_to_list :: "'v \<Rightarrow> (('o \<times> 'n) \<times> 'n) list"   and is_true :: "'n \<Rightarrow> bool"   and is_false :: "'n \<Rightarrow> bool"   and box_ubx1 :: "'t \<Rightarrow> 'n"   and unbox_ubx1 :: "'n \<Rightarrow> 't option"   and box_ubx2 :: "'u \<Rightarrow> 'n"   and unbox_ubx2 :: "'n \<Rightarrow> 'u option"   and \<OO>\<pp> :: "'p \<Rightarrow> 'n list \<Rightarrow> 'n"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'p \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'q \<Rightarrow> 'n list \<Rightarrow> 'n"   and \<II>\<nn>\<ll> :: "'p \<Rightarrow> 'n list \<Rightarrow> 'q option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'q \<Rightarrow> 'n list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'q \<Rightarrow> 'p"   and \<UU>\<bb>\<xx>\<OO>\<pp> :: "'s \<Rightarrow> ('n, 't, 'u) unboxed list \<Rightarrow> ('n, 't, 'u) unboxed option"   and \<UU>\<bb>\<xx> :: "'q \<Rightarrow> Unboxed.type option list \<Rightarrow> 's option"   and \<BB>\<oo>\<xx> :: "'s \<Rightarrow> 'q"   and \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> :: "'s \<Rightarrow> Unboxed.type option list \<times> Unboxed.type option"   and Finca_empty :: "'l"   and Finca_get :: "'l \<Rightarrow> 'm \<Rightarrow> ('n, 'o, 'm, 'p, 'q) Inca.instr fundef option"   and \<A> :: "'m \<Rightarrow> nat option"   and \<O> :: "'m \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and e :: "'l" assumes "inca_ubx_compiler Finca_add Finca_to_list Fubx_empty Fubx_get Fubx_add Fubx_to_list heap_empty heap_get heap_add heap_to_list is_true is_false box_ubx1 unbox_ubx1 box_ubx2 unbox_ubx2 \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx>\<OO>\<pp> \<UU>\<bb>\<xx> \<BB>\<oo>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> Finca_empty Finca_get" shows "inca_ubx_compiler.compile_env Finca_to_list Fubx_empty Fubx_add unbox_ubx1 unbox_ubx2 \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> \<A> \<O> e \<equiv> map_result id (env.from_list Fubx_empty Fubx_add) (Result.those (map (inca_ubx_compiler.compile_env_entry unbox_ubx1 unbox_ubx2 \<AA>\<rr>\<ii>\<tt>\<yy> \<DD>\<ee>\<II>\<nn>\<ll> \<UU>\<bb>\<xx> \<TT>\<yy>\<pp>\<ee>\<OO>\<ff>\<OO>\<pp> \<A> \<O>) (Finca_to_list e)))"
conv_abscissa_mono: fixes g :: "'a fds"   and f :: "'a fds" assumes "\<And>s. fds_converges g s \<Longrightarrow> fds_converges f s" shows "conv_abscissa f \<le> conv_abscissa g"
abs_minus_commute: fixes abs :: "'i \<Rightarrow> 'i"   and plus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and zero :: "'i"   and minus :: "'i \<Rightarrow> 'i \<Rightarrow> 'i"   and uminus :: "'i \<Rightarrow> 'i"   and less_eq :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and less :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and a :: "'i"   and b :: "'i" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (minus a b) = abs (minus b a)"
valid_frac_uminus: fixes S :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and X :: "('c \<times> 'c) set" assumes "Comm_Ring.quotient_ring S R addition multiplication zero unit"   and "quotient_ring.valid_frac S R addition multiplication zero X" shows "quotient_ring.valid_frac S R addition multiplication zero (quotient_ring.uminus_rel S R addition multiplication zero X)"
simps_2: fixes h :: "'b hp"   and hs :: "'b hp list" shows "pass\<^sub>2 (h # hs) = Some (case pass\<^sub>2 hs of None \<Rightarrow> h | Some x \<Rightarrow> link h x)"
hyps_3: shows "otherwith S (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) (oarrivemsg I) \<sigma> \<sigma>' a"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) (\<mu> \<star> \<nu> \<star> \<pi>)"
fresher_refl: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "r \<sqsubseteq> r"
inv_mult_group: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "inv (x \<otimes> y) = inv y \<otimes> inv x"
distinct_13: fixes act :: "act"   and ccs :: "ccs"   and ccs1' :: "ccs"   and ccs2' :: "ccs" shows "act.ccs \<noteq> ccs1' \<parallel> ccs2'"
isin_insert: fixes xs :: "bool list"   and t :: "trie"   and ys :: "bool list" shows "isin (insert xs t) ys = (xs = ys \<or> isin t ys)"
has_derivative_Blinfun: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "(f has_derivative f') F" shows "(f has_derivative blinfun_apply (Blinfun f')) F"
set_8: fixes x1 :: "('d, 'c) Term.term" shows "vars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (send\<langle>x1\<rangle>) = fv x1"
disc_eq_case: fixes prod :: "'b \<times> 'c" shows "(prod = prod) = (case prod of (uu_, uua_) \<Rightarrow> True)"
Radj_eq_iff: fixes a :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and b :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and x :: "'sz"   and y :: "'sz" shows "(a = b) = (Radj x y a = Radj x y b)"
f_3_def: fixes x :: "int" shows "f_3 x = (x mod int p, x div int p + 1)"
wf_prog_wwf_prog: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "wf_J_prog P" shows "wwf_J_prog P"
star_left_slide: fixes x :: "'a" shows "top \<sqinter> x \<le> x \<sqinter> top"
alg_dvd_of_int: fixes x :: "int"   and y :: "int" assumes "x dvd y" shows "of_int x alg_dvd of_int y"
decisive_on_empty: fixes f :: "'a set \<Rightarrow> 'a set" shows "decisive_on {} f"
\<Sigma>_protocol: shows "\<Sigma>_AND.\<Sigma>_protocol"
chine_composite_2: shows "\<mu>\<nu>_\<pi>.chine = \<langle>\<langle>\<mu>.chine \<cdot> \<mu>\<nu>.prj\<^sub>1 \<lbrakk>\<mu>.cod.leg0, \<nu>.cod.leg1\<rbrakk> \<nu>.chine \<cdot> \<mu>\<nu>.prj\<^sub>0\<rangle> \<cdot> \<p>\<^sub>1[\<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0, \<pi>.leg1] \<lbrakk>\<nu>.cod.leg0 \<cdot> \<p>\<^sub>0[\<mu>.cod.leg0, \<nu>.cod.leg1], \<pi>.cod.leg1\<rbrakk> \<pi>.chine \<cdot> Prj\<^sub>0\<rangle>"