hcju/isabelleps huggingface dataset:
select_convs_1: fixes pInit :: "'b \<Rightarrow> 'c"   and pTrans :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and pAct :: "'c \<Rightarrow> 'd list"   and more :: "'e" shows "pInit \<lparr>pInit = pInit, pTrans = pTrans, pAct = pAct, \<dots> = more\<rparr> = pInit"
rat_int_poly_step_agree: fixes b :: "int"   and c2 :: "int poly"   and q :: "int poly"   and a1 :: "int poly"   and a2 :: "int poly"   and c1 :: "int poly" assumes "coeff (pCons b c2) (degree q) mod lead_coeff q = 0" shows "((of_int_poly a1, of_int_poly a2) = (div_rat_poly_step (of_int_poly q) \<circ> rat_of_int) b (of_int_poly c1, of_int_poly c2)) = (Some (a1, a2) = div_int_poly_step q b (Some (c1, c2)))"
multiplicity_one: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "factorial_semiring.multiplicity times one p one = 0"
None_in_results_gpv_catch_gpv: fixes \<I> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv" shows "(None \<in> results_gpv \<I> (catch_gpv gpv)) = (\<not> colossless_gpv \<I> gpv)"
plossless_converter_of_callee: fixes \<I>1 :: "('i, 'j) \<I>"   and \<I>2 :: "('k, 'l) \<I>"   and callee :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h, 'k, 'l) gpv"   and s :: "'h" assumes "\<And>s x. x \<in> outs_\<I> \<I>1 \<Longrightarrow> plossless_gpv \<I>2 (callee s x) \<and> (\<forall>(y, s')\<in>results_gpv \<I>2 (callee s x). y \<in> responses_\<I> \<I>1 x)" shows "plossless_converter \<I>1 \<I>2 (CNV callee s)"
Pi_iff: fixes f :: "'d \<Rightarrow> 'e"   and I :: "'d set"   and X :: "'d \<Rightarrow> 'e set" shows "(f \<in> Pi I X) = (\<forall>i\<in>I. f i \<in> X i)"
pell_valuation_trivial: shows "pell_valuation (1, 0) = 1"
Limsup_bounded: fixes X :: "'c \<Rightarrow> 'd"   and C :: "'d"   and F :: "'c filter" assumes "\<forall>\<^sub>F n in F. X n \<le> C" shows "Limsup F X \<le> C"
infinite_insert_lepoll: fixes A :: "'a set"   and a :: "'a" assumes "infinite A" shows "insert a A \<lesssim> A"
ereal_min: fixes x :: "real"   and y :: "real" shows "ereal (min x y) = min (ereal x) (ereal y)"
sum_norm_le: 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 S :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x" shows "norm (comm_monoid_add.sum plus zero f S) \<le> sum g S"
cross7_nonassociative: shows "\<not> (\<forall>c a b. c \<times>\<^sub>7 a \<times>\<^sub>7 b = (c \<times>\<^sub>7 a) \<times>\<^sub>7 b)"
linear_sum: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and S :: "'e set" assumes "clinear f" shows "f (sum g S) = (\<Sum>a\<in>S. f (g a))"
dvd_add: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b"   and "a dvd c" shows "a dvd b + c"
bot_empty_eq: shows "\<bottom> = (\<lambda>x. x \<in> {})"
\<pi>_fix': fixes T :: "(symbol \<times> character list) set"   and k :: "nat"   and I :: "item set" shows "(Scan T k \<circ> Complete k \<circ> Predict k) (\<pi> k T I) = \<pi> k T I"
image_replicate_mset: fixes f :: "'d \<Rightarrow> 'c"   and n :: "nat"   and a :: "'d" shows "image_mset f (replicate_mset n a) = replicate_mset n (f a)"
path_in_nodes: fixes \<sigma> :: "'var \<Rightarrow> 'val"   and k :: "nat" shows "path \<sigma> k \<in> local.nodes"
projection_on_subset: fixes Y :: "'a set"   and X :: "'a set"   and l :: "'a list" assumes "Y \<subseteq> X"   and "l \<upharpoonleft> X = []" shows "l \<upharpoonleft> Y = []"
rev_bexI: fixes x :: "'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "x \<in> A"   and "P x" shows "\<exists>x\<in>A. P x"
hyps: shows "((v', w, v) \<in> set (rev_path xs)) = ((v, w, v') \<in> set xs)"
comp_in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a"   and g :: "'a \<times> 'a"   and c :: "'a \<times> 'a" assumes "VV.in_hom f a b"   and "VV.in_hom g b c" shows "VV.in_hom (VV.comp g f) a c"
cil_cons1: fixes w :: "'d list"   and wa :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and wb :: "'d list"   and e :: "'d" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"   and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> wb) = {}" shows "e # w \<in> e # wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"
restrict_eq: fixes A :: "'c set" shows "restrict A = filter (\<lambda>(k, v). k \<in> A)"
merge_guards_simps_5: fixes b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "merge_guards (Cond b c\<^sub>1 c\<^sub>2) = Cond b (merge_guards c\<^sub>1) (merge_guards c\<^sub>2)"
assms_3: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and m :: "'c"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "P (\<sigma>, receive m, \<sigma>')" shows "P (\<sigma>, \<tau>, \<sigma>')"
qq2: shows "q \<bullet> as \<inter> supp (q \<bullet> x) = r \<bullet> (bs \<inter> supp y)"
assms_7: shows "lp r adds lcs (lp p) (lp q)"
nth_replicate: fixes i :: "nat"   and n :: "nat"   and x :: "'g" assumes "i < n" shows "replicate n x ! i = x"
inner_commute: fixes x :: "'a"   and y :: "'a" shows "x \<bullet> y = y \<bullet> x"
aexec: shows "abc_step_l (as, lm) (Some (Inc n)) = (a, b)"
avl_del_case: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t" shows "case del x t of Same t' \<Rightarrow> avl t' \<and> height t = height t' | Diff t' \<Rightarrow> avl t' \<and> height t = height t' + 1"
max2_iff: fixes a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded"   and c :: "'a Bonk_Schramm_extension_unfolded" assumes "a \<in> Field Bonk_Schramm_extension_unfolded_wo"   and "b \<in> Field Bonk_Schramm_extension_unfolded_wo" shows "((wo.max2 a b, c) \<in> Bonk_Schramm_extension_unfolded_wo) = ((a, c) \<in> Bonk_Schramm_extension_unfolded_wo \<and> (b, c) \<in> Bonk_Schramm_extension_unfolded_wo)"
real_nonneg_mat_spectral_radius_largest_jordan_block: fixes A :: "complex mat"   and n_as :: "(nat \<times> complex) list"   and m :: "nat"   and lam :: "complex" assumes "real_nonneg_mat A"   and "jordan_nf A n_as"   and "(m, lam) \<in> set n_as"   and "cmod lam = spectral_radius A" shows "\<exists>M\<ge>m. (M, complex_of_real (spectral_radius A)) \<in> set n_as"
assms_2: shows "wfP (\<rightarrow>)\<inverse>\<inverse>"
nomatch: fixes m :: "'a simple_match"   and p :: "('a, 'b) simple_packet_scheme"   and a :: "simple_action"   and rs :: "'a simple_rule list" assumes "\<not> simple_matches m p" shows "simple_fw (SimpleRule m a # rs) p = simple_fw rs p"
A: fixes agents'' :: "'agent list" assumes "agents'' \<in> permutations_of_set agents'" shows "set_pmf (pmf_of_set (rsd_winners R alts' agents'')) \<subseteq> alts'"
intros_2: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and bs :: "'c list"   and a' :: "'b"   and b :: "'c"   and a'' :: "'b" assumes "rtrancl3p r a bs a'"   and "r a' b a''" shows "rtrancl3p r a (bs @ [b]) a''"
ct_list_eq_cons_ct_list1: fixes as :: "t list"   and xs :: "ct list"   and ys :: "t list" assumes "ct_list_eq (to_ct_list as) (xs @ to_ct_list ys)" shows "\<exists>bs. as = bs @ ys \<and> ct_list_eq (to_ct_list bs) xs"
man_exp_of_double_def: fixes d :: "double" shows "man_exp_of_double d = (if Double.is_finite d then let f = Double.of_finite d in Some (integer_of_int (mantissa f), integer_of_int (Float.exponent f)) else None)"
mult_unit_dvd_iff: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "b dvd (1::'b)" shows "(a * b dvd c) = (a dvd c)"
mult_1_right: fixes a :: "'a" shows "a ; 1' = a"
rqrm_add_def: fixes q :: "'a"   and r :: "'b"   and res :: "('a, 'b dlist) HashMap.hashmap" shows "rqrm_add q r res \<equiv> case hm.lookup q res of None \<Rightarrow> hm.update q (ls.ins r (ls.empty ())) res | Some s \<Rightarrow> hm.update q (ls.ins r s) res"
higher_differentiable_on_inner: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and g :: "'b \<Rightarrow> 'c" assumes "higher_differentiable_on S f n"   and "higher_differentiable_on S g n"   and "open S" shows "higher_differentiable_on S (\<lambda>x. f x \<bullet> g x) n"
special_cosets_bottom: shows "dual_order.bottom \<P> = W"
measurable_emeasure_Pair: fixes M :: "'c measure"   and Q :: "('d \<times> 'c) set"   and N :: "'d measure" assumes "sigma_finite_measure M"   and "Q \<in> sets (N \<Otimes>\<^sub>M M)" shows "(\<lambda>x. emeasure M (Pair x -` Q)) \<in> borel_measurable N"
Step1_2_4: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> ImpNext p \<and> \<not> unchanged (e p, c p, r p, m p, rmhist!p) \<and> $S4 rmhist p \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> S4 rmhist p$ \<and> Read rmCh mm ires p \<and> unchanged (e p, c p, r p, rmhist!p) \<or> S4 rmhist p$ \<and> (\<exists>l. Write rmCh mm ires p l) \<and> unchanged (e p, c p, r p, rmhist!p) \<or> S5 rmhist p$ \<and> MemReturn rmCh ires p \<and> unchanged (e p, c p, r p)"
fds_shift_def: fixes c :: "'a"   and f :: "'a fds" shows "fds_shift c f = fds (\<lambda>n. fds_nth f n * nat_power n c)"
Euclidean_space_def: fixes n :: "nat" shows "Euclidean_space n \<equiv> subtopology (powertop_real UNIV) {x. \<forall>i\<ge>n. x i = 0}"
\<I>_ideal_def: shows "\<I>_ideal \<equiv> \<lambda>\<eta>. \<I>_uniform UNIV {None, Some \<eta>}"
store_size: fixes m :: "mem"   and n :: "nat"   and off :: "nat"   and v :: "bytes"   and l :: "nat"   and m' :: "mem" assumes "store m n off v l = Some m'" shows "mem_size m = mem_size m'"
S: fixes n :: "nat" shows "S n \<in> lmeasurable"
simps_2: fixes oid :: "'c"   and v :: "'c" shows "insert_spec [] (oid, Some v) = []"
height_ge_one: fixes e :: "lam" shows "1 \<le> height e"
in_set_butlastI: fixes x :: "'b"   and xs :: "'b list" assumes "x \<in> set xs"   and "x \<noteq> last xs" shows "x \<in> set (butlast xs)"
simps_3: fixes a :: "char"   and as :: "char list"   and b :: "char"   and bs :: "char list" shows "(Iface (a # as) \<le> Iface (b # bs)) = (if a = b then Iface as \<le> Iface bs else a \<le> b)"
simps: fixes ao :: "'a option"   and bo :: "'a option" shows "less_option ao bo = (less_eq_option ao bo \<and> ao \<noteq> bo)"
iso_unit: fixes a :: "'a arrow_of_spans_data" assumes "obj a" shows "local.iso \<i>[a]"
test_pres1: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "pres p q"
iso_map\<^sub>1_ide: fixes f :: "'c" assumes "C.ide f" shows "D.iso (\<tau>\<^sub>1 f)"
distinct_degree_factorization_def: fixes f :: "'a mod_ring poly" shows "distinct_degree_factorization f = (if degree f = 1 then [(1, f)] else dist_degree_factorize_main f (monom 1 1) 0 [])"
sel_4: fixes x21 :: "'e"   and x22 :: "('e, 'f) Term.term list" shows "args (Fun x21 x22) = x22"
in_hd_or_tl_conv: fixes l :: "'b list"   and x :: "'b" assumes "l \<noteq> []" shows "(x = hd l \<or> x \<in> set (tl l)) = (x \<in> set l)"
wf_not_sym: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and x :: "'b" assumes "wf r"   and "(a, x) \<in> r" shows "(x, a) \<notin> r"
uMinus_def: shows "uMinus \<equiv> \<Lambda> x y. Discr (undiscr x - undiscr y)"
unit_closed: fixes G :: "('e, 'f) Group_scheme" assumes "Group G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
weakPsiCongE_3: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<doteq> Q" shows "\<Psi> \<rhd> Q \<leadsto>\<guillemotleft>weakBisim\<guillemotright> P"
integrable_f2: fixes x :: "'a" assumes "x \<in> U" shows "f x integrable_on cbox a b"
wf': shows "prefs_from_table_wf agents alts (map (\<lambda>(x, y). (x, map ((`) \<sigma>) y)) xss)"
r_result_diverg': fixes i :: "nat"   and x :: "nat"   and t :: "nat" assumes "eval r_univ [i, x] \<up>" shows "eval r_result [t, i, x] \<down>= 0"
c: assumes "c \<in> space N" shows "{c} \<in> sets N"
symmetry_functor_axioms: shows "symmetry_functor (\<cdot>\<^sub>A\<^sub>1) (\<cdot>\<^sub>A\<^sub>2)"
sup_subset_mset_def: fixes A :: "'b multiset"   and B :: "'b multiset" shows "A \<union># B = A + (B - A)"
splus_case: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" shows "sterm_sem I (Plus t1 t2) = (\<lambda>v. sterm_sem I t1 v + sterm_sem I t2 v)"
ex: shows "\<exists>n>0. fds_nth h n \<noteq> (0::'a)"
simps_6: fixes F :: "'f set"   and b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "strip_guards F (While b c) = While b (strip_guards F c)"
gfp_upperbound: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and X :: "'g"   and f :: "'g \<Rightarrow> 'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "less_eq X (f X)" shows "less_eq X (complete_lattice.gfp Sup less_eq f)"
one_fun_def: shows "1 \<equiv> rel_partial_monoid.pid"
OAlist_tc_map2_val_rneutr_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "('a, 'b) oalist_tc"   and ys :: "('a, 'c) oalist_tc" shows "OAlist_tc_map2_val_rneutr f xs ys = OAlist_tc (map2_val_pair_tc f id (map_val_pair (\<lambda>k. f k (0::'b))) (list_of_oalist_tc xs) (list_of_oalist_tc ys))"
8: shows "e * - F * e\<^sup>T \<le> bot"
div_mod_imp_dvdm: fixes b :: "int poly"   and a :: "int poly" assumes "\<exists>q r. b = q * a + smult m r" shows "a dvdm b"
cinner_ge_zero: fixes x :: "'b" shows "0 \<le> \<langle>x, x\<rangle>"
exhaust: fixes y :: "checker_type" obtains x1 :: "ct list" where "y = TopType x1"   | x2 :: "t list" where "y = Type x2"   | "y = Bot"
evalRR_sub: fixes p :: "nat \<Rightarrow> int"   and q :: "nat \<Rightarrow> int"   and x :: "int" assumes "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)"   and "x \<in> carrier R" shows "eval R R id x (p \<ominus>\<^bsub>UP R\<^esub> q) = eval R R id x p \<ominus>\<^bsub>R\<^esub> eval R R id x q"
bres_bot_bot: shows "\<bottom> \<rightarrow> \<bottom> = \<top>"
mod_minus_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus" shows "modulo (uminus (modulo a b)) b = modulo (uminus a) b"
str_del_block_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and b :: "'b set" assumes "incidence_system point_set block_collection" shows "incidence_system.str_del_block block_collection b \<equiv> {#bl - b. bl \<in># {#bl \<in># block_collection. bl \<noteq> b#}#}"
cri_less_eq_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c" assumes "dataflow_topology summary results_in" shows "dataflow_topology.cri_less_eq summary results_in = (\<lambda>(loc1, t1) (loc2, t2). \<exists>s. s \<in>\<^sub>A graph.path_weight summary loc1 loc2 \<and> results_in t1 s \<le> t2)"
1: shows "\<exists>\<epsilon>. \<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright> \<and> F \<epsilon> = unit (src\<^sub>C g) \<cdot>\<^sub>D \<psi> \<cdot>\<^sub>D (D.inv (F f) \<star>\<^sub>D g') \<cdot>\<^sub>D (F f \<star>\<^sub>D D.inv \<mu>) \<cdot>\<^sub>D D.inv (\<Phi> (f, g))"
simps_4: fixes c :: "int"   and e :: "MIR.num" shows "isrlfm (NEq (CN 0 c e)) = (0 < c \<and> numbound0 e)"
encode_operators_def: fixes \<Pi> :: "'b strips_problem"   and t :: "nat" shows "encode_operators \<Pi> t \<equiv> let ops = \<Pi>\<^sub>\<O> in encode_all_operator_preconditions \<Pi> ops t \<^bold>\<and> encode_all_operator_effects \<Pi> ops t"
monad_state_eqI: fixes A :: "'d \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and s :: "'d"   and B :: "'e \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and s' :: "'e" assumes "\<And>r t. (r, t) \<in> fst (A s) \<Longrightarrow> (r, t) \<in> fst (B s')"   and "\<And>r t. (r, t) \<in> fst (B s') \<Longrightarrow> (r, t) \<in> fst (A s)"   and "snd (A s) = snd (B s')" shows "A s = B s'"
fv_regex_alt: fixes fv :: "'b \<Rightarrow> 'a set"   and r :: "'b Regex.regex" shows "Regex.fv_regex fv r = \<Union> (fv ` regex.atms r)"
rtc_term_trans: fixes s :: "trm"   and t :: "trm"   and u :: "trm" assumes "s \<longlonglongrightarrow>\<^sup>* t"   and "t \<longlonglongrightarrow>\<^sup>* u" shows "s \<longlonglongrightarrow>\<^sup>* u"
mult_mono_nonpos_nonneg: fixes a :: "'b"   and c :: "'b"   and d :: "'b"   and b :: "'b" assumes "a \<le> c"   and "d \<le> b"   and "a \<le> (0::'b)"   and "(0::'b) \<le> d" shows "a * b \<le> c * d"
qSwapInp_preserves_qGoodInp: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "qGoodInp qinp" shows "qGoodInp (qSwapInp xs x1 x2 qinp)"
assms: shows "SimplyTyped.Pair A B \<rightarrow>\<beta> M'"
preserves_cod: fixes f :: "'a \<times> 'a" assumes "VV.arr f" shows "cod (fst f \<star> snd f) = fst (VV.cod f) \<star> snd (VV.cod f)"
has_integral_restrict_closed_subinterval: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "(f has_integral i) (cbox c d)"   and "cbox c d \<subseteq> cbox a b" shows "((\<lambda>x. if x \<in> cbox c d then f x else (0::'b)) has_integral i) (cbox a b)"
exec_gpv_oracle_bisim: fixes X :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and s1 :: "'b"   and s2 :: "'c"   and oracle1 :: "'b \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'b) spmf"   and oracle2 :: "'c \<Rightarrow> 'd \<Rightarrow> ('e \<times> 'c) spmf"   and gpv :: "('f, 'd, 'e) gpv"   and R :: "'f \<times> 'b \<Rightarrow> 'f \<times> 'c \<Rightarrow> bool" assumes "X s1 s2"   and "\<And>s1 s2 x. X s1 s2 \<Longrightarrow> rel_spmf (\<lambda>(a, s1') (b, s2'). a = b \<and> X s1' s2') (oracle1 s1 x) (oracle2 s2 x)"   and "\<And>x s1' s2'. \<lbrakk>X s1' s2'; (x, s1') \<in> set_spmf (exec_gpv oracle1 gpv s1); (x, s2') \<in> set_spmf (exec_gpv oracle2 gpv s2)\<rbrakk> \<Longrightarrow> R (x, s1') (x, s2')" shows "rel_spmf R (exec_gpv oracle1 gpv s1) (exec_gpv oracle2 gpv s2)"
AR_ANR: fixes S :: "'b set" shows "AR S = (ANR S \<and> contractible S \<and> S \<noteq> {})"
dhops_update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "ip \<noteq> dip" shows "dhops (update rt dip (dsn, dsk, flag, hops, nhip)) ip = dhops rt ip"
max_Var_floatarith_FDERIV_n_floatariths: fixes xs :: "nat list"   and d :: "floatarith list"   and fa :: "floatarith list"   and n :: "nat" assumes "length xs = length d" shows "max_Var_floatariths (FDERIV_n_floatariths fa xs d n) \<le> max (max_Var_floatariths fa) (max_Var_floatariths d)"
trivial_add_mset_remove_iff: fixes a :: "'b"   and N :: "'b multiset"   and b :: "'b" shows "(add_mset a (remove1_mset b N) = N) = (a \<in># N \<and> a = b)"
simps: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update" shows "PivotUpdateMinVars.check eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s = (if \<U> s then s else let x\<^sub>i' = min_lvar_not_in_bounds s in case x\<^sub>i' of None \<Rightarrow> s | Some x\<^sub>i \<Rightarrow> let dir = if \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i then Positive else Negative in PivotUpdateMinVars.check eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update (PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update dir x\<^sub>i s))"
get_impl_def: fixes M :: "('b SecurityInvariant \<times> 'b SecurityInvariant_configured) list" shows "get_impl M \<equiv> map fst M"
term_order_rel'_def: fixes x :: "char list list"   and y :: "char list list" shows "term_order_rel' x y = ((x, y) \<in> term_order_rel)"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "(a1 >\<^sub>t a2) = ((\<exists>ti t s. a1 = t \<and> a2 = s \<and> ti \<in> set (args t) \<and> ti \<ge>\<^sub>t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t >\<^sub>h\<^sub>d head s \<and> chkvar t s \<and> chkargs (>\<^sub>t) t s) \<or> (\<exists>t s. a1 = t \<and> a2 = s \<and> head t = head s \<and> chkargs (>\<^sub>t) t s \<and> (\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s))))"
sin_of_real_pi: shows "sin (of_real pi) = (0::'b)"
convergent_powser'_gbinomial': fixes abort :: "bool"   and n :: "nat" shows "convergent_powser' (gbinomial_series abort (real n)) (\<lambda>x. (1 + x) ^ n)"
isin_simps2_6: fixes xs :: "'c list"   and y :: "'c"   and x :: "'c" assumes "sorted_less (xs @ [y])"   and "y \<le> x" shows "x \<notin> set xs"
ground_Fvars: fixes t :: "trm" shows "ground t = (Fvars t = {})"
assms_2: fixes x :: "'c" assumes "nofail (g x)"   and "Q x" shows "P x"
funof_eq: fixes r :: "('b \<times> 'c) set"   and x :: "'b"   and y :: "'c" assumes "single_valued r"   and "(x, y) \<in> r" shows "funof r x = y"
is_elementary_monoidal_category: shows "elementary_monoidal_category (\<cdot>) (\<otimes>) \<I> \<F>C.lunit\<^sub>F\<^sub>M\<^sub>C \<F>C.runit\<^sub>F\<^sub>M\<^sub>C \<F>C.assoc\<^sub>F\<^sub>M\<^sub>C"
assms_2: shows "h' = update a i (f (get h a ! i)) h"
path_cons_elim: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and r :: "'a list"   and p :: "'b" assumes "transition_system.path execute enabled (a # r) p" obtains "enabled a p" and "transition_system.path execute enabled r (execute a p)"
refl: fixes a :: "'b list" shows "??.Sublist.prefix a a"
map_add_None: fixes m :: "'f \<Rightarrow> 'e option"   and n :: "'f \<Rightarrow> 'e option"   and k :: "'f" shows "((m ++ n) k = None) = (n k = None \<and> m k = None)"
well_ord_compare1: fixes E :: "('e, 'f) Order_scheme" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))" shows "ord_equiv D E \<or> (\<exists>c\<in>carrier E. ord_equiv D (Iod E (segment E c)))"
that: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat" assumes "is_path \<pi>"   and "\<pi> 0 = x"   and "\<pi> n = local.return"   and "\<forall>i>0. \<not> x pd\<rightarrow> \<pi> i" shows "thesis"
exec_drest_5: fixes x :: "('c, 'd) fmap"   and vs :: "'c set" assumes "fmdom' x \<subseteq> vs" shows "fmrestrict_set vs x = x"
existence_ivl_empty1: fixes x0 :: "'a" assumes "t0 \<notin> T" shows "existence_ivl t0 x0 = {}"