hcju/isabelleps huggingface dataset:
cases: fixes a :: "'a"   and \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "a \<in> SAT \<Gamma> f" obtains x :: "'a" where "a = x__" and "x__ \<in> A \<Gamma>"   | x :: "'a" where "a = x" and "weight \<Gamma> x \<le> d_IN f x"
at_right_MInf: shows "at_right (- \<infinity>) = filtermap ereal at_bot"
interval_lebesgue_integral_cong: fixes a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a"   and M :: "real measure" assumes "a \<le> b"   and "\<And>x. x \<in> einterval a b \<Longrightarrow> f x = g x"   and "einterval a b \<in> sets M" shows "interval_lebesgue_integral M a b f = interval_lebesgue_integral M a b g"
dvd_imp_gcd_dvd_gcd: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b dvd c" shows "some_gcd a b dvd some_gcd a c"
h: shows "h \<in> F \<and> h \<in> G"
vsubset_vinsertI: fixes A :: "V"   and x :: "V"   and B :: "V" assumes "A -\<^sub>\<circ> set {x} \<subseteq>\<^sub>\<circ> B" shows "A \<subseteq>\<^sub>\<circ> vinsert x B"
sub_mult_one_circ_split: fixes x :: "'a" shows "x * (1::'a)\<^sup>\<circ> \<le> x \<squnion> L"
functor_axioms: shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) F"
approx_HComplex: fixes a :: "real star"   and b :: "real star"   and c :: "real star"   and d :: "real star" assumes "a \<approx> b"   and "c \<approx> d" shows "HComplex a c \<approx> HComplex b d"
llist_induct: fixes P :: "'c llist \<Rightarrow> bool"   and xs :: "'c llist" assumes "ccpo.admissible lSup (\<sqsubseteq>) P"   and "P LNil"   and "\<And>x xs. \<lbrakk>lfinite xs; P xs\<rbrakk> \<Longrightarrow> P (LCons x xs)" shows "P xs"
mset_del_min: fixes h :: "'b heap" assumes "h \<noteq> Empty" shows "mset_heap (del_min h) = mset_heap h - {#get_min h#}"
plambda_app: fixes P :: "'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and v :: "'b" shows "plambda P f(v)\<^sub>p = (if P v then f v else undefined)"
prems_1: shows "Lyndon_mono (ws \<cdot> [a])"
assms: fixes \<sigma> :: "'a"   and p :: "('b, 'c, 'd, 'e) seqp"   and a :: "'f"   and \<sigma>' :: "'a"   and p' :: "('b, 'c, 'd, 'e) seqp"   and l :: "'e"   and l' :: "'e" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "((\<sigma>, p), a, \<sigma>', p') \<in> automaton.trans A"   and "S \<sigma> \<sigma>' a"   and "l \<in> labels \<Gamma> p"   and "l' \<in> labels \<Gamma> p'" shows "P ((\<sigma>, l), a, \<sigma>', l')"
hyperb_quad_ineq: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and t :: "'a" shows "dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z) + 2 * deltaG TYPE('a)"
le_less_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z < y" shows "z < x"
i_Exec_Comp_Stream_Acc_Output__eq_NoMsg_iAll_conv: fixes k :: "nat"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b"   and t :: "nat" assumes "0 < k" shows "(i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = NoMsg) = (\<box> t1 [t * k\<dots>,k - Suc 0]. (output_fun \<circ> i_Exec_Comp_Stream trans_fun (input \<odot> k) c) t1 = NoMsg)"
inf_level: fixes l1 :: "level"   and l2 :: "level" shows "l1 \<sqinter> l2 \<equiv> if Lo \<in> {l1, l2} then Lo else Hi"
dim_poly_of_append_vec: fixes a :: "rat vec"   and b :: "rat vec" shows "dim_poly (vec_to_lpoly (a @\<^sub>v b)) \<le> dim_vec a + dim_vec b"
congruent_triangleD_17: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "angle c1 b1 a1 = angle a2 b2 c2"
preserves_comp: fixes g :: "'a"   and f :: "'a" assumes "S.seq g f" shows "R (g \<cdot>\<^sub>S f) = R g \<cdot>\<^sub>S R f"
conseq12: fixes G :: "prog"   and A :: "'b triple set"   and P' :: "'b assn"   and t :: "Term.term"   and Q' :: "'b assn"   and P :: "'b assn"   and Q :: "'b assn" assumes "G,A\<turnstile>{P'} t\<succ> {Q'}"   and "\<forall>Y s Z. P Y s Z \<longrightarrow> (\<forall>Y' s'. (\<forall>Y Z'. P' Y s Z' \<longrightarrow> Q' Y' s' Z') \<longrightarrow> Q Y' s' Z)" shows "G,A\<turnstile>{P} t\<succ> {Q}"
Suc_2: shows "A $ i $ mod_type_class.from_nat k \<noteq> (0::'a)"
DJ_qe: fixes qe :: "fm \<Rightarrow> fm"   and bbs :: "bool list" assumes "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> Ifm bbs bs (qe p) = Ifm bbs bs (E p)" shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> Ifm bbs bs (DJ qe p) = Ifm bbs bs (E p)"
prems: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>r # rs, Undecided\<rangle> \<Rightarrow> t"
lookup_def: shows "lookup \<equiv> map_fun impl_of id map_of"
param_foldli: fixes xs :: "'f list"   and ys :: "'g list"   and Ra :: "('f \<times> 'g) set"   and c :: "'h \<Rightarrow> bool"   and d :: "'i \<Rightarrow> bool"   and Rs :: "('h \<times> 'i) set"   and f :: "'f \<Rightarrow> 'h \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i \<Rightarrow> 'i"   and a :: "'h"   and b :: "'i" assumes "(xs, ys) \<in> \<langle>Ra\<rangle>list_rel"   and "(c, d) \<in> Rs \<rightarrow> bool_rel"   and "\<And>x y. \<lbrakk>(x, y) \<in> Ra; x \<in> set xs; y \<in> set ys\<rbrakk> \<Longrightarrow> (f x, g y) \<in> Rs \<rightarrow> Rs"   and "(a, b) \<in> Rs" shows "(foldli xs c f a, foldli ys d g b) \<in> Rs"
zero: fixes i :: "nat" assumes "1 < i"   and "i \<le> n" shows "(deriv ^^ i) f z = 0"
eq_fract_3: fixes a :: "'b"   and c :: "'b" shows "Fract (0::'b) a = Fract (0::'b) c"
insert_fset_case: fixes x :: "'a"   and S :: "'a fset" assumes "x |\<notin>| S"   and "P S" shows "P (finsert x S)"
StrongEq_L_subst1: fixes P :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and \<tau> :: "'a state \<times> 'a state"   and x :: "'a state \<times> 'a state \<Rightarrow> 'b"   and y :: "'a state \<times> 'a state \<Rightarrow> 'b" assumes "cp P"   and "\<tau> \<Turnstile> x \<triangleq> y" shows "\<tau> \<Turnstile> P x \<triangleq> P y"
known_ptrs_known_ptr: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_known_ptrs\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t.a_known_ptrs known_ptr h"   and "ptr |\<in>| object_ptr_kinds h" shows "known_ptr ptr"
G_fcs_eq: fixes G :: "('c, 'd) pre_digraph"   and rev_G :: "'d \<Rightarrow> 'd"   and H :: "('c, 'd) pre_digraph"   and rev_H :: "'d \<Rightarrow> 'd"   and u :: "'c"   and v :: "'c"   and w :: "'c"   and uv :: "'d"   and uw :: "'d"   and vw :: "'d"   and HM :: "'d pre_map"   and a :: "'d" assumes "subdiv1_contr G rev_G H rev_H u v w uv uw vw HM"   and "a \<in> arcs G" shows "pre_digraph_map.face_cycle_set (subdiv1_contr.GM rev_G rev_H uv uw vw HM) a = subdiv1_contr.proj_arcs_H rev_G rev_H uv uw vw ` pre_digraph_map.face_cycle_set HM (subdiv1_contr.proj_arcs_G rev_G uv uw vw a)"
perfect_sec_P2_def: fixes R2 :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd \<times> 'e) spmf"   and S2 :: "'b \<Rightarrow> 'e \<Rightarrow> 'c spmf"   and Out2 :: "'b \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'e) spmf"   and m1 :: "'a"   and m2 :: "'b" shows "Semi_Honest_Def.sim_non_det_def.perfect_sec_P2 R2 S2 Out2 funct m1 m2 \<equiv> R2 m1 m2 = funct m1 m2 \<bind> (\<lambda>(s1, s2). Semi_Honest_Def.sim_non_det_def.Ideal2 S2 Out2 m2 m1 s2)"
rank_token_squats: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.token_squats \<Sigma> \<delta> q\<^sub>0 w x"   and "x \<le> n" shows "\<exists>i. semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = Some i"
is_Zp_cauchy_imp_has_limit: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "is_Zp_cauchy s"   and "a = res_lim s" shows "Zp_converges_to s a"
inext_append: fixes A :: "nat set"   and B :: "nat set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}"   and "B \<noteq> {}"   and "Max A < iMin B" shows "inext n (A \<union> B) = (if n \<in> B then inext n B else if n = Max A then iMin B else inext n A)"
same_seql: fixes i :: "'e"   and \<sigma>' :: "'e \<Rightarrow> 'f"   and \<sigma> :: "'e \<Rightarrow> 'f"   and P :: "'f \<times> 'g \<Rightarrow> bool"   and s :: "'g" assumes "\<forall>j\<in>{i}. \<sigma>' j = \<sigma> j"   and "seql i P (\<sigma>', s)" shows "seql i P (\<sigma>, s)"
bdia_seq: fixes x :: "'a"   and y :: "'a"   and q :: "'a" shows "\<langle>x \<cdot> y| q = \<langle>y| \<langle>x| q"
member_rec_1: fixes x :: "'d"   and xs :: "'d list"   and y :: "'d" shows "List.member (x # xs) y = (x = y \<or> List.member xs y)"
add: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and b1 :: "'d"   and b2 :: "'d" assumes "module_hom s1 s2 f" shows "f (b1 + b2) = f b1 + f b2"
induct: fixes E :: "(nat \<times> nat) set"   and x1 :: "nat list"   and x2 :: "nat"   and P :: "nat list \<Rightarrow> nat \<Rightarrow> bool" assumes "path_entry E x1 x2"   and "P [] 0"   and "\<And>u v l. \<lbrakk>(u, v) \<in> E; path_entry E l u; P l u\<rbrakk> \<Longrightarrow> P (u # l) v" shows "P x1 x2"
function_uminus_closed: fixes f :: "'c \<Rightarrow> 'a" assumes "f \<in> carrier F" shows "function_uminus S R f \<in> carrier F"
card_UNIV_length_enum: fixes enum :: "'b list"   and enum_all :: "('b \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('b \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "CARD('b) = length enum"
q11_def: shows "q11 = {0, 1, 3, 4, 5, 9}"
grid_empty_ds: fixes b :: "(nat \<times> int) list" shows "grid b {} = {b}"
Sup_distl: fixes x :: "'b"   and Y :: "'b set" shows "x \<cdot> \<Squnion> Y = \<Squnion> \<P> ((\<cdot>) x) Y"
coeff_sum: fixes p :: "'b \<Rightarrow> 'a poly"   and A :: "'b set"   and i :: "nat" shows "coeff (sum p A) i = (\<Sum>x\<in>A. coeff (p x) i)"
abs_eq: fixes x :: "'c list"   and xa :: "'c" assumes "eq_onp (\<lambda>xs. equal_base.list_distinct ceq' xs \<or> ID CEQ('c) = None) x x" shows "DList_Set.insert xa (Abs_dlist x) = Abs_dlist (equal_base.list_insert ceq' xa x)"
set_encode_inf: fixes A :: "nat set" assumes "infinite A" shows "set_encode A = 0"
filtersE: fixes F :: "'b set set"   and T :: "'b set set" assumes "F \<in> Filters\<^bsub>T\<^esub>" obtains "{} \<notin> F" and "\<Union> F \<subseteq> carrier\<^bsub>T\<^esub>" and "\<forall>A B. A \<in> F \<and> B \<in> F \<longrightarrow> A \<inter> B \<in> F" and "\<forall>A B. A \<in> F \<and> A \<subseteq> B \<and> B \<subseteq> carrier\<^bsub>T\<^esub> \<longrightarrow> B \<in> F"
1: fixes i :: "nat" shows "integrable lborel (\<lambda>x. indicat_real {l i..u i} x *\<^sub>R f x)"
U_ce: shows "{p x |x. x \<in> UNIV} \<in> ce_sets"
UTM_unhalt_lemma: fixes p :: "(action \<times> nat) list"   and i :: "nat"   and args :: "nat list" assumes "tm_wf0 p"   and "{\<lambda>tp. tp = (Bk \<up> i, <args>)} p \<up>"   and "args \<noteq> []" shows "{\<lambda>tp. tp = ([], <code p # args>)} UTM \<up>"
T_A_eq: shows "T_A (length rs) = int (T s0 rs as)"
trans: fixes K :: "'a multiset"   and M :: "'a multiset"   and N :: "'a multiset" assumes "(K, M) \<in> mult {(x, y). x < y}"   and "(M, N) \<in> mult {(x, y). x < y}" shows "(K, N) \<in> mult {(x, y). x < y}"
Runs: shows "h.mthr.if.mthr.Runs (init_fin_lift_state status (h.start_state f P C M vs)) E''"
principal_ideal_n_pow: fixes R :: "('c, 'd) Ring_scheme"   and a :: "'c"   and I :: "'c set"   and n :: "nat" assumes "Ring R"   and "a \<in> carrier R"   and "I = R \<diamondsuit>\<^sub>p a" shows "I \<^bsup>\<diamondsuit>R n\<^esup> = R \<diamondsuit>\<^sub>p (a^\<^bsup>R n\<^esup>)"
sr_max_Y: fixes y :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "y \<in> Y" shows "r y \<le> sr"
index_distinct_alt_ss: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2"   and "\<And>G. G \<in> groups \<Longrightarrow> \<not> ps \<subseteq> G" shows "int (block_collection index ps) = index"
Uniq_I: fixes P :: "'d \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x" shows "Uniq P"
vcomp_def: fixes \<nu> :: "'a arrow_of_spans_data"   and \<mu> :: "'a arrow_of_spans_data" shows "\<nu> \<bullet> \<mu> \<equiv> if arrow_of_spans (\<cdot>) \<mu> \<and> arrow_of_spans (\<cdot>) \<nu> \<and> Dom \<nu> = Cod \<mu> then \<lparr>Chn = Chn \<nu> \<cdot> Chn \<mu>, Dom = Dom \<mu>, Cod = Cod \<nu>\<rparr> else Null"
nonzeros_binop_pdevs_subset: fixes f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and x :: "nat \<Rightarrow> 'e"   and y :: "nat \<Rightarrow> 'f" shows "{i. binop_pdevs_raw f x y i \<noteq> (0::'d)} \<subseteq> {i. x i \<noteq> (0::'e)} \<union> {i. y i \<noteq> (0::'f)}"
help_lemma: fixes c :: "('b, 't, 'loc, 'c) Combined.configuration_scheme"   and p :: "'b"   and loc0 :: "'loc"   and t0 :: "'t"   and loc1 :: "'loc"   and t1 :: "'t"   and s2 :: "'sum"   and loc2 :: "'loc" assumes "0 < zcount (c_pts (prop_config c p) loc0) t0"   and "(loc0, t0)\<le>\<^sub>p(loc1, t1)"   and "s2 \<in>\<^sub>A path_summary loc1 loc2"   and "safe (prop_config c p)" shows "\<exists>t2\<le>results_in t1 s2. t2 \<in>\<^sub>A frontier (c_imp (prop_config c p) loc2)"
select_convs_1: fixes mstate_i :: "nat"   and mstate_m :: "'a mformula"   and mstate_n :: "nat"   and more :: "'b" shows "mstate_i \<lparr>mstate_i = mstate_i, mstate_m = mstate_m, mstate_n = mstate_n, \<dots> = more\<rparr> = mstate_i"
parallel_fix_ind: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'a \<rightarrow> 'a"   and G :: "'b \<rightarrow> 'b" assumes "adm (\<lambda>x. P (fst x) (snd x))"   and "P \<bottom> \<bottom>"   and "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)" shows "P (fix\<cdot>F) (fix\<cdot>G)"
line_integral_on_pair_straight_path_1: fixes \<gamma> :: "real \<Rightarrow> 'a"   and i :: "'a"   and a :: "real"   and F :: "'a \<Rightarrow> 'a" assumes "\<forall>x. \<gamma> x \<bullet> i = a"   and "\<forall>x\<in>{0..1}. \<gamma> differentiable at x" shows "line_integral F {i} \<gamma> = 0"
equivalence_preserved_by_iso_right: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a"   and \<phi> :: "'a"   and g' :: "'a" assumes "internal_equivalence f g \<eta> \<epsilon>"   and "\<guillemotleft>\<phi> : g \<Rightarrow> g'\<guillemotright>"   and "local.iso \<phi>" shows "internal_equivalence f g' ((\<phi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (f \<star> local.inv \<phi>))"
ap_spmf_def: fixes f :: "('b \<Rightarrow> 'a) spmf"   and x :: "'b spmf" shows "f \<diamondop> x = map_spmf (\<lambda>(f, x). f x) (pair_spmf f x)"
lfp_the_while_option_lattice: fixes f :: "'b \<Rightarrow> 'b" assumes "mono f"   and "finite UNIV" shows "lfp f = the (while_option (\<lambda>A. f A \<noteq> A) f bot)"
ZusammenziehungListeS: fixes \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<Theta> @ \<Theta>" shows "\<Gamma> \<Rightarrow> \<Theta>"
Pi_I: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g"   and B :: "'f \<Rightarrow> 'g set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B x" shows "f \<in> Pi A B"
paths_cross_once: fixes Q :: "'a set"   and R :: "'a set" assumes "Q \<in> \<P>"   and "R \<in> \<P>"   and "Q \<noteq> R"   and "Q \<inter> R \<noteq> {}" shows "\<exists>!a. a \<in> \<E> \<and> Q \<inter> R = {a}"
simps_43: fixes x71 :: "key"   and x72 :: "msg"   and x4 :: "key" shows "Crypt x71 x72 \<noteq> Key x4"
elims_2: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set" assumes "validate_ground_clause x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and C :: "'b literal set" where "x = I" and "xa = C" and "\<exists>L. L \<in> C \<and> validate_ground_lit I L"
some_gcd_dvd_1: fixes a :: "'a"   and b :: "'a"   and d :: "'a" shows "(some_gcd a b dvd d) = (gcd a b dvd d)"
h_col: fixes i :: "nat" assumes "i \<in> {..<l + 1}" shows "col (h 0) = col (h i)"
in_results_gpv_sub_gvps: fixes x :: "'a"   and \<I> :: "('b, 'c) \<I>"   and gpv' :: "('a, 'b, 'c) gpv"   and gpv :: "('a, 'b, 'c) gpv" assumes "x \<in> results_gpv \<I> gpv'"   and "gpv' \<in> sub_gpvs \<I> gpv" shows "x \<in> results_gpv \<I> gpv"
syzygy_module_def: fixes B :: "'b set" shows "syzygy_module B = {s. represents B s (0::'b)}"
init: fixes Mkt :: "('c, 'd) discrete_market"   and v :: "real"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and asset :: "'d" shows "remaining_qty Mkt v pf asset 0 = (\<lambda>w. 0)"
proj_successors: fixes vs :: "'c set"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and s :: "('c, 'd) fmap" shows "fmrestrict_set vs ` state_successors (prob_proj PROB vs) s \<subseteq> state_successors (prob_proj PROB vs) (fmrestrict_set vs s)"
pre1: fixes gs :: "(nat, nat list list list) trie"   and g :: "nat list list" assumes "gsopt = Some gs"   and "g \<in> set_tries gs" shows "pre_iso_test g"
inf_sup_aci_5: fixes inf :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and less_eq :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and less :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and sup :: "'h \<Rightarrow> 'h \<Rightarrow> 'h"   and x :: "'h"   and y :: "'h" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
comp_eq_elim: fixes a :: "'j \<Rightarrow> 'i"   and b :: "'h \<Rightarrow> 'j"   and c :: "'k \<Rightarrow> 'i"   and d :: "'h \<Rightarrow> 'k" assumes "a \<circ> b = c \<circ> d" obtains "\<And>v. a (b v) = c (d v)"
assms_2: shows "H1 \<lhd> G\<lparr>carrier := H\<rparr>"
sets_PiM_single: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
rel_fun_def_butlast: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and S :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and T :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'd \<Rightarrow> 'e"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'g" shows "(R ===> S ===> T) f g = (\<forall>x y. R x y \<longrightarrow> (S ===> T) (f x) (g y))"
F_Inf_have_prems: fixes \<iota> :: "'a literal multiset inference" assumes "\<iota> \<in> F_Inf" shows "prems_of \<iota> \<noteq> []"
simps_5: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "paval' (Div a\<^sub>1 a\<^sub>2) s = optdiv (paval' a\<^sub>1 s) (paval' a\<^sub>2 s)"
\<tau>red1r_preserves_len: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "length xs' = length xs"
Sigma_Int_distrib2: fixes I :: "'a set"   and A :: "'a \<Rightarrow> 'b set"   and B :: "'a \<Rightarrow> 'b set" shows "(SIGMA i:I. A i \<inter> B i) = Sigma I A \<inter> Sigma I B"
trans_constraint_creats_LEQ_only: fixes x :: "constraint" assumes "transf_constraint x \<noteq> []" shows "\<forall>x\<in>set (transf_constraint x). \<exists>a b. x = LEQ a b"
s2r_inter_hom: fixes A :: "'b set"   and B :: "'b set" shows "s2r (A \<inter> B) = s2r A \<inter> s2r B"
psubstAbs_simp: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodEnv rho"   and "freshEnv xs x rho" shows "Abs xs x X $[rho] = Abs xs x X #[rho]"
isolated_verts_def: fixes G :: "('c, 'd) pre_digraph" shows "pre_digraph.isolated_verts G \<equiv> {v \<in> verts G. out_arcs G v = {}}"
tmadd_allpolys_npoly: fixes t :: "tm"   and s :: "tm" assumes "allpolys isnpoly t"   and "allpolys isnpoly s" shows "allpolys isnpoly (tmadd t s)"
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "sum g (insert x A) = g x + sum g (A - {x})"
fds_one_abs_converges: fixes s :: "'a" shows "fds_abs_converges 1 s"
harm_aux_ineq_2: fixes k :: "real" assumes "1 \<le> k" shows "1 / (k + 1) < ln (1 + 2 / (2 * k + 1))"
dvd_gcdD1: 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 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"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a (gcd b c)" shows "dvd.dvd times a b"
regular_closed_p: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "uminus x = uminus (uminus (uminus x))"
test_bit_def: fixes m :: "nat"   and n :: "nat" shows "test_bit m n \<equiv> m div 2 ^ n mod 2 = 1"
deny_pfun_def: fixes pf :: "'e \<Rightarrow> 'f option" shows "D\<^sub>p pf \<equiv> \<lambda>x. case pf x of \<bottom> \<Rightarrow> \<bottom> | \<lfloor>y\<rfloor> \<Rightarrow> \<lfloor>deny y\<rfloor>"
assms: shows "|A| <o natLeq +c |UNIV|"
split_time_nempty: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'a list"   and t :: "nat" assumes "0 < n" shows "split_time s n (n * t) = s t"
awhd_of_awalk: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "awalk u p v" shows "awhd u p = u"
nonpos_Reals_subset_Reals: shows "\<real>\<^sub>\<le>\<^sub>0 \<subseteq> \<real>"
dgrad_setD: fixes t :: "'a"   and d :: "'a \<Rightarrow> nat"   and m :: "nat" assumes "t \<in> dgrad_set d m" shows "d t \<le> m"
max_step_mono_r: fixes s :: "real"   and j :: "nat" assumes "\<bar>(r - b) / 2\<bar> \<le> \<bar>s\<bar>"   and "0 \<le> B'"   and "0 \<le> euler_C TYPE('a)"   and "t 0 \<le> t0 + e2'"   and "0 < 1"   and "t j \<le> t0 + e2'" shows "max_stepsize j \<le> root 1 (\<bar>s\<bar> * B' / euler_C TYPE('a) / (exp (B' * (t0 + e2' - t 0) + 1) - 1))"
is_Sup_binary: fixes x :: "'b"   and y :: "'b"   and sup :: "'b" shows "is_Sup {x, y} sup = is_sup x y sup"
prems_2: shows "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"
WellDefinedFnE: fixes S :: "('a, 'b) Signature"   and A :: "'a"   and f :: "'b"   and e :: "('a, 'b) Expression"   and C :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> f E@ e : C" obtains B :: "'a" where "Sig S \<triangleright> Vx : A \<turnstile> e : B" and "f \<in> Sig S : B \<rightarrow> C"
downward_complete_def: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "dc_on P A = (\<forall>C. chain_on P C A \<and> C \<noteq> {} \<longrightarrow> (\<exists>x\<in>A. glb P C x))"
simps: fixes a :: "'a list"   and b :: "'a list" shows "delete (BUF a b) = BUF (butlast a) b"
reduc_fair_def: fixes Ns :: "'f set llist" shows "reduc_fair Ns = (Inf_from (Liminf_llist Ns - Sup_llist (lmap Red_F Ns)) \<subseteq> Sup_llist (lmap Red_I Ns))"
exhaust: fixes y :: "'b tree" obtains "y = ET"   | x21 :: "'b" and x22 :: "'b tree" and x23 :: "'b tree" and x24 :: "nat" where "y = MKT x21 x22 x23 x24"
len_lexext_rev_total: fixes B :: "'b set"   and A :: "'b set"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" assumes "\<forall>y\<in>B. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x"   and "ys \<in> lists B"   and "xs \<in> lists A" shows "len_lexext_rev gt ys xs \<or> len_lexext_rev gt xs ys \<or> ys = xs"