hcju/isabelleps huggingface dataset:
final_threads_def: fixes final :: "'b \<Rightarrow> bool"   and s :: "('c, 'a) locks \<times> (('a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option) \<times> 'd) \<times> ('a \<Rightarrow> 'e wait_set_status option) \<times> 'a set" shows "final_thread.final_threads final s \<equiv> {t. final_thread.final_thread final s t}"
divide_eq1: fixes h :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. h x \<noteq> (0::'b)" shows "(f \<in> O[F](\<lambda>x. g x / h x)) = ((\<lambda>x. f x * h x) \<in> O[F](g))"
surj_inv: fixes f :: "'a \<Rightarrow> 'a" assumes "bijection f" shows "surj (inv f)"
smsI: fixes A :: "(nat \<times> nat) multiset"   and B :: "(nat \<times> nat) multiset"   and Z :: "(nat \<times> nat) multiset" assumes "(set_mset A, set_mset B) \<in> max_strict" shows "(Z + A, Z + B) \<in> ms_strict"
measurable_sets: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and A :: "'c measure"   and S :: "'c set" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "S \<in> sets A" shows "f -` S \<inter> space M \<in> sets M"
del_invalid_point_blocks: fixes p :: "'a" assumes "p \<notin> \<V>" shows "del_point_blocks p = \<B>"
Id_on_fold: fixes A :: "'b set" assumes "finite A" shows "Id_on A = Finite_Set.fold (\<lambda>x. insert (x, x)) {} A"
shiftl_bl: fixes w :: "'a word"   and n :: "nat" shows "w << n = of_bl (to_bl w @ replicate n False)"
rep_eq: fixes x :: "('b \<Rightarrow> 'a) dnelist"   and xa :: "'b dnelist" shows "list_of_dnelist (x \<diamondop> xa) = remdups (list_of_dnelist x \<diamondop> list_of_dnelist xa)"
in_homI: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and b :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.arr f"   and "J_C.dom f = a"   and "J_C.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright>"
PiE_def: fixes S :: "'c set"   and T :: "'c \<Rightarrow> 'd set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
simps: fixes x :: "tm"   and y :: "tm" shows "x NEQ y = OrdP x AND OrdP y AND (x IN y OR y IN x)"
filterlim_within_subset: fixes "open" :: "'b set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c filter"   and x :: "'b"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "filterlim f l (topological_space.at_within open x S)"   and "T \<subseteq> S" shows "filterlim f l (topological_space.at_within open x T)"
Const\<^sub>0_add: fixes a :: "'b"   and b :: "'b" shows "Const\<^sub>0 (a + b) = Const\<^sub>0 a + Const\<^sub>0 b"
bdia_whilei_break: fixes y :: "('b \<times> 'b) set"   and p :: "('b \<times> 'b) set"   and i :: "('b \<times> 'b) set"   and t :: "('b \<times> 'b) set"   and q :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "rel_antirange_kleene_algebra.bdia y p \<subseteq> rel_antirange_kleene_algebra.ars_r i"   and "rel_antirange_kleene_algebra.ars_r i ; rel_ad t \<subseteq> rel_antirange_kleene_algebra.ars_r q"   and "rel_antirange_kleene_algebra.bdia x (rdom t ; rel_antirange_kleene_algebra.ars_r i) \<subseteq> rel_antirange_kleene_algebra.ars_r i" shows "rel_antirange_kleene_algebra.bdia (y ; rel_antidomain_kleene_algebra.whilei t i x) p \<subseteq> rel_antirange_kleene_algebra.ars_r q"
igSwapAbsIPresIGWlsAbs_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igSwapAbsIPresIGWlsAbs MOD \<equiv> \<forall>zs z1 z2 us s A. isInBar (us, s) \<and> igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD (us, s) (igSwapAbs MOD zs z1 z2 A)"
sc_a_0: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and a :: "'f" assumes "R module M"   and "a \<in> carrier R" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> \<zero>\<^bsub>M\<^esub> = \<zero>\<^bsub>M\<^esub>"
assms_2: shows "sys \<Turnstile>\<^bsub>pre\<^esub> I"
Overriding: fixes new :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides\<^sub>S old"   and "G \<turnstile>Method new member_of C" shows "thesis"
strict_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "strict_prefix a b"   and "strict_prefix b c" shows "strict_prefix a c"
bounded_two_points: fixes S :: "'a set" shows "bounded S = (\<exists>e. \<forall>x\<in>S. \<forall>y\<in>S. dist x y \<le> e)"
is_processT5_S2: fixes s :: "'a event list"   and X :: "'a event set"   and P :: "'a process"   and c :: "'a event" assumes "(s, X) \<in> F P"   and "(s @ [c], {}) \<notin> F P" shows "(s, X \<union> {c}) \<in> F P"
root_in_start_points: fixes r :: "'a"   and x :: "'a" assumes "connected_root r x"   and "is_vector r"   and "x \<noteq> 0"   and "x ; r = 0" shows "r \<le> start_points x"
Re_divide_real: fixes b :: "complex"   and a :: "complex" assumes "is_real b"   and "b \<noteq> 0" shows "Re (a / b) = Re a / Re b"
rel_carrier_Eps_in_3: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and X :: "('c set \<times> 'd) set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "X \<in> dlim \<FF> \<rho> I" shows "X = direct_lim.class_of \<FF> \<rho> I (fst (SOME x. x \<in> X)) (snd (SOME x. x \<in> X))"
shr_conv: fixes ls :: "'h"   and ts :: "'i"   and m :: "'g"   and ws :: "'j"   and "is" :: "'k" shows "shr (ls, (ts, m), ws, is) = m"
map_update_def: fixes \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a" shows "map_update \<alpha> invar update \<equiv> (\<forall>m k v. invar m \<longrightarrow> \<alpha> (update k v m) = \<alpha> m(k \<mapsto> v)) \<and> (\<forall>m k v. invar m \<longrightarrow> invar (update k v m))"
l3_inv5_derived: shows "l3_inv2 \<inter> l3_inv3 \<subseteq> l3_inv5"
star: shows "T\<^sub>p [x, y] (c @ d) (OPT2 (c @ d) [x, y]) = length c div 2 + T\<^sub>p [x, y] d (OPT2 d [x, y]) \<and> length c mod 2 = 0"
dgrad_set_leE: fixes d :: "'b \<Rightarrow> nat"   and S :: "'b set"   and T :: "'b set"   and s :: "'b" assumes "dgrad_set_le d S T"   and "s \<in> S" obtains t :: "'b" where "t \<in> T" and "d s \<le> d t"
UNIV_enum: fixes enum :: "'c list"   and enum_all :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "UNIV = set enum"
sign_simps_17: fixes aa :: "'b"   and ba :: "'b"   and ca :: "'b" shows "(aa + ba) * ca = aa * ca + ba * ca"
l_set_val_get_disconnected_nodes_axioms_def: fixes set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_disconnected_nodes_axioms set_val_locs get_disconnected_nodes_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_val_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
simps: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "gen_qe var L F = (case F of [] \<Rightarrow> case luckyFind var L [] of None \<Rightarrow> if all_degree_2 var L then list_disj (list_conj (map (substNegInfinity var) L) # map (elimVar var L []) L) else qe_eq_repeat var L [] | Some F \<Rightarrow> F | a # list \<Rightarrow> qe_eq_repeat var L F)"
simps_30: fixes x3 :: "'c" shows "set1_sum3 (In3 x3) = {}"
div_eq_dividend_iff: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "(m div n = m) = (n = 1)"
LCoset_rel_quotientI: fixes x :: "'g"   and uu :: "'g set" assumes "x \<in> G" shows "LCoset_rel uu `` {x} \<in> G // LCoset_rel uu"
code: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "siterate f x = x ## siterate f (f x)"
proj2_addpre: fixes v :: "nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set"   and pre :: "nat set" shows "\<pi>\<^sub>2 (addpre v pre) = \<pi>\<^sub>2 v"
sig_regular_top_reduced_lt_lc_unique: fixes d :: "'a \<Rightarrow> nat"   and G :: "('t \<Rightarrow>\<^sub>0 'b) set"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "dickson_grading d"   and "is_sig_GB_upt d G (lt q)"   and "p \<in> dgrad_sig_set' (length fs) d"   and "q \<in> dgrad_sig_set' (length fs) d"   and "lt p = lt q"   and "(p = 0) = (q = 0)"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G p"   and "\<not> is_sig_red (\<prec>\<^sub>t) (=) G q" shows "punit.lt (rep_list p) = punit.lt (rep_list q) \<and> lc q * punit.lc (rep_list p) = lc p * punit.lc (rep_list q)"
assms: fixes x :: "'a"   and y :: "'a" shows "poly2 p x y = poly2 q x y"
is_Receive_def: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" shows "stateful_strand_step.is_Receive stateful_strand_step = (\<exists>x2. stateful_strand_step = receive\<langle>x2\<rangle>)"
gcd_rat_def: shows "gcd = normalization_euclidean_semiring_class.gcd"
add_aform': fixes e :: "nat \<Rightarrow> real"   and x :: "real"   and X :: "(real \<times> real pdevs) \<times> real"   and y :: "real"   and Y :: "(real \<times> real pdevs) \<times> real"   and p :: "nat" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "x \<in> aform_err e X"   and "y \<in> aform_err e Y" shows "x + y \<in> aform_err e (add_aform' p X Y)"
S4: shows "c_fst (c_hd (c_cons (c_pair x y) z)) \<noteq> x1"
A_B_step: fixes a :: "'a"   and b :: "'a"   and a' :: "'b" assumes "A a b"   and "a \<sim> a'" shows "\<exists>b'. B a' b' \<and> b \<sim> b'"
Rep_uexpr_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> UNIV" obtains x :: "('b, 'a) uexpr" where "y = \<lbrakk>x\<rbrakk>\<^sub>e"
tranclp_unfold: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e" shows "r\<^sup>+\<^sup>+ a b \<equiv> (a, b) \<in> {(x, y). r x y}\<^sup>+"
convert_extTA_simps_1: fixes f :: "'i \<Rightarrow> 'e" shows "convert_extTA f \<lbrace>\<rbrace> = \<lbrace>\<rbrace>"
assms: shows "\<Psi> \<rhd> P \<approx> Q"
length_eq_val_length_tm: fixes xs :: "'a list" shows "Time_Monad.val (length_tm xs) = length xs"
LeftDerivationLadder_length_1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.LeftDerivationLadder \<NN> \<TT> \<RR> \<alpha> D L \<gamma>"   and "length L = 1" shows "LocalLexing.LeftDerivationFix \<NN> \<TT> \<RR> \<alpha> (ladder_i L 0) D (ladder_last_j L) \<gamma>"
carrier_one_not_zero: shows "(carrier K \<noteq> {\<zero>\<^bsub>K\<^esub>}) = (\<one>\<^bsub>K\<^esub> \<noteq> \<zero>\<^bsub>K\<^esub>)"
cut_greater_mono: fixes t' :: "'a"   and t :: "'a"   and I :: "'a set" assumes "t' \<le> t" shows "I \<down>> t \<subseteq> I \<down>> t'"
LowLink_le_disc: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme"   and v :: "'c" assumes "DFS_invar G param s"   and "v \<in> dom (discovered s)" shows "Min (the_discovered s ` param_DFS_defs.lowlink_set G s v) \<le> the_discovered s v"
step_1_correct: fixes p :: "prog" assumes "spec\<^sub>1 p" shows "spec\<^sub>0 p"
cfg: shows "continuous_on {a..b} (\<lambda>x. f (g x))"
isoI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.inverse_arrows f g" shows "VV.iso f"
DsC: shows "isCont Ds (poincare_map {x \<in> S. s x = 0} x)"
a0: shows "\<alpha> = \<phi> $$ (0, 0)"
assms_1: shows "A \<Turnstile> (S, U \<rightarrow>) P"
S4Hist: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> HNext rmhist p \<and> $S4 rmhist p \<and> m p$ = $m p \<longrightarrow> (rmhist!p)$ = $(rmhist!p)"
bisim1Sync10: fixes P :: "'b prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([Addr a], xs, 8 + length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
RSD_pareto_eqclassD_3: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "RSD_pareto_eqclass agents alts R A"   and "x \<in> A"   and "y \<in> alts" shows "(y \<in> A) = (\<forall>i\<in>agents. R i x y)"
arith_simps_50: fixes a :: "'a" shows "a + (0::'a) = a"
targets_char': fixes \<mu> :: "'a"   and a :: "'a" assumes "arr \<mu>" shows "(a \<in> targets \<mu>) = (some_trg \<mu> \<cong> a)"
weak_le_antisym: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c"   and y :: "'c" assumes "weak_partial_order L"   and "x \<sqsubseteq>\<^bsub>L\<^esub> y"   and "y \<sqsubseteq>\<^bsub>L\<^esub> x"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x .=\<^bsub>L\<^esub> y"
smap_into_PosetComplex: fixes X :: "'w set" assumes "X \<in> \<P>" shows "dual_order.poset_simplex_map \<P> X \<in> \<Sigma>"
exists_next_if_can_occur: fixes ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration" assumes "can_occur ev c" shows "\<exists>c'. c \<turnstile> ev \<mapsto> c'"
h1b_helper_neq: fixes neq :: "(real \<times> real \<times> real) list" assumes "\<forall>(a, b, c)\<in>set neq. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0" shows "\<exists>y. \<forall>x<y. \<forall>(a, b, c)\<in>set neq. a * x\<^sup>2 + b * x + c \<noteq> 0"
fps_nth_compose_X_power: fixes f :: "'b fps"   and d :: "nat"   and n :: "nat" shows "(f oo fps_X ^ d) $ n = (if d dvd n then f $ (n div d) else (0::'b))"
hyps_7: shows "b =\<^sub>F b\<^sub>1 @ b\<^sub>2"
min_list: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs" shows "Missing_List.min_list xs \<le> x"
product_complement_in_complements_sum: fixes x :: "i"   and y :: "i" assumes "x \<noteq> u"   and "y \<noteq> u" shows "P (\<midarrow> (x \<otimes> y)) (\<midarrow> x \<oplus> \<midarrow> y)"
Shuffle_def: fixes r :: "'a language"   and s :: "'a language" shows "Shuffle r s = Times_Plus (ShuffleLR r s)"
mat_addcol_def: fixes a :: "'a"   and k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "addcol a k l A = Matrix.mat (dim_row A) (dim_col A) (\<lambda>(i, j). if k = j then a * A $$ (i, l) + A $$ (i, j) else A $$ (i, j))"
fold_max_mono: fixes x :: "'b"   and y :: "'b"   and zs :: "'b list" assumes "x \<le> y" shows "fold max zs x \<le> fold max zs y"
simps_2: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "deep_model' Y (r # rs) = Pool (deep_model Y r rs) (deep_model Y r rs)"
bit_of_nat_iff_bit: fixes m :: "nat"   and n :: "nat" shows "bit (of_nat m) n = bit m n"
mset_subset_eq_add_mset_cancel: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(add_mset a A \<subseteq># add_mset a B) = (A \<subseteq># B)"
op_union_phantom_def: shows "op_union_phantom = (\<union>)"
trans: fixes b :: "'b"   and a :: "'b"   and c :: "'b" assumes "order.greater_eq a b"   and "order.greater_eq b c" shows "order.greater_eq a c"
BIT: shows "T\<^sub>p_on_rand' BIT (type0 [x0, y0] x y) qs = 0"
wt_App_fun_\<delta>\<^sub>h: fixes A :: "'v pvar \<Rightarrow> zhmultiset"   and s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "legal_zpassign A"   and "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt (App s t))) = eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt t))" shows "eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of (wt s)) = zhmset_of \<delta>\<^sub>h"
usubstappp_game: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and a :: "char"   and r :: "game"   and U :: "variable set" assumes "SGames \<sigma> a = Agame r" shows "usubstappp \<sigma> U (Game a) = (U \<union> BVG r, Agame r)"
assms: shows "(b, cs) |\<in>| fgroup_by f as"
smult_smult: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "smult a (smult b p) = smult (a * b) p"
s_S_defs_1: shows "s = \<lparr>inst = insts, funcs = fs, tab = clss, mem = bss, globs = gs\<rparr>"
simps_6: fixes x3 :: "'a" shows "vertex.SINK \<noteq> \<langle>x3\<rangle>"
span_union_is_sum: fixes A :: "'a vec set"   and B :: "'a vec set" assumes "A \<subseteq> carrier_vec n"   and "B \<subseteq> carrier_vec n" shows "local.span (A \<union> B) = subspace_sum (local.span A) (local.span B)"
norm_pres_extensionI2: fixes H :: "'a set"   and h :: "'a \<Rightarrow> real"   and E :: "'a set"   and F :: "'a set"   and f :: "'a \<Rightarrow> real"   and p :: "'a \<Rightarrow> real" assumes "linearform H h"   and "Subspace.subspace H E"   and "Subspace.subspace F H"   and "graph F f \<subseteq> graph H h"   and "\<forall>x\<in>H. h x \<le> p x" shows "graph H h \<in> norm_pres_extensions E p F f"
bf_between_arcs_def: fixes a :: "'a"   and b :: "'a"   and H :: "'a"   and d :: "'a" shows "bf_between_arcs a b H d \<equiv> (bijective (a * top) \<and> bijective (a[[top]])) \<and> (bijective (b * top) \<and> bijective (b[[top]])) \<and> a[[top]] \<le> (H * d)\<^sup>\<star> * H * b * top"
distr_Y_seq: fixes n :: "nat" shows "distr Omega borel (Y_seq n) = \<mu> n"
Pow_def: fixes A :: "'a set" shows "Pow A = {B. B \<subseteq> A}"
OclImplies_true2: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X implies true) = true"
remdups_adj_append'': fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<noteq> []" shows "remdups_adj (xs @ ys) = remdups_adj xs @ remdups_adj (dropWhile (\<lambda>y. y = last xs) ys)"
IH: fixes "is" :: "nat list" assumes "\<And>B. B \<in> set Bs \<Longrightarrow> Tensor.order B = 1"   and "is \<lhd> dims (prod_list Bs)" shows "lookup (prod_list Bs) is = prod_list (map2 (\<lambda>i B. lookup B [i]) is Bs)"
all_finite: shows "\<forall>x\<in>S. finite (cl_ecl x)"
ord0_le_less_trans: fixes x :: "ord0"   and y :: "ord0"   and z :: "ord0" assumes "x \<le> y"   and "y < z" shows "x < z"
poly_ext: fixes p :: "'b poly"   and q :: "'b poly" assumes "\<And>x. poly p x = poly q x" shows "p = q"
terminates_tl_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step" assumes "terminates g" shows "terminates (tl_raw g)"
interaction_bounded_by_inline_invariant: fixes consider' :: "'a \<Rightarrow> bool"   and gpv :: "('b, 'a, 'c) gpv"   and p :: "enat"   and I :: "'d \<Rightarrow> bool"   and "consider" :: "'e \<Rightarrow> bool"   and callee :: "'d \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'd, 'e, 'f) gpv"   and q :: "enat"   and s :: "'d" assumes "interaction_bounded_by consider' gpv p"   and "\<And>s x. \<lbrakk>I s; consider' x\<rbrakk> \<Longrightarrow> interaction_bounded_by consider (callee s x) q"   and "\<And>s x. \<lbrakk>I s; \<not> consider' x\<rbrakk> \<Longrightarrow> interaction_bounded_by consider (callee s x) 0"   and "I s"   and "\<And>s x y s'. \<lbrakk>(y, s') \<in> results'_gpv (callee s x); I s\<rbrakk> \<Longrightarrow> I s'" shows "interaction_bounded_by consider (inline callee gpv s) (p * q)"
cmod_diff_ge: fixes b :: "complex"   and c :: "complex" shows "sqrt (1 + (cmod b)\<^sup>2) - sqrt (1 + (cmod c)\<^sup>2) \<le> cmod (b - c)"
monotone_set_spmf: shows "monotone (ord_spmf (=)) (\<subseteq>) set_spmf"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and f :: "'fsym"   and Tl :: "'fsym trm list" shows "I.int \<xi> (Fn f Tl) = I_intF f (map (I.int \<xi>) Tl)"
permutation_bijective: fixes p :: "'b \<Rightarrow> 'b" assumes "permutation p" shows "bij p"
reflexive_inf_plus_star: fixes x :: "'a"   and y :: "'a" assumes "reflexive x" shows "coreflexive (x \<sqinter> y\<^sup>+) = (x \<sqinter> y\<^sup>\<star> = (1::'a))"
is_basis_def: fixes S :: "('b, 'c) vec set" shows "is_basis S \<equiv> vec.independent S \<and> vec.span S = UNIV"
seteq_qle_def: fixes qle :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b set"   and N :: "'b set" assumes "quasi_order qle" shows "quasi_order.seteq_qle qle M N \<equiv> quasi_order.subseteq_qle qle M N \<and> quasi_order.subseteq_qle qle N M"
abs_summable_on_insert_iff: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and A :: "'a set" shows "(f abs_summable_on insert x A) = (f abs_summable_on A)"
below0: fixes j :: "int" assumes "j \<le> i" shows "ifac j \<noteq> 0"
sup_idem: fixes x :: "'a multiset" shows "x \<union># x = x"
total_preorder_on_axioms: shows "total_preorder_on alts (R j)"
min_empty: fixes R :: "'a set"   and k :: "nat" assumes "S (min_arg R k) \<inter> R = {}" shows "\<forall>i\<in>{1..k}. S i \<inter> R = {}"
fps_inverse_def: fixes f :: "'b fps" shows "inverse f = fps_right_inverse f (inverse (f $ 0))"
UnitFunctor_def: fixes C :: "('a, 'b, 'c) Category_scheme" shows "UnitFunctor C \<equiv> ConstFunctor C UnitCategory ()"
lcm_int_def: fixes x :: "int"   and y :: "int" shows "lcm x y = int (lcm (nat \<bar>x\<bar>) (nat \<bar>y\<bar>))"
assms_10: shows "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"
is_selection: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A"   and "Tarskian_logic.remainder Cn K \<phi> \<noteq> {}" shows "selection K \<phi> \<subseteq> Tarskian_logic.remainder Cn K \<phi>"
P1_ideal_model_def: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and z :: "'e"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)" shows "P1_ideal_model M \<sigma> z \<A> = (let (\<A>1, \<A>2, \<A>3) = \<A> in sample_uniform (order \<G>) \<bind> (\<lambda>r. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>0. sample_uniform (order \<G>) \<bind> (\<lambda>\<alpha>1. let h0 = \<^bold>g [^] \<alpha>0; h1 = \<^bold>g [^] \<alpha>1; a = \<^bold>g [^] r; b0 = h0 [^] r; b1 = h1 [^] r \<otimes> \<^bold>g in \<A>1 M h0 h1 a b0 b1 z \<bind> (\<lambda>((in1, in2, in3), s). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in assert_spmf ((in1, in2, in3) = (h, a, b)) \<bind> (\<lambda>_. \<A>2 h0 h1 a b0 b1 M s \<bind> (\<lambda>(((w0, z0), w1, z1), s'). let x0' = z0 \<otimes> inv w0 [^] \<alpha>0; x1' = z1 \<otimes> inv w1 [^] \<alpha>1 in mal_def.trusted_party (x0', x1') \<sigma> \<bind> (\<lambda>(uu_, f_out2). \<A>3 s' \<bind> (\<lambda>adv_out. return_spmf (adv_out, f_out2))))))))))"
poly_0_coeff_0: fixes p :: "'a poly" shows "poly p (0::'a) = coeff p 0"
seq_if_composable: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star>\<^sub>C \<mu> \<noteq> C.null" shows "src\<^sub>C \<nu> = trg\<^sub>C \<mu>"
lm125: fixes list :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "set (takeAll P list) \<subseteq> set list"
Abs_name_inverse: fixes y :: "atom" assumes "y \<in> {a. sort_of a = Sort ''SyntaxN.name'' []}" shows "Rep_name (Abs_name y) = y"