hcju/isabelleps huggingface dataset:
cases: fixes x :: "('e, 'f, 'g) Tree \<times> ('e, 'f, 'g) Tree" obtains tset1 :: "('e, 'f, 'g) Tree set['e]" and tset2 :: "('e, 'f, 'g) Tree set['e]" where "x = (tConj tset1, tConj tset2)"   | t1 :: "('e, 'f, 'g) Tree" and t2 :: "('e, 'f, 'g) Tree" where "x = (tNot t1__, tNot t2__)"   | \<phi>1 :: "'f" and \<phi>2 :: "'f" where "x = (tPred \<phi>1, tPred \<phi>2)"   | \<alpha>1 :: "'g" and t1 :: "('e, 'f, 'g) Tree" and \<alpha>2 :: "'g" and t2 :: "('e, 'f, 'g) Tree" where "x = (tAct \<alpha>1 t1, tAct \<alpha>2 t2)"   | v :: "('e, 'f, 'g) Tree" and va :: "('e, 'f, 'g) Tree set['e]" where "x = (tNot v__, tConj va__)"   | v :: "('e, 'f, 'g) Tree" and va :: "'f" where "x = (tNot vb__, tPred vaa__)"   | v :: "('e, 'f, 'g) Tree" and va :: "'g" and vb :: "('e, 'f, 'g) Tree" where "x = (tNot vc__, tAct vab__ vba__)"   | v :: "'f" and va :: "('e, 'f, 'g) Tree set['e]" where "x = (tPred vd__, tConj vac__)"   | v :: "'f" and va :: "('e, 'f, 'g) Tree" where "x = (tPred ve__, tNot vad__)"   | v :: "'f" and va :: "'g" and vb :: "('e, 'f, 'g) Tree" where "x = (tPred vf__, tAct vae__ vbb__)"   | v :: "'g" and va :: "('e, 'f, 'g) Tree" and vb :: "('e, 'f, 'g) Tree set['e]" where "x = (tAct vg__ vaf__, tConj vbc__)"   | v :: "'g" and va :: "('e, 'f, 'g) Tree" and vb :: "('e, 'f, 'g) Tree" where "x = (tAct vh__ vag__, tNot vbd__)"   | v :: "'g" and va :: "('e, 'f, 'g) Tree" and vb :: "'f" where "x = (tAct vi__ vah__, tPred vbe__)"   | va :: "('e, 'f, 'g) Tree set['e]" and v :: "('e, 'f, 'g) Tree" where "x = (tConj vai__, tNot vj__)"   | va :: "'f" and v :: "('e, 'f, 'g) Tree" where "x = (tPred vaj__, tNot vk__)"   | va :: "'g" and vb :: "('e, 'f, 'g) Tree" and v :: "('e, 'f, 'g) Tree" where "x = (tAct vak__ vbf__, tNot vl__)"   | va :: "('e, 'f, 'g) Tree set['e]" and v :: "'f" where "x = (tConj val__, tPred vm__)"   | va :: "('e, 'f, 'g) Tree" and v :: "'f" where "x = (tNot vam__, tPred vn__)"   | va :: "'g" and vb :: "('e, 'f, 'g) Tree" and v :: "'f" where "x = (tAct van__ vbg__, tPred vo__)"   | vb :: "('e, 'f, 'g) Tree set['e]" and v :: "'g" and va :: "('e, 'f, 'g) Tree" where "x = (tConj vbh__, tAct vp__ vao__)"   | vb :: "('e, 'f, 'g) Tree" and v :: "'g" and va :: "('e, 'f, 'g) Tree" where "x = (tNot vbi__, tAct vq__ vap__)"   | vb :: "'f" and v :: "'g" and va :: "('e, 'f, 'g) Tree" where "x = (tPred vb, tAct v va)"
intros_4: shows "OclAny[?] \<sqsubset> OclSuper"
f2: fixes r :: "real"   and ra :: "real" shows "r + (- r + ra) = ra"
fun_mult_closed: fixes R :: "('c, 'd) ring_scheme"   and f :: "'e \<Rightarrow> 'c"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "ring_functions R"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "f \<otimes>\<^bsub>function_ring S R\<^esub> g \<in> carrier (function_ring S R)"
simps: fixes n :: "nat"   and k :: "nat"   and acc :: "nat" shows "catalan_aux n k acc = (if n \<le> k then acc else catalan_aux n (Suc k) (acc * (2 * (2 * k + 1)) div (k + 2)))"
w4: shows "{\<parallel>x\<parallel>\<^sup>2 |x. x \<in> M} \<noteq> {}"
hyps_2: shows "\<Gamma>\<turnstile> \<langle>s,Normal b\<rangle> =n\<Rightarrow> s'"
prime_elem_iff_irreducible: fixes x :: "'b poly" shows "prime_elem x = irreducible x"
update_convs_2: fixes trans' :: "('c \<times> 'd \<times> 'c) set \<Rightarrow> ('c \<times> 'd \<times> 'c) set"   and init :: "'c set"   and trans :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e" shows "trans_update trans' \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = \<lparr>init = init, trans = trans' trans, \<dots> = more\<rparr>"
f_differentiable_at_neg: fixes x :: "real" assumes "x < 0" shows "f differentiable at x"
lemma_3_5_ii_2: fixes a :: "'a" shows "d2 a a = (1::'a)"
thetaSeq_Sbis: shows "thetaSeq \<subseteq> Sbis"
simps_3: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly"   and c' :: "'b" shows "CN c n p +\<^sub>p (c')\<^sub>p = CN (c +\<^sub>p (c')\<^sub>p) n p"
Restrp_asymmetric: shows "asymmetric UNIV ((\<sqsubset>) \<restriction> A)"
ik\<^sub>s\<^sub>s\<^sub>t_trms\<^sub>s\<^sub>s\<^sub>t_subset: fixes A :: "('c, 'd) stateful_strand_step list" shows "ik\<^sub>s\<^sub>s\<^sub>t A \<subseteq> trms\<^sub>s\<^sub>s\<^sub>t A"
simps_4: fixes x4 :: "val"   and y4 :: "val" shows "(Val x4 = Val y4) = (x4 = y4)"
nat_sub_norm: fixes a :: "nat" shows "a = a - 0 \<and> 0 \<le> a"
Sup_bot_conv_1: fixes A :: "'a set" shows "(\<Squnion>A = \<bottom>) = (\<forall>x\<in>A. x = \<bottom>)"
l_cancel_one': fixes x :: "'a"   and a :: "'a" shows "(x = x + a) = (a = (0::'a))"
enn2real_INF: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal" assumes "A \<noteq> {}"   and "\<forall>x\<in>A. f x < \<top>" shows "enn2real (\<Sqinter> (f ` A)) = (\<Sqinter>x\<in>A. enn2real (f x))"
op_list_blit_def: fixes src :: "'a list"   and si :: "nat"   and dst :: "'a list"   and di :: "nat"   and len :: "nat" shows "op_list_blit src si dst di len \<equiv> take di dst @ take len (drop si src) @ drop (di + len) dst"
Iff_Iff: fixes p :: "'a pfm"   and p' :: "'a pfm"   and q :: "'a pfm" assumes "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')"   and "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! q)" shows "\<turnstile>\<^sub>! (p' \<^bold>\<longleftrightarrow>\<^sub>! q)"
ch2ch_Rep_cfunR: fixes Y :: "nat \<Rightarrow> 'e"   and f :: "'e \<rightarrow> 'f" assumes "chain Y" shows "chain (\<lambda>i. f\<cdot>(Y i))"
outside_bounded_nonempty: fixes S :: "'a set" assumes "bounded S" shows "outside S \<noteq> {}"
convergent_add_const_iff: fixes c :: "'a"   and f :: "nat \<Rightarrow> 'a" shows "convergent (\<lambda>n. c + f n) = convergent f"
inst\<^sub>1: shows "instance_of\<^sub>l\<^sub>s C' C"
is_extensional: fixes f :: "'a" assumes "\<not> C.arr f" shows "src\<^sub>C f = C.null"
c_manifolds_order_le: fixes l :: "enat" assumes "l \<le> k" shows "c_manifolds l charts2 charts2"
hyper_isin_def: fixes a :: "char list"   and A :: "char list set option" shows "a \<in>\<in> A \<equiv> case A of None \<Rightarrow> True | \<lfloor>A\<rfloor> \<Rightarrow> a \<in> A"
Atm: fixes atm :: "atom"   and s :: "Concrete.var \<Rightarrow> nat" shows "Example_PL.transT (Atm atm, s) (aval atm s)"
eval\<^sub>1_preserves_len: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>0 :: "nat exp"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>0 :: "val list"   and e\<^sub>1 :: "nat exp"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>1 :: "val list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>0,(h\<^sub>0, ls\<^sub>0)\<rangle> \<Rightarrow> \<langle>e\<^sub>1,(h\<^sub>1, ls\<^sub>1)\<rangle>" shows "length ls\<^sub>0 = length ls\<^sub>1"
mat_delete_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "mat_delete A i j \<equiv> mat (dim_row A - 1) (dim_col A - 1) (\<lambda>(i', j'). A $$ (if i' < i then i' else Suc i', if j' < j then j' else Suc j'))"
typeI_cube_line_integral_exists_boundary'': shows "\<forall>(k, \<gamma>)\<in>two_chain_horizontal_boundary two_chain. line_integral_exists F {i} \<gamma>"
cinner_simps_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<bullet>\<^sub>C (y - z) = x \<bullet>\<^sub>C y - x \<bullet>\<^sub>C z"
yx_2: shows "y \<notin> \<Union> (Fvars ` F)"
set_restriction_fun_is_set_restriction: fixes P :: "'a \<Rightarrow> bool" shows "set_restriction (set_restriction_fun P)"
fmupd_def: shows "fmupd \<equiv> map_fun id (map_fun id (map_fun fmlookup Abs_fmap)) map_upd"
mod_exp_def: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and e :: "nat"   and m :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "euclidean_semiring_cancel.mod_exp times one modulo b e m = modulo (power.power one times b e) m"
least_eq: shows "least_non_zero_position_of_vector (column_iarray (mod_type_class.to_nat j) (matrix_to_iarray A)) = mod_type_class.to_nat (LEAST n. A $ n $ j \<noteq> (0::'a))"
indests: fixes dests :: "nat \<Rightarrow> nat option"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and rsn :: "nat" assumes "\<forall>ip\<in>dom dests. ip \<in> kD rt \<and> sqn rt ip \<le> the (dests ip)"   and "dests ip = Some rsn" shows "ip \<in> kD rt \<and> sqn rt ip \<le> rsn"
rep_eq: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and x :: "'a"   and xa :: "('d, 'b, 'c) formula" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid satisfies transition x xa = nominal_ts.valid_Tree\<^sub>\<alpha> satisfies transition x (Rep_formula xa)"
wt_lOfWax: fixes \<sigma> :: "'tp" assumes "\<not> isRes \<sigma>" shows "TE.wt (lOfWax \<sigma>)"
simps_4: fixes disc :: "'b \<Rightarrow> bool"   and sel :: "'b \<Rightarrow> 'c"   and n :: "'c \<Rightarrow> bool"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "normalized_n_primitive (disc, sel) n (MatchAnd m1 m2) = (normalized_n_primitive (disc, sel) n m1 \<and> normalized_n_primitive (disc, sel) n m2)"
rel_G_pos_distr: fixes Co1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and Co1' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and Co2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and Co2' :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and Contra1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and Contra1' :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and Contra2' :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and uu :: "('o \<times> 'p \<times> 'q \<times> 'r \<times> 's \<times> 't \<times> 'u) itself"   and L1 :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2' :: "'s \<Rightarrow> 't \<Rightarrow> bool" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 OO rel_G L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_G (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2')"
higher_zero: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict" shows "ordered_term.higher ord_term 0 t = 0"
pure_eventual_left_append: fixes \<mu> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and u :: "'a set list" assumes "pure_eventual \<mu>"   and "w \<Turnstile>\<^sub>n \<mu>" shows "(u \<frown> w) \<Turnstile>\<^sub>n \<mu>"
lprefix_llist_of: fixes xs :: "'b list"   and ys :: "'b list" shows "lprefix (llist_of xs) (llist_of ys) = prefix xs ys"
simps: fixes ars :: "(String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "translate_ars ars = map_of (map (apsnd map_of) ars)"
max_E_in: shows "Max_mset \<epsilon> \<in># \<epsilon>"
l_coset_subset_G: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "x <# H \<subseteq> carrier G"
ipassmt_disjoint_matcheq_iifce_srcip: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface"   and i_ips :: "('a word \<times> nat) list"   and p :: "('a, 'b) simple_packet_scheme"   and p_ips :: "('a word \<times> nat) list" assumes "ipassmt_sanity_nowildcards ipassmt"   and "ipassmt_sanity_disjoint ipassmt"   and "ipassmt ifce = Some i_ips"   and "ipassmt (Iface (p_iiface p)) = Some p_ips \<and> p_src p \<in> ipcidr_union_set (set p_ips)" shows "match_iface ifce (p_iiface p) = (p_src p \<in> ipcidr_union_set (set i_ips))"
compos_aI_surj: fixes A :: "('c, 'd) aGroup_scheme"   and B :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'c" assumes "aGroup A"   and "aGroup B"   and "f \<in> aHom A B"   and "g \<in> aHom B A"   and "compos A g f = aI\<^bsub>A\<^esub>" shows "surjec\<^bsub>B,A\<^esub> g"
distinct_39: fixes x2 :: "nat"   and x11 :: "tag" shows "cNumber x2 \<noteq> cTag x11"
insert3_insert2: fixes bal_i :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and e :: "real"   and n :: "nat"   and t :: "'b tree"   and x :: "'b" assumes "RBTi2 TYPE('b) bal_i e"   and "bal_i n (height t)" shows "RBTi2.insert3 bal_i x (t, n) = RBTi2.insert2 bal_i x (t, n)"
finished_star: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "finished r x"   and "r\<^sup>*\<^sup>* x y" shows "x = y"
expunge_expunge_commute: fixes e1 :: "'c"   and mo1 :: "'d"   and i1 :: "'d"   and e2 :: "'c"   and mo2 :: "'d"   and i2 :: "'d" shows "\<langle>Expunge e1 mo1 i1\<rangle> \<rhd> \<langle>Expunge e2 mo2 i2\<rangle> = \<langle>Expunge e2 mo2 i2\<rangle> \<rhd> \<langle>Expunge e1 mo1 i1\<rangle>"
remove1_insort: fixes x :: "'a"   and xs :: "'a list" shows "remove1 x (insort x xs) = xs"
AE_in_main_part: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets (completion M)" shows "AE x in M. (x \<in> main_part M A) = (x \<in> A)"
Gram_Schmidt_matrix_iarrays_def: fixes A :: "real iarray iarray" shows "Gram_Schmidt_matrix_iarrays A = Gram_Schmidt_upt_k_iarrays A (ncols_iarray A - 1)"
plus_even: shows "{j. j + length (take 2 xs) \<in> Collect even} = Collect even"
intros_11: fixes P :: "pi" shows "P \<oplus> P \<equiv>\<^sub>e P"
node\<^sub>i_order: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "k \<le> length ts"   and "length ts \<le> 4 * k + 1"   and "\<forall>x\<in>set (subtrees ts). order k x"   and "order k t" shows "order_up\<^sub>i k (node\<^sub>i k ts t)"
label_def: fixes t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" shows "label t = fst (snd t)"
Fun: fixes x2a :: "('f, 'v) Term.term" assumes "x2a \<in> set ss" shows "x2a \<succeq> x2a"
T_insert_def: fixes x :: "'b"   and t :: "('b \<times> nat) tree" shows "T_insert x t = T_merge \<langle>\<langle>\<rangle>, (x, 1), \<langle>\<rangle>\<rangle> t + 1"
vector_component_5: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec"   and y :: "('a, 'b) Finite_Cartesian_Product.vec"   and i :: "'b" shows "(x - y) $h i = x $h i - y $h i"
rep_eq: fixes x :: "'a['c, 'b]"   and xa :: "'a['d, 'b]" shows "fromQ (x \<^bold>\<cdot> xa) = fromQ x \<cdot> fromQ xa"
Notes_imp_knows_Spy: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Notes A X \<in> set evs"   and "A \<in> bad" shows "X \<in> knows Spy evs"
tfr\<^sub>s\<^sub>s\<^sub>t_append': fixes A :: "('fun, 'var) stateful_strand_step list"   and B :: "('fun, 'var) stateful_strand_step list" assumes "tfr\<^sub>s\<^sub>s\<^sub>t (A @ B)" shows "tfr\<^sub>s\<^sub>s\<^sub>t B"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case Num_arc_delta x2 of MinInfty_delta \<Rightarrow> f1 | Num_arc_delta x \<Rightarrow> f2 x) = f2 x2"
le_utrans_refl: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "le_utrans x x"
f2: shows "\<forall>a. (\<exists>f. a = (\<Sum>a\<in>T. f a *\<^sub>C a)) \<or> (\<forall>A. (\<forall>f. a \<noteq> (\<Sum>a\<in>A. f a *\<^sub>C a)) \<or> infinite A \<or> \<not> A \<subseteq> T)"
is_valid_graph_def: fixes n :: "nat"   and S :: "(nat \<times> nat) set" shows "is_valid_graph n S = (\<forall>p\<in>S. fst p < n \<and> snd p < n)"
simps_1: fixes r :: "'a language"   and s :: "'a language" shows "\<oo> (Plus r s) = (\<oo> r \<or> \<oo> s)"
elims: fixes x :: "('a, 'b) Term.term"   and y :: "nat" assumes "num_funs x = y" obtains xa :: "'b" where "x = Var xa" and "y = 0"   | f :: "'a" and ts :: "('a, 'b) Term.term list" where "x = Fun f ts" and "y = Suc (sum_list (map num_funs ts))"
da_elim_cases_5: fixes Env :: "env"   and B :: "lname set"   and l :: "jump"   and c :: "stmt"   and A :: "assigned" assumes "Env\<turnstile> B \<guillemotright>\<langle>l\<bullet> c\<rangle>\<guillemotright> A" obtains C :: "assigned" and la :: "label" where "l = Break la" and "Env\<turnstile> B \<guillemotright>\<langle>c\<rangle>\<guillemotright> C" and "nrm A = nrm C \<inter> brk C la" and "brk A = rmlab la (brk C)"
subst_liftn: fixes i :: "nat"   and n :: "nat"   and k :: "nat"   and s :: "dB"   and t :: "dB" assumes "i \<le> n + k \<and> k \<le> i" shows "liftn (Suc n) s k[t/i] = liftn n s k"
not_le: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry" shows "(\<not> x \<preceq> y) = (y \<prec> x)"
min: shows "I \<subseteq> fst ` as \<and> (\<nexists>v. v \<Turnstile>\<^sub>t t \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as) \<and> (distinct_indices_atoms as \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. v \<Turnstile>\<^sub>t t \<and> (J, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as))"
infdist_notempty: fixes A :: "'c set"   and x :: "'c" assumes "A \<noteq> {}" shows "infdist x A = Inf (dist x ` A)"
facet: fixes c :: "'a set" shows "(c facet_of S) = (\<exists>h. h \<in> F \<and> c = S \<inter> {x. a h \<bullet> x = b h})"
AbstTermP_dbtm: fixes i :: "name"   and n :: "nat"   and u :: "dbtm" shows "{} \<turnstile> AbstTermP \<guillemotleft>Var i\<guillemotright> (ORD_OF n) (quot_dbtm u) (quot_dbtm (abst_dbtm i n u))"
t_ins_input: fixes x :: "'b"   and xt :: "'b bintree" shows "t_ins_inv x xt \<lparr>t_type.folding = False, item = x, subtrees = [xt]\<rparr>"
rabin_accept_to_Acc: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "accept\<^sub>R (semi_mojmir_def.step \<Sigma> \<up>af\<^sub>G (Abs \<psi>), semi_mojmir_def.initial (Abs \<psi>), {(mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<up>af\<^sub>G (Abs \<psi>) {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q} \<union> mojmir_to_rabin_def.merge\<^sub>R \<up>af\<^sub>G (Abs \<psi>) {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q} j, mojmir_to_rabin_def.succeed\<^sub>R \<up>af\<^sub>G (Abs \<psi>) {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q} j) |j. j < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<psi>)}) w"   and "\<pi> (G \<psi>) = mojmir_def.smallest_accepting_rank \<Sigma> \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}" shows "accepting_pair\<^sub>R (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) (Acc\<^sub>\<A> \<Sigma> \<pi> (G \<psi>)) w"
simps_2: fixes i :: "'b"   and l :: "'b sort_key extended"   and u :: "'b sort_key extended" assumes "\<lbrakk>Final i\<rbrakk> \<le> l \<and> l < u" shows "\<Psi> (l, u) i = NonFinal (\<gamma> l sort_dir.Right) (\<Psi> (derive_left (l, u)) i)"
lookup_list_to_poly: fixes ts :: "'c list"   and cs :: "'d list" shows "lookup (list_to_poly ts cs) = list_to_fun ts cs"
mult: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and a :: "nat"   and b :: "nat" assumes "dcharacter n \<chi>" shows "\<chi> (a * b) = \<chi> a * \<chi> b"
\<Psi>_sel_1: shows "edge \<Psi> = edge'"
unique_sizes_subgrp_normal: fixes q :: "nat" assumes "finite (carrier G)"   and "\<exists>!Q. Q \<in> subgroups_of_size q" shows "(THE Q. Q \<in> subgroups_of_size q) \<lhd> G"
2: shows "log 2 (real (size1 s + size1 u + size1 r1')) \<le> \<phi> t"
nat_bv_nat: fixes w :: "bit list" shows "nat_to_bv (bv_to_nat w) = norm_unsigned w"
exec_Calling_prealloc_pres: fixes h :: "heap"   and C :: "char list"   and Cs :: "char list list"   and P :: "jvm_prog"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list"   and sh :: "sheap"   and xp' :: "nat option"   and h' :: "heap"   and frs' :: "frame list"   and sh' :: "sheap" assumes "preallocated h"   and "exec_Calling C Cs P h stk loc C\<^sub>0 M\<^sub>0 pc frs sh = (xp', h', frs', sh')" shows "preallocated h'"
Compl_anti_mono: fixes A :: "'a set"   and B :: "'a set" assumes "A \<subseteq> B" shows "- B \<subseteq> - A"
SCall\<^sub>1: fixes P :: "nat exp prog"   and ps :: "nat exp list"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and vs :: "val list"   and h\<^sub>2 :: "heap"   and ls\<^sub>2 :: "val list"   and sh\<^sub>2 :: "sheap"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and body :: "nat exp"   and D :: "char list"   and sfs :: "sfields"   and ls\<^sub>2' :: "val list"   and e' :: "nat exp"   and h\<^sub>3 :: "heap"   and ls\<^sub>3 :: "val list"   and sh\<^sub>3 :: "sheap" assumes "P \<turnstile>\<^sub>1 \<langle>ps,s\<^sub>0\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h\<^sub>2, ls\<^sub>2, sh\<^sub>2)\<rangle>"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = body in D"   and "sh\<^sub>2 D = \<lfloor>(sfs, Done)\<rfloor> \<or> M = clinit \<and> sh\<^sub>2 D = \<lfloor>(sfs, Processing)\<rfloor>"   and "length vs = length Ts"   and "ls\<^sub>2' = vs @ replicate (max_vars body) undefined"   and "P \<turnstile>\<^sub>1 \<langle>body,(h\<^sub>2, ls\<^sub>2', sh\<^sub>2)\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>3, sh\<^sub>3)\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>C\<bullet>\<^sub>sM(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',(h\<^sub>3, ls\<^sub>2, sh\<^sub>3)\<rangle>"
gc: shows "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
induct: fixes P :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and a0 :: "'a option"   and a1 :: "'a option" assumes "\<And>x y. P (Some x) (Some y)"   and "\<And>y. P None y"   and "\<And>v. P (Some v) None" shows "P a0 a1"
distinct_31: fixes x91 :: "nat"   and x92 :: "'a fm" shows "FalseF \<noteq> ExN x91 x92"
simps_2: fixes v :: "char list"   and va :: "char list list" shows "mult_monoms [] (v # va) = v # va"
ab_left_minus: fixes a :: "'d" shows "- a + a = (0::'d)"
ereal_diff_nonpos: fixes a :: "ereal"   and b :: "ereal" assumes "a \<le> b"   and "a = \<infinity> \<Longrightarrow> b \<noteq> \<infinity>"   and "a = - \<infinity> \<Longrightarrow> b \<noteq> - \<infinity>" shows "a - b \<le> 0"
CS'_def: shows "CS' = map ((\<otimes>) A) CS"
assms_4: fixes e1 :: "pexpr"   and e2 :: "pexpr" assumes "e = PExpr1 (PSub e1 e2)" shows "P"
nextVertex_def: fixes f :: "face" shows "f \<bullet> \<equiv> let vs = vertices f in nextElem vs (hd vs)"
start_remove_cnt_id: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "x \<noteq> y" shows "cnt y (start_remove xs x ys) \<le> cnt y ys + cnt y xs"
atms_of_def: fixes C :: "'a literal multiset" shows "atms_of C = atm_of ` set_mset C"
tc_equiv: fixes "is" :: "'e set"   and A :: "'f set"   and f :: "'e \<Rightarrow> ('f \<times> 'f) set" assumes "\<And>i. i \<in> is \<Longrightarrow> equiv A (f i)"   and "is \<noteq> {}" shows "equiv A ((\<Union> (f ` is))\<^sup>+)"
meet_greatest: fixes z :: "'c"   and x :: "'c"   and y :: "'c" assumes "z \<sqsubseteq> x"   and "z \<sqsubseteq> y" shows "z \<sqsubseteq> x \<sqinter> y"
WTrtVar: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   and P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap" assumes "E V = \<lfloor>T\<rfloor>" shows "P,E,h,sh \<turnstile> Var V : T"
cltn2_compose_def: fixes A :: "cltn2"   and B :: "cltn2" shows "cltn2_compose A B \<equiv> cltn2_abs (cltn2_rep A ** cltn2_rep B)"
le_iff_sup: fixes x :: "'c"   and y :: "'c" shows "(x \<le> y) = (x + y = y)"
cond_pmf_singleton: fixes p :: "'c pmf"   and A :: "'c set"   and x :: "'c" assumes "set_pmf p \<inter> A = {x}" shows "cond_pmf p A = return_pmf x"