hcju/isabelleps huggingface dataset:
distinct_1: fixes condition' :: "'c"   and psi' :: "('a, 'b, 'c) psi"   and psiCase' :: "('a, 'b, 'c) psiCase" shows "\<bottom>\<^sub>c \<noteq> \<box> condition' \<Rightarrow> psi' psiCase' "
subsums_refl: fixes c :: "('a, 'b) conf" shows "c \<sqsubseteq> c"
nsets_eq_empty: fixes A :: "'b set"   and r :: "nat" assumes "finite A"   and "card A < r" shows "[A]\<^bsup>r\<^esup> = {}"
zero_less_two: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (plus one one)"
equiv_P2R: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equiv (Collect P) (P2R \<phi>) = equivP P \<phi>"
prems_1: fixes f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "Fun f T \<sqsubseteq> Fun f__ Y" shows "0 < arity f"
zero_notin_vpair: fixes x :: "V"   and y :: "V" shows "0 \<notin> elts \<langle>x, y\<rangle>"
some_NF_UNF: fixes r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "UNF r"   and "(x, y) \<in> r\<^sup>*"   and "y \<in> NF r" shows "y = some_NF r x"
incl: fixes h :: "'a"   and H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "h \<in> H" shows "h \<in> generate G H"
quorum_assm: fixes p :: "'node"   and Q :: "'node set"   and p' :: "'node" assumes "quorum_of p Q"   and "p' \<in> Q" shows "quorum_of p' Q"
cases: fixes a :: "('state, 'act, 'out) trans list" assumes "valid2 a" obtains trn :: "('state, 'act, 'out) trans" where "a = [trn__]" and "validTrans trn__"   | tr :: "('state, 'act, 'out) trans list" and trn :: "('state, 'act, 'out) trans" where "a = tr ## trn" and "valid2 tr" and "tgtOf (last tr) = srcOf trn" and "validTrans trn"
simps_4: fixes b :: "bexp"   and C\<^sub>1 :: "acom"   and C\<^sub>2 :: "acom"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "pre (IF b THEN C\<^sub>1 ELSE C\<^sub>2) Q = (\<lambda>s. eSuc (if bval b s then pre C\<^sub>1 Q s else pre C\<^sub>2 Q s))"
sur_total: fixes x :: "'a" shows "is_sur x = is_total (x\<^sup>T)"
bounded_w': fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" shows "w i \<in> \<Sigma>"
inverse_eq_divide_numeral: fixes w :: "num" shows "inverse (numeral w) = (1::'a) / numeral w"
eq_numeral_iff_iszero_9: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(numeral.numeral one plus x = zero) = ring_1.iszero zero (numeral.numeral one plus x)"
cod_char: fixes f :: "nat" shows "J.cod f = (if f \<in> {m} then f else J.null)"
i_append_nth: fixes xs :: "'d list"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" shows "(xs \<frown> f) n = (if n < length xs then xs ! n else f (n - length xs))"
gen_length_code_2: fixes n :: "nat"   and x :: "'b"   and xs :: "'b list" shows "List.gen_length n (x # xs) = List.gen_length (Suc n) xs"
g: fixes i :: "nat" assumes "i < length ss1" shows "ss2 ! i = ss1 ! g i"
take_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'g" shows "take i (replicate k x) = replicate (min i k) x"
birkhoff_lemma_L1: fixes f :: "'a \<Rightarrow> real" assumes "\<And>x. 0 \<le> f x"   and "integrable M f" shows "(\<lambda>n. \<integral>\<^sup>+ x. ennreal (norm (birkhoff_sum f n x / real n - real_cond_exp M Invariants f x)) \<partial>M) \<longlonglongrightarrow> 0"
assms_2: shows "redundant_inference (subst_ecl D \<eta>) S P \<sigma>"
higher_differentiable_on_inverse: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "higher_differentiable_on S f n"   and "(0::'c) \<notin> f ` S"   and "open S" shows "higher_differentiable_on S (\<lambda>x. inverse (f x)) n"
steps_hd_drop_suffix: fixes f :: "nat \<Rightarrow> ('s, 'p, 'f) com list \<times> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list \<times> ('s, 'f) xstate"   and c :: "('s, 'p, 'f) com"   and cs :: "('s, 'p, 'f) com list"   and css :: "(('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and s :: "('s, 'f) xstate"   and \<Gamma> :: "'p \<Rightarrow> ('s, 'p, 'f) com option"   and k :: "nat"   and pcss :: "nat \<Rightarrow> (('s, 'p, 'f) com list \<times> ('s, 'p, 'f) com list) list"   and pcs :: "nat \<Rightarrow> ('s, 'p, 'f) com list" assumes "f 0 = (c # cs, css, s)"   and "\<forall>i. \<Gamma>\<turnstile> f i \<rightarrow> f (Suc i)"   and "\<forall>i<k. \<not> (CS (f i) = cs \<and> CSS (f i) = css)"   and "\<forall>i\<le>k. if pcss i = [] then CSS (f i) = css \<and> CS (f i) = pcs i @ cs else CS (f i) = pcs i \<and> CSS (f i) = butlast (pcss i) @ [(fst (last (pcss i)) @ cs, snd (last (pcss i)) @ cs)] @ css" shows "\<forall>i<k. \<Gamma>\<turnstile> (pcs i, pcss i, S (f i)) \<rightarrow> (pcs (Suc i), pcss (Suc i), S (f (Suc i)))"
step_z_dbm_DBM: fixes A :: "('d \<times> ('e, 'f) cconstraint \<times> 'g \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry"   and v :: "'e \<Rightarrow> nat"   and n :: "nat"   and l' :: "'d"   and Z :: "('e \<Rightarrow> 'f) set" assumes "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto> \<langle>l', Z\<rangle>"   and "(\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)) \<and> (\<forall>c\<in>clk_set A. v c \<le> n) \<and> (\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k))" obtains D' :: "nat \<Rightarrow> nat \<Rightarrow> 'f DBMEntry" where "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>" and "Z = [D']\<^bsub>v,n\<^esub>"
not_less_simps_2: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "(n < Suc m) = (n = m)"
lin_vectorspace: fixes V :: "'c set"   and x :: "'c" assumes "vectorspace V"   and "x \<in> V" shows "vectorspace (lin x)"
FG_Alm_all: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln" shows "w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<phi>) = (\<forall>\<^sub>\<infinity>i. suffix i w \<Turnstile>\<^sub>n \<phi>)"
domain_eq: shows "Domainp pcr_nat_int = (\<lambda>S. \<exists>m n. {m..n} = S)"
in_notinI: fixes r :: "('d \<times> 'd) set"   and j :: "'d"   and i :: "'d" assumes "wo_rel r"   and "(j, i) \<notin> r \<or> j = i"   and "i \<in> Field r"   and "j \<in> Field r" shows "(i, j) \<in> r"
assms_3: shows "(n, v) \<in> set (zip (predecessors g n') vs)"
path_entry_append_conv: fixes G :: "nat graph_rec"   and v :: "nat"   and l :: "nat list"   and w :: "nat" assumes "cfg_doms G" shows "path_entry (g_E G) (v # l) w = (path_entry (g_E G) l v \<and> (v, w) \<in> g_E G)"
take_subseq: fixes n :: "nat"   and xs :: "'c list" shows "take n xs \<preceq> xs"
signed_of_int: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and n :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus (word_of_int n) = ring_1.of_int one plus zero minus (signed_take_bit (LENGTH('c) - Suc 0) n)"
mult_le2: fixes x :: "V"   and y :: "V" assumes "x \<noteq> 0"   and "y \<noteq> 0"   and "y \<noteq> 1" shows "x \<sqsubset> x * y"
smc_1_is_arrI: fixes a :: "V"   and \<aa> :: "V"   and b :: "V"   and f :: "V"   and \<ff> :: "V" assumes "a = \<aa>"   and "b = \<aa>"   and "f = \<ff>" shows "f : a \<mapsto>\<^bsub>smc_1 \<aa> \<ff>\<^esub> b"
merge_guards_simps_2: fixes g :: "'d \<Rightarrow> 'd" shows "merge_guards (Basic g) = Basic g"
lt_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "lt p = (if p = 0 then min_term else ord_term_lin.Max (keys p))"
gbinomial_prod_rev: fixes a :: "'c"   and k :: "nat" shows "a gchoose k = (\<Prod>i = 0..<k. a - of_nat i) div fact k"
Fiber_Relation_def: fixes \<alpha> :: "'a \<Rightarrow> 'c" shows "E(\<alpha>) = {(x, y). x \<in> S \<and> y \<in> S \<and> \<alpha> x = \<alpha> y}"
LeadsTo_wf_induct: fixes r :: "('c \<times> 'c) set"   and F :: "'d program"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and B :: "'d set" assumes "wf r"   and "\<forall>m. F \<in> A \<inter> f -` {m} \<longmapsto>w A \<inter> f -` r\<inverse> `` {m} \<union> B" shows "F \<in> A \<longmapsto>w B"
span_add_eq2: fixes y :: "'d"   and S :: "'d set"   and x :: "'d" assumes "y \<in> cspan S" shows "(x + y \<in> cspan S) = (x \<in> cspan S)"
mod_round_lt: fixes c :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < c"   and "a < b" shows "a div c < (b + c - 1) div c"
packet_independent_\<beta>_unknown_def: fixes \<beta> :: "'a \<Rightarrow> 'b \<Rightarrow> ternaryvalue" shows "packet_independent_\<beta>_unknown \<beta> \<equiv> \<forall>A. (\<exists>p. \<beta> A p \<noteq> TernaryUnknown) \<longrightarrow> (\<forall>p. \<beta> A p \<noteq> TernaryUnknown)"
hyp_port_hyps: fixes f :: "'form"   and t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list" assumes "f \<in> hyps_along t is" shows "hyps (iNodeOf (tree_at t (hyp_port_path_for t is f))) (hyp_port_h_for t is f) = Some (inPorts' (iNodeOf (tree_at t (hyp_port_path_for t is f))) ! hyp_port_i_for t is f)"
rel_compp_Grp: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_option R = (BNF_Def.Grp {x. set_option x \<subseteq> {(x, y). R x y}} (map_option fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_option x \<subseteq> {(x, y). R x y}} (map_option snd)"
not_cong_2134: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Cong A B C D" shows "\<not> Cong B A C D"
equiv_class_self: fixes A :: "'e set"   and r :: "('e \<times> 'e) set"   and a :: "'e" assumes "equiv A r"   and "a \<in> A" shows "a \<in> r `` {a}"
rep_eq: fixes x :: "'a set"   and xa :: "('a, 'b) fmap" shows "fmlookup (fmrestrict_set x xa) = map_restrict_set x (fmlookup xa)"
shadow_root_eq: fixes ptr' :: "(_) element_ptr"   and shadow_root_ptr_opt :: "(_) shadow_root_ptr option" shows "h \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt = h' \<turnstile> get_shadow_root ptr' \<rightarrow>\<^sub>r shadow_root_ptr_opt"
vertices_residual_network: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "countable_network \<Delta>" shows "vertex (countable_network.residual_network \<Delta> f) = vertex \<Delta>"
hyps_4: shows "t\<^sub>i \<in> set T"
select_min_def: fixes measure :: "'a \<Rightarrow> nat"   and xs :: "(nat \<times> 'a) list" shows "select_min measure xs = (case xs of (i, p) # ys \<Rightarrow> select_min_main measure (measure p) i ys)"
rb_aux_inv_D3_1: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rb_aux_inv (bs, ss, ps)"   and "Inl (p, q) \<in> set ps" shows "p \<in> set bs"
vector_7_7: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 7 = x7"
Val: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "P,E \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
chain_cover_witness: fixes X :: "'b set"   and C :: "nat \<Rightarrow> 'b set" assumes "finite X"   and "chain C"   and "X \<subseteq> natUnion C" shows "\<exists>n. X \<subseteq> C n"
map_prod_fun_zip: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and xs :: "'c list"   and ys :: "'d list" shows "map2 (\<lambda>x y. (f x, g y)) xs ys = zip (map f xs) (map g ys)"
upd_elim: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and upd :: "'Var ModeUpd"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c@[upd], mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w" shows "\<langle>c, update_modes upd mds, mem\<rangle>\<^sub>w \<leadsto>\<^sub>w \<langle>c', mds', mem'\<rangle>\<^sub>w"
index_size_conv: fixes xs :: "'c list"   and n :: "nat"   and x :: "'c" assumes "length xs = n" shows "(index xs x = n) = (x \<notin> set xs)"
in_mono: fixes A :: "'q set"   and B :: "'q set"   and x :: "'q" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
UNIF1_unifless: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "(E', E) \<in> unifless"
Inr_iff: fixes a :: "V"   and b :: "V" shows "(ZFC_Cardinals.Inr a = ZFC_Cardinals.Inr b) = (a = b)"
red_lcl_add_aux: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and e :: "'addr expr"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and e' :: "'addr expr"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and l0 :: "String.literal \<Rightarrow> 'addr val option" assumes "extTA,P,t \<turnstile> \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "extTA,P,t \<turnstile> \<langle>e,(hp s, l0 ++ lcl s)\<rangle> -ta\<rightarrow> \<langle>e',(hp s', l0 ++ lcl s')\<rangle>"
trans: fixes b :: "'b DBMEntry"   and a :: "'b DBMEntry"   and c :: "'b DBMEntry" assumes "b \<preceq> a"   and "c \<preceq> b" shows "c \<preceq> a"
simps_3: fixes a :: "'a" shows "lang (Atom a) = {[a]}"
simps_2: fixes ux :: "typ"   and t :: "Core.term" shows "beta_reducible (Abs ux t) = beta_reducible t"
g_def: shows "g \<equiv> \<lambda>x. norm x *\<^sub>R f (x /\<^sub>R norm x)"
2: shows "has_bochner_integral (lebesgue_on {a + c..b}) f (integral\<^sup>L (lebesgue_on {a + c..b}) f)"
literal_le_def: fixes xs :: "'a literal"   and ys :: "'a literal" shows "(xs \<le> ys) = (xs < ys \<or> xs = ys)"
assms_2: shows "r' = {(x, y). (m x, m y) \<in> r}"
main: fixes ia :: "nat" assumes "ia < n" shows "(ia \<in> I) = (k \<le> inv p ia)"
PSize2: shows "P \<in> groupP.subgroups_of_size (p ^ a)"
HEndPhase0_HInv5_q1: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase0 s s' p"   and "p \<noteq> q"   and "Inv1 s"   and "maxBalInp s (bal (dblock s q)) (inp (dblock s q))" shows "maxBalInp s' (bal (dblock s' q)) (inp (dblock s' q))"
assms_2: shows "\<forall>n. \<not> winning_position_Even ((augment_list \<sigma> ^^ n) p)"
fixed_point_imp_\<omega>_limit_set: fixes x :: "'a" assumes "x \<in> X"   and "(- f) x = (0::'a)" shows "rev.\<omega>_limit_set x = {x}"
get_child_nodes_locs_impl: shows "get_child_nodes_locs = local.a_get_child_nodes_locs"
nzero: shows "\<forall>x\<in>A. poly q2 x \<noteq> (0::'a)"
qfree_interior1: fixes \<phi> :: "atom fm" assumes "nqfree \<phi>" shows "qfree (qe_interior\<^sub>1 \<phi>)"
fact_div_fact: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "fact m div fact n = \<Prod>{n + 1..m}"
ZFfunDom: fixes A :: "ZF"   and B :: "ZF"   and f :: "ZF \<Rightarrow> ZF" shows "|dom|ZFfun A B f = A"
assms: shows "obj_terminal (smc_SemiCAT \<alpha>) \<BB>"
ghomTr4: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and K :: "'g set" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "G \<guillemotright> K " shows "F \<guillemotright> iim F G f K "
bigthetaI: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> O[F](g)"   and "f \<in> \<Omega>[F](g)" shows "f \<in> \<Theta>[F](g)"
ord_less_eq_trans: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" assumes "a <p b"   and "b = c" shows "a <p c"
ideal_prod_in_carrier: fixes I :: "'a set set"   and J :: "'a set set" assumes "ideal I (R Quot I)"   and "ideal J (R Quot I)" shows "I \<cdot>\<^bsub>R Quot I\<^esub> J \<subseteq> carrier (R Quot I)"
iMODb_div_self: fixes m :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < m" shows "[ r, mod m, c ] \<oslash> m = [r div m\<dots>,c]"
insort_key_append1: fixes ys :: "'b list"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "\<forall>y\<in>set ys. f x < f y" shows "insort_key f x (xs @ ys) = insort_key f x xs @ ys"
principal_dchar_1: fixes n :: "nat" shows "principal_dchar n 1 = 1"
bot_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "UML_Types.bot_class.bot \<equiv> Abs_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<bottom>"
simps_4: fixes uv :: "'d list"   and uw :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and F' :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "setops_list\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>uv\<langle>\<or>\<noteq>: uw \<or>\<notin>: F'\<rangle> = F'"
map2_val_compatD1: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and f :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'c) list"   and zs :: "('a \<times> 'b) list" assumes "comparator comp"   and "comparator.map2_val_compat comp f"   and "comparator.oalist_inv_raw comp zs" shows "comparator.oalist_inv_raw comp (f zs)"
carrier_direct_limE: fixes X :: "('a set \<times> 'b) set" assumes "X \<in> carrier_direct_lim" obtains U :: "'a set" and s :: "'b" where "U \<in> I" and "s \<in> \<FF> U" and "X = \<lfloor>U, s\<rfloor>"
\<alpha>_def: fixes \<mu>\<nu>\<tau> :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "\<alpha> \<mu>\<nu>\<tau> \<equiv> \<a>[fst \<mu>\<nu>\<tau>, fst (snd \<mu>\<nu>\<tau>), snd (snd \<mu>\<nu>\<tau>)]"
induct: fixes P1 :: "'a FOL_Fitting.term \<Rightarrow> bool"   and P2 :: "'a FOL_Fitting.term list \<Rightarrow> bool"   and "term" :: "'a FOL_Fitting.term" assumes "\<And>x. P1 (Var x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (App x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P1 term"
xMne: shows "{y \<in> S. x \<le> y} \<noteq> {}"
assms_3: shows "s2 \<in> d_reachable M (initial M)"
subspace_image: fixes S :: "'c set" assumes "vs2.subspace S" shows "p.subspace (Pair (0::'b) ` S)"
ordLeq_ordIso_trans: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and r'' :: "('e \<times> 'e) set" assumes "r \<le>o r'"   and "r' =o r''" shows "r \<le>o r''"
in_genby_reduced_letter_set: fixes as :: "'a list"   and A :: "'a set"   and a :: "'a" assumes "as \<in> lists A"   and "sum_list as = a" shows "a \<in> \<langle>reduced_letter_set A a\<rangle>"
insort_not_Nil: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "linorder.insort_key less_eq f a xs \<noteq> []"
while_def: fixes b :: "'d \<Rightarrow> bool"   and c :: "'d \<Rightarrow> 'd"   and s :: "'d" shows "while b c s = the (while_option b c s)"
sets_eq: fixes m :: "'a measure" assumes "m \<in> M" shows "sets m = sets A"
divides_zero: fixes a :: "'a" assumes "a \<in> carrier R" shows "a divides \<zero>"
assms_3: shows "(r, Expunge e mo r) \<in> set (node_deliver_messages xs)"
isometry_onI: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> dist (f x) (f y) = dist x y" shows "isometry_on X f"
summable_finite: fixes N :: "nat set"   and f :: "nat \<Rightarrow> 'a" assumes "finite N"   and "\<And>n. n \<notin> N \<Longrightarrow> f n = (0::'a)" shows "summable f"
insert_Diff: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "Set.insert a (A - {a}) = A"
abs_summable_on_cong: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and B :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x"   and "A = B" shows "(f abs_summable_on A) = (g abs_summable_on B)"
comp_Nil: fixes \<sigma> :: "('b \<times> 'b trm) list" shows "\<sigma> \<lozenge> [] = \<sigma>"
comm_neutral: fixes a :: "'a" shows "a \<^bold>* \<^bold>1 = a"
relpow_add: fixes m :: "nat"   and n :: "nat"   and R :: "('b \<times> 'b) set" shows "R ^^ (m + n) = R ^^ m O R ^^ n"
assert_disch4: fixes P :: "'a \<Rightarrow> bool"   and \<sigma> :: "'a" assumes "P \<sigma>" shows "\<sigma> \<Turnstile> assert\<^sub>S\<^sub>E P"
F1map_comp: fixes f1 :: "'m \<Rightarrow> 'k"   and g1 :: "'i \<Rightarrow> 'm"   and f2 :: "'n \<Rightarrow> 'l"   and g2 :: "'j \<Rightarrow> 'n" shows "F1map (f1 \<circ> g1) (f2 \<circ> g2) = F1map f1 f2 \<circ> F1map g1 g2"
assms: shows "(\<sigma>, ns) \<in> init (opnet np p)"
equivalent_on_eq: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and e :: "'b equation"   and I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" assumes "equivalent_on \<sigma> \<eta> (vars_of_eq e) I"   and "fo_interpretation I" shows "validate_ground_eq I (subst_equation e \<sigma>) = validate_ground_eq I (subst_equation e \<eta>)"
simps_4: fixes p :: "('c, 'd) tpoly"   and ps :: "('c, 'd) tpoly list" shows "poly_of (PSum (p # ps)) = poly_add (poly_of p) (poly_of (PSum ps))"
wtE_pickE_pickE: fixes xl :: "Preliminaries.var list"   and al :: "univ list" assumes "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> eintT (GE.tpOfV (xl ! i)) (al ! i)" shows "wtE (pickE xl al) \<and> (\<forall>i<length xl. pickE xl al (xl ! i) = al ! i)"
Idem_def: fixes s :: "('b \<times> 'b trm) list" shows "Idem s = s \<lozenge> s \<doteq> s"
last_fin_make_untimed_inf_truncate: fixes z :: "nat \<Rightarrow> 'd list"   and i :: "nat"   and a :: "'d" assumes "z i = [a]" shows "last (fin_make_untimed (inf_truncate z i)) = a"
simps_12: fixes accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and objRef :: "expr"   and mn :: "mname"   and pTs :: "ty list"   and args :: "expr list" shows "constVal ({accC,statT,mode}objRef\<cdot>mn( {pTs}args)) = None"
cases: fixes a1 :: "'s"   and a2 :: "'s stream" assumes "K.enabled a1 a2" obtains \<omega> :: "'s stream" and s :: "'s" where "a1 = s" and "a2 = \<omega>" and "K.enabled (shd \<omega>) (stl \<omega>)" and "shd \<omega> \<in> set_pmf (K s)"
distinct_1: fixes x11 :: "char list"   and x12 :: "'a list"   and x21 :: "char list"   and x22 :: "'a list" shows "Pos x11 x12 \<noteq> Neg x21 x22"
predicate2D: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c" assumes "P \<le> Q"   and "P x y" shows "Q x y"
assms: shows "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<bullet> A"
assms: fixes x :: "'a"   and y :: "'b" shows "ctor1 x \<noteq> ctor2 y"