hcju/isabelleps huggingface dataset:
perp_col1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and C :: "'a"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "C \<noteq> X"   and "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Col Bet C D X" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B C X"
le_infI1: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and x :: "'a"   and b :: "'a" assumes "class.semilattice_inf inf less_eq less"   and "less_eq a x" shows "less_eq (inf a b) x"
dom_map_add: fixes m :: "'c \<Rightarrow> 'd option"   and n :: "'c \<Rightarrow> 'd option" shows "dom (m ++ n) = dom n \<union> dom m"
F_v_memoryless: obtains ct :: "'s \<Rightarrow> 's pmf" where "ct \<in> Pi\<^sub>E S K" and "v \<circ> simple ct = F_sup (v \<circ> simple ct)"
lexord_fusion_def: fixes g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "lexord_fusion g1 g2 s1 s2 = lexordp (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
Compl_partition2: fixes A :: "'a set" shows "- A \<union> A = UNIV"
dropWhile_eq_Cons_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "(dropWhile P xs = y # ys) = (xs = takeWhile P xs @ y # ys \<and> \<not> P y)"
open_segment_trichotomy: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and y :: "'a" assumes "x \<in> {a<--<b}"   and "y \<in> {a<--<b}" shows "x = y \<or> y \<in> {x<--<b} \<or> y \<in> {a<--<x}"
dimension_vector: shows "finite_dimensional_vector_space.dimension cart_basis = CARD('b)"
intensional_simps_28: fixes P :: "'a \<Rightarrow> bool" shows "P \<or> \<not> P \<equiv> #True"
induct: fixes P1 :: "stmt \<Rightarrow> bool"   and P2 :: "expr \<Rightarrow> bool"   and stmt :: "stmt"   and expr :: "expr" assumes "P1 Skip"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (x1;; x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P1 x2; P1 x3\<rbrakk> \<Longrightarrow> P1 (If (x1) x2 Else x3)"   and "\<And>x1 x2. P1 x2 \<Longrightarrow> P1 (While (x1) x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (x1 :== x2)"   and "\<And>x1 x2 x3. \<lbrakk>P2 x1; P2 x3\<rbrakk> \<Longrightarrow> P1 (x1..x2:==x3)"   and "\<And>x. P1 (Meth x)"   and "\<And>x. P1 (Impl x)"   and "\<And>x. P2 (new x)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P2 (Cast x1 x2)"   and "\<And>x. P2 (LAcc x)"   and "\<And>x1 x2. P2 x1 \<Longrightarrow> P2 (x1..x2)"   and "\<And>x1 x2 x3 x4. \<lbrakk>P2 x2; P2 x4\<rbrakk> \<Longrightarrow> P2 ({x1}x2..x3(x4))" shows "P1 stmt \<and> P2 expr"
no_head_sub': fixes \<C> :: "'b list set"   and b :: "'b list" assumes "code \<C>"   and "b \<in> \<C>" shows "no_head_gen \<C> b \<subseteq> no_head \<C> b"
times_gt_left_mono: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<succ> y" shows "x * z \<succ> y * z"
vcons_in_vcpowerD_2: fixes xs :: "V"   and x :: "V"   and A :: "V"   and n :: "V" assumes "xs #\<^sub>\<circ> x \<in>\<^sub>\<circ> A ^\<^sub>\<times> n"   and "n \<in>\<^sub>\<circ> \<omega>" shows "x \<in>\<^sub>\<circ> A"
hypreal_of_real_image: shows "range hypreal_of_real = \<real>"
assms_3: fixes x :: "'a" assumes "x \<in> D" shows "ts \<subseteq> existence_ivl0 x"
y_2: shows "g $$ fls_subdegree g * y = (1::'a)"
part_equivI: fixes R :: "('b \<times> 'b) set" assumes "sym R"   and "trans R" shows "part_equiv R"
rv_branching_edges_slice_kinds_False: fixes a :: "'edge"   and ax :: "'edge"   and n :: "'node"   and n'' :: "'node"   and S :: "'node set"   and as :: "'edge list"   and s :: "'state"   and asx :: "'edge list"   and s' :: "'state" assumes "valid_edge a"   and "valid_edge ax"   and "sourcenode a = n"   and "sourcenode ax = n"   and "targetnode a = n''"   and "targetnode ax \<noteq> n''"   and "preds (slice_kinds S (a # as)) s"   and "preds (slice_kinds S (ax # asx)) s'"   and "\<forall>V\<in>rv S n. state_val s V = state_val s' V" shows "False"
card: fixes Rep :: "'a \<Rightarrow> 'b"   and Abs :: "'b \<Rightarrow> 'a"   and A :: "'b set" assumes "type_definition Rep Abs A" shows "CARD('a) = card A"
tranclE: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set" assumes "(a, b) \<in> r\<^sup>+" obtains "(a, b) \<in> r"   | c :: "'c" where "(a, c) \<in> r\<^sup>+" and "(c, b) \<in> r"
tl_append: fixes xs :: "'b list"   and ys :: "'b list" shows "tl (xs \<cdot> ys) = (case xs of \<epsilon> \<Rightarrow> tl ys | z # zs \<Rightarrow> zs \<cdot> ys)"
leg1_iso: shows "\<exists>\<nu>. \<guillemotleft>\<nu> : s\<^sub>1 \<Rightarrow> t\<^sub>1 \<star> \<pi>.chine\<guillemotright> \<and> local.iso \<nu>"
Diff_insert_absorb: fixes x :: "'k"   and A :: "'k set" assumes "x \<notin> A" shows "insert x A - {x} = A"
simps_16: fixes v :: "\<I>"   and va :: "formula Regex.regex" shows "get_and_list (MatchF v va) = [MatchF v va]"
UNIV_I: fixes x :: "'h" shows "x \<in> UNIV"
eps_match: fixes test :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat"   and r :: "'b regex" shows "eps test i r = match test r i i"
least_squares_approximation7: fixes S :: "'a set"   and v :: "'a"   and p :: "'a"   and y :: "'a" assumes "subspace S"   and "v - p \<in> orthogonal_complement S"   and "p \<in> S"   and "y \<in> S" shows "norm (v - p) \<le> norm (v - y)"
measure_return: fixes X :: "'c set"   and M :: "'c measure"   and x :: "'c" assumes "X \<in> sets M" shows "Sigma_Algebra.measure (return M x) X = indicat_real X x"
ran_upd: fixes m :: "'d \<Rightarrow> 'c option"   and k :: "'d"   and v :: "'c" shows "ran (m(k \<mapsto> v)) \<subseteq> ran m \<union> {v}"
alt_seq_def: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "padic_integers p" shows "alt_seq s = (\<lambda>k. if even k then s k else res_lim s)"
suffix_to_prefix: fixes xs :: "'b list"   and ys :: "'b list" shows "suffix xs ys = prefix (rev xs) (rev ys)"
ty_dom_subst: fixes \<Gamma> :: "binding list"   and X :: "tyvrs"   and T :: "ty" shows "ty_dom (\<Gamma>[X \<mapsto> T]\<^sub>e) = ty_dom \<Gamma>"
rreq_rrep_fresh_1: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool" shows "rreq_rrep_fresh crt (Rreq hops dip dsn dsk oip osn sip handled) = (sip \<noteq> oip \<longrightarrow> oip \<in> kD crt \<and> (osn < sqn crt oip \<or> sqn crt oip = osn \<and> the (dhops crt oip) \<le> hops \<and> the (flag crt oip) = val))"
g_2: shows "\<forall>\<^sub>F x in F. 0 < g x"
rec_t_eq_steps: fixes tp :: "(action \<times> nat) list"   and l :: "nat"   and lm :: "'b"   and stp :: "nat" assumes "tm_wf0 tp" shows "trpl_code (steps0 (Suc 0, Bk \<up> l, <lm>) tp stp) = rec_exec rec_conf [code tp, bl2wc (<lm>), stp]"
col3: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and Y :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "X \<noteq> Y"   and "Tarski_neutral_dimensionless.Col Bet X Y A"   and "Tarski_neutral_dimensionless.Col Bet X Y B"   and "Tarski_neutral_dimensionless.Col Bet X Y C" shows "Tarski_neutral_dimensionless.Col Bet A B C"
eq: fixes J :: "'b measure set" assumes "finite J" shows "sup_measure.F id J = (SUP i\<in>J. i)"
conjugI': fixes r :: "'b list"   and s :: "'b list" shows "r \<cdot> s \<sim> s \<cdot> r"
simps: fixes a :: "'a"   and b :: "'a" shows "normalization_euclidean_semiring_class.gcd a b = (if b = (0::'a) then normalize a else normalization_euclidean_semiring_class.gcd b (a mod b))"
Ams: shows "\<forall>i\<in>S. A i \<in> measurable_sets M"
upd_conv_take_nth_drop: fixes i :: "nat"   and xs :: "'c list"   and a :: "'c" assumes "i < length xs" shows "xs[i := a] = take i xs @ a # drop (Suc i) xs"
2: shows "set acc \<subseteq> set (fold (\<lambda>(ic, q_in_bs, p, q) ps'. if \<not> ic \<and> c q_in_bs ps' p q then ps' else (ic, p, q) # ps') ps ((ic', p', q') # acc))"
exhaust: fixes y :: "M" obtains "y = M1"   | "y = M2"   | "y = M3"
HEndPhase1_HInv4a_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4a s p"   and "Inv2a s" shows "HInv4a s' p"
linear_cf_def: fixes X :: "'a set" shows "linear_cf X \<equiv> set_option (local.MaxR.MaxR_opt X)"
app_invimageI2: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r\<inverse>\<^sub>\<circ> `\<^sub>\<circ> A" shows "a \<in>\<^sub>\<circ> r -`\<^sub>\<circ> A"
MOST_iff_finiteNeg: fixes P :: "'b \<Rightarrow> bool" shows "(\<forall>\<^sub>\<infinity>x. P x) = finite {x. \<not> P x}"
llist_le_def: fixes s :: "'a llist"   and t :: "'a llist" shows "(s \<le> t) = (\<exists>d. t = s @@ d)"
forest_mono: fixes Y :: "'a uprod set"   and X :: "'a uprod set" assumes "Y \<subseteq> X"   and "forest X" shows "forest Y"
HcompNml_Nml_Obj: fixes t :: "'a E.term"   and u :: "'a E.term" assumes "E.Nml t"   and "E.Obj u"   and "E.Src t = E.Trg u" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = t"
asum_multTr_pp: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier A" shows "aSum A m (aSum A n a) = aSum A (m * n) a"
surj_const_mult: fixes a :: "complex vec" assumes "a \<in> carrier_vec dA" shows "(+) a ` carrier_vec dA = carrier_vec dA"
eWlsBinp_invert: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and delta :: "'opSym"   and ebinp :: "'bindex \<Rightarrow> 'b withERR option" assumes "eWlsBinp MOD delta ebinp" shows "\<exists>binp. igWlsBinp MOD delta binp \<and> ebinp = OKI binp"
redT_updTs_None: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and tas :: "('h, 'f, 'i) new_thread_action list"   and t :: "'h" assumes "redT_updTs ts tas t = None" shows "ts t = None"
simps_3: fixes I :: "('b, 'c, 'a) interp"   and ODE1 :: "('b, 'a) ODE"   and ODE2 :: "('b, 'a) ODE" shows "ODE_sem I (OProd ODE1 ODE2) = (\<lambda>\<nu>. ODE_sem I ODE1 \<nu> + ODE_sem I ODE2 \<nu>)"
lookup_operator_def: fixes name :: "char list" shows "lookup_operator name \<equiv> find (\<lambda>(n, uu_, uu_, uu_). n = name) ast\<delta>"
Throw1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>throw e,s\<rangle> -ta\<rightarrow> \<langle>throw e',s'\<rangle>"
distinct_hd_tl: fixes xs :: "'a list" assumes "distinct xs" shows "hd xs \<notin> set (tl xs)"
simps_6: fixes t :: "'c local.term" shows "Can \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = Can t"
allg_E: fixes upds :: "(char list \<times> 'b) set"   and A :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and xs :: "(char list \<times> nat) list"   and f :: "char list \<Rightarrow> nat" assumes "\<And>l s n x. x \<in> fst ` upds \<Longrightarrow> A (l(x := n)) = A l"   and "lesvars xs \<subseteq> fst ` upds" shows "A (ListUpdateE f xs) = A f"
node_lift: fixes T :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ii :: "nat"   and I :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> bool"   and U :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and R\<^sub>i :: "nat set" assumes "T \<Turnstile> (otherwith S {ii} (orecvmsg I), other U {ii} \<rightarrow>) global P"   and "\<And>\<xi> \<xi>'. S \<xi> \<xi>' \<Longrightarrow> U \<xi> \<xi>'" shows "\<langle>ii : T : R\<^sub>i\<rangle>\<^sub>o \<Turnstile> (otherwith S {ii} (oarrivemsg I), other U {ii} \<rightarrow>) global P"
CFGExit_wf_axioms: fixes prog :: "cmd" shows "CFGExit_wf sourcenode targetnode kind (valid_edge prog) (_Entry_) (Defs prog) (Uses prog) id (_Exit_)"
simps_1: fixes P :: "'c" shows "post (SKIP {P}) = P"
Array_alloc_def': fixes xs :: "'b list"   and h :: "heap" shows "Array.alloc xs h = (let l = lim h; r = Array l in (r, Array.set r xs (h\<lparr>lim := l + 1\<rparr>)))"
phiDefNodes_aux_redirect: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and v :: "'var"   and un :: "'node list"   and n' :: "'node"   and m' :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "\<forall>n\<in>set ns. v \<notin> defs g n"   and "length (predecessors g n) \<noteq> 1"   and "unvisitedPath un ns"   and "n' \<in> set ns"   and "n' \<in> phiDefNodes_aux g v un m'"   and "m' \<in> set (\<alpha>n g)" shows "n \<in> phiDefNodes_aux g v un m'"
ns_mul_ext_def: fixes ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" shows "ns_mul_ext ns s \<equiv> (mult2_alt_ns (ns\<inverse>) (s\<inverse>))\<inverse>"
intro: fixes valid_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set) list \<Rightarrow> bool"   and init_msaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_ts_msaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and join_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and add_new_table_msaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> 'a \<Rightarrow> 'a"   and result_msaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> event_data option list set"   and valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'b"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'b"   and length_muaux :: "Monitor.args \<Rightarrow> 'b \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> event_data option list set list \<times> 'b" assumes "msaux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux"   and "muaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux" shows "maux valid_msaux init_msaux add_new_ts_msaux join_msaux add_new_table_msaux result_msaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux"
splitAt_distinct_fst_aux: fixes vs :: "'a list"   and ram :: "'a" assumes "distinct vs" shows "distinct (fst (splitAt ram vs))"
digit_diff_split: fixes n :: "nat"   and D :: "nat \<Rightarrow> nat"   and nd :: "nat" assumes "n = (\<Sum>x<nd. D x * 10 ^ x)" shows "n - sum D {..<nd} = (\<Sum>x<nd. D x * (10 ^ x - 1))"
same_append_eq: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(append xs ys = append xs zs) = (ys = zs)"
changes_poly_at_0_2: fixes p :: "'a poly"   and a :: "'a" shows "changes_poly_at [p] a = 0"
Rep_inject: fixes x :: "'a Word.word"   and y :: "'a Word.word" shows "(uint x = uint y) = (x = y)"
sscan_sdrop: fixes k :: "nat"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c stream"   and a :: "'b" shows "sdrop k (sscan f xs a) = sscan f (sdrop k xs) (fold f (stake k xs) a)"
Dxy: shows "hausdorff_distance (f ` {x--y}) {f x--f y} \<le> D"
nat_even_power_sums_real: fixes n' :: "nat" assumes "0 < n'" shows "(\<lambda>k. 1 / real (Suc k) ^ (2 * n')) sums ((- 1) ^ Suc n' * bernoulli (2 * n') * (2 * pi) ^ (2 * n') / (2 * fact (2 * n')))"
existence_ivl0_cong: fixes Y :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and x0 :: "'a" assumes "Y = Y"   and "\<And>x t. x \<in> Y \<Longrightarrow> g x = g x" shows "Y.existence_ivl0 x0 = auto_ll_on_open.existence_ivl0 g Y x0"
perm_aux_def: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" shows "perm_aux pi x = pi \<bullet> x"
mem_CD_on_Cd: fixes x :: "'x"   and ds :: "'d set"   and X :: "'x set" shows "(x \<in> CD_on ds X) = (x \<in> Cd (Xd x) X \<and> Xd x \<in> ds)"
unat_minus_one_word: shows "unat (- 1) = 2 ^ LENGTH('c) - 1"
is_slim_def: fixes V :: "'a set" shows "G.is_slim V \<equiv> (\<forall>v\<in>pverts G. v \<in> V \<or> in_degree (with_proj G) v \<le> 2 \<and> (\<exists>x p y. G.gen_iapath V x p y \<and> v \<in> set (pawalk_verts x p))) \<and> (\<forall>e\<in>parcs G. fst e \<noteq> snd e \<and> (\<exists>x p y. G.gen_iapath V x p y \<and> e \<in> set p)) \<and> (\<forall>u v p q. G.gen_iapath V u p v \<and> G.gen_iapath V u q v \<longrightarrow> p = q) \<and> V \<subseteq> pverts G"
ineM_L1: fixes ch :: "chanID"   and M :: "chanID set"   and P :: "specID"   and E :: "Expression" assumes "ch \<in> M"   and "ch \<in> ins P"   and "exprChannel ch E" shows "ineM P M E"
fold_fun_left_comm: fixes A :: "'a set"   and x :: "'a"   and z :: "'a" assumes "finite A" shows "x \<^bold>* Finite_Set.fold (\<^bold>*) z A = Finite_Set.fold (\<^bold>*) (x \<^bold>* z) A"
GEM_axioms_def: shows "GEM_axioms \<equiv> ((\<forall>x y. x \<oplus> y = (THE z. \<forall>v. O v z = (O v x \<or> O v y))) \<and> (\<forall>x y. x \<otimes> y = (THE z. \<forall>v. P v z = (P v x \<and> P v y))) \<and> (\<forall>x y. x \<ominus> y = (THE z. \<forall>w. P w z = (P w x \<and> \<not> O w y)))) \<and> ((\<forall>x. \<midarrow> x = (THE z. \<forall>w. P w z = (\<not> O w x))) \<and> u = (THE x. \<forall>y. P y x)) \<and> (\<forall>F. (\<exists>x. F x) \<longrightarrow> (\<sigma>x. F x) = (THE x. \<forall>y. O y x = (\<exists>z. F z \<and> O y z))) \<and> (\<forall>F. (\<pi>x. F x) = (\<sigma>x. \<forall>y. F y \<longrightarrow> P x y))"
c''_def: shows "c'' = (\<lambda>i. if i \<le> N then c''_aux i else 0)"
less_real_alg_def: fixes x :: "real_alg"   and y :: "real_alg" shows "(x < y) = (real_of x < real_of y)"
card_character_extensions: fixes H :: "'a set"   and \<chi> :: "'a \<Rightarrow> complex" assumes "subgroup H G"   and "character (G\<lparr>carrier := H\<rparr>) \<chi>" shows "card {\<chi>' \<in> characters G. \<forall>x\<in>H. \<chi>' x = \<chi> x} * card H = order G"
distinct7_def: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points"   and E :: "Points"   and F :: "Points"   and G :: "Points" shows "distinct7 A B C D E F G \<equiv> A \<noteq> B \<and> A \<noteq> C \<and> A \<noteq> D \<and> A \<noteq> E \<and> A \<noteq> F \<and> A \<noteq> G \<and> B \<noteq> C \<and> B \<noteq> D \<and> B \<noteq> E \<and> B \<noteq> F \<and> B \<noteq> G \<and> C \<noteq> D \<and> C \<noteq> E \<and> C \<noteq> F \<and> C \<noteq> G \<and> D \<noteq> E \<and> D \<noteq> F \<and> D \<noteq> G \<and> E \<noteq> F \<and> E \<noteq> G \<and> F \<noteq> G"
poly_mapping_plus_induct_Polys: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and X :: "'b set"   and P :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) \<Rightarrow> bool" assumes "p \<in> P[X]"   and "P 0"   and "\<And>p c t. \<lbrakk>t \<in> .[X]; p \<in> P[X]; c \<noteq> (0::'c); t \<notin> keys p; P p\<rbrakk> \<Longrightarrow> P (monomial c t + p)" shows "P p"
funpow_simp_l: fixes f :: "'c \<Rightarrow> 'c"   and n :: "nat"   and x :: "'c" shows "f ((f ^^ n) x) = (f ^^ Suc n) x"
base_exp: fixes j :: "nat" assumes "j \<in> {i..<n}" shows "base ^ j = base ^ (j - i) * base ^ i"
poly_rel_div: shows "(poly_rel ===> poly_rel ===> poly_rel) (div_field_poly_i ff_ops) (div)"
lunit_in_hom_2: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.in_hom (Span.lunit f) (Span.trg f \<circ> f) f"
simps_3: fixes n :: "nat"   and m :: "nat"   and F :: "'b formula" shows "cut_bound n m (\<^bold>\<not> F) = cut_bound m n F"
less_eq_upset_def: shows "(\<le>) \<equiv> id (\<lambda>X Y. Rep_upset Y \<subseteq> Rep_upset X)"
assms: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> A" shows "f x \<in>\<^sub>\<circ> B"
dvd_finite_2_unfold: fixes x :: "finite_2"   and y :: "finite_2" shows "(x dvd y) = (x = a\<^sub>2 \<or> y = a\<^sub>1)"
assms_3: shows "wf\<^sub>t\<^sub>r\<^sub>m (\<theta> y)"
FHSubmodule_indspace: shows "RModule Supgroup.group_ring (\<currency>) indV \<and> indV \<subseteq> addfunset FH V"
dim_mat_impl_1: fixes x :: "'b mat_impl" shows "dim_row (mat_impl x) = dim_row_impl x"
fin_g'_2: shows "finite (Range (let V_new = sorted_list_of_set (vertices (snd R) - vertices (fst R)) in set (zip V_new [n..<n + length V_new])))"
comp_char: fixes g :: "'a"   and f :: "'a" shows "L.comp g f = (if L.arr f \<and> L.arr g \<and> seq g f then g \<cdot> f else null)"
divide_numeral_1: fixes a :: "'b" shows "a / Numeral1 = a"
INF_filter_not_bot: fixes B :: "'b set"   and F :: "'b \<Rightarrow> 'c filter" assumes "\<And>X. \<lbrakk>X \<subseteq> B; finite X\<rbrakk> \<Longrightarrow> Inf (F ` X) \<noteq> bot" shows "Inf (F ` B) \<noteq> bot"
homeomorphic_map_path_connectedness_eq: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and U :: "'a set" assumes "homeomorphic_map X Y f" shows "path_connectedin X U = (U \<subseteq> topspace X \<and> path_connectedin Y (f ` U))"
mono_rule: fixes E' :: "('a \<times> 'a) set" assumes "E' \<subseteq> E_IH" shows "all_security_requirements_fulfilled M \<lparr>nodes = V, edges = E'\<rparr>"
bbw_not_dist_2: fixes x :: "int"   and y :: "int" shows "NOT (x AND y) = NOT x OR NOT y"
hyps_for_simp': fixes h :: "('form, 'var) out_port"   and n :: "('form, 'rule) graph_node"   and p :: "('form, 'var) antecedent" shows "(h \<in> fset (hyps_for n p)) = (hyps n h = Some p)"
diagonal_to_Smith_aux_PQ_append2: fixes xs :: "nat list"   and ys :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "diagonal_to_Smith_aux_PQ (xs @ [ys]) bezout (P, A, Q) = Diagonal_to_Smith_row_i_PQ ys bezout (diagonal_to_Smith_aux_PQ xs bezout (P, A, Q))"
take_map: fixes n :: "nat"   and f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "take n (map f xs) = map f (take n xs)"
iterate_add_distrib: fixes m :: "nat"   and n :: "nat"   and a :: "'a" shows "iterate_add (m + n) a = iterate_add m a + iterate_add n a"
meld_invar: fixes bq1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and bq2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.invar bq1"   and "SkewBinomialHeapStruc.invar bq2" shows "SkewBinomialHeapStruc.invar (SkewBinomialHeapStruc.meld bq1 bq2)"
cs_rotate_rule: fixes l :: "'a list"   and p :: "'a node ref option" shows "<cs_list l p> cs_rotate p <cs_list (rotate1 l)>"
ex_un_null: shows "\<exists>!n. \<forall>f. n \<bullet> f = n \<and> f \<bullet> n = n"
fmfilter_true: fixes m :: "('a, 'b) fmap"   and P :: "'a \<Rightarrow> bool" assumes "\<And>x y. fmlookup m x = Some y \<Longrightarrow> P x" shows "fmfilter P m = m"
cblinfun_apply_in_image: fixes A :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and \<psi> :: "'b" shows "A *\<^sub>V \<psi> \<in> space_as_set (A *\<^sub>S \<top>)"
adv_P1_def: fixes m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool"   and D :: "(bool \<times> bool \<times> bool \<times> bool) \<times> (bool \<times> bool \<times> bool \<times> bool \<times> bool \<times> bool) \<times> 'v_OT121 \<times> 'v_OT121 \<times> 'v_OT121 \<Rightarrow> bool spmf" shows "OT_14.adv_P1 m1 m2 D \<equiv> \<bar>spmf (R1_14 m1 m2 \<bind> D) True - spmf (funct_OT_14 m1 m2 \<bind> (\<lambda>(o1, o2). S1_14 m1 o1 \<bind> D)) True\<bar>"
prefixE: fixes xs :: "'a list"   and ys :: "'a list" assumes "prefix xs ys" obtains zs :: "'a list" where "ys = xs @ zs"
union_inter: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B" shows "F g (A \<union> B) \<^bold>* F g (A \<inter> B) = F g A \<^bold>* F g B"