hcju/isabelleps huggingface dataset:
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_set_cases_7: fixes X :: "'c list"   and F :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) list" shows "set (bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) = set X"
isTrue_exi_iff: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue (exi yy (PPf (Var yy) \<langle>\<phi>\<rangle>)) = (\<exists>n\<in>num. isTrue (PPf n \<langle>\<phi>\<rangle>))"
size1_height_if_incomplete: fixes t :: "'c tree" assumes "\<not> complete t" shows "size1 t < 2 ^ height t"
\<beta>2: shows "\<beta>2 = \<beta>0 + \<beta>2a"
rel_interior_mono: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "affine hull S = affine hull T" shows "rel_interior S \<subseteq> rel_interior T"
le_1: shows "\<I>1' \<le> \<I>1"
prefs_from_table_wfD_5: fixes agents :: "'c set"   and alts :: "'d set"   and xss :: "('c \<times> 'd set list) list"   and xs :: "'d set list" assumes "prefs_from_table_wf agents alts xss"   and "xs \<in> set (map snd xss)" shows "\<Union> (set xs) = alts"
distinct_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "[] \<noteq> x21 # x22"
valid_lists_chain1: fixes Y :: "nat \<Rightarrow> Vars.var set"   and xs :: "Vars.var list"   and G :: "CoCalls" assumes "chain Y"   and "xs \<in> valid_lists (\<Union> (range Y)) G" shows "\<exists>i. xs \<in> valid_lists (Y i) G"
leg1_uniquely_isomorphic_2: shows "\<exists>!\<nu>. \<guillemotleft>\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright>"
simps: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and a :: "('a, 'b, 'c, 'd) converter" shows "\<I>, \<I>' \<turnstile>\<^sub>C a \<surd> = (\<exists>conv. a = conv \<and> (\<forall>x. x \<in> outs_\<I> \<I> \<longrightarrow> \<I>' \<turnstile>g run_converter conv x \<surd>) \<and> (\<forall>x xa xb. x \<in> outs_\<I> \<I> \<longrightarrow> (xa, xb) \<in> results_gpv \<I>' (run_converter conv x) \<longrightarrow> xa \<in> responses_\<I> \<I> x \<and> \<I>, \<I>' \<turnstile>\<^sub>C xb \<surd>))"
simps_8: fixes bbs :: "bool list"   and bs :: "int list"   and a :: "Cooper.num" shows "Ifm bbs bs (NEq a) = (Inum bs a \<noteq> 0)"
abs_pred_term_def: fixes P :: "Term_Class.term \<Rightarrow> bool"   and t :: "Term_Class.term" shows "abs_pred P t = ((\<forall>x. t = Bound x \<longrightarrow> P t) \<and> (\<forall>t'. t = \<Lambda> t' \<longrightarrow> P t' \<longrightarrow> P t))"
right_total_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "right_total R = (\<forall>y. \<exists>x. R x y)"
soundC: assumes "\<And>\<phi>. List.member A \<phi> \<Longrightarrow> seq_sem I \<phi> = UNIV" shows "seq_sem I C = UNIV"
kruskal0_def: shows "kruskal0 \<equiv> obtain_sorted_carrier \<bind> (\<lambda>l. nfoldli l (\<lambda>_. True) (\<lambda>e T. ASSERT (e \<in> E) \<bind> (\<lambda>_. a_endpoints e \<bind> (\<lambda>(a, b). ASSERT (joins a b e \<and> forest T \<and> e \<in> E \<and> T \<subseteq> E) \<bind> (\<lambda>_. if (a, b) \<notin> connected T then ASSERT (e \<notin> T) \<bind> (\<lambda>_. RETURN (insert e T)) else RETURN T)))) {} \<bind> RETURN)"
VectorSpaceHomSetD_VectorSpaceHom: fixes T :: "'a \<Rightarrow> 'b"   and fsmult :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and V :: "'a set"   and fsmult' :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and N :: "'b set" assumes "T \<in> VectorSpaceHomSet fsmult V fsmult' N" shows "VectorSpaceHom fsmult V fsmult' T"
i_set_i_UN_set_eq: shows "i_set = i_UN_set"
orderI: fixes a :: "'d multiset"   and b :: "'d multiset" assumes "a = a \<inter># b" shows "a \<subseteq># b"
assoc_end_type_def: fixes "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" shows "assoc_end_type end \<equiv> let \<C> = assoc_end_class end in if assoc_end_max end \<le> enat 1 then if assoc_end_min end = 0 then \<langle>\<C>\<rangle>\<^sub>\<T>[?] else \<langle>\<C>\<rangle>\<^sub>\<T>[1] else if assoc_end_unique end then if assoc_end_ordered end then OrderedSet \<langle>\<C>\<rangle>\<^sub>\<T>[1] else Set \<langle>\<C>\<rangle>\<^sub>\<T>[1] else if assoc_end_ordered end then Sequence \<langle>\<C>\<rangle>\<^sub>\<T>[1] else Bag \<langle>\<C>\<rangle>\<^sub>\<T>[1]"
lex_sym_eq_iff: fixes a :: "real \<times> real"   and b :: "real \<times> real" assumes "Counterclockwise_2D_Arbitrary.lex a b" shows "Counterclockwise_2D_Arbitrary.lex b a = (a = b)"
afun_rel_intros_1: fixes x :: "'b"   and y :: "'b" assumes "x = y" shows "pure x = pure y"
double_dual_iso_apply: fixes \<chi> :: "(('a \<Rightarrow> complex) \<Rightarrow> complex) \<Rightarrow> complex"   and x :: "('a \<Rightarrow> complex) \<Rightarrow> complex" assumes "character (Characters (Characters G)) \<chi>" shows "G''.double_dual_iso x \<chi> = \<chi> x"
less_eq_synlr_def: fixes x :: "('c, 'd) synlr"   and y :: "('c, 'd) synlr" shows "x \<le> y \<equiv> unsynlr x \<subseteq> unsynlr y"
TarskiLen_def: fixes A :: "'p"   and B :: "'p"   and l :: "'p \<Rightarrow> 'p \<Rightarrow> bool" shows "TarskiLen A B l \<equiv> QCong l \<and> l A B"
filter_empty_every_not: fixes P :: "'c \<Rightarrow> bool"   and l :: "'c list" shows "(filter P l = []) = list_all (\<lambda>x. \<not> P x) l"
qmodule_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set" shows "M /\<^sub>m H = \<lparr>carrier = set_mr_cos M H, pop = mr_cospOp M H, mop = mr_cosmOp M H, zero = H, sprod = mr_cos_sprod M H\<rparr>"
D: shows "D \<in> carrier_mat (m - 1) (m - 1)"
Polys_closed_normal_form: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "F \<subseteq> P[X]"   and "p \<in> P[X]" shows "normal_form F p \<in> P[X]"
fin_succs: shows "\<forall>v\<in>set (g_V G). finite (succs v)"
rel_intros_2: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x21 :: "'g"   and y21 :: "'h"   and x22 :: "'g list"   and y22 :: "'h list" assumes "R x21 y21"   and "list_all2 R x22 y22" shows "list_all2 R (x21 # x22) (y21 # y22)"
simps_7: fixes vf :: "'b \<Rightarrow> 'b list \<Rightarrow> bool" shows "[] \<simeq> {[], [], vf} = True"
bufferd_val_filter_is_Write\<^sub>s\<^sub>b_eq: fixes sb :: "'b memref list" shows "buffered_val (filter is_Write\<^sub>s\<^sub>b sb) = buffered_val sb"
dom_ran_the: fixes G :: "'e \<Rightarrow> 'd option"   and y :: "'d"   and x :: "'e" assumes "ran G = {y}"   and "x \<in> dom G" shows "the (G x) = y"
simps_2: fixes init :: "'a set"   and trans :: "('a \<times> 'a) set"   and more :: "'b" shows "TS.trans \<lparr>init = init, trans = trans, \<dots> = more\<rparr> = trans"
solves_ode_on_subset: fixes x :: "real \<Rightarrow> 'd"   and f :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and S :: "real set"   and Y :: "'d set"   and T :: "real set"   and X :: "'d set" assumes "(x solves_ode f) S Y"   and "T \<subseteq> S"   and "Y \<subseteq> X" shows "(x solves_ode f) T X"
positive_imp_inverse_positive: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero a" shows "less zero (inverse a)"
PImp: fixes r :: "'a pfm"   and p :: "'a pfm"   and q :: "'a pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<longrightarrow>\<^sub>! [r]\<^sub>! q)"
maxsimp_def: fixes x :: "'a set" shows "chamber x \<equiv> x \<in> X \<and> (\<forall>z\<in>X. order.greater_eq z x \<longrightarrow> z = x)"
IC_trans_inv: fixes q :: "nat"   and i :: "nat"   and p :: "nat" assumes "te q (i + 1) \<le> te p (i + 1)"   and "correct p (te p (i + 1))"   and "correct q (te p (i + 1))" shows "IC q (i + 1) (te p (i + 1)) = cfn q (\<lambda>n. \<theta> q (i + 1) n + (PC q (te p (i + 1)) - PC q (te q (i + 1))))"
mn: shows "length (map of_int_hom.vec_hom fs') \<le> n"
map_add_upd_left: fixes m :: "'a"   and e2 :: "'a \<Rightarrow> 'b option"   and e1 :: "'a \<Rightarrow> 'b option"   and u1 :: "'b" assumes "m \<notin> dom e2" shows "e1(m \<mapsto> u1) ++ e2 = (e1 ++ e2)(m \<mapsto> u1)"
higher_differentiable_on_frechet_derivativeI: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and i :: "nat"   and x :: "'b"   and h :: "'b" assumes "higher_differentiable_on X f (Suc i)"   and "open X"   and "x \<in> X" shows "higher_differentiable_on X (\<lambda>x. frechet_derivative f (at x) h) i"
paste_def: fixes P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" shows "P +* Q = P outside Domain Q \<union> Q"
add_nonneg_eq_0_iff: fixes x :: "'c multiset"   and y :: "'c multiset" assumes "{#} \<subseteq># x"   and "{#} \<subseteq># y" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
IH2_2: assumes "M \<Turnstile> G" shows "M(n := False) \<Turnstile> G"
Arg2pi_unique_lemma: fixes z :: "complex"   and t :: "real"   and t' :: "real" assumes "is_Arg z t"   and "is_Arg z t'"   and "0 \<le> t"   and "t < 2 * pi"   and "0 \<le> t'"   and "t' < 2 * pi"   and "z \<noteq> 0" shows "t' = t"
cball_eq_cball_iff: fixes x :: "'a"   and d :: "real"   and y :: "'a"   and e :: "real" shows "(cball x d = cball y e) = (d < 0 \<and> e < 0 \<or> x = y \<and> d = e)"
static_to_dynamic_overriding: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides\<^sub>S old"   and "wf_prog G" shows "G \<turnstile> new overrides old"
AbGroup: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "Module smult M" shows "AbGroup M"
assms_4: shows "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"
mk_minsky_def: fixes M :: "(nat \<times> (nat, nat) Op) set"   and P :: "(nat \<Rightarrow> nat) \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool" shows "mk_minsky_wit M P \<equiv> finite M \<and> deterministic M \<and> 0 \<notin> fst ` M \<and> (\<forall>vs. \<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step M)\<^sup>* \<and> P vs vs')"
shuffles_commutes: fixes xs :: "'g list"   and ys :: "'g list" shows "shuffles xs ys = shuffles ys xs"
nres_order_simps_2: fixes M :: "'b nres" shows "(M \<le> SUCCEED) = (M = SUCCEED)"
sign_extend_bitwise_if: assumes "i < size w" shows "sign_extend e w !! i = (if i < e then w !! i else w !! e)"
is_terminal: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' y f" shows "\<exists>!g. is_coext x' f g"
less_eq_sup_top: fixes x :: "'a"   and y :: "'a" shows "(- x \<le> - y) = (- - x \<squnion> - y = top)"
fun_add_eval_closed: fixes a :: "'c"   and f :: "'c \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a" assumes "a \<in> S"   and "f \<in> carrier F"   and "g \<in> carrier F" shows "(f \<oplus>\<^bsub>F\<^esub> g) a \<in> carrier R"
observable_io_target_unique_target: fixes M :: "('d, 'e, 'f) FSM"   and q1 :: "'f"   and io :: "('d \<times> 'e) list"   and q2 :: "'f"   and tr :: "'f list" assumes "observable M"   and "io_targets M q1 io = {q2}"   and "path M (io || tr) q1"   and "length io = length tr" shows "target (io || tr) q1 = q2"
row_col_match_1: fixes A1 :: "'a list list"   and A2 :: "'a list list"   and B1 :: "'a list list"   and B2 :: "'a list list"   and i :: "nat"   and j :: "nat" assumes "mat (row_length A1) (length A1) A1"   and "mat (row_length A2) (length A2) A2"   and "mat (row_length B1) (length B1) B1"   and "mat (row_length B2) (length B2) B2"   and "length A1 = row_length A2"   and "length B1 = row_length B2"   and "A1 \<noteq> [] \<and> A2 \<noteq> [] \<and> B1 \<noteq> [] \<and> B2 \<noteq> []"   and "i < row_length A1 * row_length B1"   and "j < length A2 * length B2" shows "length (row A1 (i div row_length B1)) = length (col A2 (j div length B2))"
disc_12: fixes x21 :: "variable"   and x22 :: "typ" shows "\<not> is_Bv (Fv x21 x22)"
invertible_inverse_closed: fixes u :: "'a set" assumes "quotient.invertible u"   and "u \<in> Partition" shows "quotient.inverse u \<in> Partition"
certificate_merSK_valid_lemma: fixes i :: "nat"   and M :: "agent"   and merSK :: "nat"   and evs :: "event list" assumes "Crypt (priSK (CA i)) \<lbrace>Agent M, Key merSK, onlySig\<rbrace> \<in> parts (knows Spy evs)"   and "CA i \<notin> bad"   and "CA i \<noteq> RCA"   and "evs \<in> set_mr" shows "\<exists>X Y Z. Says (CA i) M \<lbrace>X, cert M merSK onlySig (priSK (CA i)), Y, Z\<rbrace> \<in> set evs"
ntrunc_subsetI: fixes k :: "nat"   and M :: "('f, 'g) Old_Datatype.node set" shows "ntrunc k M \<subseteq> M"
sum_le_c_pair: fixes x :: "nat"   and y :: "nat" shows "x + y \<le> c_pair x y"
Cd_Above: fixes d :: "'b"   and X :: "'a set" shows "Pdds'.Cd d X = Above (Pd_shuffle_to_top (insert d__ ds') Y d) (X \<inter> Field (Pd_shuffle_to_top (insert d__ ds') Y d)) \<inter> X"
extend_state_oracle_def: fixes callee :: "'g \<Rightarrow> 'h \<Rightarrow> ('i \<times> 'g) spmf" shows "\<dagger>callee = (\<lambda>(s', s) x. map_spmf (\<lambda>(y, s). (y, s', s)) (callee s x))"
components_idempotent: fixes x :: "'a" shows "components (components x) = components x"
plus_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "plus_hom hom \<equiv> \<forall>x y. hom (x + y) = hom x + hom y"
T_vrestr_same_measure_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat" assumes "mpt M T"   and "A \<in> sets M" shows "Sigma_Algebra.measure M (qmpt.vimage_restr M (T ^^ n) A) = Sigma_Algebra.measure M A"
count_pre_union_img: fixes B :: "'c set"   and Y :: "'d \<Rightarrow> 'c" assumes "countable B" shows "Y -` B = \<Union> (range (countable_preimages B Y))"
simps_2: fixes I :: "interp"   and x :: "variable"   and \<theta> :: "trm" shows "game_sem I (x := \<theta>) = (\<lambda>X. {\<omega>. repv \<omega> x (term_sem I \<theta> \<omega>) \<in> X})"
subtuple_to_trancl: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ym :: "'b \<rightharpoonup>\<^sub>f 'a"   and xm :: "'b \<rightharpoonup>\<^sub>f 'a" assumes "reflp r"   and "subtuple r\<^sup>+\<^sup>+ xm ym" shows "(subtuple r)\<^sup>+\<^sup>+ xm ym"
Fvars_scnj: fixes F :: "'fmla set" assumes "F \<subseteq> fmla"   and "finite F" shows "Fvars (scnj F) = \<Union> (Fvars ` F)"
coprime_crossproduct': fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and b :: "'b"   and d :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize"   and "b \<noteq> zero"   and "unit_factor b = unit_factor d"   and "algebraic_semidom.coprime times one a b"   and "algebraic_semidom.coprime times one c d" shows "(times a d = times b c) = (a = c \<and> b = d)"
deduct_inv: fixes M :: "('fun, 'var) Term.term set"   and n :: "nat"   and t :: "('fun, 'var) Term.term" assumes "\<langle>M; n\<rangle> \<turnstile>\<^sub>n t" shows "t \<in> M \<or> (\<exists>f T. t = Fun f T \<and> public f \<and> length T = arity f \<and> (\<forall>t\<in>set T. \<exists>l<n. \<langle>M; l\<rangle> \<turnstile>\<^sub>n t)) \<or> (\<exists>m\<in>subterms\<^sub>s\<^sub>e\<^sub>t M. (\<exists>l<n. \<langle>M; l\<rangle> \<turnstile>\<^sub>n m) \<and> (\<forall>k\<in>set (fst (Ana m)). \<exists>l<n. \<langle>M; l\<rangle> \<turnstile>\<^sub>n k) \<and> t \<in> set (snd (Ana m)))"
state_update_vs_allInstances_generic_noincluding': fixes pre_post :: "'d state \<times> 'd state \<Rightarrow> 'd state"   and mk :: "'d state \<Rightarrow> 'd state \<times> 'd state"   and \<sigma>' :: "nat \<Rightarrow> \<langle>'d\<rangle>\<^sub>\<bottom>"   and oid :: "nat"   and Object :: "'d"   and Type :: "'d \<Rightarrow> \<langle>'e\<rangle>\<^sub>\<bottom>"   and A :: "nat \<Rightarrow> \<langle>nat list list list\<rangle>\<^sub>\<bottom>" assumes "\<And>a. pre_post (mk a) = a"   and "\<And>x. \<sigma>' oid = \<lfloor>x\<rfloor> \<Longrightarrow> x = Object"   and "Type Object = None" shows "OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>'(oid \<mapsto> Object), assocs = A\<rparr>) = OclAllInstances_generic pre_post Type (mk \<lparr>heap = \<sigma>', assocs = A\<rparr>)"
max_c: fixes n :: "nat" shows "c n \<le> max_c"
dvd_add_triv_left_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus a b) = dvd.dvd times a b"
card_PiE: fixes S :: "'a set"   and T :: "'a \<Rightarrow> 'b set" assumes "finite S" shows "card (Pi\<^sub>E S T) = (\<Prod>i\<in>S. card (T i))"
snd_fst_flip: fixes xy :: "'c \<times> 'a" shows "snd xy = (fst \<circ> (\<lambda>(x, y). (y, x))) xy"
triangle: fixes a :: "'d"   and b :: "'d" assumes "ide a"   and "ide b" shows "(a \<otimes> \<l>[b]) \<cdot> \<a>[a, \<I>, b] = \<r>[a] \<otimes> b"
unit_factor_normalize: fixes a :: "'a" assumes "a \<noteq> (0::'a)" shows "unit_factor (normalize a) = (1::'a)"
eval_preserves_sconf: 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 e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   and "P,E \<turnstile> e :: T"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
less_fun_irrefl: fixes f :: "'b \<Rightarrow> 'c" shows "\<not> less_fun f f"
ln_lower_11_def: shows "ln_lower_11 \<equiv> \<lambda>x. - ln_upper_11 (inverse x)"
simps: fixes a1 :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and a2 :: "nat option \<times> (nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list" shows "P \<turnstile> a1 -jvm\<rightarrow>\<^sub>1 a2 = (\<exists>\<sigma> \<sigma>'. a1 = \<sigma> \<and> a2 = \<sigma>' \<and> exec (P, \<sigma>) = \<lfloor>\<sigma>'\<rfloor>)"
simps_1: fixes c :: "int" shows "neg (Pc c) = Pc (- c)"
orderE: fixes a :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># b" obtains "a = a \<inter># b"
conc_fun_RES: fixes R :: "('a \<times> 'b) set"   and X :: "'b set" shows "\<Down> R (RES X) = RES (R\<inverse> `` X)"
group_card_in_sizes: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and g :: "'b set" assumes "group_design point_set block_collection groups"   and "g \<in> groups" shows "card g \<in> group_design.group_sizes groups"
hlg_ops_def: shows "hlg_ops \<equiv> hlg_gbm.gbm_ops"
coplanar_translation_imp: fixes S :: "'a set"   and a :: "'a" assumes "coplanar S" shows "coplanar ((+) a ` S)"
assms_1: shows "e \<in> UNIV \<rightarrow> {- 1<..<1}"
prod_mono2: fixes B :: "'d set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite B"   and "A \<subseteq> B"   and "\<And>b. b \<in> B - A \<Longrightarrow> (1::'e) \<le> f b"   and "\<And>a. a \<in> A \<Longrightarrow> (0::'e) \<le> f a" shows "prod f A \<le> prod f B"
upb2_def: shows "upb2 = loc_upb (c_fst (c_snd n)) y"
openin_pullback_topology: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and T :: "'e topology"   and S :: "'d set" shows "openin (pullback_topology A f T) S = (\<exists>U. openin T U \<and> S = f -` U \<inter> A)"
iIN_def: fixes n :: "nat"   and d :: "nat" shows "[n\<dots>,d] \<equiv> {n..n + d}"
maxl_fun_func: shows "maxl_fun D \<in> carrier D \<rightarrow> carrier D"
hyp_port_path_for_def: fixes t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list"   and f :: "'form" shows "hyp_port_path_for t is f = fst (hyp_port_for' t is f)"
CptnOne: fixes P :: "'b com option"   and s :: "'b" shows "[(P, s)] \<in> cptn"
cases: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> a1 \<Rightarrow>\<^sub>r a2" obtains "a1 = []" and "a2 = Undecided"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule m__ Accept # rs__" and "a2 = Decision FinalAllow" and "matches \<gamma> m__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule ma__ Drop # rsa__" and "a2 = Decision FinalDeny" and "matches \<gamma> ma__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mb__ Reject # rsb__" and "a2 = Decision FinalDeny" and "matches \<gamma> mb__ p"   | m :: "'c match_expr" and rs :: "'c rule list" where "a1 = Rule mc__ Return # rsc__" and "a2 = Undecided" and "matches \<gamma> mc__ p"   | rs :: "'c rule list" and t :: "state" and m :: "'c match_expr" where "a1 = Rule md__ Log # rsd__" and "a2 = t__" and "\<Gamma>,\<gamma>,p\<turnstile> rsd__ \<Rightarrow>\<^sub>r t__"   | rs :: "'c rule list" and t :: "state" and m :: "'c match_expr" where "a1 = Rule me__ Empty # rse__" and "a2 = ta__" and "\<Gamma>,\<gamma>,p\<turnstile> rse__ \<Rightarrow>\<^sub>r ta__"   | m :: "'c match_expr" and rs :: "'c rule list" and t :: "state" and a :: "action" where "a1 = Rule mf__ a # rsf__" and "a2 = tb__" and "\<not> matches \<gamma> mf__ p" and "\<Gamma>,\<gamma>,p\<turnstile> rsf__ \<Rightarrow>\<^sub>r tb__"   | m :: "'c match_expr" and chain :: "char list" and rs :: "'c rule list" and X :: "final_decision" and rrs :: "'c rule list" where "a1 = Rule mg__ (Call chain__) # rrs__" and "a2 = Decision X" and "matches \<gamma> mg__ p" and "\<Gamma> chain__ = Some rsg__" and "\<Gamma>,\<gamma>,p\<turnstile> rsg__ \<Rightarrow>\<^sub>r Decision X"   | chain :: "char list" and rs :: "'c rule list" and rrs :: "'c rule list" and t :: "state" and m :: "'c match_expr" where "a1 = Rule m (Call chain) # rrs" and "a2 = t" and "\<Gamma> chain = Some rs" and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>r Undecided" and "\<Gamma>,\<gamma>,p\<turnstile> rrs \<Rightarrow>\<^sub>r t"
points_index_alt_def: fixes ps :: "'a set" shows "\<B>\<^sup>D index ps = size (filter_mset ((\<subseteq>) ps) \<B>\<^sup>D)"
Pareto_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" shows "Pareto R x y = ((\<exists>j. R j x x) \<and> (\<forall>i. R i x x \<longrightarrow> R i x y))"
split_sels_2: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "tree \<Rightarrow> tree \<Rightarrow> 'a"   and tree :: "tree" shows "P (case tree of Leaf \<Rightarrow> f1 | Branching x xa \<Rightarrow> f2 x xa) = (\<not> (tree = Leaf \<and> \<not> P f1 \<or> tree = Branching (ltree tree) (rtree tree) \<and> \<not> P (f2 (ltree tree) (rtree tree))))"