hcju/isabelleps huggingface dataset:
a: fixes h' :: "'a" assumes "h ## h'"   and "P h'" shows "Q (h + h')"
ang_vec_opposite2: fixes z2 :: "complex"   and z1 :: "complex" assumes "z2 \<noteq> 0" shows "\<angle> z1 (- z2) = \<downharpoonright>\<angle> z1 z2 + pi\<downharpoonleft>"
map_filter_def: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'b option" shows "map_filter P m = (\<lambda>x. if P x then m x else None)"
finite_lists_length_eq_iff: fixes A :: "'a set"   and n :: "nat" shows "finite {xs. set xs \<subseteq> A \<and> length xs = n} = (finite A \<or> n = 0)"
wNegU: fixes \<theta> :: "trm"   and \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word" shows "([Neg \<theta>]<>\<nu>) = (let (l, u) = [\<theta>]<>\<nu> in (wneg u, wneg l))"
singular_boundary: fixes p :: "nat"   and X :: "'b topology"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X {} c = (\<exists>d. singular_chain (Suc p) X d \<and> chain_boundary (Suc p) d = c)"
inverse_nonpositive_iff_nonpositive: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq (inverse a) zero = less_eq a zero"
submodule_def: fixes H :: "'h set"   and R :: "('i, 'j) ring_scheme"   and M :: "('i, 'h, 'k) module_scheme" shows "Module.submodule H R M \<equiv> subgroup H (add_monoid M) \<and> submodule_axioms H R M"
fder: shows "(f has_field_derivative contour_integral (circlepath z r) (\<lambda>u. f u / (u - w)\<^sup>2) / (2 * complex_of_real pi * \<i>)) (at w)"
uadmit_dterm_adjoint': fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<theta> :: "('sf, 'sz) trm" assumes "\<And>f f'. SFunctions \<sigma> f = Some f' \<Longrightarrow> dfree f'"   and "\<And>f f'. SPredicates \<sigma> f = Some f' \<Longrightarrow> fsafe f'"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT \<theta>. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"   and "dsafe \<theta>" shows "dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> = dterm_sem (local.adjoint I \<sigma> \<omega>) \<theta>"
supp_list_rev: fixes xs :: "'b list" shows "supp (rev xs) = supp xs"
circuit_min_dep: fixes X :: "'a set"   and x :: "'a" assumes "circuit X"   and "x \<in> X" shows "indep (X - {x})"
wt_ge_num_args_if_\<delta>_eq_0: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym"   and "\<delta> = 0" shows "num_args s \<le> kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s"
Pow_iff: fixes A :: "'b set"   and B :: "'b set" shows "(A \<in> Pow B) = (A \<subseteq> B)"
assms: shows "\<Gamma>(chain \<mapsto> rs),\<gamma>,p\<turnstile> \<langle>process_ret (add_match m rs), Undecided\<rangle> \<Rightarrow> Decision X"
dim_eq_span: fixes S :: "('c, 'd) vec set"   and T :: "('c, 'd) vec set" assumes "S \<subseteq> T"   and "vec.dim T \<le> vec.dim S" shows "vec.span S = vec.span T"
intros_2: fixes t :: "'b"   and n :: "nat" shows "may_lock \<lfloor>(t, n)\<rfloor> t"
simps_2: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and delta :: "'opSym"   and qinp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm option"   and qbinp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs option" shows "qInit MOD (qOp delta qinp qbinp) = igOp MOD delta (lift (qInit MOD) qinp) (lift (qInitAbs MOD) qbinp)"
insert_Id_image_Acts: fixes f :: "('e \<times> 'e) set \<Rightarrow> ('d \<times> 'd) set"   and F :: "'e program" assumes "f Id = Id" shows "insert Id (f ` Acts F) = f ` Acts F"
last_messageI: fixes m :: "'b message_af"   and i :: "nat"   and xs :: "'b message_af list" assumes "m \<noteq> NoMsg"   and "i < length xs"   and "xs ! i = m"   and "\<And>j. \<lbrakk>i < j; j < length xs\<rbrakk> \<Longrightarrow> xs ! j = NoMsg" shows "last_message xs = m"
f_aggregate_nth_eq_sublist_list: fixes n :: "nat"   and xs :: "'b list"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b" assumes "n < length xs div k" shows "f_aggregate xs k ag ! n = ag (sublist_list xs [n * k..<n * k + k])"
exec: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow> Stuck"
Heine_Borel_lemma: fixes S :: "'b set"   and \<G> :: "'b set set" assumes "compact S"   and "S \<subseteq> \<Union> \<G>"   and "\<And>G. G \<in> \<G> \<Longrightarrow> open G" obtains e :: "real" where "0 < e" and "\<And>x. x \<in> S \<Longrightarrow> \<exists>G\<in>\<G>. ball x e \<subseteq> G"
f': shows "\<guillemotleft>f' : map\<^sub>0 a \<rightarrow>\<^sub>D map\<^sub>0 b\<guillemotright> \<and> D.equivalence_map f'"
assms_3: shows "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"
mayDiverge_If_R: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and tst :: "'a"   and s :: "'b"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c2 :: "('a, 'c) com"   and c1 :: "('a, 'c) com" assumes "\<not> tval tst s"   and "PL.mayDiverge tval aval c2 s" shows "PL.mayDiverge tval aval (if tst then c1 else c2) s"
rell_F_mono_strong: fixes L1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L3 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and x :: "('p, 'r, 't, 'v, 'w, 'x, 'y, 'z, 'aa, 'ab) F"   and y :: "('q, 's, 'u, 'v, 'w, 'x, 'y, 'z, 'aa, 'ab) F"   and L1' :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and L3' :: "'t \<Rightarrow> 'u \<Rightarrow> bool" assumes "rell_F L1 L2 L3 x y"   and "\<And>a b. \<lbrakk>a \<in> set1_F x; b \<in> set1_F y; L1 a b\<rbrakk> \<Longrightarrow> L1' a b"   and "\<And>a b. \<lbrakk>a \<in> set2_F x; b \<in> set2_F y; L2 a b\<rbrakk> \<Longrightarrow> L2' a b"   and "\<And>a b. \<lbrakk>a \<in> set3_F x; b \<in> set3_F y; L3 a b\<rbrakk> \<Longrightarrow> L3' a b" shows "rell_F L1' L2' L3' x y"
alw_sconst: fixes P :: "'c stream \<Rightarrow> bool"   and x :: "'c" shows "alw P (sconst x) = P (sconst x)"
root_nat_ceiling_def: fixes p :: "nat"   and x :: "nat" shows "root_nat_ceiling p x = root_int_ceiling_pos p (int x)"
hom_boundary_chain_boundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X S c"   and "1 \<le> p" shows "hom_boundary (int p) X S (homologous_rel_set p X S c) = homologous_rel_set (p - Suc 0) (subtopology X S) {} (chain_boundary p c)"
Un_Int_distrib: fixes A :: "'d set"   and B :: "'d set"   and C :: "'d set" shows "A \<union> B \<inter> C = (A \<union> B) \<inter> (A \<union> C)"
IH: shows "finite {f. (\<forall>x. x \<notin> R \<longrightarrow> f x = c) \<and> (\<forall>x. x \<in> R \<longrightarrow> f x \<in> A)}"
div_self: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and a :: "'e" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide a a = one"
simple_function_measurable_on_UNIV: fixes f :: "'a \<Rightarrow> real" assumes "simple_function lebesgue f"   and "\<And>x. 0 \<le> f x" shows "f measurable_on UNIV"
compound_not_in_modal_multi: fixes M :: "'a"   and Ms :: "('a, 'b) form list"   and N :: "'b"   and \<Gamma> :: "('a, 'b) form multiset" shows "Compound M Ms \<notin># N \<cdot> \<Gamma>"
stepC_dec': fixes n :: "C"   and c :: "nat" assumes "n \<noteq> C []" shows "evalC (c + 3) (stepC c n) < evalC (c + 3) n"
simps_1: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "\<not> lnull (iterates f x)"
simps: fixes m :: "nat" shows "s1 m = 2 * m + 1"
lm121: fixes X :: "nat set"   and list :: "'b list" assumes "X \<subseteq> {0..<length list}" shows "(!) list ` X \<subseteq> set list"
the_\<theta>_props_1: shows "\<guillemotleft>\<pi>.the_\<theta> : t\<^sub>0 \<star> \<pi>.chine \<Rightarrow> s\<^sub>0\<guillemotright>"
fact: shows "fact = (\<lambda>(q, i). map (\<lambda>f. (f, i)) (local.main_int_poly_factorization q))"
exe_coregular_tcsigs_def: fixes cs :: "(String.literal \<times> String.literal) list"   and arss :: "('a \<times> (String.literal \<times> String.literal set list) list) list" shows "exe_coregular_tcsigs cs arss \<equiv> \<forall>ars\<in>snd ` set arss. \<forall>c\<^sub>1\<in>fst ` set ars. \<forall>c\<^sub>2\<in>fst ` set ars. class_leq (set cs) c\<^sub>1 c\<^sub>2 \<longrightarrow> list_all2 (sort_leq (set cs)) (the (lookup (\<lambda>x. x = c\<^sub>1) ars)) (the (lookup (\<lambda>x. x = c\<^sub>2) ars))"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 (trg f) f"
contra_subsetD: fixes A :: "'d set"   and B :: "'d set"   and c :: "'d" assumes "A \<subseteq> B"   and "c \<notin> B" shows "c \<notin> A"
main_int_poly_factorization: fixes f :: "int poly"   and fs :: "int poly list" assumes "local.main_int_poly_factorization f = fs"   and "square_free f"   and "0 < degree f"   and "coeff f 0 \<noteq> 0" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
keysFor_insert_Key: fixes K :: "nat"   and H :: "msg set" shows "keysFor (insert (Key K) H) = keysFor H"
distinct_17: shows "ToUpperCaseOp \<noteq> ToBooleanOp"
simps_8: fixes f11 :: "'b \<Rightarrow> 'a"   and f12 :: "'b bexp \<Rightarrow> 'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f2 :: "'b aexp \<Rightarrow> 'b aexp \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'c"   and x121 :: "'b bexp"   and x122 :: "'b aexp"   and x123 :: "'b aexp" shows "rec_aexp f11 f12 f2 (ITE x121 x122 x123) = f12 x121 x122 x123 (rec_bexp f11 f12 f2 x121) (rec_aexp f11 f12 f2 x122) (rec_aexp f11 f12 f2 x123)"
Dom_conv_Ran: fixes P :: "'b hrel" shows "Dom (P\<^sup>-) = Ran P"
matches_ipassmt_iface_replace_srcip_mexpr: fixes \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (ipassmt_iface_replace_srcip_mexpr ipassmt ifce) a p = (case ipassmt ifce of None \<Rightarrow> match_iface ifce (p_iiface p) | Some ips \<Rightarrow> p_src p \<in> ipcidr_union_set (set ips))"
uniformly_convergent_imp_convergent: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'a" assumes "uniformly_convergent_on X f"   and "x \<in> X" shows "convergent (\<lambda>n. f n x)"
map_comp: fixes g :: "'e \<Rightarrow> 'd"   and f :: "'f \<Rightarrow> 'e"   and v :: "'f cset" shows "cimage g (cimage f v) = cimage (g \<circ> f) v"
fin_string_nil: fixes n :: "nat" shows "List.member (fin_string_upto n) fin_string_empty"
prems_2: shows "\<nexists>p. epath E2 u p v"
lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S_FAnd: fixes n :: "nat"   and \<phi>\<^sub>1 :: "'a formula"   and \<phi>\<^sub>2 :: "'a formula" assumes "wf_formula n (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2)" shows "lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2) \<subseteq> lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi>\<^sub>1 \<inter> lang\<^sub>W\<^sub>S\<^sub>1\<^sub>S n \<phi>\<^sub>2 \<inter> \<Union> {enc (w, I) |w I. length I = n \<and> wf_interp_for_formula (w, I) (FAnd \<phi>\<^sub>1 \<phi>\<^sub>2)}"
negligible_outer_le: fixes S :: "'c set" shows "negligible S = (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue T \<le> e)"
relcomp_empty1: fixes R :: "('d \<times> 'c) set" shows "{} O R = {}"
J: shows "Min {i. i < n \<and> 0 < enum xs index key n mi ma ! i} \<in> {i. i < n \<and> 0 < enum xs index key n mi ma ! i}"
eq_numeral_iff_iszero_1: 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"   and y :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "(numeral.numeral one plus x = numeral.numeral one plus y) = ring_1.iszero zero (neg_numeral.sub minus plus one x y)"
P\<^sub>2_B\<^sub>2_lower_bound_P\<^sub>1: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set" assumes "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 (S \<union> L)"   and "2 * card P\<^sub>2 \<le> card (\<Union> P\<^sub>2)"   and "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))" shows "2 * card (P\<^sub>2 \<union> wrap B\<^sub>2) \<le> card P\<^sub>1 + 1"
tm_pow_range: fixes t :: "taylor_model"   and I :: "float interval list"   and e :: "nat \<Rightarrow> real"   and a :: "float interval list"   and x :: "real"   and n :: "nat"   and prec :: "nat"   and ord :: "nat" assumes "num_params (tm_poly t) \<le> length I"   and "develops_at_within e a I"   and "x \<in>\<^sub>i range_tm e t" shows "x ^ n \<in>\<^sub>i range_tm e (tm_pow prec ord I a t n)"
canon_term_pp_pair_def: fixes a :: "'t \<times> 'a" shows "canon_term_pp_pair a = (gcs (pp_of_term (fst a)) (snd a) = (0::'a))"
mult_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<parallel> x \<le> z \<parallel> y"
g_near_f: fixes x :: "'a"   and n :: "nat" assumes "x \<in> cbox a b" shows "norm (f x - g n x) \<le> 1 / (real n + 1)"
indRelRTPO_impl_TRel_is_weak_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_bisimulation (indRelRTPO TRel) (STCal Source Target)" shows "weak_reduction_bisimulation (TRel\<^sup>+) Target"
substitutiont: fixes E :: "char list \<Rightarrow> 'b"   and F :: "char list \<Rightarrow> 'b list \<Rightarrow> 'b"   and t :: "fterm"   and \<sigma> :: "char list \<Rightarrow> fterm" shows "eval\<^sub>t E F (t \<cdot>\<^sub>t \<sigma>) = eval\<^sub>t (evalsub E F \<sigma>) F t"
coboundedI2: fixes c :: "'c DBMEntry"   and b :: "'c DBMEntry"   and a :: "'c DBMEntry" assumes "c \<preceq> b" shows "c \<preceq> linordered_monoid.max a b"
Ints_diff: 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 a :: "'a"   and b :: "'a" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus"   and "b \<in> ring_1.Ints one plus zero minus" shows "minus a b \<in> ring_1.Ints one plus zero minus"
vs: shows "vs \<equiv> getFrN (map snd txs) (r # map fst txs) [] (length txs)"
ideal_prod_def: fixes R :: "('d, 'e) Ring_scheme"   and I :: "'d set"   and J :: "'d set" shows "I \<diamondsuit>\<^sub>r\<^bsub>R\<^esub> J \<equiv> \<Inter> {L. ideal R L \<and> {x. \<exists>i\<in>I. \<exists>j\<in>J. x = i \<cdot>\<^sub>r\<^bsub>R\<^esub> j} \<subseteq> L}"
q65_array_def: shows "q65_array = IArray [True, True, False, False, True, False, False, False, False, True, True, False, False, False, True, False, True, False, False, False, False, False, False, False, False, True, True, False, False, True, True, False, False, False, False, True, True, False, False, True, True, False, False, False, False, False, False, False, False, True, False, True, False, False, False, True, True, False, False, False, False, True, False, False, True, False]"
prebicategory_with_homs_axioms: shows "prebicategory_with_homs (\<cdot>) (\<star>) \<a> src trg"
delta: fixes S :: "'c set"   and a :: "'c"   and b :: "'c \<Rightarrow> 'a" assumes "finite S" shows "F.F (\<lambda>k. if k = a then b k else \<^bold>1) S = (if a \<in> S then b a else \<^bold>1)"
exec_move_ci_appD: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and E :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and fr' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "exec_move ci P t E h (stk, loc, pc, None) ta h' fr'" shows "ci_app ci (compE2 E ! pc) (compP2 P) h stk loc undefined undefined pc []"
index_mult_mat_3: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_col (A * B) = dim_col B"
G_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and p :: "'c \<Rightarrow> 'b"   and I :: "'c set" assumes "comm_monoid_set f z" shows "comm_monoid_set.G f z p I \<equiv> if finite {x \<in> I. p x \<noteq> z} then comm_monoid_set.F f z p {x \<in> I. p x \<noteq> z} else z"
mod_ring_rel_of_int: fixes x :: "int" assumes "0 \<le> x"   and "x < p" shows "mod_ring_rel x (of_int x)"
disc_41: fixes x61 :: "poscheckvariant"   and x62 :: "('e, 'f) Term.term"   and x63 :: "('e, 'f) Term.term" shows "is_InSet \<langle>x61: x62 \<in> x63\<rangle>"
\<mu>_bound_row_def: fixes fs :: "int vec list"   and bnd :: "rat"   and i :: "nat" shows "\<mu>_bound_row fs bnd i = (\<forall>j\<le>i. (gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom fs) i j)\<^sup>2 \<le> bnd)"
matchC_C_def2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchC_C theta d c = (\<forall>s t d' t'. s \<approx> t \<and> (d, t) \<rightarrow>c (d', t') \<longrightarrow> (\<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> s' \<approx> t' \<and> (d', c') \<in> theta))"
nonEmpty_image: fixes \<Gamma> :: "'a multiset"   and f :: "'a \<Rightarrow> 'a" assumes "\<Gamma> \<noteq> \<Empt>" shows "f \<cdot>\<cdot> \<Gamma> \<noteq> \<Empt>"
positively_invariant_UNIV: shows "positively_invariant UNIV"
St_def: fixes x2 :: "name"   and x3 :: "trm"   and x1 :: "stack" shows "[x2]x3\<ggreater>x1 \<equiv> Abs_stack (St_Rep ([x2].x3) (Rep_stack x1))"
assert_sat: fixes s :: "('i, 'a) state"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<not> \<U> (assert a s)" shows "\<Turnstile> (assert a s)"
ldsj: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<phi>s :: "'c list" assumes "Syntax_with_Connectives_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj"   and "set \<phi>s \<subseteq> fmla" shows "Syntax_with_Connectives_False_Disj.ldsj fls dsj \<phi>s \<in> fmla"
bst_of_list_trees: fixes ys :: "'b list"   and A :: "'b set" assumes "set ys \<subseteq> A" shows "bst_of_list ys \<in> trees A"
par: shows "par (local.comp (pr1 (cod f) (cod g)) (tuple f g)) f"
set_less_aux_empty: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "\<not> ord.set_less_aux less_eq A {}"
simps_3: fixes uu :: "'a"   and uv :: "bool" shows "restrict Falseif uu uv = Falseif"
lc_induct: fixes t :: "sterm"   and P1 :: "sterm \<Rightarrow> bool"   and P2 :: "sterm \<Rightarrow> bool" assumes "lc t"   and "\<And>x. P1 (Fvar x)"   and "\<And>t l a. \<lbrakk>lc t; P1 t; lc a; P1 a\<rbrakk> \<Longrightarrow> P1 (Call t l a)"   and "\<And>t l u. \<lbrakk>lc t; P1 t; P2 u\<rbrakk> \<Longrightarrow> P1 (Upd t l u)"   and "\<And>f T. \<forall>l\<in>dom f. P2 (the (f l)) \<Longrightarrow> P1 (Obj f T)"   and "\<And>L t. \<lbrakk>finite L; \<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> lc (t\<^bsup>[Fvar s,Fvar p]\<^esup>) \<and> P1 (t\<^bsup>[Fvar s,Fvar p]\<^esup>)\<rbrakk> \<Longrightarrow> P2 t" shows "P1 t"
emeasure_le_1: fixes s :: "'a cfg"   and S :: "'a cfg stream set" shows "emeasure (MC.T s) S \<le> 1"
induct: fixes P :: "primfun \<times> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a0 :: "primfun \<times> 'a"   and a1 :: "'a" assumes "\<And>x a e. P (x, a) e" shows "P a0 a1"
ack_nest_bound: fixes i1 :: "nat"   and i2 :: "nat"   and j :: "nat" shows "ack i1 (ack i2 j) < ack (2 + (i1 + i2)) j"
mem_read2_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and address :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (memory_read 10 address s1) \<and> t2 = snd (memory_read 10 address s2)" shows "low_equal t1 t2"
get_create: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e" assumes "bij_lens x" shows "create\<^bsub>x\<^esub> (get\<^bsub>x\<^esub> \<sigma>) = \<sigma>"
ts_length_hint2: fixes x :: "nat \<Rightarrow> 'd list"   and i :: "nat" assumes "ts x" shows "length (x i) = Suc 0"
x_6: shows "x' \<sharp> [(y, y')] \<bullet> u"
d_state: fixes state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and i :: "nat"   and fs :: "int vec list"   and upw :: "bool"   and f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and ds :: "int iarray"   and ii :: "nat" assumes "LLL_impl_inv state i fs"   and "LLL_invariant upw i fs"   and "state = (f, mu, ds)"   and "ii \<le> m" shows "d_state state ii = d fs ii"
env_restr_eq_subset: fixes S :: "'c set"   and S' :: "'c set"   and m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd" assumes "S \<subseteq> S'"   and "m1 f|` S' = m2 f|` S'" shows "m1 f|` S = m2 f|` S"
simps_9: fixes x :: "name"   and P :: "pi" shows "hnf (<\<nu>x>P) = (\<exists>a P'. a \<noteq> x \<and> P = a{x}.P')"
one_not_zero: shows "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
norm_triangle_le_diff: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and e :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm x + norm y \<le> e" shows "norm (minus x y) \<le> e"
wfGfilternew: shows "wf_graph \<lparr>nodes = hosts \<T>, edges = flows_fix \<T> \<union> filternew_flows_state \<T> \<union> backflows (filternew_flows_state \<T>)\<rparr>"
\<phi>_\<psi>: fixes b :: "'c"   and a :: "'c"   and x :: "(('c + 'd) set, 'c + 'd \<Rightarrow> 'c + 'd) concrete_category.arr" assumes "C.ide b"   and "C.ide a"   and "x \<in> HomC.set (b, a)" shows "inC (HomC.\<psi> (b, a) x) = x"