hcju/isabelleps huggingface dataset:
augment_def: fixes f' :: "nat \<times> nat \<Rightarrow> 'capacity" shows "augment f' \<equiv> \<lambda>(u, v). if (u, v) \<in> E then f (u, v) + f' (u, v) - f' (v, u) else (0::'capacity)"
ptensor_mat_comm: fixes vars1 :: "nat set"   and vars2 :: "nat set"   and m1 :: "complex mat"   and m2 :: "complex mat" assumes "vars1 \<inter> vars2 = {}" shows "partial_state2.ptensor_mat ps2_P.dims0 vars1 vars2 m1 m2 = partial_state2.ptensor_mat ps2_P.dims0 vars2 vars1 m2 m1"
i_Exec_Stream_Acc_Output_Init_eq_i_Exec_Stream_Acc_Output_Cons_output: fixes output_fun :: "'e \<Rightarrow> 'd message_af"   and c :: "'e"   and k :: "nat"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af" assumes "output_fun c = NoMsg" shows "i_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun input c = [NoMsg] \<frown> i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c"
subst_app: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and u\<^sub>1 :: "'b"   and u\<^sub>2 :: "'b"   and env :: "(name, 'b) fmap" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "subst (app u\<^sub>1 u\<^sub>2) env = app (subst u\<^sub>1 env) (subst u\<^sub>2 env)"
ccw'_imp_ccw: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' a b c" shows "ccw a b c"
on_circline_y_axis_I: fixes z :: "complex" assumes "is_imag z" shows "on_circline y_axis (of_complex z)"
lbk_option_lgeom: fixes T :: "nat"   and w :: "bool stream" shows "lbk_option T w = (MAX i\<in>{0..T}. lgeom_proc u d init (stake i w)) - lgeom_proc u d init (stake T w)"
full_meet_weak1: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K \<turnstile> \<phi>" shows "K \<div>\<^bsub>\<gamma>\<^sub>F\<^sub>C\<^esub> \<phi> = K \<inter> Cn {.\<not> \<phi>}"
VUnion_VPow: fixes A :: "V" shows "\<Union>\<^sub>\<circ> (VPow A) = A"
keys_homogenizeI: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "x \<notin> indets p"   and "t \<in> keys p" shows "monomial (poly_deg p - deg_pm t) x + t \<in> keys (homogenize x p)"
HRB_slice_realizable: fixes n :: "'node SDG_node"   and S :: "'node SDG_node set" assumes "n \<in> HRB_slice S"   and "Ball S valid_SDG_node"   and "n \<notin> S" obtains n' :: "'node SDG_node" and ns :: "'node SDG_node list" where "n' \<in> S" and "realizable (CFG_node (_Entry_)) ns n'" and "n \<in> set ns"
cgenideal_eq_rcos: fixes i :: "'a" shows "PIdl i = carrier R #> i"
path_valid_edges: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'" shows "\<forall>a\<in>set as. valid_edge a"
diff_conv_add_uminus: fixes a :: "'d"   and b :: "'d" shows "a - b = a + - b"
C_eq_Sets: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and s :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy1_strong s"   and "wellformed_policy3Pr p"   and "wellformed_policy3Pr s"   and "allNetsDistinct p"   and "set p = set s" shows "Cp (list2FWpolicy p) x = Cp (list2FWpolicy s) x"
Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and f :: "'f" shows "\<Gamma>\<turnstile>c \<down> Fault f"
eRS_deltaF: fixes e :: "'d \<rightarrow> 'd"   and R :: "('d, 'a) synlr dual"   and S :: "('d, 'a) synlr" assumes "e\<cdot>\<bottom> = \<bottom>"   and "eRSS e R S" shows "eRSS (\<delta>\<cdot>e) (dual (F (dual S, undual R))) (F (R, S))"
three_in_set3: fixes X :: "'c set" assumes "3 \<le> card X" obtains x :: "'c" and y :: "'c" and z :: "'c" where "x \<in> X" and "y \<in> X" and "z \<in> X" and "x \<noteq> y" and "x \<noteq> z" and "y \<noteq> z"
supp_Cons: fixes x :: "'b"   and xs :: "'b list" shows "supp (x # xs) = supp x \<union> supp xs"
plus_of_int: fixes x :: "real"   and n :: "int" shows "f (x + real_of_int n * p) = f x"
scf: fixes i :: "nat"   and n :: "nat"   and f :: "'f" assumes "i < n" shows "0 < scf (f, n) i"
to_nat_from_nat_k_suc: shows "mod_type_class.to_nat (mod_type_class.from_nat k) < Suc k"
rev_nth_Some: fixes xs :: "'c list"   and v :: "nat"   and y :: "'c" assumes "xs !. v = Some y" shows "v < length xs"
aform_inf_inner_refine: fixes optns :: "nat" shows "(RETURN o2 aform_inf_inner optns, Inf_inners) \<in> aforms_rel \<rightarrow> rl_rel \<rightarrow> \<langle>rnv_rel\<rangle>nres_rel"
fold_append: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list" shows "fold f (xs @ ys) = fold f ys \<circ> fold f xs"
A_less_3: fixes x' :: "real" assumes "x' \<noteq> x"   and "poly p x' = 0" shows "A < \<bar>x' - x\<bar>"
mod_mod_cancel: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "dvd.dvd times c b" shows "modulo (modulo a b) c = modulo a c"
mod_mult_left_eq: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "a mod c * b mod c = a * b mod c"
fmrel_iff: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'g \<rightharpoonup>\<^sub>f 'e"   and n :: "'g \<rightharpoonup>\<^sub>f 'f" shows "fmrel R m n = (\<forall>x. rel_option R (fmlookup m x) (fmlookup n x))"
\<I>_real_def: shows "\<I>_real \<equiv> \<lambda>\<eta>. \<I>_uniform UNIV (insert None (Some ` nlists UNIV \<eta>))"
axioms_1: fixes \<Omega> :: "'a set"   and M :: "'a set set" assumes "sigma_algebra \<Omega> M" shows "algebra \<Omega> M"
reachable_restrict: fixes r :: "'a"   and g :: "'a" assumes "vector r" shows "r\<^sup>T * g\<^sup>\<star> = r\<^sup>T * ((r\<^sup>T * g\<^sup>\<star>)\<^sup>T * (r\<^sup>T * g\<^sup>\<star>) \<sqinter> g)\<^sup>\<star>"
basis_finite: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and A :: "'b set" assumes "indep_system carrier indep"   and "indep_system.basis carrier indep A" shows "finite A"
fset_inverse: fixes x :: "'b fset" shows "Abs_fset \<langle>x\<rangle>\<^sub>f = x"
mod2_eq_if: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "modulo a (numeral.numeral one plus (num.Bit0 num.One)) = (if dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a then zero else one)"
expand: fixes resource :: "('c, 'd) resource"   and resource' :: "('c, 'd) resource" assumes "run_resource resource = run_resource resource'" shows "resource = resource'"
principal_prod_principal: fixes A :: "'b set"   and B :: "'c set" shows "principal A \<times>\<^sub>F principal B = principal (A \<times> B)"
colours_distinct_2: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "black r s" shows "\<not> white r s"
elims: fixes x :: "nat set"   and xa :: "(nat set \<times> 'a option list set) set"   and y :: "(nat set \<times> 'a option list set) set" assumes "filterQuery x xa = y" obtains V :: "nat set" and Q :: "(nat set \<times> 'a option list set) set" where "x = V" and "xa = Q" and "y = Set.filter (\<lambda>(s, uu_). \<not> Set.is_empty (s \<inter> V)) Q"
correctCompositionIn_L2: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and ch :: "chanID" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "ch \<in> ins PQ" shows "ch \<in> ins P \<or> ch \<in> ins Q"
Sum_pmf_def: fixes p :: "real"   and Da :: "'e pmf"   and Db :: "'f pmf" shows "Sum_pmf p Da Db = bernoulli_pmf p \<bind> (\<lambda>b. if b then map_pmf Inl Da else map_pmf Inr Db)"
g1: fixes z :: "real" assumes "0 < z"   and "z < 1"   and "z \<notin> s1" shows "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at (z / 2)) = 2 *\<^sub>R vector_derivative g1 (at z)"
dom_dom: fixes f :: "'a1" assumes "A1.arr f" shows "A1.dom (A1.dom f) = A1.dom f"
super: assumes "OFCLASS('b, Gcd_class)" shows "OFCLASS('b, gcd_class)"
simps_2: fixes n :: "nat"   and s :: "'a" shows "iter (Suc n) s = (if guard s then body s \<bind> iter n else return_spmf s)"
convex_imp_contractible: fixes S :: "'a set" assumes "convex S" shows "contractible S"
low_mds_eq\<^sub>A': shows "abs.low_mds_eq (mds\<^sub>A_of mds\<^sub>C') (mem\<^sub>A_of mem\<^sub>1\<^sub>C') (mem\<^sub>A_of mem\<^sub>2\<^sub>C')"
mono_neutral_left: fixes T :: "'a set"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (1::'b)" shows "prod g S = prod g T"
tObsC_abs_tObsC_eq: fixes t' :: "('a, 'es, 'as) BEState Trace"   and t :: "('a, 'es, 'as) BEState Trace" assumes "tObsC t' = tObsC t" shows "tObsC_abs t = tObsC_abs t'"
dirichlet_prod_assoc: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and h :: "nat \<Rightarrow> 'b" shows "f \<star> g \<star> h = f \<star> (g \<star> h)"
prime_harmonic_aux2: fixes A :: "nat set" assumes "finite A" shows "(\<Sum>k\<in>A. 1 / (real k)\<^sup>2) \<le> 5 / 3"
rref_upt_condition2: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" assumes "reduced_row_echelon_form_upt_k A k" shows "\<forall>i. \<not> is_zero_row_upt_k i k A \<longrightarrow> A $ i $ (LEAST k. A $ i $ k \<noteq> (0::'a)) = (1::'a)"
UP_ring: fixes R :: "('c, 'd) ring_scheme" assumes "UP_ring R" shows "ring (UP R)"
assms_2: shows "p = 2 \<or> [p = 1] (mod 4)"
ms2: fixes e1 :: "lam"   and Es1 :: "ctx list"   and e2 :: "lam"   and Es2 :: "ctx list"   and e3 :: "lam"   and Es3 :: "ctx list" assumes "<e1,Es1> \<mapsto> <e2,Es2>"   and "<e2,Es2> \<mapsto>* <e3,Es3>" shows "<e1,Es1> \<mapsto>* <e3,Es3>"
is_processT8_Pair: fixes s :: "'a event list \<times> 'a event set"   and P :: "'a process" assumes "fst s \<in> D P" shows "s \<in> F P"
nonPrincipalInvertRight: fixes R1 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R2 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M1 :: "'b"   and M2 :: "'b"   and r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and ps :: "('a, 'b) sequent list"   and c :: "('a, 'b) sequent"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and n :: "nat"   and M :: "'b"   and Ms :: "('a, 'b) form list"   and \<Gamma>' :: "('a, 'b) form multiset"   and \<Delta>' :: "('a, 'b) form multiset"   and \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset"   and S :: "('a, 'b) sequent"   and Ps :: "('a, 'b) sequent list"   and n' :: "nat" assumes "R1 \<subseteq> upRules"   and "R2 \<subseteq> modRules2"   and "R3 \<subseteq> modRules2"   and "R = Ax \<union> R1 \<union> p_e R2 M1 M2 \<union> R3"   and "r \<in> R"   and "r = (ps, c)"   and "R' = Ax \<union> R1 \<union> R2 \<union> R3"   and "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms, m) \<in> derivable (ext R R2 M1 M2) \<longrightarrow> (\<forall>r'\<in>R'. rightPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable (ext R R2 M1 M2))"   and "( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms, n) \<in> derivable (ext R R2 M1 M2)"   and "\<forall>r'\<in>R'. rightPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"   and "\<not> rightPrincipal r (Modal M Ms) R'"   and "(r \<in> Ax \<or> r \<in> upRules \<or> r \<in> modRules2) \<and> extendRule S r = (Ps,  \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms)"   and "n = n' + 1"   and "\<forall>p\<in>set Ps. \<exists>n\<le>n'. (p, n) \<in> derivable (ext R R2 M1 M2)"   and "Ps \<noteq> []" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable (ext R R2 M1 M2)"
CR2_def: fixes \<tau> :: "ty" shows "CR2 \<tau> \<equiv> \<forall>t t'. t \<in> RED \<tau> \<and>  t \<longrightarrow>\<^sub>\<beta> t' \<longrightarrow> t' \<in> RED \<tau>"
simps_2: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and n :: "nat" shows "RM M2 M1 \<Omega> V m (Suc n) = {xs' \<in> C M2 M1 \<Omega> V m (Suc n). \<not> M1 \<preceq>\<lbrakk>{xs'}\<rbrakk> M2 \<or> (\<forall>io\<in>L\<^sub>i\<^sub>n M1 {xs'}. \<exists>V''\<in>N io M1 V. \<exists>S1 vs xs. io = vs @ xs \<and> mcp (vs @ xs) V'' vs \<and> S1 \<subseteq> nodes M2 \<and> (\<forall>s1\<in>S1. \<forall>s2\<in>S1. s1 \<noteq> s2 \<longrightarrow> (\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)) \<and> m < LB M2 M1 vs xs (TS M2 M1 \<Omega> V m n \<union> V) S1 \<Omega> V'')}"
Rxa_zero: fixes R :: "('c, 'd) Ring_scheme" assumes "Ring R" shows "R \<diamondsuit>\<^sub>p \<zero>\<^bsub>R\<^esub> = {\<zero>\<^bsub>R\<^esub>}"
node\<^sub>i_def: fixes k :: "nat"   and a :: "('a btnode ref option \<times> 'a) array \<times> nat"   and ti :: "'a btnode ref option" shows "node\<^sub>i k a ti \<equiv> pfa_length a \<bind> (\<lambda>n. if n \<le> 2 * k then pfa_shrink_cap (2 * k) a \<bind> (\<lambda>a'. ref (Btnode a' ti) \<bind> (\<lambda>l. return (T\<^sub>i (Some l)))) else pfa_empty (2 * k) \<bind> (\<lambda>b. BTree_ImpSet.split_half a \<bind> (\<lambda>i. pfa_get a i \<bind> (\<lambda>m. pfa_drop a (i + 1) b \<bind> (\<lambda>b'. pfa_shrink i a \<bind> (\<lambda>a'. pfa_shrink_cap (2 * k) a' \<bind> (\<lambda>a''. let (sub, sep) = m in ref (Btnode a'' sub) \<bind> (\<lambda>l. ref (Btnode b' ti) \<bind> (\<lambda>r. return (Up\<^sub>i (Some l) sep (Some r)))))))))))"
iso_sym: fixes G :: "('d, 'e) monoid_scheme"   and H :: "('f, 'g) monoid_scheme" assumes "Group.group G"   and "G \<cong> H" shows "H \<cong> G"
I_eq: shows "I = {i. i < dim_row A \<and> i \<in> I}"
ts: shows "set_clocks cs ts (u(c' := d')) \<turnstile>\<^bsub>v,n\<^esub> M"
A_ma_k_not_zero: shows "A $ ma $ mod_type_class.from_nat k \<noteq> (0::'a)"
poly_mult_not_eq_poly_Nil: fixes p :: "'a list"   and q :: "'a list"   and x :: "'a" shows "(poly (p *** q) x \<noteq> poly [] x) = (poly p x \<noteq> poly [] x \<and> poly q x \<noteq> poly [] x)"
numeral_power_eq_of_int_cancel_iff: 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 n :: "nat"   and y :: "int" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(power.power one times (numeral.numeral one plus x) n = ring_1.of_int one plus zero minus y) = (numeral x ^ n = y)"
type_update_type: fixes ts :: "t list"   and cons :: "t list"   and prods :: "checker_type"   and ts' :: "checker_type" assumes "type_update (Type ts) (to_ct_list cons) prods = ts'"   and "ts' \<noteq> Bot" shows "ts' = prods \<and> (\<exists>ts_c. prods = TopType ts_c) \<or> (\<exists>ts_a ts_b. prods = Type ts_a \<and> ts = ts_b @ cons \<and> ts' = Type (ts_b @ ts_a))"
mat_inter_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "f \<sqinter> g \<equiv> \<lambda>i j. f i j \<cdot> g i j"
monomial_decomp_append_iff: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "monomial_decomp (ps @ qs) = (monomial_decomp ps \<and> monomial_decomp qs)"
Nml_Cod: fixes t :: "'c local.term" assumes "Nml t" shows "Nml (Cod t)"
is_lasso_prpl_def: fixes m :: "'a"   and prpl :: "'Q list \<times> 'Q list" shows "gb_graph.is_lasso_prpl (to_gbg_ext m) prpl \<equiv> graph.is_lasso_prpl_pre (to_gbg_ext m) prpl \<and> (\<forall>A\<in>gbg_F (to_gbg_ext m). set (snd prpl) \<inter> A \<noteq> {})"
intro: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" assumes "ll_on_open T f X" shows "ll_on_open_it T f X"
r_result1_saturating_the: fixes t :: "nat"   and i :: "nat"   and x :: "nat"   and t' :: "nat" assumes "0 < the (eval r_result1 [t, i, x])"   and "t \<le> t'" shows "0 < the (eval r_result1 [t', i, x])"
set_eqE: fixes R :: "('d, 'e) eq_object_scheme"   and A :: "'d set"   and B :: "'d set" assumes "A {.=}\<^bsub>R\<^esub> B" obtains "\<forall>a\<in>A. a .\<in>\<^bsub>R\<^esub> B" and "\<forall>b\<in>B. b .\<in>\<^bsub>R\<^esub> A"
power_less_imp_less_exp: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "less (power.power one times a m) (power.power one times a n)" shows "m < n"
term_ok: shows "\<Theta>,\<Gamma> \<turnstile> t"
wf_UN: fixes I :: "'d set"   and r :: "'d \<Rightarrow> ('e \<times> 'e) set" assumes "\<And>i. i \<in> I \<Longrightarrow> wf (r i)"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I; r i \<noteq> r j\<rbrakk> \<Longrightarrow> Domain (r i) \<inter> Range (r j) = {}" shows "wf (\<Union> (r ` I))"
closed_real_atMost: fixes a :: "real" shows "closed {..a}"
acyclic_irrefl: fixes r :: "('b \<times> 'b) set" shows "acyclic r = irrefl (r\<^sup>+)"
P: fixes l :: "'a" assumes "is_lub L l A" shows "P l"
simps_1: fixes C :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and x :: "'a"   and xs :: "'a list" shows "rm_MT_rules C (x # xs) = (if dom (C x) = {} then rm_MT_rules C xs else x # rm_MT_rules C xs)"
lebesgue_set_almost_fsigma: fixes S :: "'a set" assumes "S \<in> sets lebesgue" obtains C :: "'a set" and T :: "'a set" where "fsigma C" and "T \<in> null_sets lebesgue" and "C \<union> T = S" and "disjnt C T"
MutrecAux: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and n :: "nat"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "finite G \<and> card G = n \<and> verified G \<and> A \<in> G" shows "{} \<rhd> Call : A"
left_total_conversep: fixes A :: "'g \<Rightarrow> 'f \<Rightarrow> bool" shows "left_total A\<inverse>\<inverse> = right_total A"
OstrowskiTr31: fixes v :: "'a \<Rightarrow> ant"   and s :: "'a" assumes "valuation K v"   and "s \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a s)" shows "s \<noteq> \<zero>"
frag_of_eq: fixes x :: "'b"   and y :: "'b" shows "(frag_of x = frag_of y) = (x = y)"
case_2: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b" shows "(case Lt of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = f2"
impl_safe_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.impl_safe summary results_in c \<equiv> \<forall>loc1 loc2 t s. 0 < zcount (c_imp c loc1) t \<and> s \<in>\<^sub>A graph.path_weight summary loc1 loc2 \<longrightarrow> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c loc2) \<and> t' \<le> results_in t s)"
coeff_monom: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "coeff (monom a m) n = (if m = n then a else (0::'a))"
polyexp_ratfps_Cons: fixes x :: "'b \<times> nat \<times> 'b"   and xs :: "('b \<times> nat \<times> 'b) list" shows "polyexp_ratfps (x # xs) = (case x of (a, k, b) \<Rightarrow> ratfps_of_poly (smult a (fps_monom_poly b k)) / ratfps_of_poly ([:1::'b, - b:] ^ (k + 1))) + polyexp_ratfps xs"
Ball_def: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
visits_via_def: fixes P :: "'d llist"   and A :: "'d set"   and W :: "'d set" shows "visits_via P A W \<equiv> \<exists>n. enat n < llength P \<and> P $ n \<in> W \<and> lset (ltake (enat n) P) \<subseteq> A"
assms_3: fixes a :: "'a" assumes "x \<sqsubseteq> a"   and "y \<sqsubseteq> a" shows "z \<sqsubseteq> a"
inject_2: fixes x3 :: "coname"   and x4 :: "trm"   and x1 :: "name"   and x2 :: "trm"   and y3 :: "coname"   and y4 :: "trm"   and y1 :: "name"   and y2 :: "trm" shows "(Cut <x3>.x4 x1.x2 = Cut <y3>.y4 y1.y2) = ([x3].x4 = [y3].y4 \<and> [x1].x2 = [y1].y2)"
writes_singleton: fixes f :: "'d \<Rightarrow> ('a, 'b, 'c) prog"   and a :: "'d"   and h :: "'a"   and h' :: "'a" shows "writes (all_args f) (f a) h h'"
eq_general: fixes B :: "'c set"   and A :: "'d set"   and h :: "'d \<Rightarrow> 'c"   and \<gamma> :: "'c \<Rightarrow> 'e"   and \<phi> :: "'d \<Rightarrow> 'e" assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma> (h x) = \<phi> x" shows "sum \<phi> A = sum \<gamma> B"
arrI: fixes f1 :: "'a"   and f2 :: "'a \<times> 'a" assumes "C.arr f1"   and "CC.arr f2" shows "CCC.arr (f1, f2)"
cInf_less_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and X :: "'c set"   and y :: "'c" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "preorder.bdd_below less_eq X" shows "less (Inf X) y = (\<exists>x\<in>X. less x y)"
map_comp: fixes x :: "'a"   and y :: "'a" assumes "is_map x"   and "is_map y" shows "is_map (x ; y)"
approx_reachable_closed: fixes c :: "cmd"   and c0 :: "cmd"   and a :: "action + brk_ctd"   and c' :: "cmd" assumes "c \<in> approx_reachable c0"   and "cfg c a c'" shows "c' \<in> approx_reachable c0"
ex_unit_disc_fix_is_poincare_line_to_x_axis: fixes l :: "circline" assumes "is_poincare_line l" shows "\<exists>M. unit_disc_fix M \<and> moebius_circline M l = x_axis"
dtor_rel_coinduct: fixes JR :: "'c coG \<Rightarrow> 'd coG \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>ya xa. JR ya xa \<Longrightarrow> rel_pre_coG R JR (dtor_coG ya) (dtor_coG xa)" shows "JR \<le> rel_coG R"
rel_transp: fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "transp R1"   and "transp R2" shows "transp (rel_prod R1 R2)"