hcju/isabelleps huggingface dataset:
RedCast: fixes s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "hp s a = \<lfloor>(D, fs)\<rfloor>"   and "P \<turnstile> D \<preceq>\<^sup>* C" shows "P \<turnstile> \<langle>Cast C (addr a),s\<rangle> \<rightarrow> \<langle>addr a,s\<rangle>"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "C.ide (e \<cdot>\<^sub>C m)" shows "C.ide (src\<^sub>C e \<cdot>\<^sub>C src\<^sub>C m)"
S2_def: fixes \<sigma> :: "bool"   and b\<^sub>\<sigma> :: "bool" shows "S2 \<sigma> b\<^sub>\<sigma> = I \<bind> (\<lambda>(\<alpha>, \<tau>). local.etp.S \<alpha> \<bind> (\<lambda>x\<^sub>\<sigma>. local.etp.S \<alpha> \<bind> (\<lambda>y\<^sub>\<sigma>'. let x\<^sub>\<sigma>' = F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> y\<^sub>\<sigma>'; \<beta>\<^sub>\<sigma> = B \<alpha> x\<^sub>\<sigma> \<oplus> b\<^sub>\<sigma>; \<beta>\<^sub>\<sigma>' = B \<alpha> x\<^sub>\<sigma>' in return_spmf (\<sigma>, \<alpha>, \<beta>\<^sub>\<sigma>, \<beta>\<^sub>\<sigma>'))))"
mult_mv_cancel_l: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and v :: "complex \<times> complex"   and v' :: "complex \<times> complex" assumes "mat_det A \<noteq> 0"   and "A *\<^sub>m\<^sub>v v = A *\<^sub>m\<^sub>v v'" shows "v = v'"
or_def: fixes \<phi>\<^sub>1 :: "'c fm"   and \<phi>\<^sub>2 :: "'c fm" shows "or \<phi>\<^sub>1 \<phi>\<^sub>2 = (if \<phi>\<^sub>1 = FalseF then \<phi>\<^sub>2 else if \<phi>\<^sub>2 = FalseF then \<phi>\<^sub>1 else if \<phi>\<^sub>1 = TrueF \<or> \<phi>\<^sub>2 = TrueF then TrueF else Or \<phi>\<^sub>1 \<phi>\<^sub>2)"
main: fixes ii :: "nat" assumes "i + ii \<le> k" shows "(\<Sum>j\<leftarrow>[0..<Suc k - i]. local.xij_f i (i + j) * local.pprod k i (i + j)) = (\<Sum>j\<leftarrow>[0..<Suc k - (i + ii)]. local.xij_f (i + ii) (i + ii + j) * local.pprod k (i + ii) (i + ii + j))"
set_iterator_map_linord_def: fixes it :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and S0 :: "('e \<times> 'f) set" shows "set_iterator_map_linord it S0 \<equiv> set_iterator_genord it S0 (\<lambda>(k, uu_) (k', uu_). k \<le> k')"
inf_sup_ord_2: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "less_eq_prod (inf_prod x y) y"
fls_lr_inverse_subdegree_0_2: fixes f :: "'a fls"   and x :: "'a" assumes "fls_subdegree f = 0" shows "0 \<le> fls_subdegree (fls_right_inverse f x)"
scaleC_ccsubspace_def: shows "(*\<^sub>C) \<equiv> map_fun id (map_fun space_as_set Abs_clinear_space) (\<lambda>c. (`) ((*\<^sub>C) c))"
domI: fixes m :: "'g \<Rightarrow> 'f option"   and a :: "'g"   and b :: "'f" assumes "m a = \<lfloor>b\<rfloor>" shows "a \<in> dom m"
pairwise_alt: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set" shows "pairwise R S = (\<forall>x\<in>S. \<forall>y\<in>S - {x}. R x y)"
image_Collect_subsetI: fixes P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "\<And>x. P x \<Longrightarrow> f x \<in> B" shows "f ` {x. P x} \<subseteq> B"
compile_heads: fixes rs :: "(name \<times> (Term_Class.term list \<times> pterm) fset) fset" shows "fst |`| Rewriting_Pterm.compile rs = fst |`| rs"
lexordp_simps_1: fixes ys :: "'a list" shows "ord_class.lexordp [] ys = (ys \<noteq> [])"
case_prodD: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
t_stop_def: shows "other.t_stop = - v\<^sub>o / a\<^sub>o"
rep_eq: fixes x :: "('e, 'f, 'g) formula" shows "Rep_formula (Formula.Not x) = Not\<^sub>\<alpha> (Rep_formula x)"
parabolic_imp_not_elliptic_loxodromic_2: fixes f :: "'b \<Rightarrow> 'b" assumes "parabolic_isometry f" shows "\<not> loxodromic_isometry f"
R_not_trivial: shows "carrier R \<noteq> {\<zero>}"
extended_predist_self0: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "(extended_predist x y = 0) = (x = y)"
simps: fixes a :: "'transition" shows "(a \<in> executable) = (\<exists>p a. a = a \<and> p \<in> nodes \<and> enabled a p)"
ImageI: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set"   and A :: "'b set" assumes "(a, b) \<in> r"   and "a \<in> A" shows "b \<in> r `` A"
ivl_subset_existence_ivl: fixes t :: "real"   and t0 :: "real"   and x0 :: "'d" assumes "t - t0 \<in> existence_ivl0 x0" shows "{t0..t} \<subseteq> (+) t0 ` existence_ivl0 x0"
sum_zero_set: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'a set" assumes "\<forall>x\<in>s. f x = (0::'b)"   and "finite s"   and "finite t" shows "sum f (s \<union> t) = sum f t"
SUP_mult_right_ennreal: fixes f :: "'c \<Rightarrow> ennreal"   and I :: "'c set"   and c :: "ennreal" shows "\<Squnion> (f ` I) * c = (\<Squnion>i\<in>I. f i * c)"
HcompNml_Nml_Src: fixes t :: "'a E.term" assumes "E.Nml t" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> E.Src t = t"
transferFrame: fixes \<Psi>\<^sub>F :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and A\<^sub>F :: "name list"   and A\<^sub>G :: "name list"   and \<Psi>\<^sub>G :: "'b" assumes "\<Psi>\<^sub>F \<rhd> P \<longmapsto> \<alpha> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<langle>A\<^sub>F, \<Psi>\<^sub>F \<otimes> \<Psi>\<^sub>P\<rangle> \<hookrightarrow>\<^sub>F \<langle>A\<^sub>G, \<Psi>\<^sub>G \<otimes> \<Psi>\<^sub>P\<rangle>"   and "A\<^sub>F \<sharp>* P"   and "A\<^sub>G \<sharp>* P"   and "A\<^sub>F \<sharp>* subject \<alpha>"   and "A\<^sub>G \<sharp>* subject \<alpha>"   and "A\<^sub>P \<sharp>* A\<^sub>F"   and "A\<^sub>P \<sharp>* A\<^sub>G"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>F"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>G" shows "\<Psi>\<^sub>G \<rhd> P \<longmapsto> \<alpha> \<prec> P'"
interleave_Nil2: fixes xs :: "'b list" shows "xs \<otimes> [] = {xs}"
phi_Some: fixes pc :: "nat" assumes "pc < length ins"   and "c ! pc \<noteq> \<bottom>" shows "\<tau>s ! pc = c ! pc"
exhaust: fixes y :: "residual_Rep" obtains x11 :: "subject" and x12 :: "name \<Rightarrow> pi noption" where "y = BoundR_Rep x11 x12"   | x21 :: "freeRes" and x22 :: "pi" where "y = FreeR_Rep x21 x22"
Class_commutes_with_composition: fixes a :: "'a"   and b :: "'a" assumes "a \<in> M"   and "b \<in> M" shows "Class a [\<cdot>] Class b = Class (a \<cdot> b)"
blocks_nempty_alt: shows "\<forall>bl\<in>#\<B>. bl \<noteq> {}"
aff_dim_halfspace_le: fixes a :: "'b"   and r :: "real" shows "aff_dim {x. a \<bullet> x \<le> r} = (if a = (0::'b) \<and> r < 0 then - 1 else int DIM('b))"
finprod_subgroup: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'a"   and S :: "'d set"   and H :: "('c \<Rightarrow> 'a) set" assumes "f \<in> S \<rightarrow> H"   and "subgroup H (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)" shows "finprod (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) f S = finprod (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<lparr>carrier := H\<rparr>) f S"
rel_funI: fixes A :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and B :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and f :: "'p \<Rightarrow> 'r"   and g :: "'q \<Rightarrow> 's" assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" shows "rel_fun A B f g"
even_push_bit_iff: 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 bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (push_bit n a) = (n \<noteq> 0 \<or> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
homeomorphism_on_def: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "homeomorphism_on S f = (\<exists>g. homeomorphism S (f ` S) f g)"
bad_worst_chain_Suc: fixes i :: "nat" shows "local.bad (local.worst_chain (Suc i))"
open_subdiagonal: shows "open {(x, y) |x y. x < y}"
af_letter_subst: fixes \<phi> :: "'a ltln"   and \<nu> :: "'a set" shows "af_letter \<phi> \<nu> = subst \<phi> (\<lambda>\<psi>. Some (af_letter \<psi> \<nu>))"
scale_cancel_left: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c"   and y :: "'c" assumes "vector_space scale" shows "(scale a x = scale a y) = (x = y \<or> a = (0::'b))"
normalize_mult_unit_factor: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "times (normalize a) (unit_factor a) = a"
1: shows "s\<^sub>0 \<star> w \<cong> r\<^sub>0"
fls_inverse_X_intpow': fixes i :: "int" assumes "inverse (1::'a) = (1::'a)" shows "inverse (fls_X_intpow i) = fls_X_intpow (- i)"
C: shows "ipurge_tr I (c_dom D) u (c_tr step out s xs) = c_tr step out s (ipurge_tr I D u xs)"
gram_schmidt_int_def: fixes n :: "nat"   and us :: "int vec list" shows "gram_schmidt_int n us = gram_schmidt_wit n (map of_int_hom.vec_hom us)"
simple_grouptr0: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<triangleright> K"   and "K \<subseteq> H"   and "simple_Group (G / K)" shows "H = carrier G \<or> H = K"
match_simplematcher_SrcDst_1: fixes \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and X :: "'a ipt_iprange"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (Match (Src X)) a p = (p_src p \<in> ipt_iprange_to_set X)"
assms: shows "map (\<lambda>g. eval g xs) gs = map Some ys"
fold2_def: fixes s1 :: "('d, 'f, 'e) F1 \<Rightarrow> 'f"   and s2 :: "('d, 'f, 'e) F2 \<Rightarrow> 'e" shows "fold2 s1 s2 = snd (LFP.fold s1 s2)"
ennreal_divide_eq_0_iff: fixes a :: "ennreal"   and b :: "ennreal" shows "(a / b = 0) = (a = 0 \<or> b = top)"
reduced_GB_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "reduced_GB B = (THE G. finite G \<and> is_reduced_GB G \<and> pmdl G = pmdl B)"
CD_on_def: fixes ds :: "'b set"   and X :: "'a set" shows "Singleton_for_d.CD_on ds X = (\<Union>d\<in>ds. Singleton_for_d.Cd d X)"
norm_powr_real: fixes w :: "complex"   and z :: "complex" assumes "w \<in> \<real>"   and "0 < Re w" shows "cmod (w powr z) = exp (Re z * ln (Re w))"
assms: shows "((l1, v1, v2), l2, v3, v4) \<in> on_triple R"
degree_0_iff: fixes P :: "'a poly" shows "(degree P = 0) = (\<exists>a. P = [:a:])"
Times_Int_Times: fixes A :: "'c set"   and B :: "'d set"   and C :: "'c set"   and D :: "'d set" shows "A \<times> B \<inter> C \<times> D = (A \<inter> C) \<times> (B \<inter> D)"
branch_sat_fresh: fixes branch :: "(('b, 'c) fm list \<times> 'c) list"   and M :: "('d, 'b) model"   and g :: "'c \<Rightarrow> 'd"   and i :: "'c"   and v :: "'d" assumes "M, g \<Turnstile>\<^sub>\<Theta> branch"   and "i \<notin> branch_nominals branch" shows "M, g(i := v) \<Turnstile>\<^sub>\<Theta> branch"
linorder_not_less: fixes x :: "'k"   and y :: "'k" shows "(\<not> x < y) = (y \<le> x)"
diagonal_to_Smith_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "diagonal_to_Smith A bezout = diagonal_to_Smith_aux A [0..<min (nrows A) (ncols A) - 1] bezout"
perm_rem_conv: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "bij f" shows "perm_rem x f y = (if x = y then x else if f y = x then f (f y) else f y)"
exec_meth_stk_split: fixes P :: "'addr J1_prog"   and h :: "'heap"   and E :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and STK :: "'addr val list"   and t :: "'thread_id"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option" assumes "P,E,h \<turnstile> (e, xs) \<leftrightarrow> (stk, loc, pc, xcp)"   and "exec_meth_d (compP2 P) (compE2 E) (stack_xlift (length STK) (compxE2 E 0 0)) t h (stk @ STK, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "\<exists>stk''. stk' = stk'' @ STK \<and> exec_meth_d (compP2 P) (compE2 E) (compxE2 E 0 0) t h (stk, loc, pc, xcp) ta h' (stk'', loc', pc', xcp')"
Inf_lower: fixes x :: "'d \<times> 'e"   and A :: "('d \<times> 'e) set" assumes "x \<in> A" shows "less_eq_prod (Inf_prod A) x"
rbt_inter_with_key_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and t1 :: "('b, 'c) rbt"   and t2 :: "('b, 'c) rbt" shows "ord.rbt_inter_with_key less f t1 t2 = (case RBT_Impl.compare_height t1 t1 t2 t2 of compare.LT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). map_option (\<lambda>w. (k, f k v w)) (ord.rbt_lookup less t2 k)) (RBT_Impl.entries t1)) | compare.GT \<Rightarrow> rbtreeify (List.map_filter (\<lambda>(k, v). map_option (\<lambda>w. (k, f k w v)) (ord.rbt_lookup less t1 k)) (RBT_Impl.entries t2)) | compare.EQ \<Rightarrow> rbtreeify (ord.sinter_with less f (RBT_Impl.entries t1) (RBT_Impl.entries t2)))"
induct: fixes P :: "nat list \<Rightarrow> bool"   and a0 :: "nat list" assumes "P []"   and "\<And>x. P [x]"   and "\<And>x y ys. P (y # ys) \<Longrightarrow> P (x # y # ys)" shows "P a0"
init_fin_silent_moveD: fixes final :: "'c \<Rightarrow> bool"   and r :: "('e, 'f, 'c, 'd, 'g, 'h) semantics"   and t :: "'f"   and \<tau>move :: "('e, 'f, 'c, 'd, 'g, 'h) \<tau>moves"   and s :: "status"   and x :: "'c"   and m :: "'d"   and s' :: "status"   and x' :: "'c"   and m' :: "'d" assumes "\<tau>trsys.silent_move (multithreaded_base.init_fin final r t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move) ((s, x), m) ((s', x'), m')" shows "\<tau>trsys.silent_move (r t) \<tau>move (x, m) (x', m') \<and> s = s' \<and> s' = Running"
simps_1: fixes s :: "nat \<Rightarrow> 'a"   and P :: "'a \<Rightarrow> bool" shows "seq_filter s P 0 = (LEAST k. P (s k))"
minus_le_self_iff: fixes a :: "'a" shows "(- a \<le> a) = ((0::'a) \<le> a)"
toList: fixes x :: "'b odlist" shows "toList x \<in> rel_ext (\<lambda>x. sorted x \<and> distinct x)"
measure_measure_4: shows "N = Abs_measure (\<Omega>', A', \<mu>')"
bounded_linear_vec_nth: fixes i :: "'b" shows "bounded_linear (\<lambda>x. x $ i)"
mod_ring_mult64: fixes pp :: "uint64" assumes "p = int_of_uint64 pp"   and "p \<le> 4294967295" shows "(mod_ring_rel64 ===> mod_ring_rel64 ===> mod_ring_rel64) (mult_p64 pp) (*)"
bij_betw_congI: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A' :: "'d set"   and g :: "'c \<Rightarrow> 'd" assumes "bij_betw f A A'"   and "\<forall>a\<in>A. f a = g a" shows "bij_betw g A A'"
part_disjoint_from_complement: fixes x :: "i"   and y :: "i" assumes "CMC"   and "\<exists>v. \<not> O v x"   and "P y x" shows "\<not> O y (\<midarrow> x)"
star_inductl': fixes z :: "'a"   and x :: "'a"   and y :: "'a" assumes "z \<squnion> x \<cdot> y \<le> y" shows "quantale_class.star x \<cdot> z \<le> y"
p_lessI: fixes p' :: "real" assumes "(\<Sum>i<k. as ! i * bs ! i powr p') < 1" shows "p < p'"
irreducible_i_def: fixes ops :: "'b arith_ops_record"   and xs :: "'b list" shows "irreducible_i ops xs = (degree_i xs \<noteq> 0 \<and> (\<forall>q r. is_poly ops q \<longrightarrow> is_poly ops r \<longrightarrow> degree_i q < degree_i xs \<longrightarrow> degree_i r < degree_i xs \<longrightarrow> xs \<noteq> times_poly_i ops q r))"
assms_1: shows "\<forall>\<^sub>F n in sequentially. fds_nth f n = fds_nth g n"
finite_range_run: shows "finite (range (run\<^sub>t \<delta>\<^sub>\<R> q\<^sub>\<R> w))"
conjugate_exponent_ennreal_1: fixes p :: "ennreal" assumes "1 \<le> p" shows "1 / p + 1 / conjugate_exponent p = 1"
compose: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h' :: "'c \<Rightarrow> 'a"   and G :: "'c filter" assumes "f \<in> o[F](g)"   and "filterlim h' F G" shows "(\<lambda>x. f (h' x)) \<in> o[G](\<lambda>x. g (h' x))"
IH_2: assumes "\<Gamma> \<turnstile> s" shows "\<exists>a. taval a2 s a"
p\<^sub>3: shows "on_curve a b p\<^sub>3"
pderivs_append: fixes s1 :: "'a list"   and s2 :: "'a list"   and r :: "'a rexp" shows "pderivs (s1 @ s2) r = pderivs_set s2 (pderivs s1 r)"
prime_multiplicity_other: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and q :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "normalization_semidom.prime zero times one normalize q"   and "p \<noteq> q" shows "factorial_semiring.multiplicity times one p q = 0"
minus_eq: fixes x :: "'e"   and y :: "'e" shows "x \<ominus>\<^bsub>W\<^esub> y = x \<oplus>\<^bsub>W\<^esub> \<ominus>\<^bsub>W\<^esub> y"
run_one_step_basic_if_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and x71 :: "tf"   and x72 :: "b_e list"   and x73 :: "b_e list"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, $b_e.If x71 x72 x73) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
add_minus_cancel: fixes x :: "'a"   and y :: "'a" assumes "x \<in> E"   and "y \<in> E" shows "x + (- x + y) = y"
2: shows "\<forall>a b c. ideal_generated {a, b, c} = ideal_generated {1::'a} \<longrightarrow> (\<exists>p q. ideal_generated {p * a, p * b + q * c} = ideal_generated {1::'a})"
convex_bound_lt: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and x :: "'b"   and a :: "'b"   and y :: "'b"   and u :: "'b"   and v :: "'b" assumes "class.linordered_semiring_1_strict plus zero minus less_eq less times one"   and "less x a"   and "less y a"   and "less_eq zero u"   and "less_eq zero v"   and "plus u v = one" shows "less (plus (times u x) (times v y)) a"
distinct_elim: fixes v :: "'v"   and v1 :: "'v"   and w :: "'w"   and v2 :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_trail v ((v1, w, v2) # ps) v'" shows "((v1, w, v2) \<in> edges (rem_unPath ps G)) = ((v1, w, v2) \<in> E)"
sinvar_def: fixes G :: "'b graph"   and nP :: "'b \<Rightarrow> SINVAR_TaintingTrusted.taints" shows "SINVAR_TaintingTrusted.sinvar G nP \<equiv> \<forall>(v1, v2)\<in>edges G. taints (nP v1) - untaints (nP v1) \<subseteq> taints (nP v2)"
tp_mcalc: fixes \<sigma> :: "'tp" shows "prot \<sigma> \<or> protFw \<sigma> \<or> (\<forall>c\<in>\<Phi>. \<sigma> \<turnstile> c)"
group_isoI: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "Group.group G"   and "Group.group H"   and "\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>"   and "h ` carrier G = carrier H"   and "\<forall>x\<in>carrier G. \<forall>y\<in>carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y" shows "h \<in> Group.iso G H"
strict_order_var: fixes x :: "'a" shows "strict_order x = (asymmetric x \<and> transitive x)"
add_mono_thms_linordered_field_5: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i < j \<and> k < l" shows "i + k < j + l"
finite_support_pre: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and C :: "acom" assumes "finite (support Q)"   and "finite (varacom C)" shows "finite (support (pre C Q))"
path_obs_intra_subset: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "\<forall>n'\<in>set (sourcenodes as). n' \<notin> S" shows "obs_intra n' S \<subseteq> obs_intra n S"
strict_coboundedI1: fixes a :: "nat"   and c :: "nat"   and b :: "nat" assumes "a dvd c \<and> a \<noteq> c" shows "gcd a b dvd c \<and> gcd a b \<noteq> c"
lookup_eq_valueI: fixes k :: "'b"   and v :: "'c"   and xs :: "('b, 'c) oalist_ntm" assumes "(k, v) \<in> set (fst (list_of_oalist_ntm xs))" shows "OAlist_lookup_ntm xs k = v"
fail_option_def: fixes return :: "'b option \<Rightarrow> 'c" shows "fail_option return = OptionT (return None)"
assms: shows "zs \<noteq> zs' \<or> {x, y} \<inter> {x', y'} = {}"
vinsert_vempty: fixes a :: "V" shows "vinsert a []\<^sub>\<circ> = set {a}"
dghm_cn_comp_ObjMap_app: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V"   and a :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<GG>\<lparr>ObjMap\<rparr>\<lparr>\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr>\<rparr>"
CHFMT: fixes s :: "'n list" assumes "s \<in># ch" shows "\<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"
simps_1: fixes uu :: "'b match_expr \<Rightarrow> 'b match_expr list" shows "normalize_rules uu [] = []"
coprime_0_right_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one" shows "algebraic_semidom.coprime times one a zero = dvd.dvd times a one"
listassoc_refl: fixes as :: "'a list" assumes "set as \<subseteq> carrier G" shows "as [\<sim>] as"
assert_bound'_code_def: shows "assert_bound'_code = Update.assert_bound' update_code"
order_trans: fixes y :: "'b"   and x :: "'b"   and z :: "'b" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
ffb_Sup_dual_var: fixes F :: "('d \<Rightarrow> 'c set) set" shows "fb\<^sub>\<F> (\<Squnion> F) = \<Sqinter> \<P> fb\<^sub>\<F> F"
inject_2: fixes x31 :: "PreSimplyTyped.type"   and x32 :: "PreSimplyTyped.type"   and y31 :: "PreSimplyTyped.type"   and y32 :: "PreSimplyTyped.type" shows "(TArr x31 x32 = TArr y31 y32) = (x31 = y31 \<and> x32 = y32)"
keys_frag_of: fixes a :: "'b" shows "keys (monomial 1 a) = {a}"