hcju/isabelleps huggingface dataset:
Order_def: fixes D :: "('c, 'd) Order_scheme" shows "Order D \<equiv> (rel D \<subseteq> carrier D \<times> carrier D \<and> (\<forall>a. a \<in> carrier D \<longrightarrow> (a, a) \<in> rel D)) \<and> (\<forall>a b. a \<in> carrier D \<longrightarrow> b \<in> carrier D \<longrightarrow> (a, b) \<in> rel D \<longrightarrow> (b, a) \<in> rel D \<longrightarrow> a = b) \<and> (\<forall>a b c. a \<in> carrier D \<longrightarrow> b \<in> carrier D \<longrightarrow> c \<in> carrier D \<longrightarrow> (a, b) \<in> rel D \<longrightarrow> (b, c) \<in> rel D \<longrightarrow> (a, c) \<in> rel D)"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<lesssim>\<lbrakk>\<cdot>\<rbrakk>R<SRel,TRel> TargetTerm T2"
union_right_cancel: fixes M :: "'c multiset"   and K :: "'c multiset"   and N :: "'c multiset" shows "(M + K = N + K) = (M = N)"
exec_strip: shows "\<Gamma>\<turnstile> \<langle>strip_guards F (Catch c1 c2),s\<rangle> =n\<Rightarrow> t"
inc: shows "rank A < (if \<exists>a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a) then mod_type_class.to_nat (GREATEST a. (P_Gauss_Jordan A *v b) $ a \<noteq> (0::'a)) + 1 else 0)"
atLeastLessThan_empty_iff: fixes a :: "'a multiset"   and b :: "'a multiset" shows "(subset_mset.atLeastLessThan a b = {}) = (\<not> a \<subset># b)"
Vsl: shows "set Vsl \<subseteq> carrier_vec n"
weight_spmf_of_set_finite: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}" shows "weight_spmf (spmf_of_set A) = 1"
funct_OT_14_def: fixes M :: "'g \<times> 'g \<times> 'g \<times> 'g"   and C :: "bool \<times> bool" shows "funct_OT_14 M C = (let (c0, c1) = C; (m00, m01, m10, m11) = M in return_spmf ((), if c0 then if c1 then m11 else m10 else if c1 then m01 else m00))"
sameDom_def: fixes inp1 :: "'e \<Rightarrow> 'f option"   and inp2 :: "'e \<Rightarrow> 'g option" shows "sameDom inp1 inp2 \<equiv> \<forall>i. (inp1 i = None) = (inp2 i = None)"
Gr_def: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" shows "BNF_Def.Gr A f = {(a, f a) |a. a \<in> A}"
Pluso_undef: fixes \<theta> :: "trm option"   and \<eta> :: "trm option" shows "(Pluso \<theta> \<eta> = undeft) = (\<theta> = undeft \<or> \<eta> = undeft)"
measurable_lebesgue_cong: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and M :: "'c measure" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "(f \<in> lebesgue_on S \<rightarrow>\<^sub>M M) = (g \<in> lebesgue_on S \<rightarrow>\<^sub>M M)"
exhaust: fixes y :: "ccode" obtains "y = ccode.Gt"   | "y = ccode.Lt"   | "y = ccode.Eq"   | "y = Und"
two_sum_of_squares_nat_iff: fixes x :: "nat"   and y :: "nat" shows "(x\<^sup>2 + y\<^sup>2 = 2) = (x = 1 \<and> y = 1)"
trans: fixes b :: "'b list"   and a :: "'b list"   and c :: "'b list" assumes "Sublist.prefix b a"   and "Sublist.prefix c b" shows "Sublist.prefix c a"
insert_rga_insert_body_commute: fixes i2 :: "'a option"   and e1 :: "'a"   and xs :: "'a list"   and e2 :: "'a" assumes "i2 \<noteq> Some e1" shows "insert_rga (insert_body xs e1) (e2, i2) = insert_body (insert_rga xs (e2, i2)) e1"
Var: fixes xs :: "'varSort"   and x :: "'var" shows "phi (asSort xs) (gVar MOD xs x)"
vwelldefined_def: shows "vwelldefined \<equiv> value_pred.pred (\<lambda>_. list_all (\<lambda>(uu_, t). consts t |\<subseteq>| all_consts)) (\<lambda>name. name |\<in>| C) (\<lambda>dom. dom |\<subseteq>| heads)"
earlyLateBoundOutput: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'" shows "P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'"
\<Phi>a_comp: fixes g :: "'s"   and f :: "'s" assumes "S.seq g f" shows "\<Phi>a (g \<cdot> f) = restrict (\<Phi>a g \<circ> \<Phi>a f) (S'.set (\<Phi>o (S.dom f)))"
that: fixes i :: "'c" assumes "i \<in> S" shows "diff_fun k charts (f i)"
element_ptr_kinds_eq_h2: shows "element_ptr_kinds h2 = element_ptr_kinds h3"
\<tau>\<phi>_2: shows "\<tau> \<odot> \<phi> = \<eta> \<odot> \<sigma>"
Hurwitz_injective: fixes S :: "complex set"   and \<F> :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and g :: "complex \<Rightarrow> complex" assumes "open S"   and "connected S"   and "\<And>n. \<F> n holomorphic_on S"   and "g holomorphic_on S"   and "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> uniform_limit K \<F> g sequentially"   and "\<not> g constant_on S"   and "\<And>n. inj_on (\<F> n) S" shows "inj_on g S"
Q: shows "Matrix.mat 2 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then s else if i = 0 \<and> j = 1 then - b1 else if i = 1 \<and> j = 0 then t else a1) \<in> carrier_mat 2 2"
no_zero_height_Step_derivs1: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and os :: "Val list"   and S :: "(Var \<times> Val) list"   and H :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" assumes "(M, l, (os, S, H), 0, ll, t) \<in> Step" shows "False"
Ghost: fixes m :: "nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and rs :: "'c memref list"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" shows "(m, Ghost\<^sub>s\<^sub>b A L R W # rs, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (m, rs, \<O> \<union> A - R, aug (dom \<S>) R \<R>, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
hta_ensure_idx_s_def: fixes H :: "('a, 'b, 'c) hashedTa_scheme" shows "hta_ensure_idx_s H \<equiv> case hta_idx_s H of None \<Rightarrow> H\<lparr>hta_idx_s := Some (build_rule_index_s (hta_\<delta> H))\<rparr> | Some x \<Rightarrow> H"
Sup: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c set"   and f :: "'c \<Rightarrow> 'c" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq M"   and "\<forall>x\<in>M. x \<in> ccpo.iterates Sup less_eq f" shows "Sup M \<in> ccpo.iterates Sup less_eq f"
simps: fixes lx :: "'a"   and ux :: "'a"   and ly :: "'a"   and uy :: "'a" shows "Interval lx ux - Interval ly uy = Interval (lx - uy) (ux - ly)"
indexed_nominal_ts_axioms: shows "indexed_nominal_ts TYPE('idx) (\<turnstile>\<^sub>S) (\<rightarrow>\<^sub>S)"
induct: fixes x1 :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and x2 :: "Core.term"   and P :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option) \<Rightarrow> Core.term \<Rightarrow> bool" assumes "wf_term x1 x2"   and "\<And>\<Sigma> T v. wf_type \<Sigma> T \<Longrightarrow> P \<Sigma> (Fv v T)"   and "\<And>\<Sigma> n. P \<Sigma> (Bv n)"   and "\<And>\<Sigma> s ty T. \<lbrakk>const_type \<Sigma> s = Some ty; wf_type \<Sigma> T; tinstT T ty\<rbrakk> \<Longrightarrow> P \<Sigma> (Ct s T)"   and "\<And>\<Sigma> t u. \<lbrakk>wf_term \<Sigma> t; P \<Sigma> t; wf_term \<Sigma> u; P \<Sigma> u\<rbrakk> \<Longrightarrow> P \<Sigma> (t $ u)"   and "\<And>\<Sigma> T t. \<lbrakk>wf_type \<Sigma> T; wf_term \<Sigma> t; P \<Sigma> t\<rbrakk> \<Longrightarrow> P \<Sigma> (Abs T t)" shows "P x1 x2"
assms: fixes s :: "'a"   and t :: "'a"   and u :: "'a" assumes "s \<in> set ss"   and "t \<in> set ts"   and "u \<in> set us" shows "(snd (f s t) \<and> fst (f t u) \<longrightarrow> fst (f s u)) \<and> (fst (f s t) \<and> snd (f t u) \<longrightarrow> fst (f s u)) \<and> (snd (f s t) \<and> snd (f t u) \<longrightarrow> snd (f s u)) \<and> (fst (f s t) \<and> fst (f t u) \<longrightarrow> fst (f s u))"
UN_extend_simps_4: fixes A :: "'d \<Rightarrow> 'c set"   and C :: "'d set"   and B :: "'c set" shows "\<Union> (A ` C) \<inter> B = (\<Union>x\<in>C. A x \<inter> B)"
prog_configs_in2: fixes p\<^sub>1 :: "'a"   and p\<^sub>2 :: "'a"   and "is" :: "instr list"   and xs :: "'a memref list" assumes "Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 is \<in> set xs" shows "p\<^sub>2 \<in> prog_configs xs"
assms_1: shows "wf_until_aux \<sigma> V R args \<phi> \<psi> aux ne"
sum_less_suminf: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "summable f"   and "\<And>m. n \<le> m \<Longrightarrow> (0::'a) < f m" shows "sum f {..<n} < suminf f"
natural_transformation_axioms: shows "natural_transformation VVV.comp (\<bullet>) HoHV HoVH \<alpha>.map"
points_index_def: fixes B :: "'a set multiset"   and ps :: "'a set" shows "B index ps \<equiv> size (filter_mset ((\<subseteq>) ps) B)"
atLeast_Suc_atMost_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "prod g {Suc m..Suc n} = prod (g \<circ> Suc) {m..n}"
cases: fixes prog :: "cmd"   and a1 :: "w_node"   and a2 :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and a3 :: "w_node" assumes "While_CFG.path prog a1 a2 a3" obtains n :: "w_node" where "a1 = na__" and "a2 = []" and "a3 = na__" and "While_CFG.valid_node prog na__"   | n'' :: "w_node" and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list" and n' :: "w_node" and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node" and n :: "w_node" where "a1 = n" and "a2 = a # as" and "a3 = n'" and "While_CFG.path prog n'' as n'" and "valid_edge prog a" and "sourcenode a = n" and "targetnode a = n''"
left_add_zero_mat: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "0\<^sub>m nr nc + A = A"
insert_def: fixes a :: "'c"   and B :: "'c set" shows "Set.insert a B = {x. x = a} \<union> B"
last_replicate: fixes n :: "nat"   and x :: "'g" assumes "n \<noteq> 0" shows "last (replicate n x) = x"
disconnected_nodes_eq2_h: fixes doc_ptr :: "(_) document_ptr" shows "|h \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r = |h2 \<turnstile> get_disconnected_nodes doc_ptr|\<^sub>r"
oracle_flag_def: shows "UPF.oracle_flag = (\<lambda>key (flg, L) (y, t). return_spmf ((), flg \<or> t = upf_fun key y \<and> y \<notin> L, L))"
simps_2: shows "eval_ternary_Or TernaryTrue TernaryFalse = TernaryTrue"
coarser_partitions_covers: fixes Q :: "'b set set"   and elem :: "'b"   and P :: "'b set set" assumes "Q \<in> coarser_partitions_with elem P" shows "\<Union> Q = insert elem (\<Union> P)"
euler_increment'_def: fixes e :: "int"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" shows "euler_increment' e f = rk_increment' e f 1 (\<lambda>i j. 0) (\<lambda>i. 1) (\<lambda>i. 0)"
acyclic_4a_implies_3a: fixes x :: "'a" assumes "pd_kleene_allegory.acyclic_4a (\<lambda>x. top) (\<le>) (\<sqinter>) x" shows "pd_kleene_allegory.acyclic_3a (\<lambda>x. top) (\<le>) x"
g'_in_carrier: fixes \<G> :: "'a cyclic_group"   and x :: "nat" assumes "okamoto \<G>" shows "okamoto_base.g' \<G> x \<in> carrier \<G>"
new_root_impl_def: fixes v0 :: "'a"   and s :: "('a, 'e) simple_state_scheme" shows "a.c.new_root_impl v0 s \<equiv> ASSERT (v0 \<notin> visited s) \<bind> (\<lambda>_. let s = s\<lparr>ss_stack := [(v0, a.a.E `` {v0})]\<rparr>; s = s\<lparr>on_stack := {v0}\<rparr> in Let (s\<lparr>visited := insert v0 (visited s)\<rparr>) RETURN)"
assms_8: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
h14: shows "\<forall>j\<le>2 * d + (4 + k). lose (t + j) = [False]"
star_of_approx_inverse: fixes x :: "'c star"   and xa :: "'c" assumes "x \<approx> star_of xa"   and "xa \<noteq> (0::'c)" shows "inverse x \<approx> inverse (star_of xa)"
map_ident: fixes t :: "'c \<Rightarrow> 'd" shows "(\<lambda>x. x) \<circ> t = t"
propa_property1: fixes qs :: "(nat \<times> nat list) list"   and ss :: "nat list list"   and w :: "nat list" assumes "\<forall>(q, \<tau>)\<in>set qs. q < length ss \<and> \<tau> \<in> A"   and "Sorted_Less.sorted w"   and "set ss \<subseteq> A"   and "length ss = n"   and "distinct (map fst qs)" shows "\<forall>(q, \<tau>)\<in>set qs. fst (propa f qs ss w) ! q = \<tau> \<squnion>\<^bsub>f\<^esub> ss ! q"
obsf_distinguisher_def: fixes \<D> :: "('a, 'b) distinguisher" shows "obsf_distinguisher \<D> = map_gpv' (\<lambda>x. x = Some True) id option_of_exception (gpv_stop \<D>)"
simps_2: fixes uu :: "'b"   and uv :: "instr list"   and uw :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and ux :: "'c"   and uy :: "'d \<times> 'e"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and S :: "nat \<Rightarrow> bool option" shows "share_all_until_volatile_write ((uu, uv, uw, sb, ux, uy) # ts) S = share_all_until_volatile_write ts (share (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) S)"
volume_additiveI: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> ennreal" assumes "ring_of_sets \<Omega> M"   and "\<And>a. a \<in> M \<Longrightarrow> 0 \<le> \<mu> a"   and "\<mu> {} = 0"   and "\<And>a b. \<lbrakk>a \<in> M; b \<in> M; a \<inter> b = {}\<rbrakk> \<Longrightarrow> \<mu> (a \<union> b) = \<mu> a + \<mu> b" shows "volume M \<mu>"
sets_PiM_I_finite: fixes I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
path2_not_Nil: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "ns \<noteq> []"
minSpanningForest_eq_minbasis: shows "minSpanningForest = s.MSF"
rbt_lookup_map: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and t :: "('a, 'c) RBT_Impl.rbt"   and x :: "'a" assumes "class.linorder less_eq less" shows "ord.rbt_lookup less (RBT_Impl.map f t) x = map_option (f x) (ord.rbt_lookup less t x)"
harm_ge_ln: fixes n :: "nat" shows "ln (real n + 1) \<le> harm n"
restrict0_times: fixes A :: "'c set"   and x :: "'c \<Rightarrow> 'd"   and y :: "'c \<Rightarrow> 'd" shows "restrict0 A (x * y) = restrict0 A x * restrict0 A y"
pred_inject_1: fixes P :: "'b \<Rightarrow> bool" shows "list_all P []"
g_def: shows "g = (\<lambda>x. f3 x + real_cond_exp M Invariants f x)"
fg: fixes b :: "'a" assumes "b \<in> Basis" shows "f b = g b"
fds_truncate_add: fixes m :: "nat"   and f :: "'a fds"   and g :: "'a fds" shows "fds_truncate m (fds_truncate m f + fds_truncate m g) = fds_truncate m (f + g)"
der_h: fixes y :: "(real, 'n) vec" assumes "y \<in> T" shows "(h has_derivative h' y) (at y within T)"
u_3: shows "dist (x + u *\<^sub>R (y - x)) a < dist x a"
h2: shows "t2 \<noteq> Tip \<and> sortedTree h t2 \<longrightarrow> setOf (wrm h t2) = setOf t2 - {rm h t2}"
abs_summable_on_0: fixes A :: "'c set" shows "(\<lambda>_. 0::'d) abs_summable_on A"
monoid_mult_hom_axioms: shows "monoid_mult_hom (inv (inv hom))"
eq1: shows "(lcs t1 t2 - t1) \<oplus> lt p = term_of_pair (lcs t1 t2, component_of_term (lt p))"
l11_16: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "A \<noteq> B"   and "C \<noteq> B"   and "Tarski_neutral_dimensionless.Per Bet Cong A' B' C'"   and "A' \<noteq> B'"   and "C' \<noteq> B'" shows "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"
While_tE: fixes b :: "bexp"   and c :: "com"   and s :: "char list \<Rightarrow> int"   and x :: "nat"   and t :: "char list \<Rightarrow> int" assumes "(WHILE b DO c, s) \<Rightarrow> x \<Down> t" obtains "x = Suc 0" and "t = s" and "\<not> bval b s"   | xa :: "nat" and s2 :: "char list \<Rightarrow> int" and y :: "nat" where "bval b s" and "(c, s) \<Rightarrow> xa \<Down> s2" and "(WHILE b DO c, s2) \<Rightarrow> y \<Down> t" and "Suc (xa + y) = x"
ih_6: shows "InvariantWatchesEl (getF state') (getWatch1 state') (getWatch2 state')"
iT_finite_3: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "finite [ r, mod m, c ]"
step: fixes r :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set"   and m :: "nat" assumes "r \<in> R"   and "fst r \<noteq> []"   and "\<forall>p\<in>set (fst r). \<exists>n\<le>m. (p, n) \<in> derivable R" shows "(snd r, m + 1) \<in> derivable R"
of_int_le_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w \<le> z)"
opnet_tau1: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and s :: "'d net_state"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and s' :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and t :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, s), \<tau>, \<sigma>', s') \<in> S" shows "((\<sigma>, SubnetS s t), \<tau>, \<sigma>', SubnetS s' t) \<in> opnet_sos S T"
factorization_m_prod: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and g :: "int poly"   and d :: "int"   and gs :: "int poly multiset" assumes "factorization_m f (c, fs)"   and "factorization_m g (d, gs)" shows "factorization_m (f * g) (c * d, fs + gs)"
append_eq_cases: fixes x :: "'a list"   and y :: "'a list"   and m :: "'a list"   and n :: "'a list" assumes "x @ y = m @ n"   and "m \<noteq> []" shows "prefix x m \<or> strict_prefix m x"
gt_tpoly_def: fixes q :: "('v pvar, hmultiset) tpoly"   and p :: "('v pvar, hmultiset) tpoly" shows "(q >\<^sub>p p) = (\<forall>A. legal_zpassign A \<longrightarrow> eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of p) < eval_tpoly A (map_tpoly (\<lambda>x. x) zhmset_of q))"
has_vderiv_on_union: fixes f :: "real \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'b"   and s :: "real set"   and t :: "real set" assumes " D f = g on s \<union> closure s \<inter> closure t"   and " D f = g on t \<union> closure s \<inter> closure t" shows " D f = g on s \<union> t"
not_exceptional: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "Invariants.inv g"   and "final g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)"   and "\<not> exceptionalVertex g v" shows "|vertices f| \<le> 4"
lex_append_rightD: fixes xs :: "nat list"   and us :: "nat list"   and ys :: "nat list"   and vs :: "nat list" assumes "xs @ us <\<^sub>l\<^sub>e\<^sub>x ys @ vs"   and "length xs = length ys"   and "(xs, ys) \<notin> lex less_than" shows "ys = xs \<and> us <\<^sub>l\<^sub>e\<^sub>x vs"
NF_NT: fixes s :: "'c event list"   and P :: "'c process" assumes "(s, {}) \<notin> F P" shows "s \<notin> T P"
finite_essential_class_imp_recurrent: fixes C :: "'s set"   and x :: "'s" assumes "essential_class C"   and "finite C"   and "x \<in> C" shows "recurrent x"
brec_invar_inner_def: fixes Qi :: "'a set" shows "brec_invar_inner Qi \<equiv> brec_invar_add Qi \<inter> {(Q, W, uu_, uu_). set (fifo.\<alpha> W) \<subseteq> dom (hm.\<alpha> Q)}"
prf_oracle_def: fixes key :: "'key"   and \<sigma> :: "unit"   and x :: "'\<alpha>" shows "local.prf.prf_oracle key \<sigma> x = return_spmf (f key x, ())"
mem_carrier: fixes H :: "'f set"   and G :: "('f, 'g) monoid_scheme"   and x :: "'f" assumes "submonoid H G"   and "x \<in> H" shows "x \<in> carrier G"
compat_low: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and x :: "'Var" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i \<sqsubset> length cms\<^sub>1"   and "x \<in> differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "dma x = Low"
new_vars_priv: shows "- range var\<^sub>C_of = \<Union> (set priv_mem\<^sub>C)"
arr_char: fixes f :: "'c \<times> 'c" shows "C'.VV.arr f = (C'.arr (fst f) \<and> C'.arr (snd f) \<and> C'.src (fst f) = C'.trg (snd f))"
in_hhomE: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and a :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hhom \<mu> a b" obtains "Span.arr \<mu>" and "Span.obj a" and "Span.obj b" and "Span.src \<mu> = a" and "Span.trg \<mu> = b"
min_set_subset: fixes X :: "'b set" shows "min_set X \<subseteq> X"
sdrop_wait_least: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and n :: "nat" assumes "ev \<phi> xs"   and "\<phi> (sdrop n xs)" shows "wait \<phi> xs \<le> n"
assms: fixes i :: "'a" assumes "i \<in> I" shows "f i = y"
not_alw_iff: fixes P :: "'d stream \<Rightarrow> bool"   and \<omega> :: "'d stream" shows "(\<not> alw P \<omega>) = ev (\<lambda>xs. \<not> P xs) \<omega>"
eq_Nil_appendI: fixes xs :: "'a list"   and ys :: "'a list" assumes "xs = ys" shows "xs = [] @ ys"
stabilizer_subset: fixes x :: "'c" shows "stabilizer G \<phi> x \<subseteq> carrier G"
S_def: fixes r :: "'a"   and A :: "('a \<times> 'a) set" shows "Prim.S r A \<equiv> {r} \<union> fst ` A \<union> snd ` A"
int_not_simps_4: fixes w :: "num" shows "NOT (numeral w) = - numeral (w + num.One)"
vimage_fst: fixes A :: "'a set" shows "fst -` A = A \<times> UNIV"
ranked: fixes q :: "'a"   and f :: "'b"   and qs :: "'a list" assumes "q \<rightarrow> f qs \<in> ta.\<delta>" shows "A f = Some (length qs)"
preserves_seq: fixes g :: "'d"   and f :: "'d" assumes "D.seq g f" shows "D.seq (D.R g) (D.R f)"
reachable1_reachable: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>+ w" shows "v \<rightarrow>\<^sup>* w"
assms: shows "\<xi>' \<in> is_rrep \<xi>"
zcf_monom: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R" shows "zcf (monom P a n) = to_fun (monom P a n) \<zero>"
x: shows "x \<in> insert (0::'a) S \<and> x \<in> affine hull (insert (0::'a) S - {x})"
gauss_int_prime_classification: fixes x :: "gauss_int" assumes "prime x" obtains "x = 1 + \<i>\<^sub>\<int>"   | p :: "nat" where "x = of_nat p" and "prime p" and "[p = 3] (mod 4)"   | "prime (gauss_int_norm x)" and "[gauss_int_norm x = 1] (mod 4)" and "0 < ReZ x" and "0 < ImZ x" and "ReZ x \<noteq> ImZ x"
not_eventually_impI: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter"   and Q :: "'a \<Rightarrow> bool" assumes "eventually P F"   and "\<not> eventually Q F" shows "\<not> (\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x)"
simps_1: fixes uu :: "nat"   and uv :: "'c"   and uw :: "('a, 'b, 'c, 'd) itnode rose_tree list" shows "isHNode (HNode uu uv uw) = True"