hcju/isabelleps huggingface dataset:
subterms_union_finite: fixes M :: "('a, 'b) Term.term set" assumes "finite M" shows "finite (subterms\<^sub>s\<^sub>e\<^sub>t M)"
integral_swap_operativeI: fixes s :: "('a \<times> 'b) set"   and f :: "'a \<times> 'b \<Rightarrow> 'c"   and e :: "real" assumes "continuous_on s f"   and "0 < e" shows "operative (\<and>) True (\<lambda>k. \<forall>a b c d. cbox (a, c) (b, d) \<subseteq> k \<and> cbox (a, c) (b, d) \<subseteq> s \<longrightarrow> norm (integral (cbox (a, c) (b, d)) f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y)))) \<le> e * content (cbox (a, c) (b, d)))"
list_set_autoref_insert_dj: fixes x' :: "'c"   and s' :: "'c set"   and x :: "'c"   and s :: "'c list" assumes "PRIO_TAG_OPTIMIZATION"   and "SIDE_PRECOND_OPT (x' \<notin> s')"   and "(x, x') \<in> Id"   and "(s, s') \<in> \<langle>Id\<rangle>list_set_rel" shows "(x # s, (OP insert ::: Id \<rightarrow> \<langle>Id\<rangle>list_set_rel \<rightarrow> \<langle>Id\<rangle>list_set_rel) $ x' $ s') \<in> \<langle>Id\<rangle>list_set_rel"
pts_symbolic_c_preserves_Ana_invar_subst: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list" assumes "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>')"   and "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>) \<union> ik\<^sub>e\<^sub>s\<^sub>t \<A> \<union> (\<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>) \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>))" shows "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>') \<union> ik\<^sub>e\<^sub>s\<^sub>t \<A>' \<union> (\<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>') \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>'))"
all_final_except_mono': fixes ts :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and A :: "'b set"   and B :: "'b set" assumes "r2.all_final_except ts A"   and "\<And>x. x \<in> A \<Longrightarrow> x \<in> B" shows "r2.all_final_except ts B"
bangCases: fixes P :: "ccs"   and \<alpha> :: "act"   and P' :: "ccs" assumes "!P \<longmapsto>\<alpha> \<prec> P'" obtains Pa :: "ccs" where "!P = !Pa" and "Pa \<parallel> !Pa \<longmapsto>\<alpha> \<prec> P'"
Iod_segment_segment: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier (Iod D (segment D b))" shows "Iod (Iod D (segment D b)) (segment (Iod D (segment D b)) a) = Iod D (segment D a)"
nonzero_Basis: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and u :: "'a" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "u \<noteq> zero"
cong: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd"   and x :: "'b"   and y :: "'b" assumes "f = g"   and "x = y" shows "f x = g y"
c1: shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>x. h (fst x * snd x, 0))"
weakCongSimResPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and x :: "name" assumes "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "eqvt Rel'"   and "x \<sharp> \<Psi>"   and "Rel \<subseteq> Rel'"   and "\<And>\<Psi>' R S x. \<lbrakk>(\<Psi>', R, S) \<in> Rel; x \<sharp> \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>x\<rparr>R, \<lparr>\<nu>x\<rparr>S) \<in> Rel'" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>P \<leadsto>\<guillemotleft>Rel'\<guillemotright> \<lparr>\<nu>x\<rparr>Q"
simps_1: fixes dims :: "nat list"   and \<rho> :: "complex mat" shows "state_sig.denote dims SKIP \<rho> = \<rho>"
enumerate_map_upt: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and m :: "nat" shows "List.enumerate n (map f [n..<m]) = map (\<lambda>k. (k, f k)) [n..<m]"
is_partition_def: fixes cs :: "'c set list" shows "is_partition cs = (\<forall>j<length cs. \<forall>i<j. cs ! i \<inter> cs ! j = {})"
weak_transition_eqvt: fixes P :: "'state"   and \<alpha> :: "'act"   and P' :: "'state"   and p :: "perm" assumes "P \<Rightarrow>\<langle>\<alpha>\<rangle> P'" shows "p \<bullet> P \<Rightarrow>\<langle>p \<bullet> \<alpha>\<rangle> p \<bullet> P'"
that: fixes f :: "(real, 'n) vec \<Rightarrow> (real, 'n) vec" assumes "orthogonal_transformation f"   and "f a = b" shows "thesis"
size_index_conv: fixes xs :: "'a list"   and x :: "'a" shows "(length xs = index xs x) = (x \<notin> set xs)"
wf_not_sym: fixes r :: "('a \<times> 'a) set"   and a :: "'a"   and x :: "'a" assumes "wf r"   and "(a, x) \<in> r" shows "(x, a) \<notin> r"
u_simps_3: shows "trg (tab\<^sub>0 r) = src r"
exhaust_sel: fixes sum :: "'c + 'd" obtains "sum = Inl (projl sum)"   | "sum = Inr (projr sum)"
R_scalar_mult_def: fixes R :: "'a set" shows "R_scalar_mult R \<equiv> Ring1 R"
balance_list_def: fixes xs :: "'a list" shows "balance_list xs = bal_list (length xs) xs"
separating: fixes A :: "'a set"   and B :: "'a set"   and G :: "('a, 'b) graph_scheme"   and S :: "'a set" assumes "\<And>x y p. \<lbrakk>x \<in> A; y \<in> B; path G x p y\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S" shows "separating_gen G A B S"
similarity_comp: fixes A1 :: "complex \<times> complex \<times> complex \<times> complex"   and A2 :: "complex \<times> complex \<times> complex \<times> complex"   and M :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det A1 \<noteq> 0"   and "mat_det A2 \<noteq> 0" shows "similarity A1 (similarity A2 M) = similarity (A2 *\<^sub>m\<^sub>m A1) M"
swap_identical_tails: fixes i :: "nat"   and j :: "nat"   and t :: "'d list" assumes "i < j"   and "j < length t" shows "drop (j + 1) t = drop (j + 1) (swap_events i j t)"
HRB_slice_refl: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and m :: "'b"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "CFG.valid_node sourcenode targetnode valid_edge m"   and "CFG_node m \<in> S" shows "CFG_node m \<in> SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S"
exec_move_BinOp1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and bop :: "bop"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e \<guillemotleft>bop\<guillemotright> e') h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
top_greatest: fixes less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and top :: "'b::type"   and a :: "'b::type" assumes "class.order_top (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (top::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (top::'b::type)"
bij_betw_image_fixed_card_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and k :: "nat" assumes "inj_on f A" shows "bij_betw ((`) f) {X. X \<subseteq> A \<and> card X = k} {X. X \<subseteq> f ` A \<and> card X = k}"
SD_iff_expected_utilities_le: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'b pmf" assumes "finite_total_preorder_on carrier le"   and "p \<in> lotteries_on carrier"   and "q \<in> lotteries_on carrier" shows "SD le p q = (\<forall>u. vnm_utility carrier le u \<longrightarrow> measure_pmf.expectation p u \<le> measure_pmf.expectation q u)"
length_splice: fixes xs :: "'a list"   and ys :: "'a list" shows "length (splice xs ys) = length xs + length ys"
periodic_fun_axioms: shows "periodic_fun f g gm g1 gn1"
group_axioms: shows "Group_Theory.group (\<eta> ` G) (\<cdot>') \<one>'"
simps: fixes rf :: "recf" shows "rec_sigma rf = (let vl = arity rf in Pr (vl - 1) (Cn (vl - 1) rf (get_fstn_args (vl - 1) (vl - 1) @ [Cn (vl - 1) (constn 0) [recf.id (vl - 1) 0]])) (Cn (Suc vl) rec_add [recf.id (Suc vl) vl, Cn (Suc vl) rf (get_fstn_args (Suc vl) (vl - 1) @ [Cn (Suc vl) s [recf.id (Suc vl) (vl - 1)]])]))"
keys_minus_higher: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" shows "keys (p - punit.higher p v) = {u \<in> keys p. u \<preceq> v}"
span_add: fixes U :: "'a vec set"   and v :: "'a vec"   and w :: "'a vec" assumes "U \<subseteq> carrier_vec n"   and "v \<in> local.span U"   and "w \<in> carrier_vec n" shows "(w \<in> local.span U) = (v + w \<in> local.span U)"
continuous_def: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real" shows "Function_Norm.continuous V f norm \<equiv> linearform V f \<and> continuous_axioms V f norm"
has_field_def: fixes P :: "'b prog"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" shows "P \<turnstile> C has F:T (fm) in D \<equiv> \<exists>FDTs. P \<turnstile> C has_fields FDTs \<and> map_of FDTs (F, D) = \<lfloor>(T, fm)\<rfloor>"
indicator_inter_arith: fixes A :: "'c set"   and B :: "'c set"   and x :: "'c" shows "indicator (A \<inter> B) x = indicator A x * indicator B x"
sees_methods_decl_above: fixes P :: "'a prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option"   and M :: "char list"   and m :: "staticb \<times> ty list \<times> ty \<times> 'a"   and D :: "char list" assumes "P \<turnstile> C sees_methods Mm"   and "Mm M = \<lfloor>(m, D)\<rfloor>" shows "P \<turnstile> C \<preceq>\<^sup>* D"
diff_e: shows "x - real_of_float (xs ! 0) = e 0"
map_eq_conv: fixes f :: "'l \<Rightarrow> 'k"   and xs :: "'l list"   and g :: "'l \<Rightarrow> 'k" shows "(map f xs = map g xs) = (\<forall>x\<in>set xs. f x = g x)"
zip_fusion_def: fixes g1 :: "('e, 'g) generator"   and g2 :: "('f, 'h) generator"   and s1 :: "'g"   and s2 :: "'h" shows "zip_fusion g1 g2 s1 s2 = zip (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
graphI2: fixes x :: "'a"   and F :: "'a set"   and f :: "'a \<Rightarrow> real" assumes "x \<in> F" shows "\<exists>t\<in>graph F f. t = (x, f x)"
embedding_map_imp_homeomorphic_space: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "embedding_map X Y f" shows "X homeomorphic_space subtopology Y (f ` topspace X)"
is_parser_length: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list"   and s :: "char list"   and x :: "'a"   and r :: "char list" assumes "is_parser p"   and "p s = Inr (x, r)" shows "length r \<le> length s"
st_proj2_inv_ne_top: fixes xa :: "'b" shows "st_proj2_inv xa \<noteq> bot_sphere"
holomorphic_has_inverse: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set" assumes "f holomorphic_on S"   and "open S"   and "inj_on f S" obtains g :: "complex \<Rightarrow> complex" where "g holomorphic_on f ` S" and "\<And>z. z \<in> S \<Longrightarrow> deriv f z * deriv g (f z) = 1" and "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
language_def: shows "language \<equiv> {smap int (p ## trace w p) |p w. init p \<and> run w p}"
Cons_2: shows "set (a # p1) \<subseteq> carrier R"
componentI: fixes H :: "'c program"   and F :: "'c program"   and G :: "'c program" assumes "H \<le> F \<or> H \<le> G" shows "H \<le> F \<squnion> G"
asIMOD_igAbsCongS: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsDisj SEM" shows "igAbsCongS (asIMOD SEM)"
insert_def: fixes a :: "'a"   and B :: "'a set" shows "insert a B = {x. x = a} \<union> B"
dep_expl_inverse: shows "\<lfloor>\<lambda>w. universal (\<lambda>x xa. (xa explains x) w = (x dependsOn xa) w)\<rfloor>"
orth_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p" assumes "A B C Orth U V" shows "A \<noteq> B \<and> B \<noteq> C \<and> A \<noteq> C \<and> U \<noteq> V"
Confidentiality_B: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list"   and authK :: "nat"   and Ta :: "nat" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt authK \<lbrace>Key servK, Agent B, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Key authK, Agent Tgs, Number Ta\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "\<not> expiredAK Ta evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbV" shows "Key servK \<notin> analz (knows Spy evs)"
Hermite_def: fixes associates :: "'a set"   and residues :: "'a \<Rightarrow> 'a set"   and A :: "(('a, 'b) vec, 'c) vec" shows "Hermite associates residues A = (Complete_set_non_associates associates \<and> Complete_set_residues residues \<and> echelon_form A \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> associates) \<and> (\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j<i. A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> residues (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))))))"
G_validn_valid: fixes G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "\<forall>K. G_validn K G" shows "G_valid G"
S_def: shows "S = {uu_. \<exists>u v. uu_ = u \<and> (u, v) \<in> Partitions x \<and> strict_prefix u x \<and> u \<in> A\<star> \<and> v @ z \<in> A\<star>}"
pvalid_item_end: fixes p :: "(symbol \<times> character list) list"   and x :: "item" assumes "pvalid p x" shows "item_end x = charslength p"
lcos_disjoint: fixes H :: "'a set"   and a :: "'a set"   and b :: "'a set" assumes "subgroup H G"   and "a \<in> lcosets H"   and "b \<in> lcosets H"   and "a \<noteq> b" shows "a \<inter> b = {}"
simps_1: fixes i :: "nat"   and k :: "nat" shows "loose_bvar1 (Bv i) k = (i = k)"
ext_inject: fixes ISignature :: "('a, 'b) Signature"   and ICategory :: "('c, 'd) Category"   and ITypes :: "'a \<Rightarrow> 'c"   and IFunctions :: "'b \<Rightarrow> 'd"   and more :: "'e"   and ISignature' :: "('a, 'b) Signature"   and ICategory' :: "('c, 'd) Category"   and ITypes' :: "'a \<Rightarrow> 'c"   and IFunctions' :: "'b \<Rightarrow> 'd"   and more' :: "'e" shows "(\<lparr>ISignature = ISignature, ICategory = ICategory, ITypes = ITypes, IFunctions = IFunctions, \<dots> = more\<rparr> = \<lparr>ISignature = ISignature', ICategory = ICategory', ITypes = ITypes', IFunctions = IFunctions', \<dots> = more'\<rparr>) = (ISignature = ISignature' \<and> ICategory = ICategory' \<and> ITypes = ITypes' \<and> IFunctions = IFunctions' \<and> more = more')"
lossless_fast_dice_roll: fixes c :: "nat"   and v :: "nat"   and n :: "nat" assumes "c < v"   and "v \<le> n" shows "lossless_spmf (Fast_Dice_Roll.fast_dice_roll n v c)"
abs_eq: fixes x :: "'a" shows "prime_factorization x = Abs_multiset (\<lambda>p. if prime p then multiplicity p x else 0)"
residual_is_wellformed: fixes b :: "'a set" assumes "b \<in># \<B>\<^sup>R" shows "b \<subseteq> bl\<^sup>c"
IH: shows "lift_streams (xs @ ys) = smap2 (@) (lift_streams xs) (lift_streams ys)"
safe_max_Var: fixes x :: "'d" assumes "x \<in> Csafe" shows "max_Var_floatariths ode_e \<le> DIM('d)"
order_0_monom: fixes c :: "'a"   and n :: "nat" assumes "c \<noteq> (0::'a)" shows "order (0::'a) (monom c n) = n"
Fo_\<eta>o_initial: fixes y :: "'d" assumes "\<exists>x u. initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y x u" shows "initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y (Fo y) (\<eta>o y)"
w\<^sub>0_def: shows "w\<^sub>0 \<equiv> dw"
elims: fixes x :: "const"   and xa :: "const"   and y :: "const" assumes "plus_const x xa = y" obtains m :: "int" and n :: "int" where "x = Const m" and "xa = Const n" and "y = Const (m + n)"   | uv_ :: "const" where "x = Any" and "xa = uv_" and "y = Any"   | uu_ :: "const" where "x = uu_" and "xa = Any" and "y = Any"
\<tau>s: shows "\<forall>pc<length \<tau>s. \<tau>s ! pc \<in> A \<and> \<tau>s ! pc \<noteq> \<top>"
cases: fixes x :: "('a \<times> 'a) \<times> 'a \<times> 'a" obtains xa :: "'a" and y :: "'a" and u :: "'a" and v :: "'a" where "x = ((xa, y), u, v)"
matrix_mul_sum_alt: fixes A :: "(('b, 'c) vec, 'c) vec"   and B :: "(('b, 'c) vec, 'c) vec" shows "A ** B = (\<chi>i. \<Sum>k\<in>UNIV. A $ i $ k *s B $ k)"
pre_ex_unch: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "STUTINV F" shows "\<exists>v. (|~ F \<and> Unchanged v \<longrightarrow> \<circle>F)"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> F \<nabla> [x].A, m) \<in> derivable  R* \<longrightarrow> (\<forall>r'\<in>R. rightPrincipal r' (F \<nabla> [x].A) \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> \<not> multSubst \<Gamma>' \<and> \<not> multSubst \<Delta>' \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable  R*)"
is_renaming_inv_renaming_cancel_cls: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and \<rho> :: "'c"   and C :: "'b literal multiset" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms"   and "substitution_ops.is_renaming id_subst comp_subst \<rho>" shows "substitution_ops.subst_cls subst_atm (substitution_ops.subst_cls subst_atm C \<rho>) (substitution_ops.inv_renaming id_subst comp_subst \<rho>) = C"
focus_def: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "focus X p = (\<Sum>t\<in>keys p. monomial (monomial (lookup p t) (except t X)) (except t (- X)))"
equals0I: fixes A :: "'b set" assumes "\<And>y. y \<in> A \<Longrightarrow> False" shows "A = {}"
subspace_diff: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c"   and y :: "'c" assumes "module scale"   and "module.subspace scale S"   and "x \<in> S"   and "y \<in> S" shows "x - y \<in> S"
card_of_Well_order: fixes A :: "'d set" shows "Well_order (card_of A)"
fine_def: fixes d :: "'a \<Rightarrow> 'b set"   and s :: "('a \<times> 'b set) set" shows "(d fine s) = (\<forall>(x, k)\<in>s. k \<subseteq> d x)"
simps_2: fixes M :: "object atom formula set"   and \<alpha> :: "ground_action"   and \<alpha>s :: "ground_action list"   and M' :: "object atom formula set" shows "ground_action_path M (\<alpha> # \<alpha>s) M' = (M \<^sup>c\<TTurnstile>\<^sub>= ground_action.precondition \<alpha> \<and> ground_action_path (execute_ground_action \<alpha> M) \<alpha>s M')"
minus_minus: fixes x :: "nat \<times> nat" assumes "x \<in> carrier S" shows "\<ominus>\<^bsub>S\<^esub> (\<ominus>\<^bsub>S\<^esub> x) = x"
nonzero_inverse_eq_imp_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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.division_ring divide inverse one times plus zero minus uminus"   and "inverse a = inverse b"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "a = b"
openin_subtopology: fixes U :: "'a topology"   and V :: "'a set"   and S :: "'a set" shows "openin (subtopology U V) S = (\<exists>T. openin U T \<and> S = T \<inter> V)"
entails_frame: fixes P :: "'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real"   and S :: "'c \<Rightarrow> real" assumes "P \<tturnstile> R"   and "Q \<tturnstile> S" shows "P && Q \<tturnstile> R && S"
snd_to_last_max_normal: fixes G :: "('c, 'd) monoid_scheme"   and \<GG> :: "'c set list" assumes "composition_series G \<GG>"   and "finite (carrier G)"   and "1 < length \<GG>" shows "max_normal_subgroup (\<GG> ! (length \<GG> - 2)) G"
fds_deriv_def: fixes f :: "'b fds" shows "fds_deriv f = (\<chi>n. - ln (real n) *\<^sub>R f $ n)"
RG_zero_closed: shows "0 \<in> Supgroup.group_ring"
exp_not_zero_imp_exp_diff_not_zero: fixes n :: "nat"   and m :: "nat" assumes "(2::'a) ^ n \<noteq> (0::'a)" shows "(2::'a) ^ (n - m) \<noteq> (0::'a)"
strong_exhaust: fixes y :: "fmla"   and c :: "'a" obtains x11 :: "trm" and x12 :: "trm" where "y = x11 EQ x12"   | x21 :: "fmla" and x22 :: "fmla" where "y = x21 OR x22"   | x3 :: "fmla" where "y = neg x3"   | x41 :: "name" and x42 :: "fmla" where "{atom x41} \<sharp>* c" and "y = exi x41 x42"
SReal_minus_not_Infinitesimal: fixes y :: "real star" assumes "y < 0"   and "y \<in> \<real>" shows "y \<notin> Infinitesimal"
connected_components_of_def: fixes X :: "'b topology" shows "connected_components_of X \<equiv> connected_component_of_set X ` topspace X"
elims: fixes x :: "(symbol \<times> character list) list"   and y :: "nat" assumes "charslength x = y" obtains cs :: "(symbol \<times> character list) list" where "x = cs" and "y = length (chars cs)"
pre_new_root_def: fixes v0 :: "'v"   and s :: "'s" shows "pre_new_root v0 s \<equiv> gen_rwof s \<and> gds_is_empty_stack gds s \<and> \<not> gds_is_break gds s \<and> v0 \<in> V0 - gen_discovered s"
wp1n_RS3: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (rm_MT_rules Cp p)"
supset_circuit_in_imp_dep_in: fixes \<E> :: "'a set"   and C :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "circuit_in \<E> C \<and> C \<subseteq> X" shows "\<not> indep_in \<E> X"
ent_star_mono_true: fixes A :: "assn"   and A' :: "assn"   and B :: "assn"   and B' :: "assn" assumes "A \<Longrightarrow>\<^sub>A A' * true"   and "B \<Longrightarrow>\<^sub>A B' * true" shows "A * B * true \<Longrightarrow>\<^sub>A A' * B' * true"