hcju/isabelleps huggingface dataset:
npGTPowZero_sub: fixes x :: "'a"   and m :: "nat"   and n :: "nat" assumes "x \<in> carrier R"   and "x^\<^bsup>R m\<^esup> = \<zero>" shows "m \<le> n \<longrightarrow> x^\<^bsup>R n\<^esup> = \<zero>"
embedding_lemma_i: fixes I :: "'a \<Rightarrow> bool option"   and op :: "'a strips_operator" assumes "is_operator_applicable_in I op"   and "are_operator_effects_consistent op op" shows "I \<then> op = execute_parallel_operator I [op]"
correct: shows "P,\<Phi> |- (None, h, (stk, loc, C, M, pc, ics) # frs, sh) [ok]"
eq: fixes a :: "'b" assumes "a \<in> S" shows "h (j a) = g a"
assms_2: shows "lastMarked M \<notin> set (elements M)"
usum_In1I: fixes N :: "('a, 'b) node set"   and B :: "('a, 'b) node set set"   and A :: "('a, 'b) node set set" assumes "N \<in> B" shows "In1 N \<in> usum A B"
invertible_mult_column: fixes q :: "'d"   and k :: "'d"   and a :: "'e" assumes "q * k = (1::'d)"   and "k * q = (1::'d)" shows "invertible (mult_column (mat (1::'d)) a q)"
"case": fixes f :: "'b \<Rightarrow> 'b msstream \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'b msstream" shows "(case MSSCons x1 x2 of MSSCons x xa \<Rightarrow> f x xa) = f x1 x2"
cases: fixes x :: "('c, 'd) concrete_category.arr" obtains uu_ :: "'c" and uv_ :: "'c" and F :: "'d" where "x = MkArr uu_ uv_ F"   | "x = Null"
transitive_axioms: shows "transitive A (\<sqsubseteq>)"
option_rel_Some2: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d option"   and y :: "'e" shows "rel_option A x (Some y) = (\<exists>x'. x = Some x' \<and> A x' y)"
map_option_is_None: fixes f :: "'b \<Rightarrow> 'a"   and opt :: "'b option" shows "(map_option f opt = None) = (opt = None)"
sccs_finished: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" assumes "Tarjan_invar G s" shows "\<Union> (sccs s) \<subseteq> dom (finished s)"
\<section>: fixes j :: "nat" shows "Abs_fps (\<lambda>na. (- c) ^ na * fps_binomial (- of_nat n) $ na) $ j = Abs_fps (\<lambda>k. of_nat (n + k - 1 choose k) * c ^ k) $ j"
T_Tn_T_compose_1: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and n :: "nat"   and x :: "'b" assumes "qmpt M T" shows "T ((T ^^ n) x) = (T ^^ Suc n) x"
wt_jvm_prog_impl_wt_instr: fixes \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "instr list"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and pc :: "nat" assumes "wf_jvm_prog\<^bsub>\<Phi>\<^esub> P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = (mxs, mxl\<^sub>0, ins, xt) in C"   and "pc < length ins" shows "P,T,mxs,length ins,xt \<turnstile> ins ! pc,pc :: \<Phi> C M"
poly_has_sgnx_values_4: fixes p :: "real poly" assumes "p \<noteq> 0" shows "(poly p has_sgnx 1) at_bot \<or> (poly p has_sgnx - 1) at_bot"
at_within_eq: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and s :: "'b set" assumes "class.topological_space open" shows "topological_space.at_within open x s = (INF S\<in>{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
set_take_disj_set_drop_if_distinct: fixes vs :: "'c list"   and i :: "nat"   and j :: "nat" assumes "distinct vs"   and "i \<le> j" shows "set (take i vs) \<inter> set (drop j vs) = {}"
elims_3: fixes x :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool"   and xa :: "'b literal set set" assumes "\<not> validate_clause_set x xa" obtains I :: "'b trm \<Rightarrow> 'b trm \<Rightarrow> bool" and S :: "'b literal set set" where "x = I" and "xa = S" and "\<not> (\<forall>C. C \<in> S \<longrightarrow> validate_clause I C)"
LLL_bound_invariant_def: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list"   and \<alpha> :: "rat"   and outside :: "bool"   and upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "LLL.LLL_bound_invariant n m fs_init \<alpha> outside upw i fs = (LLL.LLL_invariant n m fs_init \<alpha> upw i fs \<and> LLL.f_bound m fs_init outside i fs \<and> LLL.g_bound n m fs_init fs)"
load_instr_privilege: fixes s' :: "'b sparc_state"   and i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (load_instr i s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
fixed_points_def: shows "conjP.fixed_points = {m \<in> subgroups_of_size (p ^ a). carrier (G\<lparr>carrier := P\<rparr>) \<subseteq> conjP.stabilizer m}"
distinct_7: fixes x1 :: "'b"   and x51 :: "('a, 'b) com"   and x52 :: "('a, 'b) com" shows "Atm x1 \<noteq> x51 | x52"
obtain_sum_ideal_generated: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> ideal_generated A"   and "finite A" obtains f :: "'a \<Rightarrow> 'a" where "(\<Sum>i\<in>A. f i * i) = a"
mult_1_right: fixes a :: "'c" shows "comp_inf.vector a"
rel_mode_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and SomeMode :: "Mode"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = Skip@[x -=\<^sub>m SomeMode]"   and "c\<^sub>C = Skip@[Eg2_var\<^sub>C_of_Eg1 x -=\<^sub>m SomeMode]"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop, mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
vec_real_inner_axioms_def: fixes inner_field :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inner :: "('a, 'b) vec \<Rightarrow> ('a, 'b) vec \<Rightarrow> 'a" shows "vec_real_inner_axioms inner_field inner \<equiv> \<forall>x y. inner x y = (\<Sum>i\<in>UNIV. inner_field (x $ i) (y $ i))"
simps_2: fixes c :: "'c"   and bs :: "'c list"   and a :: "'c"   and as :: "'c list" shows "splitAtRec c bs (a # as) = (if a = c then (bs, as) else splitAtRec c (bs @ [a]) as)"
irreducible_not_unit: fixes p :: "'a" assumes "irreducible p" shows "\<not> p dvd (1::'a)"
add_nonneg_pos: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "{#} \<subseteq># a"   and "{#} \<subset># b" shows "{#} \<subset># a + b"
pkt_simp: fixes no :: "nat"   and sid :: "nat" shows "pkt (no, sid) = Pkt no sid"
eq_diff0: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<ominus> y = \<zero>) = (x = y)"
ncols_not_0: fixes A :: "(('b, 'c) vec, 'd) vec" shows "0 \<noteq> ncols A"
unit_simps_1: fixes a :: "'a" assumes "B.obj a" shows "B.arr \<i>\<^sub>B[a]"
set_plus_elim: fixes x :: "'c"   and A :: "'c set"   and B :: "'c set" assumes "x \<in> A + B" obtains a :: "'c" and b :: "'c" where "x = a + b" and "a \<in> A" and "b \<in> B"
sign_def: fixes p :: "'c \<Rightarrow> 'c" shows "sign p = (if evenperm p then 1 else - 1)"
listsetI: fixes ys :: "'a list"   and xs :: "'a set list" assumes "length ys = length xs"   and "\<And>i. i < length xs \<Longrightarrow> ys ! i \<in> xs ! i" shows "ys \<in> listset xs"
prems_3: fixes x :: "'a set"   and y1 :: "'a"   and y2 :: "'a" assumes "x \<in> set (x__ # xs)"   and "y1 \<in> x"   and "y2 \<in> x" shows "f y1 y2 \<in> x"
add_inv_exists_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "\<exists>w\<in>carrier_vec n. w + v = 0\<^sub>v n \<and> v + w = 0\<^sub>v n"
inverse_unique: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "group f z inverse"   and "f a b = z" shows "inverse a = b"
circ_reflexive: fixes x :: "'a" shows "reflexive (x\<^sup>\<star>)"
pairwise_imageI: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; f x \<noteq> f y\<rbrakk> \<Longrightarrow> P (f x) (f y)" shows "pairwise P (f ` A)"
coeff: fixes x :: "'b poly" shows "coeff x \<in> {f. \<forall>\<^sub>\<infinity>n. f n = (0::'b)}"
\<epsilon>_def: shows "\<epsilon> = (\<lambda>\<Sigma> (a, bs). if a \<in> set \<Sigma> then [(a, True # bs), (a, False # bs)] else [])"
Nat_case_def: shows "Nat_case \<equiv> \<Lambda> z s n. unbox\<cdot>n >>= (\<Lambda> n'. case undiscr n' of 0 \<Rightarrow> z | Suc n'' \<Rightarrow> s\<cdot>(box\<cdot>(Discr n'')))"
add_poly_pE: fixes S :: "(char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset"   and T :: "(char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset \<times> (char list multiset \<times> int) multiset" assumes "add_poly_p S T" obtains p :: "(char list multiset \<times> int) multiset" and x :: "char list multiset \<times> int" and q :: "(char list multiset \<times> int) multiset" and r :: "(char list multiset \<times> int) multiset" where "S = (p__, add_mset x__ q__, r__)" and "T = (p__, q__, add_mset x__ r__)"   | x :: "char list multiset \<times> int" and p :: "(char list multiset \<times> int) multiset" and q :: "(char list multiset \<times> int) multiset" and r :: "(char list multiset \<times> int) multiset" where "S = (add_mset xb__ pa__, qa__, ra__)" and "T = (pa__, qa__, add_mset xb__ ra__)"   | x :: "char list multiset" and n :: "int" and p :: "(char list multiset \<times> int) multiset" and q :: "(char list multiset \<times> int) multiset" and m :: "int" and r :: "(char list multiset \<times> int) multiset" where "S = (add_mset (xc__, n__) pb__, qb__, add_mset (xc__, m__) rb__)" and "T = (pb__, qb__, add_mset (xc__, n__ + m__) rb__)"   | p :: "(char list multiset \<times> int) multiset" and x :: "char list multiset" and n :: "int" and q :: "(char list multiset \<times> int) multiset" and m :: "int" and r :: "(char list multiset \<times> int) multiset" where "S = (pc__, add_mset (xd__, n) qc__, add_mset (xd__, m) rc__)" and "T = (pc__, qc__, add_mset (xd__, n + m) rc__)"   | p :: "(char list multiset \<times> int) multiset" and q :: "(char list multiset \<times> int) multiset" and x :: "char list multiset" and r :: "(char list multiset \<times> int) multiset" where "S = (p, q, add_mset (x, 0) r)" and "T = (p, q, r)"
Elem_Elem_PFun: fixes F :: "ZF"   and U :: "ZF"   and V :: "ZF"   and p :: "ZF" assumes "Elem F (PFun U V)"   and "Elem p F" shows "isOpair p \<and> Elem (Fst p) U \<and> Elem (Snd p) V"
defNode_ex1: fixes v :: "'val"   and g :: "'g" assumes "v \<in> uninst_code.ssa.allVars uses phis g" shows "\<exists>!n. n \<in> set (\<alpha>n g) \<and> v \<in> uninst_code.ssa.allDefs phis g n"
dg_length: shows "vcard (op_dg \<BB>) = 4\<^sub>\<nat>"
assms: shows "i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
unifier_single: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and l :: "fterm literal" shows "unifier\<^sub>l\<^sub>s \<sigma> {l}"
lcoset_perm_comp: fixes e :: "'g set"   and g :: "'g set"   and h :: "'g set" assumes "Group e \<and> order.greater_eq G e"   and "normal e"   and "g \<in> G // LCoset_rel e"   and "h \<in> G // LCoset_rel e" shows "restrict1 ((+) g) (G // LCoset_rel e) \<circ> restrict1 ((+) h) (G // LCoset_rel e) = restrict1 ((+) (g + h)) (G // LCoset_rel e)"
that: fixes ub :: "real" assumes "lb < ub"   and "\<forall>z. lb < z \<and> z \<le> ub \<longrightarrow> poly p z \<noteq> 0" shows "thesis"
observable3_equiv_observable2: fixes ob :: "'a \<Rightarrow> 'b"   and P :: "'a set" shows "observable3 ob P = observable2 ob P"
varset_action_def: fixes a :: "('d, bool) fmap \<times> ('d, bool) fmap"   and varset :: "'d set" shows "varset_action a varset \<equiv> fmdom' (snd a) \<subseteq> varset"
remdups_mset_singleton_sum: fixes a :: "'a"   and A :: "'a multiset" shows "remdups_mset (add_mset a A) = (if a \<in># A then remdups_mset A else add_mset a (remdups_mset A))"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (Or p q)"   and "\<And>p q. \<lbrakk>P (NOT p); P q\<rbrakk> \<Longrightarrow> P (Imp p q)"   and "\<And>p q. \<lbrakk>P p; P q; P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (Iff p q)"   and "\<And>a. P (Lt a)"   and "\<And>a. P (Le a)"   and "\<And>a. P (Gt a)"   and "\<And>a. P (Ge a)"   and "\<And>a. P (Eq a)"   and "\<And>a. P (NEq a)"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (And p q))"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Or p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (Imp p q))"   and "\<And>p q. \<lbrakk>P p; P (NOT q); P (NOT p); P q\<rbrakk> \<Longrightarrow> P (NOT (Iff p q))"   and "\<And>p. P p \<Longrightarrow> P (NOT (NOT p))"   and "P (NOT T)"   and "P (NOT F)"   and "\<And>a. P (Ge a) \<Longrightarrow> P (NOT (Lt a))"   and "\<And>a. P (Gt a) \<Longrightarrow> P (NOT (Le a))"   and "\<And>a. P (Le a) \<Longrightarrow> P (NOT (Gt a))"   and "\<And>a. P (Lt a) \<Longrightarrow> P (NOT (Ge a))"   and "\<And>a. P (NEq a) \<Longrightarrow> P (NOT (Eq a))"   and "\<And>a. P (Eq a) \<Longrightarrow> P (NOT (NEq a))"   and "P T"   and "P F"   and "\<And>va. P (NOT (E va))"   and "\<And>va. P (NOT (A va))"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
ID1: fixes x :: "'c" shows "ID\<cdot>x = x"
1: shows "[\<^bold>\<A>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<^bold>\<A>\<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in v]"
relatesD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a stream" assumes "relates P s" shows "P (shd s) (shd (stl s))"
nonneg_power: fixes k :: "nat" shows "fds_nth (f' ^ k) n \<in> \<real>\<^sub>\<ge>\<^sub>0"
minus_geq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleRat :: "rat \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_rational_vector less_eq less plus zero minus uminus scaleRat" shows "less_eq a b = less_eq zero (minus b a)"
pHom_memc: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and f :: "'a \<Rightarrow> 'g"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "PolynRg A B Y"   and "f \<in> pHom R S X, A B Y"   and "pol_coeff S c" shows "f (polyn_expr R X (fst c) c) = polyn_expr A Y (fst c) (cf_h f c)"
\<sigma>_def: fixes P :: "(('a list \<times> nat) multiset \<Rightarrow> 'a) list" shows "Id.R.\<sigma> P = map Id.R.indexed_const P"
list_del_lo: fixes t :: "'b tree" assumes "braun t" shows "list (del_lo t) = tl (list t)"
intY1_def: shows "intY1 \<equiv> interval r (lub Y cl) (Top cl)"
fg_2: shows "g $$ fls_subdegree g * f $$ fls_subdegree f = (1::'a)"
env_restr_esing2: fixes x :: "'a"   and S :: "'a set"   and v :: "'b" assumes "x \<notin> S" shows "esing x\<cdot>v f|` S = \<bottom>"
arcs_symmetric: fixes a :: "'b"   and b :: "'b" assumes "(a, b) \<in> parcs (list_digraph G_list)" shows "(b, a) \<in> parcs (list_digraph G_list)"
fermat_4: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "x ^ 4 + y ^ 4 = z ^ 4" shows "x * y * z = 0"
assm: shows "(i2 = bot \<or> consec i1 i2) \<and> (i3 = bot \<or> consec i3 i4) \<and> consec (i1 \<squnion> i2) (i3 \<squnion> i4)"
case_prod_beta: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and p :: "'b \<times> 'c" shows "(case p of (x, xa) \<Rightarrow> f x xa) = f (cc p) (snd p)"
rtrancl_edges_sym': fixes u :: "'b"   and v :: "'b"   and g :: "'b ugraph" assumes "(u, v) \<in> (edges g)\<^sup>*" shows "(v, u) \<in> (edges g)\<^sup>*"
DirProd_subgroups: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and K :: "('e, 'f) monoid_scheme"   and I :: "'e set" assumes "Group.group G"   and "subgroup H G"   and "Group.group K"   and "subgroup I K" shows "subgroup (H \<times> I) (G \<times>\<times> K)"
ccBind_def: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and v :: "Vars.var"   and e :: "exp" shows "CoCallAnalysis.ccBind ccExp v e = (\<Lambda> (ae, G). if v--v\<notin>G \<or> \<not> isVal e then fup\<cdot>(ccExp e)\<cdot>(ae v) G|` fv e else fv e\<^sup>2)"
lincomb_Nil: fixes rs :: "'r list"   and ms :: "'n list" assumes "rs = [] \<or> ms = []" shows "rs \<bullet>\<star> ms = (0::'n)"
init_fin_\<tau>move_simps_4: fixes s :: "status"   and x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, status \<times> 'x2, 'm2, 'w, 'o action) thread_action"   and x' :: "'x2"   and m' :: "'m2" shows "r2.init_fin_\<tau>move ((s, x), m) ta ((Running, x'), m') = (s = Running \<and> (\<exists>ta'. ta = convert_TA_initial (convert_obs_initial ta') \<and> \<tau>move2 (x, m) ta' (x', m')))"
finite_sup_closed: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "\<Squnion>A \<in> carrier L"
ovalidNF_pre_imp: fixes P' :: "'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>s. P' s \<Longrightarrow> P s"   and "ovalidNF P f Q" shows "ovalidNF P' f Q"
nth_less_subdegree_zero: fixes n :: "nat"   and f :: "'b fps" assumes "n < subdegree f" shows "f $ n = (0::'b)"
one_neq_zero: fixes one :: "'e"   and zero :: "'e" assumes "class.zero_neq_one one zero" shows "one \<noteq> zero"
awalk_Cons_iff: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and e :: "'d"   and es :: "'d list"   and w :: "'c" assumes "wf_digraph G" shows "pre_digraph.awalk G u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> pre_digraph.awalk G (head G e) es w)"
\<phi>_inverse_im: shows "GRepHomSet (\<star>) W \<subseteq> \<phi> ` HRepHomSet (\<star>) W"
complete_net_preserves_subnets: fixes s :: "'c net_state"   and t :: "'c net_state"   and a :: "'d node_action"   and st' :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p1 :: "net_tree"   and p2 :: "net_tree" assumes "(SubnetS s t, a, st') \<in> cnet_sos (pnet_sos (automaton.trans (pnet np p1)) (automaton.trans (pnet np p2)))" shows "\<exists>s' t'. st' = SubnetS s' t'"
res: shows "identify_blocks_main A (Suc i_e) list = identify_blocks_main A i_b ((i_b, i_e) # list)"
rank_skew_cons_up: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" assumes "SkewBinomialHeapStruc.rank_invar (t # bq)"   and "SkewBinomialHeapStruc.rank t' \<le> SkewBinomialHeapStruc.rank t" shows "SkewBinomialHeapStruc.rank_skew_invar (t' # t # bq)"
sortedP_append_iff: fixes x :: "real \<times> real"   and xs :: "(real \<times> real) list"   and ys :: "(real \<times> real) list" shows "ccw'.sortedP x (xs @ ys) = (ccw'.sortedP x xs \<and> ccw'.sortedP x ys \<and> (\<forall>xa\<in>set xs. \<forall>y\<in>set ys. ccw' x xa y))"
Sup_upto_llist_Suc: fixes Xs :: "'a set llist"   and j :: "nat" shows "Sup_upto_llist Xs (enat (Suc j)) = Sup_upto_llist Xs (enat j) \<union> (if enat (Suc j) < llength Xs then lnth Xs (Suc j) else {})"
run_fail_writer: fixes fail :: "'d" shows "run_writer (fail_writer fail) = fail"
distinct_3: fixes x1 :: "'b"   and x31 :: "'a"   and x32 :: "('a, 'b) process_term" shows "VAR x1 \<noteq> ACT x31 x32"
nth_Cons_numeral: fixes x :: "'a"   and xs :: "'a list"   and v :: "num" shows "(x # xs) ! numeral v = xs ! (numeral v - 1)"
l_set_val_get_disconnected_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_val_get_disconnected_nodes type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_val type_wf set_val set_val_locs \<and> l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs \<and> l_set_val_get_disconnected_nodes_axioms set_val_locs get_disconnected_nodes_locs"
Suc: fixes n :: "nat" shows "(fps_exp 1 - 1) ^ m $ n * fact n = fact m * real (Stirling n m)"
rotate_A_def: shows "rotate_A = (\<lambda>v. case v of A \<Rightarrow> A | B \<Rightarrow> C | C \<Rightarrow> D | D \<Rightarrow> B)"
dc: fixes u :: "nat discr"   and v :: "nat discr" shows "undiscr u * undiscr v = undiscr v * undiscr u"
rep_eq: fixes x :: "'d"   and xa :: "'b" shows "FL_Formula.formula.Rep_formula (FL_Formula.Pred x xa) = FL_Formula.Pred\<^sub>\<alpha> x xa"
Vinv2_inductive_2: shows "{Vinv2 \<inter> Vinv1} TS.trans v_TS {> Vinv2}"
lemma_2_10_28: fixes b :: "'a"   and a :: "'a" shows "b r\<rightarrow> (a r\<rightarrow> b) l\<rightarrow> a = b r\<rightarrow> a"
join_meet_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> x \<sqinter> y = x"
sum_list_us_bounded: fixes i :: "nat"   and k :: "nat"   and y :: "nat list" assumes "i < k" shows "sum_list (\<^bold>u y i) \<le> k"
distinct_rev: fixes xs :: "'d list" shows "distinct (rev xs) = distinct xs"
prems: shows "b \<le> a \<or> p \<noteq> 0 \<and> poly p a \<noteq> 0 \<and> count_roots_between p a b = (if a < b \<and> poly p b = 0 then 1 else 0)"
is_unit_smult_iff: fixes c :: "'a"   and p :: "'a poly" shows "(Polynomial.smult c p dvd 1) = (c dvd (1::'a) \<and> p dvd 1)"
Map_in_Hom: fixes f :: "('o, 'a) arr" assumes "arr f" shows "Map f \<in> Hom (Dom f) (Cod f)"
generator: shows "carrier G \<subseteq> range (([^]) \<^bold>g)"
is_locale: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and idx :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and size :: "'a bound \<Rightarrow> nat" assumes "class.index less_eq less idx size" shows "index_locale less_eq less idx size"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "local.a_known_ptr ptr = (DocumentClass.known_ptr ptr \<or> (case cast ptr of None \<Rightarrow> False | Some shadow_root_ptr \<Rightarrow> is_shadow_root_ptr shadow_root_ptr))"
intro: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "Group.group H"   and "group_hom_axioms G H h" shows "group_hom G H h"