hcju/isabelleps huggingface dataset:
abs_subterms: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and \<alpha> :: "nat \<times> bool \<Rightarrow> 'c set" shows "subterms (t \<cdot>\<^sub>\<alpha> \<alpha>) = subterms t \<cdot>\<^sub>\<alpha>\<^sub>s\<^sub>e\<^sub>t \<alpha>"
real_dcharacter_axioms: shows "real_dcharacter n \<chi>"
hom_component_plus: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and q :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and n :: "nat" shows "hom_component (p + q) n = hom_component p n + hom_component q n"
pns: shows "(p, n, s) \<in> set (rsplit0 a) \<and> Ifm (x # bs) p"
col_lta__bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and Y :: "'a"   and Z :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet X Y Z"   and "Tarski_neutral_dimensionless.LtA Bet Cong A B C X Y Z" shows "Bet X Y Z"
Seq_transT_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and c1 :: "('a, 'c) com"   and c2 :: "('a, 'c) com"   and s :: "'b"   and s' :: "'b" shows "\<not> PL.transT tval aval (c1 ;; c2, s) s'"
generated_ring_disjoint_UNION: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'a set" assumes "finite I"   and "disjoint (A ` I)"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> generated_ring" shows "\<Union> (A ` I) \<in> generated_ring"
isomorphic_implies_hpar_2: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f f'" shows "Span.ide f'"
sorted_list_of_set_insert: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b" assumes "class.linorder less_eq less"   and "finite A" shows "linorder.sorted_list_of_set less_eq (insert x A) = linorder.insort_key less_eq (\<lambda>x. x) x (linorder.sorted_list_of_set less_eq (A - {x}))"
apply_guards_double_cons: fixes y :: "vname gexp"   and x :: "vname gexp"   and G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "apply_guards (y # x # G) s = (gval (gAnd y x) s = true \<and> apply_guards G s)"
cod_eqI: fixes b :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide b"   and "seq b f" shows "cod f = b"
compare_self_neq_LT: fixes x :: "'a" shows "compare\<cdot>x\<cdot>x \<noteq> LT"
SeqCall: fixes \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and p :: "'b"   and as :: "('a, 'b, 'c) ann list"   and n :: "nat"   and P'' :: "('a, 'b, 'c) ann"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and b :: "('a, 'b, 'c) com"   and F :: "'c set"   and P :: "'a set"   and Q :: "'a set"   and A :: "'a set"   and r :: "'a set" assumes "\<Theta> p = Some as"   and "as ! n = P''"   and "\<Gamma> p = Some b"   and "n < length as"   and "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P P'' b Q, A" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnCall r n) (Call p) Q, A"
abs_minus: fixes abs :: "'d \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (uminus a) = abs a"
Constrains_trans: fixes F :: "'b program"   and A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "F \<in> A Co B"   and "F \<in> B Co C" shows "F \<in> A Co C"
repr2: fixes A :: "'a"   and B :: "'b" assumes "(A, B) \<in> F" shows "B = f A"
emptyProc_def: shows "emptyProc = \<lparr>pid = 0, vars = lm.empty (), pc = 0, channels = [], idx = 0\<rparr>"
completeness_mat_leqb_eqc: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec" assumes "\<exists>x. [A*\<^sub>vx]\<le>b"   and "\<exists>y. [y\<^sub>v*A]=c" shows "\<exists>X. simplex (mat_leqb_eqc A b c) = Inr X"
SEval_cong: fixes s :: "form list"   and e1 :: "nat \<Rightarrow> U"   and e2 :: "nat \<Rightarrow> U"   and m :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)" shows "(\<forall>x. x \<in> set (sfv s) \<longrightarrow> e1 x = e2 x) \<longrightarrow> SEval m e1 s = SEval m e2 s"
lexs_irrefl: fixes p :: "real \<times> real" shows "\<not> lexs p p"
numer_cf6_pos: fixes x :: "real" shows "0 < numer_cf6 x"
postsplit_satisfied: shows "\<tau> \<Turnstile> POST' self a1"
dg_prod_Dom_app_component_app: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"   and "i \<in>\<^sub>\<circ> I" shows "dg_prod I \<AA>\<lparr>Dom\<rparr>\<lparr>f\<rparr>\<lparr>i\<rparr> = \<AA> i\<lparr>Dom\<rparr>\<lparr>f\<lparr>i\<rparr>\<rparr>"
assms_1: shows "l \<in> set (all_partitions_list G)"
src_port_int_TCPUDP_def: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> 'a" shows "src_port x \<equiv> (fst \<circ> snd \<circ> fst \<circ> snd) x"
Red_I_of_Inf_to_N_subset: fixes N :: "'a set" shows "{\<iota> \<in> Inf. concl_of \<iota> \<in> N} \<subseteq> Red_I N"
stream_all2_stream_rel_conv: fixes R :: "('d \<times> 'e) set" shows "stream_all2 (\<lambda>x y. (x, y) \<in> R) = (\<lambda>x y. (x, y) \<in> \<langle>R\<rangle>stream_rel)"
exhaust: fixes y :: "quat" obtains x1 :: "real" and x2 :: "real" and x3 :: "real" and x4 :: "real" where "y = Quat x1 x2 x3 x4"
length_subst_atm_mset_list: 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 AAs :: "'b multiset list"   and \<eta> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "length (substitution_ops.subst_atm_mset_list subst_atm AAs \<eta>) = length AAs"
finite_prod_fun: fixes f1 :: "'b \<Rightarrow> 'c"   and f2 :: "'b \<Rightarrow> 'c" assumes "finite {l. f1 l \<noteq> (0::'c)}"   and "finite {q. f2 q \<noteq> (0::'c)}" shows "finite {k. prod_fun f1 f2 k \<noteq> (0::'c)}"
equivalence_mapI: fixes f :: "'a"   and g :: "'a" assumes "B.quasi_inverses f g" shows "B.equivalence_map f"
hyps_2: shows "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
Suc_mod: fixes n :: "nat"   and i :: "nat" assumes "1 < n" shows "i \<noteq> Suc i mod n"
bdd_above_Ioc: fixes a :: "'a"   and b :: "'a" shows "bdd_above {a<..b}"
evaln_elim_cases_9: fixes G :: "prog"   and s :: "st"   and binop :: "binop"   and e1 :: "expr"   and e2 :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (BinOp binop e1 e2)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains v1 :: "val" and s1 :: "state" and v2 :: "val" where "v = In1 (eval_binop binop v1 v2)" and "G\<turnstile>Norm s \<midarrow>e1-\<succ>v1\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>(if need_second_arg binop v1 then In1l e2 else In1r Skip)\<succ>\<midarrow>n\<rightarrow> (In1 v2, s')"
append_eq_append_conv2: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list"   and ts :: "'b list" shows "(xs \<cdot> ys = zs \<cdot> ts) = (\<exists>us. xs = zs \<cdot> us \<and> us \<cdot> ys = ts \<or> xs \<cdot> us = zs \<and> ys = us \<cdot> ts)"
hom_lead_coeff_nonzero_imp_map_poly_hom: fixes p :: "int poly" assumes "of_int (lead_coeff p) \<noteq> (0::'a)" shows "of_int_poly p \<noteq> 0"
trg_vcomp: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "B.seq \<nu> \<mu>" shows "trg\<^sub>B (\<nu> \<cdot>\<^sub>B \<mu>) = trg\<^sub>B \<nu>"
simps: fixes v :: "'form \<times> nat list"   and p :: "('form, 'var) out_port" shows "valid_out_port (v, p) = (v |\<in>| vertices \<and> p |\<in>| outPorts (nodeOf v))"
rtype_def: fixes f :: "AttId" shows "rtype f = (case f of Dummy'dummy \<Rightarrow> NullT | Counter'dummy \<Rightarrow> NullT | _ \<Rightarrow> IntgT)"
assms: shows "Negligible.negligible (\<lambda>n. 1 / real (q n))"
eval_or_const_1: fixes \<phi> :: "tvl"   and \<psi> :: "tvl" shows "(eval_or \<phi> \<psi> = Yes) = (\<phi> = Yes \<or> \<psi> = Yes)"
Fubini': fixes M1 :: "'d measure"   and M2 :: "'e measure"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> ennreal" assumes "pair_sigma_finite M1 M2"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f x y \<partial>M1 \<partial>M2 = \<integral>\<^sup>+ x. integral\<^sup>N M2 (f x) \<partial>M1"
sgn_plus: fixes x :: "'a"   and y :: "'a" assumes "sgn x = sgn y" shows "sgn (x + y) = sgn x"
load_tmps_distinct: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "distinct_load_tmps is"
carrier_Fheap: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "carrier (Theap \<Gamma> e\<cdot>a) = edom (Aheap \<Gamma> e\<cdot>a)"
d: shows "create_insert s n \<sigma> new_id = Inr (Insert m')"
norm_14: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(\<not> (\<box>P) \<sigma>) = (\<diamond>(\<lambda>s. \<not> P s)) \<sigma>"
wf1: shows "is_density_expr ([], vs @ vs', \<Gamma>, CReal 1) t' f"
antisym: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x"   and "x \<le> y" shows "x = y"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "f \<cong> g" shows "g \<cong> f"
law_restrict_is_ring: fixes R :: "('c, 'd) ring_scheme" assumes "ring R" shows "ring (law_restrict R)"
nonzeros_trunc_err_pdevs_raw: fixes r :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "{i. trunc_err_pdevs_raw r x i \<noteq> (0::'a)} \<subseteq> {i. x i \<noteq> (0::'a)}"
ContractsWithSubstitutesAndIRC_def: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" shows "ContractsWithSubstitutesAndIRC Xd Xh Pd Ch \<equiv> ContractsWithSubstitutes Xd Xh Pd Ch \<and> ContractsWithIRC Xd Xh Pd Ch"
square_free_monic_poly: fixes as :: "('c poly \<times> nat) set"   and p :: "'c poly"   and x :: "'c" assumes "monic_factorization as p" shows "(poly (yun_gcd.square_free_monic_poly gcd p) x = (0::'c)) = (poly p x = (0::'c))"
l1: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration" assumes "computation channel trans send recv init final" shows "\<forall>t i cid. distributed_system.trace channel trans send recv init t final \<and> Marker \<in> set (msgs (distributed_system.s channel trans send recv init t i) cid) \<longrightarrow> (\<exists>j\<ge>i. Marker \<notin> set (msgs (distributed_system.s channel trans send recv init t j) cid))"
assms: shows "(red F)\<^sup>*\<^sup>* p q"
widen_i_i_i_def: shows "widen_i_i_i \<equiv> \<lambda>x xa xb. pred.Pred (\<lambda>xc. x \<turnstile> xa \<le> xb)"
bisim1AAccThrow2: fixes P :: "'b prog"   and h :: "'heap"   and i :: "'addr expr1"   and ad :: "'addr"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr expr1"   and v :: "'addr val" assumes "P,i,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>ad\<rfloor>)" shows "P,a\<lfloor>i\<rceil>,h \<turnstile> (Throw ad, xs) \<leftrightarrow> (stk @ [v], loc, length (compE2 a) + pc, \<lfloor>ad\<rfloor>)"
solves_ode_def: fixes y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set" shows "(y solves_ode f) T X = ((y has_vderiv_on (\<lambda>t. f t (y t))) T \<and> y \<in> T \<rightarrow> X)"
notin_range_Some: fixes x :: "'k option" shows "(x \<notin> range Some) = (x = None)"
xs': shows "xs' \<in>\<^sub>\<circ> A ^\<^sub>\<times> vcard xs'"
B: shows "u \<in> range D \<inter> (- I) `` range D"
poly_add: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and p :: "('d monom \<times> 'c) list"   and q :: "('d monom \<times> 'c) list" shows "eval_poly \<alpha> (poly_add p q) = eval_poly \<alpha> p + eval_poly \<alpha> q"
rel_spmf_bindI1: fixes p :: "'c spmf"   and R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd spmf"   and q :: "'e spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf R (f x) q"   and "lossless_spmf p" shows "rel_spmf R (p \<bind> f) q"
zlin_qfree: fixes p :: "fm"   and bs :: "real list" assumes "iszlfm p bs" shows "qfree p"
IU: shows "weakly_future_consistent P I D (rel_ipurge P I D)"
connected_component_maximal: fixes x :: "'b"   and T :: "'b set"   and S :: "'b set" assumes "x \<in> T"   and "connected T"   and "T \<subseteq> S" shows "T \<subseteq> connected_component_set S x"
rel_flip: fixes R :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b list"   and b :: "'c list" shows "list_all2 R\<inverse>\<inverse> a b = list_all2 R b a"
principal_ideal: fixes R :: "('e, 'f) Ring_scheme"   and a :: "'e" assumes "Ring R"   and "a \<in> carrier R" shows "ideal R (R \<diamondsuit>\<^sub>p a)"
lookup_homogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "x \<notin> indets p"   and "x \<notin> keys t" shows "lookup (homogenize x p) (monomial (poly_deg p - deg_pm t) x + t) = lookup p t"
recurrent_acc_1: fixes x :: "'s \<times> 's option"   and y :: "'s \<times> 's option" assumes "KN.recurrent x"   and "(x, y) \<in> (SIGMA x:UNIV. set_pmf (KN.Kp x))\<^sup>*" shows "KN.U y x = 1"
distinct_16: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x31@[x32] \<noteq> Skip"
simps_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.strict_sorted less [] = True"
ord_spmf_eqD_measure_spmf: fixes p :: "'c spmf"   and q :: "'c spmf" assumes "ord_spmf (=) p q" shows "measure_spmf p \<le> measure_spmf q"
left2_def: shows "left2 \<equiv> prod.fst (prod.snd (prod.snd (prod.snd (fix_syn (\<lambda>(root2, op2, rep2, left2, right2). (\<Lambda> pat. rep2\<cdot>pat\<cdot>([::], pat), \<Lambda> pat v x. Fixrec.run\<cdot>(match_Null\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(root2\<cdot>pat)) +++ match_Node\<cdot>v\<cdot>(\<Lambda> usvs l r. Fixrec.succeed\<cdot>(If prefix\<cdot>[:x:]\<cdot>(csnd\<cdot>usvs) then r else op2\<cdot>pat\<cdot>l\<cdot>x))), \<Lambda> pat usvs. Node\<cdot>usvs\<cdot>(left2\<cdot>pat\<cdot>usvs)\<cdot>(right2\<cdot>pat\<cdot>usvs), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>Null))) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> v vs. match_scons\<cdot>v\<cdot>(\<Lambda> u us. Fixrec.succeed\<cdot>(next\<cdot>vs\<cdot>(sfoldl\<cdot>(op2\<cdot>pat)\<cdot>(root2\<cdot>pat)\<cdot>us))))), \<Lambda> pat va. Fixrec.run\<cdot>(match_Pair\<cdot>va\<cdot>(\<Lambda> us v. match_snil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>Null)) +++ match_Pair\<cdot>va\<cdot>(\<Lambda> us va. match_scons\<cdot>va\<cdot>(\<Lambda> v vs. Fixrec.succeed\<cdot>(rep2\<cdot>pat\<cdot>(us :@ [:v:], vs)))))))))))"
simps_5: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and x2 :: "'d" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
monad_altc3_axioms: shows "monad_altc3 return bind altc"
function_of_def: fixes A :: "'c set"   and enum :: "nat \<Rightarrow> 'c"   and xs :: "'d list" shows "function_of A enum xs = (\<lambda>a. if a \<in> A then xs ! inv_into {0..<length xs} enum a else undefined)"
less_eq_char: fixes x :: "char"   and y :: "char" shows "x \<le> y \<equiv> int_of_char x \<le> int_of_char y"
dflt_option_rel_aux_def: fixes dflt :: "'b" shows "dflt_option_rel_aux dflt \<equiv> {(dflt, None)} \<union> {(x, Some x) |x. x \<noteq> dflt}"
set_times_mono3: fixes a :: "'b"   and C :: "'b set"   and D :: "'b set" assumes "a \<in> C" shows "a *o D \<subseteq> C * D"
lub_least: fixes leq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lub :: "'f set \<Rightarrow> 'f"   and A :: "'f set"   and z :: "'f" assumes "partial_function_definitions leq lub"   and "Complete_Partial_Order.chain leq A"   and "\<And>x. x \<in> A \<Longrightarrow> leq x z" shows "leq (lub A) z"
filter_sup_symmetric: fixes F :: "'a set"   and G :: "'a set" shows "filter_sup F G = filter_sup G F"
check_types_def: fixes P :: "'a prog"   and mxs :: "nat"   and mxl :: "nat"   and \<tau>s :: "(ty list \<times> ty err list) option err list" shows "check_types P mxs mxl \<tau>s \<equiv> set \<tau>s \<subseteq> states P mxs mxl"
l7_17_bis: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and P' :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong A P P'"   and "Tarski_neutral_dimensionless.Midpoint Bet Cong B P' P" shows "A = B"
inv_fun_scale: fixes f :: "nat \<Rightarrow> 'a"   and r :: "'a" assumes "Abstract_Linear_Poly.inv f" shows "Abstract_Linear_Poly.inv (fun_scale r f)"
integrable_reflect_real: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'c" shows "integrable (lebesgue_on {- b..- a}) (\<lambda>x. f (- x)) = integrable (lebesgue_on {a..b}) f"
eq_neg_iff_add_eq_0: fixes a :: "'a"   and b :: "'a" shows "(a = - b) = (a + b = (0::'a))"
fwi'_def: fixes m :: "nat \<times> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat"   and i :: "nat"   and j :: "nat" shows "fwi' m n k i j = REC\<^sub>T (\<lambda>fw (m, k, i, j). case (i, j) of (0, 0) \<Rightarrow> fw_upd' m k 0 0 | (0, Suc j) \<Rightarrow> fw (m, k, 0, j) \<bind> (\<lambda>m'. fw_upd' m' k 0 (Suc j)) | (Suc ia, 0) \<Rightarrow> fw (m, k, ia, n) \<bind> (\<lambda>m'. fw_upd' m' k (Suc ia) 0) | (Suc ia, Suc j) \<Rightarrow> fw (m, k, Suc ia, j) \<bind> (\<lambda>m'. fw_upd' m' k (Suc ia) (Suc j))) (m, k, i, j)"
range_lsl_iff: fixes x :: "'b" shows "(x \<in> range \<nu>) = (\<nu> x = x)"
dominates_path: fixes g :: "'g"   and n :: "'node"   and m :: "'node" assumes "dominates g n m"   and "invar g" obtains ns :: "'node list" where "g \<turnstile> n-ns\<rightarrow>m"
dense_closed_inf: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top"   and "uminus x = bot"   and "uminus y = bot" shows "uminus (inf x y) = bot"
map_simp: fixes f :: "'a \<times> 'a" assumes "B.VV.arr f" shows "B.VV.map f = f"
tm_append_second_fetch0_eq: fixes A :: "(action \<times> nat) list"   and off :: "nat"   and B :: "(action \<times> nat) list"   and s :: "nat"   and b :: "cell"   and ac :: "action" assumes "is_even (length A)"   and "off = length A div 2"   and "fetch B s b = (ac, 0)"   and "s \<noteq> 0" shows "fetch (A @ shift B off) (s + off) b = (ac, 0)"
A91_DAcc_level1: shows "DAcc level1 sA91 = {sA81}"
intro: fixes R :: "('e, 'f) ring_scheme"   and M :: "('e, 'g, 'h) module_scheme" assumes "cring R"   and "abelian_group M"   and "module_axioms R M" shows "module R M"
top_greatest: fixes a :: "'a" shows "\<bottom> \<le> a"
add_le_imp_le_right: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" assumes "a + c \<subseteq># b + c" shows "a \<subseteq># b"
Cod_cod: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "local.Cod (cod f) = local.Cod f"
homogeneous_set_homogeneous_ideal: fixes F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f" shows "homogeneous_set (ideal F)"
z_1: shows "z \<in> outside (path_image (p +++ linepath a b))"
ccField_ccProd_subset: fixes S :: "Vars.var set"   and S' :: "Vars.var set" shows "ccField (S G\<times> S') \<subseteq> S \<union> S'"
aeq_zeq: fixes m :: "int"   and n :: "int" shows "(ant m = ant n) = (m = n)"
distinct_109: fixes x101 :: "nat"   and x102 :: "msg"   and x111 :: "msg"   and x112 :: "msg" shows "Crypt x101 x102 \<noteq> \<lbrace>x111, x112\<rbrace>"
not_less_Least: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and k :: "'e"   and P :: "'e \<Rightarrow> bool" assumes "class.wellorder less_eq less"   and "less k (ord.Least less_eq P)" shows "\<not> P k"
C1_diff_components_2: fixes b :: "'a"   and f :: "real \<Rightarrow> 'a"   and s :: "real set" assumes "b \<in> Basis"   and "f C1_differentiable_on s" shows "(\<lambda>x. f x \<bullet> b) C1_differentiable_on s"
integrableD_2: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity>"
init_fin_silent_moves_RunningI: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and x' :: "'x2"   and m' :: "'m2" assumes "r2.silent_moves t (x, m) (x', m')" shows "(\<tau>trsys.silent_move (r2.init_fin t) r2.init_fin_\<tau>move)\<^sup>*\<^sup>* ((Running, x), m) ((Running, x'), m')"
valid_pre: fixes Q :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool"   and p :: "'d process_name"   and R :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool"   and P :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool" assumes "\<lbrace>Q\<rbrace> p \<lbrace>R\<rbrace>"   and "\<And>s. P s \<Longrightarrow> Q s" shows "\<lbrace>P\<rbrace> p \<lbrace>R\<rbrace>"
zero_uint8_def: shows "0 \<equiv> Abs_uint8 0"
IH: fixes t :: "('fun, 'var) Term.term" assumes "t \<in> set T" shows "intruder_synth (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>) (t \<cdot> \<delta>)"
mult_isor: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" assumes "less_eq_prod x y" shows "less_eq_prod (times_prod' x z) (times_prod' y z)"
PTrans: shows "P \<longmapsto> rev perm \<bullet> \<alpha> \<prec> P'"
PfP_extra: shows "{} \<turnstile> PfP \<guillemotleft>extra_axiom\<guillemotright>"
nat_distrib_3: fixes m :: "nat"   and n :: "nat"   and k :: "nat" shows "(m - n) * k = m * k - n * k"
subgame_ParityGame: fixes G :: "('c, 'd) ParityGame_scheme"   and V' :: "'c set" assumes "ParityGame G" shows "ParityGame (ParityGame.subgame G V')"
bounded_plus_comp: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'c" assumes "bounded (f ` S)"   and "bounded (g ` S)" shows "bounded ((\<lambda>x. f x + g x) ` S)"
absolutely_integrable_on_iff_nonneg: fixes S :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "(f absolutely_integrable_on S) = (f integrable_on S)"
mem_Ch_CH: fixes x :: "'a"   and h :: "'c"   and X :: "'a set" assumes "x \<in> BCh h X" shows "x \<in> Bossy'.CH X"