hcju/isabelleps huggingface dataset:
nnfis_add: fixes M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and a :: "real"   and f :: "'b \<Rightarrow> real"   and b :: "real"   and g :: "'b \<Rightarrow> real" assumes "measure_space M"   and "a \<in> nnfis f M"   and "b \<in> nnfis g M" shows "a + b \<in> nnfis (\<lambda>w. f w + g w) M"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'd" assumes "i < n - m" shows "map f [m..<n] ! i = f (m + i)"
iso_unit: fixes a :: "'a" assumes "B.obj a" shows "B.iso \<i>\<^sub>B[a]"
OrR_inv': fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta> \<down> n" shows "\<Gamma> \<Rightarrow> F, G, \<Delta> \<down> n"
gen_preserved2: fixes \<Gamma> :: "(W.var \<times> tyS) list"   and T :: "ty"   and \<theta> :: "(W.tvar \<times> ty) list" assumes "(ftv T - ftv \<Gamma>) \<sharp>* \<theta>" shows "ftv (\<theta><T>) - ftv (\<theta><\<Gamma>>) = ftv T - ftv \<Gamma>"
lfp_dual_gfp: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "lfp f = undual (gfp (dual \<circ> f \<circ> undual))"
frac_ge: fixes t :: "real" assumes "t \<in> {a..b}" shows "K \<le> g' t / g t"
PhiDef_defNode_eq: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> set (\<alpha>n g)"   and "n \<in> phiDefNodes g v"   and "v \<in> vars g" shows "braun_ssa.defNode g (v, n, PhiDef) = n"
vector_space_pair_on_axioms: shows "vector_space_pair_on U\<^sub>M\<^sub>_\<^sub>1 U\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>1) (*s\<^sub>2)"
higher_differentiable_on_fst_comp: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e \<times> 'f"   and k :: "nat" assumes "higher_differentiable_on S f k"   and "open S" shows "higher_differentiable_on S (\<lambda>x. fst (f x)) k"
C11: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and star :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.conway_dioid plus times one zero less_eq less star" shows "star (plus x y) = times (star (times (star x) y)) (star x)"
soldAllocationIsRestricted: fixes a :: "(integer \<times> 'c set) set"   and N :: "integer set"   and \<Omega> :: "'c set" shows "(a \<in> soldAllocations'' N \<Omega>) = (\<exists>aa. Outside' {seller} aa = a \<and> aa \<in> allAllocations'' (N \<union> {seller}) \<Omega>)"
section_retraction_compose: fixes e :: "'a"   and m :: "'a"   and e' :: "'a"   and m' :: "'a" assumes "C.ide (e \<cdot>\<^sub>C m)"   and "C.ide (e' \<cdot>\<^sub>C m')"   and "C.seq m' m" shows "C.ide ((e \<cdot>\<^sub>C e') \<cdot>\<^sub>C m' \<cdot>\<^sub>C m)"
surj_on_mult_units: fixes x :: "nat"   and q :: "nat" assumes "coprime x q"   and "inj_on (\<lambda>b. x * b mod q) ({..<q} - {0})" shows "(\<lambda>b. x * b mod q) ` ({..<q} - {0}) = {..<q} - {0}"
supp: shows "supp (aezfun (r \<delta>\<delta> g)) = supp (r \<delta> g)"
assms: shows "\<tau>exec_move ci P t e h (stk, loc, pc, xcp) s'"
C2: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and S' :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel'"   and "\<Psi>' \<rhd> S \<Longrightarrow>\<^sup>^\<^sub>\<tau> S'" shows "\<exists>T'. \<Psi>' \<rhd> T \<Longrightarrow>\<^sup>^\<^sub>\<tau> T' \<and> (\<Psi>', S', T') \<in> Rel'"
drop_eq_Nil: fixes n :: "nat"   and xs :: "'a list" shows "(drop n xs = []) = (length xs \<le> n)"
map_last: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "xs \<noteq> []" shows "last (map f xs) = f (last xs)"
set_pmf_not_empty: shows "set_pmf p \<noteq> {}"
simps_1: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and s :: "('c list \<times> 'd) pmf" shows "config'_rand A s [] = s"
Cart_lambda_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>x. f x = g x" shows "vec_lambda f = vec_lambda g"
id_union_univ_2: shows "\<Union> (range \<Sigma>n) = UNIV"
not_col_permutation_1: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet B C A"
bind_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a set" shows "A \<bind> f = {x. \<exists>B\<in>f ` A. x \<in> B}"
trace_core_eqI: fixes E :: "'a set"   and IA :: "'b set"   and IU :: "'d set"   and core1 :: "('f, 'a, 'b, 'd, 'c, 'e) core"   and p :: "'f spmf"   and core2 :: "('g, 'a, 'b, 'd, 'c, 'e) core"   and q :: "'g spmf" assumes "\<And>tr e. \<lbrakk>set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV; e \<in> E\<rbrakk> \<Longrightarrow> fst (trace_core' core1 p tr) e = fst (trace_core' core2 q tr) e"   and "\<And>tr ia. \<lbrakk>set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV; ia \<in> IA\<rbrakk> \<Longrightarrow> fst (snd (trace_core' core1 p tr)) ia = fst (snd (trace_core' core2 q tr)) ia"   and "\<And>tr iu. \<lbrakk>set tr \<subseteq> E <+> IA \<times> UNIV <+> IU \<times> UNIV; iu \<in> IU\<rbrakk> \<Longrightarrow> snd (snd (trace_core' core1 p tr)) iu = snd (snd (trace_core' core2 q tr)) iu" shows "trace_core_eq core1 core2 E IA IU p q"
wsus: shows "set (ws @ us) \<subseteq> carrier_vec n"
schroeder_1: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "(inf (composition x y) z = bot) = (inf y (composition (converse x) z) = bot)"
D: shows "\<not> (\<exists>v\<in>sinks_aux I D V (xs @ [x]). (v, w) \<in> I)"
that: fixes ops :: "('a, 'b) sas_plus_operator list"   and \<psi>' :: "('a, 'b) sas_plus_operator list list" assumes "\<psi> = ops # \<psi>'" shows "thesis"
pareto_efficient': fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool"   and x :: "'alt"   and y :: "'alt" assumes "is_pref_profile R"   and "x \<prec>[Pareto R] y" shows "x \<notin> scf R"
class_nameI: fixes dcl :: "char list"   and cl :: "cl"   and fds :: "fd list"   and meth_defs :: "meth_def list" shows "class_name (cld_def dcl cl fds meth_defs) dcl"
distinct_length_2_or_more: fixes a :: "'e"   and b :: "'e"   and xs :: "'e list" shows "distinct (a # b # xs) = (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
lfp_exp_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "lfp_exp t = Inf_exp {P. sound P \<and> t P \<tturnstile> P}"
sets_negligible_symdiff: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> sets lebesgue"   and "negligible (sym_diff S T)" shows "T \<in> sets lebesgue"
terminates_butlast_raw: fixes g :: "'e \<Rightarrow> ('f, 'e) step" assumes "terminates g" shows "terminates (butlast_raw g)"
propositional_calculus_18: fixes Q\<^sub>1 :: "fm"   and Q\<^sub>2 :: "fm"   and A1 :: "fm"   and H :: "fm set"   and Aa :: "fm"   and A :: "fm" assumes "supp Q\<^sub>1 = {}"   and "supp Q\<^sub>2 = {}"   and "insert A1 H \<turnstile> Aa" shows "insert A1 (insert (Neg Aa) H) \<turnstile> A"
degrees_leD: fixes X :: "'b pdevs list"   and K :: "nat"   and x :: "'b pdevs" assumes "degrees X \<le> K"   and "x \<in> set X" shows "degree x \<le> K"
assms_1: shows "local_flow f UNIV UNIV \<phi>"
semiconfluentp_equivclp: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "semiconfluentp r" shows "equivclp r = r\<^sup>*\<^sup>* OO r\<inverse>\<inverse>\<^sup>*\<^sup>*"
square_free_factorization_smult: fixes c :: "'c"   and p :: "'c poly"   and d :: "'c"   and bs :: "('c poly \<times> nat) list" assumes "c \<noteq> (0::'c)"   and "square_free_factorization p (d, bs)" shows "square_free_factorization (smult c p) (c * d, bs)"
cut_def: fixes f :: "'a \<Rightarrow> 'b"   and R :: "('a \<times> 'a) set"   and x :: "'a" shows "cut f R x = (\<lambda>y. if (y, x) \<in> R then f y else undefined)"
ADD_1: shows "{#s#} + ce' = csp + ({#s#} + ce)"
ctr: fixes out :: "'d"   and c :: "('b, 'd, 'e) rpv" shows "Generative_Probabilistic_Value.Pause out c = GPV (return_spmf (IO out c))"
neutral_const: fixes A :: "'e set" shows "(\<Sum>_\<in>A. (0::'d)) = (0::'d)"
linear_compose_sub: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "linear (*a) (*b) f"   and "linear (*a) (*b) g" shows "linear (*a) (*b) (\<lambda>x. f x - g x)"
U_le_1: fixes s :: "'s"   and t :: "'s" shows "U s t \<le> 1"
cadjoint_id: shows "id\<^sup>\<dagger> = id"
subst_not: fixes \<sigma> :: "'b \<Rightarrow> 'c"   and P :: "'c upred" shows "\<sigma> \<dagger> (\<not> P) = (\<not> \<sigma> \<dagger> P)"
base: fixes \<psi> :: "'a stream \<Rightarrow> bool"   and \<omega> :: "'a stream"   and \<phi> :: "'a stream \<Rightarrow> bool" assumes "\<psi> \<omega>" shows "(\<phi> suntil \<psi>) \<omega>"
isomorphic_to_zero_left: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModuleIso G smult V smult' T W"   and "V = 0" shows "W = 0"
assms: shows "ops \<in> set (\<phi>\<^sub>P \<Psi> List_Supplement.embed \<psi>)"
sorted_nth_strict_mono: fixes xs :: "'b list"   and i :: "nat"   and j :: "nat" assumes "sorted xs"   and "distinct xs"   and "i < j"   and "j < length xs" shows "xs ! i < xs ! j"
le_supI: fixes a :: "'a \<times> 'b"   and x :: "'a \<times> 'b"   and b :: "'a \<times> 'b" assumes "less_eq_prod a x"   and "less_eq_prod b x" shows "less_eq_prod (sup_prod a b) x"
discrete: fixes y :: "'a"   and x :: "'a" assumes "discrete"   and "y \<le> x" shows "\<exists>k. x = ((+) (1::'a) ^^ k) y"
nrename_AndL2': fixes R :: "trm"   and x :: "name"   and y :: "name"   and v :: "name"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = AndL2 v.N u"   and "v \<sharp> (R, u, x, y)"   and "x \<noteq> y" shows "(\<exists>N'. R = AndL2 v.N' u \<and> N'[x\<turnstile>n>y] = N) \<or> (\<exists>N'. R = AndL2 v.N' x \<and> y = u \<and> N'[x\<turnstile>n>y] = N)"
SV_inv1_inductive_1: shows "init sv_TS \<subseteq> Vinv1"
IntI: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A"   and "c \<in> B" shows "c \<in> A \<inter> B"
isl_def: fixes sum :: "'e + 'f" shows "isl sum = (\<exists>x1. sum = Inl x1)"
f4: shows "\<forall>mo. Deliver (i, Expunge e mo i) \<notin> set pre"
is_unit_gauss_cnj: fixes z :: "gauss_int" assumes "is_unit z" shows "is_unit (gauss_cnj z)"
assms_2: fixes x :: "(real, 'm) vec" assumes "x \<in> \<Union> (range F)" shows "(g has_derivative g' x) (at x within \<Union> (range F))"
sp_Cons: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v1 :: "'d \<times> nat"   and e :: "('d \<times> nat) edge"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat" assumes "wf_sub_rel_of g subs" shows "SubRel.subpath g v1 (e # es) v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs) \<and> e \<in> edges g \<and> SubRel.subpath g (tgt e) es v2 subs)"
strips_equivalent_to_sas_plus_i_a_II_1: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops :: "('c, 'd) sas_plus_operator list"   and op' :: "('c \<times> 'd) strips_operator" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<forall>op\<in>set ops. op \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "op' \<in> set (map (sasp_op_to_strips \<Psi>) ops)" shows "op' \<in> set ((\<phi> \<Psi> )\<^sub>\<O>)"
assms_2: shows "Dom f = {} \<longrightarrow> Cod f = {}"
of_complex_mult_of_complex: fixes z1 :: "complex"   and z2 :: "complex" shows "of_complex z1 *\<^sub>h of_complex z2 = of_complex (z1 * z2)"
ex_comm_monoid_add_around_imageE: fixes f :: "'f \<Rightarrow> 'e"   and S :: "'f set"   and pls :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and A :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and pls' :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero' :: "'g"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool" assumes "\<exists>C. f ` S \<subseteq> C \<and> comm_monoid_add_ow C pls zero"   and "(A ===> A ===> A) pls pls'"   and "A zero zero'"   and "Domainp (rel_set B) S"   and "\<And>x. x \<in> S \<Longrightarrow> Domainp A (f x)" obtains C :: "'e set" where "comm_monoid_add_ow C pls zero" and "f ` S \<subseteq> C" and "Domainp (rel_set A) C"
reciprocal_mult: fixes P :: "'a poly"   and Q :: "'a poly"   and p :: "nat" assumes "degree (P * Q) \<le> p"   and "degree P \<le> p"   and "degree Q \<le> p" shows "monom (1::'a) p * reciprocal_poly p (P * Q) = reciprocal_poly p P * reciprocal_poly p Q"
hcomplex_hmod_hcnj: fixes z :: "complex star" shows "hcmod (hcnj z) = hcmod z"
foldD_equality: fixes A :: "'b set"   and y :: "'a"   and e :: "'a" assumes "(A, y) \<in> foldSetD D (\<cdot>) e"   and "e \<in> D"   and "A \<subseteq> B" shows "foldD D (\<cdot>) e A = y"
sublocale_sas_plus_finite_domain_representation_ii_4: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "\<forall>v\<in>dom (\<Psi>\<^sub>I\<^sub>+). the ((\<Psi>\<^sub>I\<^sub>+) v) \<in> \<R>\<^sub>+ \<Psi> v"
select_def: fixes xs :: "'b list" shows "select xs = local.range (natural_of_nat (length xs)) \<circ>\<rightarrow> (\<lambda>k. Pair (xs ! nat_of_natural k))"
is_unit_div_mult2_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one"   and "dvd.dvd times c one" shows "divide a (times b c) = divide (divide a b) c"
simps_6: fixes f1 :: "vname \<Rightarrow> 'a"   and f2 :: "sig \<Rightarrow> 'a"   and x2 :: "sig" shows "(case mid x2 of fid x \<Rightarrow> f1 x | mid x \<Rightarrow> f2 x) = f2 x2"
awpG: fixes X :: "'a set" assumes "X \<in> wens_set F B" shows "T \<inter> X - B \<subseteq> awp G (T \<inter> X)"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. T (f, \<I>)) local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. T (f, \<I>))"
vrestr_Union: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and A :: "'b set set" assumes "qmpt M T" shows "qmpt.vimage_restr M f (\<Union> A) = \<Union> (qmpt.vimage_restr M f ` A)"
fst_convol: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "fst \<circ> <f, g> = f"
bd_def: fixes x :: "'a"   and y :: "'a" shows "range_semiring.bd (\<lambda>x y. y \<cdot> x) domain_op x y = domain_op (x \<cdot> y)"
sum_list_closed: fixes as :: "'g list" assumes "order.greater_eq G (set as)" shows "sum_list as \<in> G"
induct: fixes x1 :: "pi"   and x2 :: "pi set"   and P :: "pi \<Rightarrow> pi set \<Rightarrow> bool" assumes "(x1, x2) \<in> sumComposeSet"   and "P \<zero> {}"   and "\<And>Q S P. \<lbrakk>Q \<in> S; (P, S - {Q}) \<in> sumComposeSet; P P (S - {Q})\<rbrakk> \<Longrightarrow> P (P \<oplus> Q) S" shows "P x1 x2"
is_max_flow_val_def: fixes fv :: "'capacity" shows "is_max_flow_val fv \<equiv> \<exists>f. isMaxFlow f \<and> fv = Flow.val c s f"
of_nat_neq_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_1.of_nat one plus zero (Suc n) \<noteq> zero"
simps_4: fixes r1 :: "'b rexp"   and r2 :: "'b rexp"   and c :: "'b"   and v1 :: "'b val"   and v2 :: "'b val" shows "injval (Times r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2"
coerce_sinl: fixes x :: "'g" shows "COERCE('g \<oplus> 'h, 'e \<oplus> 'f)\<cdot>(sinl\<cdot>x) = sinl\<cdot>(COERCE('g, 'e)\<cdot>x)"
closure_bounded_linear_image_subset: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "bounded_linear f" shows "f ` closure S \<subseteq> closure (f ` S)"
acyclic_plus_asymmetric: fixes x :: "'a" shows "pd_kleene_allegory_class.acyclic x = asymmetric (x\<^sup>+)"
cpI5: fixes f :: "('a state \<times> 'a state \<Rightarrow> 'b) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'c) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'd) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'e) \<Rightarrow> ('a state \<times> 'a state \<Rightarrow> 'f) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'g"   and N :: "('a state \<times> 'a state \<Rightarrow> 'h) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'b"   and P :: "('a state \<times> 'a state \<Rightarrow> 'h) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'c"   and Q :: "('a state \<times> 'a state \<Rightarrow> 'h) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'd"   and R :: "('a state \<times> 'a state \<Rightarrow> 'h) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'e"   and S :: "('a state \<times> 'a state \<Rightarrow> 'h) \<Rightarrow> 'a state \<times> 'a state \<Rightarrow> 'f" assumes "\<forall>V W X Y Z \<tau>. f V W X Y Z \<tau> = f (\<lambda>_. V \<tau>) (\<lambda>_. W \<tau>) (\<lambda>_. X \<tau>) (\<lambda>_. Y \<tau>) (\<lambda>_. Z \<tau>) \<tau>"   and "cp N"   and "cp P"   and "cp Q"   and "cp R"   and "cp S" shows "cp (\<lambda>X. f (N X) (P X) (Q X) (R X) (S X))"
weak_paretoD: fixes scf :: "('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> ('d \<times> 'd) set"   and A :: "'d set"   and Is :: "'c set"   and Pcond :: "'d set \<Rightarrow> 'c set \<Rightarrow> ('c \<Rightarrow> ('d \<times> 'd) set) \<Rightarrow> bool"   and P :: "'c \<Rightarrow> ('d \<times> 'd) set"   and x :: "'d"   and y :: "'d" assumes "weak_pareto scf A Is Pcond"   and "Pcond A Is P"   and "x \<in> A"   and "y \<in> A"   and "\<And>i. i \<in> Is \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<prec> y" shows "x \<^bsub>(scf P)\<^esub>\<prec> y"
execlass_ex_code: fixes rel :: "('a \<times> 'a) list"   and c :: "'a" shows "class_ex (set rel) c = execlass_ex rel c"
start_def: fixes p :: "proc" shows "start p \<equiv> s0"
insertion_irrelevant_vars: fixes p :: "'a mpoly"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>v. v \<in> vars p \<Longrightarrow> f v = g v" shows "insertion f p = insertion g p"
Inf_aform_aform_of_ivl: fixes l :: "'b"   and u :: "'b" assumes "l \<le> u" shows "Inf_aform (aform_of_ivl l u) = l"
inf_sup_aci_5: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "sup_prod x y = sup_prod y x"
edges_of_walk_empty_1: shows "edges_of_walk [] = {}"
wf_hd_neq_all: fixes ss :: "nat list list"   and w :: "nat list" assumes "wf_dom ss w"   and "w \<noteq> []" shows "ss ! hd w \<noteq> rev [0..<n]"
Iar: fixes a :: "nat"   and r :: "nat" shows "u (a * N + r) x \<le> u r x + (\<Sum>i<a. u N ((T ^^ (i * N + r)) x))"
containsPropagates: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "contains f iA n" shows "contains f iA (Suc n) \<or> considers f iA n"
finite_imp_ex_iso_nat_comp: assumes "finite (Collect J.arr)" shows "\<exists>C'. isomorphic_categories C' J.comp"
Tpows_v_V: shows "set Tpows_v \<subseteq> V"
approximating_bigstep_fun_seq_semantics: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs\<^sub>1 :: "'b rule list"   and s :: "state"   and t :: "state"   and rs\<^sub>2 :: "'b rule list" assumes "\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow>\<^sub>\<alpha> t" shows "approximating_bigstep_fun \<gamma> p (rs\<^sub>1 @ rs\<^sub>2) s = approximating_bigstep_fun \<gamma> p rs\<^sub>2 t"
HomFtorObj: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>" shows "(Hom\<^bsub>C\<^esub>[X,\<midarrow>]) @@ Y = Hom\<^bsub>C\<^esub> X Y"
PRes: fixes evsPRes :: "event list"   and KP :: "nat"   and M :: "agent"   and i :: "nat"   and Transaction :: "msg"   and C :: "agent"   and OrderDesc :: "nat"   and PurchAmt :: "nat"   and P :: "agent"   and LID_M :: "nat"   and XID :: "nat"   and Chall_C :: "nat"   and KM :: "nat"   and OIData :: "msg"   and HOD :: "msg"   and P_I :: "msg" assumes "evsPRes \<in> set_pur"   and "KP \<in> symKeys"   and "M = Merchant i"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "Gets M (EncB (priSK P) KP (pubEK M) \<lbrace>Number LID_M, Number XID, Number PurchAmt\<rbrace> authCode) \<in> set evsPRes"   and "Gets M \<lbrace>Number LID_M, Nonce Chall_C\<rbrace> \<in> set evsPRes"   and "Says M P (EncB (priSK M) KM (pubEK P) \<lbrace>Number LID_M, Number XID, Hash OIData, HOD\<rbrace> P_I) \<in> set evsPRes"   and "Notes M \<lbrace>Number LID_M, Agent P, Transaction\<rbrace> \<in> set evsPRes" shows "Says M C (sign (priSK M) \<lbrace>Number LID_M, Number XID, Nonce Chall_C, Hash (Number PurchAmt)\<rbrace>) # evsPRes \<in> set_pur"
h: shows "abc_notfinal (abc_steps_l (0, lm) A (Suc n)) A"
add_numeral_left: fixes v :: "num"   and w :: "num"   and z :: "'c" shows "numeral v + (numeral w + z) = numeral (v + w) + z"
dom_char: fixes f :: "'a" shows "Right.dom f = (if Right.arr f then local.dom f else null)"
p1_6: shows "\<Union> {K. \<exists>x. (x, K) \<in> p1} = s1"
dvd_div_mult: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and c :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times c b" shows "times (divide b c) a = divide (times b a) c"
intros_13: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' c Q',A' \<and> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
left_neutral: fixes a :: "'c" shows "(1::'c) * a = a"
redT_updT'_eq_None_conv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ts' :: "'f \<Rightarrow> ('i \<times> 'j \<Rightarrow>f nat) option"   and ta :: "('f, 'i, 'k) new_thread_action"   and t :: "'f" assumes "\<And>t. (ts t = None) = (ts' t = None)" shows "(redT_updT' ts ta t = None) = (redT_updT ts' ta t = None)"
decrnum: fixes t :: "MIR.num"   and x :: "real"   and bs :: "real list" assumes "numbound0 t" shows "Inum (x # bs) t = Inum bs (decrnum t)"
unit_factor_snd_normalize_quot: fixes x :: "'a \<times> 'a" shows "unit_factor (snd (normalize_quot x)) = (1::'a)"