hcju/isabelleps huggingface dataset:
has_field_derivative_iff: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and S :: "'a set" shows "(f has_field_derivative D) (at x within S) = ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
simps_1: fixes uu :: "('f \<times> 'f) set"   and uv :: "'e \<Rightarrow> 'f"   and uw :: "'f" shows "c_ipurge uu uv uw [] = []"
has_bochner_integral_bounded_linear: fixes T :: "'c \<Rightarrow> 'd"   and M :: "'e measure"   and f :: "'e \<Rightarrow> 'c"   and x :: "'c" assumes "bounded_linear T"   and "has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. T (f x)) (T x)"
Hrel_def: fixes R1 :: "'m \<Rightarrow> 'q \<Rightarrow> bool"   and R2 :: "'n \<Rightarrow> 'r \<Rightarrow> bool" shows "Hrel R1 R2 = (BNF_Def.Grp (Hin {(x, y). R1 x y} {(x, y). R2 x y}) (Hmap fst fst))\<inverse>\<inverse> OO BNF_Def.Grp (Hin {(x, y). R1 x y} {(x, y). R2 x y}) (Hmap snd snd)"
vsv_vunion_app_right: fixes r :: "V"   and s :: "V"   and x :: "V" assumes "vsv r"   and "vsv s"   and "vdisjnt (\<D>\<^sub>\<circ> r) (\<D>\<^sub>\<circ> s)"   and "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" shows "(r \<union>\<^sub>\<circ> s)\<lparr>x\<rparr> = s\<lparr>x\<rparr>"
arg1_vsubset_Vfrom: fixes a :: "V"   and i :: "V" shows "a \<subseteq>\<^sub>\<circ> Vfrom a i"
sum: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and A :: "'e set" assumes "Modules.additive f" shows "f (sum g A) = (\<Sum>x\<in>A. f (g x))"
stable_insort_equiv: fixes cmp :: "'b comparator"   and y :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "compare cmp y x = Equiv" shows "filter (\<lambda>y. compare cmp y y = Equiv) (Sorting_Algorithms.insort cmp x xs) = x # filter (\<lambda>y. compare cmp y y = Equiv) xs"
ee_wfactorsD: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "essentially_equal G as bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a \<sim> b"
space_distr: fixes M :: "'c measure"   and N :: "'b measure"   and f :: "'c \<Rightarrow> 'b" shows "space (distr M N f) = space N"
nth_default_eq_dflt_iff: fixes dflt :: "'a"   and xs :: "'a list"   and k :: "nat" shows "(nth_default dflt xs k = dflt) = (k < length xs \<longrightarrow> xs ! k = dflt)"
subseq_of_length0: fixes xs :: "'c list"   and ys :: "'c list" shows "subseq_of_length 0 xs ys = (ys = [])"
scale_zero_left: fixes x :: "('a, 'b) vec" shows "(0::'a) *s x = 0"
uminus_cvec_def: fixes z :: "complex \<times> complex" shows "~\<^sub>v z = (let (z1, z2) = z in (- z1, z2))"
aux_lemma: fixes b :: "'a"   and n :: "nat" assumes "b \<in> carrier R" shows "local.eval (map ((\<otimes>) b) q @ replicate n \<zero>) a = local.eval (local.monom b n) a \<otimes> local.eval q a"
finiteFinestpart: fixes X :: "'a set" shows "finite (finestpart X) = finite X"
mod_mult_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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times (modulo a c) (modulo b c)) c = modulo (times a b) c"
times_hmultiset_def: fixes A :: "hmultiset"   and B :: "hmultiset" shows "A * B = HMSet {#x + y. (x, y) \<in># hmsetmset A \<times># hmsetmset B#}"
gt_imp_vars_mset: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "t >\<^sub>t s" shows "vars_mset s \<subseteq># vars_mset t"
inL2': fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "F, G, H, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "G, H, F, \<Gamma> \<Rightarrow> \<Delta> \<down> n"
pred_inject_2: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" shows "pred_option P (Some a) = P a"
splus_def: fixes t :: "'a"   and v :: "'t" shows "t \<oplus> v = term_of_pair (t + pp_of_term v, component_of_term v)"
simps_1: fixes s :: "('a, 'b) tm" shows "apps s [] = s"
u_nn: shows "AE x in M. 0 \<le> u x"
dist_eq_0_iff: fixes dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.metric_space dist uniformity open" shows "(dist x y = 0) = (x = y)"
lex_pm_lin: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "lex_pm s t \<or> lex_pm t s"
isDiagonal_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "isDiagonal A = (\<forall>a b. mod_type_class.to_nat a \<noteq> mod_type_class.to_nat b \<longrightarrow> A $ a $ b = (0::'d))"
weak_supplementation: fixes x :: "i"   and y :: "i" assumes "PP x y" shows "\<exists>z. P z y \<and> \<not> O z x"
invc2I: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "invc t" shows "invc2 t"
restrict_extensional_sub: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "A \<subseteq> B" shows "restrict f A \<in> extensional B"
extensional_funcset_ext: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> A \<rightarrow>\<^sub>E B"   and "g \<in> A \<rightarrow>\<^sub>E B"   and "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "f = g"
nth_least_not_contains: fixes n :: "nat"   and A :: "nat set"   and k :: "nat" assumes "enat (Suc n) < esize A"   and "nth_least A n < k"   and "k < nth_least A (Suc n)" shows "k \<notin> A"
D_def: fixes n :: "nat"   and fs :: "int vec list" shows "fs_int.D n fs = nat (prod (fs_int.d n fs) {..<length fs})"
simps_158: fixes x7 :: "'d" shows "set_seq_action (receive x7) = {x7}"
normalize_gcd: fixes a :: "'b"   and b :: "'b" shows "normalize (normalization_euclidean_semiring_class.gcd a b) = normalization_euclidean_semiring_class.gcd a b"
assms_2: fixes x :: "'a" shows "norm (f x) \<le> C"
assms_2: shows "M \<turnstile>\<^sub>c t"
real_differentiable_def: fixes f :: "real \<Rightarrow> real"   and x :: "real"   and s :: "real set" shows "(f differentiable at x within s) = (\<exists>D. (f has_real_derivative D) (at x within s))"
devBC_some: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cmp :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and pin :: "'b \<Rightarrow> 'a list set"   and pout :: "'b \<Rightarrow> 'a list"   and bc :: "'b \<Rightarrow> 'a list"   and mining :: "'b \<Rightarrow> bool"   and honest :: "'a \<Rightarrow> bool"   and cb :: "nat"   and nid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "Blockchain active cmp pin pout bc mining honest cb"   and "active nid (t n)" shows "Blockchain.devBC active cmp bc mining t n nid n = Some nid"
distinct_11: fixes x7 :: "'a OCL_Types.type" shows "OclSuper \<noteq> Bag x7"
modifies_join: fixes vs\<^sub>1 :: "char list set"   and a :: "char list \<Rightarrow> int \<Rightarrow> int"   and b :: "char list \<Rightarrow> int \<Rightarrow> int"   and vs\<^sub>2 :: "char list set" assumes "modifies vs\<^sub>1 a b"   and "modifies vs\<^sub>2 a b" shows "modifies (vs\<^sub>1 \<inter> vs\<^sub>2) a b"
inner_prod_cnj: fixes u :: "complex vec"   and v :: "complex vec" assumes "dim_vec u = dim_vec v" shows "\<langle>v|u\<rangle> = cnj \<langle>u|v\<rangle>"
vec_module: fixes n :: "nat" shows "module class_ring (module_vec TYPE('b) n)"
cone_elem_sum: fixes Vs :: "'b vec set"   and n :: "nat"   and x :: "'b vec"   and y :: "'b vec" assumes "Vs \<subseteq> carrier_vec n"   and "x \<in> gram_schmidt.cone n Vs"   and "y \<in> gram_schmidt.cone n Vs" shows "x + y \<in> gram_schmidt.cone n Vs"
add_decreasing2: fixes c :: "'b multiset"   and a :: "'b multiset"   and b :: "'b multiset" assumes "c \<subseteq># {#}"   and "a \<subseteq># b" shows "a + c \<subseteq># b"
infinite_iff_card_of_nat: fixes A :: "'b set" shows "infinite A = ordLeq3 (card_of UNIV) (card_of A)"
sarp_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>x y. (x, y) \<in> (rsp_on A f)\<^sup>+ \<Longrightarrow> (y, x) \<notin> rwp_on A f" shows "sarp_on A f"
irreflexive_linorder_matrix_1: fixes f :: "'c \<times> 'c \<Rightarrow> 'd"   and i :: "'c" assumes "matrix_stone_relation_algebra.irreflexive f" shows "f (i, i) = bot"
open_openin: fixes S :: "'a set" shows "open S = openin euclidean S"
D4_3: fixes F :: "\<Pi>\<^sub>3" shows "d\<^sub>3 (\<^bold>\<lambda>\<^sup>3 (\<lambda>x y z. \<lparr>F,x\<^sup>P,y\<^sup>P,z\<^sup>P\<rparr>)) = d\<^sub>3 F"
eval_S: fixes x :: "nat" shows "eval S [x] \<down>= Suc x"
less_imp_not_less: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.preorder less_eq less"   and "less x y" shows "(\<not> less y x) = True"
rel_with_target_impl_transC_TRel_is_weak_reduction_contrasimulation: fixes Rel :: "(('procS, 'procT) Proc \<times> ('procS, 'procT) Proc) set"   and TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_contrasimulation Rel (STCal Source Target)"   and "\<forall>T1 T2. (T1, T2) \<in> TRel \<longrightarrow> (TargetTerm T1, TargetTerm T2) \<in> Rel"   and "\<forall>T1 T2. (TargetTerm T1, TargetTerm T2) \<in> Rel \<longrightarrow> (T1, T2) \<in> TRel\<^sup>+" shows "weak_reduction_contrasimulation (TRel\<^sup>+) Target"
ide_char: fixes a :: "'a" shows "hhom\<^sub>C.ide a = (hhom\<^sub>C.arr a \<and> C.ide a)"
close: fixes u :: "'a"   and v :: "'a" assumes "cbox u v \<in> \<D>" shows "\<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n"
singleton: fixes x :: "'a" shows "\<Sum>\<^sub># {#x#} = x"
image_zet_rep: fixes A :: "'b set"   and g :: "'b \<Rightarrow> ZF" assumes "A \<in> zet" shows "\<exists>z. g ` A = explode z"
Lli_def: fixes i :: "nat" shows "Lli i = (SOME x. (prems_of \<iota> ! i, x) \<in> NL)"
toplevel_summands_ACI_norm: fixes r :: "'b rexp" shows "toplevel_summands \<guillemotleft>r\<guillemotright> = ACI_norm ` toplevel_summands r"
Abs_preserves_good: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort"   and x :: "'var" assumes "good X" shows "goodAbs (Abs xs x X)"
map_disc_iff: fixes f :: "'m \<Rightarrow> 'l"   and a :: "'m list" shows "(map f a = []) = (a = [])"
assms_5: shows "\<forall>\<^sub>F x in F. 0 \<le> g x"
emeasure_measure_pmf_not_zero: fixes p :: "'c pmf"   and s :: "'c set" assumes "set_pmf p \<inter> s \<noteq> {}" shows "emeasure (measure_pmf p) s \<noteq> 0"
Liminf_llist_def: fixes Xs :: "'b set llist" shows "Liminf_llist Xs = (\<Union>i\<in>{i. enat i < llength Xs}. \<Inter> (lnth Xs ` {j. i \<le> j \<and> enat j < llength Xs}))"
dropWhile_eq_drop: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "dropWhile P xs = drop (length (takeWhile P xs)) xs"
pairwise_trivial: fixes I :: "'c set" shows "pairwise (\<lambda>i j. j \<noteq> i) I"
allDefs_disjoint_inv: fixes n :: "'node"   and m :: "'node" assumes "n \<in> set (\<alpha>n g)"   and "m \<in> set (\<alpha>n g)"   and "n \<noteq> m" shows "step.allDefs g n \<inter> step.allDefs g m = {}"
assms: shows "(M, {#\<alpha>#}) \<in> mul_eq r"
binomial_ngt_0: fixes n :: "nat" shows "n choose Suc n = 0"
derived_set_of_subset_closure_of: fixes X :: "'c topology"   and S :: "'c set" shows "X derived_set_of S \<subseteq> X closure_of S"
simps_2: fixes n :: "'c"   and ps :: "('b, 'c) pform list" shows "closed (Fun n ps) = list_all closed ps"
is_nextElem_or1: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a"   and x :: "'a"   and y :: "'a" assumes "pre_between vs ram1 ram2"   and "is_nextElem vs x y"   and "before vs ram1 ram2" shows "is_sublist [x, y] (ram1 # between vs ram1 ram2 @ [ram2]) \<or> is_sublist [x, y] (ram2 # between vs ram2 ram1 @ [ram1])"
rep_eq: fixes x :: "'a vector"   and xa :: "'a vector" shows "(x < xa) = (Rep_vector x < Rep_vector xa)"
lcnj: fixes \<phi>s :: "'fmla list" assumes "set \<phi>s \<subseteq> fmla" shows "lcnj \<phi>s \<in> fmla"
beta1_def: shows "beta1 \<equiv> $pc1 = #b \<and> pc1$ = #g \<and> x$ = Suc<$x> \<and> Unchanged (y, sem, pc2)"
is_fair_run_def: fixes s :: "nat \<Rightarrow> 'c" shows "C.is_fair_run s \<equiv> \<exists>l. C.is_lrun l s \<and> C.is_lfair l s"
dirichlet_prod_assoc: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and h :: "nat \<Rightarrow> 'b" shows "dirichlet_prod (dirichlet_prod f g) h = dirichlet_prod f (dirichlet_prod g h)"
list_of_vec_vec: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" shows "list_of_vec (vec n f) = map f [0..<n]"
valid_Await: fixes c :: "('a, 'b, 'c) com"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and F :: "'c set"   and P :: "'a set"   and b :: "'a set"   and Q :: "'a set"   and A :: "'a set" assumes "atom_com c"   and "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> (P \<inter> b) c Q, A" shows "\<Gamma> \<Turnstile>\<^bsub>/F\<^esub> P Await b c Q, A"
wlp_assign_idx_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and i :: "aexp"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> (x[i] ::= a) Q s = Q (s(x := (s x)(aval i s := aval a s)))"
apex_composite: shows "SPN_fgh.\<nu>\<pi>.apex = Maps.pbdom SPN_fgh.\<nu>.leg0 SPN_fgh.\<pi>.leg1"
indicator_simps_2: fixes x :: "'b"   and S :: "'b set" assumes "x \<notin> S" shows "indicator S x = (0::'d)"
is_Inf_def: fixes A :: "'b set"   and inf :: "'b" shows "is_Inf A inf = ((\<forall>x\<in>A. inf \<sqsubseteq> x) \<and> (\<forall>z. (\<forall>x\<in>A. z \<sqsubseteq> x) \<longrightarrow> z \<sqsubseteq> inf))"
cert_A_form: fixes A :: "agent"   and NA :: "msg"   and B :: "agent"   and K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt (shrK A) \<lbrace>NA, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "A \<notin> bad"   and "evs \<in> ns_shared" shows "K \<notin> range shrK \<and> X = Crypt (shrK B) \<lbrace>Key K, Agent A\<rbrace>"
coeff_inject: fixes x :: "'d poly"   and y :: "'d poly" shows "(coeff x = coeff y) = (x = y)"
simps_1: fixes s :: "char list \<Rightarrow> int" shows "postQ SKIP s = s"
insertI2: fixes a :: "'g"   and B :: "'g set"   and b :: "'g" assumes "a \<in> B" shows "a \<in> insert b B"
RECT_rule_arb: fixes body :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"   and V :: "('a \<times> 'a) set"   and pre :: "'c \<Rightarrow> 'a \<Rightarrow> bool"   and arb :: "'c"   and x :: "'a"   and M :: "'c \<Rightarrow> 'a \<Rightarrow> 'b" assumes "trimono body"   and "wf V"   and "pre arb x"   and "\<And>f arb x. \<lbrakk>\<And>arb' x'. \<lbrakk>pre arb' x'; (x', x) \<in> V\<rbrakk> \<Longrightarrow> f x' \<le> M arb' x'; pre arb x; REC\<^sub>T body = f\<rbrakk> \<Longrightarrow> body f x \<le> M arb x" shows "REC\<^sub>T body x \<le> M arb x"
set_sel_1: fixes a :: "'d stream" shows "shd a \<in> sset a"
inv_delete: fixes join :: "('b \<times> 'c) tree \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) tree \<Rightarrow> ('b \<times> 'c) tree"   and inv :: "('b \<times> 'c) tree \<Rightarrow> bool"   and t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Set2_Join join inv"   and "inv t" shows "inv (Set2_Join.delete join x t)"
plus_const_cases: fixes a1 :: "const"   and a2 :: "const" shows "plus_const a1 a2 = (case (a1, a2) of (Const m, Const n) \<Rightarrow> Const (m + n) | (Const m, Any) \<Rightarrow> Any | (Any, b) \<Rightarrow> Any)"
dfa_axioms: shows "dfa \<lparr>dfa.states = dfa.states MS, init = dfa.init MS, final = dfa.states MS - dfa.final MS, nxt = dfa.nxt MS\<rparr>"
IH: fixes ys :: "'a list" assumes "length xs = length ys" shows "vec_of_list xs \<bullet> vec_of_list ys = sum_list (map2 (*) xs ys)"
inner: shows "fBall rs (\<lambda>(uu_, irs). Rewriting_Nterm.arity_compatibles irs \<and> is_fmap irs \<and> patterns_compatibles irs \<and> irs \<noteq> {||} \<and> fBall irs (\<lambda>(pats, rhs). linears pats \<and> abs_ish pats rhs \<and> closed_except rhs (freess pats) \<and> fdisjnt (freess pats) all_consts \<and> pre_strong_term_class.wellformed rhs \<and> \<not> shadows_consts rhs \<and> consts rhs |\<subseteq>| all_consts))"
assms: shows "fmdom' m\<^sub>1 \<inter> fmdom' m\<^sub>2 = {}"
diff_chain_at: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and g :: "'b \<Rightarrow> 'c"   and g' :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') (at x)"   and "(g has_derivative g') (at (f x))" shows "(g \<circ> f has_derivative g' \<circ> f') (at x)"
C12: 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 (times x y) = plus one (times (times x (star (times y x))) y)"
assms_2: shows "\<phi> \<in> set \<phi>s"
in_trancl_imp: fixes subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set"   and v :: "'c \<times> nat"   and v' :: "'c \<times> nat" assumes "wf_sub_rel subs"   and "(v, v') \<in> subs\<^sup>+" shows "(v, v') \<in> subs"
S1: shows "(s * (s + 1) + 2 * (s + 1)) div 2 \<le> t * (t + 1) div 2"
listToFun_def: fixes xs :: "('c \<times> 'd list) list" shows "listToFun xs \<equiv> foldr (\<lambda>(a, acts) M. concat (map (\<lambda>m. map (fun_upd m a) acts) M)) xs [\<lambda>_. undefined]"
rel_fun_curry: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and C :: "'g \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h \<Rightarrow> 'i" shows "rel_fun A (rel_fun B C) f g = rel_fun (rel_prod A B) C (\<lambda>(x, y). f x y) (\<lambda>(x, y). g x y)"
div_poly_less: fixes x :: "'a poly"   and y :: "'a poly" assumes "degree x < degree y" shows "x div y = 0"
strongdlHPPB_NDCIDCaux: fixes d :: "'d"   and PP :: "nat set"   and R :: "(('exp, 'id) MWLsCom list \<times> ('exp, 'id) MWLsCom list) set"   and V :: "('exp, 'id) MWLsCom list"   and V' :: "('exp, 'id) MWLsCom list"   and i :: "nat" assumes "SdlHPPB d PP R"   and "(V, V') \<in> R"   and "i < length V" shows "NDC d (V ! i) \<or> IDC d (V ! i) (htchLoc (pp (V ! i)))"
restriction_clm_leq_one: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "|restrict v (clm ts) c| \<le> 1"
lr_r2l_strict: fixes R :: "('a \<Rightarrow> ValD) set"   and fs :: "'a \<Rightarrow> ValD \<rightarrow> ValD" assumes "logical_relation R"   and "\<forall>xs\<in>R. (\<lambda>j. strictify\<cdot>(fs j)\<cdot>(xs j)) \<in> R" shows "(\<lambda>i. ValF\<cdot>(strictify\<cdot>(fs i))) \<in> R"
try_bind_assert_gpv: fixes b :: "bool"   and f :: "unit \<Rightarrow> ('b, 'c, 'd) gpv"   and gpv :: "('b, 'c, 'd) gpv" shows "TRY assert_gpv b \<bind> f ELSE gpv = (if b then TRY f () ELSE gpv else gpv)"
Exec_Equal_StateD: fixes localState :: "'d \<Rightarrow> 'e"   and trans_fun :: "'f message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and c1 :: "'d"   and c2 :: "'d"   and m :: "'f message_af" assumes "Exec_Equal_State localState trans_fun"   and "localState c1 = localState c2" shows "trans_fun m c1 = trans_fun m c2"
integrable_I0i_1_div_plus_square: shows "interval_lebesgue_integrable lborel 0 \<infinity> (\<lambda>x. 1 / (1 + x\<^sup>2))"
pair_loopfree_digraph_axioms: shows "pair_loopfree_digraph slim"
prf_adversary_def: fixes \<A> :: "((bool list \<times> bool list) \<times> 'b, bool list, bool list \<times> bool list) gpv \<times> (bool list \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, bool list, bool list \<times> bool list) gpv)" shows "prf_adversary \<A> = (let (\<A>1, \<A>2) = \<A> in inline prf_encrypt_oracle \<A>1 () \<bind> (\<lambda>(((p1, p2), \<sigma>), n). if valid_plain p1 \<and> valid_plain p2 then lift_spmf coin_spmf \<bind> (\<lambda>b. let pb = if b then p1 else p2 in lift_spmf (spmf_of_set (nlists UNIV len)) \<bind> (\<lambda>r. Generative_Probabilistic_Value.Pause r Generative_Probabilistic_Value.Done \<bind> (\<lambda>pad. let c = (r, pb [\<oplus>] pad) in inline prf_encrypt_oracle (\<A>2 c \<sigma>) n \<bind> (\<lambda>(b', uu_). Generative_Probabilistic_Value.Done (b' = b))))) else lift_spmf coin_spmf))"
Rxa_mult_smaller: fixes R :: "('e, 'f) Ring_scheme"   and a :: "'e"   and b :: "'e" assumes "Ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "R \<diamondsuit>\<^sub>p (a \<cdot>\<^sub>r\<^bsub>R\<^esub> b) \<subseteq> R \<diamondsuit>\<^sub>p b"
globally_sound_modes_invariant: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "globally_sound_mode_use (cms, mem)"   and "(cms, mem) \<rightarrow> (cms', mem')" shows "globally_sound_mode_use (cms', mem')"
not_one_le_zero: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
simps: fixes N :: "('a literal multiset \<times> nat) multiset"   and P :: "('a literal multiset \<times> nat) multiset"   and Q :: "('a literal multiset \<times> nat) multiset"   and n :: "nat" shows "n_of_wstate (N, P, Q, n) = n"
conj_subset_def: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" shows "(A \<subseteq> {x. P x \<and> Q x}) = (A \<subseteq> {x. P x} \<and> A \<subseteq> {x. Q x})"
postdominate_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 n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_node sourcenode targetnode valid_edge n"   and "\<not> CFGExit.method_exit sourcenode kind valid_edge Exit n" shows "Postdomination.postdominate sourcenode targetnode kind valid_edge Exit n n"
simps_87: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'c \<Rightarrow> 'b"   and f4 :: "'c \<Rightarrow> 'b"   and f5 :: "'c ltlr \<Rightarrow> 'c ltlr \<Rightarrow> 'b"   and f6 :: "'c ltlr \<Rightarrow> 'c ltlr \<Rightarrow> 'b"   and f7 :: "'c ltlr \<Rightarrow> 'b"   and f8 :: "'c ltlr \<Rightarrow> 'c ltlr \<Rightarrow> 'b"   and f9 :: "'c ltlr \<Rightarrow> 'c ltlr \<Rightarrow> 'b"   and x81 :: "'c ltlr"   and x82 :: "'c ltlr" shows "(case x81 U\<^sub>r x82 of true\<^sub>r \<Rightarrow> f1 | false\<^sub>r \<Rightarrow> f2 | prop\<^sub>r(x) \<Rightarrow> f3 x | nprop\<^sub>r(x) \<Rightarrow> f4 x | x and\<^sub>r xa \<Rightarrow> f5 x xa | x or\<^sub>r xa \<Rightarrow> f6 x xa | X\<^sub>r x \<Rightarrow> f7 x | x U\<^sub>r xa \<Rightarrow> f8 x xa | x R\<^sub>r xa \<Rightarrow> f9 x xa) = f8 x81 x82"
thetaSD_Sretr: shows "thetaSD \<subseteq> Sretr thetaSD"
ivl_subset_existence_ivl: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> existence_ivl0 x0" shows "{t0..t} \<subseteq> (+) t0 ` existence_ivl0 x0"
compose_eq: fixes x :: "'c"   and A :: "'c set"   and g :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e" assumes "x \<in> A" shows "(g \<circ> f \<down> A) x = g (f x)"
sets_stream_space_eq: fixes M :: "'b measure" shows "sets (stream_space M) = sets (\<Squnion>i. vimage_algebra (streams (space M)) (\<lambda>s. s !! i) M)"