hcju/isabelleps huggingface dataset:
enc_and_TRel_impl_indRelRTPO_respects_barbs: fixes TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>S x. (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x)) \<and> (\<forall>S x. (\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x) \<longrightarrow> (\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x))"   and "rel_preserves_binary_pred TRel (\<lambda>a. hasBarb a TWB)"   and "rel_reflects_binary_pred TRel (\<lambda>b. hasBarb b TWB)" shows "rel_respects_binary_pred (indRelRTPO TRel) (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
prefixeq_butlast: fixes xs :: "'a list" shows "prefix (butlast xs) xs"
hyps_3: shows "\<alpha> = y \<or> is_sentence \<alpha> \<and> is_sentence y"
subspace_add: fixes S :: "'b set"   and x :: "'b"   and y :: "'b" assumes "local.subspace S"   and "x \<in> S"   and "y \<in> S" shows "x + y \<in> S"
ins_list_idem_eq_isin: fixes xs :: "'b list"   and x :: "'b" assumes "sorted_less xs" shows "(x \<in> set xs) = (ins_list x xs = xs)"
abs_e1: shows "\<bar>e1\<bar> \<le> snd (trunc_bound_eucl p (alpha * fst X))"
spec_ops_ref_less: fixes xs :: "('b \<times> 'c) list"   and deps :: "'c \<Rightarrow> 'b set"   and oid :: "'b"   and oper :: "'c"   and r :: "'b" assumes "spec_ops xs deps"   and "(oid, oper) \<in> set xs"   and "r \<in> deps oper" shows "r < oid"
T: fixes u :: "real"   and x :: "nat \<Rightarrow> real" assumes "0 \<le> u"   and "u \<le> 1"   and "x \<in> S" shows "(\<lambda>i. (1 - u) * v i + u * x i) \<in> T"
simps_2: fixes a :: "'b"   and x :: "'b"   and xs :: "'b list" shows "find_first a (x # xs) = (if x = a then 0 else Suc (find_first a xs))"
mset_heap_Some: fixes hp :: "'b hp" shows "mset_heap (Some hp) = mset_hp hp"
listE_length: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set" assumes "xs \<in> list n A" shows "length xs = n"
fold_commute: fixes xs :: "'g list"   and h :: "'h \<Rightarrow> 'i"   and g :: "'g \<Rightarrow> 'h \<Rightarrow> 'h"   and f :: "'g \<Rightarrow> 'i \<Rightarrow> 'i" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<circ> g x = f x \<circ> h" shows "h \<circ> fold g xs = fold f xs \<circ> h"
dg_Rel_Cod_app: fixes \<alpha> :: "V"   and x :: "V" assumes "arr_Rel \<alpha> x" shows "dg_Rel \<alpha>\<lparr>Cod\<rparr>\<lparr>x\<rparr> = x\<lparr>ArrCod\<rparr>"
eq: shows "\<integral>\<^sup>+ x. ennreal (norm (real_cond_exp M Invariants f x)) \<partial>M = \<integral>\<^sup>+ x. ennreal (norm (f x)) \<partial>M"
init\<pi>_def: shows "Prim2.init\<pi> \<equiv> Map.empty"
chamber_D0: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0" shows "SimplicialComplex.maxsimp X (g ` C0)"
filter_inter: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> E"   and "B \<in> E" shows "A \<inter> B \<in> E"
sub3_preserves_dimensions_2: fixes d :: "nat"   and l :: "nat"   and r :: "'a"   and A :: "'a mat"   and r' :: "'a"   and A' :: "'a mat" assumes "local.sub3 d l (r, A) = (r', A')" shows "dim_col A' = dim_col A"
bit_mask_iff: fixes m :: "nat"   and n :: "nat" shows "bit ((2::'a) ^ m - (1::'a)) n = ((2::'a) ^ n \<noteq> (0::'a) \<and> n < m)"
simps_2: fixes C :: "char list"   and e :: "nat exp" shows "compE\<^sub>2 (Cast C e) = compE\<^sub>2 e @ [Checkcast C]"
c_drop_is_pr: shows "c_drop \<in> PrimRec2"
null_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool" assumes "subcategory C Arr" shows "partial_magma.null (subcategory.comp C Arr) = partial_magma.null C"
Special_Solutions_in_Solutions: fixes a :: "nat list"   and b :: "nat list"   and x :: "nat list \<times> nat list" assumes "hlde a b"   and "x \<in> hlde.Special_Solutions a b" shows "x \<in> Linear_Diophantine_Equations.Solutions a b"
simps_37: fixes vd :: "bexp"   and ve :: "'a acom"   and vf :: "'a acom"   and vg :: "'a"   and v :: "'a"   and va :: "bexp"   and vb :: "'a acom"   and vc :: "'a" shows "(IF vd THEN ve ELSE vf {vg} \<sqsubseteq> {v} WHILE va DO vb {vc}) = False"
matrix_inv_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "A\<^sup>-\<^sup>1 = (SOME A'. A ** A' = mat (1::'d) \<and> A' ** A = mat (1::'d))"
antipar_1: shows "trg\<^sub>D (F g) = src\<^sub>D (F f)"
elims_2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and x :: "symbol list"   and xa :: "(nat \<times> symbol \<times> symbol list) list"   and xb :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "CFG.Derivation \<NN> \<TT> \<RR> x xa xb" obtains a :: "symbol list" and b :: "symbol list" where "x = aa__" and "xa = []" and "xb = ba__" and "aa__ = ba__"   | a :: "symbol list" and d :: "nat \<times> symbol \<times> symbol list" and D :: "(nat \<times> symbol \<times> symbol list) list" and b :: "symbol list" where "x = a" and "xa = d # D" and "xb = b" and "\<exists>x. CFG.Derives1 \<NN> \<TT> \<RR> a (fst d) (snd d) x \<and> CFG.Derivation \<NN> \<TT> \<RR> x D b"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>n. P (At n)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (fm.And p q)"   and "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (fm.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>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (fm.And p q))"   and "\<And>p q. \<lbrakk>P (NOT p); P (NOT q)\<rbrakk> \<Longrightarrow> P (NOT (fm.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 "\<And>v. P (NOT (At v))" shows "P a0"
init_subsumed: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "RedBlack prb" shows "init_conf prb \<sqsubseteq> confs prb (root (red prb))"
stval_def: shows "stval = (\<lambda>s. {p \<in> A. p s})"
valuate_valuate_rat: fixes lp :: "linear_poly"   and vl :: "nat \<Rightarrow> rat" shows "lp \<lbrace> \<lambda>v. QDelta (vl v) 0 \<rbrace> = QDelta lp \<lbrace> vl \<rbrace> 0"
oalist_eq_rawD: fixes rep_key_order :: "'a \<Rightarrow> 'b key_order"   and ys :: "('b \<times> 'c) list \<times> 'a"   and xs :: "('b \<times> 'c) list \<times> 'a" assumes "oalist_raw.oalist_inv rep_key_order ys"   and "oalist_raw.oalist_eq_raw rep_key_order xs ys" shows "oalist_raw.lookup_raw rep_key_order xs = oalist_raw.lookup_raw rep_key_order ys"
tcopy_end_def: shows "tcopy_end \<equiv> [(L, 0), (R, 2), (W1, 3), (L, 4), (R, 2), (R, 2), (L, 5), (W0, 4), (R, 0), (L, 5)]"
sigma_finite_stock_measure: fixes t :: "pdf_type" shows "sigma_finite_measure (stock_measure t)"
isMinimalBackjumpLevel_def: fixes level :: "nat"   and l :: "Literal"   and c :: "Literal list"   and M :: "(Literal \<times> bool) list" shows "isMinimalBackjumpLevel level l c M \<equiv> isBackjumpLevel level l c M \<and> (if set c \<noteq> {l} then \<exists>ll. ll el c \<and> elementLevel (opposite ll) M = level else level = 0)"
inner_zero_right: fixes x :: "('a, 'rows) vec" shows "inner_rows x 0 = (0::'a)"
wadjust_goon_left_moving_B_Bk_Oc: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" assumes "wadjust_on_left_moving_O m rs (c, Oc # list)"   and "hd c = Bk" shows "wadjust_goon_left_moving_B m rs (tl c, Bk # Oc # list)"
monoI: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "class.order less_eq less"   and "\<And>x y. less_eq x y \<Longrightarrow> less_eq (f x) (f y)" shows "order.mono less_eq f"
flubH_le_lubH: fixes H :: "'a set" assumes "H = {x. (x, f x) \<in> r \<and> x \<in> A}" shows "(f (lub H cl), lub H cl) \<in> r"
assms: fixes e :: "real" assumes "0 < e" shows "\<forall>\<^sub>F x in F. \<bar>f x\<bar> < ereal e"
assms: shows "run_one_step d i (s, vs, ves, Label n les es) = (s', vs', RSReturn res)"
take_takefill: fixes m :: "nat"   and n :: "nat"   and fill :: "'b"   and w :: "'b list" assumes "m \<le> n" shows "take m (takefill fill n w) = takefill fill m w"
while_rule_lemma: fixes P :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and Q :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. \<lbrakk>P s; \<not> b s\<rbrakk> \<Longrightarrow> Q s"   and "wf {(t, s). P s \<and> b s \<and> t = c s}"   and "P s" shows "Q (while b c s)"
a: shows "\<forall>x. f x \<noteq> (0::'b)"
rel_interior_open_segment: fixes a :: "'a"   and b :: "'a" shows "rel_interior (open_segment a b) = open_segment a b"
homotopy_eqv_empty1: fixes S :: "'a set" shows "(S homotopy_eqv {}) = (S = {})"
gen_dfs_eq_assert: shows "a.a.gen_dfs.gen_dfs = a.a.gen_dfs.gen_dfs_assert"
prj_simps_12: shows "C.seq fgh.\<nu>\<pi>.prj\<^sub>0 \<p>\<^sub>0[fgh.\<mu>.leg0, fgh.\<nu>.leg1 \<cdot> fgh.\<nu>\<pi>.prj\<^sub>1]"
rsorted_binop: fixes xs :: "('d \<times> 'e) list"   and ys :: "('d \<times> 'f) list"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and z1 :: "'e"   and z2 :: "'f" assumes "rsorted (map fst xs)"   and "rsorted (map fst ys)" shows "rsorted (map fst (binop f z1 z2 xs ys))"
YFtorNT_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e" shows "YFtorNT C f \<equiv> MakeNT (YFtorNT' C f)"
next_plane0_final_mono: fixes g' :: "graph"   and p :: "nat"   and g :: "graph"   and f :: "face" assumes "g' \<in> set (next_plane0\<^bsub>p\<^esub> g)"   and "f \<in> \<F> g"   and "final f" shows "f \<in> \<F> g'"
the_inv_into_into: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and x :: "'d"   and B :: "'c set" assumes "inj_on f A"   and "x \<in> f ` A"   and "A \<subseteq> B" shows "the_inv_into A f x \<in> B"
closed_rel_boundary: fixes S :: "'a set" assumes "closed S" shows "closed (S - rel_interior S)"
that: fixes x :: "'a" assumes "x |\<in>| M"   and "P x y" shows "thesis"
Hadjicostas_Chapman_aux: fixes z :: "real" assumes "- 1 < z" shows "((\<lambda>u. (- ln u) powr z / (1 - u) * (- ln u + u - 1)) has_integral Gamma (z + 2) * (Re (zeta (complex_of_real (z + 2))) - 1 / (z + 1))) {0<..<1}"
interior_eq: fixes S :: "'c set" shows "(interior S = S) = open S"
ucast_scast_id: fixes x :: "'b signed word" shows "UCAST('b \<rightarrow> 'b signed) (SCAST('b signed \<rightarrow> 'b) x) = x"
reduced_word_for_0_imp_nil: fixes A :: "'a set"   and as :: "'a list" assumes "reduced_word_for A (0::'a) as" shows "as = []"
totalize_stable_iff: fixes F :: "'b program"   and A :: "'b set" shows "(totalize F \<in> stable A) = (F \<in> stable A)"
impr_times: fixes a :: "'a N"   and b :: "'a N" shows "a * (a r\<rightarrow> b) = b * (b r\<rightarrow> a)"
assms: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set ys" shows "x \<le> y"
eff: fixes I :: "nat set"   and i :: "nat"   and j :: "nat" assumes "I \<in> P"   and "i \<in> I"   and "j \<in> F I" shows "eff (fst cf1) (snd cf1) i \<approx> eff (fst cf2) (snd cf2) j"
pairwise_compat: fixes c1 :: "('c, 'd) chart"   and c2 :: "('c, 'd) chart" assumes "c1 \<in> charts2"   and "c2 \<in> charts2" shows "k-smooth_compat c1 c2"
that: shows "continuous_on (Y \<times> X) (\<lambda>(y, x). f x y)"
monotone_rel_prodD2: fixes orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ordc :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b \<times> 'c \<Rightarrow> 'd"   and a :: "'b" assumes "monotone (rel_prod orda ordb) ordc f"   and "class.preorder orda (mk_less orda)" shows "monotone ordb ordc (\<lambda>b. f (a, b))"
antisym: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry" assumes "b \<preceq> a"   and "a \<preceq> b" shows "a = b"
seqI: fixes f :: "'d"   and g :: "'d" assumes "D.arr f"   and "D.arr g"   and "D.dom g = D.cod f" shows "D.seq g f"
mult_le_mono2: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "k * i \<le> k * j"
enn2ereal_numeral: fixes a :: "num" shows "enn2ereal (numeral a) = numeral a"
infsetsum_Times': fixes A :: "'b set"   and B :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "countable A"   and "countable B"   and "(\<lambda>(x, y). f x y) abs_summable_on A \<times> B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) B) = (\<Sum>\<^sub>a(x, y)\<in>A \<times> B. f x y)"
order_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
remove: fixes x :: "'a"   and A :: "'a multiset" assumes "x \<in># A" shows "\<Sum>\<^sub># A = x + \<Sum>\<^sub># (remove1_mset x A)"
bounded: shows "\<exists>K. \<forall>a b. norm (a ** g b) \<le> norm a * norm b * K"
bigo_alt_def: fixes f :: "'a \<Rightarrow> 'b" shows "O(f) = {h. \<exists>c>0::'b. \<forall>x. \<bar>h x\<bar> \<le> c * \<bar>f x\<bar>}"
InvariantVarsQTl: fixes Q :: "Literal list"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantVarsQ Q F0 Vbl"   and "Q \<noteq> []" shows "InvariantVarsQ (tl Q) F0 Vbl"
H: shows "\<Delta> @ TVarB X Q # \<Gamma> \<turnstile> t : T"
min_dfa_wf: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat" assumes "wf_dfa M v" shows "wf_dfa (min_dfa M) v"
i_nat: shows "of_int i = - of_int (int (nat (- i)))"
max: fixes G :: "'a filter" assumes "G \<noteq> bot"   and "G \<le> F" shows "F = G"
cod_def: fixes f :: "'a2" shows "C2.cod f = (if C2.codomains f \<noteq> {} then SOME b. b \<in> C2.codomains f else C2.null)"
axioms: fixes Obj :: "'a set"   and Arr :: "'b set"   and D :: "'b \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'a" assumes "free_category Obj Arr D C" shows "graph Obj Arr D C"
simps_5: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b" shows "(case bot.Bot of bot.Value x \<Rightarrow> f1 x | bot.Bot \<Rightarrow> f2) = f2"
prf_oracle_def: fixes key :: "'key \<times> 'seed"   and \<sigma> :: "unit"   and x :: "'\<beta>" shows "prf'.prf_oracle key \<sigma> x = return_spmf (f' key x, ())"
sorted_list_of_set_unique: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set"   and l :: "'k list" assumes "class.linorder less_eq less"   and "finite A" shows "(linorder.strict_sorted less l \<and> set l = A \<and> length l = card A) = (linorder.sorted_list_of_set less_eq A = l)"
nth_prime_ge_2: fixes n :: "nat" shows "2 \<le> nth_prime n"
keys_plus_eqI: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c" assumes "keys p \<inter> keys q = {}" shows "keys (p + q) = keys p \<union> keys q"
univ_rw: shows "UNIV = {i} \<union> (UNIV - {i})"
not_irreducible_zero: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "\<not> comm_semiring_1.irreducible times one zero zero"
assms: fixes m :: "'m"   and w :: "'w" assumes "(m, w) \<in> \<mu>" shows "\<exists>w'. (m, w') \<in> \<mu>' \<and> w' \<in> m_weakly_prefers m \<mu>"
subgroup_mult_of: fixes K :: "'a set" assumes "subfield K R" shows "subgroup (K - {\<zero>}) (mult_of R)"
ext_inject: fixes pc_iiface :: "char list"   and pc_oiface :: "char list"   and pc_proto :: "8 word"   and pc_sport :: "16 word"   and pc_dport :: "16 word"   and more :: "'a"   and pc_iiface' :: "char list"   and pc_oiface' :: "char list"   and pc_proto' :: "8 word"   and pc_sport' :: "16 word"   and pc_dport' :: "16 word"   and more' :: "'a" shows "(\<lparr>pc_iiface = pc_iiface, pc_oiface = pc_oiface, pc_proto = pc_proto, pc_sport = pc_sport, pc_dport = pc_dport, \<dots> = more\<rparr> = \<lparr>pc_iiface = pc_iiface', pc_oiface = pc_oiface', pc_proto = pc_proto', pc_sport = pc_sport', pc_dport = pc_dport', \<dots> = more'\<rparr>) = (pc_iiface = pc_iiface' \<and> pc_oiface = pc_oiface' \<and> pc_proto = pc_proto' \<and> pc_sport = pc_sport' \<and> pc_dport = pc_dport' \<and> more = more')"
coeff_map_poly: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b poly"   and n :: "nat" assumes "f (0::'b) = (0::'a)" shows "poly.coeff (map_poly f p) n = f (poly.coeff p n)"
hmem_rel_iff_hmem_eclose: fixes x :: "hf"   and y :: "hf" shows "((x, y) \<in> hmem_rel) = (x \<^bold>\<in> eclose y)"
rel_map_2: fixes S1a :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S2a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "('a, 'c) tllist"   and g1 :: "'e \<Rightarrow> 'b"   and g2 :: "'f \<Rightarrow> 'd"   and y :: "('e, 'f) tllist" shows "tllist_all2 S1a S2a x (tmap g1 g2 y) = tllist_all2 (\<lambda>x y. S1a x (g1 y)) (\<lambda>x y. S2a x (g2 y)) x y"
trd_irred: fixes fs :: "('a \<Rightarrow>\<^sub>0 'd) list"   and p :: "'a \<Rightarrow>\<^sub>0 'd" shows "\<not> punit.is_red (set fs) (punit.trd fs p)"
NORMAL_def: fixes t :: "lam" shows "NORMAL t \<equiv> \<nexists>t'.  t \<longrightarrow>\<^sub>\<beta> t'"
insert_iff: fixes a :: "'b"   and b :: "'b"   and A :: "'b set" shows "((a::'b) \<in> (b::'b) \<triangleright> (A::'b set)) = (a = b \<or> a \<in> A)"
keys_eq_empty: fixes c :: "'c \<Rightarrow>\<^sub>0 'd" shows "(keys c = {}) = (c = 0)"
simps_4: fixes \<sigma> :: "nat \<Rightarrow> val"   and oper :: "pdf_operator"   and e :: "expr" shows "expr_sem_rf \<sigma> (oper $$ e) = op_sem oper (expr_sem_rf \<sigma> e)"
terminates_elim_cases_5: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate" assumes "\<Gamma>\<turnstile>Seq c1 c2 \<down> s" obtains f :: "'c" where "s = Fault f"   | sa :: "'b" where "s = Normal sa__" and "\<Gamma>\<turnstile>c1 \<down> Normal sa__" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal sa__\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> s'"   | "s = Stuck"   | sa :: "'b" where "s = Abrupt sa"
assms: shows "dirichlet_prod a ainv = (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
round_3: fixes prf_key_gen :: "'a spmf"   and prf_fun :: "'a \<Rightarrow> bool list \<Rightarrow> bool list"   and prf_domain :: "bool list set"   and prf_dlen :: "nat"   and prf_clen :: "nat"   and upf_key_gen :: "'b spmf"   and \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'c, (bool list \<times> bool list \<times> 'c) option + bool list option) gpv"   and upf_fun :: "'b \<Rightarrow> bool list \<Rightarrow> 'c" assumes "simple_cipher prf_key_gen prf_fun prf_domain prf_dlen prf_clen upf_key_gen"   and "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>" shows "\<bar>Sigma_Algebra.measure (measure_spmf (simple_cipher.game3 prf_key_gen prf_domain prf_clen upf_key_gen upf_fun \<A>)) {(b, bad). b} - spmf (simple_cipher.game2 prf_key_gen prf_domain prf_clen upf_key_gen upf_fun \<A>) True\<bar> \<le> Sigma_Algebra.measure (measure_spmf (simple_cipher.game3 prf_key_gen prf_domain prf_clen upf_key_gen upf_fun \<A>)) {(b, bad). bad}"
semidense_D: fixes x :: "'t"   and y :: "'t" assumes "x < y" shows "\<exists>b\<in>D. x \<le> b \<and> b < y"
vec_of_poly_pCons: fixes a :: "'b"   and p :: "'b poly" shows "vec_of_poly (pCons a p) = (if p = 0 then vec_of_list [a] else vec_of_poly p @\<^sub>v vec_of_list [a])"
eval_poly_with_sum_superset: fixes S :: "nat set"   and v :: "linear_poly"   and X :: "nat \<Rightarrow> rat" assumes "finite S"   and "vars v \<subseteq> S" shows "v \<lbrace> X \<rbrace> = (\<Sum>x\<in>S. Abstract_Linear_Poly.coeff v x * X x)"
T_Proj: fixes \<Gamma> :: "binding list"   and t :: "trm"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and l :: "char list"   and T :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> t : RcdT fTs"   and "fTs\<langle>l\<rangle>\<^sub>? = \<lfloor>T\<rfloor>" shows "\<Gamma> \<turnstile> t..l : T"
skl1_inv6D: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and x :: "skl1_state"   and gnx :: "msg"   and Ni :: "msg"   and Rb :: "fid_t" assumes "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "in_progressS (progress x Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}"   and "sklvl1.guessed_frame Ra xgnx = Some gnx"   and "sklvl1.guessed_frame Ra xni = Some Ni"   and "sklvl1.guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))"   and "sklvl1.guessed_frame Ra xnr = Some (NonceF (Rb $ nr))"   and "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "progress x Rb = Some {xny, xni, xnr, xgnx, xgny, xsk}"   and "can_signal x A B"   and "x \<in> skl1_inv6" shows "signalsResp x (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>) < signalsResp x (Running A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)"
A_n'_def: fixes n :: "nat" shows "A_n' n = Some ` A_n n \<union> {None}"
stream_all2_sset1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'b stream"   and ys :: "'c stream" assumes "stream_all2 P xs ys" shows "\<forall>x\<in>sset xs. \<exists>y\<in>sset ys. P x y"
normal_invE_2: fixes N :: "'a set"   and x :: "'a"   and h :: "'a" assumes "N \<lhd> add_monoid G"   and "x \<in> carrier G"   and "h \<in> N" shows "x \<oplus> h \<oplus> \<ominus> x \<in> N"
RPCReply_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCReply send rcv rst p \<equiv> \<not> $Calling rcv p \<and> $(rst!p) = #rpcB \<and> Return send p res<rcv!p> \<and> (rst!p)$ = #rpcA \<and> unchanged (caller rcv!p)"
diff_single_eq_union: fixes x :: "'k"   and M :: "'k multiset"   and N :: "'k multiset" assumes "x \<in># M" shows "(M - {#x#} = N) = (M = add_mset x N)"
parallel_wiring_def: shows "parallel_wiring = lassocr\<^sub>C \<odot> (1\<^sub>C |\<^sub>= swap_lassocr) \<odot> rassocl\<^sub>C"
right_unit: fixes a :: "'a" assumes "a \<in> R\<setminus>I" shows "a \<cdot> \<one> = a"
perzeta_conv_hurwitz_zeta_aux: fixes a :: "real"   and s :: "complex" assumes "a \<in> {0<..<1}"   and "0 < Re s" shows "rGamma (1 - s) * eval_fds (fds_perzeta a) s = complex_of_real (2 * pi) powr (s - 1) * \<i> * (\<i> powr - s * hurwitz_zeta a (1 - s) - \<i> powr s * hurwitz_zeta (1 - a) (1 - s))"
mode_le: shows "mds\<^sub>2 \<le> mds\<^sub>1"
elims_3: fixes x :: "'a option"   and xa :: "'a option" assumes "\<not> x \<le> xa" obtains x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some y" and "\<not> x \<le> y"   | uu_ :: "'a" where "xa__ = Some uu_" and "xa = None"
p_in_reachable: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "(x, y) \<notin> K1.acc" shows "K1.p x y n = 0"
inverse_arrows_runit: fixes a :: "'a" assumes "ide a" shows "inverse_arrows \<r>[a] \<r>\<^sup>-\<^sup>1[a]"
collect_clkt_def: fixes S :: "('d \<times> ('a, 'b) cconstraint \<times> ('a set \<times> 'd) pmf) set" shows "PTA.collect_clkt S = \<Union> {collect_clock_pairs (fst (snd t)) |t. t \<in> S}"
omega_sum_0_iff: fixes ns :: "nat list" assumes "normal ns" shows "(omega_sum ns = 0) = (ns = [])"
matches_rule_and_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and m' :: "'c match_expr"   and a' :: "action"   and s :: "state"   and t :: "state" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule (MatchAnd m m') a'], s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m' a'], s\<rangle> \<Rightarrow> t"
eql: shows "(\<pi> @\<^bsup>l\<^esup> \<pi>l) =\<^bsub>l\<^esub> \<pi>"