hcju/isabelleps huggingface dataset:
bij_betw_binop_G: fixes g :: "'a" assumes "g \<in> G" shows "bij_betw (binop g) G G"
hyps_3: shows "strengthen_extends prb_ rv_ e_ prb'_"
monom_degree: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" assumes "a \<noteq> \<zero>"   and "a \<in> carrier R"   and "p = monom P a m" shows "deg R (p [^]\<^bsub>P\<^esub> n) = n * m"
suboApp: shows "Subobjs P C (Cs @ tl Cs')"
cases: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'a" assumes "r\<inverse>\<inverse> a1 a2" obtains a :: "'a" and b :: "'b" where "a1 = b" and "a2 = a" and "r a b"
commute: fixes a :: "'d"   and b :: "'d" shows "a * b = b * a"
ttree_eqI: fixes t :: "'b ttree"   and t' :: "'b ttree" assumes "\<And>x xs. (x # xs \<in> paths t) = (x # xs \<in> paths t')" shows "t = t'"
Quotient3_rel_abs: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c"   and r :: "'c"   and s :: "'c" assumes "Quotient3 R Abs Rep"   and "R r s" shows "Abs r = Abs s"
pfxes_def: fixes uu :: "'b itself" shows "pfxes uu = map nat [0..int LENGTH('b)]"
verit_sko_ex': fixes P :: "'g \<Rightarrow> bool"   and A :: "bool" assumes "P (SOME x. P x) = A" shows "(\<exists>x. P x) = A"
converse_Un: fixes r :: "('b \<times> 'a) set"   and s :: "('b \<times> 'a) set" shows "(r \<union> s)\<inverse> = r\<inverse> \<union> s\<inverse>"
cosets_finite: fixes c :: "'a set"   and H :: "'a set" assumes "c \<in> rcosets H"   and "H \<subseteq> carrier G"   and "finite (carrier G)" shows "finite c"
less_linear: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less" shows "less x y \<or> x = y \<or> less y x"
not_shadows_vconsts_alt_def_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "not_shadows_vconsts (Vabs cs \<Gamma>) = (not_shadows_vconsts_env \<Gamma> \<and> list_all (\<lambda>(pat, t). fdisjnt all_consts (frees pat) \<and> \<not> shadows_consts t) cs)"
column_Gram_Schmidt_upt_k_repeated_0': fixes i :: "'a"   and j :: "'a"   and k :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "i \<noteq> j"   and "i < j"   and "j \<le> mod_type_class.from_nat k"   and "column j (Gram_Schmidt_upt_k A k) = column i (Gram_Schmidt_upt_k A k)"   and "k < ncols A" shows "column j (Gram_Schmidt_upt_k A k) = 0"
r_rs_4: shows "y = rs \<bullet>\<cdot> ms"
matrix_inv_1: fixes M :: "(('d, 'e) vec, 'f) vec" assumes "invertible M" shows "matrix_inv M ** M = mat (1::'d)"
simps_7: fixes f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd list \<Rightarrow> 'c \<Rightarrow> 'c"   and x21 :: "'d"   and x22 :: "'d list" shows "rec_list f1 f2 (x21 # x22) = f2 x21 x22 (rec_list f1 f2 x22)"
fminus_partition: fixes A :: "'a fset"   and B :: "'a fset" assumes "A |\<subseteq>| B" shows "A |\<union>| (B |-| A) = B"
intro: fixes F :: "nat \<Rightarrow> bool stream measure"   and M :: "bool stream measure" assumes "F = infinite_coin_toss_space.nat_filtration M" shows "infinite_cts_filtration_axioms M F"
reachable_mono: fixes H :: "('a, 'b) pre_digraph"   and u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v"   and "subgraph H (G \<restriction> {v. u \<rightarrow>\<^sup>* v})" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {v. u \<rightarrow>\<^sup>* v}\<^esub> v"
C_eq_Lemmas_sep_14: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "p \<noteq> []" shows "qsort p l \<noteq> []"
c2: shows "(\<exists>l\<in>set les. evalUni (elimVarUni_atom L' (LessUni l)) x) = (\<exists>(a', b', c')\<in>set les. evalUni (elimVarUni_atom L' (LessUni (a', b', c'))) x)"
mguPaaPax_mgu: shows "mgu\<^sub>l\<^sub>s mguPaaPax {Paa, Pax}"
emptyE2: fixes S :: "'b set"   and x :: "'b"   and P :: "bool" assumes "S = {}"   and "x \<in> S" shows "P"
b_EQ_d_1: fixes a :: "tm"   and c :: "tm"   and d :: "tm"   and b :: "tm" shows "{a EQ c, a EQ d, b EQ c} \<turnstile> b EQ d"
fresh_ssubst': fixes V :: "name set"   and i :: "'a"   and t :: "tm" assumes "finite V"   and "atom i \<sharp> t"   and "atom (p \<bullet> i) \<sharp> t" shows "atom i \<sharp> ssubst t V F"
mprod_exp_def: fixes K :: "('c, 'd) Ring_scheme"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "mprod_exp K e f n = e\<Pi>\<^bsub>K,n\<^esub> (\<lambda>j. f j\<^bsub>K\<^esub>\<^bsup>e j\<^esup>)"
low_deg_terms_zero: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "pol_coeff S (n, f)" shows "polyn_expr R X (n + j) (ext_cf S j (n, f)) = X^\<^bsup>R j\<^esup> \<cdot>\<^sub>r polyn_expr R X n (n, f)"
strict_mono_mono: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" assumes "class.order less_eq less"   and "order.strict_mono less f" shows "order.mono less_eq f"
mk_conn_upper: fixes H1 :: "'a hrel \<Rightarrow> 'a hrel"   and \<H>\<^sub>1 :: "'b hrel \<Rightarrow> 'a hrel"   and \<H>\<^sub>2 :: "'a hrel \<Rightarrow> 'b hrel"   and H2 :: "'b hrel \<Rightarrow> 'b hrel" shows "\<pi>\<^sup>*\<^bsub>H1 \<Leftarrow>\<langle>\<H>\<^sub>1,\<H>\<^sub>2\<rangle>\<Rightarrow> H2\<^esub> = \<H>\<^sub>2"
lemma_2_5_17: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b \<le> (b r\<rightarrow> c) l\<rightarrow> a r\<rightarrow> c"
vd: shows "valid_decomp X (snd (split 0 X L'))"
r_rtranclp_rtranclp: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "r x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
mult_zero_left: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "mbot \<otimes> a = mbot"
simple_merge_option: fixes a :: "'b option"   and b :: "'b option" shows "(New_max.merge_option (a, b) = None) = (a = None \<and> b = None)"
borel_measurable_if: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "S \<in> sets lebesgue" shows "((\<lambda>x. if x \<in> S then f x else (0::'d)) \<in> borel_measurable lebesgue) = (f \<in> borel_measurable (lebesgue_on S))"
mat_zero_r: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "A *\<^sub>m\<^sub>m mat_zero = mat_zero"
card: shows "card ((*\<^sub>v) B ` gen) = card gen"
emeasure_lborel_Icc_eq: fixes l :: "real"   and u :: "real" shows "emeasure lborel {l..u} = ennreal (if l \<le> u then u - l else 0)"
Ex_finite_integrable_function: shows "\<exists>h\<in>borel_measurable M. integral\<^sup>N M h \<noteq> \<infinity> \<and> (\<forall>x\<in>space M. 0 < h x \<and> h x < \<infinity>)"
sc_bounded_by: fixes P :: "'b \<Rightarrow> real"   and c :: "real" assumes "sound P"   and "0 \<le> c" shows "bounded_by (c * bound_of P) (\<lambda>x. c * P x)"
integrable_dominated_convergence_at_top: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and s :: "real \<Rightarrow> 'b \<Rightarrow> 'c"   and w :: "'b \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>t. s t \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. ((\<lambda>i. s i x) \<longlongrightarrow> f x) at_top"   and "\<forall>\<^sub>F i in at_top. AE x in M. norm (s i x) \<le> w x" shows "integrable M f"
distinct_21: fixes x3 :: "'b formula"   and x51 :: "'b formula"   and x52 :: "'b formula" shows "\<^bold>\<not> x3 \<noteq> x51 \<^bold>\<or> x52"
assms: shows "GE.wtA (Pr (Guard \<sigma>) Tl)"
singleE': fixes x :: "'a"   and y :: "'a" assumes "pred.eval (Predicate.single x) y" obtains "x = y"
shortestPath_is_simple: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "cf.isShortestPath s p t" shows "cf.isSimplePath s p t"
set_tag_name_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and element_ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_tag_name_locs element_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "ShadowRootClass.type_wf h = ShadowRootClass.type_wf h'"
same_awalk_by_common_arc: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and w :: "'a"   and q :: "('a \<times> 'a) list"   and x :: "'a"   and e :: "'a \<times> 'a" assumes "verts3 (with_proj G) \<subseteq> V"   and "V \<subseteq> pverts G"   and "awalk u p v"   and "awalk w q x"   and "progressing p"   and "progressing q"   and "set (inner_verts p) \<inter> V = {}"   and "set (inner_verts q) \<inter> V = {}"   and "{u, v, w, x} \<subseteq> V"   and "e \<in> set p"   and "e \<in> set q" shows "p = q"
cases: fixes S :: "'a set"   and t :: "real"   and a1 :: "'a"   and a2 :: "(real \<times> 'a) stream" assumes "trace_in S t a1 a2" obtains t' :: "real" and s :: "'a" and s' :: "'a" and \<omega> :: "(real \<times> 'a) stream" where "a1 = sa__" and "a2 = (t'__, s'__) ## \<omega>'__" and "t < t'__" and "sa__ \<in> S"   | t' :: "real" and s' :: "'a" and \<omega> :: "(real \<times> 'a) stream" and s :: "'a" where "a1 = s" and "a2 = (t', s') ## \<omega>" and "t' \<le> t" and "trace_in S t s' \<omega>"
skip_elim': fixes mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "(Skip, mem) \<leadsto>\<^sub>s (c', mem')" obtains "c' = Stop" and "mem' = mem"
comp_def: fixes f :: "'f \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f" shows "f \<circ> g = (\<lambda>x. f (g x))"
reachable_ss_V: fixes s :: "nat" assumes "s \<in> cf.V" shows "cf.reachableNodes s \<subseteq> cf.V"
nrename_interesting2: fixes x :: "name"   and z :: "name"   and u :: "name"   and y :: "name"   and M :: "trm" assumes "x \<noteq> z"   and "x \<noteq> u"   and "x \<noteq> y"   and "z \<noteq> u"   and "y \<noteq> z" shows "M[x\<turnstile>n>y][z\<turnstile>n>u] = M[z\<turnstile>n>u][x\<turnstile>n>y]"
rADnMT: fixes p :: "('a, 'b) Combinators list" assumes "p \<noteq> []" shows "removeAllDuplicates p \<noteq> []"
W_UNIV: shows "UNIV = set [W1, W2, W3]"
mojmir_accept_iff_rabin_accept: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" assumes "mojmir_to_rabin \<Sigma> \<delta> q\<^sub>0 w F" shows "mojmir_def.accept \<delta> q\<^sub>0 w F = accept\<^sub>R (semi_mojmir_def.step \<Sigma> \<delta> q\<^sub>0, semi_mojmir_def.initial q\<^sub>0, {(mojmir_to_rabin_def.fail\<^sub>R \<Sigma> \<delta> q\<^sub>0 F \<union> mojmir_to_rabin_def.merge\<^sub>R \<delta> q\<^sub>0 F j, mojmir_to_rabin_def.succeed\<^sub>R \<delta> q\<^sub>0 F j) |j. j < semi_mojmir_def.max_rank \<Sigma> \<delta> q\<^sub>0}) w"
c: shows "f \<in> mor\<^bsub>C\<^esub>"
product_lists_set: fixes xss :: "'b list list" shows "set (product_lists xss) = {xs. list_all2 (\<lambda>x ys. x \<in> set ys) xs xss}"
f_nxt_def: fixes f :: "'b \<Rightarrow> 'b ttree"   and T :: "'b set"   and x :: "'b" shows "f_nxt f T x = (if x \<in> T then f(x := TTree.empty) else f)"
single_eq_add_mset: fixes a :: "'a"   and b :: "'a"   and M :: "'a multiset" shows "({#a#} = add_mset b M) = (b = a \<and> M = {#})"
sum_distrib_left: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and r :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" assumes "class.semiring_0 plus zero times" shows "times r (comm_monoid_add.sum plus zero f A) = comm_monoid_add.sum plus zero (\<lambda>n. times r (f n)) A"
1: shows "inside (path_image c1 \<union> path_image c) \<union> inside (path_image c2 \<union> path_image c) \<union> (path_image c - {a, b}) = inside (path_image c1 \<union> path_image c2)"
prems_12: shows "InvariantVarsQ (getQ state') F0 Vbl"
corec_sel_1: fixes p :: "'c \<Rightarrow> bool"   and a :: "'c"   and g21 :: "'c \<Rightarrow> 'd"   and q22 :: "'c \<Rightarrow> bool"   and g221 :: "'c \<Rightarrow> 'd llist"   and g222 :: "'c \<Rightarrow> 'c" assumes "\<not> p a" shows "lhd (corec_llist p g21 q22 g221 g222 a) = g21 a"
represents_def: fixes scale :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'d set"   and r :: "'d \<Rightarrow>\<^sub>0 'c"   and x :: "'d" assumes "module scale" shows "module.represents scale B r x = (keys r \<subseteq> B \<and> module.rep scale r = x)"
scale_left_diff_distrib: fixes a :: "real"   and b :: "real"   and x :: "'b" shows "(a - b) *\<^sub>R x = a *\<^sub>R x - b *\<^sub>R x"
SCnotConc: fixes a :: "('b, 'c) Combinators"   and b :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "a \<oplus> b \<in> set p"   and "singleCombinators p" shows "False"
dom_char: fixes f :: "'b" shows "hhom\<^sub>D.dom f = (if hhom\<^sub>D.arr f then D.dom f else D.null)"
path_from_to_last: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto> w" shows "w \<notin> set (butlast xs)"
recv: shows "act (recvmsg rreq_rrep_sn) \<sigma> \<sigma>' a"
I_def: shows "I = {t \<in> {a..b}. \<forall>s\<in>{a..t}. dist (p a) (p s) \<le> d}"
succ_def: fixes x :: "V" shows "ZFC_in_HOL.succ x \<equiv> vinsert x x"
ennreal_ge_1: fixes x :: "real" shows "(1 \<le> ennreal x) = (1 \<le> x)"
has_derivative_at_split: fixes f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "(f has_derivative f') (at x) = ((f has_derivative f') (at_left x) \<and> (f has_derivative f') (at_right x))"
assms_1: shows "constructive_security real1 ideal1 sim1 \<I>_real1 \<I>_inner1 \<I>_common1 bound1 lossless1 w1"
bezout_ring_imp_diagonal_admits_SNF_mod_ring_admits_aux: fixes p :: "nat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<int p}"   and "1 < p"   and "class.bezout_ring (*) (1::'c) (+) (0::'c) (-) uminus" shows "Ball {A. A \<in> carrier_mat p p} admits_SNF_JNF"
fps_XD'_add: fixes a :: "'b fps"   and b :: "'b fps"   and c :: "'b fps" shows "fps_XD' a (b + c) = fps_XD' a b + fps_XD' a c"
IH: fixes i :: "nat" assumes "vs @ xs' \<in> C M2 M1 \<Omega> V m i"   and "mcp (vs @ xs') V vs" shows "Suc (length xs') = i"
BrnFT_elim: fixes ii :: "nat"   and cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('a, 'b, 'c) cmd list" assumes "ii \<in> BrnFT cl dl" obtains n :: "nat" and i :: "nat" where "n \<in> theFTOne cl dl" and "i < brn (cl ! n)" and "ii = brnL cl n + i"
assm: shows "0 < \<parallel>ext v\<parallel> \<and> len v ts c = ext v \<and> restrict v (clm ts) c = lan v \<and> |lan v| = 1"
guard_intros_1: fixes b :: "bool" assumes "b" shows "guard b = Some ()"
nonempty: shows "range (\<lambda>i. SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a))) \<noteq> {}"
isPrefixPrefixToLevel: fixes l :: "nat"   and t :: "('a \<times> bool) list" shows "isPrefix (prefixToLevel l t) t"
to_bl_to_bin: fixes w :: "'a word" shows "bl_to_bin (to_bl w) = uint w"
hausdorff: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y" shows "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
cases: fixes a1 :: "com list"   and a2 :: "state"   and a3 :: "com list"   and a4 :: "state" assumes "(a1, a2) \<rightarrow> (a3, a4)" obtains t :: "state" and f :: "state \<Rightarrow> state set" and s :: "state" and cs :: "com list" where "a1 = Do f__ # cs__" and "a2 = s__" and "a3 = cs__" and "a4 = t" and "t \<in> f__ s__"   | c1 :: "com" and c2 :: "com" and cs :: "com list" and s :: "state" where "a1 = (c1__; c2__) # csa__" and "a2 = sa__" and "a3 = c1__ # c2__ # csa__" and "a4 = sa__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" and cs :: "com list" where "a1 = (IF b__ THEN c1a__ ELSE c2a__) # csb__" and "a2 = sb__" and "a3 = c1a__ # csb__" and "a4 = sb__" and "b__ sb__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c1 :: "com" and c2 :: "com" and cs :: "com list" where "a1 = (IF ba__ THEN c1 ELSE c2) # csc__" and "a2 = sc__" and "a3 = c2 # csc__" and "a4 = sc__" and "\<not> ba__ sc__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and cs :: "com list" where "a1 = (WHILE bb__ DO c__) # csd__" and "a2 = sd__" and "a3 = csd__" and "a4 = sd__" and "\<not> bb__ sd__"   | b :: "state \<Rightarrow> bool" and s :: "state" and c :: "com" and cs :: "com list" where "a1 = (WHILE b DO ca__) # cse__" and "a2 = se__" and "a3 = ca__ # (WHILE b DO ca__) # cse__" and "a4 = se__" and "b se__"   | p :: "pname" and cs :: "com list" and s :: "state" where "a1 = CALL p # csf__" and "a2 = sf__" and "a3 = body p # csf__" and "a4 = sf__"   | f :: "state \<Rightarrow> state" and c :: "com" and g :: "state \<Rightarrow> state \<Rightarrow> state" and cs :: "com list" and s :: "state" where "a1 = (LOCAL f; c; g) # cs" and "a2 = s" and "a3 = c # Do (\<lambda>t. {g s t}) # cs" and "a4 = f s"
hyp: shows "\<forall>s. s \<in> set signs \<and> s \<notin> consistent_sign_vec_copr qs ` set (characterize_root_list_p p) \<longrightarrow> z I s * rat_of_nat (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}) = 0"
cart_eq_inner_axis: fixes a :: "(real, 'c) vec"   and i :: "'c" shows "a $ i = a \<bullet> axis i 1"
fair_price_def: fixes Mkt :: "('a, 'c) discrete_market"   and \<pi> :: "real"   and pyf :: "'a \<Rightarrow> real"   and matur :: "nat" shows "fair_price Mkt \<pi> pyf matur = (\<exists>pr. price_structure pyf matur \<pi> pr \<and> (\<forall>x Mkt2 p. x \<notin> stocks Mkt \<longrightarrow> coincides_on Mkt Mkt2 (stocks Mkt) \<and> prices Mkt2 x = pr \<and> portfolio p \<and> support_set p \<subseteq> stocks Mkt \<union> {x} \<longrightarrow> \<not> arbitrage_process Mkt2 p))"
norm_eq_sqrt_inner: fixes x :: "'b" shows "norm x = sqrt (real_of (inner x x))"
cont2cont_lambda: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'e" assumes "\<And>y. cont (\<lambda>x. f x y)" shows "cont f"
ip: shows "ip = IPv6AddrPreferred a b c d e f g h"
to_nonunique_collection_det: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma>\<^sub>1 :: "'a OCL_Types.type"   and \<sigma>\<^sub>2 :: "'a OCL_Types.type" assumes "to_nonunique_collection \<tau> \<sigma>\<^sub>1"   and "to_nonunique_collection \<tau> \<sigma>\<^sub>2" shows "\<sigma>\<^sub>1 = \<sigma>\<^sub>2"
xx_4: shows "gr = uncurry L4Src ` (case xa of None \<Rightarrow> {} | Some s \<Rightarrow> {s}) \<union> uncurry L4Dst ` (case xb of None \<Rightarrow> {} | Some s \<Rightarrow> {s}) \<union> IPv4Proto ` (case proto r of ProtoAny \<Rightarrow> {} | Proto p \<Rightarrow> {p}) \<union> IngressPort ` (case x of None \<Rightarrow> {} | Some s \<Rightarrow> {s}) \<union> IPv4Src ` (case toprefixmatch (src r) of None \<Rightarrow> {} | Some s \<Rightarrow> {s}) \<union> IPv4Dst ` (case toprefixmatch (dst r) of None \<Rightarrow> {} | Some s \<Rightarrow> {s}) \<union> {EtherType 2048}"
rel_G_pos_distr: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and uu :: "('z \<times> 'aa \<times> 'ab \<times> 'ac \<times> 'ad \<times> 'ae \<times> 'af) itself"   and L1 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and L2 :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and L1' :: "'aa \<Rightarrow> 'ab \<Rightarrow> bool"   and L2' :: "'ad \<Rightarrow> 'ae \<Rightarrow> bool" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' uu" shows "rel_G L1 L2 Co1 Co2 Contra1 Contra2 OO rel_G L1' L2' Co1' Co2' Contra1' Contra2' \<le> rel_G (L1 OO L1') (L2 OO L2') (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2')"
Frct_code_post_7: fixes a :: "int"   and b :: "int" shows "Frct (- a, b) = - Frct (a, b)"
exec_move_AAccI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and v :: "'addr val" assumes "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1\<lfloor>e2\<rceil>) h (stk @ [v], loc, length (compE2 e1) + pc, xcp) ta h' (stk' @ [v], loc', length (compE2 e1) + pc', xcp')"
shows_list_gen_def: fixes showsx :: "'a \<Rightarrow> char list \<Rightarrow> char list"   and e :: "char list"   and l :: "char list"   and s :: "char list"   and r :: "char list"   and xs :: "'a list" shows "shows_list_gen showsx e l s r xs = (if xs = [] then shows_string e else shows_string l \<circ> shows_sep showsx (shows_string s) xs \<circ> shows_string r)"
bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p_subst: fixes a :: "('b, 'a) stateful_strand_step"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<delta>) = bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p a"
self_reachable: fixes b :: "'b"   and B :: "'b set"   and R :: "('b \<times> 'b) set" assumes "b \<in> B" shows "b \<in> R\<^sup>* `` B"
has_bochner_integral_combine: fixes a :: "real"   and c :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'b"   and i :: "'b"   and j :: "'b" assumes "a \<le> c"   and "c \<le> b"   and "has_bochner_integral (lebesgue_on {a..c}) f i"   and "has_bochner_integral (lebesgue_on {c..b}) f j" shows "has_bochner_integral (lebesgue_on {a..b}) f (i + j)"
set_pps_to_list: fixes S :: "'t set" assumes "finite S" shows "set (pps_to_list S) = S"
startof_Suc2: fixes as :: "nat"   and ap :: "abc_inst list"   and n :: "nat"   and e :: "nat" assumes "abc_fetch as ap = Some (Dec n e)" shows "start_of (layout_of ap) (Suc as) = start_of (layout_of ap) as + 2 * n + 16"
insert_before_locs_def: fixes ptr :: "(_) object_ptr"   and old_parent :: "(_) object_ptr option"   and child_owner_document :: "(_) document_ptr"   and ptr_owner_document :: "(_) document_ptr" shows "insert_before_locs ptr old_parent child_owner_document ptr_owner_document = adopt_node_locs old_parent child_owner_document ptr_owner_document \<union> set_child_nodes_locs ptr \<union> set_disconnected_nodes_locs ptr_owner_document"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'c \<Rightarrow> 'a"   and f4 :: "'a"   and graph_node :: "('b, 'c) graph_node" shows "P (case graph_node of Assumption x \<Rightarrow> f1 x | Conclusion x \<Rightarrow> f2 x | Rule x \<Rightarrow> f3 x | Helper \<Rightarrow> f4) = ((\<forall>x1. graph_node = Assumption x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. graph_node = Conclusion x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. graph_node = Rule x3 \<longrightarrow> P (f3 x3)) \<and> (graph_node = Helper \<longrightarrow> P f4))"
valid_without_offending_flows: fixes F :: "('v \<times> 'v) set"   and G :: "'v graph"   and nP :: "'v \<Rightarrow> 'a" assumes "F \<in> set_offending_flows G nP" shows "sinvar (delete_edges G F) nP"
map_rel_def: fixes K :: "('a \<times> 'c) set"   and V :: "('b \<times> 'd) set" shows "\<langle>K, V\<rangle>map_rel \<equiv> (K \<rightarrow> \<langle>V\<rangle>option_rel) \<inter> {(mi, m). dom mi \<subseteq> Domain K \<and> dom m \<subseteq> Range K}"
flrestriction_fconverse: fixes r :: "V"   and A :: "V" shows "r\<inverse>\<^sub>\<bullet> \<restriction>\<^sup>l\<^sub>\<bullet> A = (r \<restriction>\<^sup>r\<^sub>\<bullet> A)\<inverse>\<^sub>\<bullet>"
c1_def: shows "c1 = real n / real (Suc n) * c"
IH: fixes PROB :: "(('a, 'b) fmap \<times> ('a, 'c) fmap) set"   and vs :: "'a set" assumes "as \<in> valid_plans PROB" shows "as_proj as vs \<in> valid_plans (prob_proj PROB vs)"
borderD_ssuf: fixes x :: "'b list"   and w :: "'b list" assumes "x \<le>b w" shows "x <s w"
CD_on_iia: fixes ds :: "'b set" shows "iia (CD_on ds)"
is_functor: shows "functor VV.comp VV.comp VV.map"