hcju/isabelleps huggingface dataset:
eq_onE: fixes X :: "'a set"   and a :: "'a"   and b :: "'a" assumes "eq_on X a b" obtains "b \<in> X" and "a = b"
OUT_jj: fixes x :: "'v" shows "d_OUT (j \<circ> map_prod Some Some \<circ> prod.swap) x = d_IN j \<langle>x\<rangle> - j (None, \<langle>x\<rangle>)"
assms_2: shows "(y, z) \<in> terminationLessState1 Vbl"
distinct_1: fixes name :: "name"   and coname :: "coname"   and coname' :: "coname"   and trm1' :: "trm"   and name' :: "name"   and trm2' :: "trm" shows "Ax name coname \<noteq> Cut <coname'>.trm1' name'.trm2'"
law1: fixes I :: "item set" assumes "wellformed_items I" shows "items_le k ((Scan T k \<circ> Complete k \<circ> Predict k) I) = (Scan (empty_tokens T) k \<circ> Complete k \<circ> Predict k) (items_le k I)"
\<gamma>': shows "\<guillemotleft>THE \<gamma>. \<guillemotleft>\<gamma> : w' \<star> w \<Rightarrow> src f\<guillemotright> \<and> local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) = g \<star> \<gamma> \<and> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w] = \<r>[f] \<cdot> (f \<star> \<gamma>) : src f \<rightarrow> src f\<guillemotright> \<and> \<guillemotleft>THE \<gamma>. \<guillemotleft>\<gamma> : w' \<star> w \<Rightarrow> src f\<guillemotright> \<and> local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) = g \<star> \<gamma> \<and> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w] = \<r>[f] \<cdot> (f \<star> \<gamma>) : w' \<star> w \<Rightarrow> src f\<guillemotright> \<and> local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) = g \<star> (THE \<gamma>. \<guillemotleft>\<gamma> : w' \<star> w \<Rightarrow> src f\<guillemotright> \<and> local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) = g \<star> \<gamma> \<and> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w] = \<r>[f] \<cdot> (f \<star> \<gamma>)) \<and> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w] = \<r>[f] \<cdot> (f \<star> (THE \<gamma>. \<guillemotleft>\<gamma> : w' \<star> w \<Rightarrow> src f\<guillemotright> \<and> local.inv ((\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]) = g \<star> \<gamma> \<and> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w] = \<r>[f] \<cdot> (f \<star> \<gamma>)))"
outsideUnion: fixes R :: "('c \<times> 'd) set"   and x :: "'c" shows "R = Outside' {x} R \<union> {x} \<times> R `` {x}"
simps_2: fixes h :: "'b \<times> bool"   and t :: "('b \<times> bool) list" shows "markedElements (h # t) = (if marked h then element h # markedElements t else markedElements t)"
trail_simps_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and e :: "'d"   and es :: "'d list"   and w :: "'c" assumes "wf_digraph G" shows "pre_digraph.trail G u (e # es) w = (e \<in> arcs G \<and> u = tail G e \<and> e \<notin> set es \<and> pre_digraph.trail G (head G e) es w)"
comp_simp: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Arr :: "'b \<Rightarrow> bool"   and g :: "'b"   and f :: "'b" assumes "subcategory C Arr"   and "partial_magma.arr (subcategory.comp C Arr) (subcategory.comp C Arr g f)" shows "subcategory.comp C Arr g f = C g f"
Sim: fixes R :: "pi"   and S :: "pi" assumes "(R, S) \<in> Rel" shows "R \<leadsto>[Rel] S"
elims: fixes x :: "('fsym, 'tp) efsym"   and y :: "'tp" assumes "GE_resOf x = y" obtains f :: "'fsym" where "x = Oldf f" and "y = resOf f"   | \<sigma> :: "'tp" where "x = Wit \<sigma>" and "y = \<sigma>"
ordered_keys_def: fixes m :: "('c, 'd) mapping" shows "Mapping.ordered_keys m = (if finite (Mapping.keys m) then sorted_list_of_set (Mapping.keys m) else [])"
Gramian_matrix_def: fixes G :: "'a vec list"   and k :: "nat" shows "gs.Gramian_matrix G k = (let M = mat k n (\<lambda>(i, j). G ! i $ j) in M * M\<^sup>T)"
add_mono_thms_linordered_field_2: fixes i :: "'b"   and j :: "'b"   and k :: "'b"   and l :: "'b" assumes "i = j \<and> order.greater l k" shows "order.greater (j + l) (i + k)"
that: fixes g :: "nat \<Rightarrow> 'a" assumes "\<And>n. P (g n)"   and "LIM n sequentially. f (g n) :> at_top" shows "thesis"
select_convs_3: fixes Obj :: "'b set"   and Mor :: "'a set"   and Dom :: "'a \<Rightarrow> 'b"   and Cod :: "'a \<Rightarrow> 'b"   and Id :: "'b \<Rightarrow> 'a"   and Comp :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and more :: "'c" shows "Dom \<lparr>Obj = Obj, Mor = Mor, Dom = Dom, Cod = Cod, Id = Id, Comp = Comp, \<dots> = more\<rparr> = Dom"
cSup_lessThan: fixes x :: "'a" shows "Sup {..<x} = x"
induct_2: fixes P :: "typ \<Rightarrow> typ \<Rightarrow> (variable \<times> String.literal set \<Rightarrow> typ option) \<Rightarrow> bool"   and Q :: "typ list \<Rightarrow> typ list \<Rightarrow> (variable \<times> String.literal set \<Rightarrow> typ option) \<Rightarrow> bool"   and a3 :: "typ list"   and a4 :: "typ list"   and a5 :: "variable \<times> String.literal set \<Rightarrow> typ option" assumes "\<And>v S T subs. P (Tv v S) T subs"   and "\<And>a Ts b Us subs. (a = b \<Longrightarrow> Q Ts Us subs) \<Longrightarrow> P (Ty a Ts) (Ty b Us) subs"   and "\<And>v va vb vc uw_. P (Ty v va) (Tv vb vc) uw_"   and "\<And>T Ts U Us subs. \<lbrakk>P T U subs; \<And>a. raw_match T U subs = Some a \<Longrightarrow> Q Ts Us a\<rbrakk> \<Longrightarrow> Q (T # Ts) (U # Us) subs"   and "\<And>subs. Q [] [] subs"   and "\<And>v va subs. Q [] (v # va) subs"   and "\<And>v va subs. Q (v # va) [] subs" shows "Q a3 a4 a5"
tail: fixes e :: "nat \<times> nat"   and u :: "nat"   and v :: "nat" assumes "e = (u, v)" shows "tail G e = u"
subcls1_def2: fixes P :: "'b prog" shows "subcls1 P = (SIGMA C:{C. is_class P C}. {D. C \<noteq> Object \<and> fst (the (class P C)) = D})"
pos_less_neg_iff: fixes A :: "'a"   and B :: "'a" shows "(Pos A < Neg B) = (A \<le> B)"
TerminationPartial: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"   and "\<forall>s\<in>P. \<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
list_induct3: fixes xs :: "'c list"   and ys :: "'d list"   and zs :: "'e list"   and P :: "'c list \<Rightarrow> 'd list \<Rightarrow> 'e list \<Rightarrow> bool" assumes "length xs = length ys"   and "length ys = length zs"   and "P [] [] []"   and "\<And>x xs y ys z zs. \<lbrakk>length xs = length ys; length ys = length zs; P xs ys zs\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys) (z # zs)" shows "P xs ys zs"
norm_empty_diag_preservation: fixes i :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "i \<le> n"   and "M i i < Le 0" shows "DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n i i < Le 0"
validn_valid: fixes G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "\<forall>n. G \<Turnstile>\<^sub>n c : A" shows "G \<Turnstile> c : A "
upred_ares_def: fixes P :: "'d upred"   and a :: "'c \<Longrightarrow> 'd" shows "P \<restriction>\<^sub>p a = (P \<restriction>\<^sub>v a) \<restriction>\<^sub>e a"
head_fun: fixes s :: "('c, 'd) tm" shows "head (fun s) = head s"
pochhammer_Suc: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "comm_semiring_1.pochhammer times one plus zero a (Suc n) = times (comm_semiring_1.pochhammer times one plus zero a n) (plus a (semiring_1.of_nat one plus zero n))"
assms_2: fixes s :: "'a net_state"   and a :: "'b node_action"   and s' :: "'a net_state"   and i :: "nat"   and d :: "nat"   and di :: "nat" assumes "(s, a, s') \<in> S" shows "a \<noteq> i:newpkt(d, di)"
NegE: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form" assumes "G \<turnstile> Neg a"   and "G \<turnstile> a" shows "G \<turnstile> FF"
interpret_floatarith_subst_floatarith: fixes fa :: "floatarith"   and D :: "nat"   and s :: "nat \<Rightarrow> floatarith"   and vs :: "real list" assumes "max_Var_floatarith fa \<le> D" shows "interpret_floatarith (subst_floatarith s fa) vs = interpret_floatarith fa (map (\<lambda>i. interpret_floatarith (s i) vs) [0..<D])"
\<Psi>o_\<Phi>o: fixes a :: "'s" assumes "S.ide a" shows "\<Psi>o (\<Phi>o a) = a"
rw: shows "\<bar>1 / 4 - x_coord y * x_coord y\<bar> = 1 / 4 - x_coord y * x_coord y"
step_1_main_similar: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat"   and n :: "nat" assumes "i \<le> j"   and "A \<in> carrier_mat n n" shows "similar_mat (step_1_main n i j A) A"
mk_minsky_dec: fixes v :: "nat" shows "mk_minsky (\<lambda>vs vs'. vs' = (\<lambda>x. if x = v then vs v - 1 else vs x))"
pi_image_filter: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd option" assumes "proper_it it it'" shows "proper_it (set_iterator_image_filter g it) (set_iterator_image_filter g it')"
Ker_image: fixes a :: "'a" assumes "a \<in> Ker" shows "\<eta> a = \<one>'"
exhaust_sel: fixes generat :: "('d, 'e, 'f) generat" obtains "generat = Pure (generat.result generat)"   | "generat = IO (generat.output generat) (continuation generat)"
one_greatest: fixes a :: "'a" shows "a \<le> (1::'a)"
frameResChainFresh'': fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and xvec :: "name list"   and yvec :: "name list"   and F :: "'d frame" assumes "assertionAux SCompose SImp SBottom SChanEq"   and "xvec \<sharp>* yvec" shows "xvec \<sharp>* (\<lparr>\<nu>*yvec\<rparr>F) = xvec \<sharp>* F"
per_union_dom: fixes R :: "('a \<times> 'a) set"   and i :: "'a"   and j :: "'a" shows "Domain (per_union R i j) = Domain R"
finite_eqvt: fixes p :: "perm"   and A :: "'a set" shows "p \<bullet> finite A = finite (p \<bullet> A)"
induct: fixes P :: "bit list \<Rightarrow> bit list \<Rightarrow> bit list \<Rightarrow> bit list \<Rightarrow> bit list \<Rightarrow> bit list \<Rightarrow> bit list \<Rightarrow> bool"   and a0 :: "bit list"   and a1 :: "bit list"   and a2 :: "bit list"   and a3 :: "bit list"   and a4 :: "bit list"   and a5 :: "bit list"   and a6 :: "bit list" assumes "\<And>b A B C D E. P b [] A B C D E"   and "\<And>b v va A B C D E. (\<And>x xa xb xc xd xe. \<lbrakk>x = sha1compressstart 79 b A B C D E; xa = addmod32 A (select x 0 31); xb = addmod32 B (select x 32 63); xc = addmod32 C (select x 64 95); xd = addmod32 D (select x 96 127); xe = addmod32 E (select x 128 159)\<rbrakk> \<Longrightarrow> P (getblock (v # va)) (delblock (v # va)) xa xb xc xd E) \<Longrightarrow> P b (v # va) A B C D E" shows "P a0 a1 a2 a3 a4 a5 a6"
JFset_minimal: fixes K1 :: "'d JF1 \<Rightarrow> 'd set"   and K2 :: "'d JF2 \<Rightarrow> 'd set"   and a :: "'d JF1"   and b :: "'d JF2" assumes "\<And>a. F1set1 (dtor1 a) \<subseteq> K1 a"   and "\<And>b. F2set1 (dtor2 b) \<subseteq> K2 b"   and "\<And>a a'. a' \<in> F1set2 (dtor1 a) \<Longrightarrow> K1 a' \<subseteq> K1 a"   and "\<And>a b'. b' \<in> F1set3 (dtor1 a) \<Longrightarrow> K2 b' \<subseteq> K1 a"   and "\<And>b a'. a' \<in> F2set2 (dtor2 b) \<Longrightarrow> K1 a' \<subseteq> K2 b"   and "\<And>b b'. b' \<in> F2set3 (dtor2 b) \<Longrightarrow> K2 b' \<subseteq> K2 b" shows "JF1set a \<subseteq> K1 a \<and> JF2set b \<subseteq> K2 b"
coprimeI_power_mod: fixes x :: "nat"   and r :: "nat"   and n :: "nat" assumes "x ^ r mod n = 1"   and "r \<noteq> 0"   and "n \<noteq> 0" shows "coprime x n"
sorted_Cons_le: fixes x :: "'c"   and xs :: "'c list"   and y :: "'c" assumes "ASSUMPTION (Sorted_Less.sorted (x # xs))"   and "y \<le> x" shows "Sorted_Less.sorted (y # xs)"
tendsF: fixes x :: "'a" assumes "x \<notin> N" shows "(\<lambda>n. F n x) \<longlonglongrightarrow> (if x \<in> S then f x else (0::'b))"
cdf_def2: fixes M :: "real measure"   and x :: "real" shows "cdf M x = Sigma_Algebra.measure M {..x}"
g'_def: shows "g' = (\<lambda>t. Im (g t) / Re (g t))"
NoReceivingNoChange: fixes cfg1 :: "('p, 'v, 's) configuration"   and m :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration"   and p :: "'p" assumes "cfg1 \<turnstile> m \<mapsto> cfg2"   and "\<not> isReceiverOf p m" shows "states cfg1 p = states cfg2 p"
erf_def: fixes z :: "'b" shows "erf z = (\<Sum>n. of_real (erf_coeffs n) * z ^ n)"
assms_1: shows "v \<Turnstile>\<^sub>l\<^sub>e c"
prj_simps_2: shows "C.seq trg_trg.prj\<^sub>0 \<p>\<^sub>1[trg.leg0 \<cdot> trg_trg.prj\<^sub>0, f.leg1]"
simps_1: fixes prog :: "cmd"   and l :: "nat"   and c :: "cmd" shows "labels_nodes prog (_ l _) c = labels prog l c"
BINDING_preserved_1: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and M' :: "trm" assumes "<a>:M \<in> BINDINGc B (\<parallel>(B)\<parallel>)"   and "M \<longrightarrow>\<^sub>a* M'" shows "<a>:M' \<in> BINDINGc B (\<parallel>(B)\<parallel>)"
cbelow_on_monom_mult_monomial: fixes c :: "'c"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and v :: "'t"   and c' :: "'c"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "c \<noteq> (0::'c)"   and "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (monomial c' v) (\<lambda>a b. red F a b \<or> red F b a) p q" shows "cbelow_on (dgrad_p_set d m) (\<prec>\<^sub>p) (monomial c (t \<oplus> v)) (\<lambda>a b. red F a b \<or> red F b a) p q"
fixp_def: fixes f :: "'b \<Rightarrow> 'b" shows "c2.fixp f = lub2 (c2.iterates f)"
SNF_2x2_works': shows "\<forall>A\<in>carrier_mat 2 2. is_SNF A (Smith_2x2_JNF A)"
distinct_27: fixes x2 :: "nat"   and x6 :: "nat" shows "Number x2 \<noteq> Passwd x6"
left_zero: fixes a :: "'a" assumes "a \<in> R" shows "\<zero> \<cdot> a = \<zero>"
eqI: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bijection f"   and "f a = f b" shows "a = b"
length_ci_inc: fixes ly :: "nat list"   and ss :: "nat"   and n :: "nat" shows "length (ci ly ss (Inc n)) = 4 * n + 18"
Abs_formula_cases: fixes x :: "'a formula" obtains y :: "'a set set" where "x = Abs_formula y" and "y \<in> fba"
distinct_9: fixes x2 :: "'a" shows "Conclusion x2 \<noteq> Helper"
bernoulli_stream_npref_prob: fixes M :: "bool stream measure"   and p :: "real"   and x :: "bool stream" assumes "M = bernoulli_stream p" shows "emeasure M {w \<in> space M. stake 0 w = stake 0 x} = 1"
alpha_tAct: fixes \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c) Tree"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c) Tree" shows "(tAct \<alpha>1 t1 =\<^sub>\<alpha> tAct \<alpha>2 t2) = (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2))"
compMb2_def: shows "compMb2 \<equiv> \<lambda>body. let ins = compE2 body @ [Return]; xt = compxE2 body 0 0 in (max_stack body, max_vars body, ins, xt)"
inj_on_image_mem_iff: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and a :: "'b"   and A :: "'b set" assumes "inj_on f B"   and "a \<in> B"   and "order.greater_eq B A" shows "(f a \<in> f ` A) = (a \<in> A)"
pmu_unfold: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f" shows "f (p\<mu> f) \<le> p\<mu> f"
le_cSup_finite: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and X :: "'a set"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "finite X"   and "x \<in> X" shows "less_eq x (Sup X)"
simps_6: fixes v :: "vname aexp"   and va :: "vname aexp" shows "enumerate_aexp_inputs (Times v va) = enumerate_aexp_inputs v \<union> enumerate_aexp_inputs va"
transitive_axioms: shows "transitive UNIV ((\<sqsubset>) \<restriction> A)"
1: shows "lpfp\<^sub>c (step' \<top>) c = Some c'"
valid_pub_AND_def: shows "valid_pub_AND = {(x0, x1). x0 \<in> valid_pub0 \<and> x1 \<in> valid_pub1}"
fut_derivs_def: fixes k :: "nat"   and \<phi> :: "('a, 'k) aformula" shows "fut_derivs k \<phi> \<equiv> ((norm \<circ> rderiv (zero idx)) ^^ k) \<phi>"
remove: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and x :: "'k"   and A :: "'k multiset" assumes "comm_monoid_mset f z"   and "x \<in># A" shows "comm_monoid_mset.F f z A = f x (comm_monoid_mset.F f z (A - {#x#}))"
v_w_1: shows "(v, w) \<in> edges_of_walk xs"
conv: shows "infsetsum'_converges (\<lambda>x. f x + g x) A"
Inf_fun_def: fixes A :: "('k \<Rightarrow> 'l) set" shows "Inf A = (\<lambda>x. INF f\<in>A. f x)"
drop_bit_eq_zero_iff_not_bit_last: fixes w :: "'b word" shows "(drop_bit (LENGTH('b) - Suc 0) w = 0) = (\<not> bit w (LENGTH('b) - Suc 0))"
Span_contains_spanset_append_right: fixes ms :: "'b list"   and ns :: "'b list" assumes "set ms \<subseteq> V"   and "set ns \<subseteq> V" shows "set ns \<subseteq> RSpan (ms @ ns)"
bounded_iff: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "(a \<le> b \<sqinter> c) = (a \<le> b \<and> a \<le> c)"
Q_exists: shows "\<exists>Q. DisjointPaths H_x v0 v1 Q \<and> card Q = sep_size"
axioms_1: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_fail_alt return bind fail alt" shows "monad_fail return bind fail"
S_W_no_deadends: fixes v :: "'a" assumes "v \<in> S - W" shows "\<not> deadend v"
Moebius_function_holomorphic: fixes w :: "complex"   and t :: "real" assumes "cmod w < 1" shows "Moebius_function t w holomorphic_on ball 0 1"
real_poly_uminus: fixes p :: "'a poly" assumes "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (- p)) \<subseteq> \<real>"
strict_mono_on_leD: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "strict_mono_on f A"   and "x \<in> A"   and "y \<in> A"   and "x \<le> y" shows "f x \<le> f y"
lg_maxr: fixes x :: "nat"   and y :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y" shows "rec_exec rec_lg [x, y] = Maxr lgR [x, y] x"
redT_updLs_preserves_lock_thread_ok: fixes ls :: "('d, 'e) locks"   and ts :: "'e \<Rightarrow> ('f \<times> 'd \<Rightarrow>f nat) option"   and t :: "'e"   and xw :: "'f \<times> 'd \<Rightarrow>f nat"   and las :: "'d lock_actions" assumes "lock_thread_ok ls ts"   and "ts t = \<lfloor>xw\<rfloor>" shows "lock_thread_ok (redT_updLs ls t las) ts"
ex_d2: shows "d \<in> wordinterval_to_set d_range"
env_restr_empty_iff: fixes S :: "'a set"   and m :: "'a \<Rightarrow> 'b" shows "(m f|` S = \<bottom>) = (edom m \<inter> S = {})"
curry_case_prod: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" shows "curry (\<lambda>(x, y). f x y) = f"
closure_open_segment: fixes a :: "'b"   and b :: "'b" shows "closure {a<--<b} = (if a = b then {} else {a--b})"
h5: shows "t mod cycleLength (nC k) mem schedule (nC k)"
match: shows "satisfy_equation_R p qs [([], []), ([0], []), ([], [0])] [[1], [0], [- 1]]"
app_vlrestrictionD_2: fixes a :: "V"   and b :: "V"   and r :: "V"   and A :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sup>l\<^sub>\<circ> A" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r"
subst_fv_unfold_ground_img: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "range_vars s = {}" shows "fv (t \<cdot> s) = fv t - subst_domain s"
distinct_35: fixes x3 :: "nat"   and x81 :: "nat"   and x82 :: "msg" shows "Nonce x3 \<noteq> Crypt x81 x82"
delta6_def: shows "delta6 = Var 0 * Var 3 * (- Var 0 + Var 1 + Var 2 - Var 3 + Var 4 + Var 5) + Var 1 * Var 4 * (Var 0 - Var 1 + Var 2 + Var 3 - Var 4 + Var 5) + Var 2 * Var 5 * (Var 0 + Var 1 - Var 2 + Var 3 + Var 4 - Var 5) - Var 1 * Var 2 * Var 3 - Var 0 * Var 2 * Var 4 - Var 0 * Var 1 * Var 5 - Var 3 * Var 4 * Var 5"
of_ocircline_o_x_axis: shows "of_ocircline o_x_axis = x_axis"
a: assumes "xs = [] \<Longrightarrow> thesis"   and "\<And>x ys. \<lbrakk>\<not> List.member ys x; xs \<approx> x # ys\<rbrakk> \<Longrightarrow> thesis" shows "thesis"
IH1: assumes "\<exists>pn. p1 \<in> subterms (\<Gamma> pn)"   and "q \<in> dterms \<Gamma> p1" shows "\<exists>pn. q \<in> subterms (\<Gamma> pn)"
funpow_swap1: fixes f :: "'d \<Rightarrow> 'd"   and n :: "nat"   and x :: "'d" shows "f ((f ^^ n) x) = (f ^^ n) (f x)"
SystemClasses_def: shows "SystemClasses \<equiv> [ObjectC, NullPointerC, ClassCastC, OutOfMemoryC]"
rsp: shows "rel_fun (=) (eq_onp (\<lambda>x. constant_term (denominator x) \<noteq> (0::'a))) (\<lambda>(x, y). let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'a) then 0 else quot_to_fract (x', y')) (\<lambda>(x, y). let (x', y') = normalize_quot (x, y) in if constant_term y' = (0::'a) then 0 else quot_to_fract (x', y'))"
less_than_bool_def: shows "less_than_bool \<equiv> {(a, b). a < b}"
cong_diff_iff_cong_0: fixes b :: "'a"   and c :: "'a"   and a :: "'a" shows "[b - c = 0::'a] (mod a) = [b = c] (mod a)"
maximal_no_deadend: fixes v :: "'a"   and Ps :: "'a llist" assumes "maximal_path (LCons v Ps)"   and "\<not> deadend v" shows "\<not> lnull Ps"
Four_Suc: shows "4 = Suc (Suc (Suc (Suc 0)))"
carrier_subgroup_generated_subset: fixes G :: "('e, 'f) monoid_scheme"   and A :: "'e set" assumes "Group.group G" shows "carrier (subgroup_generated G A) \<subseteq> carrier G"
rep_minus: fixes r :: "'c \<Rightarrow>\<^sub>0 'c"   and s :: "'c \<Rightarrow>\<^sub>0 'c" shows "ideal.rep (r - s) = ideal.rep r - ideal.rep s"
Map_cod: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" shows "Map (cod f) = Cod f"
finite_acyclic_wf: fixes r :: "('c \<times> 'c) set" assumes "finite r"   and "acyclic r" shows "wf r"
frees_app: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "frees (app u\<^sub>1 u\<^sub>2) = frees u\<^sub>1 |\<union>| frees u\<^sub>2"
induct: fixes P :: "(('b \<times> 'c) \<times> 'd) tree \<Rightarrow> 'b \<Rightarrow> bool"   and a0 :: "(('b \<times> 'c) \<times> 'd) tree"   and a1 :: "'b" assumes "\<And>x. P \<langle>\<rangle> x"   and "\<And>l a b uu_ r x. \<lbrakk>cmp x a = LT \<Longrightarrow> P l x; cmp x a = GT \<Longrightarrow> P r x\<rbrakk> \<Longrightarrow> P \<langle>l, ((a, b), uu_), r\<rangle> x" shows "P a0 a1"
l0e: fixes f2 :: "real"   and c :: "real"   and e :: "real"   and l0 :: "nat" assumes "Optimal f2 c e l0" shows "e * c / (f2 * (e - 1)) \<le> real l0"
sWlsBinp_def: fixes SEM :: "('h, 'i, 'j, 'sort, 'k, 'l, 'm) semDom_scheme"   and delta :: "'opSym"   and sbinp :: "'bindex \<Rightarrow> ('varSort, 'l) sAbs option" shows "sWlsBinp SEM delta sbinp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) sbinp \<and> liftAll2 (sWlsAbs SEM) (barOf delta) sbinp"
face_of_Times: fixes F :: "'c set"   and S :: "'c set"   and F' :: "'d set"   and S' :: "'d set" assumes "F face_of S"   and "F' face_of S'" shows "F \<times> F' face_of S \<times> S'"