hcju/isabelleps huggingface dataset:
pwd_spy_id_prik_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_id_prik n)"
B22: fixes x :: "'a" shows "x\<^sup>\<star> \<odot> x\<^sup>\<star> = x\<^sup>\<star>"
simps_2: fixes P :: "'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
abs_eq: fixes x :: "64 word" shows "uint8_of_uint64 (Abs_uint64 x) = Abs_uint8 (ucast x)"
r: shows "p x *R c + (\<Sum>y\<in>{v. p v \<noteq> 0} \<inter> (UNIV - {x}). p y *R v y) = r"
ComplexLikePoset_special_cosets: shows "ComplexLikePoset order.greater_eq order.greater \<P>"
vimage_Collect: fixes P :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P (f x) = Q x" shows "f -` Collect P = Collect Q"
rminusinf_inf: fixes p :: "fm"   and bs :: "real list" assumes "isrlfm p" shows "\<exists>z. \<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p"
assms_3: shows "x[[top]] \<le> forest_components h * e * top"
poly_map_poly_eval_poly: fixes p :: "int poly" shows "ipoly p = eval_poly of_int p"
range_eq_singletonD: fixes f :: "'d \<Rightarrow> 'c"   and a :: "'c"   and x :: "'d" assumes "range f = {a}" shows "f x = a"
inj_PairVal: shows "inj (\<lambda>(x, y). <|x, y|>)"
assms_3: fixes A :: "'a set" assumes "A \<in> sets M" shows "(T -` A \<inter> space M \<in> null_sets M) = (A \<in> null_sets M)"
wf_dom_def: fixes ss :: "nat list list"   and w :: "nat list" shows "wf_dom ss w \<equiv> (\<forall>\<tau>\<in>set ss. \<tau> \<in> A) \<and> (\<forall>p<n. Sorted_Less.sorted (rev (ss ! p)) \<and> (ss ! p \<noteq> rev [0..<n] \<longrightarrow> (\<forall>x\<in>set (ss ! p). x < p)) \<and> (ss ! p = rev [0..<n] \<longrightarrow> (\<exists>x\<in>set w. (x, p) \<in> g_E G \<and> x < p)) \<and> (p \<notin> set w \<longrightarrow> stable r step ss p)) \<and> Sorted_Less.sorted w \<and> length ss = n \<and> (\<forall>x\<in>set w. x < n)"
Equality_ax_imp_hfthm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Equality_ax" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> A"
l: shows "extreme ({p \<in> B. f p \<sim> p} \<union> P \<inter> B) (\<lambda>x y. y \<sqsubseteq> x) l"
path_trans_single_neg: fixes SES :: "('a, 'b) SES_rec"   and s1 :: "'a"   and t :: "'b list"   and s2 :: "'a"   and e :: "'b"   and sn :: "'a" assumes "s1 t\<Longrightarrow>\<^bsub>SES\<^esub> s2"   and "T\<^bsub>SES\<^esub> s2 e \<noteq> Some sn" shows "path SES s1 (t @ [e]) \<noteq> Some sn"
rewrite_neg: fixes m :: "'a common_primitive match_expr" assumes "normalized_nnf_match m" shows "matches \<gamma> (rewrite_neg m) a p = matches \<gamma> m a p"
H_on_ket_zero_is_state: shows "state 1 (H * |Deutsch.zero\<rangle>)"
get_disconnected_nodes_is_l_get_disconnected_nodes: shows "l_get_disconnected_nodes ShadowRootClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs"
prime_dvd_choose: fixes k :: "nat"   and p :: "nat" assumes "0 < k"   and "k < p"   and "prime p" shows "p dvd p choose k"
CompDefined_def: fixes CC :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" shows "f \<approx>>\<^bsub>CC\<^esub> g \<equiv> f \<in> mor\<^bsub>CC\<^esub> \<and> g \<in> mor\<^bsub>CC\<^esub> \<and> cod\<^bsub>CC\<^esub> f = dom\<^bsub>CC\<^esub> g"
wf_Nil: shows "[] \<turnstile>\<^sub>w\<^sub>f"
sub_sup_closed': fixes x :: "'a"   and y :: "'a" shows "\<exists>z. - x \<squnion> - y = - z"
ownership_distinct_nth_update: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<O>' :: "nat set"   and sb' :: "'a memref list"   and p' :: "'a"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and \<D>' :: "bool"   and \<R>' :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<forall>j<length ts\<^sub>s\<^sub>b. i \<noteq> j \<longrightarrow> (let (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j) = ts\<^sub>s\<^sub>b ! j in (\<O>' \<union> all_acquired sb') \<inter> (\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) = {})" shows "ownership_distinct (ts\<^sub>s\<^sub>b[i := (p', is', \<theta>', sb', \<D>', \<O>', \<R>')])"
biQuots_append: fixes v :: "'c list"   and u :: "'c list"   and w :: "'c list"   and A :: "'c list set" shows "(v \<in> biQuots u w A) = (u @ v @ rev w \<in> A)"
simply_connected_eq_homotopic_circlemaps2b: fixes S :: "'a set" assumes "\<And>f g. \<lbrakk>continuous_on (sphere 0 1) f; f ` sphere 0 1 \<subseteq> S; continuous_on (sphere 0 1) g; g ` sphere 0 1 \<subseteq> S\<rbrakk> \<Longrightarrow> homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f g" shows "path_connected S"
cnt_fun: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set set"   and T :: "'c set set" assumes "f \<in> cnt S T" shows "f \<in> carrier\<^bsub>S\<^esub> \<rightarrow> carrier\<^bsub>T\<^esub>"
Abs_real_int_induct: fixes P :: "real_int \<Rightarrow> bool"   and x :: "real_int" assumes "\<And>y. y \<in> {r. fst r \<le> snd r} \<Longrightarrow> P (Abs_real_int y)" shows "P x"
contf': shows "continuous_on (convex hull {b, a, c}) f"
SN_partI: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "\<And>y. r x y \<Longrightarrow> SN_partp r y" shows "SN_partp r x"
completely_multiplicative_fds_inverse: fixes f :: "nat \<Rightarrow> 'b" assumes "completely_multiplicative_function f" shows "inverse (fds f) = (\<chi>n. moebius_mu n * f n)"
MakeFtor_def: fixes F :: "('a, 'b, 'c, 'd, 'e, 'f, 'g) Functor_scheme" shows "MakeFtor F \<equiv> \<lparr>CatDom = CatDom F, CatCod = CatCod F, MapM = restrict ((##) F) (mor\<^bsub>CatDom F\<^esub>), \<dots> = Functor.more F\<rparr>"
psubstEnv_preserves_freshEnv: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "goodEnv rho"   and "goodEnv rho'"   and "freshEnv zs z rho"   and "freshEnv zs z rho'" shows "freshEnv zs z rho &[rho']"
not_col_permutation_3: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "\<not> Col A B C" shows "\<not> Col C B A"
column_Gram_Schmidt_upt_k_preserves2: fixes a :: "'a"   and i :: "nat"   and j :: "nat"   and A :: "((real, 'a) vec, 'b) vec" assumes "a \<le> mod_type_class.from_nat i"   and "i \<le> j"   and "j < ncols A" shows "column a (Gram_Schmidt_upt_k A i) = column a (Gram_Schmidt_upt_k A j)"
Conseq: fixes pre :: "'c set"   and pre' :: "'c set"   and rely :: "('c \<times> 'c) set"   and rely' :: "('c \<times> 'c) set"   and guar' :: "('c \<times> 'c) set"   and guar :: "('c \<times> 'c) set"   and post' :: "'c set"   and post :: "'c set"   and P :: "'c com" assumes "pre \<subseteq> pre'"   and "rely \<subseteq> rely'"   and "guar' \<subseteq> guar"   and "post' \<subseteq> post"   and "\<turnstile> P sat [pre', rely', guar', post']" shows "\<turnstile> P sat [pre, rely, guar, post]"
f: shows "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)"
strip_narrow_acom: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "strip C1 = strip C2" shows "strip (C1 \<triangle> C2) = strip C1"
connection_def: fixes G :: "('d, 'e, 'f, 'g, 'h) galcon_scheme" shows "connection G \<equiv> (partial_order \<X>\<^bsub>G\<^esub> \<and> partial_order \<Y>\<^bsub>G\<^esub>) \<and> \<pi>\<^sup>*\<^bsub>G\<^esub> \<in> carrier \<X>\<^bsub>G\<^esub> \<rightarrow> carrier \<Y>\<^bsub>G\<^esub> \<and> \<pi>\<^sub>*\<^bsub>G\<^esub> \<in> carrier \<Y>\<^bsub>G\<^esub> \<rightarrow> carrier \<X>\<^bsub>G\<^esub>"
minGraphProps6: fixes g :: "graph"   and v :: "nat"   and f :: "face" assumes "minGraphProps g"   and "v \<in> \<V> g"   and "f \<in> set (facesAt g v)" shows "v \<in> \<V> f"
is_iso_class_Comp: fixes G :: "'a set"   and F :: "'a set" assumes "Comp G F \<noteq> {}" shows "B.is_iso_class (Comp G F)"
real_cond_exp_mult: fixes M :: "'c measure"   and F :: "'c measure"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "f \<in> borel_measurable F"   and "g \<in> borel_measurable M"   and "integrable M (\<lambda>x. f x * g x)" shows "AEeq M (real_cond_exp M F (\<lambda>x. f x * g x)) (\<lambda>x. f x * real_cond_exp M F g x)"
hyps: fixes i :: "nat" assumes "i < length xs"   and "\<And>j. \<lbrakk>j < length xs; j \<noteq> i\<rbrakk> \<Longrightarrow> xs ! j = (0::'a)" shows "sum_list xs = xs ! i"
star_comm: fixes p :: "assertion"   and q :: "assertion" shows "p \<star> q = q \<star> p"
weak_map_of_SomeI: fixes k :: "'b"   and x :: "'c"   and l :: "('b \<times> 'c) list" assumes "(k, x) \<in> set l" shows "\<exists>x. map_of l k = \<lfloor>x\<rfloor>"
simps_5: fixes va :: "'b formula" shows "is_lit_plus (\<^bold>\<not> (\<^bold>\<not> va)) = False"
to_fract_mult: fixes x :: "'b"   and y :: "'b" shows "to_fract (x * y) = to_fract x * to_fract y"
pbeta_Upd'_3: shows "Obj f_ T_ => Obj f'_ T_"
induct: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> ('a \<times> 'b) tree \<Rightarrow> bool"   and a0 :: "'a"   and a1 :: "'b"   and a2 :: "('a \<times> 'b) tree" assumes "\<And>k p. P k p \<langle>\<rangle>"   and "\<And>k p l k1 p1 r. \<lbrakk>k < k1 \<Longrightarrow> P k p l; \<lbrakk>\<not> k < k1; k1 < k\<rbrakk> \<Longrightarrow> P k p r\<rbrakk> \<Longrightarrow> P k p \<langle>l, (k1, p1), r\<rangle>" shows "P a0 a1 a2"
assms: shows "oarrivemsg (\<lambda>\<sigma> m. msg_fresh \<sigma> m \<and> P \<sigma> m) \<sigma> m"
eq_\<I>_gpv_inline: fixes S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and \<I>' :: "('e, 'f) \<I>"   and callee1 :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a, 'e, 'f) gpv"   and callee2 :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b, 'e, 'f) gpv"   and s1 :: "'a"   and s2 :: "'b"   and A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and gpv1 :: "('g, 'c, 'd) gpv"   and gpv2 :: "('h, 'c, 'd) gpv" assumes "\<And>s1 s2 q. \<lbrakk>S s1 s2; q \<in> outs_\<I> \<I>\<rbrakk> \<Longrightarrow> eq_\<I>_gpv (rel_prod (eq_onp (\<lambda>r. r \<in> responses_\<I> \<I> q)) S) \<I>' (callee1 s1 q) (callee2 s2 q)"   and "S s1 s2"   and "eq_\<I>_gpv A \<I> gpv1 gpv2" shows "eq_\<I>_gpv (rel_prod A S) \<I>' (inline callee1 gpv1 s1) (inline callee2 gpv2 s2)"
hull_subset: fixes s :: "'b set"   and S :: "'b set \<Rightarrow> bool" shows "s \<subseteq> S hull s"
singleton_orbit_eq_fixed_point: fixes m :: "'a set" assumes "m \<in> subgroups_of_size (p ^ a)" shows "(card (conjP.orbit m) = 1) = (m \<in> conjP.fixed_points)"
add_independentS_ConsI: fixes As :: "'a set list"   and A :: "'a set" assumes "add_independentS As"   and "\<And>x a. \<lbrakk>x \<in> (\<Sum>X\<leftarrow>As. X); a \<in> A; a + x = (0::'a)\<rbrakk> \<Longrightarrow> a = (0::'a)" shows "add_independentS (A # As)"
vs1: shows "vs1 \<equiv> getFrN (map snd txs) (t1 # map fst txs) [] (length txs)"
mbisim_actions_ok_bisim_no_join_12: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'b) thread_action"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'b) thread_action"   and t :: "'t" assumes "s1 \<approx>m s2"   and "collect_cond_actions \<lbrace>ta1\<rbrace>\<^bsub>c\<^esub> = {}"   and "ta1 \<sim>m ta2"   and "r1.actions_ok s1 t ta1" shows "r2.actions_ok s2 t ta2"
less_minus_one_simps_2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (uminus one) one"
update_id: fixes s :: "('i, 'a) state"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "let s' = update x c s in \<T> s' = \<T> s \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U> s' = \<U> s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s"
partCompat_def: fixes P :: "'a set set"   and theta :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "partCompat P theta f \<equiv> \<forall>I\<in>P. Resumption_Based.compat I theta f"
llength_ltl: fixes xs :: "'b llist" shows "llength (ltl xs) = epred (llength xs)"
icd_uniq: fixes m :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and n' :: "nat" assumes "m icd\<^bsup>\<pi>\<^esup>\<rightarrow> n"   and "m icd\<^bsup>\<pi>\<^esup>\<rightarrow> n'" shows "n = n'"
transI: fixes r :: "('g \<times> 'g) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (y, z) \<in> r\<rbrakk> \<Longrightarrow> (x, z) \<in> r" shows "trans r"
self_domain_iff_ide: fixes a :: "'a1" shows "(a \<in> C1.domains a) = C1.ide a"
E_def: shows "E \<equiv> \<lambda>k. lift (\<omega>_sum (take k \<alpha>s)) (\<omega> \<up> \<alpha>s ! k)"
finite_supp_abs_residual_pair: fixes \<alpha> :: "'b"   and P :: "'c" shows "finite (supp \<langle>\<alpha>,P\<rangle>)"
basic_trans_rules_22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x < y"   and "y \<le> z" shows "x < z"
inequality_case: fixes a :: "real"   and b :: "real"   and c :: "real" shows "(\<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) = (a < 0 \<or> a = 0 \<and> (0 < b \<or> b = 0 \<and> c < 0))"
p\<^sub>1_simps: shows "trg f\<^sub>0g\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 f)"
det_Gauss_Jordan: fixes A :: "(('b, 'c) vec, 'c) vec" shows "det (Gauss_Jordan A) = (\<Prod>i\<in>UNIV. Gauss_Jordan A $ i $ i)"
endofunctor_axioms: shows "endofunctor Right_a.comp (H\<^sub>R a')"
inst_ok_imp_wf_inst: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set" assumes "inst_ok \<Theta> insts" shows "wf_inst \<Theta> (\<lambda>idn S. the_default (Tv idn S) (lookup (\<lambda>x. x = (idn, S)) insts))"
am_inj_code: fixes m :: "nat" shows "inj_on enc (cw_len_concat -` {m})"
ennreal_top_minus: fixes x :: "ennreal" shows "top - x = top"
kruskal_correct: fixes L :: "(nat \<times> int \<times> nat) list" shows "<emp> kruskal (return L) (\<lambda>(u, w, v). return (u, v)) () <\<lambda>r. \<up> (distinct r \<and> set r \<subseteq> set L \<and> minimum_spanning_forest (GD_from_list_\<alpha>_graph L r) (GD_from_list_\<alpha>_graph L L))>\<^sub>t"
return_bind: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'a Heap" shows "return x \<bind> f = f x"
prems: shows "invar \<langle>l, (ab, lv), r\<rangle>"
le_principal: fixes F :: "'a filter"   and A :: "'a set" shows "(F \<le> principal A) = (\<forall>\<^sub>F x in F. x \<in> A)"
fixing_ide_gives_functor_2: fixes a2 :: "'c" assumes "C.ide a2" shows "functor (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<lambda>f1. Hom.map (f1, a2))"
step_3: shows "\<I> \<turnstile>g gpv \<surd>"
index_smult_vec_1: fixes i :: "nat"   and v :: "'c vec"   and a :: "'c" assumes "i < dim_vec v" shows "(a \<cdot>\<^sub>v v) $ i = a * v $ i"
cfWritePreservesNextFreeBlockInvariant: fixes cfile1 :: "CFile"   and byteIndex :: "nat"   and "value" :: "byte"   and cfile2 :: "CFile" assumes "reachabilityInvariant cfile1"   and "cfWrite cfile1 byteIndex value = Some cfile2" shows "nextFreeBlockInvariant cfile2"
Red_lcl_add: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and l\<^sub>0 :: "char list \<Rightarrow> val option" assumes "P \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>" shows "P \<turnstile> \<langle>e,(h, l\<^sub>0 ++ l)\<rangle> \<rightarrow>* \<langle>e',(h', l\<^sub>0 ++ l')\<rangle>"
row_transpose: fixes M :: "'a mat"   and n :: "nat"   and i :: "nat" assumes "dim_col M = n"   and "i < n" shows "row M\<^sup>t i = col M i"
leftmost_is_nonterminal: fixes i :: "nat"   and u :: "symbol list" assumes "leftmost i u" shows "is_nonterminal (u ! i)"
eq_l: fixes a :: "MIR.num" shows "isrlfm (rsplit eq a)"
calls_chain_def: fixes \<Gamma> :: "char list \<Rightarrow> 'b rule list option" shows "calls_chain \<Gamma> = _Collect (r, s) (case \<Gamma> r of None \<Rightarrow> False | Some rs \<Rightarrow> \<exists>m. Rule m (Call s) \<in> set rs)"
CFG_SSA_axioms: shows "CFG_SSA \<alpha>e \<alpha>n invar inEdges' Entry defs uses phis"
elims_2: fixes x :: "nat"   and xa :: "floatarith"   and xb :: "real list" assumes "isDERIV x xa xb" obtains xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xc__" and "xa = Add a__ b__" and "xb = vs__" and "isDERIV xc__ a__ vs__ \<and> isDERIV xc__ b__ vs__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xca__" and "xa = Mult aa__ ba__" and "xb = vsa__" and "isDERIV xca__ aa__ vsa__ \<and> isDERIV xca__ ba__ vsa__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcb__" and "xa = Minus ab__" and "xb = vsb__" and "isDERIV xcb__ ab__ vsb__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcc__" and "xa = Inverse ac__" and "xb = vsc__" and "isDERIV xcc__ ac__ vsc__ \<and> interpret_floatarith ac__ vsc__ \<noteq> 0"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcd__" and "xa = Cos ad__" and "xb = vsd__" and "isDERIV xcd__ ad__ vsd__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xce__" and "xa = Arctan ae__" and "xb = vse__" and "isDERIV xce__ ae__ vse__"   | xc :: "nat" and vs :: "real list" where "x = xcf__" and "xa = Pi" and "xb = vsf__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcg__" and "xa = Sqrt af__" and "xb = vsg__" and "isDERIV xcg__ af__ vsg__ \<and> 0 < interpret_floatarith af__ vsg__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xch__" and "xa = Exp ag__" and "xb = vsh__" and "isDERIV xch__ ag__ vsh__"   | xc :: "nat" and a :: "floatarith" and b :: "floatarith" and vs :: "real list" where "x = xci__" and "xa = Powr ah__ b" and "xb = vsi__" and "isDERIV xci__ ah__ vsi__ \<and> isDERIV xci__ b vsi__ \<and> 0 < interpret_floatarith ah__ vsi__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcj__" and "xa = Ln ai__" and "xb = vsj__" and "isDERIV xcj__ ai__ vsj__ \<and> 0 < interpret_floatarith ai__ vsj__"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xck__" and "xa = Floor aj__" and "xb = vsk__" and "isDERIV xck__ aj__ vsk__ \<and> interpret_floatarith aj__ vsk__ \<notin> \<int>"   | xc :: "nat" and a :: "floatarith" and vs :: "real list" where "x = xcl__" and "xa = ak__ ^\<^sub>e 0" and "xb = vsl__"   | xc :: "nat" and a :: "floatarith" and n :: "nat" and vs :: "real list" where "x = xcm__" and "xa = a ^\<^sub>e Suc n__" and "xb = vsm__" and "isDERIV xcm__ a vsm__"   | xc :: "nat" and f :: "float" and vs :: "real list" where "x = xcn__" and "xa = Num f" and "xb = vsn__"   | xc :: "nat" and n :: "nat" and vs :: "real list" where "x = xc" and "xa = Var n" and "xb = vs"
tape_of_nat: fixes a :: "nat" shows "<a> = Oc \<up> Suc a"
\<sigma>_def: shows "\<sigma> = override_on (override_on succ \<sigma>2 V') \<sigma>1 (attractor p K - K)"
lookup_pair_tl: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and k :: "'a" assumes "oalist_inv_raw x xs" shows "lookup_pair x (tl xs) k = (if \<forall>k'\<in>fst ` set xs. le x k k' then 0::'c else lookup_pair x xs k)"
antisym: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "blinding_of_on A h bo"   and "bo x y"   and "bo y x"   and "x \<in> A" shows "x = y"
is_extensional: fixes f :: "'a \<times> 'a \<times> 'a" assumes "\<not> B.VVV.arr f" shows "B.HoVH f = B.null"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "weak_partial_order L"   and "weak_upper_semilattice_axioms L" shows "weak_upper_semilattice L"
lc_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.lc p = lookup p (punit.lt p)"
K_GDD_def: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set" shows "K_GDD point_set block_collection sizes groups \<equiv> K_\<Lambda>_GDD point_set block_collection sizes groups 1"
induct: fixes f :: "'d \<Rightarrow> 'd option"   and m :: "'d \<Rightarrow> nat"   and P :: "'d \<Rightarrow> bool"   and a0 :: "'d" assumes "\<And>x. (\<And>x2. \<lbrakk>f x = Some x2; m x2 < m x\<rbrakk> \<Longrightarrow> P x2) \<Longrightarrow> P x" shows "P a0"
FF_dim_3: fixes n :: "nat"   and fs :: "'b vec list" assumes "gram_schmidt_fs_Rn n fs" shows "mat_of_rows n fs \<in> carrier_mat (length fs) n"
closed_segment_def: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b = {(1 - u) *\<^sub>R a + u *\<^sub>R b |u. 0 \<le> u \<and> u \<le> 1}"
cubd_def: shows "cubd = cindex_poly_ubd (map_poly Im p) (map_poly Re p)"
arr_Rel_converse_Rel_converse_Rel: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "(T\<inverse>\<^sub>R\<^sub>e\<^sub>l)\<inverse>\<^sub>R\<^sub>e\<^sub>l = T"
assms_1: shows "p \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> w) \<and> (s \<notin> b \<longrightarrow> s \<in> w')}"
U_meas: shows "(\<Union>n. (T ^^ n) --` A) \<in> sets M"
assms_3: shows "ps (S t (Suc i)) p \<noteq> None"
inv_f_eq: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "inj f"   and "f x = y" shows "inv f y = x"
pull_up_configuration_rank: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and q :: "'d"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n"   and "y \<in> semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n" shows "semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.rank \<Sigma> \<delta> q\<^sub>0 w y n"
elims_3: fixes x :: "primfun"   and xa :: "primfun" assumes "\<not> x < xa" obtains xb :: "nat" and y :: "nat" where "x = LnChain xb" and "xa = LnChain y" and "\<not> xb < y"
proots_within_0: fixes s :: "'a set" shows "proots_within 0 s = s"
invar: shows "round_robin_invar \<sigma> (dom (thr s))"
Legendre_1_left: fixes p :: "int" assumes "prime p" shows "Legendre 1 p = 1"
not_shadows_vconsts_alt_def_1: fixes name :: "name"   and vs :: "value list" shows "not_shadows_vconsts (Vconstr name vs) = (True \<and> list_all not_shadows_vconsts vs)"
mcont_def: fixes luba :: "'d set \<Rightarrow> 'd"   and orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and lubb :: "'e set \<Rightarrow> 'e"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" shows "mcont luba orda lubb ordb f = (monotone orda ordb f \<and> cont luba orda lubb ordb f)"