hcju/isabelleps huggingface dataset:
desargues_config_2D_rkRa_rkRb_rkRc_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and R :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {P, Q, R} = 2"   and "rk {Q, R} = 2"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {R, c} = 2"
sorted_wrt2: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and zs :: "'a list" assumes "transp P" shows "sorted_wrt P (x # y # zs) = (P x y \<and> sorted_wrt P (y # zs))"
strongAppend: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi"   and Rel' :: "(pi \<times> pi) set"   and R :: "pi"   and Rel'' :: "(pi \<times> pi) set" assumes "P \<leadsto><Rel> Q"   and "Q \<leadsto>[Rel'] R"   and "eqvt Rel''"   and "Rel O Rel' \<subseteq> Rel''" shows "P \<leadsto><Rel''> R"
select_convs_7: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'a" shows "cIntensityF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cIntensityF"
hom2: shows "(\<lambda>(a, as). a # as) \<in> ring_hom (RDirProd S nil_ring) (RDirProd_list [S])"
PRE_IMP_REACH: fixes blues :: "'v set"   and reds :: "'v set"   and onstack :: "'v set"   and u0 :: "'v" assumes "pre (blues, reds, onstack, u0)" shows "u0 \<in> reachable"
map_def: fixes f :: "'j" shows "\<alpha>.map f \<equiv> if J.arr f then a else C.null"
fmeasurableD: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> fmeasurable M" shows "A \<in> sets M"
poly_pos_leq: fixes a :: "real"   and p :: "real poly" shows "(\<forall>x\<le>a. 0 < poly p x) = (p \<noteq> 0 \<and> poly_neg_inf p = 1 \<and> count_roots_below p a = 0)"
chine_composite_2: shows "H\<mu>HH\<nu>\<pi>\<rho>.chine = \<mu>.leg0 \<down>\<down> \<nu>\<pi>.leg1 \<cdot> \<nu>\<pi>_\<rho>.prj\<^sub>1"
assms_2: shows "\<^bold>|v\<^bold>| \<le> \<^bold>|y\<^bold>|"
null_part_null_sets: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets (completion M)" shows "null_part M S \<in> null_sets (completion M)"
bhvks3: fixes t :: "nat \<Rightarrow> cnf"   and p :: "PROB"   and kId :: "'kid"   and t' :: "nat \<Rightarrow> 'KS" assumes "t \<in> arch"   and "p = prob kId" shows "sb.eval kId t t' 0 (\<box>\<^sub>b[\<lambda>ks. p \<in> ksop ks]\<^sub>b \<longrightarrow>\<^sup>b \<diamond>\<^sub>b[\<lambda>ks. \<exists>P. sub P = ksrp ks]\<^sub>b)"
cupcake_single_correct: fixes env :: "v sem_env"   and e :: "exp0"   and res :: "(v, v) result"   and s :: "'b state" assumes "cupcake_evaluate_single env e res"   and "is_cupcake_all_env env"   and "is_cupcake_exp e" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate env s e (s', res)"
field_differentiable_at_Arccos: fixes z :: "complex" assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1" shows "Arccos field_differentiable at z"
pos_less_minus_divide_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less a (uminus (divide b c)) = less (times a c) (uminus b)"
sorted_list_of_set_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (linorder.sorted_list_of_set less_eq A)"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on T g"   and "g ` T \<subseteq> U"   and "\<And>x. x \<in> S \<Longrightarrow> g x = f x" shows "thesis"
absc_def: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg" shows "absc cfg = cfg_corec (abss (state cfg)) (abst \<circ> action) (\<lambda>cfg s. cont cfg (THE x. abss x = s \<and> x \<in> set_pmf (action cfg))) cfg"
d2: shows "z_R I sign = 1 \<longrightarrow> (\<forall>p\<in>set (retrieve_polys qs (fst I)). poly p x = 0) \<and> 0 < poly (prod_list (retrieve_polys qs (snd I))) x"
a: shows " t \<longrightarrow>\<^sub>\<beta> s"
least_min_algs: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and i :: "((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set" assumes "alg B1 B2 s1 s2" shows "i \<in> Field SucFbd \<longrightarrow> fst (min_algs s1 s2 i) \<subseteq> B1 \<and> snd (min_algs s1 s2 i) \<subseteq> B2"
UnD1_single: fixes c :: "'a"   and a :: "'a"   and B :: "'a set" assumes "c \<in> {a} \<union> B"   and "c \<noteq> a" shows "c \<in> B"
states_def: fixes c :: "('a, 'b) conf" shows "states c = {\<sigma>. \<exists>\<sigma>\<^sub>s\<^sub>y\<^sub>m. consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m (store c) \<and> conjunct (pred c) \<sigma>\<^sub>s\<^sub>y\<^sub>m}"
InvariantCFalseAfterDecide: fixes conflictFlag :: "bool"   and M :: "(Literal \<times> bool) list"   and C :: "Literal list"   and M' :: "(Literal \<times> bool) list"   and literal :: "Literal" assumes "InvariantCFalse conflictFlag M C"   and "M' = M @ [(literal, True)]" shows "InvariantCFalse conflictFlag M' C"
wp_validI: fixes P :: "'b \<Rightarrow> real"   and prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real" assumes "P \<tturnstile> wp prog Q" shows "\<lbrace>P\<rbrace> prog \<lbrace>Q\<rbrace>p"
hereditarily_fs_alpha_renaming: fixes f :: "'e"   and \<alpha> :: "'d"   and t\<^sub>\<alpha> :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>"   and f' :: "'e"   and \<alpha>' :: "'d"   and t\<^sub>\<alpha>' :: "('b, 'c, 'd, 'e) Tree\<^sub>\<alpha>" assumes "Act\<^sub>\<alpha> f \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> f' \<alpha>' t\<^sub>\<alpha>'" shows "hereditarily_fs t\<^sub>\<alpha> = hereditarily_fs t\<^sub>\<alpha>'"
hyps_for_Rule: fixes ip :: "('form, 'var) antecedent"   and r :: "'rule" assumes "ip |\<in>| f_antecedent r" shows "hyps_for (Rule r) ip = (\<lambda>h. Hyp h ip) |`| a_hyps ip"
suminf_f': shows "suminf f' = enn2real (\<Sum>\<^sup>+ x. ennreal (f' x))"
dec_i_def: fixes ba :: "'a list \<times> 'a list" shows "dec_i ba = (case ba of (b, a) \<Rightarrow> (tl b, hd b # a))"
mHom_def: fixes R :: "('j, 'k) Ring_scheme"   and M :: "('h, 'j, 'l) Module_scheme"   and N :: "('i, 'j, 'm) Module_scheme" shows "mHom R M N = {f \<in> aHom M N. \<forall>a\<in>carrier R. \<forall>m\<in>carrier M. f (a \<cdot>\<^sub>s\<^bsub>M\<^esub> m) = a \<cdot>\<^sub>s\<^bsub>N\<^esub> f m}"
interpret_slp_of_fas': fixes fas :: "floatarith list"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list"   and M' :: "(floatarith, nat) mapping"   and slp' :: "floatarith list"   and xs :: "real list" assumes "slp_of_fas' fas M slp = (M', slp')"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> subterms f \<subseteq> Mapping.keys M"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> the (Mapping.lookup M f) < length slp"   and "\<And>f. f \<in> Mapping.keys M \<Longrightarrow> interpret_slp slp xs ! slp_index_lookup slp M f = interpret_floatarith f xs" shows "\<Union> (subterms ` set fas) \<subseteq> Mapping.keys M' \<and> Mapping.keys M \<subseteq> Mapping.keys M' \<and> (\<forall>f\<in>Mapping.keys M'. subterms f \<subseteq> Mapping.keys M' \<and> the (Mapping.lookup M' f) < length slp' \<and> interpret_slp slp' xs ! slp_index_lookup slp' M' f = interpret_floatarith f xs)"
in_set_conv_nth: fixes x :: "'g"   and xs :: "'g list" shows "(x \<in> set xs) = (\<exists>i<length xs. xs ! i = x)"
mult_vec_distrib_left: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and r :: "'a \<Rightarrow>\<^sub>0 'd"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and q :: "'a \<Rightarrow>\<^sub>0 'd" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.mult_vec pair_of_term term_of_pair r (p + q) = term_powerprod.mult_vec pair_of_term term_of_pair r p + term_powerprod.mult_vec pair_of_term term_of_pair r q"
prv_all_neg_imp_neg_exi: fixes x :: "'var"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla" shows "prv (imp (all x (neg \<phi>)) (neg (exi x \<phi>)))"
lcm_coprime: fixes a :: "'a"   and b :: "'a" assumes "coprime a b" shows "lcm a b = normalize (a * b)"
plus_mreal_def: fixes x :: "mreal"   and y :: "mreal" shows "x + y \<equiv> mreal_max x y"
le_ennreal_iff: fixes r :: "real"   and x :: "ennreal" assumes "0 \<le> r" shows "(x \<le> ennreal r) = (\<exists>q\<ge>0. x = ennreal q \<and> q \<le> r)"
CD_on_Xd: fixes x :: "'x"   and ds :: "'d set"   and A :: "'x set" assumes "x \<in> CD_on ds A" shows "Xd x \<in> ds"
le_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<sqsubseteq> z"
norm_mult: fixes x :: "'a"   and y :: "'a" shows "norm (x * y) = norm x * norm y"
no_intersect: shows "(\<lambda>(y, d). Fun.swap x y d) ` {(y, d). y \<in> S \<and> y \<noteq> x \<and> d \<in> derangements (S - {x, y})} \<inter> (\<lambda>(y, f). f(x := y, inv f y := x)) ` ((S - {x}) \<times> derangements (S - {x})) = {}"
right_unique_mat_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "right_unique R" shows "right_unique (mat_rel R)"
continuous_within_eps_delta: fixes x :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "continuous (at x within s) f = (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
test_conv_var: fixes x :: "'a" assumes "is_test x" shows "x\<^sup>\<smile> \<le> 1'"
code_word_length_def: fixes e :: "'c list \<Rightarrow> bool list"   and l :: "'c" shows "code_word_length e l = length (e [l])"
that: fixes a :: "'a" assumes "A = a \<cdot> 1" shows "thesis"
closed: shows "is_TComp_var_instance_closed \<Gamma> M"
ffb_prop: fixes f :: "'d \<Rightarrow> 'c set" shows "fb\<^sub>\<F> f = \<partial> \<circ> bd\<^sub>\<F> (op\<^sub>K f) \<circ> \<partial>"
mat_carrier: fixes nr :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "Matrix.mat nr nc f \<in> carrier_mat nr nc"
symp_conversep: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "symp R = (R\<inverse>\<inverse> \<le> R)"
getRel_hom: fixes y :: "'d"   and z :: "'d"   and l :: "'e"   and G :: "('e, 'd) labeled_graph"   and f :: "'d \<Rightarrow> 'f" assumes "(y, z) \<in> getRel l G"   and "y \<in> vertices G"   and "z \<in> vertices G" shows "(f y, f z) \<in> getRel l (map_graph_fn G f)"
l_lagrange: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set" assumes "Group.group G"   and "finite (carrier G)"   and "subgroup H G" shows "card (lcosets\<^bsub>G\<^esub> H) * card H = order G"
hyps_4: shows "p\<^sub>4 = local.add a p\<^sub>1 p\<^sub>2"
real_le_x_sinh: fixes x :: "real" assumes "0 \<le> x" shows "x \<le> (exp x - inverse (exp x)) / 2"
full_abstraction_wrt_preorders_impl_trans_source_target_relation: fixes SRel :: "('procS \<times> 'procS) set"   and TRel :: "('procT \<times> 'procT) set" assumes "\<forall>S1 S2. ((S1, S2) \<in> SRel) = ((\<lbrakk>S1\<rbrakk>, \<lbrakk>S2\<rbrakk>) \<in> TRel)"   and "refl SRel"   and "trans TRel" shows "\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel \<and> (TargetTerm (\<lbrakk>S\<rbrakk>), SourceTerm S) \<in> Rel) \<and> SRel = _Collect (S1, S2) ((SourceTerm S1, SourceTerm S2) \<in> Rel) \<and> TRel = _Collect (T1, T2) ((TargetTerm T1, TargetTerm T2) \<in> Rel) \<and> trans Rel"
integral_h_holomorphic: fixes r :: "real"   and b :: "real"   and c :: "real"   and A :: "complex set" assumes "r \<in> {0<..<2}" shows "(\<lambda>s. integral {b..c} (h r s)) holomorphic_on A"
mintree_exists: fixes bq :: "('c, 'd) SkewBinomialTree list" shows "(bq \<noteq> []) = (getMinTree bq \<in> set bq)"
ccw_switch23: fixes P :: "real \<times> real"   and Q :: "real \<times> real" assumes "det3 0 P Q \<noteq> 0" shows "(\<not> ccw 0 Q P) = ccw 0 P Q"
distinct_load_tmps_issue_expr: fixes t :: "nat"   and e :: "expr" shows "distinct_load_tmps (issue_expr t e)"
substTrans: fixes b :: "name"   and P :: "pi"   and a :: "name"   and c :: "name" assumes "b \<sharp> P" shows "P[a::=b][b::=c] = P[a::=c]"
elims: fixes x :: "'a list"   and y :: "nat \<Rightarrow> 'a list" assumes "perm2 x = y" obtains "x = []" and "y = (\<lambda>n. [])"   | xa :: "'a" and l :: "'a list" where "x = xa # l" and "y = (\<lambda>n. rotateLeft (length (xa # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l)))) - n mod (1 + length l) mod length (xa # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l))))) (xa # rotateLeft (n mod (1 + length l)) (perm2 l (n div (1 + length l)))))"
derives1_pad: fixes \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and u :: "symbol list"   and v :: "symbol list" assumes "derives1 \<alpha> \<beta>"   and "is_sentence u"   and "is_sentence v" shows "derives1 (u @ \<alpha> @ v) (u @ \<beta> @ v)"
adds_minus: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c"   and t :: "'c" assumes "class.comm_powerprod plus minus zero"   and "plus.adds plus s t" shows "plus (minus t s) s = t"
waveD_OUT: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "wave \<Gamma> f"   and "x \<notin> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)" shows "d_OUT f x = 0"
Diff_Diff_Un: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A - B - C = A - (B \<union> C)"
max_var_def: fixes lp :: "linear_poly" shows "max_var lp \<equiv> if lp = 0 then 0 else Max (vars lp)"
subi: shows "f ` [H]\<^bsup>\<gamma>\<^esup> \<subseteq> {i}"
continuous_closedin_preimage_gen: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set"   and U :: "'c set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "closedin (top_of_set T) U" shows "closedin (top_of_set S) (S \<inter> f -` U)"
\<psi>_eqI: fixes x1 :: "'a set"   and x2 :: "'b"   and y1 :: "'a set"   and y2 :: "'b" assumes "(x1, x2) \<sim> (y1, y2)" shows "\<psi> x1 x2 = \<psi> y1 y2"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_interval_set_aux (x # xs) (y # ys) = (if x < y then False else if y < x then proper_interval (Some y) (Some x) \<or> ys \<noteq> [] \<or> \<not> exhaustive_above x xs else proper_interval_set_aux xs ys)"
that: fixes z :: "'a"   and Z :: "'a multiset" assumes "X = add_mset z Z"   and "(z, y) \<in> ns"   and "(Z, Y) \<in> multpw ns" shows "thesis"
Abs: fixes T :: "POPLmark.type"   and t :: "trm" shows "valuep (\<lambda>:T. t)"
get_min_in: fixes h :: "'b hp option" assumes "h \<noteq> None" shows "get_min h \<in> set_hp (the h)"
plusplus_empty: fixes q :: "nat"   and S :: "(nat \<times> 'b) list"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ss :: "'b list" assumes "\<forall>s'. (q, s') \<in> set S \<longrightarrow> s' \<squnion>\<^bsub>f\<^esub> ss ! q = ss ! q" shows "map snd (filter (\<lambda>(p', t'). p' = q) S) \<Squnion>\<^bsub>f\<^esub> ss ! q = ss ! q"
simps_2: fixes n :: "nat" shows "Num (Suc n) = suc (Num n)"
zeroset_eqI: fixes A :: "'a set" assumes "(0::'a) \<in> A"   and "\<And>a. a \<in> A \<Longrightarrow> a = (0::'a)" shows "A = 0"
AssertionStatEq_def: 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 \<Psi> :: "'d"   and \<Psi>' :: "'d" assumes "assertionAux SCompose SImp SBottom SChanEq" shows "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>' \<equiv> assertionAux.AssertionStatImp SImp \<Psi> \<Psi>' \<and> assertionAux.AssertionStatImp SImp \<Psi>' \<Psi>"
disj: shows "x \<in> set (vwalk_arcs p) \<or> x = (last p, hd q) \<or> x \<in> set (vwalk_arcs q)"
wordinterval_union_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_union r1 r2 = RangeUnion r1 r2"
inversions_between_permute1: fixes \<pi> :: "nat \<Rightarrow> nat"   and xs :: "'a list"   and ys :: "'a list" assumes "\<pi> permutes {..<length xs}" shows "inversions_between (permute_list \<pi> xs) ys = map_prod (inv \<pi>) id ` inversions_between xs ys"
inner_same_Basis: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set"   and u :: "'e" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis"   and "u \<in> Basis" shows "inner u u = 1"
subspace_Int: fixes I :: "'c set"   and s :: "'c \<Rightarrow> 'b set" assumes "\<And>i. i \<in> I \<Longrightarrow> subspace (s i)" shows "subspace (\<Inter> (s ` I))"
lconf_upd2: fixes P :: "'a prog"   and h :: "heap"   and l :: "sfields"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and T :: "ty"   and V :: "char list" assumes "P,h \<turnstile> l (:\<le>) E"   and "P,h \<turnstile> v :\<le> T" shows "P,h \<turnstile> l(V \<mapsto> v) (:\<le>) E(V \<mapsto> T)"
fset_of_list_subset: fixes xs :: "'a list"   and ys :: "'a list" assumes "set xs \<subseteq> set ys" shows "fset_of_list xs |\<subseteq>| fset_of_list ys"
simps_2: fixes x :: "'b"   and l :: "'b tree"   and r :: "'b tree"   and h :: "nat" shows "ht (MKT x l r h) = h"
bal_split_min: fixes t :: "'b tree234"   and x :: "'b"   and t' :: "'b up\<^sub>d" assumes "split_min t = (x, t')"   and "bal t"   and "0 < height t" shows "bal (tree\<^sub>d t')"
before_fun2: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'b \<Rightarrow> 'd"   and y :: "'b \<Rightarrow> 'e" shows "\<turnstile> $f<x, y> = f<$x, $y>"
inject: fixes x1 :: "bool"   and x2 :: "'a \<Rightarrow> 'a language"   and y1 :: "bool"   and y2 :: "'a \<Rightarrow> 'a language" shows "(Lang x1 x2 = Lang y1 y2) = (x1 = y1 \<and> x2 = y2)"
paidAnm_inbound: fixes n :: "nat"   and m :: "nat" assumes "n < length paid_A"   and "m < length (paid_A ! n)" shows "Suc (paid_A ! n ! (length (paid_A ! n) - Suc m)) < length init"
ex_eqvt: fixes p :: "perm"   and P :: "'a \<Rightarrow> bool" shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
assms: shows "half_segments_of_aform X \<noteq> []"
distance_upper_bound: fixes v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto> w" shows "distance v w \<le> length xs"
rGamma_series_complex_converges: fixes z :: "complex" shows "convergent (rGamma_series z)"
ce_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat" shows "ce_code node_eq_impl node_hash_impl \<equiv> \<lambda>x. let y' = x in case y' of (xa, xb, xca, xd, xe) \<Rightarrow> let y'a = \<lambda>xf. case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xf xd of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then nat i < as_top xca else False; y'b = \<lambda>xf. case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl xf xd of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then as_top xca \<le> nat i else False in RETURN (Some (y'a, y'b), xa, xb, xca, xd, xe)"
prod_Suc_fact: fixes n :: "nat" shows "prod Suc {0..<n} = fact n"
Basic: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e" shows "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> \<Rightarrow> Normal (f s)"
And_Not_def: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "And_Not \<phi> \<psi> = Neg (Or (Neg \<phi>) \<psi>)"
abs_summable_finiteI_converse: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and F :: "'c set" assumes "f abs_summable_on S"   and "finite F"   and "F \<subseteq> S" shows "(\<Sum>x\<in>F. norm (f x)) \<le> (\<Sum>\<^sub>ax\<in>S. norm (f x))"
simps_5: fixes x :: "Vars.var"   and S :: "stack_elem list" shows "Astack (Dummy x # S) = 0"
sorted_list_of_multiset_mset: fixes xs :: "'c list" shows "sorted_list_of_multiset (mset xs) = sort xs"
containsCall_induct: fixes P :: "(char list \<times> char list list \<times> char list list \<times> cmd) list \<Rightarrow> cmd \<Rightarrow> char list list \<Rightarrow> char list \<Rightarrow> bool"   and a0 :: "(char list \<times> char list list \<times> char list list \<times> cmd) list"   and a1 :: "cmd"   and a2 :: "char list list"   and a3 :: "char list" assumes "\<And>procs ps p. P procs Skip ps p"   and "\<And>procs V e ps p. P procs (V:=e) ps p"   and "\<And>procs c\<^sub>1 c\<^sub>2 ps p. \<lbrakk>P procs c\<^sub>1 ps p; P procs c\<^sub>2 ps p\<rbrakk> \<Longrightarrow> P procs (c\<^sub>1;; c\<^sub>2) ps p"   and "\<And>procs b c\<^sub>1 c\<^sub>2 ps p. \<lbrakk>P procs c\<^sub>1 ps p; P procs c\<^sub>2 ps p\<rbrakk> \<Longrightarrow> P procs (if (b) c\<^sub>1 else c\<^sub>2) ps p"   and "\<And>procs b c ps p. P procs c ps p \<Longrightarrow> P procs (while (b) c) ps p"   and "\<And>procs q es' rets' ps p. (\<And>x xa xb xc. \<lbrakk>ps = q # xc; (q, x, xa, xb) \<in> set procs\<rbrakk> \<Longrightarrow> P procs xb xc p) \<Longrightarrow> P procs (Call q es' rets') ps p" shows "P a0 a1 a2 a3"
path_evalF'': fixes fs :: "formula list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and A :: "formula" assumes "branch subs (pseq fs) f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "\<exists>i. EV (contains f (i, A))" shows "\<not> evalF (counterModel f) counterAssign A"
eval\<^sub>\<sigma>: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "(cms\<^sub>1 ! k, subst \<sigma> (fst (mems ! k))) \<leadsto> (cms\<^sub>1' ! k, subst \<sigma> mem\<^sub>1')"
heap_stat_upd: fixes g :: "oref \<Rightarrow> obj option"   and C :: "qtname"   and obj :: "obj"   and l :: "lname \<Rightarrow> val option" shows "heap (st (g(Inr C \<mapsto> obj)) l) = heap (st g l)"
ps_1: shows "(ps, i) \<in>. omit_branch xs branch"
assms_5: shows "hisPred t n nid n' = hisPred t n nid n''"
splus_left_canc: fixes t :: "'a"   and v :: "'t"   and u :: "'t" shows "(t \<oplus> v = t \<oplus> u) = (v = u)"
DW_def: fixes \<rho> :: "complex mat" shows "DW \<rho> = denote (While M S) \<rho>"
fadd_def: fixes m :: "roundmode"   and a :: "('a, 'b) IEEE.float"   and b :: "('a, 'b) IEEE.float" shows "fadd m a b = (if is_nan a \<or> is_nan b \<or> is_infinity a \<and> is_infinity b \<and> sign a \<noteq> sign b then some_nan else if is_infinity a then a else if is_infinity b then b else zerosign (if is_zero a \<and> is_zero b \<and> sign a = sign b then sign a else if m = To_ninfinity then 1 else 0) (IEEE.round m (valof a + valof b)))"