hcju/isabelleps huggingface dataset:
interior_rel_interior: fixes S :: "'a set" assumes "aff_dim S = int DIM('a)" shows "rel_interior S = interior S"
simps_1: fixes m :: "nat \<Rightarrow> nat" shows "flush_all_until_volatile_write [] m = m"
ih2: fixes \<Gamma> :: "(name \<times> ty) list"   and s :: "trm"   and t :: "trm"   and u :: "trm" assumes "\<Gamma> \<turnstile> s is t : T\<^sub>2"   and "\<Gamma> \<turnstile> t is u : T\<^sub>2" shows "\<Gamma> \<turnstile> s is u : T\<^sub>2"
conv_contrav: fixes x :: "'a"   and y :: "'a" shows "(x ; y)\<^sup>\<smile> = y\<^sup>\<smile> ; x\<^sup>\<smile>"
Max_wrt_nonempty: assumes "alts \<noteq> {}"   and "finite alts" shows "Max_wrt (R i) \<noteq> {}"
all_set_inter_empty_distinct: fixes js :: "'a list" assumes "\<And>xs ys. js = xs @ ys \<Longrightarrow> set xs \<inter> set ys = {}" shows "distinct js"
prems_1: fixes i :: "nat" assumes "i < DIM('a)" shows "isDERIV i (Floor fa) (list_of_eucl x @ params)"
wti: fixes c :: "'a list"   and pc :: "nat"   and s :: "'a" shows "wti c pc s = merge c pc (step pc s) (c ! (pc + 1))"
uniform_limit_bounded: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and S :: "'d set"   and l :: "'d \<Rightarrow> 'e"   and F :: "'c filter" assumes "uniform_limit S f l F"   and "\<forall>\<^sub>F i in F. bounded (f i ` S)"   and "F \<noteq> bot" shows "bounded (l ` S)"
app1: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and es :: "exp0 list"   and s2 :: "'b state"   and vs :: "v list"   and env' :: "v sem_env"   and e :: "exp0"   and bv :: "'b state \<times> (v, v) result" assumes "evaluate_list ck env s1 (rev es) (s2, Rval vs) \<and> do_opapp (rev vs) = Some (env', e) \<and> (ck \<longrightarrow> clock s2 \<noteq> 0) \<and> evaluate ck env' (if ck then update_clock (\<lambda>_. clock s2 - 1) s2 else s2) e bv" shows "evaluate ck env s1 (App Opapp es) bv"
set_bit_greater_eq: fixes k :: "int"   and n :: "nat" shows "k \<le> set_bit n k"
sign_Pls_ge_0: fixes bin :: "int" shows "(bin_sign bin = 0) = (0 \<le> bin)"
possExImpliesNecEx_v2: shows "\<lfloor>(\<lambda>w. (\<^bold>\<diamond>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x)) \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa w \<longrightarrow> xa x))) w)\<rfloor>"
Cond: fixes e :: "expr"   and \<theta> :: "nat \<Rightarrow> nat option"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt"   and t :: "nat" assumes "\<forall>sop. e \<noteq> Tmp sop" shows "\<theta>\<turnstile> (Cond e s\<^sub>1 s\<^sub>2, t) \<rightarrow>\<^sub>p ((Cond (Tmp (eval_expr t e)) s\<^sub>1 s\<^sub>2, t + used_tmps e), issue_expr t e)"
ipresIGSubst_def: fixes h :: "'c \<Rightarrow> 'd"   and MOD :: "('e, 'f, 'varSort, 'sort, 'g, 'h, 'c, 'i, 'j) model_scheme"   and MOD' :: "('k, 'l, 'varSort, 'm, 'n, 'h, 'd, 'o, 'p) model_scheme" shows "ipresIGSubst h MOD MOD' \<equiv> \<forall>ys Y y s X. igWls MOD (asSort ys) Y \<and> igWls MOD s X \<longrightarrow> h (igSubst MOD ys Y y X) = igSubst MOD' ys (h Y) y (h X)"
i_set0_mod_0_mult_in_i_set0_mult: fixes I :: "nat set"   and m :: "nat"   and k :: "nat" assumes "I \<in> i_set0"   and "m mod k = 0" shows "I \<otimes> m \<in> i_set0_mult k"
SKIP: fixes X :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and p :: "L"   and H :: "Var \<Rightarrow> L"   and G :: "Var \<Rightarrow> L"   and q :: "L" shows "X \<rhd> Skip : Sec (Q p H) (EQ G q) (EQ G q) (\<lambda>(s, t). EQ G q s t)"
subclseq_acyclic: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "G\<turnstile>D\<preceq>\<^sub>C C"   and "ws_prog G" shows "C = D"
abst_trans_fm2: fixes i :: "name"   and j :: "name"   and A :: "fm" assumes "i \<noteq> j" shows "abst_dbfm i (Suc 0) (trans_fm [j] A) = trans_fm [j, i] A"
map_snd_zip: fixes xs :: "'e list"   and ys :: "'f list" assumes "length xs = length ys" shows "map snd (zip xs ys) = ys"
vreal_in_Vset_\<omega>2: shows "\<real>\<^sub>\<circ> \<in>\<^sub>\<circ> Vset (\<omega> + \<omega>)"
assms_7: shows "x = x1 \<oplus>\<^bsub>M\<^esub> x2"
right_total_All_transfer: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_total A" shows "((A ===> (=)) ===> (=)) (Ball (Collect (Domainp A))) All"
that: fixes t :: "'a \<Rightarrow>\<^sub>0 'b" assumes "t \<in> keys p"   and "poly_deg p = deg_pm t" shows "thesis"
Inv2a_innermost_def: fixes s :: "state"   and p :: "Proc"   and bk :: "DiskBlock" shows "Inv2a_innermost s p bk = (mbal bk \<in> Ballot p \<union> {0} \<and> bal bk \<in> Ballot p \<union> {0} \<and> (bal bk = 0) = (inp bk = NotAnInput) \<and> bal bk \<le> mbal bk \<and> inp bk \<in> allInput s \<union> {NotAnInput})"
continuous_on_def: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "continuous_on s f = (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
eq3: shows "tensor_P (adjoint m1 * m3) (adjoint m2 * m4) * tensor_P m1 m2 = tensor_P (adjoint m1 * m3 * m1) (adjoint m2 * m4 * m2)"
conj_zero_right: fixes x :: "'b" shows "x AND (0::'b) = (0::'b)"
Par_Wbis_discrR2: fixes c :: "('test, 'atom) com"   and d2 :: "('test, 'atom) com"   and d1 :: "('test, 'atom) com" assumes "c \<approx>w d2"   and "discr d1" shows "c \<approx>w d1 | d2"
abs_rangeI: fixes s :: "(nat \<times> nat) list"   and v :: "nat"   and a :: "nat" assumes "wf_partial_state s"   and "(v, a) \<in> set s" shows "abs_range_map v \<noteq> None"
leg_simps_1: shows "C.arr trg_trg.leg0"
wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_def: fixes \<sigma> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<sigma> \<equiv> \<forall>v. \<Gamma> (Var v) = \<Gamma> (\<sigma> v)"
one_OrdSum_def: shows "1 \<equiv> Abs_OrdSum (1::'a, 1::'b)"
dvd_mult: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m"   and c :: "'m"   and b :: "'m" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a c" shows "dvd.dvd times a (times b c)"
simps_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq [] = True"
min_product_min_set: fixes A :: "'b fset set"   and B :: "'b fset set" shows "min_set (A \<otimes>\<^sub>m B) = A \<otimes>\<^sub>m B"
rel_reflp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "reflp R" shows "reflp (rel_fun (=) R)"
not_sym: fixes t :: "'c"   and s :: "'c" assumes "t \<noteq> s" shows "s \<noteq> t"
Jacobi_mult_left: fixes a :: "int"   and b :: "int"   and n :: "int" shows "Jacobi (a * b) n = Jacobi a n * Jacobi b n"
inverse_mod_pow: fixes x :: "int"   and p :: "int"   and n :: "nat" assumes "comm_monoid_mult_class.coprime x p"   and "1 < p"   and "n \<noteq> 0" shows "inverse_mod x (p ^ n) * x mod p ^ n = 1"
weakCongISym: fixes Prop :: "ccs \<Rightarrow> ccs \<Rightarrow> bool"   and P :: "ccs"   and Q :: "ccs"   and F :: "ccs \<Rightarrow> ccs" assumes "Prop P Q"   and "\<And>P Q. Prop P Q \<Longrightarrow> Prop Q P"   and "\<And>P Q. Prop P Q \<Longrightarrow> F P \<leadsto><weakBisimulation> F Q" shows "F P \<cong> F Q"
lossless_def: fixes \<A> :: "('pub_key \<Rightarrow> (('plain \<times> 'plain) \<times> 'b) spmf) \<times> ('cipher \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "lossless \<A> = ((\<forall>pk. lossless_spmf (fst \<A> pk)) \<and> (\<forall>cipher \<sigma>. lossless_spmf (snd \<A> cipher \<sigma>)))"
fusion_intro: fixes z :: "i"   and F :: "i \<Rightarrow> bool" assumes "\<forall>y. O y z = (\<exists>x. F x \<and> O y x)" shows "(\<sigma>x. F x) = z"
Limsup_compose_continuous_antimono: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "continuous_on UNIV f"   and "antimono f"   and "F \<noteq> \<bottom>" shows "Limsup F (\<lambda>n. f (g n)) = f (Liminf F g)"
LIMSEQ_inverse_real_of_nat: shows "(\<lambda>n. inverse (real (Suc n))) \<longlonglongrightarrow> 0"
ee_pos: fixes t :: "real" assumes "t \<in> {0..1}" shows "0 < ee t"
simps_3: fixes uu :: "('d, bool) mapping" shows "reduce uu Falseif = Falseif"
zero_polynom_imp_zero_coeffs: fixes c :: "nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat" assumes "\<And>w. (\<Sum>i\<le>n. c i * w ^ i) = (0::'b)"   and "k \<le> n" shows "c k = (0::'b)"
rcosets_carrier: fixes X :: "nat set" assumes "Group.group G"   and "X \<in> rcosets\<^bsub>G\<^esub> H" shows "X \<subseteq> carrier G"
sq_mtx_uminus_eq: fixes A :: "'a sq_mtx" shows "- A = to_mtx (\<chi>i j. - A $$ i $ j)"
\<phi>_simps_2: shows "src r\<^sub>0s\<^sub>1.\<phi> = src r\<^sub>0s\<^sub>1.p\<^sub>0"
ceiling_add_of_int: fixes x :: "'b"   and z :: "int" shows "\<lceil>x + of_int z\<rceil> = \<lceil>x\<rceil> + z"
Cons: shows "vars_partitioned (subst_cls_lists Cs (renamings_apart Cs))"
assms_1: fixes x :: "'a"   and y :: "'a" shows "f (x + y) = f x + f y"
1: shows "\<p>\<^sub>1[\<nu>.cod.leg0, \<mu>.cod.leg1] \<cdot> chine_hcomp \<nu> \<mu> = \<nu>.chine \<cdot> \<p>\<^sub>1[\<nu>.dom.leg0, \<mu>.dom.leg1]"
widens_Cons: fixes P :: "'b prog"   and x :: "ty"   and xs :: "ty list"   and ys :: "ty list" shows "P \<turnstile> (x # xs) [\<le>] ys = (\<exists>z zs. ys = z # zs \<and> subtype P x z \<and> P \<turnstile> xs [\<le>] zs)"
total: fixes T :: "'d set"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "enumerateable T less" shows "\<forall>x\<in>T. \<forall>y\<in>T. x \<noteq> y \<longrightarrow> less x y \<or> less y x"
algebraic_int_minus: fixes x :: "'a" assumes "algebraic_int x" shows "algebraic_int (- x)"
fun_left_comm: fixes y :: "'a"   and x :: "'a"   and z :: "'b" shows "f y (f x z) = f x (f y z)"
not_one_less_zero: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less one zero"
assms: shows "run_one_step d i (s, vs, ves, $Unop_i x241 x242) = (s', vs', res)"
verit_la_disequality: fixes a :: "'c"   and b :: "'c" shows "a = b \<or> \<not> a \<le> b \<or> \<not> b \<le> a"
mapCollect_cong: fixes m1 :: "'a \<Rightarrow> 'b option"   and f1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and f2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and m2 :: "'a \<Rightarrow> 'b option" assumes "\<And>k v. m1 k = Some v \<Longrightarrow> f1 k v = f2 k v"   and "m1 = m2" shows "mapCollect f1 m1 = mapCollect f2 m2"
minus_tree_def: fixes x :: "'b tree"   and y :: "'b tree" shows "x - y = pure (-) \<diamondop> x \<diamondop> y"
wf_until_aux_def: fixes valid_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> event_data option list set \<times> event_data option list set) list \<Rightarrow> bool"   and init_muaux :: "Monitor.args \<Rightarrow> 'a"   and add_new_muaux :: "Monitor.args \<Rightarrow> event_data option list set \<Rightarrow> event_data option list set \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and length_muaux :: "Monitor.args \<Rightarrow> 'a \<Rightarrow> nat"   and eval_muaux :: "Monitor.args \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> event_data option list set list \<times> 'a"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and args :: "Monitor.args"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and aux :: "'a"   and ne :: "nat" assumes "muaux valid_muaux init_muaux add_new_muaux length_muaux eval_muaux" shows "muaux.wf_until_aux valid_muaux \<sigma> V R args \<phi> \<psi> aux ne = (fv \<phi> \<subseteq> fv \<psi> \<and> (\<exists>cur auxlist. valid_muaux args cur aux auxlist \<and> cur = (if ne + length auxlist = 0 then 0 else \<tau> \<sigma> (ne + length auxlist - 1)) \<and> wf_until_auxlist \<sigma> V (args_n args) R (args_pos args) \<phi> (args_ivl args) \<psi> auxlist ne))"
nxt_both_left: fixes t :: "'a ttree"   and x :: "'a"   and t' :: "'a ttree" assumes "possible t x" shows "nxt t x \<otimes>\<otimes> t' \<sqsubseteq> nxt (t \<otimes>\<otimes> t') x"
Units_closed: fixes x :: "'a mat" assumes "x \<in> Units (ring_mat TYPE('a) n ())" shows "x \<in> carrier (ring_mat TYPE('a) n ())"
bounded_byD2: fixes b :: "real"   and P :: "'c \<Rightarrow> real" assumes "bounded_by b P" shows "P \<tturnstile> \<lambda>s. b"
set_empty2: fixes xs :: "'g list" shows "({} = set xs) = (xs = [])"
orthogonal_transformation: fixes f :: "'a \<Rightarrow> 'b" shows "orthogonal_transformation f = (linear f \<and> (\<forall>v. norm (f v) = norm v))"
SETCompE: fixes f :: "ZF"   and g :: "ZF" assumes "f \<approx>>\<^bsub>SET\<^esub> g" obtains "isZFfun f" and "isZFfun g" and "|cod|f = |dom|g"
assms_2: shows "other.s \<delta> \<le> u_max"
dual_order: fixes L :: "('a, 'b) gorder_scheme" assumes "partial_order L" shows "partial_order (inv_gorder L)"
abel_semigroup_axioms: shows "abel_semigroup (\<and>*)"
universe_difference: fixes x :: "i" assumes "CMUD"   and "\<exists>w. \<not> O w x" shows "\<forall>w. P w (u \<ominus> x) = (\<not> O w x)"
timesmod1: fixes x :: "nat"   and y :: "nat"   and n :: "nat" shows "x * (y mod n) mod n = x * y mod n"
word_not_simps_2: fixes y :: "'b word" shows "\<not> word_pred 0 < y"
weight1: fixes "oracle" :: "'b \<Rightarrow> bool list \<Rightarrow> ((bool list \<times> bool list) \<times> 'b) spmf"   and s :: "'b" assumes "\<And>s x. lossless_spmf (oracle s x)" shows "weight_spmf (exec_gpv oracle \<A>1 s) = 1"
some_sym_eq_trivial: fixes x :: "'d" shows "(SOME y. x = y) = x"
gt_trans: fixes y :: "nat"   and x :: "nat"   and z :: "nat" assumes "y < x"   and "z < y" shows "z < x"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_pairs: fixes F :: "(('fun, 'b) Term.term \<times> ('fun, 'b) Term.term) list" assumes "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F)" shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (pair ` set F)"
upclosed_set_iff: fixes X :: "'a set" shows "upclosed_set X = (\<forall>x\<in>X. \<forall>y\<ge>x. y \<in> X)"
Hermite_Lindemann_aux4: fixes X :: "complex set"   and \<beta> :: "complex \<Rightarrow> complex" assumes "finite X"   and "\<And>x. x \<in> X \<Longrightarrow> algebraic x"   and "\<And>x. x \<in> X \<Longrightarrow> algebraic (\<beta> x)"   and "\<And>x. x \<in> X \<Longrightarrow> \<beta> x \<noteq> 0"   and "(\<Sum>x\<in>X. \<beta> x * exp x) = 0" shows "X = {}"
CollectD: fixes a :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "(a::'b) \<in> {x::'b. (P::'b \<Rightarrow> bool) x}" shows "(P::'b \<Rightarrow> bool) (a::'b)"
PiE_cong: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set"   and B :: "'c \<Rightarrow> 'd set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i = B i" shows "Pi\<^sub>E I A = Pi\<^sub>E I B"
3: fixes x :: "'a" assumes "x \<in> (+) (- a) ` S" shows "open_segment (0::'a) x \<subseteq> rel_interior ((+) (- a) ` S)"
assms: shows "is_valid_problem_sas_plus \<Psi>"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "V.arr f"   and "V.dom f = a"   and "V.cod f = b" shows "V.in_hom f a b"
rep_eq: fixes x :: "\<o>" shows "Semantics.d\<^sub>0 x = Some (eval\<o> x)"
simps_7: fixes m :: "'b" shows "normalize_match (MatchNot (Match m)) = [MatchNot (Match m)]"
phiArg_def: fixes g :: "'g"   and v :: "'val"   and v' :: "'val" shows "phiArg g v v' \<equiv> \<exists>vs. phi g v = Some vs \<and> v' \<in> set vs"
prime_power_mult_nat: fixes p :: "nat"   and x :: "nat"   and y :: "nat"   and k :: "nat" assumes "prime p"   and "x * y = p ^ k" shows "\<exists>i j. x = p ^ i \<and> y = p ^ j"
simps_3: fixes tst :: "'test"   and c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" shows "SC_ZObis (if tst then c1 else c2) = (if compatTst tst then SC_ZObis c1 \<and> SC_ZObis c2 else SC_discr (if tst then c1 else c2) \<or> SC_siso (if tst then c1 else c2))"
member_of_class_relation: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_of C" shows "G\<turnstile>C\<preceq>\<^sub>C declclass m"
traces_contain_events_def: fixes ES :: "'c ES_rec" shows "traces_contain_events ES \<equiv> \<forall>l\<in>Tr\<^bsub>ES\<^esub>. \<forall>e\<in>set l. e \<in> E\<^bsub>ES\<^esub>"
simps_8: fixes var :: "nat"   and a :: "atom fm"   and b :: "atom fm" shows "grab_atoms var (Or a b) = (case grab_atoms var a of None \<Rightarrow> None | Some al \<Rightarrow> case grab_atoms var b of None \<Rightarrow> None | Some bl \<Rightarrow> Some (al @ bl))"
is_None_def: fixes a :: "'b option" shows "is_None a \<equiv> case a of None \<Rightarrow> True | Some a \<Rightarrow> False"
path_component_eq: fixes y :: "'b"   and S :: "'b set"   and x :: "'b" assumes "y \<in> path_component_set S x" shows "path_component_set S y = path_component_set S x"
C3l: fixes x :: "'a"   and y :: "'a" assumes "x \<odot> y \<le> y" shows "x\<^sup>\<star> \<odot> y \<le> y"
Sup_prod_def: fixes X :: "('c \<times> 'd) set" shows "Sup_prod X = (\<Squnion> (fst ` X), \<Squnion> (snd ` X))"
intros: fixes pq :: "('c \<times> 'd) pmf"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and q :: "'d pmf" assumes "\<And>x y. (x, y) \<in> set_pmf pq \<Longrightarrow> R x y"   and "map_pmf fst pq = p"   and "map_pmf snd pq = q" shows "rel_pmf R p q"
sup_set_def: fixes A :: "'x set"   and B :: "'x set" shows "A \<union> B = Collect (sup (\<lambda>x. x \<in> A) (\<lambda>x. x \<in> B))"
max_Var_floatarith_1: shows "max_Var_floatarith 1 = 0"
Abs_eq_iff2_3: fixes cs :: "atom list"   and x :: "'a"   and cs' :: "atom list"   and y :: "'a" shows "([cs]lst. x = [cs']lst. y) = (\<exists>p. (cs, x) \<approx>lst (=) supp p (cs', y) \<and> supp p \<subseteq> set cs \<union> set cs')"
assms_5: shows "(rs @ s # ss) \<bullet>\<cdot> (ms @ n # ns) = (0::'m)"
image_eqI: fixes b :: "'h"   and f :: "'i \<Rightarrow> 'h"   and x :: "'i"   and A :: "'i set" assumes "b = f x"   and "x \<in> A" shows "b \<in> f ` A"
intros_1: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and u :: "'f \<Rightarrow> 'g"   and a :: "'h"   and l' :: "'e"   and u' :: "'f \<Rightarrow> 'g" assumes "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsub>a\<^esub> \<langle>l', u'\<rangle>" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow> \<langle>l',u'\<rangle>"
abs_induct: fixes P :: "'b trm \<Rightarrow> bool"   and x :: "'b trm" assumes "\<And>y. P (abs_trm y)" shows "P x"
pow_A_1_def: fixes x :: "(real, 'n) Finite_Cartesian_Product.vec" shows "pow_A_1 x = A1n *v x"
normal_imp_subgroup: fixes H :: "'a set"   and G :: "('a, 'b) monoid_scheme" assumes "H \<lhd> G" shows "subgroup H G"
bo_fg: shows "bounded {x. \<not> (f x = x \<and> g x = x)}"
associative: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = x \<squnion> y \<squnion> z"
elims: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and x :: "('b \<times> nat) tree"   and xa :: "'b"   and xb :: "('b \<times> nat) tree"   and y :: "('b \<times> nat) tree" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "WBT.balanceL \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 x xa xb = y" obtains l :: "('b \<times> nat) tree" and a :: "'b" and r :: "('b \<times> nat) tree" where "x = l" and "xa = a" and "xb = r" and "y = (if WBT.balanced1 \<Delta>1 \<Delta>2 l r then N l a r else WBT.rotateL \<Gamma>1 \<Gamma>2 l a r)"
m_closed: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "x \<oplus> y \<in> carrier Zp"
dep_iff_supset_circuit: fixes X :: "'a set" assumes "X \<subseteq> carrier" shows "(\<not> indep X) = (\<exists>C. circuit C \<and> C \<subseteq> X)"
of_int_diff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and w :: "int"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (w - z) = minus (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z)"
inv_char: fixes G :: "('b, 'c) monoid_scheme"   and x :: "'b"   and y :: "'b" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x = y"
apsnd_def: fixes f :: "'b \<Rightarrow> 'c" shows "apsnd f = map_prod id f"
USdB_Strong_d_Bisimulation: fixes d :: "'d" shows "Strong_d_Bisimulation d (\<approx>\<^bsub>d\<^esub>)"
induct: fixes P :: "'a\<cdot>btree \<Rightarrow> bool"   and x :: "'a\<cdot>btree" assumes "adm P"   and "P \<bottom>"   and "\<And>a. P (Leaf\<cdot>a)"   and "\<And>app1 app2. \<lbrakk>P app1; P app2\<rbrakk> \<Longrightarrow> P (Node\<cdot>app1\<cdot>app2)" shows "P x"
vsubstEnv_compose1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y2 :: "'var"   and y :: "'var"   and y1 :: "'var" assumes "goodEnv rho" shows "rho &[y1 // y]_ys &[y2 // y]_ys = rho &[y1 @ys[y2 / y]_ys // y]_ys"
unit_simps_1: fixes a :: "'b" assumes "B.obj a" shows "C.arr (H.unit a)"
P_wrt_elem: shows "P = P - {Y} \<union> {Y}"
prime_elem_imp_coprime: fixes p :: "'a"   and n :: "'a" assumes "prime_elem p"   and "\<not> p dvd n" shows "coprime p n"
split_spec_exact_def: fixes x :: "'a set" shows "split_spec_exact x = SPEC (\<lambda>(a, b). x = a \<union> b)"
b2: shows "b2 = p / (2::'a) + m + q / ((2::'a) * a)"
assms_3: fixes i :: "'a" assumes "i \<in> I" shows "openin T (U i)"
powsum_mpoly_recurrence': fixes X :: "nat set"   and k :: "nat" assumes "card X < k"   and "finite X" shows "powsum_mpoly X k = - (\<Sum>i = 1..card X. (- 1) ^ i * sym_mpoly X i * powsum_mpoly X (k - i))"
afRead_def: fixes afile :: "nat \<times> (nat \<Rightarrow> byte)"   and byteIndex :: "nat" shows "afRead afile byteIndex \<equiv> if byteIndex < fst afile then Some (snd afile byteIndex) else None"
cdijkstra_refines: shows "cdijkstra g v0 \<le> \<Down> (br mr.\<alpha> mr.invar) mdijkstra"
s_prod_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<cdot> x \<le> z \<cdot> y"
finite_comm_group_axioms: shows "finite_comm_group (Characters (Characters G))"
AsParallelExprs: fixes i :: "nat"   and cs :: "('d, 'e, 'f) com list"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "'e \<Rightarrow> ('d, 'e, 'f) ann list option"   and as :: "(('d, 'e, 'f) ann \<times> 'd set \<times> 'd set) list"   and a :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "i < length cs"   and "assertionsR \<Gamma> \<Theta> (fst (snd (as ! i))) (snd (snd (as ! i))) (fst (as ! i)) (cs ! i) a" shows "assertionsR \<Gamma> \<Theta> Q A (AnnPar as) (Parallel cs) a"
\<sigma>2': shows "\<sigma>2' \<in> seq ars"
vbrelation_Limit_vsubset_VsetI: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (\<CC>\<lparr>Dom\<rparr>) \<subseteq>\<^sub>\<circ> Vset \<alpha>" shows "\<CC>\<lparr>Dom\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
bounded_linear_apply_blinfun: fixes b :: "'d" shows "bounded_linear (\<lambda>a. blinfun_apply a b)"
assms_2: shows "simulation bs\<^sub>0 blstep cs\<^sub>0 clstep R\<^sub>2"
mk_eqcl'_bound: fixes xs :: "nat option list"   and l :: "nat"   and x :: "nat option"   and i :: "nat"   and j :: "nat"   and T :: "bool list list"   and k :: "nat" assumes "\<And>x k. \<lbrakk>x \<in> set xs; x = Some k\<rbrakk> \<Longrightarrow> k < l"   and "x \<in> set (mk_eqcl' xs i j l T)"   and "x = Some k" shows "k \<le> l"
suffix_conc_snd: fixes u :: "'c list"   and n :: "nat"   and v :: "nat \<Rightarrow> 'c" assumes "length u \<le> n" shows "suffix n (u \<frown> v) = suffix (n - length u) v"