hcju/isabelleps huggingface dataset:
ExitBlockStuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and nrms :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and abrs :: "('e, 'd, 'f) com list" assumes "\<Gamma>\<turnstile>nrms,css \<Down> Stuck" shows "\<Gamma>\<turnstile>[],(nrms, abrs) # css \<Down> Stuck"
h3a: shows "\<forall>x. List.member (find_nonzeros_from_input_vec (solve_for_lhs p qs subsets (M_mat signs subsets))) x \<longrightarrow> x < length signs"
simplify_finite: fixes S :: "'at Literal set set" assumes "all_fulfill finite S" shows "all_fulfill finite (simplify S)"
wf_ruleset_append: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs1 :: "'b rule list"   and rs2 :: "'b rule list" shows "wf_ruleset \<gamma> p (rs1 @ rs2) = (wf_ruleset \<gamma> p rs1 \<and> wf_ruleset \<gamma> p rs2)"
emeasure_bind: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c measure"   and N :: "'c measure"   and X :: "'c set" assumes "space M \<noteq> {}"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "X \<in> sets N" shows "emeasure (M \<bind> f) X = \<integral>\<^sup>+ x. emeasure (f x) X \<partial>M"
converse_rtranclE: fixes x :: "'a"   and z :: "'a"   and r :: "('a \<times> 'a) set" assumes "(x, z) \<in> r\<^sup>*" obtains "x = z"   | y :: "'a" where "(x, y) \<in> r" and "(y, z) \<in> r\<^sup>*"
Sup_inf: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and B :: "'c set"   and a :: "'c" assumes "class.complete_distrib_lattice Inf Sup inf less_eq less sup bot top" shows "inf (Sup B) a = Sup ((\<lambda>b. inf b a) ` B)"
tarski_absolute_space_axioms: shows "tarski_absolute_space real_hyp2_C real_hyp2_B"
distinct_length_2_or_more: fixes a :: "'c"   and b :: "'c"   and xs :: "'c list" shows "distinct (a # b # xs) = (a \<noteq> b \<and> distinct (a # xs) \<and> distinct (b # xs))"
degree_poly_of_list_le: fixes ws :: "'a list" shows "degree (poly_of_list ws) \<le> length ws - 1"
less_1_mult: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and m :: "'b"   and n :: "'b" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one m"   and "less one n" shows "less one (times m n)"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "C.ide a"   and "C.seq f a" shows "f \<cdot>\<^sub>C a = f"
simps: fixes a1 :: "com"   and a2 :: "char list \<Rightarrow> val option"   and a3 :: "nat"   and a4 :: "com"   and a5 :: "char list \<Rightarrow> val option" shows "\<langle>a1,a2\<rangle> \<rightarrow>\<^bsup>a3\<^esup> \<langle>a4,a5\<rangle> = ((\<exists>c s. a1 = c \<and> a2 = s \<and> a3 = 0 \<and> a4 = c \<and> a5 = s) \<or> (\<exists>c s c'' s'' n c' s'. a1 = c \<and> a2 = s \<and> a3 = Suc n \<and> a4 = c' \<and> a5 = s' \<and> \<langle>c,s\<rangle> \<rightarrow> \<langle>c'',s''\<rangle> \<and> \<langle>c'',s''\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>c',s'\<rangle>))"
filter_sup_symmetric: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and F :: "'b set"   and G :: "'b set" assumes "class.semilattice_inf inf less_eq less" shows "semilattice_inf.filter_sup inf less_eq F G = semilattice_inf.filter_sup inf less_eq G F"
Abs_ordinal_inverse: fixes y :: "ord0 set" assumes "y \<in> UNIV // ord0rel" shows "Rep_ordinal (Abs_ordinal y) = y"
DB_step_simulation: shows "\<turnstile> [DBNext]_(inp, mid, out, q1, q2) \<longrightarrow> [Next inp qc out]_(inp, qc, out)"
DERIV_const: fixes k :: "'a"   and F :: "'a filter" shows "((\<lambda>x. k) has_field_derivative (0::'a)) F"
trace_callee_closure_start: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf" shows "trace_callee_closure callee1 callee2 A p q p q"
qPsubstInp_def: fixes qrho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option"   and qinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qTerm option" shows "qinp %[[qrho]] \<equiv> lift (qPsubst qrho) qinp"
empty_eq_map_of_iff: fixes xys :: "('b \<times> 'c) list" shows "(Map.empty = map_of xys) = (xys = [])"
hash_blindable_simps_1: fixes h :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "hash_blindable h (Unblinded x) = Content (h x)"
E9: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> G) \<longrightarrow> \<diamond>F \<longrightarrow> \<diamond>G"
RPCISpec_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState" shows "RPCISpec send rcv rst \<equiv> \<forall>p. RPCIPSpec send rcv rst p"
select_convs_2: fixes access :: "acc_modi"   and more :: "'b" shows "decl.more \<lparr>access = access, \<dots> = more\<rparr> = more"
rep_eq: fixes x :: "'a fset" shows "fthe_elem x = the_elem (fset x)"
finite_field_factorization_main_integer: fixes p :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "poly_mod_prime p"   and "finite_field_factorization_main p (finite_field_ops_integer (integer_of_int p)) f = (c, fs)"   and "poly_mod.square_free_m p f" shows "poly_mod.unique_factorization_m p f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
tllist_all2_ttlI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist" assumes "tllist_all2 P Q xs ys"   and "\<not> is_TNil xs \<or> \<not> is_TNil ys" shows "tllist_all2 P Q (ttl xs) (ttl ys)"
image_inv_into_cancel: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and A' :: "'b set"   and B' :: "'b set" assumes "f ` A = A'"   and "B' \<subseteq> A'" shows "f ` inv_into A f ` B' = B'"
has_domain_iff_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "(partial_magma.domains C f \<noteq> {}) = partial_magma.arr C f"
vec_elements_h_def: fixes v :: "('b, 'c) Finite_Cartesian_Product.vec" shows "vec_elements_h v = range (($h) v)"
intro: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "\<And>x y. \<lbrakk>less_eq x y; less_eq y x; x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> x = y" shows "antisymmetric A less_eq"
mkAutoEC_ec: fixes t :: "'s Trace"   and a :: "'a" assumes "t \<in> jkbpC" shows "runJP mkAutoEC t a = {t' \<in> jkbpC. jview a t' = jview a t}"
cases: fixes a1 :: "Core.term"   and a2 :: "Core.term" assumes "a1 \<rightarrow>\<^sub>\<eta> a2" obtains s :: "Core.term" and T :: "typ" where "a1 = Abs T__ (s__ $ Bv 0)" and "a2 = decr 0 s__" and "\<not> is_dependent s__"   | s :: "Core.term" and t :: "Core.term" and u :: "Core.term" where "a1 = sa__ $ u__" and "a2 = t__ $ u__" and "sa__ \<rightarrow>\<^sub>\<eta> t__"   | s :: "Core.term" and t :: "Core.term" and u :: "Core.term" where "a1 = u $ sb__" and "a2 = u $ ta__" and "sb__ \<rightarrow>\<^sub>\<eta> ta__"   | s :: "Core.term" and t :: "Core.term" and T :: "typ" where "a1 = Abs T s" and "a2 = Abs T t" and "s \<rightarrow>\<^sub>\<eta> t"
AnnoBlock: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and V :: "String.literal"   and T :: "ty"   and e :: "'b expr"   and e' :: "'b expr"   and vo :: "'b val option" assumes "is_lub,P,E(V \<mapsto> T) \<turnstile> e \<leadsto> e'" shows "is_lub,P,E \<turnstile> {V:T=vo; e} \<leadsto> {V:T=vo; e'}"
simps_1: fixes used :: "'a set"   and p :: "('a, 'b) fm" shows "witness used (\<^bold>\<not> \<^bold>\<forall> p) = {\<^bold>\<not> p\<langle>\<^bold>\<dagger> (SOME a. a \<notin> used) []/0\<rangle>}"
map_sel_1: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b le_constraint" shows "lec_rel (map_le_constraint f a) = lec_rel a"
prob_space_measure_pmf: fixes p :: "'a pmf" shows "prob_space (measure_pmf p)"
add_Suc_right: fixes m :: "nat"   and n :: "nat" shows "m + Suc n = Suc (m + n)"
sin_cos_squared_add3: fixes x :: "'a" shows "cos x * cos x + sin x * sin x = (1::'a)"
per_compare_def: fixes R :: "('a \<times> 'b) set"   and a :: "'a"   and b :: "'b" shows "per_compare R a b \<equiv> (a, b) \<in> R"
insert_map_inverse: fixes i :: "nat"   and x :: "'c"   and f :: "nat \<Rightarrow> 'c" shows "delete_map i (insert_map i x f) = f"
map_simp_ide: fixes a :: "'a" assumes "hom.ide a" shows "\<phi>.map a = (d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, a \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (a \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[a, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> a \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[a \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[a]"
spaceN_add: fixes x :: "'c"   and N :: "'c quasinorm"   and y :: "'c" assumes "x \<in> space\<^sub>N N"   and "y \<in> space\<^sub>N N" shows "x + y \<in> space\<^sub>N N"
non_empty_min: fixes i :: "nat_int" shows "i \<noteq> bot \<longrightarrow> (\<exists>m. minimum i = m)"
mod_neq_imp_diff_mod_neq0: fixes a :: "nat"   and m :: "nat"   and b :: "nat" assumes "a mod m \<noteq> b mod m"   and "a \<le> b" shows "0 < (b - a) mod m"
card_of_ordIso_subst: fixes A :: "'d set"   and B :: "'d set" assumes "A = B" shows "ordIso2 (card_of A) (card_of B)"
assms_1: shows "PTA_Reachability.pairwise eq_elapsed xs"
intros_8: fixes F :: "'b formula" shows "((F::'b formula) \<^bold>\<rightarrow> \<bottom>) \<^bold>\<rightarrow> \<^bold>\<not> F \<in> AX10"
comp_inj_on_iff: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and f' :: "'b \<Rightarrow> 'c" assumes "inj_on f A" shows "inj_on f' (f ` A) = inj_on (f' \<circ> f) A"
Alloc_def2: fixes G :: "prog"   and otag :: "obj_tag"   and P :: "'b assn"   and Y :: "vals"   and s :: "state"   and Z :: "'b" shows "Alloc G otag P Y s Z = (\<forall>s' a. G\<turnstile>s \<midarrow>halloc otag\<succ>a\<rightarrow> s' \<longrightarrow> P (In1 (Addr a)) s' Z)"
compass_defs_2: shows "west = proj2_abs (vector [- 1, 0, 1])"
is_poincare_line_trough_zero_is_line: fixes l :: "circline" assumes "is_poincare_line l"   and "0\<^sub>h \<in> circline_set l" shows "is_line l"
step_load_ubx_miss: fixes F :: "'fenv_ubx"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl, 'opubx, 'ubx1, 'ubx2) Ubx.instr fundef"   and pc :: "nat"   and \<tau> :: "Unboxed.type"   and x :: "'var"   and i :: "('dyn, 'ubx1, 'ubx2) unboxed"   and i' :: "'dyn"   and H :: "'henv"   and d :: "'dyn"   and F' :: "'fenv_ubx"   and \<Sigma> :: "('dyn, 'ubx1, 'ubx2) unboxed list"   and st :: "('fun, ('dyn, 'ubx1, 'ubx2) unboxed) frame list" assumes "Fubx_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = ILoadUbx \<tau> x"   and "cast_Dyn i = Some i'"   and "heap_get H (x, i') = Some d"   and "Subx.unbox \<tau> d = None"   and "Fubx_add F f (Subx.generalize_fundef fd) = F'" shows "Subx.step (State F H (Frame f pc (i # \<Sigma>) # st)) (State F' H (Subx.box_stack f (Frame f (Suc pc) (OpDyn d # \<Sigma>) # st)))"
simps_3: fixes x1a :: "'a"   and x21 :: "'a"   and x22 :: "'a" shows "Vertex x1a \<noteq> Edge x21 x22"
combine_altdef: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t1 :: "('a, 'b) RBT.rbt"   and t2 :: "('a, 'b) RBT.rbt" shows "RBT.combine f t1 t2 = RBT.combine_with_key (\<lambda>_. f) t1 t2"
setsetmapim_the_inv_into: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set set" assumes "inj_on f (\<Union> A)" shows "the_inv_into (\<Union> A) f \<turnstile> (f \<turnstile> A) = A"
zeroprimeideal: fixes R :: "('c, 'd) ring_scheme" assumes "domain R" shows "primeideal {\<zero>\<^bsub>R\<^esub>} R"
simps_2: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'b"   and x :: "'c"   and y :: "'c" shows "rel_comp_pow r [i] x y = r i x y"
simps_3: fixes x :: "'a word" shows "proper_interval (Some x) None = (x \<noteq> - 1)"
disjnt_empty1: fixes A :: "'a set" shows "disjnt {} A"
is_interval_minus_translation: fixes x :: "'b"   and X :: "'b set" shows "is_interval ((-) x ` X) = is_interval X"
Gateway_L6_induction: fixes req :: "nat \<Rightarrow> reqType list"   and x :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and k :: "nat"   and t :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and x1 :: "nat \<Rightarrow> aType list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and x2 :: "nat \<Rightarrow> aType list"   and d :: "nat"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and i :: "nat \<Rightarrow> ECall_Info list"   and m :: "nat" assumes "msg (Suc 0) req"   and "msg (Suc 0) x"   and "msg (Suc 0) stop"   and "ts lose"   and "\<forall>j\<le>k. lose (t + j) = [False]"   and "\<forall>m\<le>k. req (t + m) \<noteq> [send]"   and "ack t = [connection_ok]"   and "Sample req dt x1 stop lose ack i1 vc"   and "Delay x2 i1 d x1 i2"   and "Loss lose x i2 x2 i"   and "m \<le> k" shows "ack (t + m) = [connection_ok]"
c_hd_aux2: fixes u :: "nat" assumes "1 < c_len u" shows "c_hd u = c_fst (c_snd (u - 1))"
distr: fixes i :: "nat" shows "distr M borel (X i) = distr M borel (X 0)"
game0_alt_def: shows "game0 = lift_spmf (sample_uniform (order \<G>)) \<bind> (\<lambda>x. lift_spmf coin_spmf \<bind> (\<lambda>b. inline hash_oracle' (\<A>1 (\<^bold>g [^] x)) {} \<bind> (\<lambda>(((msg1, msg2), \<sigma>), s). TRY assert_gpv (valid_plains msg1 msg2) \<bind> (\<lambda>_. aencrypt (\<^bold>g [^] x) (if b then msg1 else msg2) \<bind> (\<lambda>cipher. inline hash_oracle' (\<A>2 cipher \<sigma>) s \<bind> (\<lambda>(guess, s'). Generative_Probabilistic_Value.Done (guess = b)))) ELSE lift_spmf coin_spmf)))"
eq_onD: fixes r :: "'b \<Rightarrow> 'c"   and s :: "'b \<Rightarrow> 'c"   and X :: "'b set"   and x :: "'b" assumes "r = s on X"   and "x \<in> X" shows "r x = s x"
sel: fixes x2 :: "'a" shows "node.inner (Inner x2) = x2"
add_set_Suc: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and n :: "nat" shows "add_set f A (Suc n) = {x. \<exists>s\<in>add_set f A n. \<exists>t\<in>A. x = s \<^sub>f+ t}"
swapAll_pres_wlsAll: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and xs :: "'varSort"   and s' :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "(wls s X \<longrightarrow> wls s (X #[z1 \<and> z2]_zs)) \<and> (wlsAbs (xs, s') A \<longrightarrow> wlsAbs (xs, s') (A $[z1 \<and> z2]_zs)) \<and> (wlsInp delta inp \<longrightarrow> wlsInp delta (inp %[z1 \<and> z2]_zs)) \<and> (wlsBinp delta binp \<longrightarrow> wlsBinp delta (binp %%[z1 \<and> z2]_zs))"
Act\<^sub>\<alpha>_eq_iff: fixes \<alpha>1 :: "'g"   and t1 :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and \<alpha>2 :: "'g"   and t2 :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "(Act\<^sub>\<alpha> \<alpha>1 t1 = Act\<^sub>\<alpha> \<alpha>2 t2) = (tAct \<alpha>1 (rep_Tree\<^sub>\<alpha> t1) =\<^sub>\<alpha> tAct \<alpha>2 (rep_Tree\<^sub>\<alpha> t2))"
p_2: shows "(\<lambda>x. ball x (d x)) fine p"
rev_nth_Some: fixes xs :: "'b list"   and v :: "nat"   and y :: "'b" assumes "xs !. v = Some y" shows "v < length xs"
nth_least_le_Max: fixes A :: "'a set"   and n :: "nat" assumes "finite A"   and "A \<noteq> {}"   and "enat n < esize A" shows "nth_least A n \<le> Max A"
agg_sum_add_bot: fixes f :: "'d \<Rightarrow> 'c" shows "aggregation.sum_0 f {k. True} = aggregation.sum_0 f {k. True} + bot"
collapse_1: fixes llist :: "'a llist" assumes "lnull llist" shows "llist = []\<^sub>l"
exhaust: fixes y :: "'b list" obtains "y = []"   | x21 :: "'b" and x22 :: "'b list" where "y = Cons x21 x22"
simps_1: fixes uu :: "'a"   and ii :: "nat set"   and ni :: "nat set"   and m :: "'b" shows "inoclosed uu (ii\<not>ni:arrive(m)) = eq_newpkt m"
Domain_iff: fixes a :: "'c"   and r :: "('c \<times> 'd) set" shows "(a \<in> Domain r) = (\<exists>y. (a, y) \<in> r)"
powr_eq_one_iff: fixes a :: "real"   and x :: "real" assumes "1 < a" shows "(a powr x = 1) = (x = 0)"
prems_1: shows "si + Suc len \<le> length lsrc"
valid_add_new_table_mmsaux_unfolded: fixes args :: "Monitor.args"   and cur :: "nat"   and nt :: "nat"   and gc :: "nat"   and maskL :: "bool list"   and maskR :: "bool list"   and data_prev :: "(nat \<times> 'b option list set) queue"   and data_in :: "(nat \<times> 'b option list set) queue"   and tuple_in :: "('b option list, nat) mapping"   and tuple_since :: "('b option list, nat) mapping"   and auxlist :: "(nat \<times> 'b option list set) list"   and X :: "'b option list set" assumes "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"   and "table (args_n args) (args_R args) X" shows "valid_mmsaux args cur (add_new_table_mmsaux args X (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since)) (case auxlist of [] \<Rightarrow> [(cur, X)] | (t, y) # ts \<Rightarrow> if t = cur then (t, y \<union> X) # ts else (cur, X) # auxlist)"
redT_updW_PostWS_imp_PostWS: fixes t :: "'b"   and ws :: "'b \<Rightarrow> 'c wait_set_status option"   and wa :: "('b, 'c) wait_set_action"   and ws' :: "'b \<Rightarrow> 'c wait_set_status option"   and t'' :: "'b"   and w :: "wake_up_status" assumes "redT_updW t ws wa ws'"   and "ws t'' = \<lfloor>PostWS w\<rfloor>"   and "t'' \<noteq> t" shows "ws' t'' = \<lfloor>PostWS w\<rfloor>"
reducedly_dynamically_complete_calculus_axioms: shows "reducedly_dynamically_complete_calculus Bot Inf (\<Turnstile>) Red_I Red_F"
suma_right_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "A B C D E F SumA G H I" shows "A B C D E F SumA I H G"
restrict_idle: fixes n :: "nat"   and A :: "nat set"   and v :: "'b option list" assumes "wf_tuple n A v" shows "restrict A v = v"
sum_integral_diff_series_Bseq: shows "Bseq sum_integral_diff_series"
enumFrom_def: fixes toEnum :: "Integer \<rightarrow> 'b"   and fromEnum :: "'b \<rightarrow> Integer" shows "Enum.enumFrom toEnum fromEnum = (\<Lambda> x. map\<cdot>toEnum\<cdot>(intsFrom\<cdot>(fromEnum\<cdot>x)))"
simps: fixes a1 :: "'b list"   and a2 :: "'c sequent" shows "upRulesp a1 a2 = (\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> SRCTransforms.mset c = \<LM> Compound R Fs  \<RM> \<and> ps \<noteq> [])"
\<delta>0s': fixes i :: "'a" shows "0 < (SOME \<delta>. 0 < \<delta> \<and> (\<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)))"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and infty :: "'b infty" shows "P (case infty of infty.Infty \<Rightarrow> f1 | Num x \<Rightarrow> f2 x) = ((infty = infty.Infty \<longrightarrow> P f1) \<and> (\<forall>x2. infty = Num x2 \<longrightarrow> P (f2 x2)))"
diff_single_eq_union: fixes x :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "x \<in># M" shows "(remove1_mset x M = N) = (M = add_mset x N)"
pow_mult_distrib: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<otimes> y = y \<otimes> x"   and "x \<in> carrier Zp"   and "y \<in> carrier Zp" shows "(x \<otimes> y) [^] n = x [^] n \<otimes> y [^] n"
m_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "subgroup H G"   and "x \<in> H"   and "y \<in> H" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> H"
less_2: fixes xs' :: "'a list" assumes "rtrancl_path r x xs' y"   and "distinct (x # xs')"   and "set xs' \<subseteq> set xs" shows "thesis"
E: shows "E = (\<lambda>s. 1 + 3 * (nat (s ''a'') + 1 - nat (s ''z'')))"
rbd_disj: fixes R :: "('c \<times> 'd) set" shows "Sup_pres (bd\<^sub>\<R> R)"
arcs_ends_conv: fixes G :: "('a, 'b) pre_digraph" shows "arcs_ends G = (\<lambda>e. (tail G e, head G e)) ` arcs G"
coeff_map_poly_hom: fixes p :: "int poly"   and i :: "nat" shows "poly.coeff (Ring_Hom_Poly.of_int_poly p) i = of_int (poly.coeff p i)"
get_all_matching_src_ips_executable: fixes iface :: "iface"   and m :: "'b common_primitive match_expr" shows "wordinterval_to_set (get_all_matching_src_ips_executable iface m) = get_all_matching_src_ips iface m"
commute: fixes a :: "'g"   and b :: "'g" shows "min a b = min b a"
enabled_iff: fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a"   and \<omega> :: "'a stream" shows "MC_syntax.enabled K s \<omega> = (MC_syntax.enabled K (shd \<omega>) (stl \<omega>) \<and> shd \<omega> \<in> set_pmf (K s))"
hom_zero: shows "to_fract (0::'a) = 0"
is_expansion_cos_ms_zero_exp: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and C :: "'a"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and Sin :: "'a"   and Cos :: "'a" assumes "basis_wf (b # basis)"   and "is_expansion (MS (MSLCons (C, 0) xs) f) (b # basis)"   and "((\<lambda>x. sin (eval C x)) expands_to Sin) basis"   and "((\<lambda>x. cos (eval C x)) expands_to Cos) basis" shows "is_expansion (MS (minus_ms_aux (scale_shift_ms_aux (Cos, 0) (cos_ms_aux' xs)) (scale_shift_ms_aux (Sin, 0) (sin_ms_aux' xs))) (\<lambda>x. cos (f x))) (b # basis)"
WT_callee: fixes callee :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'a) spmf"   and I :: "'a \<Rightarrow> bool"   and \<I> :: "('b, 'c) \<I>"   and s :: "'a" assumes "callee_invariant_on callee I \<I>"   and "I s" shows "\<I> \<turnstile>c callee s \<surd>"
vector_derivative_def: fixes f :: "real \<Rightarrow> 'a"   and net :: "real filter" shows "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"
qmodule_module: fixes H :: "'a set" assumes "submodule R M H" shows "R module M /\<^sub>m H"
fTo_Order_sub: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> carrier fTo D"   and "B \<in> carrier fTo D" shows "A \<preceq>\<^bsub>fTo D\<^esub> B = (A \<subseteq> B)"
separating_eq: shows "separating_gen \<Gamma> = separating_gen \<Gamma>'"
wod_backward_slice_valid_node: fixes n :: "'node"   and S :: "'node set" assumes "n \<in> wod_backward_slice S" shows "valid_node n"
simps_7: fixes v :: "ct list" shows "produce Bot (TopType v) = Bot"
hyps: assumes "is_tangle_diagram (zs \<circ> y)" shows "is_tangle_diagram zs"
in_homI: fixes f :: "'a1"   and a :: "'a1"   and b :: "'a1" assumes "C1.arr f"   and "C1.dom f = a"   and "C1.cod f = b" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>1 b\<guillemotright>"
read_tmps_le: fixes i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "\<forall>i\<in>read_tmps sb. i < t"
igSubstIGVar2_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstIGVar2 MOD \<equiv> \<forall>ys y Y. igWls MOD (asSort ys) Y \<longrightarrow> igSubst MOD ys Y y (igVar MOD ys y) = Y"
star_rtc3: fixes x :: "'a list set" shows "(p_one \<union> p_prod x x = x) = (x = p_star x)"