hcju/isabelleps huggingface dataset:
d\<mu>_def: fixes n :: "nat"   and gs :: "int vec list"   and i :: "nat"   and j :: "nat" shows "LLL.d\<mu> n gs i j = int_of_rat (rat_of_int (LLL.d n gs (Suc j)) * gram_schmidt_fs.\<mu> n (map of_int_hom.vec_hom gs) i j)"
Nil_tl: fixes xs :: "'c list" shows "([] = tl xs) = (xs = [] \<or> (\<exists>x. xs = [x]))"
fundamental_theorem_algebra_factorized': fixes p :: "complex poly" shows "p = Polynomial.smult (Polynomial.lead_coeff p) (\<Prod>x\<in>#poly_roots p. [:- x, 1:])"
mult_subdistl: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; (y \<cdot> z) \<le> x ; y"
mset_merge: fixes h1 :: "'b hp option"   and h2 :: "'b hp option" shows "mset_heap (merge h1 h2) = mset_heap h1 + mset_heap h2"
periodic_arg_plus_0: fixes x :: "real" shows "x + 0 = x"
sup_eq_bot_iff: fixes x :: "'d multiset"   and y :: "'d multiset" shows "(x \<union># y = {#}) = (x = {#} \<and> y = {#})"
simps: fixes A :: "rat mat"   and signs :: "rat list list"   and subsets :: "nat list list"   and lhs_vec :: "rat vec" shows "reduction_step A signs subsets lhs_vec = (let reduce_cols_A = reduce_mat_cols A lhs_vec; rows_keep = rows_to_keep reduce_cols_A in (take_rows_from_matrix reduce_cols_A rows_keep, take_indices subsets rows_keep, take_indices signs (find_nonzeros_from_input_vec lhs_vec)))"
new_shadow_root_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) document_ptr"   and getter :: "('c, 'a, 'b, 'j option) RDocument_scheme \<Rightarrow> 'm" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
simps: fixes k :: "nat"   and t :: "'b btree" shows "invar_inorder k t = (bal t \<and> root_order k t)"
image_mset_respects_domain_permutation: fixes A :: "'c set"   and B :: "'d set" shows "(\<lambda>f. image_mset f (mset_set A)) respects domain_permutation A B"
perm_is_lub_simp: fixes \<pi> :: "perm"   and S :: "'c set"   and x :: "'c" shows "\<pi> \<bullet> S <<| \<pi> \<bullet> x = S <<| x"
ForallI: fixes G :: "('a, 'b) form list"   and a :: "('a, 'b) form"   and n :: "'a" assumes "G \<turnstile> a[App n []/0]"   and "news n G"   and "new n a" shows "G \<turnstile> Forall a"
assms_1: shows "(\<sigma>, SubnetS s t) \<in> oreachable (opnet onp (p\<^sub>1 \<parallel> p\<^sub>2)) (\<lambda>\<sigma> _. oarrivemsg I \<sigma>) (other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)))"
set_lebesgue_integral_def: fixes M :: "'c measure"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" shows "set_lebesgue_integral M A f \<equiv> LINT x|M. indicat_real A x *\<^sub>R f x"
Identity: fixes \<Psi> :: "'b" shows "\<Psi> \<otimes> \<bottom> \<simeq> \<Psi>"
2: shows "{} \<turnstile>\<^sub>c fresh_pgwt (\<Union> (funs_term ` T)) (\<Gamma> s)"
length_product: fixes xs :: "'b list"   and ys :: "'c list" shows "length (List.product xs ys) = length xs * length ys"
WhileTReds: fixes b :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and c :: "char list exp"   and v\<^sub>1 :: "val"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and e :: "char list exp"   and s\<^sub>3 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>3 :: "bool" assumes "P \<turnstile> \<langle>b,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>c,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v\<^sub>1,s\<^sub>2,b\<^sub>2\<rangle>"   and "P \<turnstile> \<langle>while (b) c,s\<^sub>2,b\<^sub>2\<rangle> \<rightarrow>* \<langle>e,s\<^sub>3,b\<^sub>3\<rangle>" shows "P \<turnstile> \<langle>while (b) c,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e,s\<^sub>3,b\<^sub>3\<rangle>"
square_integrable_imp_integrable_product: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "integrable (lebesgue_on S) (\<lambda>x. f x * g x)"
viterbi_ix_drop_one: fixes o :: "'t"   and os :: "'t list"   and s :: "'s"   and t :: "'s"   and n :: "nat" shows "viterbi_ix (o # os) s t (n + 1) = viterbi_ix os s t n"
assms_1: shows "G \<subseteq> \<langle>\<C>\<rangle>"
coinduct: fixes X :: "'s \<Rightarrow> 's cfg \<Rightarrow> bool"   and x1 :: "'s"   and x2 :: "'s cfg" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>cfg s. x1 = s \<and> x2 = cfg \<and> state cfg = s \<and> action cfg \<in> K s \<and> (\<forall>x. x \<in> set_pmf (action cfg) \<longrightarrow> X x (cont cfg x) \<or> cfg_onp x (cont cfg x))" shows "cfg_onp x1 x2"
truncate_down: fixes prec :: "nat"   and x :: "real" shows "truncate_down prec x \<le> x"
mult_strict_mono': fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "a < b"   and "c < d"   and "(0::'b) \<le> a"   and "(0::'b) \<le> c" shows "a * c < b * d"
simps_6: fixes A :: "ty" shows "default_val (A\<lfloor>\<rceil>) = Null"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P P'"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (P' \<parallel> Q)"
polybound0_I: fixes a :: "poly"   and b :: "'a"   and bs :: "'a list"   and b' :: "'a" assumes "polybound0 a" shows "\<lparr>a\<rparr>\<^sub>p\<^bsup>b # bs\<^esup> = \<lparr>a\<rparr>\<^sub>p\<^bsup>b' # bs\<^esup>"
permutes_bij: fixes p :: "'d \<Rightarrow> 'd"   and s :: "'d set" assumes "p permutes s" shows "bij p"
nn_integral_count_space_indicator: fixes X :: "'b set"   and f :: "'b \<Rightarrow> ennreal" assumes "NO_MATCH UNIV X" shows "integral\<^sup>N (count_space X) f = set_nn_integral (count_space UNIV) X f"
sorted_list_of_set_inj_aux: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "sorted_list_of_set A = sorted_list_of_set B" shows "A = B"
part12: fixes V :: "'b set" assumes "infinite V" shows "part12_pred V (part12 V)"
first_half_middle: shows "arr_mset mod_arr l m = arr_mset arr l m"
2: shows "\<not> prv (exi x (cnj fls (eql (Var x) (Var x))))"
ennreal_minus_cancel: fixes c :: "ennreal"   and a :: "ennreal"   and b :: "ennreal" assumes "c \<noteq> top"   and "a \<le> c"   and "b \<le> c"   and "c - a = c - b" shows "a = b"
ex_un_null: shows "\<exists>!n. \<forall>f. local.comp n f = n \<and> local.comp f n = n"
lower_ivl_of_aform_err: fixes p :: "nat"   and Rd :: "(real \<times> real pdevs) \<times> real" shows "lower (ivl_of_aform_err p Rd) = Inf_aform_err p Rd"
bisimSubstInputPushRes: fixes x :: "name"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> N" shows "\<Psi> \<rhd> \<lparr>\<nu>x\<rparr>(M\<lparr>\<lambda>*xvec N\<rparr>.P) \<sim>\<^sub>s M\<lparr>\<lambda>*xvec N\<rparr>.\<lparr>\<nu>x\<rparr>P"
C_extension: fixes M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and n :: "nat" shows "C M2 M1 \<Omega> V m (Suc n) \<subseteq> append_sets V (lists_of_length (inputs M2) n)"
Abs_bit1_inverse: fixes y :: "int" assumes "y \<in> {0..<1 + 2 * int CARD('b)}" shows "Rep_bit1 (Abs_bit1 y) = y"
non_trivial_combine_imp_dependent: fixes K :: "'c set"   and Ks :: "'c list"   and Us :: "'c list" assumes "subfield K S"   and "set Ks \<subseteq> K"   and "S.combine Ks Us = \<zero>\<^bsub>S\<^esub>"   and "\<not> set (take (length Us) Ks) \<subseteq> {\<zero>\<^bsub>S\<^esub>}" shows "S.dependent K Us"
elims: fixes x :: "MIR.num"   and y :: "int" assumes "maxcoeff x = y" obtains i :: "int" where "x = C i" and "y = \<bar>i\<bar>"   | n :: "nat" and c :: "int" and t :: "MIR.num" where "x = CN n c__ t__" and "y = max \<bar>c__\<bar> (maxcoeff t__)"   | c :: "int" and t :: "MIR.num" and s :: "MIR.num" where "x = CF c t s" and "y = max \<bar>c\<bar> (maxcoeff s)"   | v :: "nat" where "x = Bound v__" and "y = 1"   | v :: "MIR.num" where "x = Neg va__" and "y = 1"   | v :: "MIR.num" and va :: "MIR.num" where "x = Add vb__ vaa__" and "y = 1"   | v :: "MIR.num" and va :: "MIR.num" where "x = Sub vc__ vab__" and "y = 1"   | v :: "int" and va :: "MIR.num" where "x = Mul vd__ va" and "y = 1"   | v :: "MIR.num" where "x = Floor v" and "y = 1"
fps_X_compose_fps_exp: fixes a :: "'a" shows "fps_X oo fps_exp a = fps_exp a - 1"
costp_def: fixes p :: "prog" shows "costp p \<equiv> coste (body p)"
1: shows "(d, t) \<rightarrow>*c (d', t')"
assms_1: shows "coincides_on Mkt Mkt2 A"
toSink: fixes s :: "('St, 'U, 'C) state" assumes "s \<in> UNIV" shows "toSink s \<in> \<delta> s \<inter> Sink"
POS_INF_def: shows "POS_INF = 2 ^ 31 - 1"
adjust_paras': fixes xs :: "nat list"   and n :: "nat"   and x :: "nat"   and y :: "nat"   and anything :: "nat list" assumes "length xs = n" shows "{\<lambda>nl. nl = xs @ x # y # anything} [Inc n] [+] [Dec (Suc n) 2, Goto 0] {\<lambda>nl. nl = xs @ Suc x # 0 # anything}"
simple_Entry_path: fixes g :: "'g"   and n :: "'node" assumes "invar g"   and "n \<in> set (\<alpha>n g)" obtains ns :: "'node list" where "g \<turnstile> Entry g-ns\<rightarrow>n" and "n \<notin> set (butlast ns)"
StepWhileFalse: fixes b :: "expr"   and s :: "char list \<Rightarrow> val option"   and c :: "cmd" assumes "interpret b s = Some false" shows "while (b) c \<turnstile> \<langle>while (b) c,s,0\<rangle> \<leadsto> \<langle>Skip,s,1\<rangle>"
ccard_same: fixes p :: "nat \<Rightarrow> bool"   and n' :: "nat"   and n :: "nat" assumes "\<not> p (Suc n')" shows "card {i. n < i \<and> i \<le> n' \<and> p i} = card {i. n < i \<and> i \<le> Suc n' \<and> p i}"
set_remove_nth_subset: fixes j :: "nat"   and xs :: "'b list" shows "set (remove_nth j xs) \<subseteq> set xs"
rel_def: shows "rel \<equiv> \<lparr>carrier = carrier R \<times> S, eq = \<lambda>(r, s) (r', s'). \<exists>t\<in>S. t \<otimes> (s' \<otimes> r \<ominus> s \<otimes> r') = \<zero>\<rparr>"
NoMessageLossStep: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and m :: "('a, 'c) message"   and cfg2 :: "('a, 'c, 'b) configuration"   and p :: "'a"   and m' :: "('a, 'c) message" assumes "asynchronousSystem.steps trans sends cfg1 m cfg2"   and "isReceiverOf p m"   and "\<not> isReceiverOf p m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
nbr_is_hbl: shows "Height_Bounded_Labeling c s t f (relabel_effect f l u)"
K_eq: fixes t :: "real"   and s :: "'a" shows "K (t, s) = distr (exponential (escape_rate s) \<Otimes>\<^sub>M measure_pmf (J s)) S (\<lambda>(t', s). (t + t', s))"
nGt_le_conv: fixes x :: "'a"   and y :: "'a" shows "(comp x y \<noteq> Gt) = le x y"
mod_ring_mult_integer: fixes pp :: "integer" assumes "p = int_of_integer pp" shows "(mod_ring_rel_integer ===> mod_ring_rel_integer ===> mod_ring_rel_integer) (mult_p_integer pp) (*)"
elims: fixes x :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and y :: "('a literal multiset \<times> nat) multiset" assumes "wN_of_wstate x = y" obtains N :: "('a literal multiset \<times> nat) multiset" and P :: "('a literal multiset \<times> nat) multiset" and Q :: "('a literal multiset \<times> nat) multiset" and n :: "nat" where "x = (N, P, Q, n)" and "y = N"
Spec: fixes s :: "'d"   and t :: "'d"   and r :: "('d \<times> 'd) set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option" assumes "(s, t) \<in> r" shows "\<Gamma>\<turnstile> \<langle>Spec r,Normal s\<rangle> \<Rightarrow> Normal t"
rel_refl_strong: fixes x :: "('d, 'e) nondetT"   and R1a :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>z1. z1 \<in> phantom_nondetT x \<Longrightarrow> R1a z1 z1"   and "\<And>z2. z2 \<in> set_nondetT x \<Longrightarrow> R2a z2 z2" shows "rel_nondetT' R1a R2a x x"
summable_norm_comparison_test: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> real" assumes "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n"   and "summable g" shows "summable (\<lambda>n. norm (f n))"
WHILEIT_weaken: fixes I :: "'c \<Rightarrow> bool"   and I' :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and x :: "'c" assumes "\<And>x. I x \<Longrightarrow> I' x" shows "WHILE\<^sub>T\<^bsup>I'\<^esup> b f x \<le> WHILE\<^sub>T\<^bsup>I\<^esup> b f x"
IH_2: assumes "wbt r"   and "r \<noteq> \<langle>\<rangle>" shows "2 \<le> (1 - \<alpha>) ^ (height r - 1) * real (size1 r)"
a2: shows "hd (s (Suc j)) = (if hd (x j) = Zero then hd (s j) - r else hd (s j) + r)"
cases: fixes a1 :: "'a list"   and a2 :: "'b sequent" assumes "(a1, a2) \<in> upRules" obtains c :: "'b sequent" and R :: "'b" and Fs :: "'b form list" and ps :: "'a list" where "a1 = ps" and "a2 = c" and "MultiSequents.mset c \<equiv> \<LM> Compound R Fs  \<RM>" and "ps \<noteq> []"
a_ptr_kinds_M_def: fixes ptr_kinds :: "'a \<Rightarrow> 'b fset" shows "l_ptr_kinds_M.a_ptr_kinds_M ptr_kinds = Heap_Error_Monad.bind get_heap (\<lambda>h. return (sorted_list_of_set (fset (ptr_kinds h))))"
punit_mult_scalar: shows "MPoly_Type_Class.punit.mult_scalar = (*)"
FIN': fixes A :: "'Q set" shows "finite (A \<inter> range r)"
Sum_any_product: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite {a. f a \<noteq> (0::'b)}"   and "finite {b. g b \<noteq> (0::'b)}" shows "Sum_any f * Sum_any g = (\<Sum>a. \<Sum>b. f a * g b)"
simps: fixes x :: "'a ext" shows "x\<^sup>T = x"
suffix_imp_sublist: fixes xs :: "'b list"   and ys :: "'b list" assumes "suffix xs ys" shows "sublist xs ys"
cases: fixes a1 :: "'c \<Rightarrow> ('d, 'c) Term.term"   and a2 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and a3 :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 a1 a2 a3" obtains t :: "('d, 'c) Term.term" and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" where "a1 = Var" and "a2 = add_mset (ta__, ta__) E__" and "a3 = E__"   | ss :: "('d, 'c) Term.term list" and ts :: "('d, 'c) Term.term list" and f :: "'d" and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" where "a1 = Var" and "a2 = add_mset (Fun f ss, Fun f ts) Ea__" and "a3 = Ea__ + mset (zip ss ts)" and "length ss = length ts"   | x :: "'c" and t :: "('d, 'c) Term.term" and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" where "a1 = subst x__ tb__" and "a2 = add_mset (Var x__, tb__) Eb__" and "a3 = subst_mset (subst x__ tb__) Eb__" and "x__ \<notin> vars_term tb__"   | x :: "'c" and t :: "('d, 'c) Term.term" and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" where "a1 = subst x t" and "a2 = add_mset (t, Var x) E" and "a3 = subst_mset (subst x t) E" and "x \<notin> vars_term t"
\<omega>_limit_set_invariant: fixes x :: "'a" shows "rev.invariant (rev.\<omega>_limit_set x)"
alg_F1set: 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 x :: "('f, 'd, 'e) F1" assumes "alg B1 B2 s1 s2"   and "F1set2 x \<subseteq> B1"   and "F1set3 x \<subseteq> B2" shows "s1 x \<in> B1"
norm_ge_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "0 \<le> norm x"
axioms_2: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>" shows "tiny_digraph \<alpha> \<AA>"
Q_3: shows "Q pdivides\<^bsub>M\<^esub> \<sigma> P"
inner_verts_conv: fixes G :: "('b, 'c) pre_digraph"   and p :: "'c list"   and u :: "'b" shows "pre_digraph.inner_verts G p = butlast (tl (pre_digraph.awalk_verts G u p))"
fin_dim_subspace_sum: fixes W1 :: "'a vec set"   and W2 :: "'a vec set" assumes "subspace class_ring W1 V"   and "subspace class_ring W2 V"   and "vectorspace.fin_dim class_ring (vs W1)"   and "vectorspace.fin_dim class_ring (vs W2)" shows "vectorspace.fin_dim class_ring (vs (subspace_sum W1 W2))"
lm015: fixes P :: "('a \<times> 'b) set"   and X :: "'a set"   and Y :: "'a set" shows "P || (X \<inter> Y) \<subseteq> P || X \<and> Outside' (X \<union> Y) P \<subseteq> Outside' X P"
distinct_5: fixes x11 :: "floatarith"   and x12 :: "floatarith"   and x13 :: "floatarith"   and x14 :: "form"   and x41 :: "floatarith"   and x42 :: "floatarith" shows "Bound x11 x12 x13 x14 \<noteq> LessEqual x41 x42"
evaln_elim_cases_8: fixes G :: "prog"   and s :: "st"   and unop :: "unop"   and e :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (UnOp unop e)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains va :: "val" where "v = In1 (eval_unop unop va)" and "G\<turnstile>Norm s \<midarrow>e-\<succ>va\<midarrow>n\<rightarrow> s'"
boolean_axioms_hold: fixes A :: "fm"   and e :: "(name, hf) finfun" assumes "A \<in> boolean_axioms" shows "eval_fm e A"
narrow2_acom: fixes y :: "'a acom"   and x :: "'a acom" assumes "y \<sqsubseteq> x" shows "x \<triangle>\<^sub>c y \<sqsubseteq> x"
minus_prod_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "x - y = (fst x - fst y, snd x - snd y)"
pp_of_term_of_pair: fixes t :: "'a"   and k :: "'k" shows "pp_of_term (term_of_pair (t, k)) = t"
tvar_not_in_term_imp_free_not_in_term: fixes y :: "variable"   and S :: "String.literal set"   and \<tau> :: "typ"   and t :: "Core.term"   and a :: "variable" assumes "(y, S) \<in> tvsT \<tau>"   and "(y, S) \<notin> tvs t" shows "(a, \<tau>) \<notin> fv t"
subst_subst: fixes \<sigma> :: "'f \<Rightarrow> 'g"   and \<rho> :: "'g \<Rightarrow> 'h"   and e :: "('e, 'h) uexpr" shows "\<sigma> \<dagger> \<rho> \<dagger> e = (\<rho> \<circ> \<sigma>) \<dagger> e"
valof_normal_of_Float: fixes x :: "Float.float" assumes "is_normal_Float TYPE(('a, 'b) IEEE.float) x" shows "valof (normal_of_Float x) = real_of_float x"
overlappingI: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b"   and u :: "'b"   and t :: "'b" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred"   and "term.matches subst s u"   and "term.matches subst t u" shows "term.overlapping subst s t"
prems_5: shows "mod_type_class.to_nat a \<le> Suc n"
drop_0: fixes xs :: "'a list" shows "drop 0 xs = xs"
abs_ereal_uminus: fixes x :: "ereal" shows "\<bar>- x\<bar> = \<bar>x\<bar>"
ipv6_unparsed_compressed_to_preferred_identity2: fixes ls :: "16 word option list"   and ipv6prferred :: "ipv6addr_syntax" shows "(ipv6_unparsed_compressed_to_preferred ls = Some ipv6prferred) = (\<exists>ipv6compressed. parse_ipv6_address_compressed ls = Some ipv6compressed \<and> ipv6addr_c2p ipv6compressed = ipv6prferred)"
less_view_ext_def: fixes V :: "'a view_scheme"   and V' :: "'a view_scheme" shows "V < V' \<equiv> ext V \<le> ext V' \<and> lan V \<le> lan V' \<and> own V' = own V \<and> more V \<le> more V' \<and> \<not> (ext V' \<le> ext V \<and> lan V' \<le> lan V \<and> own V' = own V \<and> more V' \<le> more V)"
fBex_triv_one_point2: fixes A :: "'b fset"   and a :: "'b" shows "fBex A ((=) a) = (a |\<in>| A)"
l: shows "set l = {(k, the (subs k)) |k. k \<in> dom subs}"
Iff_MP2_same: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A IFF B"   and "H \<turnstile> B" shows "H \<turnstile> A"
mstep_sound: fixes Cs :: "fterm literal set set"   and Cs' :: "fterm literal set set"   and F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "mresolution_step Cs Cs'"   and "eval\<^sub>c\<^sub>s F G Cs" shows "eval\<^sub>c\<^sub>s F G Cs'"
val_t2p: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K" shows "v (x \<cdot>\<^sub>r y) = v x + v y"
less_irrefl: fixes x :: "('x, nat) pp" shows "\<not> ord_strict x x"
next_not_member: fixes S :: "'a set"   and s :: "'a" assumes "finite S"   and "Next S \<le> s" shows "next s \<notin> S"
term_p_subid: fixes x :: "'a" shows "(x \<cdot> 1\<^sub>\<pi> = x) = (x \<le> 1\<^sub>\<pi>)"
baseI: fixes x :: "(nat \<times> int) list"   and dm :: "nat"   and ds :: "nat set"   and p :: "(nat \<times> int) list" assumes "x \<in> grid (start dm) ({0..<dm} - ds)"   and "p \<in> grid x ds" shows "base ds p = x"
lincomb_scalar_prod_left: fixes W :: "'a vec set"   and v :: "'a vec"   and a :: "'a vec \<Rightarrow> 'a" assumes "W \<subseteq> carrier_vec n"   and "v \<in> carrier_vec n" shows "lincomb a W \<bullet> v = (\<Sum>w\<in>W. a w * (w \<bullet> v))"
compatible_ordering_axioms: shows "compatible_ordering A (\<lambda>x y. y \<sqsubseteq> x) (\<lambda>x y. y \<sqsubset> x)"
orelse_thms_1: fixes y :: "tr" shows "(TT orelse y) = TT"
no_loops_in_iapath: fixes V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and a :: "'a \<times> 'a" assumes "G.gen_iapath V u p v"   and "a \<in> set p" shows "fst a \<noteq> snd a"
wfP_eq_minimal: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "wfP r = (\<forall>Q x. x \<in> Q \<longrightarrow> (\<exists>z\<in>Q. \<forall>y. r y z \<longrightarrow> y \<notin> Q))"
card_of_Times_ordLess_infinite: fixes C :: "'a set"   and A :: "'b set"   and B :: "'c set" assumes "infinite C"   and "|A| <o |C|"   and "|B| <o |C|" shows "|A \<times> B| <o |C|"
root_0: fixes x :: "real" shows "root 0 x = 0"
well_order_induct_imp: fixes r :: "('a \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "wo_rel r"   and "\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x" shows "a \<in> Field r \<longrightarrow> P a"
is_reduced_GB_unique': fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "is_reduced_GB A"   and "is_reduced_GB B"   and "pmdl A = pmdl B" shows "A \<subseteq> B"
well_def_sdp: fixes prog :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "well_def prog" shows "sub_distrib_pconj prog"