hcju/isabelleps huggingface dataset:
simps_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v1 :: "'a"   and w :: "'b"   and v2 :: "'a"   and p :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" shows "is_path_undir G v ((v1, w, v2) # p) v' = (v = v1 \<and> ((v1, w, v2) \<in> edges G \<or> (v2, w, v1) \<in> edges G) \<and> is_path_undir G v2 p v')"
perm_vars_inv_nth: fixes n :: "nat"   and vs :: "nat list"   and i :: "nat" assumes "perm_vars n vs"   and "i < n" shows "perm_vars_inv n vs (vs ! i) = i"
fps_unit_factor_uminus: fixes f :: "'a fps" shows "unit_factor (- f) = - unit_factor f"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c llist" shows "lmap g (lmap f v) = lmap (g \<circ> f) v"
U_2: shows "\<forall>i<length T. P (T ! i) (U ! i)"
prog1_def: shows "prog1 \<equiv> IF Less (N 0) (V ''x'') THEN ''low'' ::= N 1 ELSE ''low'' ::= N 0"
ev: shows "\<forall>\<^sub>F x in F. g x = (0::'b) \<longrightarrow> h x = (0::'b)"
CondT: fixes s :: "'b"   and b :: "'b set"   and P1 :: "'b com"   and P2 :: "'b com" assumes "s \<in> b" shows "(Some (Cond b P1 P2), s) -c\<rightarrow> (Some P1, s)"
is_Ghost\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_Ghost\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = True"
variant_names_fresh: fixes S :: "variable set"   and n :: "nat"   and s :: "variable" assumes "finite S" shows "\<forall>s\<in>set (fst (variant_variables n s S)). s \<notin> S"
distr_uminus_real: fixes M :: "real measure"   and f :: "real \<Rightarrow> ennreal" assumes "has_density M lborel f" shows "has_density (distr M borel uminus) lborel (\<lambda>x. f (- x))"
k: shows "Legendre (a * b) p = Legendre a p * Legendre b p + p * k"
rel_Class_iff: fixes x :: "'a set \<times> 'b"   and y :: "'a set \<times> 'b" shows "x \<sim> y = (y \<in> Sigma I \<FF> \<and> x \<in> equivalence.Class (Sigma I \<FF>) {(x, y). x \<sim> y} y)"
precise_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> assn" shows "precise R \<equiv> \<forall>a a' h p F F'. h \<Turnstile> R a p * F \<and>\<^sub>A R a' p * F' \<longrightarrow> a = a'"
assms_3: shows "ats \<doteq> \<B> s"
hoaret_sound: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>t {P} c {Q}" shows "\<Turnstile>\<^sub>t {P} c {Q}"
in_rangeC_singleton: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "f x \<in> rangeC {f}"
17: shows "infs (gaccepting A) (gtrace (fromN 1 ||| r) (0, p))"
clinear_continuous_within: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set" assumes "bounded_clinear f" shows "continuous (at x within s) f"
sgn_poly_mapping_def: shows "sgn \<equiv> \<lambda>x. x /\<^sub>R norm x"
inv_1: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"
dc_prop2: fixes x :: "'a" shows "d (x \<cdot> 1\<^sub>\<pi>) = d x"
f_aggregate_0: fixes xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" shows "f_aggregate xs 0 ag = []"
op_set_insert_dj_def: shows "op_set_insert_dj \<equiv> insert"
arr_eqI: fixes f :: "'c local.term set"   and g :: "'c local.term set"   and t :: "'c local.term"   and u :: "'c local.term" assumes "par f g"   and "t \<in> f"   and "u \<in> g"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "f = g"
that: fixes a :: "'edge"   and n :: "'node" assumes "sourcenode a = (_Entry_)"   and "targetnode a = n"   and "n -tl as\<rightarrow>* n'"   and "valid_edge a"   and "a = hd as" shows "thesis"
Sup_set_def: fixes A :: "'a set set" shows "\<mu> A = {x. \<Squnion>B\<in>A. s2p B x}"
closure_of_def: fixes X :: "'d topology"   and S :: "'d set" shows "X closure_of S \<equiv> {x \<in> topspace X. \<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y\<in>S. y \<in> T)}"
mult_def: fixes r :: "('a \<times> 'a) set" shows "mult r = (mult1 r)\<^sup>+"
cases: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "exp0 list"   and a5 :: "'b state \<times> (v list, v) result" assumes "evaluate_list a1 a2 a3 a4 a5" obtains ck :: "bool" and env :: "v sem_env" and s :: "'b state" where "a1 = ck__" and "a2 = envb__" and "a3 = sb__" and "a4 = []" and "a5 = (sb__, Rval [])"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and v1 :: "v" and vs :: "v list" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" where "a1 = cka__" and "a2 = envc__" and "a3 = s1__" and "a4 = ea__ # es__" and "a5 = (s3__, Rval (v1__ # vs))" and "evaluate cka__ envc__ s1__ ea__ (s2__, Rval v1__) \<and> evaluate_list cka__ envc__ s2__ es__ (s3__, Rval vs)"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and err :: "v error_result" and s :: "'b state" and s' :: "'b state" where "a1 = ckb__" and "a2 = envd__" and "a3 = s" and "a4 = eb__ # esa__" and "a5 = (s', Rerr err__)" and "evaluate ckb__ envd__ s eb__ (s', Rerr err__)"   | ck :: "bool" and env :: "v sem_env" and e :: "exp0" and es :: "exp0 list" and v1 :: "v" and err :: "v error_result" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" where "a1 = ck" and "a2 = env" and "a3 = s1" and "a4 = e # es" and "a5 = (s3, Rerr err)" and "evaluate ck env s1 e (s2, Rval v1) \<and> evaluate_list ck env s2 es (s3, Rerr err)"
two_plus_two: shows "(2::'a) + (2::'a) = (4::'a)"
cols_id_unit_vecs: fixes d :: "nat" shows "cols (1\<^sub>m d) = unit_vecs d"
concatMapS_defined: fixes a :: "('b, 'c) Stream"   and f :: "'b \<rightarrow> ('d, 'e) Stream" assumes "a \<noteq> \<bottom>" shows "concatMapS\<cdot>f\<cdot>a \<noteq> \<bottom>"
1: shows "\<forall>g. (\<Sum>v\<in>set_of_vector X. g v *s v) = 0 \<longrightarrow> (\<forall>v\<in>set_of_vector X. g v = (0::'a))"
fst_mismatch_def: shows "c\<^sub>0 = (u\<^sub>0 \<cdot> u\<^sub>1) ! \<^bold>|\<alpha>\<^bold>|"
finite_rtrancl_delta_Image: fixes A :: "'a set"   and r :: "'a rexp"   and s :: "'a rexp" shows "finite ({((r, s), delta a r, delta a s) |r s a. a \<in> A}\<^sup>* `` {(init r, init s)})"
subseteq_qle_trans: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq>\<^sub>\<preceq> B"   and "B \<subseteq>\<^sub>\<preceq> C" shows "A \<subseteq>\<^sub>\<preceq> C"
rbt_fromlist_def: fixes ls :: "'a list" shows "rbt_fromlist ls \<equiv> RBT.bulkload (map (\<lambda>l. (l, ())) ls)"
trans: fixes x :: "'o"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "le x a b"   and "le x b c" shows "le x a c"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree" assumes "\<And>l a uu_ r. (r \<noteq> \<langle>\<rangle> \<Longrightarrow> P r) \<Longrightarrow> P \<langle>l, (a, uu_), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
f_Exec_eq_f_Exec_Stream_last2: fixes xs :: "'a list"   and trans_fun :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and c :: "'b" assumes "xs \<noteq> []" shows "f_Exec_Comp trans_fun xs c = last (f_Exec_Comp_Stream trans_fun xs c)"
fpxs_nth_mult_val: fixes f :: "'b fpxs"   and g :: "'b fpxs" shows "fpxs_nth (f * g) (fpxs_val f + fpxs_val g) = fpxs_nth f (fpxs_val f) * fpxs_nth g (fpxs_val g)"
nat_int_comparison_2: fixes a :: "nat"   and b :: "nat" shows "(a < b) = (int a < int b)"
execute_parallel_operator_equals_execute_sequential_strips_if: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list" assumes "are_all_operators_applicable s ops"   and "are_all_operator_effects_consistent ops"   and "are_all_operators_non_interfering ops" shows "execute_parallel_operator s ops = execute_serial_plan s ops"
assms: shows "v \<prec>\<^sub>t lt (lower p u)"
semiring_norm_2: shows "num.One + num.One = num.Bit0 num.One"
p2s2r: shows "p2r = s2r \<circ> p2s"
iFROM_iIN_neq: fixes n' :: "nat"   and n :: "nat"   and d :: "nat" shows "[n'\<dots>] \<noteq> [n\<dots>,d]"
ennreal_INF_const_minus: fixes I :: "'b set"   and c :: "ennreal"   and f :: "'b \<Rightarrow> ennreal" assumes "I \<noteq> {}" shows "(SUP x\<in>I. c - f x) = c - Inf (f ` I)"
sel_1: fixes x1 :: "'a" shows "the_Var (Var x1) = x1"
cp_dot\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<B>\<O>\<S>\<S>: fixes X :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(X.boss) \<tau> = (\<lambda>_. X \<tau>.boss) \<tau>"
le: fixes A :: "nat set \<times> nat set set"   and B :: "nat set \<times> nat set set" assumes "nonempty_graph A"   and "finite_graph A"   and "A \<simeq> B" shows "max_density A \<le> max_density B"
assms_1: shows "det2 a11 a12 a21 a22 \<noteq> (0::'a)"
list2FWpolicyconc: fixes a :: "('a, 'b) Combinators list"   and xa :: "('a, 'b) Combinators" assumes "a \<noteq> []" shows "list2FWpolicy (xa # a) = xa \<oplus> list2FWpolicy a"
n_valid_RmiI: fixes na :: "nat"   and b :: "nat ifex"   and bdd :: "bdd" assumes "(na, b) \<in> Rmi bdd" shows "bdd_node_valid bdd na"
np_base_def: fixes \<G> :: "'a cyclic_group" shows "np_base \<G> \<equiv> finite (carrier \<G>) \<and> 0 < order \<G> \<and> prime (order \<G>)"
t_in_2: shows "t \<in> GSMP (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
const_trm_def: fixes n :: "name" shows "const n = trm.Const (of_name n)"
distinct_43: fixes x4 :: "('a, 'b, 'c) expr"   and x8 :: "('a, 'b, 'c) expr" shows "Ref x4 \<noteq> Rjoin x8"
comm_neutral: fixes a :: "'c \<Rightarrow> bool" shows "(a \<and>* \<box>) = a"
inheritable_instance_methd_cases: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and sig :: "sig"   and old :: "qtname \<times> methd" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "is_class G D"   and "wf_prog G"   and "methd G D sig = Some old"   and "Protected \<le> accmodi old"   and "\<not> is_static old" obtains "methd G C sig = Some old"   | new :: "qtname \<times> methd" where "methd G C sig = Some new" and "G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old"
coboundedI1: fixes c :: "'t"   and a :: "'t"   and b :: "'t" assumes "c \<preceq>\<^sub>t a" shows "c \<preceq>\<^sub>t ord_term_lin.max a b"
roofed_eq: fixes B :: "'v set"   and S :: "'v set" shows "roofed_gen \<Gamma> B S = roofed_gen \<Gamma>' B S"
insert_commutes: fixes e1 :: "'c \<times> 'd \<times> bool"   and e2 :: "'c \<times> 'd \<times> bool"   and i1 :: "'c option"   and i2 :: "'c option"   and xs :: "('c \<times> 'd \<times> bool) list" assumes "fst e1 \<noteq> fst e2"   and "i1 = None \<or> i1 \<noteq> Some (fst e2)"   and "i2 = None \<or> i2 \<noteq> Some (fst e1)" shows "Ordered_List.insert xs e1 i1 \<bind> (\<lambda>ys. Ordered_List.insert ys e2 i2) = Ordered_List.insert xs e2 i2 \<bind> (\<lambda>ys. Ordered_List.insert ys e1 i1)"
mono_if_cont: fixes f :: "'a set \<Rightarrow> 'b set" assumes "cont f" shows "mono f"
Int_UN_distrib: fixes B :: "'b set"   and A :: "'c \<Rightarrow> 'b set"   and I :: "'c set" shows "B \<inter> \<Union> (A ` I) = (\<Union>i\<in>I. B \<inter> A i)"
bounded_Int: fixes S :: "'a set"   and T :: "'a set" assumes "bounded S \<or> bounded T" shows "bounded (S \<inter> T)"
thetaWhileZOT_ZOretrT: shows "thetaWhileZOT \<subseteq> ZOretrT (thetaWhileZOT \<union> ZObisT)"
parallel_fix_ind: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and F :: "'c \<rightarrow> 'c"   and G :: "'d \<rightarrow> 'd" assumes "adm (\<lambda>x. P (fst x) (snd x))"   and "P \<bottom> \<bottom>"   and "\<And>x y. P x y \<Longrightarrow> P (F\<cdot>x) (G\<cdot>y)" shows "P (fix\<cdot>F) (fix\<cdot>G)"
prob_space_distr: fixes f :: "'a \<Rightarrow> 'd"   and M' :: "'d measure" assumes "random_variable M' f" shows "prob_space (distr M M' f)"
assms_1: fixes as :: "'a list" assumes "\<forall>a\<in>set as. depends\<^sub>0 a \<and> anormal a" shows "is_dnf_qe qe as"
preserves_iso: fixes f :: "'a \<times> 'a \<times> 'a" assumes "CCC.iso f" shows "local.iso (\<alpha> f)"
abs_eq: fixes x :: "'a multiset \<times> 'a multiset" shows "mset_neg (abs_zmultiset x) = Abs_multiset (case x of (Mp, Mn) \<Rightarrow> count (Mn - Mp))"
is_class_intros: fixes P :: "'a prog"   and C :: "String.literal" assumes "class P C \<noteq> None" shows "is_class P C"
sub_instr_result: fixes i :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "fst i \<in> {arith_type SUB, arith_type SUBcc, arith_type SUBX}" shows "snd (sub_instr i s) = False"
rtrancl_image_advance: fixes q :: "'a"   and R :: "('a \<times> 'a) set"   and Q0 :: "'a set"   and x :: "'a" assumes "q \<in> R\<^sup>* `` Q0"   and "(q, x) \<in> R" shows "x \<in> R\<^sup>* `` Q0"
proper_it_def: fixes it :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and it' :: "('g \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g" shows "proper_it it it' \<equiv> \<exists>l. it = foldli l \<and> it' = foldli l"
Tpolymap_def: shows "Tpolymap = VectorSpaceEnd.polymap (\<cdot>) V T"
order_le_imp_less_or_eq: fixes x :: "'b"   and y :: "'b" assumes "x \<le> y" shows "x < y \<or> x = y"
section_and_retraction_eq_homeomorphic_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "(section_map X Y f \<and> retraction_map X Y f) = homeomorphic_map X Y f"
sub_branch_def: fixes f :: "'d \<Rightarrow> 'c"   and blocks :: "(('b, 'd) fm list \<times> 'd) list" shows "sub_branch f blocks \<equiv> map (sub_block f) blocks"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "\<ll>.map f = S.null"
Obj_implies_Ide: fixes t :: "'a local.term" assumes "Obj t" shows "Ide t"
hyps: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a"   and y :: "'b"   and ys :: "'b list" shows "f(x \<mapsto> y, xs [\<mapsto>] ys) = (if x \<in> set (take (length ys) xs) then f(xs [\<mapsto>] ys) else f(xs [\<mapsto>] ys, x \<mapsto> y))"
sup_ge2: fixes sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and y :: "'f"   and x :: "'f" assumes "class.semilattice_sup sup less_eq less" shows "less_eq y (sup x y)"
oInv_le: fixes a :: "ordinal"   and x :: "ordinal" assumes "a < F (oSuc x)" shows "oInv F a \<le> x"
ded_left: fixes a :: "'a G"   and b :: "'a G"   and c :: "'a G" shows "a * b l\<rightarrow> c = a l\<rightarrow> b l\<rightarrow> c"
0: shows "a = Outside' {seller} aa \<and> aa \<in> allAllocations'' (N \<union> {seller}) \<Omega>"
R1: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set"   and Y :: "('b, 'c) ipurge_rec" assumes "\<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = None, Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y" shows "\<lparr>Pol = I, Map = D, Doms = U, List = xs, ListOp = Some [], Set = ipurge_ref_aux I D U [] X\<rparr> \<in> ipurge_fail_aux_t_set Y"
coeff_0_denom: shows "constant_term (lr_fps_denominator cs) \<noteq> (0::'a)"
in_lin_tss: fixes t :: "nat" assumes "t \<in> set (linearize tss)" shows "t \<le> cur \<and> enat cur \<le> enat t + right I"
is_regular_sequence_snocD_2: fixes fs :: "('c \<Rightarrow>\<^sub>0 'd) list"   and f :: "'c \<Rightarrow>\<^sub>0 'd" assumes "is_regular_sequence (fs @ [f])" shows "is_regular_sequence fs"
F1_RV: shows "of_int_hom.vec_hom (fs' ! i) = map of_int_hom.vec_hom fs' ! i"
not_eq_extremum: fixes a :: "'c list" shows "(a \<noteq> []) = strict_prefix [] a"
f1: fixes j :: "nat" shows "j < length (mat_to_cols_list (A * B)) \<longrightarrow> mat_to_cols_list (A * B) ! j = M ! j"
S_1: shows "S \<subseteq> {x \<in> space M. (\<forall>n\<in>I. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. n \<in> I \<and> P n x))}"
Tarjan_invar_def: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state" shows "Tarjan_invar G s \<equiv> Tarjan G \<and> DFS_invar G (Tarjan_def.tarjan_params G) s"
intCase: assumes "w \<noteq> NEG_INF"   and "w \<noteq> POS_INF" shows "\<exists>r. Rep_bword bw \<equiv>\<^sub>E r"
intros_1: fixes s :: "dB"   and dummy :: "dB" assumes "\<not> free s 0" shows "Abs (s \<degree> Var 0) \<rightarrow>\<^sub>\<eta> s[dummy/0]"
that: fixes \<xi> :: "'a"   and p :: "'b"   and q :: "'c"   and R :: "nat set" assumes "st = NodeS ii ((\<xi>, p), q) R" shows "thesis"
harm_aux_ineq_1: fixes k :: "real" assumes "1 < k" shows "1 / k < ln (1 + 1 / (k - 1))"
sig_red_single_regular_lc: fixes top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "sig_red_single (\<prec>\<^sub>t) top_tail p q f t" shows "lc q = lc p"
prv_nprv1I: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (imp \<phi> \<psi>)" shows "nprv {\<phi>} \<psi>"
a_coset_add_zero: fixes G :: "('c, 'd) ring_scheme"   and M :: "'c set" assumes "abelian_group G"   and "M \<subseteq> carrier G" shows "M +>\<^bsub>G\<^esub> \<zero>\<^bsub>G\<^esub> = M"
cf_get_def: fixes cf :: "nat \<times> nat \<Rightarrow> int"   and e :: "nat \<times> nat" shows "cf_get cf e \<equiv> assert (case e of (u, v) \<Rightarrow> u \<in> V \<and> v \<in> V) \<bind> (\<lambda>_. return (cf e))"
not_col_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C \<and> A \<noteq> B \<and> B \<noteq> C \<and> A \<noteq> C"
pders_coeffs_sgn_eq: fixes p :: "real poly" shows "map (\<lambda>p. sgn (poly p 0)) (pders p) = map sgn (coeffs p)"
SAT: shows "\<E> (TER f) \<subseteq> SAT \<Gamma> k"
OrI2: fixes \<Gamma> :: "'b formula set"   and G :: "'b formula"   and F :: "'b formula" assumes "\<Gamma> \<turnstile> G" shows "\<Gamma> \<turnstile> F \<^bold>\<or> G"
purgeIdle_getGMUserCom_purge: fixes sl :: "('St, 'U, 'C) state list"   and sl' :: "('St, 'U, 'C) state list" assumes "wffp sl"   and "wffp sl'"   and "list_all2 f sl sl'" shows "purge GH (map getGMUserCom (purgeIdle (tl sl))) = purge GH (map getGMUserCom (purgeIdle (tl sl')))"
repstate_simps: fixes \<nu> :: "fin_string + fin_string \<Rightarrow> 32 Word.word"   and \<nu>' :: "fin_string \<Rightarrow> real" shows "(\<nu> REP \<nu>') = (\<forall>x. (\<nu> (Inl x) \<equiv>\<^sub>L \<nu>' x) \<and> (\<nu> (Inr x) \<equiv>\<^sub>U \<nu>' x))"
literalFalse_def: fixes literal :: "Literal"   and valuation :: "Literal list" shows "literalFalse literal valuation \<equiv> opposite literal el valuation"
\<tau>Exect_induct: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option"   and Q :: "'addr val list \<Rightarrow> 'addr val list \<Rightarrow> nat \<Rightarrow> 'addr option \<Rightarrow> bool" assumes "\<tau>Exec_movet ci P t e h (stk, loc, pc, xcp) (stk'', loc'', pc'', xcp'')"   and "\<And>stk' loc' pc' xcp'. \<tau>exec_move ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp') \<Longrightarrow> Q stk' loc' pc' xcp'"   and "\<And>stk' loc' pc' xcp' stk'' loc'' pc'' xcp''. \<lbrakk>\<tau>Exec_movet ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp'); \<tau>exec_move ci P t e h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp''); Q stk' loc' pc' xcp'\<rbrakk> \<Longrightarrow> Q stk'' loc'' pc'' xcp''" shows "Q stk'' loc'' pc'' xcp''"
lconf_def: fixes P :: "'b prog"   and h :: "heap"   and l :: "sfields"   and E :: "char list \<Rightarrow> ty option" shows "P,h \<turnstile> l (:\<le>) E \<equiv> \<forall>V v. l V = \<lfloor>v\<rfloor> \<longrightarrow> (\<exists>T. E V = \<lfloor>T\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
dmu_array_row_cost_def: fixes dmus :: "int iarray iarray"   and i :: "nat" shows "dmu_array_row_cost dmus i = (let fi = fs !! i; sp = fi \<bullet> fs !! 0; local_cost = 2 * n; (res, main_cost) = dmu_array_row_main_cost fi i (iarray_append dmus (IArray [sp])) 0 in (res, local_cost + main_cost))"
that: fixes z :: "complex" assumes "z \<in> ball \<xi> r"   and "f z = 0" shows "thesis"
less_le_trans: fixes x :: "'d word"   and y :: "'d word"   and z :: "'d word" assumes "x <s y"   and "y \<le>s z" shows "x <s z"
term_sym: fixes s :: "dB"   and t :: "dB" assumes "s \<leftrightarrow> t" shows "t \<leftrightarrow> s"
le_real_sqrt_sumsq': fixes y :: "real"   and x :: "real" shows "y \<le> sqrt (x * x + y * y)"
rref_upt_condition1_explicit: fixes A :: "(('b, 'c) vec, 'd) vec"   and k :: "nat"   and i :: "'d"   and j :: "'d" assumes "reduced_row_echelon_form_upt_k A k"   and "is_zero_row_upt_k i k A"   and "i < j" shows "is_zero_row_upt_k j k A"
le1234_lt__lt: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D"   and "Tarski_neutral_dimensionless.Lt Bet Cong C D E F" shows "Tarski_neutral_dimensionless.Lt Bet Cong A B E F"
a: shows " t \<longrightarrow>\<^sub>\<beta> t'"
one_closed: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
T3: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and v :: "'b \<Rightarrow> 'c" shows "\<turnstile> \<box>[_actrans P v]_v = \<box>[P]_v"
axioms_2: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equivalence A equiv" shows "reflexive A equiv"
StrongEq_sym: fixes X :: "'a state \<times> 'a state \<Rightarrow> 'b"   and Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(X \<triangleq> Y) = (Y \<triangleq> X)"
distinct_3: fixes x1 :: "agent"   and x3 :: "fresh_t" shows "msg.Agent x1 \<noteq> Nonce x3"
less_sets_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<lless> B \<equiv> \<forall>x\<in>A. \<forall>y\<in>B. x < y"
M_def: fixes n :: "nat"   and m :: "nat"   and fs_init :: "int vec list" shows "LLL.M n m fs_init = Max ({\<bar>fs_init ! i $ j\<bar> |i j. i < m \<and> j < n} \<union> {0})"
ext_cf_pol_coeff: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and n :: "nat" assumes "pol_coeff S c" shows "pol_coeff S (ext_cf S n c)"
iso_pullback_ide: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "cospan \<mu> \<nu>"   and "ide \<mu>" shows "local.iso \<p>\<^sub>0[\<mu>, \<nu>]"
no_end_point_char: fixes p :: "'a"   and x :: "'a" assumes "point p" shows "(x\<^sup>T ; p \<noteq> 0) = (p \<le> x ; 1)"
final: shows "nfa.final \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr> \<subseteq> nfa.states \<lparr>nfa.states = HF.Inl ` dfa.states MS \<union> HF.Inr ` dfa.states MT, init = {HF.Inl (dfa.init MS)}, final = HF.Inr ` dfa.final MT, nxt = \<lambda>q x. sum_case (\<lambda>qs. {HF.Inl (dfa.nxt MS qs x)}) (\<lambda>qt. {HF.Inr (dfa.nxt MT qt x)}) q, eps = (\<lambda>q. (HF.Inl q, HF.Inr (dfa.init MT))) ` dfa.final MS\<rparr>"
SIGMA_image_vimage: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set" shows "snd ` (SIGMA x:f ` X. f -` {x} \<inter> X) = X"
iwlsFSwSTR_qInit_respectsP_alpha: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "qInit MOD respectsP (#=)"