hcju/isabelleps huggingface dataset:
conj_bot_right: fixes c :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice) \<iinter> \<bottom> = \<bottom>"
assms_2: shows "y \<notin> \<int>\<^sub>\<le>\<^sub>0"
closed_set_weak_conv_usc: fixes S :: "real set"   and l :: "real" assumes "closed S"   and "prob S < l" shows "\<forall>\<^sub>F n in sequentially. Sigma_Algebra.measure (\<mu> n) S < l"
less_sets_weaken1: fixes A' :: "'a set"   and B :: "'a set"   and A :: "'a set" assumes "A' \<lless> B"   and "A \<subseteq> A'" shows "A \<lless> B"
IH_2: fixes \<Gamma> :: "(name, name) fmap" assumes "fmpred (=) \<Gamma>" shows "alpha_equiv \<Gamma> t2_ t2_"
in_Edges_in_set: fixes x :: "nat"   and y :: "nat"   and vs :: "nat list" assumes "(x, y) \<in> Edges vs" shows "x \<in> set vs \<and> y \<in> set vs"
5: shows "(top * e)\<^sup>T * (F * e\<^sup>T * top \<sqinter> w\<^sup>T\<^sup>\<star>) \<le> F"
hd_in_awalk_verts_1: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "awalk u p v" shows "u \<in> set (pawalk_verts u p)"
set_der_stack_lt: fixes k :: "nat"   and xs :: "int list"   and i\<^sub>0 :: "nat" assumes "k \<in> set (der_stack xs i\<^sub>0)" shows "k < i\<^sub>0"
angle_le_pi: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "angle a b c \<le> pi"
z_def: shows "z = cis (2 * pi / real (ord x))"
ereal_mono2: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "ereal x \<le> ereal y"
distinct_5: fixes x1 :: "'a" shows "Assumption x1 \<noteq> Helper"
exhaust: fixes y :: "('b, 'a) label" obtains x1 :: "'a" where "y = LBase x1"   | x2 :: "'b" where "y = LCall x2"   | "y = LRet"   | x4 :: "'b" where "y = LSpawn x4"
weak_ranking_unique: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and As :: "'b set list" assumes "finite_total_preorder_on carrier le"   and "is_weak_ranking As"   and "of_weak_ranking As = le" shows "As = weak_ranking le"
select_convs_3: fixes ip :: "nat"   and sn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and rreqs :: "(nat \<times> nat) set"   and store :: "nat \<Rightarrow> (p \<times> nat list) option"   and msg :: "msg"   and data :: "nat"   and dests :: "nat \<Rightarrow> nat option"   and dip :: "nat"   and oip :: "nat"   and hops :: "nat"   and dsn :: "nat"   and dsk :: "k"   and osn :: "nat"   and sip :: "nat"   and handled :: "bool"   and more :: "'a" shows "rt \<lparr>ip = ip, sn = sn, rt = rt, rreqs = rreqs, store = store, msg = msg, data = data, dests = dests, dip = dip, oip = oip, hops = hops, dsn = dsn, dsk = dsk, osn = osn, sip = sip, handled = handled, \<dots> = more\<rparr> = rt"
inf_K_next_stimulus: fixes a :: "'a"   and b :: "'a"   and s :: "'b" assumes "a \<le>\<^sub>\<K> b" shows "\<lambda> (s, a) \<le>\<^sub>\<S> \<lambda> (s, b)"
mult_isol: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<cdot> x \<le> z \<cdot> y"
graph_union_semantics: fixes A :: "('e, 'd) labeled_graph"   and e :: "'e allegorical_term"   and B :: "('e, 'd) labeled_graph" shows "(:A:\<lbrakk>e\<rbrakk>) \<union> (:B:\<lbrakk>e\<rbrakk>) \<subseteq> :graph_union A B:\<lbrakk>e\<rbrakk>"
termi_call_steps_def: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" shows "termi_call_steps \<Gamma> = {((t, q), s, p). \<Gamma>\<turnstile>Call p \<down> Normal s \<and> (\<exists>c. \<Gamma>\<turnstile> (Call p, Normal s) \<rightarrow>\<^sup>+ (c, Normal t) \<and> redex c = Call q)}"
le_prod_encode_2: fixes b :: "nat"   and a :: "nat" shows "b \<le> prod_encode (a, b)"
compat_mult2_1: fixes s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "s O ns \<subseteq> s"   and "refl ns"   and "trans ns" shows "mult2_ns ns s O mult2_s ns s \<subseteq> mult2_s ns s"
indep: shows "\<forall>s\<^sub>1 s\<^sub>2. Tr\<^sub>F n s\<^sub>1 \<inter> Tr\<^sub>f n s\<^sub>2 = {}"
lang_dual_subset_lists: fixes n :: "nat"   and r :: "'b rexp_dual" assumes "wf_dual n r" shows "lang_dual n r \<subseteq> lists (\<Sigma> n)"
scope_def: fixes outPorts :: "'c \<Rightarrow> 'd fset"   and vertices :: "'a fset"   and nodeOf :: "'a \<Rightarrow> 'c"   and edges :: "(('a \<times> 'd) \<times> 'a \<times> 'b) set" shows "Pre_Port_Graph.scope outPorts vertices nodeOf edges \<equiv> \<lambda>ps. {x. Pre_Port_Graph.scopep outPorts vertices nodeOf edges ps x}"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>L f : L a \<Rightarrow> L b\<guillemotright>"
simps_3: fixes sys :: "'b \<Rightarrow> ('a, 'b) process_term"   and a :: "'a"   and T :: "('a, 'b) process_term" shows "CONT sys (ACT a T) = T"
eint3_cases: fixes y :: "eint"   and ya :: "eint"   and yb :: "eint" obtains int :: "int" and inta :: "int" and intb :: "int" where "y = eint int__" and "ya = eint inta__" and "yb = eint intb"   | int :: "int" and inta :: "int" where "y = eint intc__" and "ya = eint intaa__" and "yb = \<infinity>"   | int :: "int" and inta :: "int" where "y = eint intd__" and "ya = \<infinity>" and "yb = eint intab__"   | int :: "int" where "y = eint inte__" and "ya = \<infinity>" and "yb = \<infinity>"   | int :: "int" and inta :: "int" where "y = \<infinity>" and "ya = eint intf__" and "yb = eint inta"   | int :: "int" where "y = \<infinity>" and "ya = eint intg__" and "yb = \<infinity>"   | int :: "int" where "y = \<infinity>" and "ya = \<infinity>" and "yb = eint int"   | "y = \<infinity>" and "ya = \<infinity>" and "yb = \<infinity>"
vimage_Union: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'c set set" shows "f -` \<Union> A = \<Union> ((-`) f ` A)"
abs_eq: fixes xa :: "((real, 'a) vec, 'a) vec"   and x :: "((real, 'a) vec, 'a) vec" shows "to_mtx xa * to_mtx x = to_mtx (xa ** x)"
class_of_zero_rng_of_frac: fixes s :: "'a" assumes "s \<in> S" shows "(\<zero> |\<^bsub>rel\<^esub> s) = \<zero>\<^bsub>rec_rng_of_frac\<^esub>"
that: fixes n :: "nat" assumes "rotate n ys \<in> set (cyc_permutations xs)" shows "thesis"
sq_norm_pos: fixes j :: "nat" assumes "j < length (map of_int_hom.vec_hom fs)" shows "0 < \<parallel>fs.gs.gso j\<parallel>\<^sup>2"
Red_Red_I_def: fixes N :: "'f set" shows "Red_Red_I N = Red_I N \<union> Inf_between UNIV (Red_F N)"
countable_acc: fixes X :: "'a set" assumes "countable X" shows "countable ((SIGMA x:UNIV. set_pmf (J x))\<^sup>* `` X)"
subterm_is_var: fixes x :: "'b"   and t :: "('a, 'b) Term.term" assumes "Var x \<sqsubseteq> t" shows "x \<in> fv t"
def_hashmap_size_assoc_list_def: fixes uu :: "('a, 'b) assoc_list itself" shows "def_hashmap_size uu \<equiv> 10"
scalar_vec_plus_distrib_right: fixes nr :: "nat"   and u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "vec nr u"   and "vec nr v"   and "vec nr w" shows "scalar_prod u (vec_plus v w) = scalar_prod u v + scalar_prod u w"
lincomb_is_mod_hom: fixes S :: "'c set" assumes "finite S"   and "S \<subseteq> carrier V" shows "mod_hom K (func_space S) V (\<lambda>a. lincomb a S)"
arr_cod: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr \<times> 'a" assumes "A_BxA.arr f" shows "A_BxA.arr (A_BxA.cod f)"
the_ext_def: fixes y' :: "'c"   and g :: "'d" shows "\<eta>y.the_ext y' g = (THE f. \<eta>y.is_ext y' g f)"
Axiom_of_Powers: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> Vset \<alpha>" shows "VPow a \<in>\<^sub>\<circ> Vset \<alpha>"
sigma_third_divisor: fixes a :: "nat"   and n :: "nat" assumes "1 < a"   and "a < n"   and "a dvd n" shows "1 + a + n \<le> sigma n"
T_A_A'_leq: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init"   and "n \<le> length (BIT_Off.paid_A' acts)" shows "BIT_Off.T_A acts qs init n \<le> BIT_Off.T_A' acts qs init n"
intros_3: fixes m1 :: "real mat convnet"   and m2 :: "real mat convnet" assumes "shared_weight_net m1"   and "shared_weight_net m2"   and "count_weights True (remove_weights m1) = count_weights True (remove_weights m2)"   and "\<And>x. x < count_weights True (remove_weights m1) \<Longrightarrow> extract_weights True m1 x = extract_weights True m2 x" shows "shared_weight_net (Pool m1 m2)"
kE_int_toFull: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and T :: "'a trm" assumes "InfModel wtFsym wtPsym arOf resOf parOf \<Phi> intT intF intP"   and "Struct.wtE intT (InfModel.kE intT \<xi>)"   and "Signature.wt wtFsym arOf resOf T" shows "InfModel.toFull intT (Signature.tpOf resOf T) (Struct.int intF (InfModel.kE intT \<xi>) T) = Struct.int (InfModel.intFF arOf resOf intT intF) \<xi> T"
full_functor_axioms_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "full_functor_axioms A B F \<equiv> \<forall>a a' g. partial_magma.ide A a \<longrightarrow> partial_magma.ide A a' \<longrightarrow> partial_magma.in_hom B g (F a') (F a) \<longrightarrow> (\<exists>f. partial_magma.in_hom A f a' a \<and> F f = g)"
basic_monos_1: fixes A :: "'d set" shows "A \<subseteq> A"
continuous_open_preimage: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_on S f"   and "open S"   and "open T" shows "open (S \<inter> f -` T)"
dirichlet_prod_neutral_right_neutral: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex" assumes "0 < n" shows "dirichlet_prod f dirichlet_prod_neutral n = f n"
setdist_eq_0_compact_closed: fixes S :: "'a set"   and T :: "'a set" assumes "compact S"   and "closed T" shows "(setdist S T = 0) = (S = {} \<or> T = {} \<or> S \<inter> T \<noteq> {})"
lex_append_left_iff: fixes r :: "('g \<times> 'g) set"   and xs :: "'g list"   and ys :: "'g list"   and zs :: "'g list" assumes "\<forall>x. (x, x) \<notin> r" shows "((xs @ ys, xs @ zs) \<in> lex r) = ((ys, zs) \<in> lex r)"
Id_def: shows "stack.Id \<equiv> Abs_stack Id_Rep"
is_finite_Float_eq: fixes f :: "Float.float" shows "is_finite_Float TYPE(('a, 'b) IEEE.float) f = (let e = Float.exponent f; bm = bitlen \<bar>mantissa f\<bar> in bm \<le> int (Suc LENGTH('b)) \<and> bm \<le> 2 ^ (LENGTH('a) - 1) - e \<and> 1 - 2 ^ (LENGTH('a) - 1) - int LENGTH('b) < e)"
restrict_map_remerge: fixes S :: "'c set"   and T :: "'c set"   and m :: "'c \<Rightarrow> 'd option" assumes "S \<inter> T = {}" shows "m |` S ++ m |` T = m |` (S \<union> T)"
has_line_integral_on_reverse_path: fixes g :: "real \<Rightarrow> 'a"   and F :: "'a \<Rightarrow> 'a"   and basis :: "'a set"   and c :: "real" assumes "valid_path g"   and "((\<lambda>x. \<Sum>b\<in>basis. F (g x) \<bullet> b * (vector_derivative g (at x within {0..1}) \<bullet> b)) has_integral c) {0..1}" shows "((\<lambda>x. \<Sum>b\<in>basis. F (reversepath g x) \<bullet> b * (vector_derivative (reversepath g) (at x within {0..1}) \<bullet> b)) has_integral - c) {0..1}"
unlabel_def: fixes S :: "('e \<times> 'd) list" shows "unlabel S \<equiv> map snd S"
Sigma'_smaller: shows "Sigma A' B' \<subseteq> Sigma A B"
nodes_in_wf_ll: fixes pret :: "dag"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and i :: "nat"   and no :: "ref" assumes "wf_ll pret levellista var"   and "i < length levellista"   and "no \<in> set (levellista ! i)" shows "var no = i \<and> no \<in> set_of pret"
Gamma_series_euler'_def: fixes z :: "'a"   and n :: "nat" shows "Gamma_series_euler' z n = inverse z * (\<Prod>k = 1..n. exp (z * of_real (ln (1 + inverse (real k)))) / ((1::'a) + z / of_nat k))"
reflexive: fixes w :: "'b stream"   and A :: "('b, 'a) nba"   and V :: "(nat \<times> 'a) set"   and p :: "nat \<times> 'a" shows "transition_system.reachablep (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V) p p"
hyps: fixes E :: "'a set" assumes "dimension m F E" shows "dimension (n * m) K E"
succ_inv_2: fixes s :: "('i, 'a) state"   and s' :: "('i, 'a) state" assumes "\<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s" shows "\<nabla> s'"
get_disconnected_document_pure: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap" shows "pure (get_disconnected_document node_ptr) h"
V'_def: shows "V' \<equiv> {Z. Z \<subseteq> V \<and> (\<exists>M. Z = [M]\<^bsub>v,n\<^esub> \<and> dbm_int M n)}"
A_singleton: fixes nm :: "'a stream measure \<times> 'a stream measure" shows "A `` {nm} = (\<lambda>x. (n (fst nm) (snd nm) x, m (fst nm) (snd nm) x)) ` set_pmf (p (fst nm) (snd nm))"
C6: shows "rel_respects_binary_pred Rel (\<lambda>a. hasBarb a (STCalWB SWB TWB))"
stable: shows "\<mu>_stable (af \<phi> (prefix i w)) (suffix i w)"
nonneg_vel_ego: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "0 \<le> v\<^sub>e"
assms_3: shows "abs_conv_abscissa (fds_deriv f / f) < ereal (s \<bullet> (1::'a))"
prec_after_EV_SIGNAL_step: fixes s :: "rstate_t"   and aseq :: "int_point_t list"   and n :: "nat" assumes "rprecondition s (rcurrent s) (aseq ! n)"   and "Suc n < length aseq"   and "is_SK_EV_SIGNAL aseq"   and "\<not> raborting s (rcurrent s) (aseq ! n)"   and "\<not> rwaiting s (rcurrent s) (aseq ! n)" shows "rprecondition (rstep s (aseq ! n)) (rcurrent s) (aseq ! Suc n)"
chooseNext: fixes g :: "'g" assumes "redundant g" shows "chooseNext g \<in> allVars g \<and> trivial g (chooseNext g)"
simps_2: fixes p :: "perm"   and f :: "'b" shows "p \<bullet> Eff f = Eff (p \<bullet> f)"
unique_row_col_2: fixes nr1 :: "nat"   and nc1 :: "nat"   and M :: "'a list list"   and nr2 :: "nat"   and nc2 :: "nat" assumes "mat nr1 nc1 M"   and "mat nr2 nc2 M"   and "M \<noteq> []" shows "nc1 = nc2"
left_cancel: fixes a :: "'ag"   and b :: "'ag"   and c :: "'ag" assumes "a \<in> U"   and "b \<in> U"   and "c \<in> U" shows "(a +\<^sub>o\<^sub>w b = a +\<^sub>o\<^sub>w c) = (b = c)"
symbS_def: shows "symbS = SymSystem initS localPrecS localRelS"
wU: shows "set (w # us) \<subseteq> carrier_vec n"
domain_eq: shows "Domainp pcr_real_alg_3 = (\<lambda>x. x \<in> Collect invariant_2)"
is_idirprod_def: fixes G :: "('c, 'd) monoid_scheme"   and A :: "'c set"   and S :: "'e \<Rightarrow> 'c set"   and I :: "'e set" assumes "Group.group G" shows "group.is_idirprod G A S I = ((\<forall>i\<in>I. S i \<lhd> G) \<and> A = IDirProds G S I \<and> group.compl_fam G S I)"
all_left_def: fixes xs :: "int list" shows "all_left xs \<equiv> nfoldli [0..<length xs] (\<lambda>_. True) (\<lambda>i (stk, lf). ASSERT (set stk \<subseteq> {0..<length xs}) \<bind> (\<lambda>_. let stk = pop xs stk (xs ! i) in ASSERT (stk = der_stack xs i) \<bind> (\<lambda>_. ASSERT (i < length lf) \<bind> (\<lambda>_. if stk = [] then let lf = lf[i := None] in RETURN (i # stk, lf) else let lf = lf[i := Some (hd stk)] in RETURN (i # stk, lf))))) ([], replicate (length xs) None) \<bind> (\<lambda>(uu_, lf). RETURN lf)"
model_to_dimacs_model_complete: fixes f :: "nat formula"   and n_vars :: "nat"   and dimacs_M :: "int list" assumes "\<forall>v\<in>atoms f. 0 < v \<and> v < n_vars"   and "is_cnf f"   and "distinct (map dimacs_lit_to_var dimacs_M)"   and "dimacs_model dimacs_M (cnf_to_dimacs f)" shows "dimacs_model_to_abs dimacs_M (\<lambda>_. False) \<Turnstile> f"
diff_union_single_conv2: fixes a :: "'a"   and J :: "'a multiset"   and I :: "'a multiset" assumes "a \<in># J" shows "J + I - {#a#} = J - {#a#} + I"
simps_1: fixes y :: "'a" shows "count_list [] y = 0"
Lp_2: fixes p :: "real"   and M :: "'a measure" assumes "1 \<le> p" shows "defect (\<LL> (ennreal p) M) = 1"
int_of_integer_inv: fixes x :: "int" shows "int_of_integer (integer_of_int x) = x"
godd_def: shows "PROP godd \<equiv> (\<And>n kvs t k v kvs'. \<lbrakk>0 < n; n \<le> length kvs; P n kvs; rbtreeify_f n kvs = (t, (k, v) # kvs'); n \<le> Suc (length kvs'); Q n kvs'\<rbrakk> \<Longrightarrow> Q (Suc (2 * n)) kvs)"
hyps: shows "Q \<oplus> T \<longmapsto> \<alpha> \<prec> Q'"
reachable_imp_reachable_tot: fixes s :: "'b"   and F :: "'b program" assumes "s \<in> reachable F" shows "s \<in> reachable (totalize F)"
right_id: fixes x :: "int poly fract" shows "rat_poly_times x 1 = x"
seq_char: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" shows "CC.seq g f = (C.seq (fst g) (fst f) \<and> C.seq (snd g) (snd f))"
phl_cons1: fixes x :: "'a"   and w :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> w"   and "w \<cdot> y \<le> y \<cdot> z" shows "x \<cdot> y \<le> y \<cdot> z"
bigc_def: shows "bigc = (\<lambda>n. contour_integral (part_circlepath 0 (R n) 0 pi) (f s) - cnj (contour_integral (part_circlepath 0 (R n) 0 pi) (f (cnj s))))"
Reception: fixes evsr :: "event list"   and A :: "agent"   and B :: "agent"   and X :: "msg" assumes "evsr \<in> set_cr"   and "Says A B X \<in> set evsr" shows "Gets B X # evsr \<in> set_cr"
map_axioms: shows "Set_Theory.map (\<lambda>\<pp>\<in>Spec. undefined) {} {}"
curry_case_prod: fixes f :: "'k \<Rightarrow> 'l \<Rightarrow> 'm" shows "curry (\<lambda>(x, y). f x y) = f"
Greatest_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.order less_eq less" shows "order.Greatest less_eq P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> less_eq y x))"
2: shows "make_mat n (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i))))) $$ (Suc i, Suc i) = snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))) $$ (1, 1)"
syz_cofactors: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> pmdl (set (init_syzygy_list bs))" shows "proj_poly_syz (length bs) p = sum_list (map2 (\<odot>) (cofactor_list_syz (length bs) p) bs)"
unit_simps_2: shows "local.dom \<zeta> = src f"
prod_emb_Un: fixes M :: "'b set"   and L :: "'b \<Rightarrow> 'c measure"   and K :: "'b set"   and A :: "('b \<Rightarrow> 'c) set"   and B :: "('b \<Rightarrow> 'c) set" shows "prod_emb M L K (A \<union> B) = prod_emb M L K A \<union> prod_emb M L K B"
thmD10_ladder: fixes p :: "(symbol \<times> character list) list"   and k :: "nat"   and X :: "symbol \<times> character list"   and T :: "(symbol \<times> character list) set"   and N :: "symbol"   and \<alpha> :: "symbol list"   and \<beta> :: "symbol list"   and r :: "nat"   and \<gamma> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and k' :: "nat"   and x :: "item"   and I :: "item set" assumes "p \<in> \<PP>"   and "charslength p = k"   and "X \<in> T"   and "T \<subseteq> \<X> k"   and "(N, \<alpha> @ \<beta>) \<in> \<RR>"   and "r \<le> length p"   and "leftderives [\<SS>] (terminals (take r p) @ [N] @ \<gamma>)"   and "LeftDerivationLadder \<alpha> D L (terminals (drop r p @ [X]))"   and "ladder_last_j L = length (drop r p)"   and "k' = k + length (chars_of_token X)"   and "x = Item (N, \<alpha> @ \<beta>) (length \<alpha>) (charslength (take r p)) k'"   and "I = items_le k' (\<pi> k' {} (Scan T k (Gen (Prefixes p))))" shows "x \<in> I"
less_imp_le: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec> b" shows "a \<preceq> b"
split_relation_alt: fixes as :: "'a list"   and ls :: "'a list"   and rs :: "'a list"   and i :: "nat" shows "split_relation as (ls, rs) i = (as = ls @ rs \<and> i = length ls)"
that: fixes xo :: "'b list" assumes "xo \<in> lists Q"   and "x = map f xo" shows "thesis"
dist_def: fixes G :: "graph"   and m :: "nat"   and n :: "nat" shows "dist G m n = path_weight G (SOME p. is_shortest_path G m n p)"
int_Zcarr: fixes k :: "int" shows "k \<in> carrier \<Z>"
3: shows "fv (t \<cdot> \<theta>2) \<inter> insert v (subst_domain (\<sigma>1 \<circ>\<^sub>s \<theta>2)) = {}"
ceiling_divide_upper: fixes q :: "'a"   and p :: "'a" assumes "(0::'a) < q" shows "p \<le> of_int \<lceil>p / q\<rceil> * q"
isRefV_simps_4: fixes "by" :: "int" shows "isRefV (byteV by) = False"
fst_map_prod: fixes f :: "'m \<Rightarrow> 'k"   and g :: "'n \<Rightarrow> 'l"   and x :: "'m \<times> 'n" shows "fst (map_prod f g x) = f (fst x)"
scalar_prod: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and v1 :: "'a list"   and v2 :: "'a list" assumes "class.semiring_0 plus zero times" shows "scalar_prodI zero plus times v1 v2 = monoid_add.sum_list plus zero (map2 times v1 v2)"
enc_eqI: fixes x :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "x \<in> enc (w, I)"   and "x \<in> enc (w', I')"   and "wf_interp_for_formula (w, I) \<phi>"   and "wf_interp_for_formula (w', I') \<phi>"   and "length I = length I'" shows "enc (w, I) = enc (w', I')"
zero_sub_left: fixes x :: "complex" shows "0\<^sub>h -\<^sub>h of_complex x = of_complex (- x)"
equivalence_functor_axioms: shows "equivalence_functor (\<cdot>) (\<cdot>) (\<lambda>f. T (f, cod \<iota>))"
mkClockAuto_implements: shows "MC.Clock.implements (mkClockAuto agents jkbp envInit envAction envTrans envVal envObs)"
trancl_into_trancl: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
meas_M: shows "M \<in> state_measure (V \<union> V') \<Gamma> \<rightarrow>\<^sub>M subprob_algebra (stock_measure t)"
mbr_simp: fixes C :: "qtname"   and m :: "memberdecl" shows "mbr (C, m) = m"
sub_branch_upd_fresh: fixes i :: "'b"   and branch :: "(('c, 'b) fm list \<times> 'b) list"   and f :: "'b \<Rightarrow> 'd"   and j :: "'d" assumes "i \<notin> branch_nominals branch" shows "sub_branch (f(i := j)) branch = sub_branch f branch"
simps_2: fixes v :: "nat" shows "polypow (Suc v) = (\<lambda>p. let q = p ^\<^sub>p Suc v div 2; d = q *\<^sub>p q in if even (Suc v) then d else p *\<^sub>p d)"
abs_eq: fixes xb :: "nat \<Rightarrow> bool"   and xa :: "dflt_size word"   and x :: "nat" shows "uint_set_bits xb (Abs_uint xa) x = Abs_uint (set_bits_aux xb xa x)"
range_to_int_mod_ring: shows "range to_int_mod_ring = {0..<int CARD('a)}"
le_divide_eq_numeral1_1: fixes a :: "'b"   and b :: "'b"   and w :: "num" shows "(a \<le> b / numeral w) = (a * numeral w \<le> b)"
simps_1: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and x :: "variable" shows "usubstappt \<sigma> U (Var x) = Aterm (Var x)"
xfer_run: fixes p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'capacity"   and l' :: "nat \<Rightarrow> nat" assumes "((f'__, l'__), p, f', l') \<in> trcl pr_algo_lts" obtains p' :: "pr_operation' list" where "((f'__, l'__), p', f', l') \<in> trcl pr_algo_lts'" and "p = map project_operation p'"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "subseqs (x # xs) = (let xss = subseqs xs in map ((#) x) xss @ xss)"
times_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "xs \<cdot> ys \<equiv> xs @ ys"
evaln_elim_cases_26: fixes G :: "prog"   and s :: "st"   and va :: "Term.var"   and e :: "expr"   and n :: "nat"   and v :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1l (va:=e)\<succ>\<midarrow>n\<rightarrow> (v, s')" obtains w :: "val" and f :: "val \<Rightarrow> state \<Rightarrow> state" and s1 :: "state" and vb :: "val" and s2 :: "state" where "v = In1 vb" and "s' = assign f vb s2" and "G\<turnstile>Norm s \<midarrow>va=\<succ>(w, f)\<midarrow>n\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>e-\<succ>vb\<midarrow>n\<rightarrow> s2"