hcju/isabelleps huggingface dataset:
terminates_Normal_elim_cases_5: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "'b" assumes "\<Gamma>\<turnstile>Seq c1 c2 \<down> Normal s" obtains "\<Gamma>\<turnstile>c1 \<down> Normal s" and "\<forall>s'. \<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> s' \<longrightarrow> \<Gamma>\<turnstile>c2 \<down> s'"
t_height: shows "height t = 1 + max (height (MKT ln ll lr lh)) (height (MKT rn rl rr rh))"
sc: shows "non_speculative P (w_values P (\<lambda>_. {}) (map snd E)) (llist_of \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)"
rk_acA'C'\<beta>: fixes Q :: "Points"   and A' :: "Points"   and a :: "Points"   and C' :: "Points"   and c :: "Points"   and \<beta> :: "Points"   and R :: "Points"   and A :: "Points" assumes "rk {Q, A', a} = 2"   and "rk {Q, C', c} = 2"   and "rk {A', C'} = 2"   and "rk {A', C', \<beta>} = 2"   and "rk {R, A, a} = 2"   and "4 \<le> rk {A', A, C', a}" shows "rk {a, c, A', C', \<beta>} = 3"
NF_Id_on_fst_image: fixes A :: "('b \<times> 'b) set" shows "NF (Id_on (fst ` A)) = NF A"
6: shows "bprv (eqv (softSubst \<chi> \<langle>\<chi>\<rangle> xx) (subst \<phi> \<langle>\<psi>\<rangle> xx))"
keys_lowerPoly\<^sub>0: fixes b :: "nat"   and i :: "nat"   and mp :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "keys (lowerPoly\<^sub>0 b i mp) = higherPowers b i ` (keys mp \<inter> {x. \<forall>j\<in>{b..<b + i}. lookup x j = 0})"
assms: shows "inv_check_left_moving_in_middle (as, lm) (s, Bk # list, Oc # r) ires"
elims: fixes x :: "'b up\<^sub>i"   and y :: "'b tree234" assumes "tree\<^sub>i x = y" obtains t :: "'b tree234" where "x = T\<^sub>i t" and "y = t"   | l :: "'b tree234" and a :: "'b" and r :: "'b tree234" where "x = Up\<^sub>i l a r" and "y = \<langle>l, a, r\<rangle>"
assms: shows "\<S>\<bullet>\<C> \<turnstile> [Label n es0 es] : ts _> ts'"
disc_58: fixes x8 :: "'c ltl" shows "is_LTLGlobal (G x8)"
\<pi>_\<theta>_bound: fixes x :: "real" assumes "4 \<le> x" shows "\<pi> x - \<theta> x / ln x \<in> {0..2 / ln 2 * sqrt x + 8 * ln 2 * x / (ln x)\<^sup>2}"
insertI1: fixes a :: "'c"   and B :: "'c set" shows "a \<in> insert a B"
PiE_cancel_merge: fixes I :: "'d set"   and J :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e"   and B :: "'d \<Rightarrow> 'e set" assumes "I \<inter> J = {}" shows "(merge I J (x, y) \<in> Pi\<^sub>E (I \<union> J) B) = (x \<in> Pi I B \<and> y \<in> Pi J B)"
a: shows "\<Gamma> \<turnstile> t \<equiv> s : T"
fromFull_def: fixes \<sigma> :: "'tp" shows "fromFull \<sigma> \<equiv> inv_into {a. intT \<sigma> a} (toFull \<sigma>)"
bounded_path_connected_Compl_real: fixes S :: "real set" assumes "bounded S"   and "path_connected (- S)" shows "S = {}"
ucast_s: fixes w :: "word32" assumes "get_S w = 0" shows "w AND 128 = 0"
zip_Nil: fixes ys :: "'d list" shows "zip [] ys = []"
zero_in_Field: assumes "Field r \<noteq> {}" shows "local.zero \<in> Field r"
assms: shows "\<tau> \<Turnstile> OclAllInstances_generic pre_post H->includes\<^sub>S\<^sub>e\<^sub>t(x)"
m: shows "(derived K ^^ m) (g ` J) = {\<one>\<^bsub>K\<^esub>}"
HStartBallot_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc" shows "HStartBallot s s' p = (StartBallot s s' p \<and> HNextPart s s')"
hs_delete_def: fixes x :: "'a"   and ht :: "('a, unit) hashtable" shows "hs_delete x ht \<equiv> hm_delete x ht"
chainTransitionAppend_5: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P'' :: "pi"   and P' :: "pi" assumes "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P''"   and "P'' \<Longrightarrow>\<^sub>\<tau> P'" shows "P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P'"
preserves_arr: fixes f :: "'d" assumes "D.arr f" shows "D.arr (trg\<^sub>D f)"
sum_moebius_mu_ln_eq: shows "H = (\<lambda>x. - dirichlet_prod' moebius_mu \<psi> x)"
det_I: shows "det (mat (1::'a)) = (1::'a)"
simps_2: fixes x21 :: "'g"   and x22 :: "'h \<Rightarrow> ('f, 'g, 'h) env option"   and y21 :: "'g"   and y22 :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "(Env x21 x22 = Env y21 y22) = (x21 = y21 \<and> x22 = y22)"
vsubset_vsingleton_rightD: fixes A :: "V"   and a :: "V" assumes "A \<subseteq>\<^sub>\<circ> set {a}" shows "A = set {a} \<or> A = 0"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Cop.in_hom f a b" shows "\<guillemotleft>\<Psi>.map f : Cones.map a \<rightarrow>\<^sub>S Y a b\<guillemotright>"
assms: shows "z \<notin> \<real>\<^sub>\<ge>\<^sub>0"
iso_iff: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "(h \<in> Group.iso G H) = (carrier H \<subseteq> h ` carrier G \<and> (\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>))"
assms_2: shows "[xs [\<mapsto>] As] x = Some Ai"
zeroi_def: fixes R :: "('e, 'f) Ring_scheme" shows "zeroi R = (\<lambda>j. \<zero>\<^bsub>R\<^esub>)"
better_Cut_substc1: fixes a :: "coname"   and P :: "trm"   and b :: "coname"   and N :: "trm"   and M :: "trm"   and x :: "name"   and y :: "name" assumes "a \<sharp> (P, b)"   and "b \<sharp> N" shows "Cut <a>.M x.N{b:=y.P} = Cut <a>.M{b:=y.P} x.N"
SA_arrow: fixes \<Gamma> :: "binding list"   and T\<^sub>1 :: "POPLmarkRecord.type"   and S\<^sub>1 :: "POPLmarkRecord.type"   and S\<^sub>2 :: "POPLmarkRecord.type"   and T\<^sub>2 :: "POPLmarkRecord.type" assumes "\<Gamma> \<turnstile> T\<^sub>1 <: S\<^sub>1"   and "\<Gamma> \<turnstile> S\<^sub>2 <: T\<^sub>2" shows "\<Gamma> \<turnstile> S\<^sub>1 \<rightarrow> S\<^sub>2 <: T\<^sub>1 \<rightarrow> T\<^sub>2"
pow_comm: fixes t :: "'b list"   and k :: "nat"   and m :: "nat" shows "t \<^sup>@ k \<cdot> t \<^sup>@ m = t \<^sup>@ m \<cdot> t \<^sup>@ k"
adds_pp_refl: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.adds_pp pair_of_term (term_powerprod.pp_of_term pair_of_term v) v"
Pfunc_def: fixes A :: "'d set"   and B :: "'e set" shows "Pfunc A B \<equiv> {f. (\<forall>a. f a \<noteq> None \<longrightarrow> a \<in> A) \<and> (\<forall>a. case f a of None \<Rightarrow> True | Some b \<Rightarrow> b \<in> B)}"
braun_size_merge: fixes l :: "'b tree"   and x :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, x, r\<rangle>" shows "braun (merge l r) \<and> size (merge l r) = size l + size r"
A: fixes U :: "'a set" shows "U \<subseteq> unaffected_domains I D (D ` set ys) zs \<longrightarrow> ipurge_tr_rev_aux I D U (xs @ zs) = ipurge_tr_rev_aux I D U (xs @ ys @ ipurge_tr_aux I D (D ` set ys) zs)"
eventually_not_equal: fixes a :: "'a" shows "\<forall>\<^sub>F x in at_top. x \<noteq> a"
cases: fixes a1 :: "'a"   and a2 :: "'a" assumes "rel_12_21 a1 a2" obtains "a1 = \<^bold>1" and "a2 = \<^bold>2"   | "a1 = \<^bold>2" and "a2 = \<^bold>1"
type_bexpr: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and e :: "'BExp" shows "\<Gamma> \<turnstile>\<^sub>b e \<in> max_dom (to_total \<Gamma> ` bexp_vars e) "
invertible_mat_transpose: fixes A :: "'a mat" assumes "invertible_mat A" shows "invertible_mat A\<^sup>T"
prems: shows "\<forall>i\<in>insert s S. a i \<in> sfis (f i) M"
cone_axioms: shows "D'.cone a' (D'.cones_map f \<tau>o\<chi>.map)"
getPos_NegPos_map_simp2: fixes X :: "'b \<Rightarrow> 'a"   and src :: "'b negation_type list" shows "getPos (NegPos_map X src) = map X (getPos src)"
convex_imp_ANR: fixes S :: "'a set" assumes "convex S" shows "ANR S"
all_countings_def: fixes a :: "nat"   and b :: "nat" shows "all_countings a b = card {f \<in> {1..a + b} \<rightarrow>\<^sub>E {A, B}. card {x \<in> {1..a + b}. f x = A} = a \<and> card {x \<in> {1..a + b}. f x = B} = b}"
raw_subst_drop: fixes x :: "name"   and t :: "'a" shows "abs_pred (\<lambda>t. x |\<notin>| frees t \<longrightarrow> (\<forall>env. subst t (fmdrop x env) = subst t env)) t"
l0f2e: shows "1 / (f2 * (e - 1)) \<le> l0"
frontier_of_subset_subtopology: fixes X :: "'c topology"   and S :: "'c set"   and T :: "'c set" shows "subtopology X S frontier_of T \<subseteq> S"
subterms_subst: fixes t :: "('a, 'b) Term.term"   and d :: "'b \<Rightarrow> ('a, 'b) Term.term" shows "subterms (t \<cdot> d) = (subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t d) \<union> subterms\<^sub>s\<^sub>e\<^sub>t (d ` (fv t \<inter> subst_domain d))"
simps_1: fixes uu :: "nat"   and i :: "int"   and \<beta> :: "nat \<Rightarrow> contour option"   and ve :: "(nat \<times> char list) \<times> contour \<Rightarrow> d option" shows "\<A> (C uu i) \<beta> ve = DI i"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/{} \<^esub>P c' Q,A"
natural_transformation_axioms: shows "natural_transformation C C Adj.FG.map FGFG.map (F \<circ> \<eta> \<circ> G)"
length_takeWhile_le: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "length (takeWhile P xs) \<le> length xs"
invertible: fixes u :: "'a" assumes "u \<in> G" shows "sub.invertible u"
le_def: fixes c :: "int"   and t :: "MIR.num" shows "le c t = (if c = 0 then Le t else if 0 < c then Le (CN 0 c t) else Ge (CN 0 (- c) (Neg t)))"
rOpen: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and yvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and x :: "name"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> M\<lparr>\<nu>*(xvec @ yvec)\<rparr>\<langle>N\<rangle> \<prec> P'"   and "x \<in> supp N"   and "\<And>C. Prop C \<Psi> P M (xvec @ yvec) N P'"   and "x \<sharp> \<Psi>"   and "x \<sharp> M"   and "x \<sharp> xvec"   and "x \<sharp> yvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "yvec \<sharp>* \<Psi>"   and "yvec \<sharp>* P"   and "yvec \<sharp>* M"   and "yvec \<sharp>* C"   and "x \<sharp> C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (\<lparr>\<nu>x\<rparr>P) M (xvec @ x # yvec) N P'"
split_n_m_x_abbrev_dims_2: fixes n :: "nat"   and m :: "nat"   and x :: "(nat, 'a) mapping"   and a :: "'a vec"   and b :: "'a vec" assumes "split_n_m_x n m x = (a, b)" shows "dim_vec b = m"
times_uexpr_def: fixes u :: "('c, 'd) uexpr"   and v :: "('c, 'd) uexpr" shows "u * v = bop (*) u v"
add_shift_minus: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point"   and p\<^sub>3 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "local.add a p\<^sub>1 p\<^sub>2 = p\<^sub>3" shows "p\<^sub>1 = local.add a p\<^sub>3 (opp p\<^sub>2)"
TSTD_def: fixes c :: "nat \<times> cell list \<times> cell list" shows "TSTD c = (let (st, l, r) = c in st = 0 \<and> (\<exists>m. l = Bk \<up> m) \<and> (\<exists>rs n. r = Oc \<up> Suc rs @ Bk \<up> n))"
distinct_27: shows "EqualsIgnoreCaseOp \<noteq> AtOp"
fst_mult_sv: fixes k :: "complex"   and v :: "complex \<times> complex" shows "fst (k *\<^sub>s\<^sub>v v) = k * fst v"
sg6: shows "ack (Suc t) = [st (Suc t)]"
affine_translation_aux: fixes a :: "'a"   and S :: "'a set" assumes "affine ((+) a ` S)" shows "affine S"
complete_defs: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "complete_on r A = (SN_on r A \<and> CR_on r A)"
edge_to_valid_in_port: fixes p :: "nat list"   and c :: "'form" assumes "p \<in> it_paths (it' c)"   and "c \<in> set conclusions" shows "valid_in_port (edge_to c p)"
sees_field_i_i_i_o_iE: fixes x :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and xa :: "char list"   and xb :: "char list"   and xd :: "char list"   and xc :: "ty" assumes "pred.eval (sees_field_i_i_i_o_i x xa xb xd) xc" obtains "x \<turnstile> xa sees xb:xc in xd"
sorted_list_of_set_set_of: fixes l :: "'a list" assumes "strict_sorted l" shows "list_of (list.set l) = l"
assms_1: fixes i :: "'i" shows "f i + y \<le> z"
atLeastLessThan_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c"   and u :: "'c" shows "ord.atLeastLessThan less_eq less l u \<equiv> ord.atLeast less_eq l \<inter> ord.lessThan less u"
nMTeqSet: fixes p :: "'d list"   and s :: "'d list"   and C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option" assumes "set p = set s" shows "none_MT_rules C p = none_MT_rules C s"
axis_index: fixes v :: "(real, 'a) vec" assumes "v \<in> Basis" shows "v = axis (axis_index v) 1"
bind_strict: fixes k :: "'g \<rightarrow> 'e\<cdot>'f" assumes "k\<cdot>\<bottom> = \<bottom>" shows "\<bottom> \<bind> k = \<bottom>"
MGF_deriv: fixes G :: "prog"   and t :: "Term.term" assumes "wf_prog G" shows "G,{}\<turnstile>{\<doteq>} t\<succ> {G\<rightarrow>}"
dvd_mult_left: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times a c"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'b list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
L2_set_nonneg: fixes f :: "'a \<Rightarrow> real"   and A :: "'a set" shows "0 \<le> L2_set f A"
simple_linux_router_nol12_def: fixes rt :: "'a routing_rule list"   and fw :: "'a simple_rule list"   and p :: "('a, 'b) simple_packet_scheme" shows "simple_linux_router_nol12 rt fw p \<equiv> let rd = routing_table_semantics rt (p_dst p); p = p\<lparr>p_oiface := output_iface rd\<rparr>; fd = simple_fw fw p in (case fd of Decision FinalAllow \<Rightarrow> Some () | Decision FinalDeny \<Rightarrow> None) \<bind> (\<lambda>_. Some p)"
ptensor_mat_def: fixes m1 :: "'b mat"   and m2 :: "'b mat" shows "tensor_P m1 m2 = ps_P.tensor_mat m1 m2"
map_of_map: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "('a \<times> 'c) list" shows "map_of (map (\<lambda>(k, v). (k, f v)) xs) = map_option f \<circ> map_of xs"
buffon_def: fixes l :: "real"   and d :: "real" shows "buffon l d = uniform_measure lborel ({- d / 2..d / 2} \<times> {- pi..pi}) \<bind> (\<lambda>(x, \<phi>). return (count_space UNIV) (needle l x \<phi> \<inter> {- d / 2, d / 2} \<noteq> {}))"
lift_imp_neg: fixes A :: "'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> bool" assumes "\<turnstile> A \<longrightarrow> B" shows "\<turnstile> \<not> B \<longrightarrow> \<not> A"
range_of'_def: fixes \<Psi> :: "('d, 'c) sas_plus_problem"   and v :: "'d" shows "\<R>\<^sub>+ \<Psi> v \<equiv> case range_of \<Psi> v of None \<Rightarrow> {} | Some as \<Rightarrow> set as"
assms_3: shows "automaton.trans A = seqp_sos \<Gamma>"
simple_incidence_system_axioms_def: fixes block_collection :: "'b set multiset" shows "simple_incidence_system_axioms block_collection \<equiv> \<forall>bl. bl \<in># block_collection \<longrightarrow> count block_collection bl = 1"
emeasure_buffon_set'_long: shows "emeasure lborel (buffon_set' l d) = ennreal (l * (1 - sqrt (1 - (d / l)\<^sup>2)) + arccos (d / l) * d)"
division_to_div: fixes x :: "int"   and y :: "int"   and z :: "int" assumes "of_int x = of_int y / of_int z" shows "x = y div z"
v_in: shows "v \<in> fv\<^sub>s\<^sub>t (S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S')"
cobounded2: fixes a :: "'c"   and b :: "'c" shows "a \<sqinter> b \<le> b"
get_tag_name_pure: fixes element_ptr :: "(_) element_ptr"   and h :: "(_) heap" shows "pure (get_tag_name element_ptr) h"
less_le_not_le: fixes y :: "'c"   and x :: "'c" shows "(y < x) = (y \<le> x \<and> \<not> x \<le> y)"
less_eq_sat_def: fixes x :: "'b sat"   and y :: "'b sat" shows "(x \<le> y) = (nat_of x \<le> nat_of y)"
less_imp_le: fixes x :: "'a list"   and y :: "'a list" assumes "x <lex y" shows "x \<le>lex y"
singular_relcycle_add: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X S a"   and "singular_relcycle p X S b" shows "singular_relcycle p X S (a + b)"
StaticCastRedsNull: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>null,s'\<rangle>"
ker_def: fixes F :: "('c, 'd) aGroup_scheme"   and G :: "('e, 'f) aGroup_scheme"   and f :: "'c \<Rightarrow> 'e" shows "ker\<^bsub>F,G\<^esub> f = {a \<in> carrier F. f a = \<zero>\<^bsub>G\<^esub>}"
prems_6: shows "Re (trace (\<rho>\<^sub>1 + \<rho>\<^sub>2)) \<le> 1"
map_upt_Suc: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "map f [0..<Suc n] = f 0 # map (\<lambda>i. f (Suc i)) [0..<n]"
measurement2_leq_one_mat: fixes P :: "complex mat"   and Q :: "complex mat"   and M :: "nat \<Rightarrow> complex mat" assumes "P \<in> carrier_mat d d"   and "Q \<in> carrier_mat d d"   and "P \<le>\<^sub>L 1\<^sub>m d"   and "Q \<le>\<^sub>L 1\<^sub>m d"   and "measurement d 2 M" shows "adjoint (M 0) * P * M 0 + adjoint (M 1) * Q * M 1 \<le>\<^sub>L 1\<^sub>m d"
cartesian_product: fixes g :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and B :: "'e set"   and A :: "'d set" shows "F.F (\<lambda>x. F.F (g x) B) A = F.F (\<lambda>(x, y). g x y) (A \<times> B)"
simps_2: fixes sr :: "'d \<Rightarrow> 'c \<times> 'e"   and s :: "'d net_state"   and t :: "'d net_state" shows "netlift sr (SubnetS s t) = netlift sr s ++ netlift sr t"
ia: shows "ia \<equiv> if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1"
Class_unit_normal_subgroup: shows "pre.additive.Class zeroA = f \<^sup>\<inverse> A I"
\<gamma>_fun_def: fixes \<gamma> :: "'c \<Rightarrow> 'b set"   and F :: "'a \<Rightarrow> 'c" shows "\<gamma>_fun \<gamma> F = {f. \<forall>x. f x \<in> \<gamma> (F x)}"
in_hom_char: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" shows "VoV.in_hom f a b = (VoV.arr a \<and> VoV.arr b \<and> VoV.arr f \<and> VxV.in_hom f a b)"
f'_nonneg: fixes n :: "nat" shows "0 \<le> fds_nth f' n"
esqrt_ereal_ereal_sqrt: fixes t :: "real" assumes "0 \<le> t" shows "esqrt (ereal t) = ereal (sqrt t)"
concat_parts_eq: fixes a :: "nat"   and d :: "nat"   and v :: "'b list" assumes "a * d = length v" shows "concat (map (fixed_length_sublist v d) [0..<a]) = v"
arev_in_arcs: fixes a :: "'b \<times> 'b" assumes "a \<in> arcs (with_proj (list_digraph G_list))" shows "edge_rev (to_map' (snd G_list) xss) a \<in> arcs (with_proj (list_digraph G_list))"
substitution_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" shows "uninst.substitution u p g \<equiv> THE v'. uninst.isTrivialPhi p g (uninst.chooseNext u p g) v'"
fps_nth_deriv_linear: fixes n :: "nat"   and a :: "'a"   and f :: "'a fps"   and b :: "'a"   and g :: "'a fps" shows "fps_nth_deriv n (fps_const a * f + fps_const b * g) = fps_const a * fps_nth_deriv n f + fps_const b * fps_nth_deriv n g"
notpos: shows "\<nexists>xs ys. qs = xs @ ys \<and> xs \<in> Lxx x y"
mapping_of_lift_add: fixes x :: "nat"   and y :: "nat"   and a :: "'b mpoly"   and b :: "'b mpoly" shows "mapping_of (liftPoly x y (a + b)) = mapping_of (liftPoly x y a) + mapping_of (liftPoly x y b)"
assms_1: shows "type_update (TopType ts) (to_ct_list cons) (Type prods) = TopType ts'"
strategyproofness: shows "(pmf (sds R7) d + pmf (sds R7) b < pmf (sds R15) d + pmf (sds R15) b \<or> pmf (sds R7) d + (pmf (sds R7) b + pmf (sds R7) a) < pmf (sds R15) d + (pmf (sds R15) b + pmf (sds R15) a)) \<or> pmf (sds R7) d + pmf (sds R7) b = pmf (sds R15) d + pmf (sds R15) b \<and> pmf (sds R7) d + (pmf (sds R7) b + pmf (sds R7) a) = pmf (sds R15) d + (pmf (sds R15) b + pmf (sds R15) a)"
minarc_meet_bot: fixes x :: "'a" assumes "minarc x \<sqinter> x = bot" shows "minarc x = bot"
has_as_product_def: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'a \<Rightarrow> 'j"   and a :: "'j" shows "J.has_as_product J D a \<equiv> \<exists>\<pi>. product_cone J (\<cdot>\<^sub>J) D a \<pi>"
wls_skel_swap: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x1 :: "'var"   and x2 :: "'var" assumes "wls s X" shows "skel (X #[x1 \<and> x2]_xs) = skel X"
rel_frontier_nonempty_interior: fixes S :: "'b set" assumes "interior S \<noteq> {}" shows "rel_frontier S = frontier S"
valid_SE_def: fixes \<sigma> :: "'b"   and m :: "'b \<Rightarrow> (bool \<times> 'b) option" shows "(\<sigma> \<Turnstile> m) = (m \<sigma> \<noteq> None \<and> fst (the (m \<sigma>)))"
order_iff_strict: fixes b :: "'a word"   and a :: "'a word" shows "(b \<le>s a) = (b <s a \<or> a = b)"
surj_to_test: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "\<forall>b\<in>B. \<exists>a\<in>A. f a = b" shows "surj_to f A B"
is_proper: shows "proper_design (\<V> \<times> I) (\<Sum>\<^sub># (image_mset f \<B>))"
plus_assoc: fixes a :: "int poly fract"   and b :: "int poly fract"   and c :: "int poly fract" shows "rat_poly_plus (rat_poly_plus a b) c = rat_poly_plus a (rat_poly_plus b c)"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_vars (IF v t e) = v # ifex_vars t @ ifex_vars e"
G_Inf_have_prems: fixes \<iota> :: "'a literal multiset inference"   and M :: "'a literal multiset set" assumes "\<iota> \<in> G_Inf M" shows "prems_of \<iota> \<noteq> []"
fps_compose_Lambert_W': shows "fps_X * fps_exp 1 oo fps_Lambert_W = fps_X"
H_def: shows "H \<equiv> sum_upto (\<lambda>n. moebius_mu n * ln (real n))"
ofilter_Under: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "wo_rel r"   and "A \<subseteq> Field r" shows "ofilter r (Under r A)"
if_type: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'BExp"   and t :: "'BExp set"   and P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and P'' :: "'BExp set"   and \<Gamma>''' :: "'Var \<Rightarrow> 'BExp set option"   and P''' :: "'BExp set" assumes "\<Gamma> \<turnstile>\<^sub>b e \<in> t "   and "P \<turnstile> t"   and "\<turnstile> \<Gamma>,\<S>,P +\<^sub>\<S> e {c\<^sub>1} \<Gamma>',\<S>',P'"   and "\<turnstile> \<Gamma>,\<S>,P +\<^sub>\<S> bexp_neg e {c\<^sub>2} \<Gamma>'',\<S>',P''"   and "\<Gamma>' =:\<^sub>P' \<Gamma>'''"   and "\<Gamma>'' =:\<^sub>P'' \<Gamma>'''"   and "P' \<turnstile> P'''"   and "P'' \<turnstile> P'''"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>' \<S>' P' \<longrightarrow> tyenv_wellformed mds \<Gamma>''' \<S>' P'''"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>'' \<S>' P'' \<longrightarrow> tyenv_wellformed mds \<Gamma>''' \<S>' P'''" shows "\<turnstile> \<Gamma>,\<S>,P {Stmt.If e c\<^sub>1 c\<^sub>2} \<Gamma>''',\<S>',P'''"
neutr_eq_iff: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and z :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and a :: "'g"   and b :: "'g" assumes "semilattice_neutr_order f z less_eq less" shows "(z = f a b) = (a = z \<and> b = z)"
simps_5: fixes v :: "'b"   and va :: "'b"   and vb :: "'c"   and xs :: "('b, 'c) Combinators list" shows "singleCombinators (AllowPortFromTo v va vb # xs) = singleCombinators xs"