hcju/isabelleps huggingface dataset:
REFPHI0: shows "\<Phi>'' S \<sigma>0 (\<alpha> ` S) \<sigma>0'"
sorted_sublist_map_entire: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and xs :: "'c list" assumes "sorted_sublist_map R h xs 0 (length xs - 1)" shows "sorted_wrt (\<lambda>x y. R (h x) (h y)) xs"
intervals_decomposition: fixes A :: "'b set"   and a :: "'b"   and b :: "'b" assumes "finite A"   and "A \<subseteq> {a..b}"   and "a \<in> A"   and "b \<in> A"   and "a < b" shows "(\<Union>U\<in>{{u..next_in A u} |u. u \<in> A - {b}}. U) = {a..b}"
quadratic_hensel_step_def: fixes q :: "int"   and S :: "int poly"   and T :: "int poly"   and D :: "int poly"   and H :: "int poly" shows "local.quadratic_hensel_step q S T D H = local.hensel_step q q S T D H S T D H"
intensional_simps_27: fixes y :: "'a \<Rightarrow> bool" shows "y \<or> y \<equiv> y"
zero_row: fixes i :: "'n"   and A :: "((real, 'n) vec, 'n) vec" assumes "row i A = 0" shows "P A"
assms_2: shows "\<forall>\<^sub>F x in F. g2 x \<noteq> (0::'b)"
cancel_comp1: fixes l :: "'a literal" shows "l\<^sup>c\<^sup>c = l"
card_of_Pow_Func: fixes A :: "'c set" shows "|Pow A| =o |Func A UNIV|"
simp_and1: fixes Q :: "bool"   and P :: "bool"   and P' :: "bool"   and PP :: "bool" assumes "Q \<Longrightarrow> CNV P P'"   and "PP \<longrightarrow> P' \<and> Q" shows "PP \<longrightarrow> P \<and> Q"
Rep_data_tuple: fixes D :: "'a data" shows "Rep_data D = (DataValue D, Data.DataSpace D)"
WbisT_def: shows "WbisT \<equiv> bis WretrT"
degree_mod_less: fixes y :: "'c poly"   and x :: "'c poly" assumes "y \<noteq> 0" shows "x mod y = 0 \<or> degree (x mod y) < degree y"
standard_liouville_def: fixes p :: "nat \<Rightarrow> int"   and q :: "int" shows "standard_liouville p q = (\<Sum>k. real_of_int (p k) / real_of_int q ^ fact (Suc k))"
env_dom_def: fixes e :: "char list \<Rightarrow> 'b option" shows "env_dom (Env e) = dom e"
Qg_top: fixes N :: "'a set" assumes "G \<triangleright> N" shows "c_top G N \<in> set_rcs G N \<rightarrow> set_rcs G N \<rightarrow> set_rcs G N"
distinct_1: fixes x1 :: "'b"   and x2 :: "'b" shows "x1\<^sup>+ \<noteq> x2\<inverse>"
obj_def: fixes a :: "'a" shows "obj a \<equiv> arr a \<and> src a = a"
changes_filter_empty_2: fixes xs :: "'b list"   and a :: "'b" assumes "filter (\<lambda>x. x \<noteq> (0::'b)) xs = []" shows "changes (a # xs) = 0"
group_axioms: shows "Groups.group (+) (0::'b) uminus"
lcoeff_monic_mult: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic p" shows "coeff (p * q) (degree p + degree q) = lead_coeff q"
rrsmult_supp: fixes R :: "'a set"   and r :: "'a"   and f :: "'a \<Rightarrow> 'b" assumes "Ring1 R" shows "supp (Ring1.rightreg_scalar_mult R r f) \<subseteq> R"
to_hma\<^sub>m_def: fixes a :: "'b mat" shows "to_hma\<^sub>m a = (\<chi>i j. a $$ (Bij_Nat.to_nat i, Bij_Nat.to_nat j))"
sPP2: shows "subst (PP (encF N (Var xx))) t xx = PP (encF N t)"
measurable_Node: fixes M :: "'d measure" shows "(\<lambda>(l, x, r). \<langle>l, x, r\<rangle>) \<in> tree_sigma M \<Otimes>\<^sub>M M \<Otimes>\<^sub>M tree_sigma M \<rightarrow>\<^sub>M tree_sigma M"
axis_eq_axis: fixes i :: "'b"   and x :: "'a"   and j :: "'b"   and y :: "'a" shows "(axis i x = axis j y) = (x = y \<and> i = j \<or> x = (0::'a) \<and> y = (0::'a))"
that: assumes "HT' \<pi> (\<lambda>s. JOIN_VARS (f s) (P s) (\<lambda>v P. BB_PROTECT ((v, f\<^sub>0 s\<^sub>0) \<in> R \<and> P))) c Q"   and "ASSUME_\<Theta> \<pi> f\<^sub>0 s\<^sub>0 R \<Theta>" shows "thesis"
wf_terminates_while: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and b :: "'d set"   and c :: "('d, 'e, 'f) com" shows "wf {(t, s). \<Gamma>\<turnstile>While b c \<down> Normal s \<and> s \<in> b \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> Normal t}"
continuous_on_product_then_coordinatewise: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and i :: "'d" assumes "continuous_on S f" shows "continuous_on S (\<lambda>x. f x i)"
rg_succ_ref1: fixes am :: "nat \<Rightarrow> nat list"   and u :: "nat" assumes "is_adj_map am" shows "(rg_succ am cf u, cf.E `` {u}) \<in> \<langle>nat_rel\<rangle>list_set_rel"
wf_rcv_fv_single: fixes V :: "'a set"   and t :: "('b, 'a) Term.term"   and S' :: "('b, 'a) strand_step list" assumes "wf\<^sub>s\<^sub>t V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S')" shows "fv t \<subseteq> V"
Chains_Field: fixes C :: "'a set"   and r :: "('a \<times> 'a) set"   and x :: "'a" assumes "C \<in> Chains r"   and "x \<in> C" shows "x \<in> Field r"
assms: shows "g1 +++ g2 piecewise_C1_differentiable_on {0..1}"
eNorm_zero: fixes N :: "'a quasinorm" shows "eNorm N (0::'a) = 0"
enum_n_top: fixes a :: "nat \<Rightarrow> nat" assumes "a \<in> s" shows "(a = s.enum n) = (\<forall>a'\<in>s. a' \<le> a)"
hpair_neq_Ord': fixes k :: "hf"   and x :: "hf"   and y :: "hf" assumes "Ord k" shows "k \<noteq> \<langle>x, y\<rangle>"
pw: shows "pairwise (\<lambda>i j. disjnt (cball (a i) (r i)) (cball (a j) (r j))) C"
set_drop_subset: fixes n :: "nat"   and xs :: "'b list" shows "set (drop n xs) \<subseteq> set xs"
ofilter_subset_ordLeq: fixes r :: "('c \<times> 'c) set"   and A :: "'c set"   and B :: "'c set" assumes "Well_order r"   and "ofilter r A"   and "ofilter r B" shows "(A \<subseteq> B) = (Restr r A \<le>o Restr r B)"
Rdual: fixes A :: "'b set"   and po :: "'b potype"   and S :: "'b set" assumes "\<And>S. S \<subseteq> A \<Longrightarrow> \<exists>L. isLub S po L"   and "S \<subseteq> A"   and "A = pset po" shows "\<exists>G. isGlb S po G"
inject_2: fixes x2 :: "int"   and y2 :: "int" shows "(Intv' x2 = Intv' y2) = (x2 = y2)"
prefs_from_table_wf_def: fixes agents :: "'a set"   and alts :: "'b set"   and xss :: "('a \<times> 'b set list) list" shows "prefs_from_table_wf agents alts xss = (agents \<noteq> {} \<and> alts \<noteq> {} \<and> distinct (map fst xss) \<and> set (map fst xss) = agents \<and> (\<forall>xs\<in>set (map snd xss). \<Union> (set xs) = alts \<and> is_finite_weak_ranking xs))"
csubset_cinsertI: fixes B :: "'d cset"   and a :: "'d" shows "csubset_eq B (cinsert a B)"
idEnv_preserves_wls: shows "wlsEnv idEnv"
differentiable_imp_continuous_within: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and s :: "'a set" assumes "f differentiable at x within s" shows "continuous (at x within s) f"
similar_mat_sym: fixes A :: "'b mat"   and B :: "'b mat" assumes "similar_mat A B" shows "similar_mat B A"
basis_subset: shows "basis \<subseteq> dest.tangent_space (f p)"
seqI: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.arr f"   and "Maps.arr g"   and "Maps.dom g = Maps.cod f" shows "Maps.seq g f"
P_Suc: fixes i :: "nat"   and j :: "nat" assumes "P i j"   and "Q i j" shows "P (Suc i) j"
3: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A>)"
gr_to_geq: assumes "0x80000000 < w" shows "0x80000001 \<le> w"
lfinite_sswSts_iff: shows "lfinite sswSts = lfinite wSts"
mset_eq: shows "mset xs = mset ls + mset es + mset gs"
bernoulli_AE_cond_exp: fixes N :: "bool stream measure"   and q :: "real"   and X :: "bool stream \<Rightarrow> real"   and n :: "nat" assumes "N = bernoulli_stream q"   and "0 < q"   and "q < 1"   and "integrable N X" shows "AEeq N (real_cond_exp N (fct_gen_subalgebra N (stream_space borel) (proj_stoch_proc geom_proc n)) X) (expl_cond_expect N (proj_stoch_proc geom_proc n) X)"
infsetsum_reindex: fixes g :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and f :: "'b \<Rightarrow> 'c" assumes "inj_on g A" shows "infsetsum f (g ` A) = (\<Sum>\<^sub>ax\<in>A. f (g x))"
sc_completionD: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and vs :: "'addr \<times> addr_loc \<Rightarrow> ('addr val \<times> bool) option"   and ttas :: "('thread_id \<times> ('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action) list"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and x :: "'x"   and ta :: "('l, 'thread_id, 'x, 'm, 'w, ('addr, 'thread_id) obs_event action) thread_action"   and x' :: "'x"   and m' :: "'m" assumes "sc_completion s vs"   and "s -\<triangleright>ttas\<rightarrow>* s'"   and "ta_seq_consist P vs (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas)))"   and "thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>"   and "actions_ok s' t ta" shows "\<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> ta_seq_consist P (mrw_values P vs (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>)"
element_exists: fixes A :: "('a set \<times> 'b) set" assumes "A \<in> rel.Partition" shows "\<exists>a\<in>Sigma I \<FF>. a \<in> A"
swap_msgs_Send_Send: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isSend ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
finite_Int: fixes F :: "'b set"   and G :: "'b set" assumes "finite F \<or> finite G" shows "finite (F \<inter> G)"
smc_SemiCAT_components_1: fixes \<alpha> :: "V" shows "smc_SemiCAT \<alpha>\<lparr>Obj\<rparr> = ZFC_in_HOL.set {\<CC>. semicategory \<alpha> \<CC>}"
singleton_inject: fixes a :: "'d"   and b :: "'d" assumes "{a} = {b}" shows "a = b"
inv_insert: fixes t :: "('a \<times> 'b) tree"   and x :: "'a" assumes "inv t" shows "inv (local.insert x t)"
sorted_wrt_mono_rel: fixes xs :: "'d list"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> set xs; y \<in> set xs; P x y\<rbrakk> \<Longrightarrow> Q x y"   and "sorted_wrt P xs" shows "sorted_wrt Q xs"
A: fixes w :: "'a"   and f :: "'b" assumes "\<And>v. v < w \<Longrightarrow> hoare (p v) f q" shows "hoare (p w) (F f) q"
plain_nres_rel_internal_def: fixes R :: "('a \<times> 'b) set" shows "plain_nres_rel R \<equiv> {(c, a). RETURN c \<le> \<Down> R a}"
eqpoll_refl: fixes A :: "'c set" shows "A \<approx> A"
cases: fixes succs :: "'a \<Rightarrow> 'a list"   and a1 :: "'a"   and a2 :: "'a" assumes "RTranClp succs a1 a2" obtains g :: "'a" where "a1 = ga__" and "a2 = ga__"   | g :: "'a" and g' :: "'a" and g'' :: "'a" where "a1 = g" and "a2 = g''" and "g' \<in> set (succs g)" and "RTranClp succs g' g''"
comm_monoid_add_hom_axioms: shows "comm_monoid_add_hom hom"
transp_on_def: fixes U :: "'b set" shows "transp_on U = (\<lambda>r. \<forall>x\<in>U. \<forall>y\<in>U. \<forall>z\<in>U. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
sum_up_index_split: fixes f :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum f {..m + n} = sum f {..m} + sum f {Suc m..m + n}"
arrI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "J'.in_hom f a b" shows "J'.arr f"
dom_inv: fixes f :: "'a" assumes "C.iso f" shows "C.dom (C.inv f) = C.cod f"
res_fold_A_A_chamber_image: fixes F :: "'a set" assumes "SimplicialComplex.maxsimp A F" shows "restrict1 fold_A (\<Union> A) ` F = fold_A ` F"
eq_sum: fixes I :: "'c set"   and p :: "'c \<Rightarrow> 'd" assumes "finite I" shows "sum' p I = sum p I"
assms_1: shows "f (lead_coeff p) \<noteq> (0::'a)"
lang_ENC: fixes FO :: "nat set"   and n :: "nat"   and SO :: "nat set" assumes "FO \<subseteq> {0..<n}"   and "SO \<subseteq> {0..<n} - FO" shows "lang n (ENC n FO) = \<Union> {enc (w, I) |w I. length I = n \<and> wf_interp w I \<and> (\<forall>i\<in>FO. case I ! i of Inl x \<Rightarrow> True | Inr x \<Rightarrow> False) \<and> (\<forall>i\<in>SO. case I ! i of Inl x \<Rightarrow> False | Inr x \<Rightarrow> True)}"
inf_bool_def: fixes P :: "bool"   and Q :: "bool" shows "inf P Q = (P \<and> Q)"
fBallE: fixes A :: "'d fset"   and P :: "'d \<Rightarrow> bool"   and x :: "'d" assumes "fBall A P" obtains "P x"   | "x |\<notin>| A"
normalize_rules_1: fixes P :: "'b \<Rightarrow> assn"   and f :: "'c Heap"   and Q :: "'c \<Rightarrow> assn" assumes "\<And>x. <P x> f <Q>" shows "<\<exists>\<^sub>Ax. P x> f <Q>"
VPow_def: fixes x :: "V" shows "VPow x \<equiv> set (set ` Pow (elts x))"
is_unit_Jacobi_aux: fixes x :: "int"   and a :: "int" assumes "is_unit x" shows "Jacobi a x = 1"
assms_4: shows "(x, y) = split_i_j_x (dim_col A) l X"
eq_class_of_Submodule: fixes H :: "'a set"   and N :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N" shows "carrier (mdl M N /\<^sub>m H) = N \<^sub>s/\<^sub>M H"
card_le_inj: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card A \<le> card B" shows "\<exists>f. f ` A \<subseteq> B \<and> inj_on f A"
dbm_abstr_soundness: fixes u :: "'c \<Rightarrow> 'd"   and cc :: "('c, 'd) cconstraint"   and v :: "'c \<Rightarrow> nat"   and n :: "nat" assumes "u \<turnstile> cc"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>collect_clks cc. v c \<le> n" shows "u \<turnstile>\<^bsub>v,n\<^esub> abstr cc (\<lambda>i j. \<infinity>) v"
ns_def: shows "ns \<equiv> addpre s (\<pi>\<^sub>7 r)"
fin: fixes u :: "'c \<Rightarrow>\<^sub>0 int" assumes "Poly_Mapping.keys u \<subseteq> S" shows "finite {x \<in> S. f x [^] poly_mapping.lookup u x \<noteq> \<one>}"
times_Quantity_ext_def: fixes x :: "('a, 'b, 'c) Quantity_scheme"   and y :: "('a, 'b, 'c) Quantity_scheme" shows "x \<cdot> y = \<lparr>mag = mag x \<cdot> mag y, dim = Quantity.dim x \<cdot> Quantity.dim y, \<dots> = Quantity.more x \<cdot> Quantity.more y\<rparr>"
simps_5: fixes v :: "'b formula"   and va :: "'b formula" shows "is_disj (v \<^bold>\<and> va) = is_lit_plus (v \<^bold>\<and> va)"
rbt_insert_with_key_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and v :: "'c"   and t :: "('b, 'c) RBT_Impl.rbt" shows "ord.rbt_insert_with_key less f k v t = paint color.B (ord.rbt_ins less f k v t)"
rcond_rassume_expand: fixes b :: "'b upred"   and P :: "('b, 'c) urel"   and Q :: "('b, 'c) urel" shows "P \<triangleleft> b \<triangleright>\<^sub>r Q = (?[b] ;; P) \<sqinter> (?[\<not> b] ;; Q)"
ffb_g_ode_subset: fixes f :: "'a \<Rightarrow> 'a"   and T :: "real set"   and S :: "'a set"   and \<phi> :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and U :: "'a \<Rightarrow> real set"   and G :: "'a \<Rightarrow> bool"   and Q :: "'a set" assumes "local_flow f T S \<phi>"   and "\<And>s. s2p S s \<Longrightarrow> s2p (U s) 0 \<and> is_interval (U s) \<and> U s \<subseteq> T" shows "fb\<^sub>\<F> x\<acute>=\<lambda>t. f & G on U S @ 0 Q = {s. s2p S s \<longrightarrow> (\<forall>t\<in>U s. (\<forall>\<tau>\<in>down (U s) t. G (\<phi> \<tau> s)) \<longrightarrow> s2p Q (\<phi> t s))}"
WellDefinedEqE: fixes S :: "('a, 'b) Signature"   and A :: "'a"   and e1 :: "('a, 'b) Expression"   and e2 :: "('a, 'b) Expression"   and B :: "'a" assumes "Sig S \<triangleright> Vx : A \<turnstile> e1\<equiv>e2 : B" obtains "Sig S \<triangleright> Vx : A \<turnstile> e1 : B" and "Sig S \<triangleright> Vx : A \<turnstile> e2 : B"
Rolle: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real" assumes "a < b"   and "f a = f b"   and "continuous_on {a..b} f"   and "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable at x" shows "\<exists>z>a. z < b \<and> (f has_real_derivative 0) (at z)"
eval\<Pi>\<^sub>3_inverse: fixes x :: "\<Pi>\<^sub>3" shows "make\<Pi>\<^sub>3 (eval\<Pi>\<^sub>3 x) = x"
set_val_locs_def: shows "set_val_locs \<equiv> l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_val_locs"
preserves_comp_2: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd"   and f2' :: "'c"   and f2 :: "'c" assumes "binary_functor A1 A2 B F"   and "partial_magma.arr A2 (A2 f2' f2)" shows "(\<lambda>f1. F (f1, A2 f2' f2)) = vertical_composite.map A1 B (\<lambda>f1. F (f1, f2)) (\<lambda>f1. F (f1, f2'))"
simps_22: fixes vc :: "'a"   and v :: "char list"   and va :: "aexp"   and vb :: "'a" shows "(SKIP {vc} \<sqsubseteq> v ::= va {vb}) = False"
NList_def: fixes A :: "'a set"   and N :: "nat set" shows "NList A N \<equiv> \<Union> (nList A ` N)"
A_def: shows "A = F \<union> set (Macaulay_red (Keys_to_list fs) fs)"
lin_dep_def: fixes S :: "'c set" shows "lin_dep S = (\<exists>A a v. finite A \<and> A \<subseteq> S \<and> a \<in> A \<rightarrow> carrier R \<and> lincomb a A = \<zero>\<^bsub>M\<^esub> \<and> v \<in> A \<and> a v \<noteq> \<zero>)"
hyperset_defs_5: fixes A :: "'f set option"   and B :: "'f set option" shows "A \<sqsubseteq> B \<equiv> case B of None \<Rightarrow> True | \<lfloor>B\<rfloor> \<Rightarrow> case A of None \<Rightarrow> False | \<lfloor>A\<rfloor> \<Rightarrow> A \<subseteq> B"
mem'_def: shows "mem' = mem(x := ev\<^sub>A mem e)"
inj_on_one_iff: shows "inj_on (int.lift f) (carrier \<F>\<^bsub>{()}\<^esub>) = (\<forall>x. x \<in> carrier \<F>\<^bsub>{()}\<^esub> \<longrightarrow> int.lift f x = \<one>\<^bsub>int_group\<^esub> \<longrightarrow> x = \<one>\<^bsub>\<F>\<^bsub>{()}\<^esub>\<^esub>)"
parCases: fixes P :: "ccs"   and Q :: "ccs"   and \<alpha> :: "act"   and R :: "ccs"   and Prop :: "act \<Rightarrow> ccs \<Rightarrow> bool" assumes "P \<parallel> Q \<longmapsto>\<alpha> \<prec> R"   and "\<And>P'. P \<longmapsto>\<alpha> \<prec> P' \<Longrightarrow> Prop \<alpha> (P' \<parallel> Q)"   and "\<And>Q'. Q \<longmapsto>\<alpha> \<prec> Q' \<Longrightarrow> Prop \<alpha> (P \<parallel> Q')"   and "\<And>P' Q' a. \<lbrakk>P \<longmapsto>a \<prec> P'; Q \<longmapsto>coAction a \<prec> Q'; a \<noteq> \<tau>; \<alpha> = \<tau>\<rbrakk> \<Longrightarrow> Prop (\<tau>) (P' \<parallel> Q')" shows "Prop \<alpha> R"
zero: shows "dirichlet_prod f g 0 = (0::'a)"
normalized_n_primitive_MatchAnd_combine_map: fixes disc_sel :: "('b \<Rightarrow> bool) \<times> ('b \<Rightarrow> 'c)"   and f :: "'c \<Rightarrow> bool"   and rst :: "'b match_expr"   and C :: "'d \<Rightarrow> 'b"   and pts :: "'d list"   and m' :: "'b match_expr" assumes "normalized_n_primitive disc_sel f rst"   and "\<forall>m'\<in>(\<lambda>spt. Match (C spt)) ` set pts. normalized_n_primitive disc_sel f m'"   and "m' \<in> (\<lambda>spt. MatchAnd (Match (C spt)) rst) ` set pts" shows "normalized_n_primitive disc_sel f m'"
is_category: fixes Obj :: "'b set"   and Null :: "'b" assumes "discrete_category Obj Null" shows "category (DiscreteCategory.discrete_category.comp Obj Null)"
path_nonneg_weight: fixes v :: "'V"   and p :: "('V \<times> 'W \<times> 'V) list"   and v' :: "'V" assumes "is_path v p v'" shows "(0::'W) \<le> path_weight p"
r2f_f2r_galois_var2: fixes f :: "('c \<times> 'd) set \<Rightarrow> 'e"   and R :: "('c \<Rightarrow> 'd set) \<Rightarrow> 'e" shows "(f \<circ> \<R> = R) = (R \<circ> \<F> = f)"
lconjby_set_closed: fixes g :: "'g"   and A :: "'g set" assumes "g \<in> G"   and "order.greater_eq G A" shows "order.greater_eq G (lconjby g ` A)"
assms: shows "(X, rhs) \<in> Init (UNIV // \<approx>A)"
einterpret_euler_incr_fas: fixes X0 :: "floatarith list"   and CX :: "floatarith list"   and vs :: "real list"   and h :: "floatarith" assumes "length ode_e = DIM('c)"   and "length X0 = DIM('c)"   and "length CX = DIM('c)"   and "DIM('c) \<le> length vs"   and "max_Var_floatariths ode_e \<le> DIM('c)" shows "einterpret (euler_incr_fas X0 h CX) vs = einterpret X0 vs + interpret_floatarith h vs *\<^sub>R ode (einterpret CX vs)"
brouwer_pred_def: shows "brouwer_pred = (\<Union>i. {(m, n). n = Succ m \<or> (\<exists>f. n = Lim f \<and> m = f i)})"
clearjunk0_def: fixes m :: "('a, 'b) fmap" shows "clearjunk0 m = fmfilter (\<lambda>k. fmlookup m k \<noteq> Some (0::'b)) m"
MClkNext_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkNext send rcv cst p = (MClkFwd send rcv cst p \<or> MClkRetry send rcv cst p \<or> MClkReply send rcv cst p)"
B_def: shows "B = filter (\<lambda>i. \<not> P i) [0..<n]"
dist_in_open_segment: fixes x :: "'b"   and a :: "'b"   and b :: "'b" assumes "x \<in> {a<--<b}" shows "dist x a < dist a b \<and> dist x b < dist a b"
sp_one: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and v1 :: "'d \<times> nat"   and e :: "('d \<times> nat) edge"   and v2 :: "'d \<times> nat" assumes "wf_sub_rel_of g subs" shows "SubRel.subpath g v1 [e] v2 subs = (sub_rel_of g subs \<and> (v1 = src e \<or> (v1, src e) \<in> subs) \<and> e \<in> edges g \<and> (tgt e = v2 \<or> (tgt e, v2) \<in> subs))"
elims_2: fixes x :: "atom set \<times> 'a"   and xa :: "atom set \<times> 'a" assumes "x \<approx>abs_set xa" obtains bs :: "atom set" and xb :: "'a" and cs :: "atom set" and y :: "'a" where "x = (bs, xb)" and "xa = (cs, y)" and "\<exists>p. (bs, xb) \<approx>set (=) supp p (cs, y)"
normalized_dst_ips_def2: fixes ms :: "'b common_primitive match_expr" shows "normalized_dst_ips ms = normalized_n_primitive (is_Dst, dst_sel) normalized_cidr_ip ms"
vwUS: shows "adjuster n w us + w \<notin> set (us @ ws)"
evals_cases_1: fixes 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 s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>[],s\<rangle> [\<Rightarrow>] \<langle>e',s'\<rangle>" obtains "e' = []" and "s' = s"
typing_ok: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "ty" assumes "\<Gamma> \<turnstile> t : T" shows "\<turnstile> \<Gamma> ok"
DgrDataRefinement2_def: fixes P :: "'e \<Rightarrow> 'f"   and D :: "'e \<times> 'e \<Rightarrow> 'f \<Rightarrow> 'f"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g"   and D' :: "'e \<times> 'e \<Rightarrow> 'g \<Rightarrow> 'g" shows "DgrDataRefinement2 P D R D' = (\<forall>i j. DataRefinement ({. P i .} \<circ> D (i, j)) (R i) (R j) (D' (i, j)))"
boundedByAdd2: fixes M :: "nat"   and B :: "(nat \<times> 'b) set"   and N :: "nat" assumes "boundedBy M B" shows "boundedBy (N + M) B"
single_n_in_A: fixes G :: "nat graph_rec"   and p :: "nat" assumes "cfg_doms G"   and "p < length (cfg_doms.nodes G)" shows "[p] \<in> (rev \<circ> sorted_list_of_set) ` Pow (set (cfg_doms.nodes G))"
simps_6: fixes T :: "Arraytype" shows "isprimitive (ArrT T) = False"
mod_ring_divide32: shows "(mod_ring_rel32 ===> mod_ring_rel32 ===> mod_ring_rel32) (divide_p32 pp) (/)"
bounded_linear_inner_right: fixes a :: "'d" shows "bounded_linear ((\<bullet>) a)"
F2_pos: fixes x :: "'a" shows "0 \<le> F2 x"
entails_def: fixes S :: "'at Literal set set"   and C :: "'at Literal set" shows "entails S C \<equiv> \<forall>I. I \<Turnstile> S \<longrightarrow> I \<Turnstile> C"
subgroupE_4: fixes H :: "'c set"   and a :: "'c"   and b :: "'c" assumes "H \<le> H"   and "a \<in> H"   and "b \<in> H" shows "a \<otimes>\<^bsub>H\<^esub> b \<in> H"
mset_neg_uminus: fixes A :: "'a zmultiset" shows "mset_neg (- A) = mset_pos A"