hcju/isabelleps huggingface dataset:
prime_imp_coprime: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b"   and n :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "normalization_semidom.prime zero times one normalize p"   and "\<not> dvd.dvd times p n" shows "algebraic_semidom.coprime times one p n"
normalized_poly_mono: fixes B :: "(char list multiset \<times> int) multiset"   and A :: "(char list multiset \<times> int) multiset" assumes "normalized_poly B"   and "A \<subseteq># B" shows "normalized_poly A"
suffix_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A" shows "(list_all2 A ===> list_all2 A ===> (=)) suffix suffix"
rep_eq: fixes x :: "('a, 'b) ffun" shows "ffun_app x = pfun_app (pfun_of x)"
lookup_except_when: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "lookup (except p S) = (\<lambda>t. lookup p t when t \<notin> S)"
d_s_id_prop: shows "d 1\<^sub>\<sigma> = 1\<^sub>\<sigma>"
assms: shows "P \<turnstile> C sees F:T (fm) in D"
1: shows "bij_betw ((`) \<psi>) (Pow (w `\<rightarrow> C0)) (Pow z)"
arrow_of_spans_of_maps_axioms: shows "arrow_of_spans_of_maps (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>0 (local.dom \<mu>)) (tab\<^sub>1 (local.dom \<mu>)) (tab\<^sub>0 (cod \<mu>)) (tab\<^sub>1 (cod \<mu>)) \<mu>.chine"
notnot_PL: fixes A :: "'a set"   and p :: "'a" shows "(A \<turnstile> .\<not> (.\<not> p)) = (A \<turnstile> p)"
equal_dlist_def: fixes dxs :: "'a dlist"   and dys :: "'a dlist" shows "equal_class.equal dxs dys = equal_class.equal (list_of_dlist dxs) (list_of_dlist dys)"
mon_c_unconc_2: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and x :: "'b list"   and b :: "'b list multiset" shows "mon_c fg (add_mset x b) = mon_s fg x \<union> mon_c fg b"
assms_2: shows "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
prems_2: shows "Suc n = 1 + length r - length d"
hyps_2: fixes s :: "('a, 'b) xstate" assumes "\<Gamma>\<turnstile>sequence Seq (flatten c2) \<down> s" shows "\<Gamma>\<turnstile>c2 \<down> s"
tangent_space_restrict: fixes X :: "('c \<Rightarrow> real) \<Rightarrow> real"   and p :: "'c" assumes "X \<in> manifold_eucl.dest.tangent_space p" shows "extensional0 manifold_eucl.dest.diff_fun_space X"
mult_not_zero: fixes a :: "'b"   and b :: "'b" assumes "a * b \<noteq> bot" shows "a \<noteq> bot \<and> b \<noteq> bot"
bisims1List1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and es :: "'addr expr1 list" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e # es,h \<turnstile> (e' # es, xs) [\<leftrightarrow>] (stk, loc, pc, xcp)"
refl: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "Order_Predicates.preorder_on carrier le"   and "x \<in> carrier" shows "le x x"
preds_split: fixes ets :: "'a edge_kind list"   and ets' :: "'a edge_kind list"   and s :: "'a" shows "preds (ets @ ets') s = (preds ets s \<and> preds ets' (transfers ets s))"
poly_divide_tendsto_0_at_infinity: fixes q :: "'a poly"   and p :: "'a poly" assumes "degree q < degree p" shows "((\<lambda>x. poly q x / poly p x) \<longlongrightarrow> (0::'a)) at_infinity"
poly_eval_zero_left: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "poly_eval 0 p = lookup p 0"
finite_countable_subset: fixes A :: "'c set" shows "finite {X. X \<subseteq> A \<and> countable X} = finite A"
index_div_eq: fixes i :: "nat"   and a :: "nat"   and b :: "nat" assumes "i \<in> {a * b..<(a + 1) * b}" shows "i div b = a"
right_idem: fixes a :: "'a"   and b :: "'a" shows "sup (sup a b) b = sup a b"
iso_tail_inv_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "iso_tail (inv_iso hom) = tail G"
A_def: fixes i :: "nat" shows "A i = f -` (case i of 0 \<Rightarrow> {\<infinity>} | Suc n \<Rightarrow> {..ennreal (real (Suc n))}) \<inter> space M"
Sup_empty: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
assms_2: shows "h' \<turnstile> iterate_M ys \<rightarrow>\<^sub>h h''"
parts_Un_subset2: fixes G :: "msg set"   and H :: "msg set" shows "parts (G \<union> H) \<subseteq> parts G \<union> parts H"
WT_implies_WTrt: 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 e :: "expr"   and T :: "ty"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "P,E \<turnstile> e :: T" shows "P,E,h \<turnstile> e : T"
zero_assm: shows "\<forall>x\<in>{mod_type_class.to_nat i..<length (IArray.list_of (vec_to_iarray A))}. IArray.list_of (vec_to_iarray A) ! x = (0::'a)"
simps_2: fixes accC :: "qtname"   and statDeclC :: "qtname"   and stat :: "bool"   and objRef :: "expr"   and fn :: "vname" shows "assignsV ({accC,statDeclC,stat}objRef..fn) = assignsE objRef"
is_cadjoint_def: fixes F :: "'b \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'b" shows "is_cadjoint F G = (\<forall>x y. \<langle>F x, y\<rangle> = \<langle>x, G y\<rangle>)"
FAccThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
fbox_C: shows "P \<le> |x\<acute>=f & G on U S @ t\<^sub>0] C"
comp_assoc: fixes h :: "'a"   and g :: "'a"   and f :: "'a" shows "C (C h g) f = C h (C g f)"
rep_eq: fixes x :: "nat"   and xa :: "'b poly" shows "mapping_of (mpoly_of_poly x xa) = mpoly_of_poly_aux x xa"
echelon_form_upt_k_condition1': fixes A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and i :: "'f"   and j :: "'f" assumes "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"
tdghm_op_tdghm_op_tdghm: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "op_tdghm (op_tdghm \<NN>) = \<NN>"
all_not_fin_conv: fixes A :: "'b fset" shows "(\<forall>x. x |\<notin>| A) = (A = {||})"
induct: fixes P :: "v sem_env \<Rightarrow> 'b state \<Rightarrow> v \<Rightarrow> (pat \<times> exp0) list \<Rightarrow> v \<Rightarrow> bool"   and a0 :: "v sem_env"   and a1 :: "'b state"   and a2 :: "v"   and a3 :: "(pat \<times> exp0) list"   and a4 :: "v" assumes "\<And>uu_ uv_ uw_ err_v. P uu_ uv_ uw_ [] err_v"   and "\<And>env s v0 p e pes err_v. (\<lbrakk>allDistinct (pat_bindings p []); pmatch (c env) (refs s) p v0 [] = No_match\<rbrakk> \<Longrightarrow> P env s v0 pes err_v) \<Longrightarrow> P env s v0 ((p, e) # pes) err_v" shows "P a0 a1 a2 a3 a4"
direct_decomp_direct_decomp: fixes A :: "'a set"   and s :: "'a set"   and ss :: "'a set list"   and rs :: "'a set list" assumes "direct_decomp A (s # ss)"   and "direct_decomp s rs" shows "direct_decomp A (ss @ rs)"
A2: shows "\<forall>S T. (SourceTerm S, TargetTerm T) \<in> Rel \<longrightarrow> (\<lbrakk>S\<rbrakk>, T) \<in> TRel\<^sup>*"
of_nat_le_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "nat"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n) = (m \<le> n)"
correct_fin_inf_append2: fixes x :: "'a"   and s :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "fin_inf_append [x] s (Suc i) = s i"
tendsto_of_nat: shows "filterlim of_nat at_infinity sequentially"
Lcm_0_iff': fixes A :: "'a set" shows "(Lcm A = (0::'a)) = (\<nexists>l. l \<noteq> (0::'a) \<and> (\<forall>a\<in>A. a dvd l))"
strict_trans1: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "prefix b a"   and "strict_prefix c b" shows "strict_prefix c a"
subset_image_iff: fixes B :: "'d set"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(B \<subseteq> f ` A) = (\<exists>AA. AA \<subseteq> A \<and> B = f ` AA)"
bounded_clinear_def: fixes f :: "'c \<Rightarrow> 'd" shows "bounded_clinear f \<equiv> clinear f \<and> bounded_clinear_axioms f"
stream_space_def: fixes M :: "'b measure" shows "stream_space M = distr (Pi\<^sub>M UNIV (\<lambda>i. M)) (vimage_algebra (streams (space M)) (!!) (Pi\<^sub>M UNIV (\<lambda>i. M))) to_stream"
assms_2: shows "op \<in> set ((\<Phi>\<inverse> \<Pi> \<A> t) ! k)"
det_assoc: fixes M :: "'a process"   and P :: "'a process"   and Q :: "'a process" shows "(M [+] P [+] Q) = (M [+] (P [+] Q))"
ex_mono: fixes Q :: "'a upred"   and P :: "'a upred"   and x :: "'b \<Longrightarrow> 'a" assumes "P \<sqsubseteq> Q" shows "(\<exists> x \<bullet> P) \<sqsubseteq> (\<exists> x \<bullet> Q)"
PO_l3_inv10: shows "reach l3 \<subseteq> l3_inv10"
neq_Black: fixes c :: "color" shows "(c \<noteq> Black) = (c = Red)"
a_star: fixes x :: "('a \<times> 'a) set" shows "rel_ad (x\<^sup>*) = {}"
guard_def: fixes p :: "xml \<Rightarrow> bool"   and p1 :: "xml \<Rightarrow> char list +\<^sub>\<bottom> 'a"   and p2 :: "xml \<Rightarrow> char list +\<^sub>\<bottom> 'a"   and x :: "xml" shows "Xmlt.guard p p1 p2 x = (if p x then p1 x else p2 x)"
pprt_def: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "lattice_ab_group_add.pprt zero sup x = sup x zero"
Enat: fixes n :: "nat"   and \<rho> :: "val list" shows "E (ENat n) \<rho> = return (VNat n)"
lborel_integrable_real_affine: fixes f :: "real \<Rightarrow> 'b"   and c :: "real"   and t :: "real" assumes "integrable lborel f"   and "c \<noteq> 0" shows "integrable lborel (\<lambda>x. f (t + c * x))"
eval\<^sub>1_eval: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and Vs :: "char list list"   and ls :: "val list" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l)\<rangle> \<Rightarrow> \<langle>e',(h', l')\<rangle>"   and "fv e \<subseteq> set Vs"   and "l \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"   and "length Vs + max_vars e \<le> length ls" shows "\<exists>ls'. compP\<^sub>1 P \<turnstile>\<^sub>1 \<langle>compE\<^sub>1 Vs e,(h, ls)\<rangle> \<Rightarrow> \<langle>fin\<^sub>1 e',(h', ls')\<rangle> \<and> l' \<subseteq>\<^sub>m [Vs [\<mapsto>] ls']"
merge_F_def: fixes m :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and mL :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option" shows "merge_F m mL = merge_sum merge_unit (merge_prod m mL)"
axioms_2: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map" assumes "digraph_map G M" shows "bidirected_digraph G (edge_rev M)"
fmfilter_comm: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and m :: "('d, 'e) fmap" shows "fmfilter P (fmfilter Q m) = fmfilter Q (fmfilter P m)"
substT_compose_diff: fixes s :: "'trm"   and t1 :: "'trm"   and t2 :: "'trm"   and x1 :: "'var"   and x2 :: "'var" assumes "s \<in> atrm"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x1 \<in> var"   and "x2 \<in> var"   and "x1 \<noteq> x2"   and "x1 \<notin> FvarsT t2" shows "substT (substT s t1 x1) t2 x2 = substT (substT s t2 x2) (substT t1 t2 x2) x1"
residue_id: fixes x :: "int"   and n :: "int" assumes "x \<in> carrier (residue_ring n)"   and "0 \<le> n" shows "residue n x = x"
rg_def: shows "rg \<equiv> component_of (G_\<alpha>g g) r"
chain2: shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. \<Squnion> (x ` Y)) ` Z)"
lengthAux: fixes xvec :: "name list"   and y :: "name"   and yvec :: "name list" assumes "length xvec = length (y # yvec)" obtains z :: "name" and zvec :: "name list" where "xvec = z # zvec" and "length zvec = length yvec"
strict_trans2: fixes b :: "'a multiset"   and a :: "'a multiset"   and c :: "'a multiset" assumes "b \<subset># a"   and "c \<subseteq># b" shows "c \<subset># a"
that: fixes r :: "V"   and A :: "V" assumes "x = \<langle>\<langle>r, A\<rangle>, r \<restriction>\<^sup>r\<^sub>\<circ> A\<rangle>"   and "r \<in>\<^sub>\<circ> R"   and "A \<in>\<^sub>\<circ> X" shows "thesis"
fact_asymp_equiv: shows "fact \<sim>[sequentially] (\<lambda>n. sqrt (2 * pi * real n) * (real n / exp 1) ^ n)"
language_defs_5: fixes r :: "'a language"   and s :: "'a language" shows "(r \<le> s) = (Plus r s = s)"
mkArr_Fun: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "mkArr (Dom f) (Cod f) (Fun f) = f"
uGraph_from_list_\<alpha>_inj_on: fixes E :: "(nat \<times> int \<times> nat) list" assumes "uGraph_from_list_invar E" shows "inj_on \<alpha> (set E)"
cos_times_sin: fixes w :: "'a"   and z :: "'a" shows "cos w * sin z = (sin (w + z) - sin (w - z)) / (2::'a)"
notFinished_WtFT: fixes n :: "nat"   and cl :: "('a, 'b, 'c) cmd list" assumes "n < length cl"   and "\<not> finished (cl ! n)" shows "1 / real (length cl) \<le> 1 - WtFT cl"
sff_1: shows "p = smult c (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
DisN: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and p :: "('c, 'b) fm"   and q :: "('c, 'b) fm" assumes "Hintikka A H"   and "(\<^bold>\<not> (p \<^bold>\<or> q)) at i in' H" shows "(\<^bold>\<not> p) at i in' H \<and> (\<^bold>\<not> q) at i in' H"
vconcat_def: fixes xss :: "V" shows "vconcat xss = vfsequence_of_vlist (concat (map vlist_of_vfsequence (vlist_of_vfsequence xss)))"
inf_le1: fixes x :: "'d multiset"   and y :: "'d multiset" shows "x \<inter># y \<subseteq># x"
nc_prod_aux: fixes x :: "'a" shows "(x \<sqinter> nc) \<cdot> nc \<cdot> nc = (x \<sqinter> nc) \<cdot> nc"
pick_def: fixes X :: "'a set" shows "pick X \<equiv> SOME x. x \<in> X"
d2: assumes "\<not> d" shows "(t, \<alpha>, s) \<in> ars"
Abs_swap1_2: fixes a :: "atom"   and x :: "'a"   and bs :: "atom set"   and b :: "atom" assumes "a \<notin> supp x - bs"   and "b \<notin> supp x - bs" shows "[bs]res. x = [(a \<rightleftharpoons> b) \<bullet> bs]res. (a \<rightleftharpoons> b) \<bullet> x"
e_append_zeros_def: fixes b :: "nat"   and z :: "nat" shows "e_append_zeros b z \<equiv> e_append b (Partial_Recursive.list_encode (replicate z 0))"
am1: fixes x :: "'a" shows "ad x \<cdot> x = ad (1::'a)"
product_category_axioms: shows "product_category (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>A\<^sub>1)"
intros_12: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty" assumes "\<Gamma> \<turnstile> e, eP, eV : Syntax.Prod \<tau>\<^sub>1 \<tau>\<^sub>2" shows "\<Gamma> \<turnstile> Prj2 e, Prj2 eP, Prj2 eV : \<tau>\<^sub>2"
not_outside: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set"   and u :: "'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" assumes "ordinal_utility carrier relation u"   and "x \<succeq>[relation] y" shows "x \<in> carrier"
map_ran_simps_1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "map_ran f [] = []"
hyps: fixes u :: "real \<times> real"   and v :: "real \<times> real"   and w :: "real \<times> real"   and x0 :: "real \<times> real" assumes "ccw'.sortedP 0 (u # v # w # ys)" shows "ccw'.sortedP x0 (map snd (polychain_of x0 (u # v # w # ys)))"
normF_negate: fixes F :: "'a set"   and x :: "'a \<Rightarrow> complex" assumes "finite F" shows "normF F (\<lambda>a. - x a) = normF F x"
simps: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a2 :: "'Var \<Rightarrow> 'Val"   and a3 :: "nat"   and a4 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a5 :: "'Var \<Rightarrow> 'Val" shows "mevalp a1 a2 a3 a4 a5 = (\<exists>cms n mem cm' mem'. a1 = cms \<and> a2 = mem \<and> a3 = n \<and> a4 = cms[n := cm'] \<and> a5 = mem' \<and> (cms ! n, mem) \<leadsto> (cm', mem') \<and> n < length cms)"
simps_6: fixes x :: "WS1S_Nameful.fo"   and \<phi> :: "WS1S_Nameful.ws1s" shows "sos (ws1s.Ex1 x \<phi>) = sos \<phi>"
finish_def: fixes u :: "'v"   and s :: "('v, 'es) state_scheme" shows "finish u s \<equiv> let f = finished s(u \<mapsto> counter s); c = counter s + 1; st = tl (stack s) in s\<lparr>finished := f, counter := c, stack := st\<rparr>"
sumlist_def: fixes xs :: "'b list" shows "class_semiring.sumlist xs \<equiv> foldr (+) xs (0::'b)"
in_Chains_subset: fixes M :: "'b set"   and r :: "('b \<times> 'b) set"   and M' :: "'b set" assumes "M \<in> Chains r"   and "M' \<subseteq> M" shows "M' \<in> Chains r"
assms_4: shows "\<pi> 0 \<noteq> local.return"
ldrop_all: fixes xs :: "'b llist"   and m :: "enat" assumes "llength xs \<le> m" shows "ldrop m xs = LNil"
chain: shows "Complete_Partial_Order.chain (\<sqsubseteq>) A"
induct: fixes x :: "'h"   and \<Gamma> :: "('i, 'j) \<I>"   and xa :: "('h, 'i, 'j) gpv"   and P :: "('h, 'i, 'j) gpv \<Rightarrow> bool" assumes "x \<in> results_gpv \<Gamma> xa"   and "\<And>gpv. Pure x \<in> set_spmf (the_gpv gpv) \<Longrightarrow> P gpv"   and "\<And>out c gpv input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<Gamma> out; x \<in> results_gpv \<Gamma> (c input); P (c input)\<rbrakk> \<Longrightarrow> P gpv" shows "P xa"
carr_prodag_def: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" shows "carr_prodag I A = {f \<in> extensional I. f \<in> I \<rightarrow> Un_carrier I A \<and> (\<forall>i\<in>I. f i \<in> carrier (A i))}"
simps_4: fixes x :: "nat"   and e :: "cexpr"   and oper :: "pdf_operator"   and e' :: "cexpr" shows "cexpr_subst x e (oper $$\<^sub>c e') = oper $$\<^sub>c (cexpr_subst x e e')"
simps: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a1 :: "'b"   and a2 :: "'b"   and a3 :: "nat" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.distance sourcenode targetnode kind valid_edge a1 a2 a3 = (\<exists>n as n' x. a1 = n \<and> a2 = n' \<and> a3 = x \<and> CFG.intra_path sourcenode targetnode kind valid_edge n as n' \<and> length as = x \<and> (\<forall>as'. CFG.intra_path sourcenode targetnode kind valid_edge n as' n' \<longrightarrow> x \<le> length as'))"
elims_2: fixes x :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray"   and xa :: "nat"   and xb :: "int vec list" assumes "LLL_impl_inv x xa xb" obtains f :: "int vec list \<times> int vec list" and mu :: "int iarray iarray" and ds :: "int iarray" and i :: "nat" and fs :: "int vec list" where "x = (f, mu, ds)" and "xa = i" and "xb = fs" and "list_repr i f (map ((!) fs) [0..<m]) \<and> d_repr ds fs \<and> mu_repr mu fs"
primepow_odd_mult: fixes d :: "nat" assumes "Suc 0 < d" shows "primepow_odd (aprimedivisor d * d) = primepow_even d"
integrable: shows "set_integrable lborel {g a..g b} f"
mustT_Par_R: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com"   and s :: "'state" assumes "mustT (c1 | c2) s" shows "mustT c2 s"
mat_det_mult: fixes A :: "complex \<times> complex \<times> complex \<times> complex"   and B :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det (A *\<^sub>m\<^sub>m B) = mat_det A * mat_det B"
hcomp_in_vhomE: fixes \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.in_hom (\<nu> \<circ> \<mu>) f g" obtains "Span.arr \<mu>" and "Span.arr \<nu>" and "Span.src \<nu> = Span.trg \<mu>" and "Span.src \<mu> = Span.src f" and "Span.src \<mu> = Span.src g" and "Span.trg \<nu> = Span.trg f" and "Span.trg \<nu> = Span.trg g"
mset_lit_subst: fixes L :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list" shows "mset_lit (subst_lit L \<sigma>) = {#x \<lhd> \<sigma>. x \<in># mset_lit L#}"
snd_conv: fixes x1 :: "'c"   and x2 :: "'b" shows "snd (x1, x2) = x2"
lhs_step_SPEC: fixes \<Phi> :: "'b \<Rightarrow> bool"   and m :: "'b nres" assumes "\<And>x. \<Phi> x \<Longrightarrow> RETURN x \<le> m" shows "SPEC \<Phi> \<le> m"
reachable_transition: fixes a :: "'c"   and A :: "('c, 'd) nba"   and q :: "'d"   and p :: "'d"   and r :: "'d" assumes "a \<in> alphabet A"   and "q \<in> reachable A p"   and "r \<in> transition A a q" shows "r \<in> reachable A p"
inv_inv_notame: fixes p :: "nat" shows "invariant (\<lambda>g. Invariants.inv g \<and> notame7 g) next_plane\<^bsub>p\<^esub>"
induced_automorphism_bij_betw_vertexset: shows "bij_betw \
 (\<Union> X) (\<Union> X)"
pull_up_configuration_state_rank: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and n :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "semi_mojmir_def.configuration \<delta> q\<^sub>0 w q n = {}" shows "semi_mojmir_def.state_rank \<Sigma> \<delta> q\<^sub>0 w q n = None"
and_mask_lt_2p: fixes w :: "'a word"   and n :: "nat" shows "uint (w && mask n) < 2 ^ n"
def_l: shows "L = Pos (Eq t s) \<or> L = Neg (Eq t s)"
le_ref2T: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<le> Q" shows "T Q \<subseteq> T P"
induct: fixes P :: "val \<Rightarrow> bool"   and val :: "val" assumes "P Unit"   and "\<And>x. P (Bool x)"   and "\<And>x. P (Intg x)"   and "P Null"   and "\<And>x. P (Addr x)" shows "P val"
some: fixes b :: "'b"   and \<sigma>' :: "'a" assumes "ioprog a \<sigma> = Some (b, \<sigma>')"   and "\<sigma>' \<Turnstile> _bind_SE s (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>P\<^sub>u\<^sub>r\<^sub>g\<^sub>e S ioprog) (P (b # s))" shows "Q"
dishonest_def: fixes bc :: "nat \<Rightarrow> 'a list"   and mining :: "nat \<Rightarrow> bool" shows "dishonest bc mining \<equiv> \<forall>n. prefix (bc (Suc n)) (bc n) \<or> (\<exists>b. bc (Suc n) = bc n @ [b]) \<and> mining (Suc n)"