hcju/isabelleps huggingface dataset:
valid_weak_action_modality_fresh: fixes \<alpha> :: "'act"   and P :: "'state"   and x :: "('idx, 'pred, 'act) formula" assumes "bn \<alpha> \<sharp>* P" shows "P \<Turnstile> \<langle>\<langle>\<alpha>\<rangle>\<rangle>x = (\<exists>P'. P \<Rightarrow>\<langle>\<alpha>\<rangle> P' \<and> P' \<Turnstile> x)"
simps: fixes a1 :: "'b \<Rightarrow> ('c, 'd) trm"   and a2 :: "('c + 'b, 'e, 'd) hp" shows "NPadmit a1 a2 = ((\<exists>\<sigma> a. a1 = \<sigma> \<and> a2 = $\<alpha> a) \<or> (\<exists>\<sigma> a b. a1 = \<sigma> \<and> a2 = (a ;; b) \<and> NPadmit \<sigma> a \<and> NPadmit \<sigma> b \<and> PUadmitFO \<sigma> b (BVP (PsubstFO a \<sigma>)) \<and> hpsafe (PsubstFO a \<sigma>)) \<or> (\<exists>\<sigma> a. a1 = \<sigma> \<and> a2 = a** \<and> NPadmit \<sigma> a \<and> PUadmitFO \<sigma> a (BVP (PsubstFO a \<sigma>)) \<and> hpsafe (PsubstFO a \<sigma>)) \<or> (\<exists>\<sigma> ODE \<phi>. a1 = \<sigma> \<and> a2 = EvolveODE ODE \<phi> \<and> OadmitFO \<sigma> ODE (BVO ODE) \<and> NFadmit \<sigma> \<phi> \<and> FUadmitFO \<sigma> \<phi> (BVO ODE) \<and> fsafe (FsubstFO \<phi> \<sigma>) \<and> osafe (OsubstFO ODE \<sigma>)) \<or> (\<exists>\<sigma> a b. a1 = \<sigma> \<and> a2 = (a \<union>\<union> b) \<and> NPadmit \<sigma> a \<and> NPadmit \<sigma> b) \<or> (\<exists>\<sigma> \<theta> x. a1 = \<sigma> \<and> a2 = (x := \<theta>) \<and> TadmitFO \<sigma> \<theta>) \<or> (\<exists>\<sigma> \<theta> x. a1 = \<sigma> \<and> a2 = DiffAssign x \<theta> \<and> TadmitFO \<sigma> \<theta>) \<or> (\<exists>\<sigma> \<phi>. a1 = \<sigma> \<and> a2 = ? \<phi> \<and> NFadmit \<sigma> \<phi>))"
fun_left_idem: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'a"   and z :: "'b" assumes "comp_fun_idem f" shows "f x (f x z) = f x z"
uminus_uminus_mat: fixes A :: "'a mat" shows "- (- A) = A"
h: shows "|~ F \<and> Unchanged v \<longrightarrow> \<circle>F"
simps_1: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "iinput \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = iinput"
set_eq_class_of_rng_of_frac_def: fixes rel :: "('c \<times> 'd, 'e) eq_object_scheme" shows "set_class_of\<^bsub>rel\<^esub> \<equiv> {r |\<^bsub>rel\<^esub> s |r s. (r, s) \<in> carrier rel}"
exists_base: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K R"   and "R.dimension n K E" shows "\<exists>Vs. set Vs \<subseteq> carrier R \<and> R.independent K Vs \<and> length Vs = n \<and> R.Span K Vs = E"
path_linepath: fixes a :: "'b"   and b :: "'b" shows "path (linepath a b)"
iIN_subset_conv: fixes n' :: "nat"   and d' :: "nat"   and n :: "nat"   and d :: "nat" shows "([n'\<dots>,d'] \<subseteq> [n\<dots>,d]) = (n \<le> n' \<and> n' + d' \<le> n + d)"
liftT_liftT': fixes i :: "nat"   and m :: "nat"   and j :: "nat"   and n :: "nat"   and T :: "POPLmark.type" assumes "i + m \<le> j" shows "\<up>\<^sub>\<tau> n j (\<up>\<^sub>\<tau> m i T) = \<up>\<^sub>\<tau> m i (\<up>\<^sub>\<tau> n (j - m) T)"
dconvfactor_compose: fixes c\<^sub>1 :: "'e \<Rightarrow>\<^sub>U 'd"   and c\<^sub>2 :: "'c \<Rightarrow>\<^sub>U 'e"   and d :: "(int, sdim) dimvec" shows "dconvfactor (c\<^sub>1 \<circ>\<^sub>C c\<^sub>2) d = dconvfactor c\<^sub>1 d \<cdot> dconvfactor c\<^sub>2 d"
finite_measure_eq_AE: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c pmf" assumes "AE x in measure_pmf M. (x \<in> A) = (x \<in> B)"   and "A \<in> measure_pmf.events M"   and "B \<in> measure_pmf.events M" shows "measure_pmf.prob M A = measure_pmf.prob M B"
P1: shows "Bet R D A \<and> Cong S C R D"
hta_\<alpha>_is_ta: shows "tree_automaton (hta_\<alpha> H)"
singleton_inject: fixes a :: "'g"   and b :: "'g" assumes "{a} = {b}" shows "a = b"
subseq_drop_many: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "subseq xs ys" shows "subseq xs (zs @ ys)"
mult_coprime: fixes a :: "nat"   and b :: "nat"   and x :: "'b" assumes "coprime a b" shows "divisor_sigma x (a * b) = divisor_sigma x a * divisor_sigma x b"
f4: shows "Deliver (i, Append i e) \<notin> set pre"
termFSwImorph_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'a"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'b"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b, 'c) model_scheme" shows "termFSwImorph h hA MOD \<equiv> ipresWlsAll h hA MOD \<and> ipresCons h hA MOD \<and> ipresFreshAll h hA MOD \<and> ipresSwapAll h hA MOD"
invertible_mult: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and B :: "(('a, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" assumes "invertible A"   and "invertible B" shows "invertible (A ** B)"
that: fixes q :: "nat"   and r :: "real" assumes "t = real q * T + r"   and "0 \<le> r"   and "r < T" shows "thesis"
test_absorb_1: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "x \<squnion> x \<sqinter> y = x"
diff_union_single_conv: fixes a :: "'b"   and J :: "'b multiset"   and I :: "'b multiset" assumes "a \<in># J" shows "I + J - {#a#} = I + (J - {#a#})"
floor_neg_numeral: fixes v :: "num" shows "\<lfloor>- numeral v\<rfloor> = - numeral v"
surj_imp_inj_inv: fixes f :: "'c \<Rightarrow> 'b" assumes "surj f" shows "inj (inv f)"
frontier_ic: fixes a :: "real"   and b :: "real" assumes "a < b" shows "frontier {a<..b} = {a, b}"
simps: fixes cenum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and P :: "'a list \<Rightarrow> bool"   and n :: "nat" shows "ex_n_lists cenum_ex P n = (if n = 0 then P [] else cenum_ex (\<lambda>x. ex_n_lists cenum_ex (\<lambda>xs. P (x # xs)) (n - 1)))"
rel_setD1: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'b set"   and B :: "'c set"   and x :: "'b" assumes "rel_set R A B"   and "x \<in> A" shows "\<exists>y\<in>B. R x y"
assigns_Extend_2: fixes ord :: "WS1S_Prelim.order"   and ord' :: "WS1S_Prelim.order"   and m :: "nat"   and i :: "nat"   and \<AA> :: "interp"   and P :: "nat fset" assumes "ord \<noteq> ord'" shows "m\<^bsup>Extend ord i \<AA> P\<^esup>ord' = m\<^bsup>\<AA>\<^esup>ord'"
excd_impl_exicd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a" assumes "IFC prob"   and "\<exists>k. IFC_def.is_cdi prob i \<pi> k" shows "\<exists>k. IFC_def.is_icdi prob i \<pi> k"
LabelFun_def: shows "LabelFun \<equiv> snd \<circ> snd \<circ> snd \<circ> Rep_kripke"
coincides_on_self_financing: fixes Mkt :: "('c, 'd) discrete_market"   and Mkt2 :: "('c, 'd) discrete_market"   and A :: "'d set"   and p :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real" assumes "coincides_on Mkt Mkt2 A"   and "support_set p \<subseteq> A"   and "self_financing Mkt p" shows "self_financing Mkt2 p"
f_def: shows "f = (\<lambda>y. (y - a) / b)"
cos_arcsin: fixes x :: "real" assumes "- 1 \<le> x"   and "x \<le> 1" shows "cos (arcsin x) = sqrt (1 - x\<^sup>2)"
decs_sub_from_counter: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s d res" shows "\<forall>extra count count' s' r'. clock s = count + extra \<and> clock s' = count' + extra \<and> res = (s', r') \<and> ck = True \<longrightarrow> evaluate_decs ck mn env (update_clock (\<lambda>_. count) s) d (update_clock (\<lambda>_. count') s', r')"
top_prod_def: shows "top_prod = (\<top>, \<top>)"
mult_zero_ow_axioms_def: fixes U :: "'a set"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" shows "mult_zero_ow_axioms U times zero \<equiv> (\<forall>a. a \<in> U \<longrightarrow> times zero a = zero) \<and> (\<forall>a. a \<in> U \<longrightarrow> times a zero = zero)"
simps_7: fixes uu :: "'b"   and pr1 :: "('a, 'b) Digit"   and m1 :: "('a, 'b) FingerTreeStruc"   and sf1 :: "('a, 'b) Digit"   and ts :: "('a, 'b) Node list"   and uv :: "'b"   and pr2 :: "('a, 'b) Digit"   and m2 :: "('a, 'b) FingerTreeStruc"   and sf2 :: "('a, 'b) Digit" shows "FingerTreeStruc.app3 (Deep uu pr1 m1 sf1) ts (Deep uv pr2 m2 sf2) = FingerTreeStruc.deep pr1 (FingerTreeStruc.app3 m1 (FingerTreeStruc.nodes (FingerTreeStruc.digitToNlist sf1 @ ts @ FingerTreeStruc.digitToNlist pr2)) m2) sf2"
SS_DR_iff_source_target_rel: fixes success :: "'barbs" shows "(((\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x))) \<and> (\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)))) \<and> enc_reflects_divergence) = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_barb_set Rel (STCalWB SWB TWB) {success} \<and> rel_reflects_divergence Rel (STCal Source Target))"
distinct_11: fixes x4 :: "int" shows "Null \<noteq> Intg x4"
Fiber_equality: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set"   and a :: "'c"   and b :: "'c" assumes "fiber_relation \<alpha> S T"   and "a \<in> S"   and "b \<in> S" shows "(equivalence.Class S (fiber_relation_notation.Fiber_Relation S \<alpha>) a = equivalence.Class S (fiber_relation_notation.Fiber_Relation S \<alpha>) b) = (\<alpha> a = \<alpha> b)"
dc: shows "d - c \<in> (+) (- c) ` S"
\<Upsilon>_cong: fixes p :: "fm"   and x :: "real"   and bs :: "real list"   and U' :: "(MIR.num \<times> int) set"   and U :: "(MIR.num \<times> int) set" assumes "isrlfm p"   and "(\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ` U' = (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (U \<times> U)"   and "\<forall>(t, n)\<in>U. numbound0 t \<and> 0 < n"   and "\<forall>(t, n)\<in>U'. numbound0 t \<and> 0 < n" shows "(\<exists>(t, n)\<in>U. \<exists>(s, m)\<in>U. Ifm (x # bs) (\<upsilon> p (Add (Mul m t) (Mul n s), 2 * n * m))) = (\<exists>(t, n)\<in>U'. Ifm (x # bs) (\<upsilon> p (t, n)))"
bis_Frel: fixes B1 :: "'f set"   and B2 :: "'g set"   and s1 :: "'f \<Rightarrow> ('h, 'f, 'g) F1"   and s2 :: "'g \<Rightarrow> ('i, 'f, 'g) F2"   and B1' :: "'j set"   and B2' :: "'k set"   and s1' :: "'j \<Rightarrow> ('h, 'j, 'k) F1"   and s2' :: "'k \<Rightarrow> ('i, 'j, 'k) F2"   and R1 :: "('f \<times> 'j) set"   and R2 :: "('g \<times> 'k) set" shows "bis B1 B2 s1 s2 B1' B2' s1' s2' R1 R2 = ((R1 \<subseteq> B1 \<times> B1' \<and> R2 \<subseteq> B2 \<times> B2') \<and> (\<forall>b1 b1'. (b1, b1') \<in> R1 \<longrightarrow> F1rel (=) (in_rel R1) (in_rel R2) (s1 b1) (s1' b1')) \<and> (\<forall>b2 b2'. (b2, b2') \<in> R2 \<longrightarrow> F2rel (=) (in_rel R1) (in_rel R2) (s2 b2) (s2' b2')))"
compact_interval: fixes a :: "'b"   and b :: "'b" shows "compact {a..b}"
dec_def: fixes k :: "nat"   and m :: "nat" shows "dec k m = (if k < m then m - Suc 0 else m)"
1: shows "set as \<inter> {x. a \<noteq> x} = set as"
coset_hom_2: fixes h :: "'a \<Rightarrow> 'b"   and G :: "('a, 'c) monoid_scheme"   and H :: "('b, 'd) monoid_scheme"   and I :: "'a set"   and a :: "'a" assumes "h \<in> hom G H"   and "I \<subseteq> carrier G"   and "a \<in> carrier G" shows "h ` (I #>\<^bsub>G\<^esub> a) = h ` I #>\<^bsub>H\<^esub> h a"
ideal_sub_vp: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "ideal (Vr K v) I"   and "I \<noteq> carrier (Vr K v)" shows "I \<subseteq> vp K v"
set_intros_2: fixes yb :: "'a"   and x31 :: "('a, 'b) fm"   and x32 :: "('a, 'b) fm" assumes "yb \<in> params_fm x31" shows "yb \<in> params_fm (x31 \<^bold>\<longrightarrow> x32)"
safety_prop_constrains: fixes A :: "'c set"   and B :: "'c set" shows "safety_prop (A co B) = (A \<subseteq> B)"
oracle_encrypt2'_def: shows "oracle_encrypt2' = (\<lambda>(k_prf, k_upf) b (bad, D) (msg1, msg0). case length msg1 = prf_clen \<and> length msg0 = prf_clen of True \<Rightarrow> spmf_of_set prf_domain \<bind> (\<lambda>x. spmf_of_set (nlists UNIV prf_clen) \<bind> (\<lambda>P. let (p, F) = case D x of None \<Rightarrow> (P, False) | Some r \<Rightarrow> (r, True); c = p [\<oplus>] (if b then msg1 else msg0); t = upf_fun k_upf (x @ c) in return_spmf (Some (x, c, t), bad \<or> F, D(x \<mapsto> p)))) | False \<Rightarrow> return_spmf (None, bad, D))"
IH: shows "(\<sigma>, s) \<in> oreachable (opnet onp p\<^sub>1) (otherwith S (net_tree_ips p\<^sub>1) (oarrivemsg I)) (other U (net_tree_ips p\<^sub>1)) \<and> (\<sigma>, t) \<in> oreachable (opnet onp p\<^sub>2) (otherwith S (net_tree_ips p\<^sub>2) (oarrivemsg I)) (other U (net_tree_ips p\<^sub>2)) \<and> net_tree_ips p\<^sub>1 \<inter> net_tree_ips p\<^sub>2 = {}"
assms_1: shows "chaos \<sqsubseteq> (c::'a::refinement_lattice)"
60: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> (x \<sqinter> y \<squnion> x \<sqinter> (y \<squnion> top)) = x \<sqinter> (y \<squnion> top)"
true_disj_zero_2: fixes P :: "'a upred" shows "(true \<or> P) = true"
H1: fixes x :: "'a" assumes "x \<in> U - S" shows "supp_sum (\<lambda>W. H W x) \<C> = 1"
less_gexp_antisym: fixes x :: "'b gexp"   and y :: "'b gexp" shows "(x < y) = (\<not> y < x \<and> x \<noteq> y)"
assms_4: shows "\<forall>p\<in>set (p\<^sub>2 # ps). real_of_int l - \<delta> \<le> real_of_int (fst p) \<and> real_of_int (fst p) \<le> real_of_int l + \<delta>"
unit_dvdE: fixes a :: "'a"   and b :: "'a" assumes "is_unit a" obtains c :: "'a" where "a \<noteq> (0::'a)" and "b = a * c"
object_ptr_kinds_M_eq3_h: shows "object_ptr_kinds h = object_ptr_kinds h2"
const: fixes c :: "'a"   and F :: "'b filter" assumes "c \<noteq> (0::'a)" shows "\<Theta>[F](\<lambda>_. c) = \<Theta>[F](\<lambda>_. 1::'a)"
closed_connected_component: fixes S :: "'a set"   and x :: "'a" assumes "closed S" shows "closed (connected_component_set S x)"
Q'_def: shows "Q' \<equiv> {q. not_blocked p (Q \<inter> W) q}"
bit_double_iff: fixes a :: "'a"   and n :: "nat" shows "bit ((2::'a) * a) n = (bit a (n - 1) \<and> n \<noteq> 0 \<and> (2::'a) ^ n \<noteq> (0::'a))"
F_RV: shows "of_int_hom.vec_hom (fs ! i) = map of_int_hom.vec_hom fs ! i"
assms_3: shows "(a \<rightleftharpoons> c) \<bullet> x = x"
homotopic_on_emptyI: fixes X :: "'c topology"   and P :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and X' :: "'d topology" assumes "topspace X = {}"   and "P f"   and "P g" shows "homotopic_with P X X' f g"
update_simps_4: fixes up :: "(nat, 'b \<times> 'a) mapping \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and s :: "('b, 'a) state" shows "\<V> (\<B>\<^sub>i\<^sub>u_update up s) = \<V> s"
uadmit_ode_adjoint': fixes \<sigma> :: "('sf, 'sc, 'sz) subst"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and ODE :: "('sf, 'sz) ODE" assumes "ssafe \<sigma>"   and "is_interp I"   and "Vagree \<nu> \<omega> (\<Union>i\<in>{i |i. Inl i \<in> SIGO ODE}. case SFunctions \<sigma> i of None \<Rightarrow> {} | Some x \<Rightarrow> FVT x)"   and "osafe ODE" shows "ODE_sem (local.adjoint I \<sigma> \<nu>) ODE = ODE_sem (local.adjoint I \<sigma> \<omega>) ODE"
is_const_simps_1: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_const unconst (const name)"
pred_inject_2: fixes P :: "'g \<Rightarrow> bool"   and a :: "'g"   and aa :: "'g list" shows "list_all P (a # aa) = (P a \<and> list_all P aa)"
Abs_freelist_inverse: fixes as :: "'a list" shows "freeword (Abs_freelist as) = map pairtrue as"
lpoly_of_v_equals_v_append0: fixes v :: "rat vec"   and a :: "nat" shows "vec_to_lpoly v = vec_to_lpoly (v @\<^sub>v 0\<^sub>v a)"
sorted_list_of_set_UN_lessThan: fixes k :: "nat"   and A :: "nat \<Rightarrow> 'a set" assumes "strict_mono_sets {..<k} A"   and "\<And>i. i < k \<Longrightarrow> finite (A i)" shows "list_of (\<Union> (A ` {..<k})) = concat (map (list_of \<circ> A) (list_of {..<k}))"
del_block_def: fixes b :: "'a set" shows "fin.del_block b \<equiv> remove1_mset b (add_block bl)"
induct: fixes P :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b list" assumes "\<And>uu_. P uu_ []"   and "\<And>x xs. P 0 (x # xs)"   and "\<And>n x xs. P n xs \<Longrightarrow> P (Suc n) (x # xs)" shows "P a0 a1"
H_def: shows "H \<equiv> {a \<in> {1..<n}. coprime a n \<and> ([a ^ (2 ^ (j - 1) * m) = 1] (mod n) \<or> [a ^ (2 ^ (j - 1) * m) = n - 1] (mod n))}"
step_invar: fixes s :: "'S" assumes "s \<in> dwa_invar WA"   and "dwa_cond WA s" shows "dwa_step WA s \<in> dwa_invar WA"
in_hhom_def: fixes \<mu> :: "'c"   and a :: "'c"   and b :: "'c" shows "C.in_hhom \<mu> a b \<equiv> C.arr \<mu> \<and> src\<^sub>C \<mu> = a \<and> trg\<^sub>C \<mu> = b"
aclassical_1: fixes v :: "i" shows "[\<^bold>\<forall>R. \<^bold>\<exists>x y. \<lparr>A!,x\<^sup>P\<rparr> \<^bold>& \<lparr>A!,y\<^sup>P\<rparr> \<^bold>& x \<^bold>\<noteq> y \<^bold>& (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,x\<^sup>P\<rparr>) \<^bold>= (\<^bold>\<lambda>z. \<lparr>R,z\<^sup>P,y\<^sup>P\<rparr>) in v]"
word_and_not: fixes x :: "'b word" shows "x AND NOT x = 0"
Done: fixes r :: "('a, 'b, 'c) Resumption.resumption"   and r' :: "('a, 'b, 'c) Resumption.resumption" assumes "X r r'"   and "Resumption.resumption.is_Done r'" shows "Resumption.resumption.is_Done r \<and> option_ord (Resumption.resumption.result r) (Resumption.resumption.result r')"
0: shows "pderiv f = monom (UP Zp) ([n] \<cdot> \<one>) (n - 1)"
group_add_ow_axioms: shows "group_add_ow U (-\<^sub>o\<^sub>w) (+\<^sub>o\<^sub>w) 0\<^sub>o\<^sub>w uminus"
nat_intv_frac_decomp: fixes c :: "nat"   and d :: "real" assumes "real c < d"   and "d < real (c + 1)" shows "d = real c + frac d"
hs_it_init_def: shows "hs_it_init \<equiv> hm_it_init"
x: shows "x \<in> A \<Gamma> \<or> Inr x \<in> S"
E_def: shows "E = C \<inter> {(int (nat p) - 1) div 2<..}"
ideduct_synth_mono: fixes M :: "('fun, 'var) Term.term set"   and t :: "('fun, 'var) Term.term"   and M' :: "('fun, 'var) Term.term set" assumes "M \<turnstile>\<^sub>c t"   and "M \<subseteq> M'" shows "M' \<turnstile>\<^sub>c t"
comm_monoid_axioms: shows "comm_monoid (\<cdot>) (1::'e)"
bigo_useful_add: fixes f :: "'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> O(h)"   and "g \<in> O(h)" shows "f + g \<in> O(h)"
0: shows "\<forall>X\<in>\<Union> ` PP. \<forall>Y\<in>\<Union> ` PP. X \<inter> Y = {} \<longrightarrow> X \<noteq> Y"
Nset_have_two: fixes j :: "nat"   and n :: "nat" assumes "j \<in> {h. h \<le> Suc n}" shows "\<exists>m\<in>{h. h \<le> Suc n}. j \<noteq> m"
\<omega>: shows "\<omega> \<in> space (Pi\<^sub>M {0..<n} M)"
roots: fixes P :: "'a list" assumes "P \<in> carrier (poly_ring R)" shows "Id.S.splitted (\<sigma> P)"
radical_dvd: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "dvd.dvd times (factorial_semiring.radical zero times one normalize x) x"
comm_neutral: fixes a :: "'a" shows "a * (1::'a) = a"
rtrancl_on_rtranclI: fixes a :: "'b"   and b :: "'b"   and F :: "'b set"   and r :: "('b \<times> 'b) set" assumes "(a, b) \<in> rtrancl_on F r" shows "(a, b) \<in> r\<^sup>*"
hyps: shows "p \<noteq> [] \<longrightarrow> (\<forall>x s. s \<in> set p \<and> x \<in> dom (C A) \<longrightarrow> x \<notin> dom (C s)) \<longrightarrow> y \<in> dom (C A) \<longrightarrow> y \<notin> dom (C (list2FWpolicy p))"
surjective: fixes r :: "('c, 'd) Group_scheme" shows "r = \<lparr>carrier = carrier r, Group.top = (\<cdot>\<^bsub>r\<^esub>), iop = iop r, one = \<one>\<^bsub>r\<^esub>, \<dots> = Group.more r\<rparr>"
open_inv_chart_image: fixes X :: "'b set"   and c :: "('c, 'b) chart" assumes "open X"   and "X \<subseteq> codomain c" shows "open (inv_chart c ` X)"
wr_CC: shows "\<forall>C\<subseteq>A. well_related_set C (\<sqsubseteq>) \<longrightarrow> C \<in> CC"
ts_invI: fixes ts :: "'c \<Rightarrow> ('d \<times> 'e \<Rightarrow>f nat) option"   and I :: "'c \<Rightarrow> 'f option"   and P :: "'f \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'g \<Rightarrow> bool"   and m :: "'g" assumes "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> \<exists>i. I t = \<lfloor>i\<rfloor> \<and> P i t x m" shows "ts_inv P I ts m"
split_sel_asm: fixes P :: "'d \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> 'd"   and prod :: "'e \<times> 'f" shows "P (case prod of (x, xa) \<Rightarrow> f x xa) = (\<not> (prod = (fst prod, snd prod) \<and> \<not> P (f (fst prod) (snd prod))))"
cobounded1: fixes a :: "'b::semilattice_inf"   and b :: "'b::semilattice_inf" shows "(a::'b::semilattice_inf) \<sqinter> (b::'b::semilattice_inf) \<le> a"
map_def: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d Predicate.pred" shows "Predicate.map f P = Predicate.bind P (Predicate.single \<circ> f)"
SMP_Union: fixes f :: "'b \<Rightarrow> ('fun, 'var) Term.term set"   and M :: "'b set" shows "SMP (\<Union> (f ` M)) = (\<Union>m\<in>M. SMP (f m))"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>) (\<cdot>) trg local.map \<phi>" shows "equivalence_functor (\<cdot>) (\<cdot>) trg"
red2_Nml: fixes a :: "'a local.term"   and b :: "'a local.term" assumes "Nml (a \<^bold>\<star> b)" shows "a \<^bold>\<Down> b = a \<^bold>\<star> b"
vimage_def: fixes f :: "'e \<Rightarrow> 'f"   and B :: "'f set" shows "f -` B \<equiv> {x. f x \<in> B}"
coefficients_Nil: fixes p :: "poly" shows "coefficients p \<noteq> []"
assms_3: shows "e = [$Br i] \<or> e = []"
child_owner_document: shows "h \<turnstile> get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child) \<rightarrow>\<^sub>r child_owner_document"
\<a>'_def: fixes \<mu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<nu> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<tau> :: "('a, 'a set) Maps.arr arrow_of_spans_data" shows "Span.\<a>' \<mu> \<nu> \<tau> \<equiv> Span.\<alpha>' (\<mu>, \<nu>, \<tau>)"
rangeU_def: shows "rangeU = U *\<^sub>S \<top>"
cases: fixes a1 :: "nat"   and a2 :: "nat list set" assumes "Form a1 a2" obtains xs :: "nat list" and ys :: "nat list" where "a1 = 0" and "a2 = {xs__, ys__}" and "length xs__ = length ys__" and "xs__ \<noteq> ys__"   | k :: "nat" and xs :: "nat list" and ys :: "nat list" and zs :: "nat list" where "a1 = 2 * k__ - 1" and "a2 = {xsa__, ysa__}" and "Form_Body k__ k__ xsa__ ysa__ zs__" and "0 < k__"   | k :: "nat" and xs :: "nat list" and ys :: "nat list" and zs :: "nat list" where "a1 = 2 * k" and "a2 = {xs, ys}" and "Form_Body (Suc k) k xs ys zs" and "0 < k"
zero_order_2: fixes n :: "'d multiset" shows "(n \<subseteq># {#}) = (n = {#})"
r_one: fixes x :: "nat \<times> nat" assumes "x \<in> carrier S" shows "x \<otimes>\<^bsub>S\<^esub> \<one>\<^bsub>S\<^esub> = x"