hcju/isabelleps huggingface dataset:
djf_Or: fixes bbs :: "bool list"   and bs :: "int list"   and f :: "'a \<Rightarrow> fm"   and p :: "'a"   and q :: "fm" shows "Ifm bbs bs (djf f p q) = Ifm bbs bs (Or (f p) q)"
insert_def: shows "DList.insert \<equiv> (id ---> rep_dlist ---> dlist) (#)"
continuous_on_enn2ereal: shows "continuous_on UNIV enn2ereal"
fixing_ide_gives_functor_2: fixes a2 :: "'a" assumes "ide a2" shows "functor (\<cdot>) (\<cdot>) (\<lambda>f1. local.Prod (f1, a2))"
assms: shows "P \<Longrightarrow>\<alpha> \<prec> P'"
interior_of_eq: fixes X :: "'d topology"   and S :: "'d set" shows "(X interior_of S = S) = openin X S"
sas_plus_problem_to_strips_problem_variable_set_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem" assumes "is_valid_problem_sas_plus \<Psi>" shows "set ((\<phi> \<Psi> )\<^sub>\<V>) = (\<Union>v\<in>set (\<Psi>\<^sub>\<V>\<^sub>+). {(v, a) |a. a \<in> \<R>\<^sub>+ \<Psi> v})"
finite_UNIV_option_def: shows "finite_UNIV = Phantom('a option) (of_phantom finite_UNIV)"
map\<^sub>0_in_hom_1: fixes a :: "'c" assumes "C.obj a" shows "\<guillemotleft>\<epsilon>'.map\<^sub>0 a : I\<^sub>C.map\<^sub>0 a \<rightarrow>\<^sub>C FG.map\<^sub>0 a\<guillemotright>"
add_diff_inverse: fixes a :: "'d"   and b :: "'d" assumes "a \<le> b" shows "a + (b - a) = b"
F_j_analytically_valid: shows "analytically_valid Dx_pair (\<lambda>p. F p \<bullet> j) i"
UN_extend_simps_3: fixes A :: "'c set"   and B :: "'d \<Rightarrow> 'c set"   and C :: "'d set" shows "A \<union> \<Union> (B ` C) = (if C = {} then A else \<Union>x\<in>C. A \<union> B x)"
norm_triangle_eq: fixes x :: "'b"   and y :: "'b" shows "(norm (x + y) = norm x + norm y) = (norm x *\<^sub>R y = norm y *\<^sub>R x)"
atom_prm_atom_def: fixes pi :: "(atom \<times> atom) list"   and a :: "atom" shows "pi \<bullet> a \<equiv> atom_prm_atom pi a"
reflexive: fixes Rel :: "(pi \<times> pi) set"   and P :: "pi" assumes "Id \<subseteq> Rel" shows "P \<leadsto><Rel> P"
minus_minus: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<ominus> (\<ominus> x) = x"
pCons_0_add: fixes p :: "'b poly"   and q :: "'b poly" shows "pCons (0::'b) (p + q) = pCons (0::'b) p + pCons (0::'b) q"
assms: fixes s :: "'a"   and x :: "'b" shows "ord_spmf (=) (callee1 s x) (callee2 s x)"
out_var_def: fixes x :: "'b \<Longrightarrow> 'd" shows "$x\<acute> = x ;\<^sub>L snd\<^sub>L"
hensel_fpxs2_def: shows "hensel_fpxs2 = Abs_fps (snd \<circ> hensel_fpxs_aux)"
in_totatives_iff: fixes k :: "nat"   and n :: "nat" shows "(k \<in> totatives n) = (0 < k \<and> k \<le> n \<and> coprime k n)"
errMOD_igFreshClsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igConsIPresIGWls MOD"   and "igFreshCls MOD" shows "igFreshClsSTR (errMOD MOD)"
guardedEqvt_1: fixes p :: "(name \<times> name) list"   and P :: "('a, 'b, 'c) psi" shows "p \<bullet> guarded P = guarded (p \<bullet> P)"
pow_Suc_list: fixes u :: "'c list"   and n :: "nat" shows "u \<^sup>@ Suc n = u \<cdot> u \<^sup>@ n"
frequently_going_toI: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c"   and F :: "'b filter"   and G :: "'a filter"   and P :: "'c \<Rightarrow> bool"   and A :: "'c set" assumes "LIM n G. f (g n) :> F"   and "\<forall>\<^sub>F n in G. P (g n)"   and "\<forall>\<^sub>F n in G. g n \<in> A"   and "G \<noteq> bot" shows "frequently P (f going_to F within A)"
add_mset: fixes x :: "'a"   and N :: "'a multiset" shows "F (add_mset x N) = x \<^bold>* F N"
simps_2: fixes S :: "'c ltl set"   and \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "eval\<^sub>G S (\<phi> or \<psi>) = eval\<^sub>G S \<phi> or eval\<^sub>G S \<psi>"
5: shows "f\<^sup>\<star> * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
variableDefinedImpliesLiteralDefined: fixes literal :: "Literal"   and valuation :: "Literal list" shows "(var literal \<in> vars valuation) = (literalTrue literal valuation \<or> literalFalse literal valuation)"
dy_fake_msg_monotone: fixes c :: "msg"   and b :: "agent set"   and I :: "msg set"   and C :: "chan set"   and b' :: "agent set"   and I' :: "msg set"   and C' :: "chan set" assumes "c \<in> dy_fake_msg b I C"   and "b \<subseteq> b'"   and "I \<subseteq> I'"   and "C \<subseteq> C'" shows "c \<in> dy_fake_msg b' I' C'"
intro: fixes fs :: "'a vec list"   and n :: "nat" assumes "set fs \<subseteq> carrier_vec n" shows "gram_schmidt_fs_Rn n fs"
enum_distinct: fixes enum :: "'c list"   and enum_all :: "('c \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('c \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "distinct enum"
content_ff_def: fixes p :: "'b fract poly" shows "content_ff p = some_gcd_ff_list (coeffs p)"
insertll_len2: fixes n :: "nat"   and vs :: "bool list list"   and x :: "bool list"   and k :: "nat" assumes "list_all (is_alph n) vs"   and "length x = length vs" shows "list_all (is_alph (Suc n)) (insertll k x vs)"
xu: fixes n :: "nat" shows "x n \<in> sfis (u n) M"
map_resource_resource_of_oracle: fixes f :: "'h \<Rightarrow> 'j"   and g :: "'k \<Rightarrow> 'i"   and "oracle" :: "'l \<Rightarrow> 'j \<Rightarrow> ('k \<times> 'l) spmf"   and s :: "'l" shows "map_resource f g (RES oracle s) = RES (map_fun id (map_fun f (map_spmf (map_prod g id))) oracle) s"
Top_def: fixes po :: "'b potype" shows "Top po = greatest (\<lambda>x. True) po"
forw_subst: fixes a :: "'a"   and b :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "a = b"   and "P b" shows "P a"
\<phi>_stream: fixes xs :: "('s \<times> ('c \<Rightarrow> real)) stream"   and x :: "'s \<times> ('c \<Rightarrow> real)" assumes "\<And>x y. \<lbrakk>x \<in> S; x ~ y\<rbrakk> \<Longrightarrow> x \<in> S"   and "\<And>x y. \<lbrakk>x \<in> S; x ~ y\<rbrakk> \<Longrightarrow> x ~ y"   and "pred_stream (\<lambda>s. s \<in> S) xs"   and "x \<in> S" shows "pred_stream (\<lambda>s. \<phi> (reps (abss s)) = \<phi> s) (x ## xs)"
trivial_diagram: fixes \<sigma> :: "'a \<times> ('b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "\<sigma> \<in> seq ars" shows "diagram ars (\<sigma>, (fst \<sigma>, []), (lst \<sigma>, []), \<sigma>)"
\<eta>_def: shows "\<eta> = (SOME \<eta>. \<exists>\<epsilon>. adjoint_equivalence hhom\<^sub>C.comp hhom\<^sub>D.comp G\<^sub>1 F\<^sub>1 \<eta> \<epsilon>)"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> B" shows "x \<otimes> y = y \<otimes> x"
last_i_take: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "last (f \<Down> Suc n) = f n"
assms: shows "ordered_sel_resolvent UNIV P1 P2 C"
not_in_iff: fixes x :: "'b"   and M :: "'b multiset" shows "(x \<notin># M) = (count M x = 0)"
succK_def: shows "succK \<equiv> \<Lambda> nK. bindK\<cdot>nK\<cdot>(\<Lambda> (ValKN\<cdot>n). unitK\<cdot>(ValKN\<cdot>(n + 1)))"
table_of_map2_SomeI: fixes t :: "('b \<times> 'a) list"   and k :: "'b"   and x :: "'a"   and C :: "'c" assumes "table_of t k = Some x" shows "table_of (map (\<lambda>(k, y). ((k, C), y)) t) (k, C) = Some x"
S_transition_Act_iff: fixes P :: "'state"   and \<alpha> :: "('act, 'pred) S_action"   and P' :: "'state" shows "S_transform.S_transition P \<langle>S_action.Act \<alpha>,P'\<rangle> = P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle>"
deg_non_zero: fixes ua :: "int poly"   and lua :: "int"   and vsa :: "int poly multiset" assumes "ua =m Polynomial.smult lua (\<Prod>\<^sub># vsa)"   and "comm_monoid_mult_class.coprime lua p"   and "size vsa \<noteq> 0"   and "\<And>v. v \<in># vsa \<Longrightarrow> irreducible\<^sub>d_m v \<and> monic v" shows "0 < degree ua"
isinp_def: fixes evs :: "event list"   and r :: "room"   and g :: "guest" shows "isinp evs r g = (g \<in> isin evs r)"
induct: fixes P1 :: "freeExp \<Rightarrow> bool"   and P2 :: "freeExp list \<Rightarrow> bool"   and freeExp_list :: "freeExp list" assumes "\<And>x. P1 (VAR x)"   and "\<And>x1 x2. \<lbrakk>P1 x1; P1 x2\<rbrakk> \<Longrightarrow> P1 (PLUS x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (FNCALL x1 x2)"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)" shows "P2 freeExp_list"
fls_divide_X: fixes f :: "'a fls" shows "f / fls_X = fls_shift 1 f"
ex_creds_def: shows "ex_creds \<equiv> \<lambda>x. if x = Actor ''Patient'' then ({''PIN'', ''skey''}, {}) else if x = Actor ''Doctor'' then ({''PIN''}, {}) else ({}, {})"
simps_2: fixes x21 :: "'b"   and x22 :: "'b list" shows "[] \<noteq> x21 # x22"
not_empty_eff_in_as_proj: fixes vs :: "'d set"   and a :: "('d, 'f) fmap \<times> ('d, 'e) fmap"   and as :: "(('d, 'f) fmap \<times> ('d, 'e) fmap) list" assumes "fmdom' (fmrestrict_set vs (snd a)) \<noteq> {}" shows "as_proj (a # as) vs = action_proj a vs # as_proj as vs"
ringE_4: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y"
div_smult_left: fixes a :: "'a"   and x :: "'a poly"   and y :: "'a poly" shows "smult a x div y = smult a (x div y)"
safe_def': fixes s :: "v_state"   and r :: "nat"   and v :: "val" shows "safe s r v \<equiv> \<forall>r'<r. \<forall>Q\<in>Quorum. \<forall>w. votes s r' ` Q = {Some w} \<longrightarrow> v = w"
sheaf_spec_is_presheaf: shows "presheaf_of_rings Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec"
rep_eq: fixes x :: "'a\<^sup>+"   and xa :: "'a\<^sup>+" shows "Rep_pos (x / xa) = Rep_pos x / Rep_pos xa"
Self_not_in_closed: fixes sa :: "char list"   and n :: "nat"   and pa :: "char list"   and t :: "sterm" shows "sa \<notin> FV ({n \<leftarrow> [sa,pa]} t)"
simps_11: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x2 :: "'e"   and y2 :: "'f" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
path_connected_space_def: fixes X :: "'a topology" shows "path_connected_space X \<equiv> \<forall>x\<in>topspace X. \<forall>y\<in>topspace X. \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
simps: fixes k :: "real"   and a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "k *\<^sub>s\<^sub>v\<^sub>4 (a, b, c, d) = (k * a, k * b, k * c, k * d)"
"1_2": shows "u \<preceq>\<^sub>F w"
ENF_notevalmodel_imp_ex_offending_min: shows "\<exists>F\<subseteq>edges G. is_offending_flows_min_set F G nP"
assoc: fixes a :: "'b list"   and b :: "'b list"   and c :: "'b list" shows "(a @ b) @ c = a @ b @ c"
cut_le_empty_iff: fixes I :: "'a set"   and t :: "'a" shows "(I \<down>\<le> t = {}) = (\<forall>x\<in>I. t < x)"
elims: fixes x :: "('a, 'b) S_action"   and y :: "atom set" assumes "bn x = y" obtains \<alpha> :: "'a" where "x = S_action.Act \<alpha>" and "y = bn \<alpha>"   | uu_ :: "'b" where "x = S_action.Pred uu_" and "y = {}"
ih: fixes x :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat"   and y :: "'a literal list list" assumes "self_call x = Some y" shows "lfinite (derivation_from x)"
le: fixes X :: "'a set" assumes "X \<subseteq> A" shows "integral\<^sup>N (count_space X) f \<le> integral\<^sup>N (count_space (R `` X)) g"
star_left_induct_var: fixes y :: "'a"   and x :: "'a" shows "y * x \<le> x \<longrightarrow> y\<^sup>\<star> * x \<le> x"
\<Delta>_naturality: shows "(\<a>[f, g, h] \<star> tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> TTfgh.tab = ((f \<star> g \<star> h) \<star> TTfgh_TfTgh.the_\<theta>) \<cdot> \<a>[f \<star> g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0, TTfgh_TfTgh.chine] \<cdot> (TfTgh.tab \<star> TTfgh_TfTgh.chine) \<cdot> TTfgh_TfTgh.the_\<nu>"
map2_val_compat'D2: fixes f :: "('a \<times> 'd) list \<times> 'o \<Rightarrow> ('a \<times> 'e) list \<times> 'o"   and zs :: "('a \<times> 'd) list \<times> 'o" assumes "map2_val_compat' f"   and "oalist_inv zs" shows "snd (f zs) = snd zs"
set_partitions_itself: fixes A :: "'c set" assumes "A \<noteq> {}" shows "{A} partitions A"
true_clss_def: fixes I :: "'b set"   and CC :: "'b literal multiset set" shows "(I \<Turnstile>s CC) = (\<forall>C\<in>CC. I \<Turnstile> C)"
mdef: shows "m = 2 ^ multiplicity 2 m * (m div 2 ^ multiplicity 2 m) \<and> coprime 2 (m div 2 ^ multiplicity 2 m)"
induct: fixes P :: "nat \<Rightarrow> bool"   and a0 :: "nat" assumes "\<And>n. \<lbrakk>\<not> n \<le> 20 \<Longrightarrow> P (nat \<lceil>2 / 10 * real n\<rceil>); \<not> n \<le> 20 \<Longrightarrow> P (nat \<lceil>7 / 10 * real n + 3\<rceil>)\<rbrakk> \<Longrightarrow> P n" shows "P a0"
r: shows "bl2wc (Oc \<up> rs) = nat"
assms_3: fixes d :: "'a" assumes "a dvd d"   and "b dvd d" shows "c dvd d"
Map_in_Hom: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)"
surj_pair: fixes p :: "'d \<times> 'e" shows "\<exists>x y. p = (x, y)"
assms: shows "x \<in> EX' (f \<union> EX' (EF f))"
is_faithful: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "f \<otimes> cod \<iota> = f' \<otimes> cod \<iota>" shows "f = f'"
smap2_szip: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and s1 :: "'c stream"   and s2 :: "'d stream" shows "smap2 f s1 s2 = smap (\<lambda>(x, y). f x y) (szip s1 s2)"
antisym_conv1: fixes x :: "'t"   and y :: "'t" assumes "\<not> x \<prec>\<^sub>t y" shows "(x \<preceq>\<^sub>t y) = (x = y)"
hoare_r_conseq: fixes p\<^sub>1 :: "'a upred"   and p\<^sub>2 :: "'a upred"   and S :: "'a hrel"   and q\<^sub>2 :: "'a upred"   and q\<^sub>1 :: "'a upred" assumes "`p\<^sub>1 \<Rightarrow> p\<^sub>2`"   and "\<lbrace>p\<^sub>2\<rbrace> S \<lbrace>q\<^sub>2\<rbrace>\<^sub>u"   and "`q\<^sub>2 \<Rightarrow> q\<^sub>1`" shows "\<lbrace>p\<^sub>1\<rbrace> S \<lbrace>q\<^sub>1\<rbrace>\<^sub>u"
closest_point_dot: fixes S :: "'b set"   and x :: "'b"   and a :: "'b" assumes "convex S"   and "closed S"   and "x \<in> S" shows "(a - closest_point S a) \<bullet> (x - closest_point S a) \<le> 0"
iso_hom_relboundary_contractible: fixes X :: "'a topology"   and T :: "'a set"   and S :: "'a set"   and p :: "int" assumes "contractible_space X"   and "contractible_space (subtopology X T)"   and "T \<subseteq> S"   and "topspace X \<inter> T \<noteq> {}" shows "hom_relboundary p X S T \<in> Group.iso (relative_homology_group p X S) (relative_homology_group (p - 1) (subtopology X S) T)"
intros_8: fixes i :: "nat"   and \<S> :: "s_context"   and tvs :: "t list"   and vs :: "v list"   and \<C> :: "t_context"   and rs :: "t list option"   and es :: "e list"   and ts :: "t list" assumes "i < length (s_inst \<S>)"   and "tvs = map typeof vs"   and "\<C> = (s_inst \<S> ! i)\<lparr>local := local (s_inst \<S> ! i) @ tvs, return := rs\<rparr>"   and "\<S>\<bullet>\<C> \<turnstile> es : [] _> ts"   and "rs = Some ts \<or> rs = None" shows "\<S>\<bullet>rs \<tturnstile>_ i vs;es : ts"
Seq_def: fixes f :: "unit \<Rightarrow> 'a seq" shows "Seq f = pred_of_seq (f ())"
antidom_def: fixes x :: "'a" shows "a x = 1' \<cdot> - (x ; 1)"
cast\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>2\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_some: fixes obj :: "(_) Object"   and document :: "('a, 'b, 'c, 'd option) RDocument_scheme" shows "(cast obj = Some document) = (cast document = obj)"
WTrtSFAcc: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option" assumes "P \<turnstile> C has F,Static:T in D" shows "P,E,h,sh \<turnstile> C\<bullet>\<^sub>sF{D} : T"
neq: fixes i :: "'n"   and j :: "'n" assumes "i \<noteq> j" shows "A i j = (0::'a)"
Map_comp: fixes f :: "('a E.term, 'a) arr"   and g :: "('a E.term, 'a) arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Map (g \<cdot> f) = Map g \<cdot>\<^sub>B Map f"
v: fixes a :: "rat"   and n :: "'a ns_constraint" assumes "(a, n) \<in> set C" shows "v \<Turnstile>\<^sub>l\<^sub>e lec_of_nsc (a *R n)"
eval_elim_cases_20: fixes G :: "prog"   and s :: "st"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt"   and x :: "vals"   and s' :: "state" assumes "G\<turnstile>Norm s \<midarrow>In1r (If(e) c1 Else c2)\<succ>\<rightarrow> (x, s')" obtains b :: "val" and s1 :: "state" where "x = \<diamondsuit>" and "G\<turnstile>Norm s \<midarrow>e-\<succ>b\<rightarrow> s1" and "G\<turnstile>s1 \<midarrow>(if the_Bool b then c1 else c2)\<rightarrow> s'"
Cons: 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))"
closerToC_def: shows "closerToC \<equiv> {B \<in> folding.\<C>. order.greater (chamber_distance B D) (chamber_distance B C)}"
awalk_verts_reachable_from: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "H.awalk u p v"   and "w \<in> set (H.awalk_verts u p)" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> w"
WW_seg_def: fixes n :: "nat" shows "WW_seg n \<equiv> {l \<in> WW. length l = n}"
unifiers_occur_left_is_Fun: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "x \<in> vars_term t"   and "is_Fun t" shows "unifiers (insert (Var x, t) E) = {}"
offs_num_cons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat" shows "offs_num n (x # xs) index key mi ma i = (if index key x n mi ma = i then Suc else id) (offs_num n xs index key mi ma i)"
set_rec: fixes xs :: "'b list" shows "set xs = rec_list {} (\<lambda>x _. insert x) xs"
VSigma_def: fixes A :: "V"   and B :: "V \<Rightarrow> V" shows "VSigma A B \<equiv> set (\<Union>x\<in>elts A. \<Union>y\<in>elts (B x). {\<langle>x, y\<rangle>})"
simps_4: shows "sch_to_pr Base_snd = c_snd"
2: shows "rv \<in> fringe prb \<and> rv \<noteq> src re"
Iset_def: fixes A :: "'b set star" shows "Iset A = {x. (*p2* (\<in>)) x A}"
assms: shows "A \<subseteq># image_mset \<eta> C'"
x_lt_2_x: fixes x :: "nat" shows "x < 2 ^ x"
eq: shows "B = A \<inter> B \<union> (B - A)"
ideal_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set" shows "ideal R I = (R +> I \<and> (\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r\<^bsub>R\<^esub> x \<in> I))"
l_get_attribute_axioms: shows "l_get_attribute type_wf get_attribute get_attribute_locs"
NSDERIV_minus: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" assumes "NSDERIV f x :> D" shows "NSDERIV (\<lambda>x. - f x) x :> - D"
p2_2: fixes x :: "'a"   and K :: "'a set" assumes "(x, K) \<in> p2" shows "x \<in> K"
rHom_0_0: fixes A :: "('d, 'e) Ring_scheme"   and R :: "('f, 'g) Ring_scheme"   and f :: "'d \<Rightarrow> 'f" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f \<zero>\<^bsub>A\<^esub> = \<zero>\<^bsub>R\<^esub>"
l_set_child_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs \<and> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs"
set_sel_2: fixes a :: "'b list"   and x :: "'b" assumes "a \<noteq> []"   and "x \<in> set (tl a)" shows "x \<in> set a"
image_func1: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "\<P> (f \<circ> g) = \<P> f \<circ> \<P> g"
S5Basic_12: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<diamond>(\<phi> \<^bold>& \<^bold>\<diamond>\<psi>) \<^bold>\<equiv> \<^bold>\<diamond>\<phi> \<^bold>& \<^bold>\<diamond>\<psi> in v]"
freshSets_1: fixes C :: "'a" shows "[] \<sharp>* C"
Cons_6: shows "indexed_eval_aux (P # Ps) i = indexed_eval_aux Qs i"
is_ub_def: fixes S :: "'b set"   and x :: "'b" shows "S <| x = (\<forall>y\<in>S. y \<sqsubseteq> x)"
branch_instr_def: fixes instr :: "sparc_operation \<times> inst_operand list" shows "branch_instr instr \<equiv> let instr_name = fst instr; op_list = snd instr; disp22 = get_operand_w22 (op_list ! 1); flaga = get_operand_flag (op_list ! 0) in do icc_val \<leftarrow> gets (branch_instr_sub1 instr_name); npc_val \<leftarrow> gets (cpu_reg_val nPC); pc_val \<leftarrow> gets (cpu_reg_val PC); write_cpu npc_val PC; if icc_val = 1 then do write_cpu (pc_val + sign_ext24 (ucast disp22 << 2)) nPC; if instr_name = bicc_type BA \<and> flaga = 1 then do set_annul True; return () od else return () od else do write_cpu (npc_val + 4) nPC; if flaga = 1 then do set_annul True; return () od else return () od od"
prems_9: shows "(ah_, a'h_) \<in> \<langle>R\<rangle>ltlr_rel \<rightarrow> \<langle>R\<rangle>ltlr_rel \<rightarrow> Rv \<rightarrow> Rv \<rightarrow> Rv"
assms_2: shows "\<forall>a b. b < mod_type_class.from_nat k \<longrightarrow> A $ a $ b = B $ a $ b"
assms_1: shows "\<forall>x\<in>s. f x \<noteq> y"
psubsetD: fixes A :: "'e set"   and B :: "'e set"   and c :: "'e" assumes "A \<subset> B"   and "c \<in> A" shows "c \<in> B"
assms: shows "X \<in>\<^sub>\<circ> Vset i"
A: shows "\<forall>x y z. x \<sqinter> z = y \<sqinter> z \<and> x \<squnion> z = y \<squnion> z \<longrightarrow> x = y"
finite_Pow_iff: fixes A :: "'d set" shows "finite (Pow A) = finite A"
preserves_id_def: fixes G :: "('o, 'a, 'a set, 'a set_arrow) functor" shows "preserves_id G \<equiv> \<forall>A\<in>Ob. G \<^bsub>\<a>\<^esub> (Id A) = Id\<^bsub>BB\<^esub> G \<^bsub>\<o>\<^esub> A"
star_inductl_var_eq2: fixes y :: "'a"   and x :: "'a" assumes "y = x \<cdot> y" shows "y = x\<^sup>\<star> \<cdot> y"
fempty_is_fimage: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e fset" shows "({||} = f |`| A) = (A = {||})"
valid_path_compose: fixes g :: "real \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" assumes "valid_path g"   and "\<And>x. x \<in> path_image g \<Longrightarrow> f field_differentiable at x"   and "continuous_on (path_image g) (deriv f)" shows "valid_path (f \<circ> g)"
lset_intros_2: fixes x :: "'e"   and xs :: "'e llist"   and x' :: "'e" assumes "x \<in> lset xs" shows "x \<in> lset (LCons x' xs)"
indicator_mult_ennreal: fixes A :: "'b set"   and x :: "'b"   and r :: "real" shows "indicator A x * ennreal r = ennreal (indicat_real A x * r)"
map_gpv'_comp: fixes f :: "'j \<Rightarrow> 'g"   and g :: "'k \<Rightarrow> 'h"   and h :: "'i \<Rightarrow> 'l"   and f' :: "'m \<Rightarrow> 'j"   and g' :: "'n \<Rightarrow> 'k"   and h' :: "'l \<Rightarrow> 'o"   and gpv :: "('m, 'n, 'o) gpv" shows "map_gpv' f g h (map_gpv' f' g' h' gpv) = map_gpv' (f \<circ> f') (g \<circ> g') (h' \<circ> h) gpv"
3: fixes y :: "'a" assumes "y \<in> rel_interior S" shows "a \<bullet> x < a \<bullet> y"
less_eintE: fixes n :: "eint"   and m :: "int" assumes "n < eint m" obtains k :: "int" where "n = eint k" and "k < m"
conv_radius_inftyI': fixes c :: "real"   and f :: "nat \<Rightarrow> 'b" assumes "\<And>r. c < r \<Longrightarrow> \<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)" shows "conv_radius f = \<infinity>"