hcju/isabelleps huggingface dataset:
igSubstClsSTR_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstClsSTR MOD \<equiv> igSubstIGVar1STR MOD \<and> igSubstIGVar2STR MOD \<and> igSubstIGAbsSTR MOD \<and> igSubstIGOpSTR MOD"
rep_eq: fixes x :: "(char list \<times> 'a list) prefix" shows "msteps x = msteps0 (Rep_prefix x)"
comp_left_subdist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<sqinter> y) * z \<le> x * z \<sqinter> y * z"
relH_set_ref: fixes r :: "'d ref"   and as :: "nat set"   and h :: "heap"   and x :: "'d" assumes "addr_of_ref r \<notin> as"   and "in_range (h, as)" shows "relH as h (Ref.set r x h)"
prime_elem_mono: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "'a"   and q :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero p"   and "\<not> dvd.dvd times q one"   and "dvd.dvd times q p" shows "comm_semiring_1.prime_elem times one zero q"
simps_3: fixes env :: "v sem_env"   and es :: "(char list \<times> char list \<times> exp0) list"   and ux :: "char list" shows "is_cupcake_value (Recclosure env es ux) = (list_all (\<lambda>(_, uu_, y). is_cupcake_exp y) es \<and> is_cupcake_all_env env)"
counterexample_not_gu_condition_aux: shows "([a\<^sub>c, b\<^sub>c, c\<^sub>c], [b\<^sub>c, a\<^sub>c, c\<^sub>c]) \<in> rel_induct P\<^sub>c I\<^sub>c id a\<^sub>c"
tbisim_SomeI: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1"   and x' :: "'x2"   and m' :: "'m2"   and nw :: "bool"   and ln :: "'l \<Rightarrow>f nat" assumes "t \<turnstile> (x, m) \<approx> (x', m')"   and "nw \<or> x \<approx>w x'" shows "tbisim nw t \<lfloor>(x, ln)\<rfloor> m \<lfloor>(x', ln)\<rfloor> m'"
stop_no_eval: fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mds :: "Mode \<Rightarrow> 'a set"   and mem :: "'a \<Rightarrow> 'b"   and c' :: "('a, 'c, 'd) Stmt"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b" assumes "sifum_lang eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars" shows "(((Stop, mds), mem), (c', mds'), mem') \<notin> sifum_lang.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
intros_6: fixes X :: "msg"   and H :: "msg set"   and K :: "key" assumes "X \<in> synth H"   and "Key K \<in> H" shows "Crypt K X \<in> synth H"
insert: fixes g :: "'c \<Rightarrow> 'b"   and x :: "'c"   and A :: "'c multiset" shows "\<Prod>\<^sub># (image_mset g (add_mset x A)) = g x * \<Prod>\<^sub># (image_mset g A)"
eval\<^sub>G_G_nested: fixes \<G> :: "'b ltl set"   and \<phi> :: "'b ltl" shows "\<^bold>G (eval\<^sub>G \<G> \<phi>) \<subseteq> \<^bold>G \<phi>"
bounded_subset_cbox_symmetric: fixes S :: "'b set" assumes "bounded S" obtains a :: "'b" where "S \<subseteq> cbox (- a) a"
Under_Field: fixes r :: "('c \<times> 'c) set"   and A :: "'c set" shows "Under r A \<subseteq> Field r"
val_by_adj_map: fixes am :: "nat \<Rightarrow> nat list" assumes "is_adj_map am" shows "f.val = (\<Sum>v\<in>set (am s). cf (v, s))"
quasi_inverses_some_quasi_inverse_2: fixes f :: "'a" assumes "equivalence_map f" shows "quasi_inverses f\<^sup>~ f"
orderE: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" obtains "a = sup a b"
ms_ci_in: shows "(mset C, i) \<in># image_mset (apfst mset) (mset P)"
coset_join3: fixes x :: "'a"   and H :: "'a set" assumes "x \<in> carrier R"   and "subgroup H (add_monoid R)"   and "x \<in> H" shows "x <#\<^bsub>add_monoid R\<^esub> H = H"
oiface_rewrite_matches_Primitive: fixes \<alpha> :: "action \<Rightarrow> ('c, 'd) tagged_packet_scheme \<Rightarrow> bool"   and ipassmt :: "iface \<Rightarrow> ('c word \<times> nat) list option"   and x :: "'c common_primitive"   and a :: "action"   and p :: "('c, 'd) tagged_packet_scheme" shows "(matches (common_matcher, \<alpha>) (MatchNot (oiface_rewrite ipassmt (Match x))) a p = matches (common_matcher, \<alpha>) (MatchNot (Match x)) a p) = (matches (common_matcher, \<alpha>) (oiface_rewrite ipassmt (Match x)) a p = matches (common_matcher, \<alpha>) (Match x) a p)"
cong_cop_per2: fixes A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "A \<noteq> B"   and "Per A B X"   and "Per A B Y"   and "Cong B X B Y"   and "Coplanar A B X Y" shows "X = Y \<or> X Y ReflectL A B"
inject_2: fixes x21 :: "'a process"   and x22 :: "'a process"   and y21 :: "'a process"   and y22 :: "'a process" shows "(Choice x21 x22 = Choice y21 y22) = (x21 = y21 \<and> x22 = y22)"
id_o: fixes g :: "'b \<Rightarrow> 'c" shows "id \<circ> g = g"
frag_len_filter_le: fixes l' :: "'a list"   and l :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "sublist l' l" shows "length (filter P l') \<le> length (filter P l)"
chain1: shows "Complete_Partial_Order.chain (\<le>) ((\<lambda>x. \<Squnion> (f x ` Y)) ` Y)"
prems_3: shows "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"
impossible_Cons: fixes xs :: "'d list"   and ys :: "'d list"   and x :: "'d" assumes "length xs \<le> length ys" shows "(xs = x # ys) = False"
trdsp_in_pmdl: fixes bs :: "('t \<Rightarrow>\<^sub>0 'd) list"   and p :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e"   and q :: "('t \<Rightarrow>\<^sub>0 'd) \<times> nat \<times> 'e" shows "trdsp bs (p, q) \<in> pmdl (insert (fst p) (insert (fst q) (set bs)))"
iT_Minus_Min: fixes I :: "nat set"   and k :: "nat" assumes "I \<noteq> {}"   and "iMin I \<le> k" shows "iMin (k \<ominus> I) = k - Max (I \<down>\<le> k)"
Ipol_Pc_2: fixes l :: "'a list" shows "Ipol l (Pc 1) = \<one>"
circ_isotone: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "x \<preceq> y" shows "x\<^sup>\<odot> \<preceq> y\<^sup>\<odot>"
add_commute: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "ab_semigroup_add_ow U plus"   and "a \<in> U"   and "b \<in> U" shows "plus a b = plus b a"
atS_def: fixes p :: "'a"   and ls :: "'b set"   and s :: "('c, 'b, 'a, 'd, 'e) system_state" shows "atS p ls s = (\<exists>l\<in>ls. at p l s)"
r_neg: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> \<ominus>\<^bsub>P\<^esub> x = \<zero>\<^bsub>P\<^esub>"
\<theta>_simps_5: shows "cod \<theta>' = src r"
finite_config_BIT: fixes init :: "'a list"   and qs :: "'a list" assumes "distinct init" shows "finite (set_pmf (Partial_Cost_Model.config'_rand BIT (fst BIT init \<bind> (\<lambda>is. return_pmf (init, is))) qs))"
subset_antisym: fixes A :: "'g set"   and B :: "'g set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
union_is_single: fixes M :: "'b multiset"   and N :: "'b multiset"   and a :: "'b" shows "(M + N = {#a#}) = (M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#})"
shiftr_mask_le: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "mask n >> m = 0"
compose_eq': fixes A :: "'a set"   and G :: "'c \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'c" shows "compose A G F = restrict (G \<circ> F) A"
ex_mono: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "(\<exists>x. P x) \<longrightarrow> (\<exists>x. Q x)"
mod_mult_right_eq: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a (modulo b c)) c = modulo (times a b) c"
subgroup_self: shows "subgroup (carrier (G\<lparr>carrier := P\<rparr>)) (G\<lparr>carrier := P\<rparr>)"
mono_les: fixes s :: "'a"   and S :: "'a set"   and K :: "'a \<Rightarrow> 'a pmf"   and N :: "'a set"   and l1 :: "'a \<Rightarrow> real"   and l2 :: "'a \<Rightarrow> real"   and c :: "'a \<Rightarrow> real" assumes "s \<in> S"   and "(\<Union>s\<in>S. set_pmf (K s)) \<subseteq> S \<union> N"   and "\<And>s. s \<in> S \<Longrightarrow> integrable (measure_pmf (K s)) l1"   and "\<And>s. s \<in> S \<Longrightarrow> integrable (measure_pmf (K s)) l2"   and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>N. (s, t) \<in> (SIGMA x:UNIV. set_pmf (K x))\<^sup>*"   and "\<And>s. s \<in> S \<Longrightarrow> measure_pmf.expectation (K s) l1 + c s \<le> l1 s"   and "\<And>s. s \<in> S \<Longrightarrow> l2 s \<le> measure_pmf.expectation (K s) l2 + c s"   and "\<And>s. s \<in> N \<Longrightarrow> l2 s \<le> l1 s"   and "finite ((\<lambda>x. l2 x - l1 x) ` (S \<union> N))" shows "l2 s \<le> l1 s"
perm_rem_commutes: fixes f :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "bij f" shows "perm_rem a (perm_rem b f) = perm_rem b (perm_rem a f)"
mod_add_right_eq: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus a (modulo b c)) c = modulo (plus a b) c"
assms_2: shows "F2 f = map_option Subx.generalize_fundef (F1 f)"
sup_commute: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "semilattice_sup sup less_eq less" shows "sup x y = sup y x"
mult_pred: fixes m :: "nat"   and n :: "nat" shows "(m - Suc 0) * n = m * n - n"
CondF: 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 s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>2 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e\<^sub>1 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>false,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
SFAssRedsNonStatic: fixes e\<^sub>2 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v :: "val"   and h\<^sub>2 :: "heap"   and l\<^sub>2 :: "sfields"   and sh\<^sub>2 :: "sheap"   and b\<^sub>2 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list" assumes "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),b\<^sub>2\<rangle>"   and "P \<turnstile> C has F,NonStatic:t in D" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sF{D} := e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>THROW IncompatibleClassChangeError,(h\<^sub>2, l\<^sub>2, sh\<^sub>2),False\<rangle>"
A_def: shows "A = {x. \<exists>y. q x y = 0}"
weakStepSimAct_def: fixes P :: "pi"   and Rs :: "residual"   and C :: "'a"   and Rel :: "(pi \<times> pi) set" shows "weakStepSimAct P Rs C Rel \<equiv> (\<forall>Q' a x. Rs = a<\<nu>x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l a<\<nu>x> \<prec> P' \<and> (P', Q') \<in> Rel)) \<and> (\<forall>Q' a x. Rs = a<x> \<prec> Q' \<longrightarrow> x \<sharp> C \<longrightarrow> (\<exists>P''. \<forall>u. \<exists>P'. P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P' \<and> (P', Q'[x::=u]) \<in> Rel)) \<and> (\<forall>Q' \<alpha>. Rs = \<alpha> \<prec> Q' \<longrightarrow> (\<exists>P'. P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel))"
simps_6: fixes unop :: "unop"   and e :: "expr" shows "assignsE (UnOp unop e) = assignsE e"
append_rest: shows "(m'', sb'' @ xs, \<O>'', \<R>'', \<S>'') \<rightarrow>\<^sub>f\<^sup>* (m', sb' @ xs, \<O>', \<R>', \<S>')"
skl1_inv5_init: shows "init skl1 \<subseteq> skl1_inv5"
gt_hd_def: fixes \<xi> :: "('s, 'v) hd"   and \<zeta> :: "('s, 'v) hd" shows "(\<xi> >\<^sub>h\<^sub>d \<zeta>) = (\<forall>g\<in>local.ground_heads \<xi>. \<forall>f\<in>local.ground_heads \<zeta>. g >\<^sub>s f)"
arrI: fixes f :: "('a, 'a set) Maps.arr" assumes "f \<noteq> Maps.Null"   and "Maps.Dom f \<in> Collect obj"   and "Maps.Cod f \<in> Collect obj"   and "Maps.Map f \<in> Maps.Hom (Maps.Dom f) (Maps.Cod f)" shows "Maps.arr f"
map_cong: fixes x :: "'a llist"   and ya :: "'a llist"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = ya"   and "\<And>z. z \<in> lset ya \<Longrightarrow> f z = g z" shows "lmap f x = lmap g ya"
f_correct: fixes as :: "'b negation_type list" assumes "f as = None" shows "\<not> matches \<gamma> (alist_and (NegPos_map C as)) a p"
square_free_factorization_prod_listI: fixes p :: "'a poly"   and c :: "'a"   and bs1 :: "('a poly \<times> nat) list"   and bs :: "'a poly list"   and i :: "nat"   and bs2 :: "('a poly \<times> nat) list" assumes "square_free_factorization p (c, bs1 @ (prod_list bs, i) # bs2)"   and "\<And>b. b \<in> set bs \<Longrightarrow> 0 < degree b" shows "square_free_factorization p (c, bs1 @ map (\<lambda>b. (b, i)) bs @ bs2)"
starfun_ln_less_zero: fixes x :: "real star" assumes "0 < x"   and "x < 1" shows "(*f* real_ln) x < 0"
comp_ide_self: fixes a :: "'c" assumes "C.ide a" shows "a \<cdot> a = a"
OclForall_rep_set_true: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and P :: "('d state \<times> 'd state \<Rightarrow> 'e) \<Rightarrow> 'd state \<times> 'd state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> \<delta> X" shows "(\<tau> \<Turnstile> UML_Set.OclForall X P) = (\<forall>x\<in>\<lceil>\<lceil>Rep_Set\<^sub>b\<^sub>a\<^sub>s\<^sub>e (X \<tau>)\<rceil>\<rceil>. \<tau> \<Turnstile> P (\<lambda>\<tau>. x))"
stream_space_single_def: fixes X :: "'d \<Rightarrow> 'c stream"   and r :: "'c stream" shows "stream_space_single X r = (if \<exists>U. U \<in> sets (stream_space borel) \<and> U \<inter> range X = {r} then SOME U. U \<in> sets (stream_space borel) \<and> U \<inter> range X = {r} else {})"
classes_translate: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "exe_osig_conds oss" shows "subclass (translate_osig oss) = set (fst oss)"
case_None: assumes "simple_match_and m1 m2 = None" shows "\<not> (simple_matches m1 p \<and> simple_matches m2 p)"
simps_2: fixes f :: "'b" shows "\<lbrace>\<^bold>\<langle>f\<^bold>\<rangle>\<rbrace> = (if D.arr f then f else D.null)"
ide_def: fixes a :: "'a arrow_of_spans_data" shows "V.ide a \<equiv> a \<bullet> a \<noteq> V.null \<and> (\<forall>f. (f \<bullet> a \<noteq> V.null \<longrightarrow> f \<bullet> a = f) \<and> (a \<bullet> f \<noteq> V.null \<longrightarrow> a \<bullet> f = f))"
eval_Lunit: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = D.lunit \<lbrace>Cod t\<rbrace> \<cdot>\<^sub>D D.tensor D.unity \<lbrace>t\<rbrace>"
rintmod_def: fixes x :: "real"   and y :: "real" shows "rintmod x y = real_of_int (\<lfloor>x\<rfloor> mod \<lfloor>y\<rfloor>)"
packet_assume_new: fixes p :: "('a, 'b) tagged_packet_scheme"   and \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and rs :: "'a common_primitive rule list"   and s :: "state" assumes "match_tcp_flags ipt_tcp_syn (p_tcp_flags (p::('a, 'b) tagged_packet_scheme))"   and "p_tag_ctstate (p::('a, 'b) tagged_packet_scheme) = CT_New" shows "approximating_bigstep_fun (common_matcher, \<alpha>::action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool) (p::('a, 'b) tagged_packet_scheme) (packet_assume_new (rs::'a common_primitive rule list)) (s::state) = approximating_bigstep_fun (common_matcher, \<alpha>) p rs s"
mset_single_iff_right: fixes x :: "'b"   and xs :: "'b list" shows "({#x#} = mset xs) = (xs = [x])"
C1_differentiable_compose: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> 'b" assumes "f C1_differentiable_on S"   and "g C1_differentiable_on f ` S"   and "\<And>x. finite (S \<inter> f -` {x})" shows "g \<circ> f C1_differentiable_on S"
kD_update_unchanged: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "rt = update rt dip (dsn, dsk, flag, hops, nhip)" shows "dip \<in> kD rt"
thm_cont_prop_2: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[Contingent F \<^bold>\<equiv> \<^bold>\<diamond>(\<^bold>\<exists>x. \<lparr>F,x\<^sup>P\<rparr>) \<^bold>& \<^bold>\<diamond>(\<^bold>\<exists>x. \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr>) in v]"
set_map_2: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'a"   and f3 :: "'f \<Rightarrow> 'c"   and v :: "('d, 'e, 'f) val" shows "LID\<^sub>V (map_val f1 f2 f3 v) = f2 ` LID\<^sub>V v"
closure_of_UNIV: fixes X :: "'c topology" shows "X closure_of UNIV = topspace X"
lin_indpt_take_cols_id: fixes inds :: "nat list" shows "lin_indpt (set (cols (take_cols (1\<^sub>m n) inds)))"
converse_rtranclp_induct2: fixes r :: "'a \<times> 'b \<Rightarrow> 'a \<times> 'b \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and bx :: "'a"   and "by" :: "'b"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay) (bx, by)"   and "P bx by"   and "\<And>a b aa ba. \<lbrakk>r (a, b) (aa, ba); r\<^sup>*\<^sup>* (aa, ba) (bx, by); P aa ba\<rbrakk> \<Longrightarrow> P a b" shows "P ax ay"
system_automaton_simps_3: fixes prog :: "program" shows "li.sa.V0 prog = li.init prog"
mh: fixes i :: "nat" assumes "Suc i \<le> Suc n" shows "w !! i = x !! i"
size_2: fixes x :: "'a list" shows "size (IArray x) = List.length x + Suc 0"
of_bl_drop: fixes n :: "nat"   and xs :: "bool list" shows "of_bl (drop n xs) = of_bl xs AND mask (length xs - n)"
hom_power: fixes x :: "'a"   and n :: "nat" shows "[:x ^ n:] = [:x:] ^ n"
measurable_distr_eq2: fixes Mg' :: "'a measure"   and Mg :: "'c measure"   and Ng :: "'b measure"   and g :: "'c \<Rightarrow> 'b" shows "Mg' \<rightarrow>\<^sub>M distr Mg Ng g = Mg' \<rightarrow>\<^sub>M Ng"
split_asm: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd llist \<Rightarrow> 'c"   and llist :: "'d llist" shows "P (case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = (\<not> (llist = LNil \<and> \<not> P f1 \<or> (\<exists>x21 x22. llist = LCons x21 x22 \<and> \<not> P (f2 x21 x22))))"
extendContain: fixes r :: "'a sequent list \<times> 'a sequent"   and ps :: "'a sequent list"   and c :: "'a sequent"   and Ps :: "'a sequent list"   and C :: "'a sequent"   and S :: "'a sequent"   and p :: "'a sequent" assumes "r = (ps, c)"   and "(Ps, C) = extendRule S r"   and "p \<in> set ps" shows "extend S p \<in> set Ps"
asymI: fixes R :: "('b \<times> 'b) set" assumes "\<And>a b. (a, b) \<in> R \<Longrightarrow> (b, a) \<notin> R" shows "asym R"
comp_fun_commute_axioms: shows "comp_fun_commute (\<^bold>*)"
proj2_line_rep_non_zero: fixes l :: "proj2_line" shows "proj2_line_rep l \<noteq> 0"
alt_fail2: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and fail :: "'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "'b" assumes "monad_fail_alt return bind fail alt" shows "alt m fail = m"
refl_onPD1: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "refl_onP A r"   and "r a b" shows "a \<in> A"
prems_1: shows "avl (MKT n l (MKT rn rl rr rh) h)"
finprod_closed: fixes f :: "'c \<Rightarrow> 'a vec"   and A :: "'c set" assumes "f \<in> A \<rightarrow> carrier_vec n" shows "finsum V f A \<in> carrier_vec n"
Max_ge_iff: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>s signed.Max A) = (\<exists>a\<in>A. x \<le>s a)"
wf_ps_chain_def: shows "wf_ps_chain \<equiv> chain_all wf_ps_triple"
major: shows "final_thread.cond_action_oks final1 s1 t cas"
zhmset_of_enat_eq_\<omega>\<^sub>z_iff: fixes n :: "enat" shows "(zhmset_of_enat n = \<omega>\<^sub>z) = (n = \<infinity>)"
Set_filter_insert: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and A :: "'b set" shows "Set.filter P (insert x A) = (if P x then insert x (Set.filter P A) else Set.filter P A)"
prod_pos: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "\<forall>a\<in>A. less zero (f a)" shows "less zero (comm_monoid_mult.prod times one f A)"
NotActualBoxD: fixes v :: "i"   and \<phi> :: "\<o>" assumes "\<not> [\<^bold>\<A>(\<^bold>\<box>\<phi>) in v]" shows "\<not> [\<^bold>\<box>\<phi> in v]"
extract_None_iff: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "(List.extract P xs = None) = (\<not> (\<exists>x\<in>set xs. P x))"
extreme_bound_iff: fixes A :: "'b set"   and X :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'b" shows "extreme_bound A less_eq X s = (s \<in> A \<and> (\<forall>c\<in>A. (\<forall>x\<in>X. less_eq x c) \<longrightarrow> less_eq s c) \<and> (\<forall>x\<in>X. less_eq x s))"
subst_fm_in_se_fm: fixes A :: "fm"   and k :: "name"   and x :: "tm" assumes "se_fm A" shows "se_fm (A(k::=x))"
Crypt_parts_imp_used: fixes K :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Crypt K X \<in> parts (knows Spy evs)"   and "K \<in> symKeys"   and "evs \<in> set_cr" shows "Key K \<in> used evs"
END_END_before: fixes i :: "'b"   and j :: "'b" assumes "\<I> i"   and "\<I> j" shows "END i \<lless> END j = ((i, j) \<in> b \<union> m \<union> ov \<union> s \<union> d)"
t_add_closed: fixes x :: "'a"   and y :: "'a" shows "\<tt>\<tt> (!x + !y) = !x + !y"
eexp_special_values_iff_3: fixes x :: "ereal" shows "(eexp x = \<infinity>) = (x = \<infinity>)"
mult1: shows "(N, add_mset a M0) \<in> mult1 r"
a_set_attribute_locs_def: fixes element_ptr :: "(_) element_ptr" shows "a_set_attribute_locs element_ptr \<equiv> all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr attrs_update)"
vsv_is_VLambda: fixes r :: "V" assumes "vsv r" shows "VLambda (\<D>\<^sub>\<circ> r) (app r) = r"
inf_assoc: fixes x :: "'a multiset"   and y :: "'a multiset"   and z :: "'a multiset" shows "x \<inter># y \<inter># z = x \<inter># (y \<inter># z)"