hcju/isabelleps huggingface dataset:
abs_add_one_gt_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (plus one (abs x))"
nat_filtration_borel_measurable_integrable: fixes f :: "bool stream \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> borel_measurable (nat_filtration n)" shows "integrable M f"
distinct_17: fixes x21 :: "char list"   and x22 :: "aexp"   and x61 :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> ((char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int) \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)"   and x62 :: "bexp"   and x63 :: "acom" shows "x21 ::= x22 \<noteq> {x61} WHILE x62 DO x63"
DisjunctiveD: fixes x :: "'c \<Rightarrow> 'd"   and X :: "'c set" assumes "x \<in> Apply.Disjunctive" shows "x (Sup X) = Sup (x ` X)"
distincts_orbits_list: fixes as :: "'b list"   and f :: "'b \<Rightarrow> 'b" assumes "distinct as"   and "permutation f" shows "distincts (orbits_list f as)"
refl: fixes A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and D :: "nat \<Rightarrow> nat \<Rightarrow> 'g DBMEntry"   and v :: "'f \<Rightarrow> nat"   and n :: "nat" shows "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>*\<^bsub>v,n\<^esub> \<langle>l, D\<rangle>"
get_dense_char: fixes pf :: "('c, 'd) lifted_pair dense"   and f :: "('c, 'd) phi" shows "Rep_lifted_pair (Rep_dense pf) f = (top, up_filter (get_dense pf f))"
chain_ldropn: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and n :: "nat" assumes "chain R xs"   and "enat n < llength xs" shows "chain R (ldropn n xs)"
drop_update_cancel: fixes n :: "nat"   and m :: "nat"   and xs :: "'g list"   and x :: "'g" assumes "n < m" shows "drop m (xs[n := x]) = drop m xs"
carrier_local_ring_at_def: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "pr_ideal R I addition multiplication zero unit" shows "R \<^bsub>I addition multiplication zero\<^esub> \<equiv> R\<setminus>I \<^sup>\<inverse> R \<^bsub>addition multiplication zero\<^esub>"
SubstForm_imp_subst_fm: fixes v :: "hf"   and e :: "(name, hf) finfun"   and u :: "tm"   and x :: "hf"   and x' :: "hf" assumes "SubstForm v \<lbrakk>\<guillemotleft>u\<guillemotright>\<rbrakk>e x x'"   and "Form x" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e" and "x' = \<lbrakk>\<guillemotleft>A(decode_Var v::=u)\<guillemotright>\<rbrakk>e"
assms_2: shows "InvariantClCharacterization (getCl state) (getC state) (getM state)"
assign_simp: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and s' :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<Rightarrow> s' = (s' = s(x := aval a s))"
not_bet_distincts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Bet A B C" shows "A \<noteq> B \<and> B \<noteq> C"
pCons2_3: shows "polymap (p + q) = polymap p + polymap q"
assms_2: shows "s\<^sub>2 = s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>(l \<mapsto> v), \<E> [VE (CV Unit)]))"
h4: shows "lincomb a A = \<zero>\<^bsub>M\<^esub>"
at_le: fixes s :: "'a set"   and t :: "'a set"   and x :: "'a" assumes "s \<subseteq> t" shows "at x within s \<le> at x within t"
x: shows "x \<in> Collect (root_cond (p, l, r))"
comp_functor_identity: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah" assumes "functor A B F" shows "F \<circ> identity_functor.map A = F"
simps_3: fixes f :: "'c"   and v :: "'c" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<Down> \<^bold>\<langle>v\<^bold>\<rangle> = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> \<^bold>\<langle>v\<^bold>\<rangle>"
hyps_5: shows "lookup_pair (map f xs) (fst (f (k, v))) = snd (f (k, lookup_pair xs k))"
curry_in_hom: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f1 :: "'a"   and F :: "'a \<times> 'b \<Rightarrow> 'c"   and G :: "'a \<times> 'b \<Rightarrow> 'c"   and \<tau> :: "'a \<times> 'b \<Rightarrow> 'c" assumes "currying A1 A2 B"   and "partial_magma.arr A1 f1"   and "natural_transformation (product_category.comp A1 A2) B F G \<tau>" shows "partial_magma.in_hom (concrete_category.COMP (Collect (functor A2 B)) (\<lambda>F G. Collect (natural_transformation A2 B F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map A2 B \<sigma> \<tau>)) (currying.curry A1 A2 B F G \<tau> f1) (currying.curry A1 A2 B F F F (partial_magma.dom A1 f1)) (currying.curry A1 A2 B G G G (partial_magma.cod A1 f1))"
ps_prod_D_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "ps_prod_D x y \<equiv> D (fst x) (fst y) \<and> D (snd x) (snd y)"
ide_dom: fixes f :: "'a1" assumes "A1.arr f" shows "A1.ide (A1.dom f)"
Send: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "send\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (??.local.update\<^sub>s\<^sub>t \<S> (send\<langle>t\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"
inheritable_instance_methd_props: fixes C :: "qtname"   and D :: "qtname"   and G :: "prog"   and sig :: "sig"   and old :: "qtname \<times> methd" assumes "G\<turnstile>C\<preceq>\<^sub>C D"   and "is_class G D"   and "wf_prog G"   and "methd G D sig = Some old"   and "Protected \<le> accmodi old"   and "\<not> is_static old" shows "\<exists>new. methd G C sig = Some new \<and> \<not> is_static new \<and> G\<turnstile>resTy new\<preceq>resTy old \<and> accmodi old \<le> accmodi new"
lo_lemma'': fixes x :: "nat"   and y :: "nat" assumes "\<not> Suc 0 < x" shows "rec_exec rec_lo [x, y] = lo x y"
1: shows "s \<in> V \<and> p = {s} \<union> neighbors s"
cases: fixes P1 :: "'c \<Rightarrow> bool"   and P2 :: "'d \<Rightarrow> bool"   and a :: "'c \<times> 'd" assumes "pred_prod P1 P2 a" obtains aa :: "'c" and b :: "'d" where "a = (aa, b)" and "P1 aa" and "P2 b"
simps_2: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "filter P (x # xs) = (if P x then x # filter P xs else filter P xs)"
left_inverse: fixes a :: "'a" shows "inverse a \<cdot> a = (1::'a)"
residue_of_prod: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and k :: "nat" shows "(a \<otimes> b) k = a k \<otimes>\<^bsub>residue_ring (p ^ k)\<^esub> b k"
measurable_compose: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'b \<Rightarrow> 'c"   and L :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "g \<in> N \<rightarrow>\<^sub>M L" shows "(\<lambda>x. g (f x)) \<in> M \<rightarrow>\<^sub>M L"
main_step: shows "\<exists>a. \<forall>l. 0 < Re (\<Sum>i\<in>I. a i * Lam i ^ l)"
Skip: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "'e" shows "\<Gamma>\<turnstile>Skip \<down> Normal s"
intros_2: fixes \<tau> :: "'a basic_type"   and \<sigma> :: "'a basic_type" assumes "\<tau> \<sqsubset>\<^sub>B \<sigma>" shows "\<tau>[?] \<sqsubset> \<sigma>[?]"
interceptor_def: fixes \<sigma> :: "'grp set"   and x :: "'grp" shows "interceptor \<sigma> x = hash x \<bind> (\<lambda>h. Done (h, {x} \<union> \<sigma>))"
rep_eq: fixes x :: "rat"   and xa :: "'b fpxs" shows "fpxs_nth (fpxs_shift x xa) = (\<lambda>n. fpxs_nth xa (n + x))"
affine_basis_exists: fixes V :: "'a set" shows "\<exists>B\<subseteq>V. \<not> affine_dependent B \<and> affine hull V = affine hull B"
invar_rel_internal: fixes a :: "'h \<Rightarrow> 'g set"   and X :: "('f \<times> 'h) set"   and S :: "('e \<times> 'g set) set" shows "invar_rel a X S = {((x, s'), y). \<exists>s. (s', s) \<in> X \<and> (x, y) \<in> S \<and> y \<subseteq> a s}"
repL_geq: fixes w :: "32 Word.word"   and r :: "real"   and r' :: "real" assumes "w \<equiv>\<^sub>L r"   and "r \<le> r'" shows "w \<equiv>\<^sub>L r'"
mon_env_empty: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme" shows "mon_env fg [] = {}"
cas_with_proj_eq: fixes G :: "'b pair_pre_digraph" shows "pre_digraph.cas (with_proj G) = pcas"
C1: fixes S :: "ccs"   and T :: "ccs"   and U :: "ccs" assumes "(S, T) \<in> Rel" shows "(S \<oplus> U, T) \<in> Rel'"
continuous_oVeblen_0: shows "continuous (\<lambda>x. oVeblen F x 0)"
assms: shows "(\<lambda>(x, y). f x y) abs_summable_on A \<times> B"
sup_state_opt_Some_Some: fixes P :: "'b prog"   and a :: "ty list \<times> ty err list"   and b :: "ty list \<times> ty err list" shows "P \<turnstile> \<lfloor>a\<rfloor> \<le>' \<lfloor>b\<rfloor> = P \<turnstile> a \<le>\<^sub>i b"
heap_WHILET_rule': fixes R :: "('b \<times> 'b) set"   and P :: "assn"   and I :: "'b \<Rightarrow> 'c \<Rightarrow> assn"   and s :: "'b"   and si :: "'c"   and F :: "assn"   and bi :: "'c \<Rightarrow> bool Heap"   and b :: "'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c Heap"   and Q :: "'b \<Rightarrow> 'c \<Rightarrow> assn" assumes "wf R"   and "P \<Longrightarrow>\<^sub>A I s si * F"   and "\<And>si s. <I s si * F> bi si <\<lambda>r. I s si * F * \<up> (r = b s)>\<^sub>t"   and "\<And>si s. b s \<Longrightarrow> <I s si * F> f si <\<lambda>r. \<exists>\<^sub>As'. I s' r * F * \<up> ((s', s) \<in> R)>\<^sub>t"   and "\<And>si s. \<not> b s \<Longrightarrow> I s si * F \<Longrightarrow>\<^sub>A Q s si" shows "<P> heap_WHILET bi f si <\<lambda>r. \<exists>\<^sub>As. Q s r>\<^sub>t"
S1: fixes x :: "i" shows "E x \<^bold>\<leftarrow> E (dom x)"
rep_eq: fixes x :: "('c, 'd) mapping" shows "Mapping.keys x = dom (mapping.rep x)"
assms_5: fixes w :: "'a vec"   and x :: "'a vec" assumes "w \<in> W"   and "x \<in> set Ls" shows "w \<bullet> x = (0::'a)"
self_codomain_iff_ide: fixes a :: "('obj, 'obj) concrete_category.arr" shows "(a \<in> codomains a) = ide a"
compileProg_def: fixes F :: "((Class \<times> Method) \<times> Var list \<times> Expr) list" shows "compileProg F = ((\<forall>C m par e. F\<down>(C, m) = Some (par, e) \<longrightarrow> (\<exists>code l0 l. mbody_is C m (rev par, code, l0) \<and> (l0, [], e, code, l) \<in> compileExpr)) \<and> (\<forall>C m. (\<exists>M. mbody_is C m M) = (\<exists>fdecl. F\<down>(C, m) = Some fdecl)))"
arrI: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "Fop.Bop.in_hom f a b" shows "Fop.Bop.arr f"
analz_image_freshK: fixes evs :: "event list"   and KK :: "nat set"   and K :: "nat" assumes "evs \<in> certified_mail"   and "priEK TTP \<notin> KK" shows "(Key K \<in> analz (Key ` KK \<union> knows Spy evs)) = (K \<in> KK \<or> Key K \<in> analz (knows Spy evs))"
vbrelation_Limit_in_VsetI: fixes \<alpha> :: "V"   and U :: "V"   and f :: "V \<Rightarrow> V" assumes "Limit \<alpha>"   and "\<D>\<^sub>\<circ> (VLambda U f) \<in>\<^sub>\<circ> Vset \<alpha>"   and "\<R>\<^sub>\<circ> (VLambda U f) \<in>\<^sub>\<circ> Vset \<alpha>" shows "VLambda U f \<in>\<^sub>\<circ> Vset \<alpha>"
eqsig: fixes A :: "'a set" assumes "A \<in> sigma_sets (space M) {f -` B \<inter> space M |B. B \<in> sets N}" shows "(x \<in> A) = (w \<in> A)"
distinct_1: fixes x1 :: "'a"   and x2 :: "'a" shows "Assumption x1 \<noteq> Conclusion x2"
indep_var_lebesgue_integral: fixes X1 :: "'a \<Rightarrow> 'b"   and X2 :: "'a \<Rightarrow> 'b" assumes "indep_var borel X1 borel X2"   and "integrable M X1"   and "integrable M X2" shows "expectation (\<lambda>\<omega>. X1 \<omega> * X2 \<omega>) = expectation X1 * expectation X2"
refl: fixes A :: "'e set"   and h :: "'e \<Rightarrow> 'f"   and bo :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "blinding_of_on A h bo"   and "x \<in> A" shows "bo x x"
no_pd_path: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a"   and y :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob"   and "\<not> IFC_def.is_pd prob y x" obtains \<pi> :: "nat \<Rightarrow> 'a" and n :: "nat" where "IFC_def.is_path prob \<pi>" and "\<pi> 0 = x" and "\<pi> n = IFC_def.return prob" and "\<forall>k\<le>n. \<pi> k \<noteq> y"
valid_extractable: fixes I :: "msg set"   and K :: "msg set" assumes "I \<subseteq> valid" shows "extractable K I \<subseteq> valid"
fall_elim: fixes F :: "('state \<times> 'aprop set) stream \<Rightarrow> ('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "fall F \<pi>l" obtains "\<And>\<pi>. \<lbrakk>wfp AP' \<pi>; \<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l); \<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0\<rbrakk> \<Longrightarrow> F \<pi> \<pi>l"
app1: fixes env :: "v sem_env"   and es :: "exp0 list"   and rs :: "(v, v) result list"   and vs :: "v list"   and env' :: "v sem_env"   and e :: "exp0"   and bv :: "(v, v) result" assumes "list_all2_shortcircuit (cupcake_evaluate_single env) (rev es) rs"   and "sequence_result rs = Rval vs"   and "do_opapp (rev vs) = Some (env', e)"   and "cupcake_evaluate_single env' e bv" shows "cupcake_evaluate_single env (App Opapp es) bv"
limit_cone_some_limit_cone: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b" assumes "diagram J C D"   and "\<exists>a \<chi>. limit_cone J C D a \<chi>" shows "limit_cone J C D (diagram.some_limit J C D) (diagram.some_limit_cone J C D)"
gis_on_stack_code_def: fixes node_eq_impl :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and node_hash_impl :: "nat \<Rightarrow> 'c \<Rightarrow> nat" shows "gis_on_stack_code node_eq_impl node_hash_impl \<equiv> \<lambda>x xa. case Impl_Array_Hash_Map.ahm_lookup node_eq_impl node_hash_impl x (case snd xa of (xb, xc, xd, xe) \<Rightarrow> xd) of None \<Rightarrow> False | Some i \<Rightarrow> if 0 \<le> i then True else False"
matrix_left_invertible_ker: fixes A :: "(('a, 'c) vec, 'b) vec" shows "(\<exists>B. B ** A = mat (1::'a)) = (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
BitM_inc_eq: fixes n :: "num" shows "Num.BitM (Num.inc n) = num.Bit1 n"
fmap_induct3: fixes F1 :: "'a \<Rightarrow> 'b option"   and F2 :: "'a \<Rightarrow> 'b option"   and F3 :: "'a \<Rightarrow> 'b option"   and P :: "('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> ('a \<Rightarrow> 'b option) \<Rightarrow> bool" assumes "dom F1 = dom F2"   and "dom F3 = dom F1"   and "P Map.empty Map.empty Map.empty"   and "\<And>x a b c F1 F2 F3. \<lbrakk>P F1 F2 F3; dom F1 = dom F2; dom F3 = dom F1; x \<notin> dom F1\<rbrakk> \<Longrightarrow> P (F1(x \<mapsto> a)) (F2(x \<mapsto> b)) (F3(x \<mapsto> c))" shows "P F1 F2 F3"
Cod_x: shows "J_C.Cod x = J_C.Map d"
has_ivl_integral_ivl_integral: fixes f :: "real \<Rightarrow> 'a"   and a :: "real"   and b :: "real" shows "(f integrable_on closed_segment a b) = (f has_ivl_integral ivl_integral a b f) a b"
simps_14: fixes Vs :: "char list list"   and e :: "char list exp" shows "compE\<^sub>1 Vs (throw e) = throw (compE\<^sub>1 Vs e)"
inject_1: fixes x1 :: "('b ann_com option \<times> 'b set) list"   and y1 :: "('b ann_com option \<times> 'b set) list" shows "(Parallel x1 = Parallel y1) = (x1 = y1)"
n_anti: fixes x :: "'a"   and y :: "'a" assumes "!x \<le> !y" shows "t y \<le> t x"
minus_mod_eq_div_mult: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_modulo plus minus zero times one divide modulo" shows "minus a (modulo a b) = times (divide a b) b"
wfzf_implies_wf: fixes R :: "(ZF \<times> ZF) set" assumes "wfzf R" shows "wf R"
rec_lg_def: shows "rec_lg = (let rec_lgR = Cn 3 rec_le [Cn 3 rec_power [recf.id 3 1, recf.id 3 2], recf.id 3 0]; conR1 = Cn 2 rec_conj [Cn 2 rec_less [Cn 2 (constn 1) [recf.id 2 0], recf.id 2 0], Cn 2 rec_less [Cn 2 (constn 1) [recf.id 2 0], recf.id 2 1]]; conR2 = Cn 2 rec_not [conR1] in Cn 2 rec_add [Cn 2 rec_mult [conR1, Cn 2 (rec_maxr rec_lgR) [recf.id 2 0, recf.id 2 1, recf.id 2 0]], Cn 2 rec_mult [conR2, Cn 2 (constn 0) [recf.id 2 0]]])"
1: shows "rawpsubstT (rawpsubstT r1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = rawpsubstT (rawpsubstT r1 (zip (map Var vs1) (map snd txs))) (zip (map fst txs) vs1)"
sum_mult_scalar_distrib_right: fixes f :: "'d \<Rightarrow> 'a \<Rightarrow>\<^sub>0 'c"   and A :: "'d set"   and v :: "'t \<Rightarrow>\<^sub>0 'c" shows "sum f A \<odot> v = (\<Sum>a\<in>A. f a \<odot> v)"
le_trans: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "c \<in> carrier D"   and "a \<preceq>\<^bsub>D\<^esub> b"   and "b \<preceq>\<^bsub>D\<^esub> c" shows "a \<preceq>\<^bsub>D\<^esub> c"
assms_4: shows "(ir, Rem is e2) \<in> set (node_deliver_messages xs)"
Anno_cases_12: fixes is_lub' :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'b expr"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'b expr" assumes "is_lub',P,E \<turnstile> e\<bullet>F{D} \<leadsto> e'" obtains e'a :: "'b expr" and U :: "ty" and C :: "String.literal" and T :: "ty" and fm :: "fmod" and Da :: "String.literal" where "D = STR ''''" and "e' = e'a__\<bullet>F{Da__}" and "is_lub',P,E \<turnstile> e \<leadsto> e'a__" and "is_lub',P,E \<turnstile> e'a__ :: U" and "class_type_of' U = \<lfloor>C__\<rfloor>" and "P \<turnstile> C__ sees F:T__ (fm__) in Da__" and "(\<exists>Ua. U = Ua\<lfloor>\<rceil>) \<longrightarrow> F \<noteq> length"   | e'a :: "'b expr" and T :: "ty" where "e' = e'a\<bullet>length" and "is_lub',P,E \<turnstile> e \<leadsto> e'a" and "is_lub',P,E \<turnstile> e'a :: Ta__\<lfloor>\<rceil>" and "F = length" and "D = STR ''''"   | C :: "String.literal" and Da :: "String.literal" and fs :: "fdecl list" and ms :: "(String.literal list \<times> 'b expr) option mdecl list" and T :: "ty" and fm :: "fmod" and D' :: "String.literal" where "e' = Cast (Class Da) (Var this)\<bullet>F{D'}" and "E this = \<lfloor>Class C\<rfloor>" and "C \<noteq> Object" and "class P C = \<lfloor>(Da, fs, ms)\<rfloor>" and "P \<turnstile> Da sees F:T (fm) in D'" and "e = Var Annotate.super" and "D = STR ''''"
sifum_security_axioms: shows "sifum_security Stop eval\<^sub>w some_val some_val'"
right_unit: fixes a :: "'b" assumes "a \<in> f ` A" shows "addB a zeroB = a"
IH2: fixes s :: "'a net_state"   and s' :: "'a net_state" assumes "s \<in> reachable (pnet np p2) TT"   and "(s, disconnect(i, i'), s') \<in> automaton.trans (pnet np p2)" shows "netmap s' = netmap s"
hom_uminus: fixes x :: "'a" shows "[:- x:] = - [:x:]"
zero_prod_root_unit: fixes ns :: "nat list" shows "(prod_root_unity ns = 0) = (0 \<in> set ns)"
S\<^sub>0_def: shows "rs.S\<^sub>0 = \<lbrakk>\<lbrakk>tab\<^sub>0 s\<rbrakk>\<rbrakk>"
step_2: fixes shadow_root :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr"   and ancestors :: "(_) object_ptr list" assumes "child = cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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 shadow_root"   and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host"   and "h \<turnstile> get_ancestors_si (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^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 host) \<rightarrow>\<^sub>r ancestors"   and "ancestor \<in> set ancestors" shows "set ancestor_ancestors \<subseteq> set ancestors"
consistentE: fixes prognosis :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<Rightarrow> Arity list \<Rightarrow> Arity \<Rightarrow> (Vars.var \<times> exp) list \<times> exp \<times> stack_elem list \<Rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and cHeap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list \<times> Vars.var list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "CardinalityPrognosisSafe prognosis cHeap Aheap Aexp"   and "CardinalityPrognosisSafe.consistent prognosis Aexp (ae, ce, a, as) (\<Gamma>, e, S)" obtains as :: "Arity list" and r :: "Vars.var list" where "asb__ = (as, r)" and "ArityAnalysisLetSafe.a_consistent Aexp (ae, a, as) (restrictA (- set r) \<Gamma>, e, restr_stack (- set r) S)" and "edom ae = edom ce" and "prognosis ae as a (\<Gamma>, e, S) \<sqsubseteq> ce" and "\<And>x. \<lbrakk>x \<in> thunks \<Gamma>; many \<sqsubseteq> ce x\<rbrakk> \<Longrightarrow> ae x = up\<cdot>0" and "set r \<subseteq> domA \<Gamma> \<union> upds S - edom ce"
Par2B: fixes Q :: "pi"   and a :: "subject"   and x :: "name"   and Q' :: "pi"   and P :: "pi" assumes "Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> Q'"   and "x \<sharp> P"   and "x \<sharp> Q"   and "x \<sharp> a" shows "P \<parallel> Q \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P \<parallel> Q'"
Suc_less_le: fixes x :: "nat"   and y :: "nat"   and n :: "nat" assumes "x < y - n" shows "x \<le> y - Suc n"
sasp_op_to_strips_effect_consistent_2: fixes op :: "('c \<times> 'd) strips_operator"   and \<Psi> :: "('c, 'd) sas_plus_problem"   and op' :: "('c, 'd) sas_plus_operator"   and v :: "'c"   and a :: "'d" assumes "op = \<phi>\<^sub>O \<Psi> op'"   and "op' \<in> set (\<Psi>\<^sub>\<O>\<^sub>+)"   and "is_valid_operator_sas_plus \<Psi> op'" shows "(v, a) \<in> set (delete_effects_of op) \<longrightarrow> (v, a) \<notin> set (add_effects_of op)"
isCont_artanh: fixes x :: "real" assumes "- 1 < x"   and "x < 1" shows "isCont artanh x"
emeasure_space_1: fixes M :: "'b measure" assumes "prob_space M" shows "emeasure M (space M) = 1"
size_4: fixes x1 :: "nat" shows "size (Input x1) = 0"
l1_inv6D: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and x :: "l1_state"   and gnx :: "msg"   and Rb :: "fid_t" assumes "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "in_progressS (progress x Ra) {xnx, xgnx, xgny, xsk, xEnd}"   and "guessed_frame Ra xgnx = Some gnx"   and "guessed_frame Ra xgny = Some (Exp Gen (NonceF (Rb $ ny)))"   and "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr>"   and "progress x Rb = Some {xny, xgnx, xgny, xsk}"   and "can_signal x A B"   and "x \<in> l1_inv6" shows "signalsResp x (Commit A B (Exp gnx (NonceF (Rb $ ny)))) < signalsResp x (Running A B (Exp gnx (NonceF (Rb $ ny))))"
iso_verts_head: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and e :: "'d" assumes "iG.digraph_isomorphism hom"   and "e \<in> arcs (app_iso hom G)" shows "iso_head hom (iso_arcs hom e) = iso_verts hom (head (app_iso hom G) e)"