hcju/isabelleps huggingface dataset:
swap_cs_Recv_Snapshot: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev d"   and "distributed_system.next channel trans send recv d ev' e"   and "isRecv ev"   and "isSnapshot ev'"   and "distributed_system.next channel trans send recv c ev' d'"   and "distributed_system.next channel trans send recv d' ev e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "cs e i = cs e' i"
s: shows "(\<Sum>i\<le>m - k. p i * i) = m - k"
lemma_1_reachability_s: fixes s :: "('a, bool) fmap"   and PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and as :: "(('a, bool) fmap \<times> ('a, bool) fmap) list" assumes "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB" shows "last ` state_set (state_list s as) \<subseteq> reachable_s PROB s"
div_Tr3: fixes r :: "nat"   and s :: "nat"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "l < s * r" shows "Suc (l div s) \<le> r"
ring_mat_def: fixes ty :: "'a itself"   and n :: "nat"   and b :: "'b" shows "ring_mat ty n b \<equiv> \<lparr>carrier = carrier_mat n n, mult = (*), one = 1\<^sub>m n, zero = 0\<^sub>m n n, add = (+), \<dots> = b\<rparr>"
varDefs_names_def: shows "varDefs_names = map vdName"
right_closed: fixes a :: "'a"   and b :: "'a" assumes "(a, b) \<in> E(\<eta>)" shows "b \<in> M"
winning_path_ltl: fixes p :: "Player"   and P :: "'a llist" assumes "winning_path p P"   and "\<not> lnull P"   and "\<not> lnull (ltl P)" shows "winning_path p (ltl P)"
between_trivial: 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" assumes "Tarski_neutral_dimensionless Bet Cong" shows "Bet A B B"
is_ortho_set_antimono: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B"   and "is_ortho_set B" shows "is_ortho_set A"
spec1_assoc: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point"   and p\<^sub>3 :: "'b point" assumes "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3"   and "is_generic p\<^sub>1 p\<^sub>2"   and "is_generic p\<^sub>2 p\<^sub>3"   and "is_generic (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"   and "is_generic p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3)" shows "ell_field_class.add a p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3) = ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
mem_upI: fixes f :: "nat \<Rightarrow> 'a"   and R :: "('a, 'b) ring_scheme" assumes "\<And>n. f n \<in> carrier R"   and "\<exists>n. bound \<zero>\<^bsub>R\<^esub> n f" shows "f \<in> up R"
arrI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "C.arr f"
cnj_mult: fixes x :: "quat"   and y :: "quat" shows "Quaternions.cnj (x * y) = Quaternions.cnj y * Quaternions.cnj x"
Lp_domination_2: fixes g :: "'b \<Rightarrow> real"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and p :: "ennreal" assumes "g \<in> borel_measurable M"   and "f \<in> space\<^sub>N (\<LL> p M)"   and "AE x in M. \<bar>g x\<bar> \<le> \<bar>f x\<bar>" shows "Norm (\<LL> p M) g \<le> Norm (\<LL> p M) f"
lrev_work2_lrev_work1_eq: shows "lrev_work2 = lrev_work1"
simps_9: fixes e :: "char list exp"   and M :: "char list"   and es :: "char list exp list" shows "fv (e\<bullet>M(es)) = fv e \<union> fvs es"
rel_symp: fixes R1 :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and R2 :: "'h \<Rightarrow> 'h \<Rightarrow> bool" assumes "symp R1"   and "symp R2" shows "symp (tllist_all2 R1 R2)"
simps_3: fixes \<nu> :: "'c"   and u :: "'c local.term" shows "Nml (\<^bold>\<langle>\<nu>\<^bold>\<rangle> \<^bold>\<star> u) = (C.arr \<nu> \<and> Nml u \<and> \<not> is_Prim\<^sub>0 u \<and> \<^bold>\<langle>src\<^sub>C \<nu>\<^bold>\<rangle>\<^sub>0 = Trg u)"
invertible_column_add: fixes a :: "'a"   and b :: "'a"   and q :: "'b" assumes "a \<noteq> b" shows "invertible (column_add (mat (1::'b)) a b q)"
weakly_preserves_hcomp: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "F f \<star>\<^sub>D F g \<cong>\<^sub>D F (f \<star>\<^sub>C g)"
rep_eq: fixes x :: "integer"   and xa :: "integer" shows "int_of_integer (lcm x xa) = lcm (int_of_integer x) (int_of_integer xa)"
cases: fixes a1 :: "nat \<Rightarrow> classDef option"   and a2 :: "nat \<Rightarrow> nat option"   and a3 :: "exp"   and a4 :: "nat" assumes "a1;a2 \<turnstile> a3 : a4" obtains \<Gamma> :: "nat \<Rightarrow> nat option" and x :: "nat" and C :: "nat" and CT :: "nat \<Rightarrow> classDef option" where "a1 = CTa__" and "a2 = \<Gamma>__" and "a3 = Var x" and "a4 = Ca__" and "\<Gamma>__ x = Some Ca__"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and C0 :: "nat" and Cf :: "varDef list" and fi :: "nat" and fDef :: "varDef" and Ci :: "nat" where "a1 = CTb__" and "a2 = \<Gamma>'__" and "a3 = FieldProj e0__ fi" and "a4 = Ci" and "CTb__;\<Gamma>'__ \<turnstile> e0__ : C0__" and "fields(CTb__,C0__) = Cf" and "lookup Cf (\<lambda>fd. vdName fd = fi) = Some fDef" and "vdType fDef = Ci"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and C0 :: "nat" and m :: "nat" and Ds :: "nat list" and C :: "nat" and es :: "exp list" and Cs :: "nat list" where "a1 = CTc__" and "a2 = \<Gamma>''__" and "a3 = MethodInvk e0a__ m es__" and "a4 = Cb__" and "CTc__;\<Gamma>''__ \<turnstile> e0a__ : C0" and "mtype(CTc__,m,C0) = Ds__ \<rightarrow> Cb__" and "CTc__;\<Gamma>''__ \<turnstile>+ es__ : Cs__" and "CTc__ \<turnstile>+ Cs__ <: Ds__" and "length es__ = length Ds__"   | CT :: "nat \<Rightarrow> classDef option" and C :: "nat" and Df :: "varDef list" and es :: "exp list" and Ds :: "nat list" and \<Gamma> :: "nat \<Rightarrow> nat option" and Cs :: "nat list" where "a1 = CTd__" and "a2 = \<Gamma>'''__" and "a3 = New Cc__ es" and "a4 = Cc__" and "fields(CTd__,Cc__) = Df" and "length es = length Df" and "varDefs_types Df = Ds" and "CTd__;\<Gamma>'''__ \<turnstile>+ es : Cs" and "CTd__ \<turnstile>+ Cs <: Ds"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and D :: "nat" and C :: "nat" where "a1 = CTe__" and "a2 = \<Gamma>''''__" and "a3 = Cast Cd__ e0b__" and "a4 = Cd__" and "CTe__;\<Gamma>''''__ \<turnstile> e0b__ : D__" and "CTe__ \<turnstile> D__ <: Cd__"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and D :: "nat" and C :: "nat" where "a1 = CTf__" and "a2 = \<Gamma>'''''__" and "a3 = Cast Ce__ e0c__" and "a4 = Ce__" and "CTf__;\<Gamma>'''''__ \<turnstile> e0c__ : Da__" and "CTf__ \<turnstile> Ce__ <: Da__" and "Ce__ \<noteq> Da__"   | CT :: "nat \<Rightarrow> classDef option" and \<Gamma> :: "nat \<Rightarrow> nat option" and e0 :: "exp" and D :: "nat" and C :: "nat" where "a1 = CT" and "a2 = \<Gamma>" and "a3 = Cast C e0" and "a4 = C" and "CT;\<Gamma> \<turnstile> e0 : D" and "CT \<turnstile> C \<not><: D" and "CT \<turnstile> D \<not><: C"
A: shows "{} \<turnstile> \<alpha>1 IMP \<alpha>2 IMP \<beta>"
cases: fixes x :: "'d DBMEntry" obtains d :: "'d" where "x = Le d__"   | d :: "'d" where "x = Lt d"   | "x = \<infinity>"
scopeExtPar': fixes x :: "name"   and Q :: "pi"   and P :: "pi" assumes "x \<sharp> Q" shows "<\<nu>x>(P \<parallel> Q) \<sim>\<^sub>l <\<nu>x>P \<parallel> Q"
simps_1: fixes uu :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "distinct_on uu [] = True"
index_zero_implies_one_group: fixes ps :: "'a set" assumes "ps \<subseteq> \<V>"   and "card ps = 2"   and "\<B> index ps = 0" shows "size (filter_mset ((\<subseteq>) ps) (mset_set \<G>)) = 1"
l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order_si Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs get_shadow_root get_shadow_root_locs ShadowRootClass.type_wf ShadowRootClass.known_ptr"
p1: shows "directly_subsumes e1 e2 s1 s2 t1 t2"
simps_1: shows "bool_to_ternary True = TernaryTrue"
assms_1: shows "is_tangle_diagram (w1 * ws1)"
bisim_invD1: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s1' :: "'x1 \<times> 'm1" assumes "bisim_inv"   and "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s1 -1-ta1\<rightarrow> s1'" shows "\<exists>s2'. t \<turnstile> s1' \<approx> s2'"
horizontal_homs_axioms: shows "horizontal_homs (\<cdot>\<^sub>C) src\<^sub>C trg\<^sub>C"
\<pi>_def: fixes k :: "nat"   and T :: "(symbol \<times> character list) set"   and I :: "item set" shows "\<pi> k T I = limit (\<lambda>I. Scan T k (Complete k (Predict k I))) I"
cd_iso: fixes x :: "'a" shows "d (x \<cdot> 1\<^sub>\<pi>) \<cdot> 1\<^sub>\<pi> = x \<cdot> 1\<^sub>\<pi>"
Free: fixes \<alpha> :: "freeRes"   and Q' :: "pi" assumes "Q \<longmapsto> \<alpha> \<prec> Q'" shows "\<exists>P'. P \<longmapsto> \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
compact_Affine: fixes X :: "'a \<times> 'a pdevs" shows "compact (Affine X)"
exec_move_AAssI3: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e3 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and v' :: "'addr val"   and v :: "'addr val" assumes "exec_move ci P t e3 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1\<lfloor>e2\<rceil> := e3) h (stk @ [v', v], loc, length (compE2 e1) + length (compE2 e2) + pc, xcp) ta h' (stk' @ [v', v], loc', length (compE2 e1) + length (compE2 e2) + pc', xcp')"
exec_gpv_parametric: fixes S :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and CALL :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and A :: "'h \<Rightarrow> 'i \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun CALL (rel_spmf (rel_prod (=) S)))) (rel_fun (rel_gpv A CALL) (rel_fun S (rel_spmf (rel_prod A S)))) exec_gpv exec_gpv"
Inverse_relI: fixes f :: "'b"   and g :: "'b" assumes "f \<approx>> g"   and "f ;; g = id (dom f)"   and "g ;; f = id (cod f)" shows "cinv f g"
dual_inv_inj: shows "inj \<partial>\<^sup>-"
assms: shows "\<exists>x. x \<in> s \<and> x < k"
poly_iff: fixes t :: "real" shows "(poly g t = 0) = (poly pc (complex_of_real t) = 0)"
eq2: shows "post * (1\<^sub>m N - post) = 0\<^sub>m N N"
jkbpC_def: shows "SPR.jkbpC \<equiv> \<Union> (range SPR.jkbpCn)"
comp_injective_below_complement: fixes y :: "'a"   and x :: "'a" assumes "y \<sqinter> id y \<le> top" shows "- x \<sqinter> y \<le> - (x \<sqinter> y)"
isRefV_simps_6: fixes T :: "Arraytype"   and a :: "nat" shows "isRefV (arrV T a) = True"
project_leadsTo_D: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "F \<squnion> project h UNIV G \<in> A \<longmapsto> B" shows "extend h F \<squnion> G \<in> extend_set h A \<longmapsto> extend_set h B"
single_zero: fixes k :: "'c" shows "monomial (0::'d) k = 0"
set_intros_1: fixes x21 :: "'e"   and x22 :: "'e list" shows "x21 \<in> set (x21 # x22)"
count_repeat_mset: fixes i :: "nat"   and A :: "'b multiset"   and a :: "'b" shows "count (repeat_mset i A) a = i * count A a"
primerew: fixes m :: "nat"   and p :: "nat" assumes "m dvd p"   and "m \<noteq> 1"   and "m \<noteq> p" shows "\<not> prime p"
classes_changed_cons: fixes t :: "'b cdecl"   and P :: "'b prog"   and P' :: "'b prog" shows "classes_changed (t # P) (t # P') = classes_changed P P' - {fst t}"
relpow_fun_conv: fixes a :: "'c"   and b :: "'c"   and n :: "nat"   and R :: "('c \<times> 'c) set" shows "((a, b) \<in> R ^^ n) = (\<exists>f. f 0 = a \<and> f n = b \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> R))"
dvd: shows "\<Prod>\<^sub># A dvd \<Prod>\<^sub># B * \<Prod>\<^sub># C"
comp_null_2: fixes f :: "'a" shows "A f A.null = A.null"
simps_1: fixes a :: "'f"   and xs :: "'h list" shows "lookup (Val a) xs = (if xs = [] then Some (Val a) else None)"
ws_class_induct'': fixes G :: "prog"   and C :: "qtname"   and c :: "class"   and P :: "qtname \<Rightarrow> class \<Rightarrow> bool" assumes "class G C = Some c"   and "ws_prog G"   and "\<And>co. class G Object = Some co \<Longrightarrow> P Object co"   and "\<And>C c sc. \<lbrakk>class G C = Some c; class G (super c) = Some sc; C \<noteq> Object; P (super c) sc\<rbrakk> \<Longrightarrow> P C c" shows "P C c"
m1_trans_def: shows "m1_trans \<equiv> \<Union>A B Ra Rb Rs Na Nb Kab. (m1_step1 Ra A B Na \<union> m1_step2 Rb A B \<union> m1_step3 Rs A B Na Kab \<union> m1_step4 Ra A B Na Kab \<union> m1_step5 Rb A B Nb Kab \<union> m1_step6 Ra A B Na Nb Kab \<union> m1_step7 Rb A B Nb Kab \<union> m1_leak Rs Ra Rb A B)\<^sup>="
2: shows "tfr\<^sub>s\<^sub>e\<^sub>t (trms\<^sub>l\<^sub>s\<^sub>t \<A>)"
add_to_nat_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "plus x y = mod_type.from_nat Abs (mod_type.to_nat Rep x + mod_type.to_nat Rep y)"
elims_2: fixes x :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and xa :: "acom"   and xb :: "'a \<Rightarrow> nat"   and xc :: "char list \<Rightarrow> int" assumes "preList x xa xb xc" obtains C :: "acom" and l :: "'a \<Rightarrow> nat" and s :: "char list \<Rightarrow> int" where "x = []" and "xa = Ca__" and "xb = la__" and "xc = sa__"   | xd :: "'a" and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" and xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list" and C :: "acom" and l :: "'a \<Rightarrow> nat" and s :: "char list \<Rightarrow> int" where "x = (xd, e) # xs" and "xa = C" and "xb = l" and "xc = s" and "l xd = preT C e s \<and> preList xs C l s"
weakBisimulationE_1: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<approx> Q" shows "P \<leadsto>\<^sup>^<weakBisimulation> Q"
\<eta>_in_terms_of_\<phi>: fixes y :: "'b" assumes "Adj.D.ide y" shows "Adj.\<eta> y = Adj.\<eta>\<epsilon>.\<phi> y (F y)"
safe: shows "safe (s\<lparr>isin := (isin s)(r1 := isin s r1 \<union> {g1}), roomk := (roomk s)(r1 := k2), safe := (safe s)(r1 := owns s r1 = \<lfloor>g1\<rfloor> \<and> isin s r1 = {} \<and> k2 = currk s r1 \<or> safe s r1)\<rparr>) r"
back_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat" shows "back cl n \<equiv> (`) (\<lambda>ii. ii - brnL cl n)"
ex_match: fixes n :: "nat"   and ts :: "'a MFOTL.trm list"   and v :: "'a option list" assumes "wf_tuple n (\<Union> (fv_trm ` set ts)) v"   and "\<forall>t\<in>set ts. \<forall>x\<in>fv_trm t. x < n" shows "\<exists>f. match ts (map (MFOTL.eval_trm (map the v)) ts) = Some f \<and> v = tabulate f 0 n"
UN_upper: fixes a :: "'a"   and A :: "'a set"   and B :: "'a \<Rightarrow> 'b set" assumes "a \<in> A" shows "B a \<subseteq> \<Union> (B ` A)"
in_orbit_affected: fixes b :: "'b"   and f :: "'b perm"   and a :: "'b" assumes "b \<in> orbit f a"   and "a \<in> affected f" shows "b \<in> affected f"
prime: fixes p :: "nat" assumes "prime p" shows "totient p = p - 1"
tl1_to_tl2_def: fixes s2 :: "'s2"   and stls1 :: "('s1 \<times> 'tl1 \<times> 's1) llist" shows "tl1_to_tl2 s2 stls1 = unfold_llist (\<lambda>(s2, stls1). lnull stls1) (\<lambda>(s2, stls1). let (s1, tl1, s1') = lhd stls1; (tl2, s2') = SOME (tl2, s2'). s2 -2-tl2\<rightarrow> s2' \<and> s1' \<approx> s2' \<and> tl1 \<sim> tl2 in (s2, tl2, s2')) (\<lambda>(s2, stls1). let (s1, tl1, s1') = lhd stls1; (tl2, s2') = SOME (tl2, s2'). s2 -2-tl2\<rightarrow> s2' \<and> s1' \<approx> s2' \<and> tl1 \<sim> tl2 in (s2', ltl stls1)) (s2, stls1)"
prime_elem_multD: 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 a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "comm_semiring_1.prime_elem times one zero (times a b)" shows "dvd.dvd times a one \<or> dvd.dvd times b one"
Un_set_offending_flows_bound_minus_subseteq': fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set"   and X :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "E' \<subseteq> E"   and "\<Union> (get_offending_flows M \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> X" shows "\<Union> (get_offending_flows M \<lparr>nodes = V, edges = E - E'\<rparr>) \<subseteq> X - E'"
ground_fm_aux_simps_2: fixes t :: "tm"   and u :: "tm"   and S :: "atom set" shows "ground_fm_aux (t IN u) S = (ground_aux t S \<and> ground_aux u S)"
hyperpow_hrabs: fixes r :: "'a star"   and n :: "nat star" shows "\<bar>r\<bar> pow n = \<bar>r pow n\<bar>"
valid_add_new_table_mmsaux: fixes args :: "Monitor.args"   and cur :: "nat"   and aux :: "nat \<times> nat \<times> bool list \<times> bool list \<times> (nat \<times> 'a option list set) queue \<times> (nat \<times> 'a option list set) queue \<times> ('a option list, nat) mapping \<times> ('a option list, nat) mapping"   and auxlist :: "(nat \<times> 'a option list set) list"   and X :: "'a option list set" assumes "valid_mmsaux args cur aux auxlist"   and "table (args_n args) (args_R args) X" shows "valid_mmsaux args cur (add_new_table_mmsaux args X aux) (case auxlist of [] \<Rightarrow> [(cur, X)] | (t, y) # ts \<Rightarrow> if t = cur then (t, y \<union> X) # ts else (cur, X) # auxlist)"
Acc_to_mojmir_accept: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> (G \<psi>)) w"   and "the (\<pi> (G \<psi>)) < semi_mojmir_def.max_rank \<Sigma> \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>)" shows "mojmir_def.accept \<delta>\<^sub>M (q\<^sub>0\<^sub>M \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"
simps_12: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "rec_latin_type f1 f2 f3 Num = f3"
lunit_agreement: fixes a :: "'a" assumes "ide a" shows "MC.lunit a = \<l>[a]"
mod_minus_right: fixes a :: "'a"   and b :: "'a" shows "a mod - b = - (- a mod b)"
bdd_above_multiset_imp_finite_support: fixes A :: "'k multiset set" assumes "A \<noteq> {}"   and "subset_mset.bdd_above A" shows "finite (\<Union>X\<in>A. {x. 0 < count X x})"
rewrite_prog_map_f: fixes rewrite_instr :: "nat \<Rightarrow> 'l \<Rightarrow> 'm \<Rightarrow> ('o, 'n \<times> 'm) result"   and f :: "'n \<Rightarrow> 'l"   and n :: "nat"   and xs :: "'l list"   and \<Sigma>1 :: "'m"   and ys :: "'n list"   and \<Sigma>2 :: "'m" assumes "\<And>x \<Sigma>1 n x' \<Sigma>2. rewrite_instr n x \<Sigma>1 = Ok (x', \<Sigma>2) \<Longrightarrow> f x' = x"   and "rewrite_prog rewrite_instr n xs \<Sigma>1 = Ok (ys, \<Sigma>2)" shows "map f ys = xs"
that: fixes u' :: "'a"   and v' :: "'a" assumes "(u', v') \<in> edges g"   and "\<not> P u'"   and "P v'" shows "thesis"
type': shows "\<forall>S\<in>set Ts. is_type P S"
relcomppE: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'c"   and c :: "'e" assumes "(r OO s) a c" obtains b :: "'d" where "r a b" and "s b c"
assms_2: shows "\<forall>(k2, \<gamma>2)\<in>one_chain2. \<gamma>2 piecewise_C1_differentiable_on {0..1}"
insert_def: fixes x :: "'b"   and xs :: "'b list" shows "List.insert x xs = (if x \<in>. xs then xs else x # xs)"
Read: fixes P :: "'a prog"   and al :: "addr_loc"   and a :: "'b"   and v :: "'b val" assumes "is_volatile P al" shows "saction P (NormalAction (ReadMem a al v))"
typing_varE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and \<tau> :: "PreSimplyTyped.type" assumes "\<Gamma> \<turnstile> Var x : \<tau>" shows "\<Gamma> x = Some \<tau>"
conj_involution: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "conjugate (conjugate f) = f"
tsubstT_fun_typ_dist: fixes T :: "typ"   and T1 :: "typ"   and \<rho> :: "variable \<Rightarrow> String.literal set \<Rightarrow> typ" shows "tsubstT (T \<rightarrow> T1) \<rho> = tsubstT T \<rho> \<rightarrow> tsubstT T1 \<rho>"
linorder_not_less: fixes x :: "'b"   and y :: "'b" shows "(\<not> order.greater y x) = order.greater_eq x y"
conv: shows "ltl_convert \<phi> = (APs, \<phi>\<^sub>i)"
IO: fixes out :: "'d"   and \<I> :: "('d, 'e) \<I>"   and D :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and c :: "'e \<Rightarrow> 'f"   and c' :: "'e \<Rightarrow> 'g"   and A :: "'h \<Rightarrow> 'i \<Rightarrow> bool" assumes "out \<in> outs_\<I> \<I>"   and "\<And>input. input \<in> responses_\<I> \<I> out \<Longrightarrow> D (c input) (c' input)" shows "eq_\<I>_generat A \<I> D (IO out c) (IO out c')"
not_imp: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" shows "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F::'b formula) \<^bold>\<rightarrow> F \<^bold>\<rightarrow> (G::'b formula)"
centre_in_cball: fixes x :: "'a"   and e :: "real" shows "(x \<in> cball x e) = (0 \<le> e)"
inner2_aform_def: fixes X :: "'a \<times> 'a pdevs"   and a :: "'a"   and b :: "'a" shows "inner2_aform X a b = (inner2 (fst X) a b, pdevs_inner2 (snd X) a b)"
aux_lemma: fixes Ja :: "'a set"   and Ka :: "'a set"   and s :: "'a" assumes "ideal Ja R"   and "ideal Ka R"   and "s \<in> I \<cdot> Ja" shows "s \<in> I \<cdot> (Ja <+> Ka) \<and> s \<in> I \<cdot> (Ka <+> Ja)"
single_one_def: shows "local.single_one \<equiv> \<lambda>l k A i j. (i \<notin> {k, l} \<longrightarrow> A $$ (i, k) = (0::'a)) \<and> A $$ (l, k) = (1::'a)"
LCons: fixes xs :: "'c llist"   and x :: "'c" assumes "lfinite xs" shows "lfinite (LCons x xs)"
suminf_ereal: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "(\<Sum>i. ereal (f i)) \<noteq> \<infinity>" shows "(\<Sum>i. ereal (f i)) = ereal (suminf f)"
IdInj: fixes X :: "'a"   and Y :: "'a" assumes "X \<in> obj"   and "Y \<in> obj"   and "id X = id Y" shows "X = Y"
preserves_reflects_arr: fixes f :: "'a" shows "arr (src f) = arr f"
assms_1: fixes i :: "nat" assumes "i \<in> set (fst xs)" shows "i < length as"
Pdrop_valid: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and n :: "nat" assumes "vm_path G P v0" shows "Digraph.valid_path G (ldropn n P)"
times_u_def: fixes x :: "'b\<^sub>\<bottom>"   and y :: "'b\<^sub>\<bottom>" shows "x * y = (case x of up\<cdot>a \<Rightarrow> \<Lambda> (up\<cdot>b). up\<cdot>(a * b))\<cdot>y"
neq_Leaf_iff: fixes kdt :: "'b kdt" shows "(\<nexists>p. kdt = Leaf p) = (\<exists>k v l r. kdt = Node k v l r)"