hcju/isabelleps huggingface dataset:
is_pr_Hom_rev: fixes \<phi> :: "'a \<Rightarrow> 'b"   and A :: "'a list set"   and B :: "'b list set" assumes "is_pr_Hom \<phi> A B" shows "is_pr_Hom \<phi> (rev ` A) (rev ` B)"
in_hom_char: fixes f :: "'a"   and a :: "'a"   and b :: "'a" shows "Right_a.in_hom f a b = (Right_a.arr a \<and> Right_a.arr b \<and> Right_a.arr f \<and> \<guillemotleft>f : a \<rightarrow> b\<guillemotright>)"
ring_n_mprod_mprodR: fixes n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set"   and m :: "nat"   and e :: "nat \<Rightarrow> int"   and f :: "nat \<Rightarrow> 'a" assumes "distinct_pds K n P"   and "\<forall>j\<le>m. e j \<in> Zset"   and "\<forall>j\<le>m. 0 \<le> e j"   and "\<forall>j\<le>m. f j \<in> carrier (O\<^bsub>K P n\<^esub>) - {\<zero>\<^bsub>O\<^bsub>K P n\<^esub>\<^esub>}" shows "mprod_exp K e f m = mprod_expR (O\<^bsub>K P n\<^esub>) (nat \<circ> e) f m"
content_decompose: fixes p :: "'a poly" obtains p' :: "'a poly" where "p = Polynomial.smult (Polynomial.content p) p'" and "Polynomial.content p' = (1::'a)"
xor_zero_left: fixes x :: "'c" shows "(0::'c) XOR x = x"
join_connected: fixes g\<^sub>1 :: "'b ugraph"   and g\<^sub>2 :: "'b ugraph"   and u :: "'b"   and v :: "'b" assumes "Undirected_Graph.connected g\<^sub>1"   and "Undirected_Graph.connected g\<^sub>2"   and "u \<in> nodes g\<^sub>1"   and "v \<in> nodes g\<^sub>2" shows "Undirected_Graph.connected (ins_edge (u, v) (graph_join g\<^sub>1 g\<^sub>2))"
phi_top: shows "phi top = Filters.filter.Abs_filter UNIV"
ctstate_is_UNIV: fixes c :: "ctstate set" shows "ctstate_is_UNIV c = (c = UNIV)"
red_simps_4: fixes t :: "'a local.term" assumes "Ide t" shows "Cod (t\<^bold>\<down>) = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
omega_zero_equiv_1: fixes x :: "'a"   and y :: "'a" shows "(\<Omega> x y = (zero_class.zero::'a)) = (d y \<le>  |x\<rangle> y)"
fsmult_def: fixes a :: "'f"   and v :: "'v" shows "a \<sharp>\<cdot> v \<equiv> (a \<delta>\<delta> (0::'g)) \<cdot> v"
IH: fixes x :: "('a, 'b) Term.term literal"   and x2 :: "'b \<Rightarrow> ('a, 'b) Term.term option" assumes "x \<in> set Ks"   and "match_term_list [(atm_of L, atm_of x)] \<sigma> = Some x2" shows "subsumes_list Ls (remove1 x Ks) x2 = subsumes_list_modulo Ls (remove1 x Ks) x2"
bit_iff_odd: 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 bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a n = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (divide a (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n)))"
v_round_def: fixes r :: "nat"   and r_votes :: "process \<Rightarrow> val option"   and r_decisions :: "process \<Rightarrow> val option" shows "v_round r r_votes r_decisions = {(s, s'). r = next_round s \<and> no_defection s r_votes r \<and> d_guard r_decisions r_votes \<and> s' = s\<lparr>next_round := Suc r, votes := (votes s)(r := r_votes), decisions := decisions s ++ r_decisions\<rparr>}"
retain_TCons_conv: fixes A :: "('c, 'b) tllist set" shows "local.retain_TCons A = A \<inter> {xs. \<not> is_TNil xs}"
fact_cancel: fixes n :: "nat"   and c :: "'a" shows "of_nat (Suc n) * c / fact (Suc n) = c / fact n"
sq_def: fixes x :: "'a" shows "sq x \<equiv> x * x"
hcomplex_defs_8: shows "hrcis = *f2* rcis"
1: fixes \<pi> :: "('state \<times> 'aprop set) stream"   and \<pi>' :: "('state \<times> 'aprop set) stream"   and i :: "nat" shows "wfp AP' \<pi> \<and> wfp AP' \<pi>' \<longrightarrow> f (fst (\<pi> !! i)) (fst (\<pi>' !! i)) = \<phi> [] [sdrop i \<pi>, sdrop i \<pi>'] \<and> g (fst (\<pi> !! i)) (fst (\<pi>' !! i)) = \<psi> [] [sdrop i \<pi>, sdrop i \<pi>']"
inc_predict_support_trading_strat': fixes pf1 :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and asset :: "'d"   and B :: "'d set" assumes "trading_strategy pf1"   and "asset \<in> support_set pf1 \<union> B" shows "borel_predict_stoch_proc F (pf1 asset)"
if_pred_holds: fixes P :: "unit pred" shows "if_pred (holds P) = P"
cases: fixes a1 :: "nat"   and a2 :: "'b"   and subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b" assumes "(a1, a2) \<in> tree subs gamma" obtains "a1 = 0" and "a2 = gamma"   | n :: "nat" and delta :: "'b" and sigma :: "'b" where "a1 = Suc n" and "a2 = sigma" and "(n, delta) \<in> tree subs gamma" and "sigma \<in> subs delta"
k: shows "\<forall>i e pre. (\<exists>mo. Deliver (i, Store e mo i) \<in> set pre) = (Deliver (i, Store e (k i e pre) i) \<in> set pre)"
opt_less_def: fixes x :: "int option"   and y :: "int option" shows "opt_less x y = (case x of None \<Rightarrow> True | Some a \<Rightarrow> case y of None \<Rightarrow> True | Some b \<Rightarrow> a < b)"
HInit_HInv1: fixes s :: "state" shows "HInit s \<longrightarrow> HInv1 s"
add_diff_cancel_right: fixes plus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and minus :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and a :: "'r"   and c :: "'r"   and b :: "'r" assumes "class.cancel_ab_semigroup_add plus minus" shows "minus (plus a c) (plus b c) = minus a b"
jkbpC_represents_uniquely: fixes T :: "'s Trace set" assumes "represents T" shows "T = jkbpC"
mmulti_join_correct: fixes A_pos :: "nat set list"   and n :: "nat"   and A_neg :: "nat set list"   and L :: "'a option list set list"   and z :: "'a option list" assumes "A_pos \<noteq> []"   and "list_all2 (\<lambda>A X. table n A X \<and> wf_set n A) (A_pos @ A_neg) L" shows "(z \<in> mmulti_join n A_pos A_neg L) = (wf_tuple n (\<Union>A\<in>set A_pos. A) z \<and> list_all2 (\<lambda>A. (\<in>) (restrict A z)) A_pos (take (length A_pos) L) \<and> list_all2 (\<lambda>A. (\<notin>) (restrict A z)) A_neg (drop (length A_pos) L))"
keys_monom_multE: fixes v :: "'a"   and c :: "'b"   and t :: "'a"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys (MPoly_Type_Class.punit.monom_mult c t p)" obtains u :: "'a" where "u \<in> keys p" and "v = MPoly_Type_Class.punit.splus t u"
colossless_gpv_1: shows "colossless_gpv \<I> (local.restrict_gpv gpv)"
eigenvalue_imp_nonzero_dim: fixes A :: "'b mat"   and n :: "nat"   and ev :: "'b" assumes "A \<in> carrier_mat n n"   and "eigenvalue A ev" shows "0 < n"
simps_2: fixes c :: "nat \<times> cell list \<times> cell list"   and p :: "(action \<times> nat) list \<times> nat"   and n :: "nat" shows "steps c p (Suc n) = steps (step c p) p n"
C_eq_RAD_aux: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []" shows "Cp (list2FWpolicy p) x = Cp (list2FWpolicy (removeAllDuplicates p)) x"
prv_exi_inst: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm" shows "prv (imp (subst \<phi> t x) (exi x \<phi>))"
hnorm_triangle_ineq4: fixes a :: "'a star"   and b :: "'a star" shows "hnorm (a - b) \<le> hnorm a + hnorm b"
nn_integral_le_const: fixes c :: "ennreal"   and f :: "'a \<Rightarrow> ennreal"   and p :: "'a spmf" assumes "0 \<le> c"   and "AE x in measure_spmf p. f x \<le> c" shows "integral\<^sup>N (measure_spmf p) f \<le> c"
simps_1: fixes L :: "'b language" shows "in_language L [] = \<oo> L"
sup_commute: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = y \<sqinter> x"
pp_ub1: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and ls :: "'b list"   and y :: "'b"   and x :: "'b" assumes "Semilat A r f"   and "set ls \<subseteq> A"   and "y \<in> A"   and "x \<in> set ls" shows "x \<sqsubseteq>\<^bsub>r\<^esub> ls \<Squnion>\<^bsub>f\<^esub> y"
reds_preserves_hconf: fixes es :: "char list exp list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and es' :: "char list exp list"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and 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 \<Rightarrow> ty option"   and Ts :: "ty list" assumes "P \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',(h', l')\<rangle>"   and "P,E,h \<turnstile> es [:] Ts"   and "P \<turnstile> h \<surd>" shows "P \<turnstile> h' \<surd>"
map_restrict_axioms: shows "map_restrict \<alpha>1 invar1 \<alpha>2 invar2 restrict"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_has_all_failures: fixes P :: "'a process"   and A :: "'a set" assumes "events_of P \<subseteq> A" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P A \<sqsubseteq>\<^sub>F P"
simps_3: fixes x :: "char" shows "proper_interval (Some x) None = (x \<noteq> CHR 0xFF)"
sigma_finite_measure_axioms: shows "sigma_finite_measure (count_space A)"
calculate_data_satisfies_properties_R: fixes p :: "real poly"   and qs :: "real poly list" shows "p \<noteq> 0 \<and> 0 < length qs \<longrightarrow> satisfies_properties_R p qs (get_subsets_R (calculate_data_R p qs)) (get_signs_R (calculate_data_R p qs)) (get_matrix_R (calculate_data_R p qs))"
extend_measure_cong: fixes \<Omega> :: "'d set"   and \<Omega>' :: "'d set"   and I :: "'e set"   and I' :: "'e set"   and G :: "'e \<Rightarrow> 'd set"   and G' :: "'e \<Rightarrow> 'd set"   and \<mu> :: "'e \<Rightarrow> ennreal"   and \<mu>' :: "'e \<Rightarrow> ennreal" assumes "\<Omega> = \<Omega>'"   and "I = I'"   and "G = G'"   and "\<And>i. i \<in> I' \<Longrightarrow> \<mu> i = \<mu>' i" shows "extend_measure \<Omega> I G \<mu> = extend_measure \<Omega>' I' G' \<mu>'"
maybe_or_valid: fixes x :: "trilean"   and y :: "trilean" assumes "x \<or>? y \<noteq> invalid" shows "x \<noteq> invalid \<and> y \<noteq> invalid"
issue_expr_load_tmps_range': fixes t :: "nat"   and e :: "expr" shows "load_tmps (issue_expr t e) = {i. t \<le> i \<and> i < t + used_tmps e}"
set_disconnected_nodes_get_tag_name: fixes ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
subresultant_prs_impl: fixes f :: "'c poly \<times> 'c \<Rightarrow> 'b"   and G1 :: "'c poly"   and G2 :: "'c poly" shows "subresultant_prs_impl f G1 G2 = f (subresultant_prs dichotomous_Lazard G1 G2)"
wf_digraph_del_arc: fixes G :: "('c, 'd) pre_digraph"   and a :: "'d" assumes "wf_digraph G" shows "wf_digraph (pre_digraph.del_arc G a)"
in_set_conv_decomp: fixes x :: "'b"   and xs :: "'b list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs)"
HARoot_def: fixes HA :: "('d, 'e, 'f) hierauto" shows "HARoot HA \<equiv> Root (SAs HA) (CompFun HA)"
tw_def: shows "tw \<equiv> takeWhile ((\<noteq>) u) tjs"
dtor1_def: fixes x :: "'d JF1" shows "dtor1 x = F1map id Abs_JF1 Abs_JF2 (str_final1 (Rep_JF1 x))"
zero_neq_one_ow_def: fixes U :: "'b set"   and one :: "'b"   and zero :: "'b" shows "zero_neq_one_ow U one zero \<equiv> zero_ow U zero \<and> one_ow U one \<and> zero_neq_one_ow_axioms one zero"
take_hd_drop: fixes n :: "nat"   and xs :: "'k list" assumes "n < length xs" shows "take n xs @ [hd (drop n xs)] = take (Suc n) xs"
of_nat_primes_omega_primorial: fixes x :: "real" shows "real (primes_omega (primorial x)) = primes_pi x"
simps_5: fixes v1 :: "'b val"   and v2 :: "'b val" shows "flat (Seq v1 v2) = flat v1 @ flat v2"
A: shows "finite (pverts G) \<and> (\<exists>U V. pverts G = U \<union> V \<and> U \<inter> V = {} \<and> card U = m \<and> card V = n \<and> parcs G = U \<times> V \<union> V \<times> U)"
InitPhase_HInv3_q: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and p :: "Proc"   and d :: "Disk" assumes "InitializePhase s s' q"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
tr_filter_conc_eq1: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list"   and cs :: "'c set" shows "a @ b = tr_filter (a @ c) cs \<longrightarrow> b = tr_filter c cs"
mem_permute_iff: fixes p :: "perm"   and x :: "'c"   and X :: "'c set" shows "(p \<bullet> x \<in> p \<bullet> X) = (x \<in> X)"
f_Img_ex_set: fixes f :: "'e \<Rightarrow> 'd"   and P :: "'e \<Rightarrow> bool" shows "{f x |x. P x} = f ` {x. P x}"
SEQ_assoc_D: fixes P :: "'a process"   and Q :: "'a process"   and S :: "'a process" shows "D (P `;` (Q `;` S)) = D (P `;` Q `;` S)"
mult_le_cancel2_hmset: fixes m :: "hmultiset"   and k :: "hmultiset"   and n :: "hmultiset" shows "(m * k \<le> n * k) = (0 < k \<longrightarrow> m \<le> n)"
cases: fixes a :: "'a"   and r :: "('b \<times> 'a) set" assumes "a \<in> Range r" obtains a :: "'b" and b :: "'a" where "ab__ = b" and "(a, b) \<in> r"
lunit_coherence1: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and T\<^sub>C :: "'d \<times> 'd \<Rightarrow> 'd"   and \<alpha>\<^sub>C :: "'d \<times> 'd \<times> 'd \<Rightarrow> 'd"   and \<iota>\<^sub>C :: "'d"   and D :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and T\<^sub>D :: "'e \<times> 'e \<Rightarrow> 'e"   and \<alpha>\<^sub>D :: "'e \<times> 'e \<times> 'e \<Rightarrow> 'e"   and \<iota>\<^sub>D :: "'e"   and F :: "'d \<Rightarrow> 'e"   and \<phi> :: "'d \<times> 'd \<Rightarrow> 'e"   and a :: "'d" assumes "monoidal_functor C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F \<phi>"   and "partial_magma.ide C a" shows "D (monoidal_category.lunit D T\<^sub>D \<alpha>\<^sub>D (D (F \<iota>\<^sub>C) (\<phi> (monoidal_category.unity C \<iota>\<^sub>C, monoidal_category.unity C \<iota>\<^sub>C))) (F a)) (T\<^sub>D (monoidal_functor.\<psi> C \<iota>\<^sub>C D T\<^sub>D \<iota>\<^sub>D F \<phi>, F a)) = monoidal_category.lunit D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D (F a)"
arcs_ends_blocks_wf: fixes bl :: "'a set" assumes "bl \<in># arcs_blocks" shows "bl \<subseteq> verts G"
dpd4: fixes x :: "'a"   and y :: "'a" shows "ad (ad (x + y)) = ad (ad x) + ad (ad y)"
in_domain_list_relE: fixes xs :: "'d list"   and R :: "('d \<times> 'e) set" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> Domain R" obtains ys :: "'e list" where "(xs, ys) \<in> \<langle>R\<rangle>list_rel"
mult_eq': shows "(\<otimes>\<^bsub>G\<^esub>) = (\<lambda>x y. x * y mod n)"
Collect_restrict: fixes X :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "{x \<in> X. P x} \<subseteq> X"
mat_def: shows "Matrix.mat \<equiv> (id ---> id ---> id ---> Abs_mat) (\<lambda>nr nc f. (nr, nc, Matrix.mk_mat nr nc f))"
simps_2: fixes e :: "'b \<times> 'b"   and es :: "('b \<times> 'b) list" shows "rev_path (e # es) = rev_path es @ [(snd e, fst e)]"
append_self_conv2: fixes xs :: "'b list"   and ys :: "'b list" shows "(append xs ys = ys) = (xs = [])"
saturated_def: fixes N :: "'f set" shows "local.lifted_q_calc.saturated N = (Inf_from N \<subseteq> lifted_q_calc.Red_I_\<G> N)"
M_def: shows "M = card {i. i < N \<and> f i}"
step_mono: fixes wf_mb :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool" assumes "wf_prog wf_mb P" shows "SemilatAlg.mono r step (length is) A"
ilt: fixes i :: "nat" assumes "i < n" shows "inv p i < n"
assms_1: shows "f \<in> V\<^sub>0\<^sub>1"
measure_space_def: fixes \<Omega> :: "'c set"   and A :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal" shows "measure_space \<Omega> A \<mu> = (sigma_algebra \<Omega> A \<and> positive A \<mu> \<and> countably_additive A \<mu>)"
not_outside_2: fixes i :: "'agent"   and x :: "'alt"   and y :: "'alt" assumes "R i x y" shows "x \<in> alts"
test_absorb_4: fixes x :: "'a"   and y :: "'a" assumes "test x"   and "test y" shows "(x \<squnion> y) \<sqinter> x = x"
no_move1_to_no_move2: fixes s1 :: "'s1"   and s2 :: "'s2" assumes "s1 \<approx> s2"   and "\<And>tl1 s1'. \<not> s1 -1-tl1\<rightarrow> s1'" shows "\<exists>s2'. s2 -\<tau>2\<rightarrow>* s2' \<and> (\<forall>tl2 s2''. \<not> s2' -2-tl2\<rightarrow> s2'') \<and> s1 \<approx> s2'"
vals_cons: fixes vh :: "exp"   and vt :: "exp list" assumes "val(vh)"   and "vals(vt)" shows "vals((vh # vt))"
parameters_def: fixes m :: "methd" shows "parameters m = set (map (EName \<circ> VNam) (pars m)) \<union> (if static m then {} else {This})"
Red_F_of_Red_F_subset: fixes N' :: "'g set"   and N :: "'g set" assumes "N' \<subseteq> Red_F_G N" shows "Red_F_G N \<subseteq> Red_F_G (N - N')"
antecedent: shows "P v (x \<oplus> y) \<and> P v (x \<oplus> z)"
length_trace_parallel_plan_strips_lte_length_plan_plus_one: fixes I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" shows "length (trace_parallel_plan_strips I \<pi>) \<le> length \<pi> + 1"
enumBase_def: fixes nmax :: "nat" shows "enumBase nmax \<equiv> map (\<lambda>i. [i]) [0..<Suc nmax]"
\<phi>: shows "local.iso \<phi> \<and> \<guillemotleft>\<phi> : f \<star> a \<Rightarrow> g \<star> a\<guillemotright>"
Red1CallNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and M :: "String.literal"   and vs :: "'addr val list"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>null\<bullet>M(map Val vs),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
assms: shows "pair \<in> N \<times> (Pow \<Omega> - {{}})"
rconjby_order2: fixes s :: "'a"   and x :: "'a" assumes "s + s = (0::'a)" shows "rconjby x s + rconjby x s = (0::'a)"
simps_6: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'c \<Rightarrow> 'b"   and x2 :: "'c" shows "(case Diff x2 of Same x \<Rightarrow> f1 x | Diff x \<Rightarrow> f2 x) = f2 x2"
se_Assume_eq: fixes c :: "('c, 'd) conf"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> bool"   and c' :: "('c, 'd) conf" shows "se c (Assume e) c' = (c' = \<lparr>store = store c, pred = pred c \<union> {adapt_bexp e (store c)}\<rparr>)"
g_def: shows "g = (\<lambda>x. if g0 x = \<infinity> then 0 else g0 x)"
ref_map_execs: fixes f :: "'b \<Rightarrow> 'c"   and B :: "('b, 'd) ioa"   and A :: "('c, 'd) ioa"   and e_B :: "'b \<times> ('d \<times> 'b) list" assumes "is_ref_map f B A"   and "is_exec_of B e_B" shows "\<exists>e_A. is_exec_of A e_A \<and> trace (ioa.asig A) e_A = trace (ioa.asig A) e_B"
insertAssertionFreshChain_1: fixes xvec :: "name list"   and F :: "'b frame"   and \<Psi> :: "'b" assumes "xvec \<sharp>* F"   and "xvec \<sharp>* \<Psi>" shows "xvec \<sharp>* insertAssertion F \<Psi>"
lookup_one_zero: shows "lookup 1 (0::'d) = (1::'c)"
sxalloc_mono_no_abrupt: fixes G :: "prog"   and s0 :: "state"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>sxalloc\<rightarrow> s1"   and "normal s1" shows "normal s0"
product_category_def: fixes C1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "product_category C1 C2 \<equiv> category C1 \<and> category C2"
inv_ins: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "invc t"   and "invh t" shows "invc2 (ins x t) \<and> (color t = Black \<longrightarrow> invc (ins x t)) \<and> invh (ins x t) \<and> bheight (ins x t) = bheight t"
nodes_target: fixes p :: "'c"   and A :: "('d, 'c) nba"   and r :: "('d \<times> 'c) list" assumes "p \<in> nodes A"   and "path A r p" shows "target r p \<in> nodes A"
cylinder_empty_prob: fixes A :: "nat set set" assumes "A \<subseteq> S_edges" shows "prob (cylinder S_edges A {}) = p ^ card A"
Ch_range: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set" assumes "Contracts Xd Xh Pd Ch" shows "\<forall>h X. Ch h X \<subseteq> {x \<in> X. Xh x = h}"
f1: shows "(U\<^sub>f * U\<^sub>f) $$ (i, j) = of_nat (1 - f (i div 2)) * U\<^sub>f $$ (i, j) + of_nat (f (i div 2)) * U\<^sub>f $$ (i + 1, j)"
InvGlobVacantUptoImpliesNrec_def: fixes c :: "('c, 'd) configuration" shows "InvGlobVacantUptoImpliesNrec c = (\<forall>t q. GlobVacantUpto c q t \<longrightarrow> NrecVacantUpto c t)"
separating_essential: fixes G :: "('a, 'b) graph_scheme"   and A :: "'a set"   and B :: "'a set"   and S :: "'a set" assumes "separating_gen G A B S" shows "separating_gen G A B {x \<in> S. essential G B S x}"
mat_add_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" shows "f \<oplus> g \<equiv> \<lambda>i j. plus (f i j) (g i j)"
vimage_Collect: fixes P :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> bool" assumes "\<And>x. P (f x) = Q x" shows "f -` Collect P = Collect Q"
cong: fixes Theap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var ttree" shows "TTreeAnalysisCardinalityHeap.cHeap Theap \<equiv> TTreeAnalysisCardinalityHeap.cHeap Theap"
face_cycle_set_altdef: fixes a :: "'b" shows "H.face_cycle_set a = {(H.face_cycle_succ ^^ n) a |n. True}"
discr_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "discr c"   and "(c, s) \<rightarrow>c (c', s')" shows "discr c'"
eventually_nonneg_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" shows "eventually_nonneg F f = (\<forall>\<^sub>F x in F. (0::'d) \<le> f x)"
ts': shows "ts' = ts[i := (p, is, \<theta>, sb', \<D>, \<O>', \<R>')]"
simps: fixes i :: "nat"   and l :: "nat"   and acc :: "int vec" shows "gso_int_tail' i l acc = (if i \<le> l then acc else let v = \<mu>' l l \<cdot>\<^sub>v acc - \<mu>' i l \<cdot>\<^sub>v gso_int l l in Let (map_vec (\<lambda>k. k div \<mu>' (l - 1) (l - 1)) v) (gso_int_tail' i (l + 1)))"
gso_int_tail: fixes i :: "nat" shows "gso_int_tail i = gso_int i i"
Disjunctuous_Monotonic: fixes H :: "'d upred \<Rightarrow> 'd upred" assumes "Disjunctuous H" shows "Monotonic H"
SN_Un_conv: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "trans (r \<union> s)" shows "SN (r \<union> s) = (SN r \<and> SN s)"
F: shows "set xs \<subseteq> range p \<union> range q"
finite_imp_closed: fixes S :: "'b set" assumes "finite S" shows "closed S"
l_set_mode_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_mode_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs"
finite_cartesian_product: fixes A :: "'b set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "finite (A \<times> B)"
insert_def: fixes a :: "'c"   and B :: "'c set" shows "insert a B = {x. x = a} \<union> B"
inject: fixes x2 :: "'m"   and y2 :: "'m" shows "(Some x2 = Some y2) = (x2 = y2)"
sep_conj_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.sep_conj plus sep_disj P Q = sep_algebra.sep_conj plus sep_disj Q P"