hcju/isabelleps huggingface dataset:
take_bit_eq_0_iff: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat"   and a :: "'k" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "(take_bit n a = zero) = dvd.dvd times (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) a"
eq_iff: fixes a :: "'a"   and b :: "'a" shows "(a = b) = (a \<sqsubseteq> b \<and> b \<sqsubseteq> a)"
initial: fixes initial :: "'a \<Rightarrow> bool"   and p :: "'a"   and execute :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and enabled :: "'b \<Rightarrow> 'a \<Rightarrow> bool" assumes "initial p" shows "p \<in> transition_system_initial.nodes execute enabled initial"
assms_8: shows "sum f1 C1 = sum f2 C2"
lit_ueval: fixes x :: "'a"   and b :: "'b" shows "\<lbrakk>\<guillemotleft>x\<guillemotright>\<rbrakk>\<^sub>e b = x"
exH_k_mult_pre_is_psi: shows "exH_n *\<^sub>v ket_pre = \<psi>"
closed_segment_subset_domainI: fixes t0 :: "real"   and t :: "real"   and s :: "real" assumes "t0 \<in> S"   and "t \<in> S"   and "s \<in> {t0--t}" shows "s \<in> S"
set_antichain1: fixes x :: "'a"   and X :: "'a antichain" assumes "x \<in> set_antichain X" shows "x \<in>\<^sub>A X"
max_block: fixes k :: "nat"   and la :: "complex" assumes "(k, la) \<in> set n_as" shows "cmod la \<le> 1 \<and> (cmod la = 1 \<longrightarrow> k \<le> m)"
homotopic_with_symD: fixes P :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "homotopic_with P X Y f g" shows "homotopic_with P X Y g f"
induct: fixes P :: "'V \<Rightarrow> ('V \<times> 'W \<times> 'V) list \<Rightarrow> 'V \<Rightarrow> bool"   and a0 :: "'V"   and a1 :: "('V \<times> 'W \<times> 'V) list"   and a2 :: "'V" assumes "\<And>v v'. P v [] v'"   and "\<And>v v1 w v2 p v'. P v2 p v' \<Longrightarrow> P v ((v1, w, v2) # p) v'" shows "P a0 a1 a2"
get_shadow_root_locs_impl: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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" assumes "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" shows "get_shadow_root_locs = get_shadow_root_locs"
product_in_second_factor: fixes x :: "i"   and y :: "i" assumes "CM"   and "O x y" shows "P (x \<otimes> y) y"
bij_betw_imp_inj_on: fixes f :: "'h \<Rightarrow> 'i"   and A :: "'h set"   and B :: "'i set" assumes "bij_betw f A B" shows "inj_on f A"
induct: fixes x :: "('a, 'b) form list"   and P :: "('a, 'b) form list \<Rightarrow> bool" assumes "\<turnstile> x"   and "\<And>i l G. P (Pred i l # Neg (Pred i l) # G)"   and "\<And>G. P (Neg \<bottom> # G)"   and "\<And>G. P (\<top> # G)"   and "\<And>A G. \<lbrakk>\<turnstile> A # G; P (A # G)\<rbrakk> \<Longrightarrow> P (Neg (Neg A) # G)"   and "\<And>A B G. \<lbrakk>\<turnstile> Neg A # Neg B # G; P (Neg A # Neg B # G)\<rbrakk> \<Longrightarrow> P (Neg (And A B) # G)"   and "\<And>A B G. \<lbrakk>\<turnstile> A # B # G; P (A # B # G)\<rbrakk> \<Longrightarrow> P (Or A B # G)"   and "\<And>A B G. \<lbrakk>\<turnstile> Neg A # B # G; P (Neg A # B # G)\<rbrakk> \<Longrightarrow> P (Impl A B # G)"   and "\<And>A G B. \<lbrakk>\<turnstile> A # G; P (A # G); \<turnstile> B # G; P (B # G)\<rbrakk> \<Longrightarrow> P (And A B # G)"   and "\<And>A G B. \<lbrakk>\<turnstile> Neg A # G; P (Neg A # G); \<turnstile> Neg B # G; P (Neg B # G)\<rbrakk> \<Longrightarrow> P (Neg (Or A B) # G)"   and "\<And>A G B. \<lbrakk>\<turnstile> A # G; P (A # G); \<turnstile> Neg B # G; P (Neg B # G)\<rbrakk> \<Longrightarrow> P (Neg (Impl A B) # G)"   and "\<And>A t G. \<lbrakk>\<turnstile> A[t/0] # G; P (A[t/0] # G)\<rbrakk> \<Longrightarrow> P (Exists A # G)"   and "\<And>A t G. \<lbrakk>\<turnstile> Neg (A[t/0]) # G; P (Neg (A[t/0]) # G)\<rbrakk> \<Longrightarrow> P (Neg (Forall A) # G)"   and "\<And>A n G. \<lbrakk>\<turnstile> A[App n []/0] # G; P (A[App n []/0] # G); news n (A # G)\<rbrakk> \<Longrightarrow> P (Forall A # G)"   and "\<And>A n G. \<lbrakk>\<turnstile> Neg (A[App n []/0]) # G; P (Neg (A[App n []/0]) # G); news n (A # G)\<rbrakk> \<Longrightarrow> P (Neg (Exists A) # G)"   and "\<And>G G'. \<lbrakk>\<turnstile> G; P G; set G = set G'\<rbrakk> \<Longrightarrow> P G'" shows "P x"
summable_rabs_cancel: fixes f :: "nat \<Rightarrow> real" assumes "summable (\<lambda>n. \<bar>f n\<bar>)" shows "summable f"
equiv_trans_def: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and u :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "equiv_trans t u = (le_trans t u \<and> le_trans u t)"
Vr_local: fixes v :: "'a \<Rightarrow> ant"   and I :: "'a set" assumes "valuation K v"   and "maximal_ideal (Vr K v) I" shows "vp K v = I"
wfp_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "wfp_on P A = (\<nexists>f. \<forall>i. f i \<in> A \<and> P (f (Suc i)) (f i))"
simps_2: fixes x :: "nat" assumes "gx3 \<le> x" shows "f' x = g' (real x) + (\<Sum>i<k. as ! i * f' ((ts ! i) x))"
sg5: shows "\<forall>ka\<le>Suc d. lose (d + (t + (3 + k)) + ka) = [False]"
Abs_quot_inject: fixes x :: "'b set"   and y :: "'b set" assumes "x \<in> quot"   and "y \<in> quot" shows "(Abs_quot x = Abs_quot y) = (x = y)"
samequot_exec_samequot: fixes n :: "nat"   and as :: "'a"   and r :: "'b rexp" shows "lang n (samequot_exec as r) = lang n (samequot as r)"
dfs_correct: fixes E :: "('b \<times> 'b) set"   and src :: "'b"   and tgt :: "'b" assumes "finite (E\<^sup>* `` {src})" shows "dfs E src tgt \<le> SPEC (\<lambda>r. r = ((src, tgt) \<in> E\<^sup>*))"
upd_row_upd_row_cancel: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d"   and x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" shows "upd_row (upd_row M i x) i y = upd_row M i y"
carrier_stalk_def: shows "stx.carrier_stalk \<equiv> dlim \<O>\<^sub>X \<rho>\<^sub>X (neighborhoods x)"
open_bijective_linear_image_eq: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "linear f"   and "bij f" shows "open (f ` A) = open A"
pr_yields_binary_product: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "category_with_binary_products C"   and "partial_magma.ide C a"   and "partial_magma.ide C b" shows "category.has_as_binary_product C a b (category_with_binary_products.pr1 C a b) (category_with_binary_products.pr0 C a b)"
f0: shows "gate 3 (Quantum.Id 2 \<Otimes> Z * X)"
res_def: shows "A_CHAN.res \<equiv> RES (A_CHAN.auth_oracle \<oplus>\<^sub>O A_CHAN.send_oracle \<oplus>\<^sub>O A_CHAN.recv_oracle) Void"
Units_one_closed: shows "0\<^sub>v n \<in> Units \<lparr>carrier = carrier_vec n, mult = (+), one = 0\<^sub>v n, \<dots> = undefined\<rparr>"
rGamma_eq_zero_iff_aux: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and rGamma :: "'a \<Rightarrow> 'a"   and z :: "'a" assumes "class.Gamma dist uniformity open divide inverse times one plus zero minus uminus scaleR norm sgn rGamma" shows "(rGamma z = zero) = (\<exists>n. z = uminus (semiring_1.of_nat one plus zero n))"
\<oo>\<^sub>P_in_cfl: fixes \<alpha> :: "('t + 'n) list" assumes "local.\<oo>\<^sub>P \<alpha>" shows "local.in_cfl \<alpha> []"
split_sel: fixes P :: "'g \<Rightarrow> bool"   and f1 :: "'g"   and f2 :: "'h \<Rightarrow> 'g"   and option :: "'h option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = Some (the option) \<longrightarrow> P (f2 (the option))))"
rderivs_append: fixes ws :: "'a list"   and ws' :: "'a list"   and r :: "'b rexp" shows "rderivs (ws @ ws') r = rderivs ws' (rderivs ws r)"
finite_Diff2: fixes B :: "'c set"   and A :: "'c set" assumes "finite B" shows "finite (A - B) = finite A"
d_def: shows "d = c \<circ> (*) (1 / a)"
Int_empty_left: fixes B :: "'a set" shows "{} \<inter> B = {}"
ceq_vector_fraction_iff: fixes x :: "'b"   and u :: "complex"   and v :: "complex"   and a :: "'b" shows "(x = (u / v) *\<^sub>C a) = (if v = 0 then x = (0::'b) else v *\<^sub>C x = u *\<^sub>C a)"
l_inv_ex: fixes x :: "'a" assumes "x \<in> carrier G" shows "\<exists>y\<in>carrier G. y \<otimes> x = \<one>"
trivial_relative_homology_group_gen: fixes X :: "'b topology"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and p :: "int" assumes "continuous_map X (subtopology X S) f"   and "homotopic_with (\<lambda>h. True) (subtopology X S) (subtopology X S) f id"   and "homotopic_with (\<lambda>k. True) X X f id" shows "trivial_group (relative_homology_group p X S)"
2: shows "{(x, y) |y. x < y in xs \<and> y < x in ys} \<subseteq> Inv xs ys \<union> {(y, x) |y. y < x in xs \<and> y < x in ys}"
select_convs_2: fixes E_ES :: "'c set"   and I_ES :: "'c set"   and O_ES :: "'c set"   and Tr_ES :: "'c list set"   and more :: "'d" shows "I_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = I_ES"
option_while_def: fixes C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option"   and r :: "'d" shows "option_while C B r \<equiv> if \<exists>s. (Some r, s) \<in> option_while' C B then THE s. (Some r, s) \<in> option_while' C B else None"
monom_mult_def: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult c t p = Abs_poly_mapping (\<lambda>v. if t adds\<^sub>p v then c * lookup p (v \<ominus> t) else (0::'c))"
distinct_map_scaleRI: fixes xs :: "'a list"   and c :: "real" assumes "distinct xs"   and "c \<noteq> 0" shows "distinct (map ((*\<^sub>R) c) xs)"
assms_3: shows "coincide_on \<sigma> \<sigma>' (vars_of_cl (cl_ecl C))"
induct: fixes P1 :: "('a, 'b) form \<Rightarrow> bool"   and P3 :: "('a, 'b) form list \<Rightarrow> bool"   and P2 :: "('a, 'b) form list \<Rightarrow> bool"   and form_list :: "('a, 'b) form list" assumes "\<And>x. P1 (At x)"   and "\<And>x1 x2. P3 x2 \<Longrightarrow> P1 (Compound x1 x2)"   and "\<And>x1 x2. P2 x2 \<Longrightarrow> P1 (Modal x1 x2)"   and "P1 ff"   and "P2 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P2 x2\<rbrakk> \<Longrightarrow> P2 (x1 # x2)"   and "P3 []"   and "\<And>x1 x2. \<lbrakk>P1 x1; P3 x2\<rbrakk> \<Longrightarrow> P3 (x1 # x2)" shows "P2 form_list"
simps_2: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list" shows "pawalk_verts u (e # es) = fst e # pawalk_verts (snd e) es"
resComm: fixes x :: "name"   and y :: "name"   and P :: "pi" shows "<\<nu>x><\<nu>y>P \<sim>\<^sub>l <\<nu>y><\<nu>x>P"
derived_S5_rules_1_a: fixes \<chi> :: "\<o>"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<chi> in v] \<Longrightarrow> [\<^bold>\<diamond>\<phi> \<^bold>\<rightarrow> \<psi> in v]"   and "[\<^bold>\<box>\<chi> in v]" shows "[\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
cases: fixes a :: "nat \<Rightarrow> classDef option" assumes "a OK" obtains CT :: "nat \<Rightarrow> classDef option" where "a = CT" and "Object \<notin> dom CT" and "\<forall>C CDef. CT C = Some CDef \<longrightarrow> CT \<turnstile> CDef OK \<and> cName CDef = C"
crel_vs_executeD: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "['d]"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "Q heap"   and "state_dp_consistency.cmem heap" obtains x :: "'d" and heap' :: "heap" where "execute b heap = Some (x, heap')" and "P heap'" and "Q heap'" and "state_dp_consistency.cmem heap'" and "R a x"
assms_7: shows "last (msgs (S t j) cid) \<noteq> Marker"
card_Diff_singleton: fixes A :: "'q set"   and x :: "'q" assumes "finite A"   and "x \<in> A" shows "card (A - {x}) = card A - 1"
chamber_system_into: shows "order.greater_eq folding.\<C> (f \<turnstile> folding.\<C>)"
inc_BitM_eq: fixes n :: "num" shows "Num.inc (Num.BitM n) = num.Bit0 n"
existence_ivl_reverse: fixes t :: "real"   and x0a :: "'a"   and t0 :: "real"   and x0 :: "'a \<Rightarrow>\<^sub>L 'a" assumes "t \<in> Y.mvar.existence_ivl x0a t0 x0" shows "t0 \<in> Y.mvar.existence_ivl x0a t (Y.mvar.flow x0a t0 x0 t)"
border_def: fixes xs :: "'b list"   and ys :: "'b list" shows "border xs ys = (prefix xs ys \<and> suffix xs ys)"
dvd_0_left_iff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times zero a = (a = zero)"
2: fixes A :: "'a set" assumes "i \<notin> A" shows "(A \<in> Rep_formula y) = (A \<in> Rep_formula y')"
CSHOnextConfig_def: fixes A :: "('d, 'e, 'f, 'g) CHOAlgorithm_scheme"   and r :: "nat"   and cfg :: "'d \<Rightarrow> 'e"   and HO :: "'d \<Rightarrow> 'd set"   and SHO :: "'d \<Rightarrow> 'd set"   and coord :: "'d \<Rightarrow> 'd"   and cfg' :: "'d \<Rightarrow> 'e" shows "CSHOnextConfig A r cfg HO SHO coord cfg' \<equiv> \<forall>p. \<exists>\<mu>\<in>SHOmsgVectors A r p cfg (HO p) (SHO p). CnextState A r p (cfg p) \<mu> (coord p) (cfg' p)"
mono2mono_lfilter: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "'c \<Rightarrow> 'd llist"   and P :: "'d \<Rightarrow> bool" assumes "monotone orda (\<sqsubseteq>) t" shows "monotone orda (\<sqsubseteq>) (\<lambda>x. lfilter P (t x))"
distrib_left: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring plus times" shows "times a (plus b c) = plus (times a b) (times a c)"
simps: fixes I :: "('e, 'f, 'd) interp"   and ODE :: "('e, 'd) ODE"   and sol :: "(real, 'd) vec" shows "mk_xode I ODE sol = (sol, ODE_sem I ODE sol)"
sel: fixes gpv :: "('b, 'd, 'e) gpv" shows "the_gpv (gpv_stop gpv) = map_spmf (map_generat Some id (\<lambda>rpv input. case input of None \<Rightarrow> Generative_Probabilistic_Value.Done None | Some input' \<Rightarrow> gpv_stop (rpv input'))) (the_gpv gpv)"
ln: shows "lookup \<Gamma> n = Some (shift (snd \<Gamma> - k) 0 (fst (fst \<Gamma> ! n)))"
inherited_def: fixes subs :: "'b \<Rightarrow> 'b set"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool" shows "inherited subs P = ((\<forall>A B. (P A \<and> P B) = P (A \<union> B)) \<and> (\<forall>A. P A = P (incLevel ` A)) \<and> (\<forall>n Gamma A. \<not> terminal subs Gamma \<longrightarrow> P A = P (insert (n, Gamma) A)) \<and> P {})"
witness_propD: fixes \<delta> :: "('d, 'e) ta_rule set"   and m :: "'d \<Rightarrow> 'e tree option"   and q :: "'d"   and t :: "'e tree" assumes "witness_prop \<delta> m"   and "m q = Some t" shows "accs \<delta> t q"
4: assumes "\<lbrakk>M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>; \<A>\<rbrakk>\<^sub>d \<I>" shows "\<lbrakk>M; \<A>\<rbrakk>\<^sub>d' \<I>"
dg_prod_2_ObjE: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and ab :: "V" assumes "digraph \<alpha> \<AA>"   and "digraph \<alpha> \<BB>"   and "ab \<in>\<^sub>\<circ> (\<AA> \<times>\<^sub>D\<^sub>G \<BB>)\<lparr>Obj\<rparr>" obtains a :: "V" and b :: "V" where "ab = [a, b]\<^sub>\<circ>" and "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
simps_2: fixes v :: "variable"   and T :: "typ"   and \<rho> :: "variable \<Rightarrow> typ \<Rightarrow> Core.term" shows "subst (Fv v T) \<rho> = \<rho> v T"
sorted_on_insort_key_on: fixes r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list" assumes "Linear_order r"   and "f ` set (x # xs) \<subseteq> Field r" shows "sorted_on r (map f (insort_key_on r f x xs)) = sorted_on r (map f xs)"
mult_assoc2: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> E" shows "a \<cdot> b \<cdot> x = (a * b) \<cdot> x"
right_idem: fixes a :: "'d"   and b :: "'d" shows "min (min a b) b = min a b"
mat_size: shows "M_mat_R signs subsets \<in> carrier_mat (length signs) (length signs)"
Compose: fixes S :: "('fun, 'var) strand_step list"   and T :: "('fun, 'var) Term.term list"   and f :: "'fun"   and S' :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "simple S"   and "length T = arity f"   and "public f" shows "(S @ send\<langle>Fun f T\<rangle>\<^sub>s\<^sub>t # S', \<theta>) \<leadsto> (S @ map Send T @ S', \<theta>)"
arr_char: fixes f :: "'c \<times> 'c \<times> 'c" shows "C'.VVV.arr f = (C'.arr (fst f) \<and> C'.VV.arr (snd f) \<and> C'.src (fst f) = C'.trg (fst (snd f)))"
a: shows "x:M \<in> NOTLEFT (NOT B) (\<parallel><B>\<parallel>)"
bisimE_2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "\<Psi> \<rhd> P \<leadsto>[bisim] Q"
image_func1: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "(`) (f \<circ> g) = (`) f \<circ> (`) g"
more_arith_simps_10: fixes a :: "'e" shows "- (- a) = a"
openin_contains_cball: fixes T :: "'b set"   and S :: "'b set" shows "openin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x\<in>S. \<exists>e>0. cball x e \<inter> T \<subseteq> S))"
reflT: fixes F :: "behavior \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
has_bochner_integral_divide_zero: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x / c) (x / c)"
witness_valid: fixes Y :: "nat"   and r :: "nat"   and rs :: "nat list" shows "valid_net' (witness Y r rs)"
rcos_mult_step3: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "H <#> (H #> x) #> y = H #> x \<otimes> y"
B: shows "round index key p q r' (v, ns, drop (Suc (Suc n)) xs) = (u', ns', xs')"
glb_eqI: fixes M :: "'a set"   and l :: "'a" assumes "M >>| l" shows "\<Sqinter>M = l"
CONSTRAINT_def: fixes f :: "'c"   and R :: "('d \<times> 'c) set" shows "CONSTRAINT f R \<equiv> True"
mod_h_bot_indep: fixes P :: "assn"   and h :: "heap"   and h' :: "heap" shows "((h, {}) \<Turnstile> P) = ((h', {}) \<Turnstile> P)"
sgn_fract_def: fixes q :: "'a fract" shows "sgn q = (if q = 0 then 0 else if 0 < q then 1 else - 1)"
power_commutes: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "class.monoid_mult one times" shows "times (power.power one times a n) a = times a (power.power one times a n)"
FDERIV_floatariths_def: fixes fas :: "floatarith list"   and xs :: "nat list"   and d :: "floatarith list" shows "FDERIV_floatariths fas xs d = map (\<lambda>fa. FDERIV_floatarith fa xs d) fas"
Int_stableD: fixes M :: "'b set set"   and a :: "'b set"   and b :: "'b set" assumes "Int_stable M"   and "a \<in> M"   and "b \<in> M" shows "a \<inter> b \<in> M"
rel_4: shows "int_of_uint32 x' \<le> p"
one_closed: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.monoid G" shows "\<one>\<^bsub>G\<^esub> \<in> carrier G"
elims: fixes x :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and xa :: "v"   and xb :: "(pat \<times> exp0) list"   and xc :: "v"   and y :: "(exp0 \<times> pat \<times> (char list \<times> v) list, v) result" assumes "cupcake_match_result x xa xb xc = y" obtains uu_ :: "(char list, char list, nat \<times> tid_or_exn) namespace" and uv_ :: "v" and err_v :: "v" where "x = uu_" and "xa = uv_" and "xb = []" and "xc = err_v__" and "y = Rerr (Rraise err_v__)"   | cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace" and v0 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and err_v :: "v" where "x = cenv" and "xa = v0" and "xb = (p, e) # pes" and "xc = err_v" and "y = (if distinct (pat_bindings p []) then case cupcake_pmatch cenv p v0 [] of No_match \<Rightarrow> cupcake_match_result cenv v0 pes err_v | Match_type_error \<Rightarrow> Rerr (Rabort Rtype_error) | Match env' \<Rightarrow> Rval (e, p, env') else Rerr (Rabort Rtype_error))"
foundation4: fixes \<tau> :: "'a state \<times> 'a state" shows "\<tau> |\<noteq> null"
INFM_EX: fixes P :: "'b \<Rightarrow> bool" assumes "\<exists>\<^sub>\<infinity>x. P x" shows "\<exists>x. P x"
k: shows "k = (LEAST k. \<pi> k = x \<or> \<pi> k = y)"
ring_mat_simps_1: fixes ty :: "'b itself"   and n :: "nat"   and b :: "'c" shows "(\<otimes>\<^bsub>ring_mat ty n b\<^esub>) = (*)"
RD_bounds_sublistD: fixes Pi :: "(('b, bool) fmap \<times> ('b, bool) fmap) set" assumes "finite Pi" shows "problem_plan_bound Pi \<le> RD Pi"
tagged_partial_division_of_trivial: fixes p :: "('a \<times> 'a set) set" shows "(p tagged_partial_division_of {}) = (p = {})"
\<tau>move2While5': fixes pc :: "nat"   and c :: "'addr expr1"   and P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1" assumes "pc = length (compE2 c)" shows "\<tau>move2 P h stk (while (c) e) pc None"
less_imp_neq: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less"   and "less x y" shows "x \<noteq> y"
that_2: fixes i :: "'a" assumes "i \<in> Basis"   and "y = - i" shows "thesis"
absolutely_integrable_scaleR_right: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real"   and c :: "'d" assumes "f absolutely_integrable_on S" shows "(\<lambda>x. f x *\<^sub>R c) absolutely_integrable_on S"
psuper_not_sc: fixes T :: "'a set" assumes "S \<subset> T"   and "T \<subseteq> verts G"   and "strongly_connected (G \<restriction> T)" shows "False"
eq3: shows "ind_in_set A ` ({0..<M} \<inter> A) \<inter> ind_in_set A ` (A - B) = ind_in_set A ` ({0..<M} \<inter> A) \<inter> ind_in_set A ` ({0..<M} \<inter> (A - B))"
of_bool_eq_2: fixes one :: "'c"   and zero :: "'c" assumes "class.zero_neq_one one zero" shows "zero_neq_one.of_bool one zero True = one"
m_cf_def: fixes S :: "('e, 'f) Ring_scheme"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "m_cf S c = (fst c, \<lambda>j. -\<^sub>a\<^bsub>S\<^esub> snd c j)"
is_ipd_in_pdt: fixes y :: "'n"   and x :: "'n" assumes "y ipd\<rightarrow> x" shows "(x, y) \<in> pdt"
ll_on_open_rev: shows "ll_on_open (preflect t0 ` T) (\<lambda>t. - f (preflect t0 t)) X"
action_loc_addr_fun: fixes ad :: "'a"   and al :: "addr_loc"   and P :: "'b prog"   and E :: "('c \<times> ('a, 'c) obs_event action) llist"   and a :: "nat"   and ad' :: "'a"   and al' :: "addr_loc" assumes "(ad, al) \<in> action_loc P E a"   and "(ad', al') \<in> action_loc P E a" shows "ad = ad'"
lookup_true: fixes l :: "'a list"   and f :: "'a \<Rightarrow> bool"   and r :: "'a" assumes "lookup l f = Some r" shows "f r"
tc_extract_function: fixes p :: "'b set"   and c :: "'b com"   and q :: "'b set" assumes "[p]   c   [q]" shows "\<exists>f. \<forall>s. s \<in> p \<longrightarrow> f s \<in> q"
distinct_3: fixes x1 :: "'e" shows "mutator x1 \<noteq> sys"
assms_1: shows "strict_monoidal_functor B T\<^sub>B \<alpha>\<^sub>B \<iota>\<^sub>B C T\<^sub>C \<alpha>\<^sub>C \<iota>\<^sub>C F"
eq_iff_2: fixes name :: "name"   and namea :: "name" shows "(Var name = Var namea) = (name = namea)"
dlist_not_memb_remove: fixes dl :: "'b dlist"   and x :: "'b" assumes "\<not> member dl x" shows "remove x dl = dl"