hcju/isabelleps huggingface dataset:
nres_rel_def: fixes R :: "('e \<times> 'g) set" shows "\<langle>R\<rangle>nres_rel \<equiv> {(c, a). c \<le> \<Down> R a}"
C_eq_None: 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> []"   and "applied_rule_rev Cp x p = \<bottom>" shows "Cp (list2FWpolicy p) x = \<bottom>"
effect_to_assignments_def: fixes op :: "'a strips_operator" shows "effect_to_assignments op \<equiv> effect__strips op"
orbit_contains_rep: fixes m :: "'a set" assumes "m \<in> subgroups_of_size (p ^ a)" shows "m \<in> conjP.orbit m"
cases: fixes x :: "arctic" obtains n :: "int" where "x = Num_arc n"   | "x = MinInfty"
length_map_index': fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'c list" shows "length (map_index' n f xs) = length xs"
size_6: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "size (IF x31 x32 x33) = size x32 + size x33 + Suc 0"
bi_unique_def: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "bi_unique R = ((\<forall>x y z. R x y \<longrightarrow> R x z \<longrightarrow> y = z) \<and> (\<forall>x y z. R x z \<longrightarrow> R y z \<longrightarrow> x = y))"
simps_6: fixes \<iota> :: "nat"   and V :: "('a, 'b) MWLsCom list" shows "pp (spawn\<^bsub>\<iota>\<^esub> V) = \<iota>"
transp_def: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
pre_if_regular_and_not_post: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and i :: "nat" assumes "computation channel trans send recv init final"   and "regular_event (t ! i)"   and "\<not> computation.postrecording_event channel trans send recv init t i"   and "i < length t" shows "computation.prerecording_event channel trans send recv init t i"
pwd_spy_enc_sign_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_enc_sign n)"
tl_self_iff_empty: fixes xs :: "'b list" shows "(tl xs = xs) = (xs = [])"
pickFT_finished: fixes cl :: "('d, 'e, 'f) cmd list" assumes "WtFT cl = 1" shows "finished (cl ! pickFT cl)"
A71_DAcc_level1: shows "DAcc level1 sA71 = {sA81}"
euler_increment: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and h :: "real"   and t :: "real"   and x :: "'b" shows "euler_increment f h t x = f t x"
ret_path_none_pd: fixes x :: "'n" assumes "x \<in> local.nodes"   and "x \<noteq> local.return" obtains \<pi> :: "nat \<Rightarrow> 'n" and n :: "nat" where "is_path \<pi>" and "\<pi> 0 = x" and "\<pi> n = local.return" and "\<forall>i>0. \<not> x pd\<rightarrow> \<pi> i"
finite_Set_filter: fixes A :: "'b set"   and P :: "'b \<Rightarrow> bool" assumes "finite A" shows "finite (Set.filter P A)"
assms_1: shows "dim_row A = prod_list (nths ds rmodes)"
exp_dvdE: 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 push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit"   and "dvd.dvd times (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) a" obtains b :: "'a" where "a = push_bit n b"
simps_12: fixes f11 :: "'a"   and f12 :: "X \<Rightarrow> 'a \<Rightarrow> 'a"   and f13 :: "Y \<Rightarrow> 'b \<Rightarrow> 'a"   and f21 :: "X \<Rightarrow> 'a \<Rightarrow> 'b"   and f22 :: "Y \<Rightarrow> 'b \<Rightarrow> 'b"   and f23 :: "'b" shows "rec_X f11 f12 f13 f21 f22 f23 X.A = f11"
inner_ge_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and x :: "'a" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "0 \<le> inner x x"
i_expand_i_take_mult_Suc: fixes i :: "nat"   and k :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a message_af" assumes "i < k" shows "f \<odot> k \<Down> (n * k + Suc i) = f \<Down> n \<odot> k @ f n # NoMsg\<^bsup>i\<^esup>"
pp_surjective: fixes x :: "'a" shows "(top \<sqinter> - - x = top) = dense (top \<sqinter> x)"
append: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and xs :: "'b list"   and ys :: "'b list" assumes "monoid_list f z" shows "monoid_list.F f z (xs @ ys) = f (monoid_list.F f z xs) (monoid_list.F f z ys)"
cong: fixes S_raw :: "'a \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'd) spmf" shows "\<Sigma>_protocols_to_commitments.commit S_raw \<equiv> \<Sigma>_protocols_to_commitments.commit S_raw"
sig_c: shows "sigma_array_cost dmus (dmus !! i) (dmus !! Suc j) (dmus !! j !! j) j = (sig, c_sig)"
tendsto_Ln: fixes f :: "'a \<Rightarrow> complex"   and L :: "complex"   and F :: "'a filter" assumes "(f \<longlongrightarrow> L) F"   and "L \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "((\<lambda>x. Ln (f x)) \<longlongrightarrow> Ln L) F"
isconstant_polybound0: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "isconstant p = polybound0 p"
countableE_infinite: fixes S :: "'f set" assumes "countable S"   and "infinite S" obtains e :: "'f \<Rightarrow> nat" where "bij_betw e S UNIV"
C: shows "n = 0 \<or> n = Suc 0"
simps_1: fixes n :: "nat"   and l :: "nat" shows "nxt Ins (n, l) = (n + 1, if real n + 1 \<le> f2 * real l then l else nat \<lceil>e * real l\<rceil>)"
wf_A: shows "\<forall>T. statT = ArrayT T \<longrightarrow> invmode statM e \<noteq> SuperM"
NatStar_hypnat_of_nat: fixes A :: "nat set" assumes "finite A" shows "*s* A = hypnat_of_nat ` A"
wf_eff_pddl_ground_act_is_sound_opr: fixes g_opr :: "ground_action" assumes "wf_effect_inst (ground_action.effect g_opr)" shows "sound_opr g_opr (pddl_opr_to_act g_opr)"
bl_cast_long_short_long_ingoreLeadingZero_generic: fixes w :: "'b word" assumes "length (dropWhile Not (to_bl w)) \<le> LENGTH('c)"   and "LENGTH('c) \<le> LENGTH('b)" shows "of_bl (to_bl (of_bl (to_bl w))) = w"
ALG'_det_def: fixes Strat :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and i :: "nat"   and x :: "'c" shows "ALG'_det Strat qs init i x = ALG x qs i (swaps (snd (Strat ! i)) (steps' init qs Strat i), ())"
widest_spread_calc_spread: fixes ks :: "'a list"   and k :: "'a"   and s :: "real"   and ps :: "(real, 'a) vec list" assumes "ks \<noteq> []"   and "(k, s) = widest_spread ks ps" shows "s = calc_spread k ps"
in_unifiersE: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "\<sigma> \<in> unifiers E" obtains "\<And>e. e \<in> E \<Longrightarrow> fst e \<cdot> \<sigma> = snd e \<cdot> \<sigma>"
space_PiM_empty: fixes M :: "'b \<Rightarrow> 'c measure" shows "space (Pi\<^sub>M {} M) = {\<lambda>k. undefined}"
widens_refl: fixes P :: "'a prog"   and xs :: "ty list" shows "P \<turnstile> xs [\<le>] xs"
execn_Normal_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Basic f,Normal s\<rangle> =n\<Rightarrow> t" obtains "t = Normal (f s)"
iso_is_section: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.section f"
measure_bounded_differentiable_image: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and B :: "real" assumes "S \<in> lmeasurable"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> \<bar>det (matrix (f' x))\<bar> \<le> B" shows "Sigma_Algebra.measure lebesgue (f ` S) \<le> B * Sigma_Algebra.measure lebesgue S"
prems_1: shows "is_minimal_basis (set ((a # xs) @ ys))"
is2_replicate: shows "is2 \<lhd> replicate (2 ^ (length (tl rs) - 2)) (last rs)"
typing_ctxt_correct2: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and E :: "ctxt"   and T :: "Syntax.type"   and U :: "Syntax.type"   and r :: "trm" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>c\<^sub>t\<^sub>x\<^sub>t E : T \<Leftarrow> U"   and "\<Gamma> , \<Delta> \<turnstile>\<^sub>T r : U" shows "\<Gamma> , \<Delta> \<turnstile>\<^sub>T ctxt_subst E r : T"
islimpt_eq_acc_point: fixes l :: "'b"   and S :: "'b set" shows "l islimpt S = (\<forall>U. l \<in> U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
interchange_rows_j: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'d" shows "interchange_rows A i j $ j = A $ i"
up_one_closed: shows "(\<lambda>n. if n = 0 then \<one> else \<zero>) \<in> up R"
assms_5: shows "f : c \<mapsto>\<^bsub>dg_prod K \<AA>\<^esub> d"
\<sigma>\<tau>zz: shows "subst \<sigma>\<tau> (Var zz) out = exi z (cnj (subst \<tau> (Var z) out) (subst (subst \<sigma> (Var zz) out) (Var z) inp))"
assms_3: shows "is_valid_operator_sas_plus \<Psi> op'"
d_min'_def: shows "d_min' = truncate_down p (real_divl p 1 b - alpha * b)"
TC': fixes x :: "V" shows "TC x = x \<squnion> TC (\<Squnion> (elts x))"
dom_mkArr: fixes A :: "'t set"   and B :: "'t set"   and F :: "'t \<Rightarrow> 't" assumes "S'.arr (S'.mkArr A B F)" shows "S'.dom (S'.mkArr A B F) = S'.mkIde A"
incl_subgroup: fixes J :: "'a set"   and I :: "'a set" assumes "subgroup J G"   and "subgroup I (G\<lparr>carrier := J\<rparr>)" shows "subgroup I G"
disjE_realizer2: fixes P :: "bool"   and Q :: "'a \<Rightarrow> bool"   and x :: "'a option"   and R :: "'b \<Rightarrow> bool"   and f :: "'b"   and g :: "'a \<Rightarrow> 'b" assumes "case x of None \<Rightarrow> P | Some q \<Rightarrow> Q q"   and "P \<Longrightarrow> R f"   and "\<And>q. Q q \<Longrightarrow> R (g q)" shows "R (case x of None \<Rightarrow> f | Some q \<Rightarrow> g q)"
S: fixes \<psi> :: "'a ltl"   and j :: "nat" assumes "\<psi> \<in> \<^bold>G \<phi> \<union> \<G>" shows "suffix l w \<Turnstile> \<psi> = suffix (l + j) w \<Turnstile> \<psi>"
linear_upper2: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and l :: "real"   and x :: "real" assumes "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_real_derivative f' x) (at x within {a..b})"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> l \<le> f' x"   and "x \<in> {a..b}" shows "f x \<le> f b + l * (x - b)"
vsubset_vsingleton_rightI: fixes A :: "V"   and a :: "V" assumes "A = set {a} \<or> A = []\<^sub>\<circ>" shows "A \<subseteq>\<^sub>\<circ> set {a}"
Standard_diff: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> Standard"   and "y \<in> Standard" shows "x - y \<in> Standard"
is_valid_fun_call_def: fixes get :: "'a \<Rightarrow> 'b fundef option"   and fd2 :: "('c, 'd, 'a, 'e, 'f, 'g, 'h, 'i) Ubx.instr fundef"   and n :: "nat"   and \<Sigma>2 :: "('j, 'k, 'l) unboxed list"   and g :: "'a" shows "is_valid_fun_call get fd2 n \<Sigma>2 g \<equiv> n < length (body fd2) \<and> body fd2 ! n = Ubx.instr.ICall g \<and> (\<exists>gd. get g = Some gd \<and> list_all is_dyn_operand (take (arity gd) \<Sigma>2))"
conjunction_sup_nonaborting: fixes conj :: "'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice"   and chaos :: "'b::refinement_lattice"   and c :: "'b::refinement_lattice"   and d :: "'b::refinement_lattice" assumes "conj_distrib (conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (chaos::'b::refinement_lattice)"   and "(chaos::'b::refinement_lattice) \<sqsubseteq> (c::'b::refinement_lattice)"   and "(chaos::'b::refinement_lattice) \<sqsubseteq> (d::'b::refinement_lattice)" shows "(conj::'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice \<Rightarrow> 'b::refinement_lattice) (c::'b::refinement_lattice) (d::'b::refinement_lattice) = c \<squnion> d"
rb_spp_aux_def: shows "rb_spp_aux = tailrec.fun (\<lambda>x. snd (snd (fst x)) = []) (\<lambda>x. x) rb_spp_body"
MaxI_ex: fixes A :: "'a set" assumes "\<exists>x. x \<in> A"   and "finite A" shows "Max A \<in> A"
Cons_2: shows "\<exists>a\<in>set (x # xs). P a"
vfsequence_vcons: fixes x :: "V" shows "vfsequence (xs #\<^sub>\<circ> x)"
zeta_tail_bigo: fixes s :: "real" assumes "1 < s" shows "(\<lambda>n. Re (hurwitz_zeta (real n + 1) (complex_of_real s))) \<in> O(\<lambda>x. real x powr (1 - s))"
ao: shows "\<forall>j\<in>I. \<one>\<^bsub>G j\<^esub> \<in> carrier (G j)"
a1d_ua: shows "Q $$ (1, 1) * (Matrix.mat 1 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else b) * Q) $$ (0, 0) = u * a"
list_to_lpoly_coeff_nth: fixes i :: "nat"   and cs :: "rat list" assumes "i < length cs" shows "Abstract_Linear_Poly.coeff (list_to_lpoly cs) i = cs ! i"
mult_sound: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (\<lambda>s. P s * Q s)"
orderI: fixes a :: "'a upred"   and b :: "'a upred" assumes "a = (a \<and> b)" shows "b \<sqsubseteq> a"
distinct_23: fixes ccs1 :: "ccs"   and ccs2 :: "ccs"   and ccs' :: "ccs" shows "ccs1 \<oplus> ccs2 \<noteq> !ccs'"
len_lt: shows "len m i j xs < len m i j (start_remove xs k [])"
eval_eqI: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Par t u"   and "\<^bold>\<lfloor>t\<^bold>\<rfloor> = \<^bold>\<lfloor>u\<^bold>\<rfloor>" shows "\<lbrace>t\<rbrace> = \<lbrace>u\<rbrace>"
subnetsOfAdr_def: fixes a :: "'a" shows "subnetsOfAdr a = {x. a \<sqsubset> x}"
8: shows "nth_least (liset A w \<inter> {..<Suc (nth_least (liset A w) k')}) i = nth_least (liset A w) i"
cong_def: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "BNF_Corec.cong rel eval retr \<equiv> (\<forall>R S. R \<le> S \<longrightarrow> rel R \<le> rel S) \<and> (\<forall>R. equivp R \<longrightarrow> equivp (retr R)) \<and> (\<forall>R x y. rel_fun (rel R) R eval eval \<longrightarrow> rel (inf R (retr R)) x y \<longrightarrow> retr R (eval x) (eval y))"
LPDs_trans: fixes r :: "mregex"   and s :: "mregex"   and t :: "mregex" assumes "r \<in> LPD s"   and "s \<in> LPDs t" shows "r \<in> LPDs t"
dims_eq: shows "dims (mat_tensorlist_mult A Ts ds $ i) = dims (listsum ds (map (\<lambda>j. A $$ (i, j) \<cdot> Ts $ j) [0..<dim_vec Ts]))"
assms_4: shows "\<pp>\<^sub>1[b, c] \<cdot> f = f1"
finprod_one: fixes A :: "'c set" shows "(\<Sum>i\<in>A. (0::'b)) = (0::'b)"
sum_distrib_right: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and r :: "'c" shows "sup_monoid.sum f A * r = sup_monoid.sum (\<lambda>n. f n * r) A"
scaleR: fixes r :: "real"   and b :: "'a" shows "h (r *\<^sub>R b) = r *\<^sub>R h b"
pivot_set: shows "\<exists>f. pivot_fun A f (dim_col A) \<and> set (pivot_positions A) = {(i, f i) |i. i < dim_row A \<and> f i \<noteq> dim_col A}"
regular: fixes \<sigma> :: "heap option"   and c :: "'b Heap"   and r :: "'b"   and h' :: "heap" assumes "\<not> is_exn \<sigma>"   and "execute c (the_state \<sigma>) = Some (r, h')" shows "run c \<sigma> (Some h') r"
less_divide_eq_1_neg: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and 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 a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less one (divide b a) = less b a"
AddSA_CompFun_the: fixes SA :: "('d, 'e, 'f) seqauto"   and A :: "('d, 'e, 'f) hierauto"   and S :: "'d" assumes "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "the (CompFun (A [++] (S, SA)) S) = insert SA (the (CompFun A S))"
cases: fixes a :: "'e"   and g :: "'e \<Rightarrow> ('f, 'e) step" assumes "a \<in> terminates_on g" obtains s :: "'e" where "a = sb__" and "g sb__ = Done"   | s :: "'e" and s' :: "'e" where "a = sc__" and "g sc__ = Skip s'a__" and "s'a__ \<in> terminates_on g"   | s :: "'e" and aa :: "'f" and s' :: "'e" where "a = s" and "g s = Yield aa s'" and "s' \<in> terminates_on g"
simps_1: fixes P :: "'b"   and Cal :: "'b processCalculus"   and Q :: "'b" shows "P \<longmapsto>Cal\<^bsup>0\<^esup> Q = (P = Q)"
valid_split2: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and c :: "'f list multiset"   and c' :: "'f list multiset" assumes "valid fg c"   and "valid fg c'"   and "mon_c fg c \<inter> mon_c fg c' = {}" shows "valid fg (c + c')"
lookup_map_entry: fixes x :: "'c"   and f :: "'b \<Rightarrow> 'b"   and m :: "('c, 'b) mapping" shows "lookup (map_entry x f m) x = map_option f (lookup m x)"
degree_diff_less: fixes p :: "'a poly"   and n :: "nat"   and q :: "'a poly" assumes "degree p < n"   and "degree q < n" shows "degree (p - q) < n"
surjective: fixes r :: "('a, 'b, 'c) lens_scheme" shows "r = \<lparr>lens_get = get\<^bsub>r\<^esub>, lens_put = put\<^bsub>r\<^esub>, \<dots> = lens.more r\<rparr>"
simps_2: fixes G :: "graph"   and x :: "nat"   and xs :: "nat list" shows "path_weight G (x # xs) = (if xs = [] then 0 else weight G x (hd xs) + path_weight G xs)"
set_mult_def: fixes A :: "('c \<Rightarrow> 'd) set"   and B :: "('c \<Rightarrow> 'd) set" shows "set_mult A B = {\<lambda>x. f x * g x |f g. f \<in> A \<and> g \<in> B}"
ZObisT_transC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>01T d"   and "s \<approx> t"   and "(c, s) \<rightarrow>c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>c (d', t')" and "s' \<approx> t'" and "c' \<approx>01T d'"   | "s' \<approx> t" and "c' \<approx>01T d"
not_less_iff_gr_or_eq: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
rep_eq: fixes x :: "int_poly_factorization_algorithm" shows "int_poly_factorization_algorithm x = Rep_int_poly_factorization_algorithm x"
Ker_def: shows "Ker = \<eta> -` {\<one>'} \<inter> G"
adjoint_pair_preserved_by_iso: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<phi> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and \<psi> :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g"   and "Span.in_hom \<phi> f f'"   and "Span.iso \<phi>"   and "Span.in_hom \<psi> g g'"   and "Span.iso \<psi>" shows "Span.adjoint_pair f' g'"
intros_9: fixes vi :: "v list"   and j :: "nat"   and s :: "s"   and v :: "v"   and vs :: "v list"   and v' :: "v"   and i :: "nat" assumes "length vi = j" shows "\<lparr>s;vi @ [v] @ vs;[$C v', $Set_local j]\<rparr> \<leadsto>_ i \<lparr>s;vi @ [v'] @ vs;[]\<rparr>"
ccSup_upper2: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and u :: "'b"   and v :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A"   and "u \<in> A"   and "less_eq v u" shows "less_eq v (Sup A)"
constr_rep_6: fixes x1 :: "ccs" shows "Rep_ccs !x1 = Bang_Rep (Rep_ccs x1)"
inf_commutative: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - y \<sqinter> - x"
update_convs_4: fixes IntAgrK' :: "rat option \<Rightarrow> rat option"   and NonceS :: "rat option"   and IntMapK :: "rat option"   and ExtMapK :: "rat option"   and IntAgrK :: "rat option"   and ExtAgrK :: "rat option"   and more :: "'a" shows "IntAgrK_update IntAgrK' \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr> = \<lparr>NonceS = NonceS, IntMapK = IntMapK, ExtMapK = ExtMapK, IntAgrK = IntAgrK' IntAgrK, ExtAgrK = ExtAgrK, \<dots> = more\<rparr>"
univ_poly_is_monoid: fixes K :: "'a set" assumes "subring K R" shows "Group.monoid (K [X])"
intro: fixes x :: "'d \<Longrightarrow> 'e" assumes "weak_lens x"   and "wb_lens_axioms x" shows "wb_lens x"
card_Diff2_less: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "finite A"   and "x \<in> A"   and "y \<in> A" shows "card (A - {x} - {y}) < card A"
eq: shows "x * M x = 1 + sum_upto (\<lambda>n. moebius_mu n * frac (x / real n)) x"
sinvar_valid_remove_flattened_offending_flows: fixes m :: "'b SecurityInvariant_configured"   and nodesG :: "'b set"   and edgesG :: "('b \<times> 'b) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = nodesG, edges = edgesG\<rparr>" shows "c_sinvar m \<lparr>nodes = nodesG, edges = edgesG - \<Union> (c_offending_flows m \<lparr>nodes = nodesG, edges = edgesG\<rparr>)\<rparr>"
ngba_g_V0: fixes A :: "('d, 'c) ngba" shows "g_V0 (ngba_g A) = initial A"
\<Omega>: shows "\<Omega> = t -` trees_cyl (map_tree (\<lambda>_. space M) s)"
take_rows_from_matrix_def: fixes matr :: "'a mat"   and indices_to_keep :: "nat list" shows "take_rows_from_matrix matr indices_to_keep = mat_of_rows (dim_col matr) (take_indices (rows matr) indices_to_keep)"
RETURN_rule: fixes \<Phi> :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "\<Phi> x" shows "return x \<le> SPEC \<Phi>"
eq: fixes s :: "'a" assumes "s \<in> N" shows "l2 s \<le> l1 s"
init_set: fixes x :: "'node" assumes "x \<in> reach root"   and "x \<noteq> root" shows "x \<in> path {root} {root}"
rHom_func: fixes f :: "'d \<Rightarrow> 'e"   and A :: "('d, 'f) Ring_scheme"   and R :: "('e, 'g) Ring_scheme" assumes "f \<in> rHom A R" shows "f \<in> carrier A \<rightarrow> carrier R"
Inl_inject: fixes x :: "'e"   and y :: "'e" assumes "Inl x = Inl y" shows "x = y"
a_consistent_thunk_once: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and S :: "stack_elem list"   and e :: "exp"   and u :: "Arity" assumes "a_consistent (ae, a, as) (\<Gamma>, Var x, S)"   and "map_of \<Gamma> x = Some e"   and "ae x = up\<cdot>u"   and "heap_upds_ok (\<Gamma>, S)" shows "a_consistent (env_delete x ae, u, as) (delete x \<Gamma>, e, S)"