hcju/isabelleps huggingface dataset:
restrict_to_rtranclp: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "transp_on P A"   and "x \<in> A"   and "y \<in> A" shows "(restrict_to P A)\<^sup>*\<^sup>* x y = P\<^sup>=\<^sup>= x y"
prime_power_inj'_2: fixes p :: "'a"   and q :: "'a"   and m :: "nat"   and n :: "nat" assumes "prime p"   and "prime q"   and "p ^ m = q ^ n"   and "0 < m"   and "0 < n" shows "m = n"
assms_2: shows "\<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>"
all_nodes_same_var: shows "\<forall>no1\<in>set (prx @ node # sfx). \<forall>no2\<in>set (prx @ node # sfx). var no1 = var no2"
inject_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(\<langle>\<epsilon>, x1\<rangle> = \<langle>\<epsilon>, y1\<rangle>) = (x1 = y1)"
simps_1: fixes l :: "'a list"   and c :: "int" shows "Ipol l (Pc c) = \<guillemotleft>c\<guillemotright>"
prf_codomain_length: fixes key_prf :: "'prf_key"   and m :: "bool list" assumes "key_prf \<in> set_spmf prf_key_gen"   and "m \<in> prf_domain" shows "length (prf_fun key_prf m) = prf_clen"
induct: fixes P :: "'b \<Rightarrow> ('b \<times> bal) tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "('b \<times> bal) tree" assumes "\<And>uu_. P uu_ \<langle>\<rangle>"   and "\<And>x l a ba r. \<lbrakk>cmp x a = LT \<Longrightarrow> P x l; cmp x a = GT \<Longrightarrow> P x r\<rbrakk> \<Longrightarrow> P x \<langle>l, (a, ba), r\<rangle>" shows "P a0 a1"
naturality: fixes f :: "'j" assumes "J.arr f" shows "\<chi> (J.cod f) \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] \<chi>.A.map f = D f \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] \<chi> (J.dom f)"
simps: fixes s1 :: "('a, 'e, 'f, 'g) SLin_state_scheme"   and s2 :: "'h" shows "P15 (s1, s2) = Ball (abortVals s1) ((\<preceq>) (dstate s1))"
Chernoff: fixes X :: "'a set" shows "Chernoff_on X linear_cf"
admits_tupling: shows "\<forall>I. I \<noteq> UNIV \<longrightarrow> admits_tupling I"
enum_finite_1_def: shows "enum = [a\<^sub>1]"
div_Tr1: fixes r :: "nat"   and s :: "nat"   and l :: "nat" assumes "0 < r"   and "0 < s"   and "l \<le> s * r" shows "l div s \<le> r"
assms_3: shows "is_parallel_solution_for_problem \<Psi> \<psi>"
cyclic_set_eq: fixes xs :: "'d list"   and ys :: "'d list" assumes "cyclic xs ys" shows "set xs = set ys"
pi_product: fixes it_a :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it_a' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it_b :: "'b \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it_b' :: "'b \<Rightarrow> ('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "proper_it it_a it_a'"   and "\<And>x. proper_it (it_b x) (it_b' x)" shows "proper_it (set_iterator_product it_a it_b) (set_iterator_product it_a' it_b')"
ab_semigroup_add_hom_axioms: shows "ab_semigroup_add_hom (map_poly hom)"
integrable_weighted_\<theta>: fixes a :: "real"   and x :: "real" assumes "2 \<le> a"   and "a \<le> x" shows "(\<lambda>t. \<theta> t / (t * (ln t)\<^sup>2)) integrable_on {a..x}"
eta_into_beta_eta_reds: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<eta> t" shows "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t"
notIn_Un_bij_betw: fixes b :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and A' :: "'b set" assumes "b \<notin> A"   and "f b \<notin> A'"   and "bij_betw f A A'" shows "bij_betw f (A \<union> {b}) (A' \<union> {f b})"
finite_Inf_in: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "finite A"   and "A \<noteq> {}"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> inf x y \<in> A" shows "Inf A \<in> A"
span_diff: fixes x :: "'t \<Rightarrow>\<^sub>0 'c"   and S :: "('t \<Rightarrow>\<^sub>0 'c) set"   and y :: "'t \<Rightarrow>\<^sub>0 'c" assumes "x \<in> pmdl S"   and "y \<in> pmdl S" shows "x - y \<in> pmdl S"
Span_append_eq_set_add: fixes K :: "'a set"   and Us :: "'a list"   and Vs :: "'a list" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "set Vs \<subseteq> carrier R" shows "R.Span K (Us @ Vs) = R.Span K Us <+> R.Span K Vs"
outs_gpv_map_gpv_id: fixes \<I> :: "('d, 'e) \<I>"   and f :: "'g \<Rightarrow> 'f"   and gpv :: "('g, 'd, 'e) gpv" shows "outs_gpv \<I> (map_gpv f id gpv) = outs_gpv \<I> gpv"
one_lane: shows "|lan v1| = 1 \<and> |lan v2| = 1"
NA_Crypt_imp_Server_msg: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NA :: "msg"   and K :: "nat" assumes "A \<notin> bad"   and "A \<noteq> B"   and "evs \<in> otway"   and "Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace> \<in> parts (knows Spy evs)" shows "\<exists>NB. Says Server B \<lbrace>Crypt (shrK A) \<lbrace>NA, Agent A, Agent B, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Agent A, Agent B, Key K\<rbrace>\<rbrace> \<in> set evs"
scaleC_left: fixes r :: "complex"   and a :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'a"   and b :: "'b" shows "r *\<^sub>C a *\<^sub>V b = r *\<^sub>C (a *\<^sub>V b)"
simps_1: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and e1 :: "'c match_expr"   and e2 :: "'c match_expr"   and p :: "'d" shows "matches \<gamma> (MatchAnd e1 e2) p = (matches \<gamma> e1 p \<and> matches \<gamma> e2 p)"
cobounded2: fixes b :: "'x"   and a :: "'x" shows "b \<le> sup a b"
mirror_point_def: fixes x :: "'a"   and y :: "'a" shows "mirror_point x y = 2 *\<^sub>R x - y"
cod_inv: fixes f :: "'a \<times> 'a" assumes "CC.iso f" shows "CC.cod (CC.inv f) = CC.dom f"
trading_strategy_def: fixes p :: "'b \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real" shows "trading_strategy p = (portfolio p \<and> (\<forall>asset\<in>support_set p. borel_predict_stoch_proc G (p asset)))"
rep_eq: fixes x :: "'d ccsubspace set" shows "space_as_set (\<Sqinter> x) = \<Inter> (space_as_set ` x)"
run_owner_iv_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s" shows "PriKey (fst (snd (run_owner_iv n s))) \<notin> used s"
assms: fixes x :: "'a" assumes "x \<in> vars e" shows "E x = E' x"
append_replicate_commute: fixes n :: "nat"   and x :: "'b"   and k :: "nat" shows "replicate n x @ replicate k x = replicate k x @ replicate n x"
trans: fixes ba :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "flat_ord ba b a"   and "flat_ord ba c b" shows "flat_ord ba c a"
rel_dds_mono: fixes A' :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B' :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "A' \<le> A"   and "B \<le> B'" shows "rel_dds A B \<le> rel_dds A' B'"
vset_neq_1: fixes b :: "V"   and A :: "V"   and a :: "V" assumes "b \<notin>\<^sub>\<circ> A"   and "a \<in>\<^sub>\<circ> A" shows "b \<noteq> a"
assms: shows "(NodeS i s R, ii\<not>ni:arrive(m), NodeS i' s' R') \<in> node_sos S"
time_stuttering: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and n :: "nat"   and c :: "clock"   and \<tau> :: "'b tag_const"   and m :: "nat" assumes "dilating f sub r"   and "time (Rep_run sub n c) = \<tau>"   and "\<And>k. f n < k \<and> k \<le> m \<Longrightarrow> \<nexists>k\<^sub>0. f k\<^sub>0 = k"   and "f n < m" shows "time (Rep_run r m c) = \<tau>"
op_norm_triangle: fixes A :: "(('a, 'b) vec, 'c) vec"   and B :: "(('a, 'b) vec, 'c) vec" shows "\<parallel>A + B\<parallel>\<^sub>o\<^sub>p \<le> (\<parallel>A\<parallel>\<^sub>o\<^sub>p) + (\<parallel>B\<parallel>\<^sub>o\<^sub>p)"
dBox_seq: fixes P :: "'b hrel"   and Q :: "'b hrel"   and \<Phi> :: "'b upred" shows "\<^bold>[P ;; Q\<^bold>]\<Phi> = \<^bold>[P\<^bold>]\<^bold>[Q\<^bold>]\<Phi>"
state_and_event_determine_next: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and c'' :: "('p, 's, 'm) configuration" assumes "c \<turnstile> ev \<mapsto> c'"   and "c \<turnstile> ev \<mapsto> c''" shows "c' = c''"
repeatable_def: fixes t :: "'b ttree" shows "repeatable t = (\<forall>x. possible t x \<longrightarrow> nxt t x = t)"
fiter_induct_eq: fixes c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "(c::'a::refinement_lattice)\<^sup>\<star> ; (d::'a::refinement_lattice) = gfp (\<lambda>x::'a::refinement_lattice. c ; x \<sqinter> d)"
cond: fixes \<sigma>' :: "'state"   and coll :: "'coll"   and P :: "'prog"   and \<sigma> :: "'state" assumes "(\<sigma>', coll) \<in> csmall P \<sigma>"   and "R P coll"   and "Q P \<sigma>" shows "R' P \<sigma> \<sigma>' coll"
Iy: shows "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Py (snd x)))"
Compose: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi"   and O :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel"   and "(\<Psi>', T, U) \<in> Rel''"   and "(\<Psi>', U, O) \<in> Rel'" shows "(\<Psi>', S, O) \<in> Rel''"
dist_les_9: fixes nat' :: "nat" shows "ValFF \<notsqsubseteq> ValN\<cdot>nat'"
ESPEC_def: fixes \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool" shows "ESPEC \<Phi> \<Psi> \<equiv> SPEC (\<lambda>x. case x of Inl e \<Rightarrow> \<Phi> e | Inr r \<Rightarrow> \<Psi> r)"
assms_2: fixes x :: "real" assumes "x \<in> A" shows "1 \<le> f x"
characteriseAx: fixes r :: "'a list \<times> 'b sequent" assumes "r \<in> Ax" shows "r = ([],  \<LM> ff  \<RM> \<Rightarrow>* Em) \<or> (\<exists>i. r = ([],  \<LM> At i  \<RM> \<Rightarrow>* At i))"
InfiniteEnsures: fixes N :: "state \<times> state \<Rightarrow> bool"   and sigma :: "behavior"   and A :: "state \<times> state \<Rightarrow> bool"   and P :: "state \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>N"   and "sigma \<Turnstile> \<box>\<diamond>A"   and "\<turnstile> A \<and> N \<longrightarrow> P$" shows "sigma \<Turnstile> \<box>\<diamond>P"
\<phi>_left_support': fixes x :: "real"   and i :: "int"   and l :: "nat" assumes "x \<in> {real_of_int (i - 1) / 2 ^ (l + 1)..real_of_int i / 2 ^ (l + 1)}" shows "\<phi> (l, i) x = 1 + x * 2 ^ (l + 1) - real_of_int i"
elims_1: fixes x :: "('fsym, 'tp) efsym"   and y :: "bool" assumes "GE_wtFsym x = y" obtains f :: "'fsym" where "x = Oldf f" and "y = wtFsym f"   | \<sigma> :: "'tp" where "x = Wit \<sigma>" and "y = (\<not> isRes \<sigma> \<or> protCl \<sigma>)"
hyps_1: shows "(\<Union>a\<in>toplevel_summands r1. toplevel_summands \<guillemotleft>lderiv as \<guillemotleft>a\<guillemotright>\<guillemotright>) = toplevel_summands \<guillemotleft>lderiv as \<guillemotleft>r1\<guillemotright>\<guillemotright>"
\<tau>exec_moveE: fixes addr2thread_id :: "'a \<Rightarrow> 'b"   and thread_id2addr :: "'b \<Rightarrow> 'a"   and spurious_wakeups :: "bool"   and empty_heap :: "'c"   and allocate :: "'c \<Rightarrow> htype \<Rightarrow> ('c \<times> 'a) set"   and typeof_addr :: "'c \<Rightarrow> 'a \<Rightarrow> htype option"   and heap_read :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> bool"   and heap_write :: "'c \<Rightarrow> 'a \<Rightarrow> addr_loc \<Rightarrow> 'a val \<Rightarrow> 'c \<Rightarrow> bool"   and ci :: "('a, 'c) check_instr"   and P :: "'a J1_prog"   and t :: "'b"   and e :: "'a expr1"   and h :: "'c"   and stk :: "'a val list"   and loc :: "'a val list"   and pc :: "nat"   and xcp :: "'a option"   and s' :: "'a val list \<times> 'a val list \<times> nat \<times> 'a option" assumes "JVM_heap_base.\<tau>exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e h (stk, loc, pc, xcp) s'" obtains "JVM_heap_base.exec_move addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t e h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'" and "JVM_heap_base.\<tau>move2 typeof_addr P h stk e pc xcp"
1: shows "finite (fv (mk_eq (Abs T (subst_bvs1' s 1 (map (\<lambda>(x, y). Fv x y) vs))) (Abs T (subst_bvs1' t 1 (map (\<lambda>(x, y). Fv x y) vs)))) \<union> FV \<Gamma> \<union> fv s)"
ide_char: fixes a :: "'a" shows "Right_a.ide a = (Right_a.arr a \<and> ide a)"
"3_1": assumes "x = y"   and "\<And>x. \<lbrakk>x \<in> set xs; P x\<rbrakk> \<Longrightarrow> False" shows "filter P (merge_wrt rel xs ys) = filter P ys"
vec_hom_smult: fixes ev :: "'a"   and v :: "'a Matrix.vec" shows "vec\<^sub>h (ev \<cdot>\<^sub>v v) = hom ev \<cdot>\<^sub>v vec\<^sub>h v"
ffieldI1: fixes a :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> r \<union>\<^sub>\<circ> \<R>\<^sub>\<bullet> r" shows "a \<in>\<^sub>\<circ> \<F>\<^sub>\<bullet> r"
liset_prefix: fixes i :: "nat"   and A :: "'b set"   and v :: "'b llist"   and u :: "'b llist" assumes "i \<in> liset A v"   and "u \<le> v"   and "enat i < llength u" shows "i \<in> liset A u"
Vset_trans: fixes x :: "V"   and X :: "V"   and i :: "V" assumes "x \<in>\<^sub>\<circ> X"   and "X \<in>\<^sub>\<circ> Vset i" shows "x \<in>\<^sub>\<circ> Vset i"
positive_cong_eq: fixes \<mu>' :: "'a set \<Rightarrow> ennreal"   and \<mu> :: "'a set \<Rightarrow> ennreal" assumes "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu>' a = \<mu> a" shows "positive (sigma_sets \<Omega> A) \<mu>' = positive (sigma_sets \<Omega> A) \<mu>"
hyps_3: shows "\<Gamma>\<^sub>2 \<sqsubseteq>\<^sub>c \<Gamma>\<^sub>1"
is_reduced_GB_def: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "is_reduced_GB B \<equiv> is_Groebner_basis B \<and> is_auto_reduced B \<and> is_monic_set B \<and> 0 \<notin> B"
FieldI2: fixes i :: "'d"   and j :: "'d"   and R :: "('d \<times> 'd) set" assumes "(i, j) \<in> R" shows "j \<in> Field R"
wf_progress_satisfiable: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x" assumes "s \<in> wf_state"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x" shows "\<exists>taa x' m' s'. t \<turnstile> \<langle>x, shr s\<rangle> -taa\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s' t taa"
rev_zip: fixes xs :: "'d list"   and ys :: "'e list" assumes "length xs = length ys" shows "rev (zip xs ys) = zip (rev xs) (rev ys)"
intro: fixes hom :: "'b \<Rightarrow> 'c" assumes "monoid_add_hom hom" shows "comm_monoid_add_hom hom"
hyps: fixes xa :: "'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "xa \<in> fmran' \<pi>"   and "\<tau> \<le> xa"   and "\<sigma> \<le> xa" shows "\<tau> \<squnion> \<sigma> \<le> xa"
test_bit_over: fixes x :: "'b word"   and n :: "nat" assumes "size x \<le> n" shows "x !! n = False"
dhops_update_changed: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dip :: "nat"   and osn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt dip (osn, kno, val, hops, nhip)" shows "the (dhops (update rt dip (osn, kno, val, hops, nhip)) dip) = hops"
that: fixes A :: "((real, 'n) vec, 'n) vec" assumes "rotation_matrix A"   and "A *v axis k 1 = a" shows "thesis"
idiff_ile_mono: fixes m :: "enat"   and n :: "enat"   and l :: "enat" assumes "m \<le> n" shows "m - l \<le> n - l"
obtain_positive_integrable_function: fixes M :: "'b measure" assumes "sigma_finite_measure M" obtains f :: "'b \<Rightarrow> real" where "f \<in> borel_measurable M" and "\<And>x. 0 < f x" and "\<And>x. f x \<le> 1" and "integrable M f"
assms_4: shows "has_least_prefixpoint (l \<circ> h)"
eventually_subadditiveI: fixes N0 :: "nat"   and u :: "nat \<Rightarrow> real" assumes "\<And>m n. \<lbrakk>N0 < m; N0 < n\<rbrakk> \<Longrightarrow> u (m + n) \<le> u m + u n" shows "eventually_subadditive u N0"
notstutstep_nexnat1: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<not> stutstep s n" shows "nextnat (s |\<^sub>s n) = 1"
arev_o_arev: shows "arev \<circ> arev = id"
map_spmf_of_set_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "inj_on f A" shows "map_spmf f (spmf_of_set A) = spmf_of_set (f ` A)"
ftf_syn1: fixes a :: "'c event"   and u :: "'c event list"   and t :: "'c event list"   and s :: "'c event list"   and A :: "'c event set" shows "a \<notin> set u \<and> a \<notin> set t \<and> s setinterleaves ((t, u), A) \<longrightarrow> a \<notin> set s"
aq: shows "\<forall>p. prime p \<and> p dvd q \<longrightarrow> (\<exists>b. [a ^ ((n - 1) div p) = b] (mod n) \<and> coprime (b - 1) n)"
shared_share_acquired: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat" assumes "sharing_consistent \<S> \<O> sb"   and "a \<in> dom \<S>" shows "a \<in> dom (share sb \<S>) \<union> acquired True sb \<O>"
Norm_Distr2: fixes r :: "'c \<times> 'd \<Rightarrow> 'b"   and P :: "'e \<Rightarrow> 'c decision option"   and Q :: "('f \<Rightarrow> 'd decision option) list"   and d :: "'a \<Rightarrow> 'e \<times> 'f" shows "r o\<^sub>f (P \<Otimes>\<^sub>2 list2policy Q \<circ> d) = list2policy ((P \<Otimes>\<^sub>L Q) (\<Otimes>\<^sub>2) r d)"
isPBij: fixes \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "s \<equiv>\<^sub>\<beta> t" shows "\<beta> \<in> Pbij"
\<beta>_eval: fixes i :: "nat" assumes "i < N" shows "\<beta> $ i = complex_of_real (if f i then 1 / sqrt (real M) else 0)"
proj_list_Cons_1: fixes i :: "'a"   and d :: "'b"   and D :: "('a strand_label \<times> 'b) list" shows "proj i ((ln i, d) # D) = (ln i, d) # proj i D"
\<phi>_def: fixes f :: "'a" shows "\<phi> f \<equiv> \<a>\<^sub>D[\<tau>\<^sub>0 (trg\<^sub>C f), F f, \<sigma>.map\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D (\<tau>\<^sub>1 f \<star>\<^sub>D \<sigma>.map\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[G f, \<tau>\<^sub>0 (src\<^sub>C f), \<sigma>.map\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D (G f \<star>\<^sub>D D.inv (\<sigma>.counit (src\<^sub>C f))) \<cdot>\<^sub>D \<r>\<^sub>D\<^sup>-\<^sup>1[G f]"
cong_refl: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b b a"
converse_def: shows "converse \<equiv> \<lambda>r. _Collect (xa, x) ((\<lambda>x xa. (x, xa) \<in> r)\<inverse>\<inverse> xa x)"
assms: shows "p \<leadsto>\<^bsub>\<Gamma>\<^esub> q"
not_getbase_repeats: fixes D :: "char list"   and xs :: "base list" assumes "D \<notin> set (map getbase xs)" shows "Repeats D \<notin> set xs"
simps_3: fixes iface :: "iface"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and m :: "'a common_primitive match_expr"   and rs :: "'a common_primitive rule list"   and allowed :: "'a word set"   and denied1 :: "'a word set" shows "no_spoofing_algorithm iface ipassmt (Rule m Drop # rs) allowed denied1 = no_spoofing_algorithm iface ipassmt rs allowed (denied1 \<union> (get_all_matching_src_ips iface m - allowed))"
A_def: shows "A = (\<lambda>d. {k \<in> {0<..n}. gcd k n = d})"
disc_18: fixes x31 :: "nat"   and x32 :: "'d"   and x33 :: "'d"   and x34 :: "'e"   and x35 :: "'e"   and x36 :: "'f" shows "\<not> isSnapshot (Recv x31 x32 x33 x34 x35 x36)"
convex_hull_3_alt: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "convex hull {a, b, c} = {a + u *\<^sub>R (b - a) + v *\<^sub>R (c - a) |u v. 0 \<le> u \<and> 0 \<le> v \<and> u + v \<le> 1}"
g1: shows "g1 \<in> carrier G \<and> \<phi> g1 x = y"
iMODb_cut_greater_aux: fixes m :: "nat"   and t :: "nat"   and r :: "nat"   and c :: "nat" assumes "0 < m"   and "t < r + m * c"   and "r \<le> t" shows "(r + m * c - (t + m - (t - r) mod m)) div m = c - Suc ((t - r) div m)"
toplevel_summands_flatten: fixes X :: "'b rexp set" assumes "X \<noteq> {}"   and "finite X"   and "\<forall>x\<in>X. \<nexists>r s. x = Plus r s" shows "toplevel_summands (flatten PLUS X) = X"
sub1_triangulizes_column: fixes A :: "'a mat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "mute_fun mf"   and "local.sub1 (A $$ (l, l)) (dim_row A - Suc l) l (r, A) = (r', A')"   and "local.triangular_to l A"   and "0 < dim_row A"   and "l < dim_row A"   and "l < dim_col A" shows "local.triangular_column l A'"
safe_free_flowing_implies_safe_delayed': fixes ts\<^sub>s\<^sub>b :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S>\<^sub>s\<^sub>b :: "nat \<Rightarrow> bool option"   and m\<^sub>s\<^sub>b :: "nat \<Rightarrow> nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option" assumes "initial\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b \<S>\<^sub>s\<^sub>b"   and "(ts\<^sub>s\<^sub>b, m\<^sub>s\<^sub>b, \<S>\<^sub>s\<^sub>b) \<sim> (ts, m, \<S>)"   and "ini.safe_reach_direct program_step safe_free_flowing (ts, m, \<S>)" shows "ini.safe_reach_direct program_step safe_delayed (ts, m, \<S>)"
Cast_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and T :: "ty" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (Cast T e) h s s'"
Frame'_def: fixes P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> bool" shows "Frame' P P' Q F \<equiv> \<forall>s. (P' \<and>* P) s \<longrightarrow> (Q \<and>* F) s"
iMin_insert: fixes I :: "'b set"   and n :: "'b" assumes "I \<noteq> {}" shows "iMin (insert n I) = min n (iMin I)"
assms_2: shows "lookup p ` keys p \<subseteq> P[- X]"
Uint64_def: shows "Uint64 \<equiv> map_fun int_of_integer Abs_uint64 word_of_int"
elliptic_isometryD_2: fixes f :: "'b \<Rightarrow> 'b" assumes "elliptic_isometry f" shows "isometry f"
UN_le_eq_Un0: fixes M :: "nat \<Rightarrow> 'c set"   and n :: "nat" shows "\<Union> (M ` {..n}) = \<Union> (M ` {1..n}) \<union> M 0"
join_meet_distrib1: fixes L :: "('c, 'd) boolean_algebra_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "boolean_algebra L"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> y \<sqinter>\<^bsub>L\<^esub> z = (x \<squnion>\<^bsub>L\<^esub> y) \<sqinter>\<^bsub>L\<^esub> (x \<squnion>\<^bsub>L\<^esub> z)"
psubsetE: fixes A :: "'e set"   and B :: "'e set" assumes "A \<subset> B" obtains "A \<subseteq> B" and "\<not> B \<subseteq> A"
isTrue_PPf_implies_\<phi>R: assumes "isTrue (all yy (neg (PPf (Var yy) \<langle>\<phi>R\<rangle>)))" shows "isTrue \<phi>R"
P_def: shows "P = (\<not> (LIM x at_top. poly q x / poly p x :> at_bot) \<and> \<not> (LIM x at_top. poly q x / poly p x :> at_top))"