hcju/isabelleps huggingface dataset:
rcosetsI: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set"   and x :: "'a" assumes "Group.group G"   and "H \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #>\<^bsub>G\<^esub> x \<in> rcosets\<^bsub>G\<^esub> H"
add_mset_remove_trivial: fixes x :: "'b"   and M :: "'b multiset" shows "add_mset x M - {#x#} = M"
eval_inv_character: fixes \<chi> :: "'a \<Rightarrow> complex"   and j :: "'a" shows "inv_character \<chi> j = cnj (\<chi> j)"
that: fixes g :: "'a \<Rightarrow> 'b" assumes "continuous_on S g"   and "g ` S \<subseteq> U"   and "\<And>x. x \<in> C \<Longrightarrow> g x = f x" shows "thesis"
take_bit_drop_bit: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (drop_bit n a) = drop_bit n (take_bit (m + n) a)"
CLS_chine: shows "\<lbrakk>\<lbrakk>TTfgh_THfgh.chine\<rbrakk>\<rbrakk> = Maps.tuple \<lbrakk>\<lbrakk>Tfg_Hfg.chine \<star> TTfgh.p\<^sub>1\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>0 (f \<star> g)\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 h\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>h.chine \<star> TTfgh.p\<^sub>0\<rbrakk>\<rbrakk>"
dagger_of_dagger_is_id: fixes M :: "complex mat" shows "M\<^sup>\<dagger>\<^sup>\<dagger> = M"
seqSubstPar: fixes P :: "pi"   and Q :: "pi"   and \<sigma> :: "(name \<times> name) list" shows "(P \<parallel> Q)[<\<sigma>>] = P[<\<sigma>>] \<parallel> Q[<\<sigma>>]"
take_minus_one_conv_butlast: fixes n :: "nat"   and l :: "'a list" assumes "n \<le> length l" shows "take (n - Suc 0) l = butlast (take n l)"
lift_transform_fun_cong: fixes t1 :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and e1 :: "exp"   and t2 :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and a1 :: "'b\<^sub>\<bottom>"   and a2 :: "'b\<^sub>\<bottom>"   and e2 :: "exp" assumes "\<And>a. t1 a e1 = t2 a e1"   and "a1 = a2"   and "e1 = e2" shows "lift_transform t1 a1 e1 = lift_transform t2 a2 e2"
vfsequence_vlrestriction_succ: fixes k :: "V" assumes "k \<in>\<^sub>\<circ> vcard xs" shows "xs \<restriction>\<^sup>l\<^sub>\<circ> ZFC_in_HOL.succ k = xs \<restriction>\<^sup>l\<^sub>\<circ> k #\<^sub>\<circ> xs\<lparr>k\<rparr>"
False: shows "\<not> a \<noteq> \<infinity>\<^sub>h"
fg: fixes i :: "'a"   and y :: "real" assumes "i \<in> Basis" shows "a \<bullet> i < g i y \<and> g i y < b \<bullet> i \<and> f i (g i y) = y"
\<psi>D_im: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "map (map (\<psi> T)) indVfbasis = negHorbit_homVfbasis T"
Quotient_transaction\<^sub>h: shows "Quotient (=) Transaction\<^sub>h the_Transaction\<^sub>h transaction_merkle.cr_transaction\<^sub>h"
zero_less_dist_iff: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.metric_space dist uniformity open" shows "(0 < dist x y) = (x \<noteq> y)"
coprime_imp_coprime: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "algebraic_semidom.coprime times one c d"   and "\<And>e. \<lbrakk>\<not> dvd.dvd times e one; dvd.dvd times e a; dvd.dvd times e b\<rbrakk> \<Longrightarrow> dvd.dvd times e c"   and "\<And>e. \<lbrakk>\<not> dvd.dvd times e one; dvd.dvd times e a; dvd.dvd times e b\<rbrakk> \<Longrightarrow> dvd.dvd times e d" shows "algebraic_semidom.coprime times one a b"
pair_list_eqI: fixes xs :: "('g \<times> 'h) list"   and ys :: "('g \<times> 'h) list" assumes "map fst xs = map fst ys"   and "map snd xs = map snd ys" shows "xs = ys"
pw_rwof_init_1: fixes m0 :: "'c nres"   and cond :: "'c \<Rightarrow> bool"   and step :: "'c \<Rightarrow> 'c nres"   and s :: "'c" assumes "nofail (m0 \<bind> WHILE cond step)"   and "inres m0 s" shows "rwof m0 cond step s"
A'_in_rows: shows "A' \<subseteq> set (cols A)"
RModuleHom_restrict0_submodule: fixes N :: "'v set" assumes "RModule FG (\<cdot>) N \<and> N \<subseteq> V" shows "RModuleHom FG (\<cdot>) N (\<star>) (T \<down> N)"
isArrV_simps_6: fixes T :: "Arraytype"   and a :: "nat" shows "isArrV (arrV T a) = True"
isomorphic_def: fixes a :: "'a"   and a' :: "'a" shows "A.isomorphic a a' = (\<exists>f. \<guillemotleft>f : a \<rightarrow>\<^sub>A a'\<guillemotright> \<and> A.iso f)"
curry_K: fixes c :: "'e" shows "curry (\<lambda>x. c) = (\<lambda>x y. c)"
simps_3: fixes x :: "'a"   and l1 :: "'a list"   and y :: "'a"   and l2 :: "'a list" shows "sublist_HOL4 (x # l1) (y # l2) = (x = y \<and> sublist_HOL4 l1 l2 \<or> sublist_HOL4 (x # l1) l2)"
upper_times: fixes A :: "'b interval"   and B :: "'b interval" shows "upper (A * B) = Max {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
suppa2__conga123: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "A B C SuppA D E F "   and "A' B' C' SuppA D E F " shows "A B C CongA A' B' C'"
trms\<^sub>s\<^sub>s\<^sub>t_proj_subset_1: fixes n :: "'g"   and A :: "('g strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (proj n A) \<subseteq> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
Local: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and f :: "state \<Rightarrow> state"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool"   and g :: "state \<Rightarrow> state \<Rightarrow> state" assumes "\<forall>s'. C \<turnstile>\<^sub>t {\<lambda>z s. P z s' \<and> s = f s'} c {\<lambda>z t. Q z (g s' t)}" shows "C \<turnstile>\<^sub>t {P} LOCAL f; c; g {Q}"
solves_ode_singleton: fixes y :: "real \<Rightarrow> 'b"   and t0 :: "real"   and X :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b" assumes "y t0 \<in> X" shows "(y solves_ode f) {t0} X"
until_constant_simp: fixes \<psi> :: "'a ltln"   and w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltln" assumes "is_constant \<psi>" shows "w \<Turnstile>\<^sub>n \<phi> U\<^sub>n \<psi> = w \<Turnstile>\<^sub>n \<psi>"
Wbis_Sym: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>w d" shows "d \<approx>w c"
abelian_monoid_axioms: shows "abelian_monoid (R Quot I)"
t_1: shows "A \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile> t"
eq_fin_concat_eq_fin_start: fixes u :: "'action list"   and v\<^sub>1 :: "'action list"   and v\<^sub>2 :: "'action list" assumes "u @ v\<^sub>1 =\<^sub>F u @ v\<^sub>2" shows "v\<^sub>1 =\<^sub>F v\<^sub>2"
vsv_vimageI2: fixes r :: "V"   and a :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r"
bot_in: shows "{#} \<in> N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)"
A_def: shows "A = {x \<in> {a..b}. dist (f x) (f a) \<le> M * (x - a)}"
add_less_cancel_right: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and c :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less (plus a c) (plus b c) = less a b"
ide_in_hom: fixes a :: "'a" shows "ide a = in_hom a a a"
Hv_sel_1: fixes v :: "(real, 7) vec" shows "Ree (Hv v) = 0"
lincomb_eq_mat_mult: fixes A :: "'a mat"   and nc :: "nat"   and v :: "'a vec" assumes "A \<in> carrier_mat n nc"   and "v \<in> carrier_vec nc"   and "distinct (cols A)" shows "lincomb (\<lambda>a. v $ find_first a (cols A)) (set (cols A)) = A *\<^sub>v v"
Suc_1: fixes \<rho> :: "complex mat" assumes "\<rho> \<in> carrier_mat m m"   and "partial_density_operator \<rho>"   and "M1 \<in> carrier_mat m m"   and "adjoint M1 * M1 \<le>\<^sub>L 1\<^sub>m m"   and "\<And>\<rho>. \<rho> \<in> carrier_mat m m \<and> partial_density_operator \<rho> \<Longrightarrow> partial_density_operator (DS \<rho>)"   and "\<And>\<rho>. \<rho> \<in> carrier_mat m m \<and> partial_density_operator \<rho> \<Longrightarrow> DS \<rho> \<in> carrier_mat m m" shows "partial_density_operator (denote_while_n_iter M0 M1 DS n \<rho>)"
has: shows "P \<turnstile> C has F:T (fm) in E"
fold_replicate: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and x :: "'b" shows "fold f (replicate n x) = f x ^^ n"
simps_7: fixes b :: "expr"   and e1 :: "expr"   and e2 :: "expr" shows "free_vars_expr_code (IF b THEN e1 ELSE e2) = free_vars_expr_code b \<union> free_vars_expr_code e1 \<union> free_vars_expr_code e2"
inject: fixes x51 :: "pdf_type"   and x52 :: "pdf_type"   and y51 :: "pdf_type"   and y52 :: "pdf_type" shows "(PRODUCT x51 x52 = PRODUCT y51 y52) = (x51 = y51 \<and> x52 = y52)"
skl1_step4_refines_a0i_commit_skip_r: fixes A :: "agent"   and B :: "agent"   and Ni :: "msg"   and Rb :: "fid_t"   and gnx :: "msg" shows "{R0sk1iar \<inter> UNIV \<times> skl1_inv6} (a0i_commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)\<^sup>=, skl1_step4 Rb A B Ni gnx {> R0sk1iar}"
subset_imp_ds_subset: fixes r :: "('a \<times> 'a) set"   and S :: "'a set"   and T :: "'a set" assumes "trans r"   and "S \<subseteq> r \<down>s T" shows "r \<down>s S \<subseteq> r \<down>s T"
elt_set_plus_def: fixes a :: "'c"   and B :: "'c set" shows "a +o B = {c. \<exists>b\<in>B. c = a + b}"
steps_closure_is_weak_reduction_simulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" shows "weak_reduction_simulation (stepsClosure Rel Cal) Cal"
change_env_le: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list"   and v' :: "val"   and \<rho>' :: "(nat \<times> val) list" assumes "v \<in> E e \<rho>"   and "v' \<sqsubseteq> v"   and "FV e \<turnstile> \<rho> \<sqsubseteq> \<rho>'" shows "v' \<in> E e \<rho>'"
prems_2: shows "(x # xs, yys) \<in> lexord r"
lc_monomial: fixes t :: "'t"   and c :: "'c" shows "lc (monomial c t) = c"
estep_Z: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Z, xs, ls) # fs, rv)" shows "estep (encode_config c) = encode_config (step c)"
cInf_eq_minimum: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and X :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "s2p X z"   and "\<And>x. s2p X x \<Longrightarrow> less_eq z x" shows "Inf X = z"
cases: fixes \<delta> :: "real \<Rightarrow> real"   and a1 :: "real \<times> real"   and a2 :: "(real \<times> real \<times> real) list" assumes "fine \<delta> a1 a2" obtains a :: "real" where "a1 = (aa__, aa__)" and "a2 = []"   | b :: "real" and c :: "real" and D :: "(real \<times> real \<times> real) list" and a :: "real" and x :: "real" where "a1 = (a, c)" and "a2 = (a, x, b) # D" and "fine \<delta> (b, c) D" and "a < b" and "a \<le> x" and "x \<le> b" and "b - a < \<delta> x"
sel: fixes x :: "unit + 'b \<times> 'b list_R1" shows "unR (list_R1 x) = x"
trancl_rtrancl_absorb: fixes R :: "('b \<times> 'b) set" shows "(R\<^sup>+)\<^sup>* = R\<^sup>*"
Decision_approximating_bigstep_fun: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list"   and X :: "final_decision" shows "approximating_bigstep_fun \<gamma> p rs (Decision X) = Decision X"
transaction_projection_for_def: fixes participant :: "participant"   and t :: "transaction" shows "Canton_Transaction_Tree.transaction_projection_for participant t = (let tvs = Canton_Transaction_Tree.transaction_views_for participant t in if tvs = [] then blind_transaction t else the (transaction.Merge (set tvs)))"
rep_eq: fixes x :: "'b interval" shows "lower x = fst (bounds_of_interval x)"
a: shows "\<forall>\<^sub>F K in sequentially. ennreal (1 - d) < emeasure M (Ogood \<inter> {x \<in> space M. \<forall>n\<in>{1..Nf 1}. \<forall>l\<in>{1..n}. - (real K * real l) < u n x - u (n - l) x})"
image_iff: fixes z :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "s2p (\<P> f A) z = (\<exists>x\<in>A. z = f x)"
group_isomorphisms_imp_iso: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and f :: "'a \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'a" assumes "group_isomorphisms G H f g" shows "f \<in> Group.iso G H"
resultant_impl_start_def: shows "resultant_impl_start = subresultant_prs_impl (\<lambda>(Gk, hk). if degree Gk = 0 then hk else (0::'b))"
neg_one_power_add_eq_neg_one_power_diff: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and k :: "nat"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus"   and "k \<le> n" shows "power.power one times (uminus one) (n + k) = power.power one times (uminus one) (n - k)"
set_size_list: fixes x :: "'a"   and xs :: "'a list"   and f :: "'a \<Rightarrow> nat" assumes "x \<in> set xs" shows "f x \<le> size_list f xs"
f3: shows "\<forall>x0 x1 x2. (\<exists>v3. Suc v3 < Suc x2 \<and> x1 !! v3 \<noteq> x0 !! v3) = (Suc (nna x0 x1 x2) < Suc x2 \<and> x1 !! nna x0 x1 x2 \<noteq> x0 !! nna x0 x1 x2)"
Dup_omit_branch_mem: fixes p :: "('b, 'c) fm"   and i :: "'c"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and xs :: "(nat \<times> nat) set"   and a :: "'c"   and q :: "('b, 'c) fm" assumes "Dup p i branch xs"   and "q at a in' set branch" shows "q at a in' set (omit_branch xs branch)"
solution_def: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "solution A n x = (\<forall>i<n. (\<Sum>j = 0..<n. A i j * x j) = A i n)"
run_tick_count_sub: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and n :: "nat" assumes "dilating f sub r" shows "#\<^sub>\<le> sub c n = #\<^sub>\<le> r c f n"
inv_equality: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "\<And>x. g (f x) = x"   and "\<And>y. f (g y) = y" shows "inv f = g"
mult_isol_var: fixes u :: "'a"   and x :: "'a"   and v :: "'a"   and y :: "'a" assumes "u \<le> x"   and "v \<le> y" shows "u \<cdot> v \<le> x \<cdot> y"
volume_empty: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" assumes "volume M f" shows "f {} = 0"
assms: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n"   and "j < m_j \<or> j = m_j \<and> m_i \<le> i" shows "p A i j"
block_size_lt_v: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size" shows "u_block_size \<le> int (card point_set)"
assms_12: shows "isUIP (opposite (getCl state)) (getC state) (getM state)"
is_cupcake_all_envD_1: fixes env :: "v sem_env" assumes "is_cupcake_all_env env" shows "is_cupcake_ns (sem_env.v env)"
ConstP_imp_TermP: fixes t :: "tm" shows "{ConstP t} \<turnstile> TermP t"
rconjby_inj: fixes x :: "'a" shows "inj (rconjby x)"
Units_r_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
solvable: shows "\<forall>x\<in>set (snd (yun_factorization gcd p)). degree (fst x) \<le> 2 \<or> (\<forall>x\<in>set (coeffs (fst x)). x \<in> \<rat>)"
assms_3: shows "((\<xi>, p), a, \<xi>', q) \<in> seqp_sos \<Gamma>"
assms: fixes m :: "nat"   and n :: "nat" assumes "0 < m"   and "0 < n" shows "fds_nth f m * fds_nth g n = fds_nth g n * fds_nth f m"
focus_coeffs_subset_Polys: fixes X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "range (lookup (focus X p)) \<subseteq> P[- X]"
weak_bounded_lattice_def: fixes L :: "('a, 'b) gorder_scheme" shows "weak_bounded_lattice L \<equiv> weak_lattice L \<and> weak_partial_order_bottom L \<and> weak_partial_order_top L"
simps_1: fixes P :: "'d \<Rightarrow> bool" shows "filter P [] = []"
r_cancel_one': fixes x :: "'a list"   and a :: "'a list" assumes "x \<in> carrier (poly_ring R)"   and "a \<in> carrier (poly_ring R)" shows "(x = a \<oplus>\<^bsub>poly_ring R\<^esub> x) = (a = \<zero>\<^bsub>poly_ring R\<^esub>)"
Liminf_def: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'b" shows "Liminf F f = (\<Squnion>P\<in>{P. eventually P F}. \<Sqinter> (f ` {x. P x}))"
inv_mult_group: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and y :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))"   and "y \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "inv\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> (x \<otimes>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> y) = inv\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> y \<otimes>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> inv\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> x"
failures_def: fixes P :: "'a process" shows "failures P \<equiv> fst (Rep_process P)"
multiset_inter_commute: fixes A :: "'c multiset"   and B :: "'c multiset" shows "A \<inter># B = B \<inter># A"
subst_simp_fn: fixes x :: "'a"   and y :: "'a"   and M :: "'a trm"   and T :: "PreSimplyTyped.type"   and A :: "'a trm" assumes "x \<noteq> y"   and "y \<notin> fvs M" shows "Fn y T A[x ::= M] = Fn y T A[x ::= M]"
measure_of_subset: fixes M :: "'c set set"   and \<Omega> :: "'c set"   and M' :: "'c set set"   and \<mu> :: "'c set \<Rightarrow> ennreal"   and \<mu>' :: "'c set \<Rightarrow> ennreal" assumes "M \<subseteq> Pow \<Omega>"   and "M' \<subseteq> M" shows "sets (measure_of \<Omega> M' \<mu>) \<subseteq> sets (measure_of \<Omega> M \<mu>')"
the_elem_eq: fixes x :: "'f" shows "the_elem {x} = x"
mem_compact_implies_subset_existence_interval: fixes x0 :: "'a"   and K :: "'a set" assumes "0 \<in> UNIV"   and "x0 \<in> X"   and "\<And>t. \<lbrakk>t \<in> UNIV; t \<in> existence_ivl0 x0\<rbrakk> \<Longrightarrow> flow0 x0 t \<in> K"   and "compact K"   and "K \<subseteq> X" shows "UNIV \<subseteq> existence_ivl0 x0"
equivalence_of_categories_axioms: shows "equivalence_of_categories (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F G \<eta> \<epsilon>"
return_wp_rule: fixes Q :: "'d \<Rightarrow> assn"   and x :: "'d" shows "<Q x> return x <Q>"
diff_ge_0_iff_ge: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (minus a b) = less_eq b a"
irreducible\<^sub>dE: fixes p :: "'a poly" assumes "irreducible\<^sub>d p" obtains "0 < degree p" and "\<And>q r. \<lbrakk>degree q < degree p; degree r < degree p\<rbrakk> \<Longrightarrow> p \<noteq> q * r"
shift_snth_less: fixes p :: "nat"   and xs :: "'d list"   and s :: "'d stream" assumes "p < length xs" shows "(xs @- s) !! p = xs ! p"
SV_inv4I: fixes x :: "v_state" assumes "\<And>v a r. votes x r a = Some v \<Longrightarrow> safe x (Suc r) v" shows "x \<in> SV_inv4"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>xs. P xs []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
open_Collect_less_Int: fixes s :: "'c set"   and f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "continuous_on s f"   and "continuous_on s g" shows "\<exists>A. open A \<and> A \<inter> s = {x \<in> s. f x < g x}"
representsD1: fixes B :: "'b set"   and r :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b" assumes "represents B r x" shows "keys r \<subseteq> B"
start_heap_def: fixes G :: "'a prog" shows "start_heap G \<equiv> [addr_of_sys_xcpt NullPointer \<mapsto> blank G NullPointer, addr_of_sys_xcpt ClassCast \<mapsto> blank G ClassCast, addr_of_sys_xcpt OutOfMemory \<mapsto> blank G OutOfMemory, addr_of_sys_xcpt NoClassDefFoundError \<mapsto> blank G NoClassDefFoundError, addr_of_sys_xcpt IncompatibleClassChangeError \<mapsto> blank G IncompatibleClassChangeError, addr_of_sys_xcpt NoSuchFieldError \<mapsto> blank G NoSuchFieldError, addr_of_sys_xcpt NoSuchMethodError \<mapsto> blank G NoSuchMethodError]"
Sup_less_def: 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 x :: "'c \<Rightarrow> 'b"   and w :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "complete_lattice.Sup_less Sup x w = Sup {y. \<exists>v<w. y = x v}"
nat_term_comp_nat_term_compare: fixes to :: "'a nat_term_order" shows "nat_term_comp (nat_term_compare to)"
simps: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" shows "abs_split.node\<^sub>i k ts t = (if length ts \<le> 2 * k then abs_split.T\<^sub>i (Node ts t) else case BTree_Set.split_half ts of (ls, (sub, sep) # rs) \<Rightarrow> abs_split.Up\<^sub>i (Node ls sub) sep (Node rs t))"
lookup_default_def: fixes d :: "'b"   and m :: "('c, 'b) mapping"   and k :: "'c" shows "lookup_default d m k = (case lookup m k of None \<Rightarrow> d | Some v \<Rightarrow> v)"
nil_in: shows "[] \<in> fst ` set (P @ Q)"
OUTfromVCorrect2_def: fixes x :: "chanID" shows "OUTfromVCorrect2 x \<equiv> OUTfromV x = {} \<longrightarrow> (\<forall>v. x \<notin> VARto v)"
seq_char: fixes i :: "'b"   and j :: "'b" shows "C'.seq i j = (i \<in> S \<and> j \<in> S \<and> seq (\<phi> i) (\<phi> j))"
coclop_lsl_lsu: shows "coclop (\<nu> \<circ> \<nu>\<^sup>\<natural>)"
post_meas_1_is_state: fixes n :: "nat"   and v :: "complex mat"   and i :: "nat" assumes "state n v"   and "prob1 n v i \<noteq> 0" shows "state n (post_meas1 n v i)"
eo: shows "(\<forall>p\<in>set qs. degree p = 0) \<or> (\<exists>p\<in>set qs. 0 < degree p)"
put_M_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and setter :: "('aj \<Rightarrow> 'aj) \<Rightarrow> ('z, 'x, 'y, 'ag option) RShadowRoot_scheme \<Rightarrow> ('z, 'x, 'y, 'ag option) RShadowRoot_scheme"   and v :: "'aj" assumes "h \<turnstile> ok put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr setter v" shows "ptr |\<in>| shadow_root_ptr_kinds h"