hcju/isabelleps huggingface dataset:
planning_dimacs_sound_code': fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and h :: "nat"   and dimacs_M :: "int list" assumes "ast_problem.well_formed prob"   and "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op"   and "\<forall>\<pi>\<in>set (ast_problem.ast\<delta> prob). is_standard_operator \<pi>" shows "let cnf_formula = SASP_to_DIMACS' h prob; decoded_plan = decode_DIMACS_model' dimacs_M h prob in dimacs_model dimacs_M cnf_formula \<longrightarrow> ast_problem.valid_plan prob decoded_plan"
arith_simps_56: fixes a :: "'a" shows "(0::'a) - a = - a"
fls_X_intpow_times_comm: fixes i :: "int"   and f :: "'a fls" shows "fls_X_intpow i * f = f * fls_X_intpow i"
dyad_rec_termination: fixes x :: "real"   and b :: "nat \<Rightarrow> 'b"   and l :: "'b \<Rightarrow> 'b"   and r :: "'b \<Rightarrow> 'b" assumes "x \<in> dyadics" shows "dyad_rec_dom (b, l, r, x)"
assms_5: shows "lv p d \<le> lv p' d"
fpr_target_is_head: fixes P :: "cld list"   and ctx :: "ctx"   and fqn :: "fqn"   and path :: "(ctx \<times> cld) list" assumes "find_path_f P ctx (cl_fqn fqn) = Some path" shows "\<exists>ctxcld. find_cld_f P ctx fqn = Some ctxcld \<and> (\<exists>path''. path = ctxcld # path'')"
Cauchy_def: fixes X :: "nat \<Rightarrow> 'c" shows "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
pm_of_poly_prod: fixes x :: "'c"   and f :: "'e \<Rightarrow> 'd poly"   and I :: "'e set" shows "pm_of_poly x (prod f I) = (\<Prod>i\<in>I. pm_of_poly x (f i))"
in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "\<guillemotleft>fst f : fst a \<rightarrow> fst b\<guillemotright>"   and "\<guillemotleft>snd f : snd a \<rightarrow> snd b\<guillemotright>" shows "CC.in_hom f a b"
OstrowskiTr10: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "\<not> 0 \<le> v x" shows "0 < v (x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> x \<cdot>\<^sub>r (1\<^sub>r \<plusminus> -\<^sub>a x))\<^bsup>\<hyphen> K\<^esup>)"
map_replicate_trivial: fixes x :: "'e"   and i :: "nat" shows "map (\<lambda>i. x) [0..<i] = replicate i x"
DEFL_eq_writerT: shows "DEFL('e\<cdot>('f, 'g) writerT) = DEFL('e\<cdot>'g writer\<cdot>'f)"
IH: fixes xa :: "nat" assumes "xa \<in> {..<k}"   and "x\<^sub>0 \<le> (ts ! xa) x" shows "gc2 * f2' ((ts ! xa) x) \<le> f ((ts ! xa) x)"
simps_4: fixes x2 :: "nat"   and x1 :: "nat" shows "R x2 \<noteq> I x1"
evals_finals_same: fixes es :: "char list exp list"   and P :: "(char list list \<times> char list exp) prog"   and s :: "heap \<times> sfields \<times> sheap"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap" assumes "finals es"   and "P \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>" shows "es = es' \<and> s = s'"
step_empty_stack: fixes x :: "(recf \<times> nat list \<times> nat list) list \<times> nat option" assumes "fst x = []" shows "fst (step x) = []"
simple_closed_path_wn3: fixes p :: "real \<Rightarrow> complex" assumes "simple_path p"   and "pathfinish p = pathstart p" obtains z :: "complex" where "z \<in> inside (path_image p)" and "cmod (winding_number p z) = 1"
med12s_def: fixes t :: "l2_state" shows "med12s t \<equiv> \<lparr>ik = ik t, secret = secret t, progress = progress t, signalsInit = signalsInit t, signalsResp = signalsResp t\<rparr>"
bounded_def: fixes S :: "'b set" shows "bounded S = (\<exists>x eee. \<forall>y\<in>S. dist x y \<le> eee)"
psi_bounds_sustained: fixes k :: "nat"   and c :: "real"   and n :: "nat" assumes "\<And>n. n \<le> 2 ^ k \<Longrightarrow> psi n \<le> c * ln 2 * real n"   and "4 * ln (1 + 2 ^ k) + 3 \<le> (c / 2 - 1) * ln 2 * (1 + 2 ^ k)"   and "4 / (1 + 2 ^ k) \<le> (c / 2 - 1) * ln 2"   and "0 \<le> c" shows "psi n \<le> c * ln 2 * real n"
solutions_def: fixes a :: "nat list"   and b :: "nat list" shows "solutions a b = (let A = Max (set b); B = Max (set a); m = length a; n = length b in minimize (check a b (generate A B m n)))"
one_more_paths_exists: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" assumes "ProofStepInduct G v0 v1 paths P_new sep_size" shows "\<exists>paths. DisjointPaths G v0 v1 paths \<and> card paths = Suc sep_size"
apex_def: shows "trg_f.apex \<equiv> C.dom trg_f.leg0"
ars_r_def: fixes x :: "'a" shows "antirange_semiring.ars_r ad x = ad (ad x)"
mult_p64_def: fixes p :: "uint64"   and x :: "uint64"   and y :: "uint64" shows "mult_p64 p x y = x * y mod p"
result_range_overflow: shows "(sint a sdiv sint b = 2 ^ (size a - 1)) = (a = - (2 ^ (size a - 1)) \<and> b = - 1)"
is_equality_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "is_equality R = (R = (=))"
rel_mono_strong: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b stream"   and y :: "'c stream"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "stream_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> sset x; yb \<in> sset y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "stream_all2 Ra x y"
succ_in_Sup_Ord: fixes \<beta> :: "V"   and A :: "V set" assumes "ZFC_in_HOL.succ \<beta> = \<Squnion> A"   and "small A"   and "A \<subseteq> ON"   and "Ord \<beta>" shows "ZFC_in_HOL.succ \<beta> \<in> A"
Cd_Above: fixes d :: "'b"   and X :: "'a set" shows "PdXXX.Cd d X = Above (Pd' d) (X \<inter> Field (Pd' d)) \<inter> X"
a_host_shadow_root_rel_def: fixes h :: "(_) heap" shows "local.a_host_shadow_root_rel h = (\<lambda>(x, y). (cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r x, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r y)) ` {(host, shadow_root). host |\<in>| element_ptr_kinds h \<and> |h \<turnstile> get_shadow_root host|\<^sub>r = Some shadow_root}"
div_mult_mod_eq_minus: fixes i :: "nat"   and n :: "nat"   and j :: "nat" shows "i div 2 ^ n * 2 ^ n + i mod 2 ^ n - j div 2 ^ n * 2 ^ n - j mod 2 ^ n = i - j"
sel_1: fixes x1 :: "'a"   and x2 :: "'a stream" shows "shd (x1 ## x2) = x1"
eval\<^sub>wp_eval\<^sub>w_eq: shows "C.eval\<^sub>wp = eval_abv\<^sub>C"
that: fixes \<epsilon> :: "real" assumes "0 < \<epsilon>"   and "\<epsilon> < k"   and "\<And>z. z \<in> ball 0 \<epsilon> - {0} \<Longrightarrow> cmod (f z) \<le> B" shows "thesis"
g2_path_continuous: shows "continuous_on (cbox a b) (\<lambda>x. (x, g2 x))"
outs'_gpv_Cont: fixes generat :: "('d, 'e, ('d, 'e, 'f) rpv) generat"   and gpv :: "('d, 'e, 'f) gpv"   and c :: "('d, 'e, 'f) rpv"   and x :: "'e"   and input :: "'f" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "c \<in> generat_conts generat"   and "x \<in> outs'_gpv (c input)" shows "x \<in> outs'_gpv gpv"
fs_cont: fixes k :: "nat" assumes "k \<le> 2 * N + 1" shows "continuous_on {real_of_int a..} (fs k)"
a_r1_q1: shows "a = m * q1 + r1"
has_derivative_transform_within_open: fixes f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'c"   and x :: "'b"   and t :: "'b set"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "(f has_derivative f') (at x within t)"   and "open s"   and "x \<in> s"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "(g has_derivative f') (at x within t)"
Left_CosetE: fixes A :: "'a set"   and P :: "'a set \<Rightarrow> bool" assumes "A \<in> contained.Partition"   and "\<And>a. a \<in> G \<Longrightarrow> P (a \<cdot>| L)" shows "P A"
zunion: fixes x :: "'a"   and a :: "'a zet"   and b :: "'a zet" shows "zin x (zunion a b) = (zin x a \<or> zin x b)"
substEnv_idEnv: fixes xs :: "'varSort"   and X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and x :: "'var" shows "idEnv &[X / x]_xs = idEnv [x \<leftarrow> X]_xs"
reduced_word_for_sum_list: fixes A :: "'a set"   and a :: "'a"   and as :: "'a list" assumes "reduced_word_for A a as" shows "sum_list as = a"
inf_sup_aci_5: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y = y \<sqinter> x"
p1: shows "{OrdP (Var k), VarP v} \<turnstile> (SyntaxN.All j (SyntaxN.All j' (SyntaxN.All k' (SeqStTermP v a (Var i) (Var j) s (Var k) IMP SeqStTermP v a (Var i) (Var j') s' (Var k') IMP Var j' EQ Var j))))(i::=t)"
inducts: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "integrable M f" obtains x :: "'d" where "has_bochner_integral M f x"
assoc'_simps_1: fixes f :: "'d"   and g :: "'d"   and h :: "'d" assumes "D.ide f"   and "D.ide g"   and "D.ide h"   and "src\<^sub>D f = trg\<^sub>D g"   and "src\<^sub>D g = trg\<^sub>D h" shows "D.arr \<a>\<^sub>D\<^sup>-\<^sup>1[f, g, h]"
Sups_dual_def: shows "\<partial> \<circ> Supi = Inf \<circ> (`) \<partial>"
x_split_def: fixes f :: "'a poly" shows "x_split f = (let fs = coeffs f; zs = takeWhile ((=) (0::'a)) fs in case zs of [] \<Rightarrow> (0, f) | a # list \<Rightarrow> (length zs, poly_of_list (dropWhile ((=) (0::'a)) fs)))"
intro: 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 "type_wf = ShadowRootClass.type_wf"   and "get_shadow_root = get_shadow_root"   and "get_shadow_root_locs = get_shadow_root_locs" shows "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"
REQ_RESP_def: shows "REQ_RESP = [: \<box> (\<lambda>xs ys. xs 0 \<longrightarrow> (\<diamond> (\<lambda>ys. ys 0)) ys) :]"
restr_trancl_under: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "Restr (r\<^sup>+) ((r\<^sup>*)\<inverse> `` A) = (Restr r ((r\<^sup>*)\<inverse> `` A))\<^sup>+"
kat_1_equiv_opp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(r x \<cdot> y \<le> y \<cdot> r z) = (y \<cdot> ar z \<le> ar x \<cdot> y)"
emptyset_is_bot: fixes S :: "'d set" shows "{} \<sqsubseteq> S"
ct_suffix_extend_ct_list_eq: fixes xs :: "ct list"   and ys :: "ct list"   and xs' :: "ct list"   and ys' :: "ct list" assumes "ct_suffix xs ys"   and "ct_list_eq xs' ys'" shows "ct_suffix (xs @ xs') (ys @ ys')"
xs_def: shows "xs = map (snd \<circ> quotient_of) (coeffs (of_int_poly p))"
weight_eq: fixes x :: "'v" assumes "x \<notin> A \<Gamma>'" shows "weight \<Gamma> x = weight \<Gamma>' x"
lang_rderivs: fixes n :: "nat"   and r :: "'b rexp"   and w :: "'a list" assumes "local.wf n r"   and "wf_word n w" shows "lang n (rderivs w r) = rQuots w (lang n r)"
delta': fixes S :: "'a set"   and a :: "'a"   and b :: "'a \<Rightarrow> 'b" assumes "finite S" shows "(\<Sum>k\<in>S. if a = k then b k else (0::'b)) = (if a \<in> S then b a else (0::'b))"
inverse_arrows_sym: fixes f :: "'c"   and g :: "'c" shows "C.inverse_arrows f g = C.inverse_arrows g f"
Cond_not_s\<^sub>2: fixes s\<^sub>2 :: "stmt"   and e :: "expr"   and s\<^sub>1 :: "stmt" shows "s\<^sub>2 \<noteq> Cond e s\<^sub>1 s\<^sub>2"
empty_eq_map_of_iff: fixes xys :: "('c \<times> 'd) list" shows "(Map.empty = map_of xys) = (xys = [])"
H2_J_2: fixes P :: "'b alpha_d_scheme \<times> 'b alpha_d_scheme \<Rightarrow> bool" assumes "\<forall>A. case A of (A, A') \<Rightarrow> P (A, A'\<lparr>ok := False\<rparr>) \<longrightarrow> P (A, A'\<lparr>ok := True\<rparr>)" shows "P is H2 healthy"
Lcg_def: fixes R :: "('h, 'i) Ring_scheme"   and M :: "('j, 'h, 'k) Module_scheme"   and j :: "nat" shows "Lcg R M j = (\<exists>H. finite_generator R M H \<and> j = card H)"
f2: shows "\<forall>n s sa na. (nn sa s n < Suc (Suc n) \<and> geom_proc (nn sa s n) s \<noteq> geom_proc (nn sa s n) sa \<or> \<not> na < Suc n) \<or> s !! na = sa !! na"
out_of_scope: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent"   and v' :: "'vertex" assumes "valid_in_port (v, p)"   and "v' |\<in>| vertices"   and "v' \<notin> scope (v, p)" shows "freshenLC (vidx v) ` local_vars (nodeOf v) p \<inter> subst_lconsts (inst v') = {}"
content_1_imp_nonconstant_prime_factors: fixes p :: "int poly"   and q :: "int poly" assumes "Polynomial.content p = 1"   and "q \<in># prime_factorization p" shows "0 < Polynomial.degree q"
bij_betw_imageI: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "inj_on f A"   and "f ` A = B" shows "bij_betw f A B"
i_aggregate_nth_eq_map: fixes f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and ag :: "'b list \<Rightarrow> 'b"   and n :: "nat" shows "i_aggregate f k ag n = ag (map f [n * k..<n * k + k])"
HF: fixes i :: "nat" shows "healthy (((\<lambda>x. wp (body ;; Embed x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) ^^ i) (\<lambda>P s. 0))"
endomorph: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b" assumes "VectorSpaceEnd smult V T" shows "T ` V \<subseteq> V"
listall_conj1: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "listall (\<lambda>x. P x \<and> Q x) xs" shows "listall P xs"
extensional_selection: fixes Cn :: "'b set \<Rightarrow> 'b set"   and selection :: "'b set \<Rightarrow> 'b \<Rightarrow> 'b set set"   and K :: "'b set"   and A :: "'b set"   and \<phi> :: "'b"   and \<psi> :: "'b" assumes "PartialMeetContraction Cn selection"   and "K = Cn A"   and "Cn {\<phi>} = Cn {\<psi>}" shows "selection K \<phi> = selection K \<psi>"
ker_f: shows "kernel (G\<lparr>carrier := H\<rparr>) (G\<lparr>carrier := N <#> H\<rparr> Mod N) f = N \<inter> H"
Eps_Opt_def: fixes P :: "'b \<Rightarrow> bool" shows "Eps_Opt P = (if \<exists>x. P x then Some (SOME x. P x) else None)"
write_actions_actions: fixes a :: "nat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist" assumes "a \<in> write_actions E" shows "a \<in> actions E"
distinct_1: fixes x1 :: "'a"   and x2 :: "'b" shows "SourceTerm x1 \<noteq> TargetTerm x2"
uminus_interval_vector: fixes a :: "'a"   and b :: "'a" shows "uminus ` cbox a b = cbox (- b) (- a)"
coprime_iff_gcd_eq_1: fixes one :: "'d"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and gcd :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and lcm :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit_factor :: "'d \<Rightarrow> 'd"   and normalize :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one a b = (gcd a b = one)"
zero_neq_one: fixes one :: "'d"   and zero :: "'d" assumes "class.zero_neq_one one zero" shows "zero \<noteq> one"
simps_1: fixes x :: "('a set set, 'b) Combinators"   and xs :: "('a set set, 'b) Combinators list" shows "separated (x # xs) = ((\<forall>s. s \<in> set xs \<longrightarrow> disjSD_2 x s) \<and> separated xs)"
mzeromap_mHom: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and N :: "('i, 'e, 'j) Module_scheme" assumes "Ring R"   and "R module M"   and "R module N" shows "mzeromap M N \<in> mHom R M N"
b: shows "divisor_set 48 = {1, 2, 3, 4, 6, 8, 12, 16, 24, 48}"
connected_closed: fixes s :: "'c set" shows "connected s = (\<nexists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
subst_redex_redex: fixes c :: "('a, 'b, 'c) com" shows "subst_redex c (redex c) = c"
rep_eq: fixes x :: "ennreal"   and xa :: "ennreal" shows "enn2ereal (x \<sqinter> xa) = enn2ereal x \<sqinter> enn2ereal xa"
set_eqvt: fixes pi :: "('e \<times> 'e) list"   and xs :: "'d list" shows "pi \<bullet> set xs = set (pi \<bullet> xs)"
is_a_intersec_def: fixes P :: "Points"   and A :: "Points"   and B :: "Points"   and C :: "Points"   and D :: "Points" shows "is_a_intersec P A B C D \<equiv> col P A B \<and> col P C D"
final_thread_def: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f" shows "final_thread s t \<equiv> case thr s t of None \<Rightarrow> False | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> final x \<and> ln = no_wait_locks \<and> wset s t = None"
ind: fixes xs :: "cell list"   and ys :: "cell list" assumes "x = length xs" shows "bl2nat (xs @ ys) 0 = bl2nat xs 0 + bl2nat ys (length xs)"
RRR_test: fixes f :: "real set \<Rightarrow> real set"   and g :: "real set \<Rightarrow> real set" assumes "f RRR = g RRR" shows "test f = test g"
simps_3: fixes v :: "t" shows "ct_eq (TSome v) TAny = True"
complete_leaves: fixes t :: "'a tree23"   and as :: "'a list" assumes "t \<in> trees (leaves as)" shows "complete t \<and> height t = 0"
less_imp_neq: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "x \<prec> y" shows "x \<noteq> y"
lq_triv: fixes u :: "'b list"   and z :: "'b list" shows "u\<inverse>\<^sup>>(u \<cdot> z) = z"
sup_atom: fixes x :: "'a" assumes "x \<noteq> \<bottom>"   and "\<forall>y z. (x \<le> y \<or> x \<le> z) = (x \<le> y \<squnion> z)" shows "atom x"
implies_inf_identity: fixes x :: "'a"   and y :: "'a" shows "(x \<leadsto> y) \<sqinter> y = y"
D1D: shows "poly_mod.Mp p D1 = poly_mod.Mp p D"
Rats_int_div_natE: fixes x :: "'a" assumes "x \<in> \<rat>" obtains m :: "int" and n :: "nat" where "0 < n" and "x = of_int m / of_nat n" and "coprime m (int n)"
extractFrameFresh_1: fixes x :: "name"   and P :: "('a, 'b, 'c) psi" assumes "x \<sharp> P" shows "x \<sharp> extractFrame P"
geod: shows "\<exists>g. isometry_on {0..dist x0 y0} g \<and> g 0 = x0 \<and> g (dist x0 y0) = y0"
rcos_self: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and H :: "'c set" assumes "Group.group G"   and "x \<in> carrier G"   and "subgroup H G" shows "x \<in> H #>\<^bsub>G\<^esub> x"
preserves_iso: fixes f :: "'a" assumes "local.iso f" shows "local.iso (f \<otimes> \<I>)"
YMap2: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and F :: "('d, ZF, 'e, ZF, \<lparr>mor2ZF :: 'e \<Rightarrow> ZF, \<dots> :: 'f\<rparr>, unit) Functor"   and X :: "'d"   and x :: "ZF" assumes "LSCategory C"   and "Ftor F : Op C \<longrightarrow> SET"   and "X \<in> obj\<^bsub>C\<^esub>"   and "x |\<in>| (F @@ X)" shows "YMap C X (YMapInv C X F x) = x"
d: shows "\<forall>i. A $ i $ i \<noteq> 0"
aneg: fixes n :: "nat" shows "- real n * T \<in> existence_ivl0 x \<and> flow0 x (- real n * T) = x"
extreme_point_exists_convex: fixes S :: "'b set" assumes "compact S"   and "convex S"   and "S \<noteq> {}" obtains x :: "'b" where "x extreme_point_of S"
semi_mojmir_axioms: shows "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"
UP_l_distr: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a"   and r :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P"   and "r \<in> carrier P" shows "(p \<oplus>\<^bsub>P\<^esub> q) \<otimes>\<^bsub>P\<^esub> r = p \<otimes>\<^bsub>P\<^esub> r \<oplus>\<^bsub>P\<^esub> q \<otimes>\<^bsub>P\<^esub> r"
"4_3": shows "(\<forall>l''. sublist l'' l''' \<and> list_all P1 l'' \<longrightarrow> length (filter P2 l'') \<le> k1) \<and> subseq l''' l2"
A: shows "min (esqrt (extended_Gromov_distance x y)) (eexp (ereal (- epsilonG TYPE('a)) * extended_Gromov_product_at basepoint x y)) \<le> 1"
hd_append2: fixes xs :: "'g list"   and ys :: "'g list" assumes "xs \<noteq> []" shows "hd (xs @ ys) = hd xs"
init_segment_insert_iff: fixes S :: "nat set"   and n :: "nat"   and T :: "nat set" assumes "S \<lless> {n}"   and "\<And>x. x \<in> T - S \<Longrightarrow> n \<le> x" shows "init_segment (insert n S) T = (init_segment S T \<and> n \<in> T)"
all_zero: shows "(\<forall>m\<ge>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'a)) = vector_all_zero_from_index (i, column_iarray k (matrix_to_iarray A))"
vint_of_int_in_vintI: fixes a :: "int" shows "a\<^sub>\<int> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
L2_set_mul: fixes F :: "'a set" shows "L2_set (cmod \<circ> (\<lambda>i. c * x i)) F = cmod c * L2_set (cmod \<circ> x) F"
sorted_iff_nth_mono: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq xs = (\<forall>i j. i \<le> j \<longrightarrow> j < length xs \<longrightarrow> less_eq (xs ! i) (xs ! j))"
set1_F_bound: fixes x :: "('p, 'x, 'y, 'q, 'r, 's, 't, 'u, 'v, 'w) F" shows "|set1_F x| \<le>o bd_F"
KBasic2_10: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<not>\<^bold>\<box>\<^bold>\<diamond>\<^bold>\<not>\<phi> in v]"
stable_def: fixes P :: "state \<Rightarrow> bool" shows "stable P \<equiv> \<box>($P \<longrightarrow> P$)"
that: fixes q :: "int mpoly"   and r :: "int mpoly"   and s :: "int mpoly" assumes "p = MPoly_Type.monom (monomial (Suc 0) x') 1 * q + r + s * ((MPoly_Type.monom (monomial (Suc 0) x') 1)\<^sup>2 - MPoly_Type.monom (monomial (Suc 0) x') 1)"   and "x' \<notin> vars r"   and "x' \<notin> vars q" shows "thesis"
info_disc_filtr: shows "disc_filtr M G \<and> sets (G \<bottom>) = {{}, space M}"
sup_ge2: fixes y :: "'a"   and x :: "'a" shows "y \<le> x \<squnion> y"
EM_remainder'_def: fixes n :: "nat"   and f :: "real \<Rightarrow> 'b"   and a :: "real"   and b :: "real" shows "EM_remainder' n f a b = ((- 1) ^ Suc n / fact n) *\<^sub>R integral {a..b} (\<lambda>t. pbernpoly n t *\<^sub>R f t)"
contains_predI: fixes A :: "'c set"   and x :: "'c" assumes "Predicate_Compile.contains A x" shows "pred.eval (Predicate_Compile.contains_pred A x) ()"
idinv: shows "optoy i \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) onl \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>(\<sigma>, l). state.id (\<sigma> i) = i)"
cbd: shows "b O d \<subseteq> b \<union> m \<union> ov \<union> s \<union> d"
assms_2: shows "y \<in> state_set (state_list s as)"
Inr_def: shows "Sum_Type.Inr = Abs_sum \<circ> Inr_Rep"
prv_sdsj_imp: fixes \<chi> :: "'fmla"   and F :: "'fmla set" assumes "\<chi> \<in> fmla"   and "F \<subseteq> fmla"   and "finite F"   and "\<And>\<phi>. \<phi> \<in> F \<Longrightarrow> prv (imp \<phi> \<chi>)" shows "prv (imp (sdsj F) \<chi>)"
simps_3: fixes bbs :: "bool list"   and bs :: "int list"   and a :: "Cooper.num" shows "Ifm bbs bs (Lt a) = (Inum bs a < 0)"
total_onPI: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> r x y \<or> x = y \<or> r y x" shows "total_onP A r"
trancl_into_trancl2: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>+" shows "(a, c) \<in> r\<^sup>+"
hom_dvd: fixes p :: "'a poly"   and q :: "'a poly" assumes "p dvd q" shows "poly_lift p dvd poly_lift q"
adjoin_subgroup: fixes H :: "'a set"   and a :: "'a" assumes "subgroup H G"   and "a \<in> carrier G" shows "subgroup (adjoin G H a) G"
val_1mx_pos: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "0 < v (1\<^sub>r \<plusminus> -\<^sub>a x)" shows "v x = 0"
map2_mult_scalar_proj_poly_syz: fixes xs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and n :: "nat"   and ys :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "map2 (\<odot>) xs (map (proj_poly_syz n) ys) = map (proj_poly_syz n \<circ> (\<lambda>(x, y). x \<odot> y)) (zip xs ys)"
perm_rem_simps_1: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "bij f"   and "x = y" shows "perm_rem x f y = x"
to_nat_b: shows "mod_type_class.to_nat b \<le> nrows A - Suc 0"
tickFree_def: fixes s :: "'c event list" shows "tickFree s = (tick \<notin> set s)"