hcju/isabelleps huggingface dataset:
cases: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "('b, 'c) com" assumes "PL_Indis indis"   and "PL_Indis.discr tval aval indis a" obtains c :: "('b, 'c) com" where "a = c" and "\<And>s c' s'. PL.transC tval aval (c, s) (c', s') \<Longrightarrow> (s, s') \<in> indis \<and> PL_Indis.discr tval aval indis c'" and "\<And>s s'. PL.transT tval aval (c, s) s' \<Longrightarrow> (s, s') \<in> indis"
\<I>_uniform_UNIV: shows "\<I>_uniform UNIV UNIV = \<I>_full"
SeqHRP_imp_OrdP: fixes x :: "tm"   and y :: "tm"   and s :: "tm"   and k :: "tm" shows "{SeqHRP x y s k} \<turnstile> OrdP k"
v'_def: shows "v' \<equiv> hd (tl (Q_hit_pre @ [y]))"
edge'_simps_3: fixes y :: "'v vertex ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.local.\<Psi>.edge' y ??.MFMC_Flow_Attainability.flow_attainability.node.SOURCE = False"
inner_sum_right: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and f :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner x (comm_monoid_add.sum plus zero f A) = (\<Sum>y\<in>A. inner x (f y))"
add_closed: fixes g :: "('b, 'a) aezfun"   and h :: "('b, 'a) aezfun" assumes "g \<in> FG"   and "h \<in> FG" shows "g + h \<in> FG"
fns4: fixes fa :: "'a \<Rightarrow> 'a"   and fr :: "'b \<Rightarrow> 'b" assumes "Rep \<circ> fa \<circ> Abs = fr" shows "Rep \<circ> fa = fr \<circ> Rep \<and> fa \<circ> Abs = Abs \<circ> fr"
set_less_eq_aux_Compl2_conv_set_less_eq_aux_Compl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder_proper_interval less_eq less proper_interval"   and "finite UNIV"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys" shows "ord.set_less_eq_aux less_eq (set xs) (- set ys) = proper_intrvl.set_less_eq_aux_Compl less proper_interval None xs ys"
substL_star: fixes L1 :: "nat list set"   and \<sigma> :: "nat \<Rightarrow> nat rexp option" shows "L (substL (star L1) \<sigma>) = L (starS (substL L1 \<sigma>))"
add_strict_increasing: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "{#} \<subset># a"   and "b \<subseteq># c" shows "b \<subset># a + c"
fInf_unfold: fixes f :: "nat \<Rightarrow> 'b \<times> 'c" shows "inf_prod (f 0) (Inf_prod (range (\<lambda>n. f (Suc n)))) = Inf_prod (range f)"
matches_dest_1: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and m' :: "'c match_expr"   and a :: "action"   and p :: "'d" assumes "matches \<gamma> (MatchAnd m m') a p"   and "\<not> matches \<gamma> m a p" shows "False"
eventually_mono: fixes P :: "'d \<Rightarrow> bool"   and F :: "'d filter"   and Q :: "'d \<Rightarrow> bool" assumes "eventually P F"   and "\<And>x. P x \<Longrightarrow> Q x" shows "eventually Q F"
subst_lit: fixes \<sigma> :: "'c \<Rightarrow> 'd"   and v :: "'b" shows "\<sigma> \<dagger> \<guillemotleft>v\<guillemotright> = \<guillemotleft>v\<guillemotright>"
nrows_def: fixes A :: "'b matrix" shows "nrows A \<equiv> if nonzero_positions (Rep_matrix A) = {} then 0 else Suc (Max (fst ` nonzero_positions (Rep_matrix A)))"
infinite_cardSuc_regularCard: fixes r :: "('b \<times> 'b) set" assumes "infinite (Field r)"   and "Card_order r" shows "regularCard (cardSuc r)"
osum_Well_order: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "Well_order r"   and "Well_order r'" shows "Well_order (r +o r')"
gcsrp_l_i: fixes N :: "'a set"   and a :: "'a" assumes "G \<triangleright> N"   and "a \<in> gcsrp_map G N ` set_rcs G N" shows "gcsrp_top G N (gcsrp_iop G N a) a = gcsrp_one G N"
min_galleryD_gallery: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X"   and "SimplicialComplex.min_maxsimpchain X xs" shows "SimplicialComplex.maxsimpchain X xs"
z': shows "z \<notin> \<int>\<^sub>\<le>\<^sub>0"
cr_option_def: shows "cr_option \<equiv> \<lambda>x y. x = return_pmf y"
assoc_oo: fixes f :: "'t \<rightarrow> 's"   and g :: "'u \<rightarrow> 't"   and h :: "'r \<rightarrow> 'u" shows "f oo g oo h = (f oo g) oo h"
lt_of_nat_term_order: fixes to :: "'a nat_term_order"   and x :: "'a"   and y :: "'a" shows "lt_of_nat_term_order to x y = (nat_term_compare to x y = Lt)"
fBexCI: fixes A :: "'a fset"   and P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<forall>x|\<in>|A. \<not> P x \<Longrightarrow> P a"   and "a |\<in>| A" shows "fBex A P"
Base: fixes R :: "'a language \<Rightarrow> 'a language \<Rightarrow> bool"   and x :: "'a language"   and y :: "'a language" assumes "R x y" shows "Plus_cong R x y"
subcls1_acyclic: fixes G :: "prog" assumes "ws_prog G" shows "acyclic (subcls1 G)"
set_of_subdag: fixes p :: "ref"   and low :: "ref \<Rightarrow> ref"   and high :: "ref \<Rightarrow> ref"   and t :: "dag"   and no :: "ref"   and not :: "dag" assumes "Dag p low high t"   and "Dag no low high not"   and "no \<in> set_of t" shows "not \<le> t"
rep_eq: fixes x :: "'f \<Rightarrow> bool"   and xa :: "('f, 'e) generator" shows "generator (takeWhile_trans x xa) = takeWhile_raw x (generator xa)"
Nml_VcompNml_2: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term"   and u :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Nml V src trg t"   and "bicategorical_language.Nml V src trg u"   and "bicategorical_language.Dom V src trg t = bicategorical_language.Cod V src trg u" shows "bicategorical_language.Dom V src trg (bicategorical_language.VcompNml V t u) = bicategorical_language.Dom V src trg u"
degree_least_nonzero: fixes f :: "'b pdevs" assumes "degree f \<noteq> 0" shows "pdevs_apply f (degree f - 1) \<noteq> (0::'b)"
div_mult_self1: 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 b :: "'c"   and a :: "'c"   and c :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus a (times c b)) b = plus c (divide a b)"
NTCatCod: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "\<eta>1 \<approx>>\<bullet> \<eta>2" shows "NTCatCod \<eta>1 = NTCatCod \<eta>2"
fls_const_nth: fixes c :: "'b"   and n :: "int" shows "fls_nth (fls_const c) n = (if n = 0 then c else (0::'b))"
CondFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c\<^sub>2 :: "('d, 'e, 'f) com"   and t :: "('d, 'f) xstate"   and c\<^sub>1 :: "('d, 'e, 'f) com" assumes "s \<notin> b"   and "\<Gamma>\<turnstile> \<langle>c\<^sub>2,Normal s\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>Cond b c\<^sub>1 c\<^sub>2,Normal s\<rangle> \<Rightarrow> t"
mat_delete_dim_2: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" shows "dim_col (mat_delete A i j) = dim_col A - 1"
conj_ac_1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a AND b) AND c = a AND b AND c"
less_imp_not_less: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.preorder less_eq less"   and "less x y" shows "(\<not> less y x) = True"
p: shows "urel_integer (pp - 2) (int (nat (p - 2)))"
inner_vec_def: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec" shows "inner x y = (\<Sum>i\<in>UNIV. inner (x $ i) (y $ i))"
atomize_not: fixes A :: "bool" shows "(A \<Longrightarrow> False) \<equiv> \<not> A"
G'_fields_2: shows "G'.V0 = f ` V0"
delta_bound_def: fixes d :: "'a"   and x :: "'a" shows "delta_bound d x = nat \<lceil>x * of_int \<lceil>(1::'a) / d\<rceil>\<rceil>"
\<sigma>\<^sub>3_def: shows "\<sigma>\<^sub>3 = \<sigma>\<^sub>2(''highIn'' \<mapsto> highIn)"
meet_self: fixes r :: "'a" shows "r \<sqinter> r = r"
simps_2: fixes uv :: "'b" shows "(extB.Val uv \<le> extB.Bot) = False"
simpdvd: fixes t :: "MIR.num"   and d :: "int"   and bs :: "real list" assumes "nozerocoeff t"   and "d \<noteq> 0" shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
set_insert_list: fixes x :: "'a"   and xs :: "'a list" shows "set (insert_list x xs) = insert x (set xs)"
BS1: fixes r :: "'a" assumes "ide r" shows "\<exists>f g. is_left_adjoint f \<and> is_left_adjoint g \<and> r \<cong> g \<star> f\<^sup>*"
inj_ring_hom_axioms: shows "inj_ring_hom of_real"
numsub_def: fixes s :: "Cooper.num"   and t :: "Cooper.num" shows "numsub s t = (if s = t then C 0 else numadd s (numneg t))"
summable: shows "(\<lambda>(x, y). f x y) abs_summable_on A \<times> B"
simps_1: shows "imbal \<langle>\<rangle> = 0"
nonneg_lincomb_def: fixes c :: "'a vec \<Rightarrow> 'a"   and Vs :: "'a vec set"   and b :: "'a vec" shows "n.nonneg_lincomb c Vs b = (n.lincomb c Vs = b \<and> c ` Vs \<subseteq> {x. (0::'a) \<le> x})"
ccw'_sortedP_uminus: fixes xs :: "(real \<times> real) list" assumes "ccw'.sortedP 0 xs" shows "ccw'.sortedP 0 (map uminus xs)"
assms_2: shows "R2 \<subseteq> modRules2"
max_v_ev_1: shows "eigen_vector A local.max_v local.max_ev"
cm_preserves_inv_imps_work_sum: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int" assumes "next_change_multiplicity' c0 c1 loc t n"   and "inv_imps_work_sum c0" shows "inv_imps_work_sum c1"
berlekamp_resulting_mat_closed_1: fixes u :: "'b mod_ring poly" shows "berlekamp_resulting_mat u \<in> carrier_mat (degree u) (degree u)"
tensor_preserves_ide: fixes a :: "'c local.term set"   and b :: "'c local.term set" assumes "ide a"   and "ide b" shows "ide (a \<otimes> b)"
keys_add: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and g :: "'b \<Rightarrow>\<^sub>0 'c" assumes "keys f \<inter> keys g = {}" shows "keys f \<union> keys g = keys (f + g)"
transrec3_def: fixes a :: "V"   and b :: "V \<Rightarrow> V \<Rightarrow> V"   and c :: "V \<Rightarrow> (V \<Rightarrow> V) \<Rightarrow> V" shows "transrec3 a b c \<equiv> transrec (\<lambda>r x. if x = 0 then a else if Limit x then c x (restrict r (elts x)) else b (ZFC_in_HOL.pred x) (r (ZFC_in_HOL.pred x)))"
f_join_UNIV: fixes xs :: "'b list" shows "xs \<Join>\<^sub> UNIV = xs"
satC_append: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and c1 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list"   and c2 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" shows "GE.satC \<xi> (c1 @ c2) = (GE.satC \<xi> c1 \<or> GE.satC \<xi> c2)"
star_idemp: fixes A :: "'b list set" shows "star (star A) = star A"
Nil_sp: fixes g :: "('c \<times> nat, 'd) rgraph_scheme"   and v1 :: "'c \<times> nat"   and v2 :: "'c \<times> nat"   and subs :: "(('c \<times> nat) \<times> 'c \<times> nat) set" shows "SubRel.subpath g v1 [] v2 subs = (sub_rel_of g subs \<and> v1 \<in> Graph.vertices g \<and> (v1 = v2 \<or> (v1, v2) \<in> subs\<^sup>+))"
remove1_sorted_poly_rel_mset_poly_rel_minus: fixes r :: "(char list list \<times> int) list"   and r' :: "int mpoly"   and a :: "char list" assumes "(r, r') \<in> sorted_poly_rel O mset_poly_rel"   and "([a], - 1) \<in> set r" shows "(remove1 ([a], - 1) r, r' + Var (\<phi> a)) \<in> sorted_poly_rel O mset_poly_rel"
pred_envirI: fixes p :: "'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e" assumes "\<And>x. p (f x)" shows "pred_envir p f"
all_not_in_conv: fixes A :: "'a set" shows "(\<forall>x. x \<notin> A) = (A = {})"
max_reg_Minus: fixes a1 :: "vname aexp"   and a2 :: "vname aexp" shows "max_reg (Minus a1 a2) = max (max_reg a1) (max_reg a2)"
dickson_le_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" shows "dickson_le d m s t = (d s \<le> m \<and> d t \<le> m \<and> s \<preceq> t)"
leftrec_base: fixes u :: "'b"   and v :: "'b"   and lc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and rc :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat" shows "leftrec u v lc rc (real m / 2) = u"
2: shows "((regular r \<and> regular q \<and> vector q \<and> asymmetric t \<and> t \<le> r \<and> t \<le> q \<and> q = t\<^sup>T\<^sup>\<star> * s \<and> p = - q \<sqinter> r\<^sup>T * q) \<and> p \<noteq> bot) \<and> card {x. regular x \<and> x \<le> - - (- q \<sqinter> r\<^sup>T\<^sup>\<star> * s)} = n"
little_Fermat_nat: fixes p :: "nat"   and a :: "nat" assumes "prime p"   and "\<not> p dvd a" shows "[a ^ p = a] (mod p)"
x: shows "real_of_1 (q, l, r) = real_of_1 (q, l', r')"
DERIV_cmult_right: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set"   and c :: "'a" assumes "(f has_field_derivative D) (at x within s)" shows "((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
measurable_cexpr_sem: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and V :: "nat set" assumes "\<Gamma> \<turnstile>\<^sub>c e : t"   and "free_vars e \<subseteq> V" shows "(\<lambda>\<sigma>. cexpr_sem \<sigma> e) \<in> state_measure V \<Gamma> \<rightarrow>\<^sub>M stock_measure t"
identity_map: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and U :: "'c set"   and x :: "'d" assumes "presheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"   and "is_open U"   and "x \<in> \<FF> U" shows "\<rho> U U x = x"
fold_const_fa: fixes fa :: "floatarith"   and xs :: "real list" shows "interpret_floatarith (fold_const_fa fa) xs = interpret_floatarith fa xs"
assms_1: fixes b1 :: "'a"   and b2 :: "'a" shows "f (b1 + b2) = f b1 + f b2"
that: fixes as :: "ct list"   and zs :: "ct list" assumes "ys = as @ zs"   and "ct_list_eq as xs" shows "thesis"
eqv: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "eqv \<phi> \<chi> \<in> fmla"
zero_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'b" shows "blinfun_apply a (0::'c) = (0::'b)"
primeidealCE: assumes "cring R"   and "\<not> primeideal I R" obtains "carrier R = I"   | "\<exists>a b. a \<in> carrier R \<and> b \<in> carrier R \<and> a \<otimes> b \<in> I \<and> a \<notin> I \<and> b \<notin> I"
swap_lens_idem: shows "swap\<^sub>L ;\<^sub>L swap\<^sub>L = \<Sigma>"
\<phi>\<^sub>0_in_hom_2: fixes f :: "'a" assumes "\<guillemotleft>f : src e\<^sub>0 \<rightarrow> src e\<^sub>1\<guillemotright>"   and "ide f" shows "\<guillemotleft>(d\<^sub>1 \<star> \<a>\<^sup>-\<^sup>1[e\<^sub>1, f \<star> d\<^sub>0, e\<^sub>0]) \<cdot> \<a>[d\<^sub>1, e\<^sub>1, (f \<star> d\<^sub>0) \<star> e\<^sub>0] \<cdot> ((d\<^sub>1 \<star> e\<^sub>1) \<star> \<a>\<^sup>-\<^sup>1[f, d\<^sub>0, e\<^sub>0]) \<cdot> (\<eta>\<^sub>1 \<star> f \<star> \<eta>\<^sub>0) \<cdot> \<l>\<^sup>-\<^sup>1[f \<star> src e\<^sub>0] \<cdot> \<r>\<^sup>-\<^sup>1[f] : f \<Rightarrow> d\<^sub>1 \<star> (e\<^sub>1 \<star> f \<star> d\<^sub>0) \<star> e\<^sub>0\<guillemotright>"
elims_2: fixes x :: "real_min_top"   and xa :: "real_min_top" assumes "x < xa" obtains uv_ :: "real" where "xa__ = real_min_top.R uv_" and "xa = PInfty"   | x :: "real" and y :: "real" where "xa__ = real_min_top.R x" and "xa = real_min_top.R y" and "x < y"
Py: shows "distributed M T Y (\<lambda>x. ennreal (Py x))"
PiE_def: fixes S :: "'d set"   and T :: "'d \<Rightarrow> 'e set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
cases: fixes r :: "'a graph" obtains nodes :: "'a set" and edges :: "('a \<times> 'a) set" where "r = \<lparr>nodes = nodes, edges = edges\<rparr>"
dot_norm: fixes x :: "'b"   and y :: "'b" shows "x \<bullet> y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
permute_cfun_eq: fixes p :: "perm" shows "permute p = (\<lambda>f. Abs_cfun (permute p) oo f oo Abs_cfun (permute (- p)))"
continuous_map_into_subtopology: fixes X :: "'a topology"   and X' :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set" assumes "continuous_map X X' f"   and "f ` topspace X \<subseteq> T" shows "continuous_map X (subtopology X' T) f"
order_G: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat" assumes "sylow G p a m" shows "order G = p ^ a * m"
locally_Lipschitz_negl_bounded: fixes B :: "real"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "DIM('c) \<le> DIM('d)"   and "0 < B"   and "bounded S"   and "negligible S"   and "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> (\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> B * norm (y - x))" shows "negligible (f ` S)"
uminus_real_alg: fixes x :: "real_alg" shows "- real_of x = real_of (- x)"
comm_monoid_mset_axioms: shows "comm_monoid_mset (\<^bold>*) \<^bold>1"
rmshadow'_sound: fixes P :: "'i simple_packet set"   and rs :: "'i simple_rule list" assumes "p \<notin> P" shows "simple_fw (rmshadow' rs P) p = simple_fw rs p"
elem_exists_non_empty_set: fixes A :: "'b set" assumes "0 < card A" obtains x :: "'b" where "x \<in> A"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (load_sub2 address 10 rd curr_win w s1)) \<and> t2 = snd (fst (load_sub2 address 10 rd curr_win w s2))"
cod_MkArr: fixes A :: "'c \<Rightarrow> 's"   and B :: "'c \<Rightarrow> 's"   and F :: "'c \<Rightarrow> 's" assumes "Cop_S.arr (Cop_S.MkArr A B F)" shows "Cop_S.cod (Cop_S.MkArr A B F) = Cop_S.MkIde B"
tableau_completeness: fixes G :: "(nat, nat) form list"   and p :: "(nat, nat) form" assumes "\<forall>e f g. list_all (eval e f g) G \<longrightarrow> eval e f g p" shows "tableauproof G p"
lnths_cong: fixes xs :: "'c llist"   and ys :: "'c llist"   and A :: "nat set"   and B :: "nat set" assumes "xs = ys"   and "\<And>n. enat n < llength ys \<Longrightarrow> (n \<in> A) = (n \<in> B)" shows "lnths xs A = lnths ys B"
l_to_tree_order_wf_get_root_node_wf_axioms_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool" shows "l_to_tree_order_wf_get_root_node_wf_axioms type_wf known_ptrs to_tree_order get_root_node heap_is_wellformed \<equiv> (\<forall>h ptr to ptr' root_ptr ptr''. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> to_tree_order ptr \<rightarrow>\<^sub>r to \<longrightarrow> ptr' \<in> set to \<longrightarrow> h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root_ptr \<longrightarrow> ptr'' \<in> set to \<longrightarrow> h \<turnstile> get_root_node ptr'' \<rightarrow>\<^sub>r root_ptr) \<and> (\<forall>h ptr root_ptr to ptr'. heap_is_wellformed h \<longrightarrow> type_wf h \<longrightarrow> known_ptrs h \<longrightarrow> h \<turnstile> get_root_node ptr \<rightarrow>\<^sub>r root_ptr \<longrightarrow> h \<turnstile> to_tree_order root_ptr \<rightarrow>\<^sub>r to \<longrightarrow> ptr' \<in> set to \<longrightarrow> h \<turnstile> get_root_node ptr' \<rightarrow>\<^sub>r root_ptr)"
collapse: fixes hp :: "'b hp" shows "Hp (un_Hp1 hp) (hps hp) = hp"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a ; b ; c = a ; (b ; c)"
scaleR_scaleR: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "real"   and b :: "real"   and x :: "'a" assumes "class.real_vector plus zero minus uminus scaleR" shows "scaleR a (scaleR b x) = scaleR (a * b) x"
that: fixes xs' :: "'a list" assumes "rtrancl_path r x xs' y"   and "distinct (x # xs')"   and "set xs' \<subseteq> set xs" shows "thesis"
max_procs_def: shows "max_procs \<equiv> 255::'a"
cobounded1: fixes a :: "'c"   and b :: "'c" shows "a \<le> max a b"
iT_Div_card_inj_on: fixes k :: "nat"   and I :: "nat set" assumes "inj_on (\<lambda>n. n div k) I" shows "card (I \<oslash> k) = card I"
vector_derivative_of_real_right: fixes f :: "'a \<Rightarrow> 'a"   and x :: "real" assumes "f field_differentiable at (of_real x)" shows "vector_derivative (\<lambda>x. f (of_real x)) (at x) = deriv f (of_real x)"
try_spmf_lossless: fixes p :: "'c spmf"   and q :: "'c spmf" assumes "lossless_spmf p" shows "TRY p ELSE q = p"
induct: fixes P :: "('b \<times> nat) tree \<Rightarrow> bool"   and a0 :: "('b \<times> nat) tree" assumes "\<And>l a uu_ r. (l \<noteq> \<langle>\<rangle> \<Longrightarrow> P l) \<Longrightarrow> P \<langle>l, (a, uu_), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
deterministic_RP_SomeD: fixes N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and n :: "nat"   and R :: "'a literal list list" assumes "deterministic_RP (N, P, Q, n) = Some R" shows "\<exists>N' P' Q' n'. (\<exists>k. (deterministic_RP_step ^^ k) (N, P, Q, n) = (N', P', Q', n')) \<and> is_final_dstate (N', P', Q', n') \<and> R = map fst Q'"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "atom"   and y :: "atom fm" assumes "asubst\<^sub>2 x xa xb = y" obtains l :: "nat" and u :: "nat" where "x = l__" and "xa = u__" and "xb = Less 0 0" and "y = FalseF"   | l :: "nat" and u :: "nat" and j :: "nat" where "x = la__" and "xa = ua__" and "xb = Less 0 (Suc j__)" and "y = Or (Atom (Less ua__ j__)) (Atom (Eq ua__ j__))"   | l :: "nat" and u :: "nat" and i :: "nat" where "x = lb__" and "xa = ub__" and "xb = Less (Suc i__) 0" and "y = Or (Atom (Less i__ lb__)) (Atom (Eq i__ lb__))"   | l :: "nat" and u :: "nat" and i :: "nat" and j :: "nat" where "x = lc__" and "xa = uc__" and "xb = Less (Suc ia__) (Suc ja__)" and "y = Atom (Less ia__ ja__)"   | l :: "nat" and u :: "nat" where "x = ld__" and "xa = ud__" and "xb = Eq 0 0" and "y = TrueF"   | l :: "nat" and u :: "nat" and v :: "nat" where "x = le__" and "xa = ue__" and "xb = Eq 0 (Suc v__)" and "y = FalseF"   | l :: "nat" and u :: "nat" and v :: "nat" where "x = lf__" and "xa = uf__" and "xb = Eq (Suc v) 0" and "y = FalseF"   | l :: "nat" and u :: "nat" and i :: "nat" and j :: "nat" where "x = l" and "xa = u" and "xb = Eq (Suc i) (Suc j)" and "y = Atom (Eq i j)"
columns_rw: shows "columns A = range (\<lambda>i. column i A)"
Skip_no_step: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and s :: "('b, 'c) xstate"   and cfg :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> (Skip, s) \<rightarrow> cfg" shows "P"
diff_is_0_eq: fixes m :: "nat"   and n :: "nat" shows "(m - n = zero_class.zero) = (m \<le> n)"
split_paired_Ex: fixes P :: "'d \<times> 'e \<Rightarrow> bool" shows "(\<exists>x. P x) = (\<exists>a b. P (a, b))"
reciprocal_reciprocal: fixes P :: "'a poly"   and p :: "nat" assumes "degree P \<le> p" shows "reciprocal_poly p (reciprocal_poly p P) = P"
internal_iface_name_to_set2: fixes ifce :: "char list" shows "internal_iface_name_to_set ifce = {i. internal_iface_name_match ifce i}"