hcju/isabelleps huggingface dataset:
cycle_removal_lemma: fixes s :: "('a, 'b) fmap"   and as1 :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and as2 :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and as3 :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list" assumes "exec_plan s (as1 @ as2) = exec_plan s as1" shows "exec_plan s (as1 @ as2 @ as3) = exec_plan s (as1 @ as3)"
complex_of_real_def: fixes r :: "real" shows "complex_of_real r = Complex r 0"
update_since_1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and args :: "Monitor.args"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and aux :: "'msaux"   and ne :: "nat"   and n :: "nat"   and rel1 :: "event_data option list set"   and rel2 :: "event_data option list set"   and rel :: "event_data option list set"   and aux' :: "'msaux"   and I :: "\<I>"   and pos :: "bool" assumes "wf_since_aux \<sigma> V R args \<phi> \<psi> aux ne"   and "qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<phi>) rel1"   and "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) ne \<psi>) rel2"   and "(rel, aux') = update_since args rel1 rel2 (\<tau> \<sigma> ne) aux"   and "fv \<phi> \<subseteq> fv \<psi>"   and "args_ivl args = I"   and "args_n args = n"   and "args_L args = fv \<phi>"   and "args_R args = fv \<psi>"   and "args_pos args = pos" shows "wf_since_aux \<sigma> V R args \<phi> \<psi> aux' (Suc ne)"
lowner_lub_is_limit: fixes dim :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "matrix_seq dim f" shows "limit_mat f (matrix_seq.lowner_lub f) dim"
Quotient_\<kappa>: shows "Quotient (=) make\<kappa> eval\<kappa> cr_\<kappa>"
ord_isom_onto: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "f ` carrier D = carrier E"
finite_Keys: fixes A :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "finite A" shows "finite (Keys A)"
slast_strict: shows "slast\<cdot>\<bottom> = \<bottom>"
cnj_eqvt: fixes p :: "perm"   and A :: "fmla"   and B :: "fmla" shows "p \<bullet> A AND B = (p \<bullet> A) AND (p \<bullet> B)"
Inv_preserves_Diag: fixes t :: "'a local.term" assumes "Can t"   and "Diag t" shows "Diag (Inv t)"
simps_2: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list" shows "wellformed_policy1 (x # xs) = (DenyAll \<notin> set xs)"
10: shows "(\<Sum>y\<in>{x. poly p x = 0 \<and> 0 < poly q x}. 1) = card {x. poly p x = 0 \<and> 0 < poly q x}"
hyps_1: assumes "\<not> getConflictFlag state'"   and "getQ state' \<noteq> []" shows "exhaustiveUnitPropagate_dom (applyUnitPropagate state')"
xor_left_self: fixes x :: "'a"   and y :: "'a" shows "x XOR x XOR y = y"
inf1D1: fixes A :: "'a \<Rightarrow> bool"   and B :: "'a \<Rightarrow> bool"   and x :: "'a" assumes "(A \<sqinter> B) x" shows "A x"
inv_mult_group: fixes x :: "'a"   and y :: "'a" shows "- (x + y) = - y + - x"
wf_dia_inv_4: fixes V :: "node fset"   and \<Lambda> :: "node \<Rightarrow> assertion_gadget"   and E :: "(node fset \<times> command_gadget \<times> node fset) list" assumes "wf_dia (Graph V \<Lambda> E)" shows "linearity E"
dbm_lt_asym: fixes e :: "'c DBMEntry"   and f :: "'c DBMEntry" assumes "e \<prec> f" shows "\<not> f \<prec> e"
in_hom_char: fixes f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and b :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" shows "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright> = (J_C.arr f \<and> J_C.ide a \<and> J_C.ide b \<and> J_C.Dom f = J_C.Dom a \<and> J_C.Cod f = J_C.Dom b)"
simps_2: fixes \<Gamma> :: "name list"   and name :: "name" shows "nterm_to_term \<Gamma> (Nvar name) = (case find_first name \<Gamma> of None \<Rightarrow> Free name | Some x \<Rightarrow> Bound x)"
cos_npi_int: fixes n :: "int" shows "cos (pi * real_of_int n) = (if even n then 1 else - 1)"
fi_match_entails: fixes m :: "(assn \<times> assn) list" assumes "fi_m_match m" shows "fi_m_fst m \<Longrightarrow>\<^sub>A fi_m_snd m"
simps_10: fixes v :: "'b" shows "restricted_subformulas prop\<^sub>n(v) = {}"
Span_finite_dimension: fixes K :: "'a list set set"   and Us :: "'a list set list" assumes "subfield K (Rupt A P)"   and "set Us \<subseteq> carrier (Rupt A P)" shows "Rupt.finite_dimension K (Rupt.Span K Us)"
mset_seqeuences: fixes key :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "mset (concat (sequences key xs)) = mset xs"
design_points_nempty: shows "\<V>\<^sup>+ \<noteq> {}"
i_in_F: shows "to_F \<iota> \<in> Inf_F"
frontier_of_restrict: fixes X :: "'c topology"   and S :: "'c set" shows "X frontier_of S = X frontier_of (topspace X \<inter> S)"
i_shrink_eq_NoMsg_iAll_conv: fixes k :: "nat"   and s :: "nat \<Rightarrow> 'b message_af"   and t :: "nat" assumes "0 < k" shows "((s \<div> k) t = NoMsg) = (\<box> t1 [t * k\<dots>,k - Suc 0]. s t1 = NoMsg)"
App: fixes env :: "nat \<Rightarrow> LambdaType.type"   and s :: "dB"   and T :: "LambdaType.type"   and U :: "LambdaType.type"   and t :: "dB" assumes "env \<turnstile> s : T \<Rightarrow> U"   and "env \<turnstile> t : T" shows "env \<turnstile> s \<degree> t : U"
psubst_fresh_name: fixes x :: "name"   and \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and M :: "trm" assumes "x \<sharp> \<theta>_n"   and "x \<sharp> \<theta>_c"   and "x \<sharp> M" shows "x \<sharp> \<theta>_n,\<theta>_c<M>"
bounded_linear_add: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f x + g x)"
HAInitState_notmem_Range_ChiRel: fixes A :: "('a, 'b, 'c) hierauto" shows "HAInitState A \<notin> Range (ChiRel A)"
sum_assoc_eqpoll: fixes a :: "V"   and b :: "V"   and c :: "V" shows "elts (a \<Uplus> b \<Uplus> c) \<approx> elts (a \<Uplus> (b \<Uplus> c))"
facts_8: fixes n :: "nat" assumes "2 * card (uverts H) \<le> n" shows "prob_space_with_indicators.\<Delta>\<^sub>d (MGn p n) {S. S \<subseteq> {1..n} \<and> card S = card (uverts H)} (\<lambda>S. {es \<in> space (MGn p n). subgraph (f S) (induced_subgraph S (edge_space.edge_ugraph n es))}) \<le> (\<Sum>k = 1..card (uverts H). real (card (uverts H) ^ k) * (real n ^ (2 * card (uverts H) - k) * p n powr (real (2 * card (uedges H)) - max_density H * real k)))"
monom_one: shows "monom (UP Zp) \<one> 0 = \<one>\<^bsub>UP Zp\<^esub>"
trapped_backward_iff_rev_trapped_forward: fixes x :: "'a"   and K :: "'a set" shows "trapped_backward x K = rev.trapped_forward x K"
add_assoc: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b point"   and p\<^sub>2 :: "'b point"   and p\<^sub>3 :: "'b point" assumes "nonsingular a b"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2"   and "on_curve a b p\<^sub>3" shows "ell_field_class.add a p\<^sub>1 (ell_field_class.add a p\<^sub>2 p\<^sub>3) = ell_field_class.add a (ell_field_class.add a p\<^sub>1 p\<^sub>2) p\<^sub>3"
simps: fixes a :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "asymp a = (\<exists>R. a = R \<and> (\<forall>x xa. R x xa \<longrightarrow> \<not> R xa x))"
is_natural_2: fixes f :: "'d" assumes "D.arr f" shows "F (D.cod f) \<cdot>\<^sub>C F f = F f"
Pd_shuffle_to_top_simps_3: fixes d :: "'d"   and ds' :: "'d set"   and Y :: "'x set" assumes "d \<notin> ds'" shows "Pd_shuffle_to_top ds' Y d = Pd d"
0: shows "rawpsubst (rawpsubst (cnj \<phi>1 \<phi>2) (zip (map Var us) (map snd txs))) (zip (map fst txs) us) = cnj (rawpsubst (rawpsubst \<phi>1 (zip (map Var us) (map snd txs))) (zip (map fst txs) us)) (rawpsubst (rawpsubst \<phi>2 (zip (map Var us) (map snd txs))) (zip (map fst txs) us))"
ring_iso_memE_3: fixes h :: "'e \<Rightarrow> 'f"   and R :: "('e, 'g) ring_scheme"   and S :: "('f, 'h) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "h \<in> ring_iso R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
vsv_vinsertD: fixes x :: "V"   and r :: "V" assumes "vsv (vinsert x r)" shows "vsv r"
least_overrider_is_type: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and m :: "char list list \<times> expr"   and Cs' :: "char list list"   and wf_md :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> (C, Cs) has overrider M = (Ts, T, m) via Cs'"   and "wf_prog wf_md P" shows "is_type P T"
functor_comp: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and C :: "'ai \<Rightarrow> 'ai \<Rightarrow> 'ai"   and G :: "'ah \<Rightarrow> 'ai" assumes "functor A B F"   and "functor B C G" shows "functor A C (G \<circ> F)"
bind_returns_heap_E2: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and h' :: "'l" assumes "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>h h'"   and "pure f h" obtains x :: "'o" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> g x \<rightarrow>\<^sub>h h'"
omega_def: shows "Omega_Algebra_Models.omega \<equiv> \<lambda>R. {(xa, x). omegap (\<lambda>x xa. (x, xa) \<in> R) xa x}"
assoc_is_natural_1: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and \<nu> :: "'c"   and \<tau> :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a> \<mu> \<nu> \<tau> = V (H \<mu> (H \<nu> \<tau>)) (\<a> (partial_magma.dom V \<mu>) (partial_magma.dom V \<nu>) (partial_magma.dom V \<tau>))"
mk_doubleton_image_veqpoll: fixes X :: "V"   and Y :: "V" assumes "X \<subseteq>\<^sub>\<circ> Y" shows "X \<approx>\<^sub>\<circ> mk_doubleton_image X Y"
cases: fixes x :: "('b \<times> color) tree \<times> 'b \<times> ('b \<times> color) tree" obtains t1 :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (R (R t1 aa__ t2__) ba__ t3__, c__, t4__)"   | a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (R \<langle>\<rangle> ab__ (R t2a__ bb__ t3a__), ca__, t4a__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" and b :: "'b" and t3 :: "('b \<times> color) tree" and c :: "'b" and t4 :: "('b \<times> color) tree" where "x = (R (B v__ vc__ vb__) ac__ (R t2b__ b t3), c, t4)"   | a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>\<rangle>, ad__, t2c__)"   | v :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (B \<langle>\<rangle> va__ vba__, ae__, t2d__)"   | va :: "'b \<times> color" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>\<langle>\<rangle>, vaa__, \<langle>\<rangle>\<rangle>, af__, t2e__)"   | va :: "'b \<times> color" and v :: "('b \<times> color) tree" and ve :: "'b" and vd :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>\<langle>\<rangle>, vab__, B vd__ ve__ vd\<rangle>, ag__, t2f__)"   | vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and v :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (B (B vca__ vf__ vea__) vg__ vbb__, ah__, t2g__)"   | vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and va :: "'b \<times> color" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>B vcb__ vfa__ veb__, vac__, \<langle>\<rangle>\<rangle>, ai__, t2h__)"   | vc :: "('b \<times> color) tree" and vf :: "'b" and ve :: "('b \<times> color) tree" and va :: "'b \<times> color" and v :: "('b \<times> color) tree" and vh :: "'b" and vg :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (\<langle>B vcc__ vf ve, va, B vh__ vh vg\<rangle>, aj__, t2i__)"   | v :: "('b \<times> color) tree" and vc :: "'b" and vb :: "('b \<times> color) tree" and a :: "'b" and t2 :: "('b \<times> color) tree" where "x = (B v vc vb, a, t2)"
mult_cancellation_lemma: fixes a :: "V"   and x :: "V"   and r :: "V"   and y :: "V"   and s :: "V" assumes "a * x + r = a * y + s"   and "r \<sqsubset> a"   and "s \<sqsubset> a" shows "x = y \<and> r = s"
not_prime_unit: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times x one" shows "\<not> normalization_semidom.prime zero times one normalize x"
map_key_PP_monom_mult_punit: fixes c :: "'c"   and t :: "('a, 'b) pp"   and p :: "('a, 'b) pp \<Rightarrow>\<^sub>0 'c" shows "Poly_Mapping.map_key PP (monom_mult_punit c t p) = monom_mult_punit c (pp.mapping_of t) (Poly_Mapping.map_key PP p)"
mono_neutral_right: fixes T :: "'b set"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite T"   and "S \<subseteq> T"   and "\<forall>i\<in>T - S. g i = (0::'c)" shows "sum g T = sum g S"
all_maps_list_def: fixes G_list :: "'b list \<times> ('b \<times> 'b) list" shows "all_maps_list G_list = (cyc_permutationss \<circ> grouped_out_arcs) G_list"
assms: shows "\<not> rt1 \<sqsubseteq>\<^bsub>dip\<^esub> rt2"
s_prod_assoc1: fixes R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set"   and T :: "('a \<times> 'a set) set" shows "R \<cdot> S \<cdot> T \<subseteq> R \<cdot> (S \<cdot> T)"
inf_sup_distrib2: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and y :: "'e"   and z :: "'e"   and x :: "'e" assumes "class.distrib_lattice inf less_eq less sup" shows "inf (sup y z) x = sup (inf y x) (inf z x)"
take_auxlist_takeWhile: shows "take (length done) auxlist = takeWhile (check_before I nt) auxlist"
l4_18: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and C' :: "'p" assumes "A \<noteq> B"   and "Col A B C"   and "Cong A C A C'"   and "Cong B C B C'" shows "C = C'"
nat_le_dec: fixes m :: "nat"   and n :: "nat" shows "m < n \<or> \<not> m < n"
bl_not_Nil: fixes x :: "'b word" shows "to_bl x \<noteq> []"
UPFDefs_24: fixes p :: "'a" shows "D\<^sub>U p = \<lfloor>deny ()\<rfloor>"
f1: fixes x :: "atom" assumes "x \<in> atom ` xs"   and "x \<in> supp T" shows "x \<sharp> f xs T"
orthogonal_commute: fixes x :: "'a"   and y :: "'a" shows "orthogonal x y = orthogonal y x"
tSup_def: shows "local.tSup \<equiv> map_fun ((`) (\<lambda>ys. (llist_of_tllist ys, terminal ys))) (\<lambda>(xs, a). tllist_of_llist a xs) (\<lambda>A. (lSup (fst ` A), flat_lub b (snd ` (A \<inter> {(xs, uu_). lfinite xs}))))"
gbinomial_minus: fixes a :: "'a"   and k :: "nat" shows "- a gchoose k = (- (1::'a)) ^ k * (a + of_nat k - (1::'a) gchoose k)"
lens_equiv_refl: fixes X :: "'a \<Longrightarrow> 'b" shows "X \<approx>\<^sub>L X"
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 "satC \<xi> (c1 @ c2) = (satC \<xi> c1 \<or> satC \<xi> c2)"
atomize_eq: fixes x :: "'c"   and y :: "'c" shows "(x \<equiv> y) \<equiv> x = y"
hv_props_4: shows "common_sudiv_exists (two_chain_horizontal_boundary two_chain) \<H> \<or> common_reparam_exists \<H> (two_chain_horizontal_boundary two_chain)"
more_than_one: shows "\<nexists>m. obs (sourcenode a) (wod_backward_slice S) = {m} \<or> obs (sourcenode a) (wod_backward_slice S) = {}"
branch_def: fixes subs :: "'b \<Rightarrow> 'b set"   and Gamma :: "'b"   and f :: "nat \<Rightarrow> 'b" shows "branch subs Gamma f = (f 0 = Gamma \<and> (\<forall>n. terminal subs (f n) \<longrightarrow> f (Suc n) = f n) \<and> (\<forall>n. \<not> terminal subs (f n) \<longrightarrow> f (Suc n) \<in> subs (f n)))"
arctan_upper_32_def: shows "arctan_upper_32 \<equiv> \<lambda>x. 7 * (33 * x ^ 4 + 170 * x\<^sup>2 + 165) * x / (5 * (5 * x ^ 6 + 105 * x ^ 4 + 315 * x\<^sup>2 + 231))"
simplex_like_has_bottom: fixes P :: "'a set" assumes "simplex_like P" shows "has_bottom P"
id_eq_1: fixes v :: "i"   and x :: "'a" shows "[x \<^bold>= x in v]"
ax_psubst: fixes \<theta>_n :: "(name \<times> coname \<times> trm) list"   and \<theta>_c :: "(coname \<times> name \<times> trm) list"   and M :: "trm"   and x :: "name"   and a :: "coname" assumes "\<theta>_n,\<theta>_c<M> = Ax x a"   and "a \<sharp> (\<theta>_n, \<theta>_c)"   and "x \<sharp> (\<theta>_n, \<theta>_c)" shows "M = Ax x a"
map_gpv_conv_bind: fixes f :: "'g \<Rightarrow> 'd"   and x :: "('g, 'e, 'f) gpv" shows "map_gpv f (\<lambda>x. x) x = x \<bind> (\<lambda>x. Generative_Probabilistic_Value.Done (f x))"
plus_of_nat_eq_0_imp: fixes z :: "'a"   and n :: "nat" assumes "z + of_nat n = (0::'a)" shows "z \<in> \<int>\<^sub>\<le>\<^sub>0"
realistic_AS_partial_def: fixes aseq :: "'action_t list" shows "realistic_AS_partial aseq \<equiv> \<exists>n aseq'. n \<le> length aseq' \<and> aseq' \<in> AS_set \<and> aseq = lastn n aseq'"
e_take_init: fixes f :: "nat \<Rightarrow> nat option"   and k :: "nat"   and n :: "nat" assumes "f \<in> \<R>"   and "k < Suc n" shows "e_take (Suc k) (f \<triangleright> n) = f \<triangleright> k"
simps_4: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and w :: "nat \<Rightarrow> 'b set" shows "\<F>\<G> (\<phi> U\<^sub>n \<psi>) w = \<F>\<G> \<phi> w \<union> \<F>\<G> \<psi> w"
pseudoinverse: fixes \<omega> :: "real"   and n :: "nat"   and x :: "real" assumes "0 < \<omega>"   and "\<omega> < 1" shows "(\<omega> \<le> \<mu>.C n x) = (\<mu>.I n \<omega> \<le> x)"
hd_drop_conv_nth: fixes n :: "nat"   and xs :: "'e list" assumes "n < length xs" shows "hd (drop n xs) = xs ! n"
m_c_def: fixes C :: "(char list \<Rightarrow> 'av) option acom" shows "m\<^sub>c C = (\<Sum>a\<leftarrow>annos C. m\<^sub>o a (vars C))"
reachable_awalkI: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v"
Suc: fixes k :: "nat"   and s :: "'a"   and t :: "'a" shows "alternating_list (Suc k) s t = alternating_list k s t @ [if gcd_nat.greater_eq k 2 then s else t]"
poly_fcompose: fixes q2 :: "'b poly"   and x :: "'b"   and p :: "'b poly"   and q1 :: "'b poly" assumes "poly q2 x \<noteq> (0::'b)" shows "poly (fcompose p q1 q2) x = poly p (poly q1 x / poly q2 x) * poly q2 x ^ degree p"
DisjCont: fixes A :: "fm" shows "A OR A IMP A \<in> boolean_axioms"
set_sort_on: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "set (sort_key_on r f xs) = set xs"
intro: fixes G_\<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and G_\<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and G_invar :: "'a \<Rightarrow> bool"   and G_adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and G_empty :: "'a"   and G_add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and M_empty :: "'c"   and M_update :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_delete :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and M_lookup :: "'c \<Rightarrow> 'b \<Rightarrow> 'b option"   and M_invar :: "'c \<Rightarrow> bool"   and Q_empty :: "'d"   and Q_update :: "'b \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_delete :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and Q_invar :: "'d \<Rightarrow> bool"   and Q_lookup :: "'d \<Rightarrow> 'b \<Rightarrow> nat option"   and Q_is_empty :: "'d \<Rightarrow> bool"   and Q_getmin :: "'d \<Rightarrow> 'b \<times> nat" assumes "Prim_Impl_Adts G_\<alpha>w G_\<alpha>g G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin" shows "Prim_Impl_Defs G_invar G_adj G_empty G_add_edge M_empty M_update M_delete M_lookup M_invar Q_empty Q_update Q_delete Q_invar Q_lookup Q_is_empty Q_getmin G_\<alpha>w G_\<alpha>g"
gt_same: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t = head s"   and "chkargs (>\<^sub>t) t s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "t >\<^sub>t s"
assoc: fixes g :: "'a"   and h :: "'a"   and k :: "'a" assumes "g \<in> G"   and "h \<in> G"   and "k \<in> G" shows "binop (binop g h) k = binop g (binop h k)"
sort_all_coeffs_def: fixes xs :: "(char list list \<times> int) list" shows "sort_all_coeffs xs = monadic_nfoldli xs (\<lambda>_. RETURN True) (\<lambda>(a, n) b. sort_coeff a \<bind> (\<lambda>a. RETURN ((a, n) # b))) []"
inv_left: fixes a :: "'a" shows "inv ((\<langle>$\<rangle>) f) (f \<langle>$\<rangle> a) = a"
entail_set_all_formulas: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N1 :: "'a set"   and N2 :: "'a set" assumes "consequence_relation Bot entails" shows "entails N1 N2 = (\<forall>C\<in>N2. entails N1 {C})"
order_refl: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.preorder less_eq less" shows "less_eq x x"
assms_2: shows "Deliver (i1, Expunge e mo i2) \<in> set xs"
simps_1: fixes uu :: "'b \<Rightarrow> bool" shows "dnf_to_bool uu [] = False"
field_cont_on_top_edge_image: shows "continuous_on {p. \<exists>x. x \<in> cbox a b \<and> p = (x, g1 x)} F"
HEndPhase1_Inv6: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv6 s"   and "Inv1 s"   and "Inv2a s"   and "Inv2b s"   and "Inv2c s" shows "HInv6 s'"
mult_empty: fixes M :: "'b multiset"   and R :: "('b \<times> 'b) set" shows "(M, {#}) \<notin> mult R"
leg0_is_map: shows "is_left_adjoint (tab\<^sub>0 r)"
nth_eq_iff_index: fixes xs :: "'a list"   and n :: "nat"   and i :: "'a" assumes "distinct xs"   and "n < length xs" shows "(xs ! n = i) = (n = index xs i)"
abs: fixes C_info :: "(name, dt_def) fmap"   and rs :: "(name \<times> value) list"   and \<Gamma> :: "(name, value) fmap"   and cs :: "(Term_Class.term \<times> sterm) list" shows "special_constants.veval C_info rs \<Gamma> (Sabs cs) (Vabs cs \<Gamma>)"
left_inverse: fixes a :: "'b perm" shows "inverse a * a = 1"
mk_deriv_fix_def: fixes i :: "nat"   and n :: "nat"   and j :: "nat" shows "mk_deriv_fix i n j = (n, j, i)"
cblinfun_scaleC_right_def: shows "cblinfun_scaleC_right \<equiv> map_fun id CBlinfun (*\<^sub>C)"
lemma_st_part_gt_ub: fixes r :: "real star" assumes "x < r"   and "r \<in> \<real>" shows "isUb \<real> {s \<in> \<real>. s < x} r"