hcju/isabelleps huggingface dataset:
LNil_eq_ltake_iff: fixes n :: "enat"   and xs :: "'a llist" shows "(LNil = ltake n xs) = (xs = LNil \<or> n = 0)"
r_apow_def: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and a :: "ant" shows "I\<^bsup> R a\<^esup> = (if a = \<infinity> then {\<zero>\<^bsub>R\<^esub>} else if a = 0 then carrier R else I \<^bsup>\<diamondsuit>R na a\<^esup>)"
length_twcmpser_def: fixes G :: "'e"   and n :: "nat"   and f :: "nat \<Rightarrow> 'f set" shows "length_twcmpser G n f = card (f ` {i. i \<le> n}) - Suc 0"
R\<^sub>\<MM>_def: shows "R\<^sub>\<MM> = (\<lambda>x. \<MM> x - ln x)"
simps_2: fixes f :: "'c \<Rightarrow> 'a" shows "psubst f TT = TT"
simps_2: fixes equal :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list"   and y :: "'c" shows "equal_base.list_member equal (x # xs) y = (equal x y \<or> equal_base.list_member equal xs y)"
approx_slp_outer_plain: fixes p :: "nat"   and n :: "nat"   and slp :: "floatarith list"   and XS :: "(real \<times> real pdevs) list"   and RS :: "(real \<times> real pdevs) list"   and fas :: "floatarith list"   and xs :: "real list" assumes "approx_slp_outer p n slp XS = Some RS"   and "slp = slp_of_fas fas"   and "n = length fas"   and "xs \<in> Joints XS" shows "interpret_floatariths fas xs \<in> Joints RS"
assms: shows "\<lbrace>\<alpha> p \<cdot> \<tau> s\<rbrace>x\<lbrace>\<alpha> p\<rbrace>"
simps_1: fixes i :: "int" shows "|DI i| = {}"
adv_nonneg: fixes \<eta> :: "nat" shows "0 \<le> adv \<eta>"
key_sets_mono: fixes H :: "msg set"   and H' :: "msg set"   and X :: "msg" assumes "H \<subseteq> H'" shows "key_sets X H \<subseteq> key_sets X H'"
multi_nonempty_split: fixes M :: "'k multiset" assumes "M \<noteq> {#}" shows "\<exists>A a. M = add_mset a A"
simps_2: fixes x :: "'b equation" shows "atom (Neg x) = x"
closedin_closed_trans: fixes T :: "'b set"   and S :: "'b set" assumes "closedin (top_of_set T) S"   and "closed T" shows "closed S"
scale_right_diff_distrib: fixes a :: "'b"   and x :: "'b"   and y :: "'b" shows "a * (x - y) = a * x - a * y"
1: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "p \<in> P[X]" shows "poly_subst (\<lambda>x. if x \<in> X then monomial (1::'a) (monomial 1 x) else 1) p = p"
precond: shows "(0 < iter \<and> tsN = TS M2 M1 \<Omega> V m (iter - 1) \<and> cN = C M2 M1 \<Omega> V m iter \<and> rmN = RM M2 M1 \<Omega> V m (iter - 1) \<and> obs = L\<^sub>i\<^sub>n M2 (tsN \<union> cN) \<and> obsI = L\<^sub>i\<^sub>n M1 (tsN \<union> cN) \<and> obs\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M2 (tsN \<union> cN). {io} \<times> B M2 io \<Omega>) \<and> obsI\<^sub>\<Omega> = (\<Union>io\<in>L\<^sub>i\<^sub>n M1 (tsN \<union> cN). {io} \<times> B M1 io \<Omega>) \<and> OFSM M1 \<and> OFSM M2 \<and> asc_fault_domain M2 M1 m \<and> test_tools M2 M1 FAIL PM V \<Omega>) \<and> \<not> (cN \<noteq> {} \<and> obsI \<subseteq> obs \<and> obsI\<^sub>\<Omega> \<subseteq> obs\<^sub>\<Omega>)"
hyps: assumes "\<not> c \<le> d"   and "shift2_inv (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))" shows "Max (poly_deg ` fst ` set (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))) \<le> Max (poly_deg ` fst ` set (shift2 c (Suc d) (fst (shift1 (qs, {q \<in> set qs. poly_deg (fst q) = d \<and> m < card (snd q)})))))"
assms_2: shows "lambda C -quasi_isometry_on {A..B} d"
swapEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs' :: "'varSort"   and x' :: "'var"   and y' :: "'var"   and zs :: "'varSort"   and x :: "'var"   and y :: "'var" assumes "goodEnv rho" shows "rho &[x \<and> y]_zs &[x' \<and> y']_zs' = rho &[x' \<and> y']_zs' &[x @zs[x' \<and> y']_zs' \<and> y @zs[x' \<and> y']_zs']_zs"
WT_converter_map_converter: fixes f :: "'l \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'm"   and \<I> :: "('l, 'm) \<I>"   and f' :: "'j \<Rightarrow> 'n"   and g' :: "'o \<Rightarrow> 'k"   and \<I>' :: "('n, 'o) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter" assumes "map_\<I> (inv f) (inv g) \<I>, map_\<I> f' g' \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "inj f"   and "surj g" shows "\<I>, \<I>' \<turnstile>\<^sub>C map_converter f g f' g' conv \<surd>"
simps: fixes p :: "nat"   and q :: "nat" shows "FW_word p q = (if q < p then FW_word q p else if p = 0 \<or> p = q then \<epsilon> else if gcd p q = q - p then fw_base p q else take p (FW_word p (q - p)) \<cdot> FW_word p (q - p))"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (set_annul True s1)) \<and> t2 = snd (fst (set_annul True s2))"
stream_space_def: fixes M :: "'a measure" shows "stream_space M = distr (Pi\<^sub>M UNIV (\<lambda>i. M)) (vimage_algebra (streams (space M)) (!!) (Pi\<^sub>M UNIV (\<lambda>i. M))) to_stream"
elims: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat"   and xc :: "nat list"   and y :: "pratt list" assumes "build_fpc x xa xb xc = y" obtains p :: "nat" and a :: "nat" and r :: "nat" where "x = p__" and "xa = a__" and "xb = r__" and "xc = []" and "y = [Triple p__ a__ r__]"   | p :: "nat" and a :: "nat" and r :: "nat" and ya :: "nat" and ys :: "nat list" where "x = p" and "xa = a" and "xb = r" and "xc = ya # ys" and "y = Triple p a r # build_fpc p a (r div ya) ys"
fold_code_2: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'c \<Rightarrow> 'c"   and c :: "color"   and lt :: "('d, 'e) rbt"   and k :: "'d"   and v :: "'e"   and rt :: "('d, 'e) rbt"   and x :: "'c" shows "RBT_Impl.fold f (Branch c lt k v rt) x = RBT_Impl.fold f rt (f k v (RBT_Impl.fold f lt x))"
get_disconnected_nodes_locs_impl: shows "get_disconnected_nodes_locs = a_get_disconnected_nodes_locs"
assms_4: shows "kind a' = Q':r'\<hookrightarrow>\<^bsub>p\<^esub>fs'"
r_null: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "x \<otimes>\<^bsub>F\<^esub> \<zero>\<^bsub>F\<^esub> = \<zero>\<^bsub>F\<^esub>"
lift_opt_def: fixes m :: "'b option"   and e :: "'a" shows "lift_opt m e \<equiv> case m of None \<Rightarrow> Inl e | Some x \<Rightarrow> Inr x"
until_unfold: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" shows "until \<phi> \<psi> \<pi>l = (\<psi> \<pi>l \<or> \<phi> \<pi>l \<and> until \<phi> \<psi> (map stl \<pi>l))"
pX: shows "(\<Sum>x | x \<in> keys (mapping_of p) \<and> x' \<in> keys x. MPoly_Type.monom x (MPoly_Type.coeff p x)) = MPoly_Type.monom (monomial (Suc 0) x') (1::'a) * (\<Sum>x | x \<in> keys (mapping_of p) \<and> x' \<in> keys x. MPoly_Type.monom (decrease_key x' x) (MPoly_Type.coeff p x))"
imp_contrapos: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<longrightarrow>. q) = (A \<turnstile> .\<not> q .\<longrightarrow>. .\<not> p)"
assms: shows "Ik.satPB \<xi> \<Phi>"
e: assumes "\<Gamma>\<turnstile>bdy \<down> Normal (init s)"   and "\<forall>t. \<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t \<longrightarrow> \<Gamma>\<turnstile>c s t \<down> Normal (return s t)" shows "P"
int: shows "list_all is_int_rat (concat (mat_to_list A))"
less': shows "\<bar>sgn (real_of_int (sgn ai) * b) / 2\<bar> < 1"
the_map_option: fixes x :: "'a option"   and f :: "'a \<Rightarrow> 'b" assumes "\<not> Option.is_none x" shows "the (map_option f x) = f (the x)"
fund_sol_minimal''_2: fixes z :: "nat \<times> nat" assumes "nontriv_solution z" shows "snd fund_sol \<le> snd z"
secure_termination_def: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c" shows "secure_termination I D \<equiv> \<forall>x. (D x, D None) \<in> I \<and> x \<noteq> None \<longrightarrow> (\<forall>u\<in>range D. (D x, u) \<in> I)"
select_convs_3: fixes cx :: "int"   and cy :: "int"   and more :: "'a" shows "point.more \<lparr>cx = cx, cy = cy, \<dots> = more\<rparr> = more"
sorted_inorder_separators: fixes ts :: "('b btree \<times> 'b) list"   and t :: "'b btree" assumes "sorted_less (inorder (Node ts t))" shows "sorted_less (separators ts)"
fm_image_measurable: fixes M :: "'d measure"   and X :: "('a \<Rightarrow> 'd) set" assumes "space M = UNIV"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. M))" shows "fm ` X \<in> sets (Pi\<^sub>F {f ` J} (\<lambda>_. M))"
domain_eq: shows "Domainp pcr_float = (\<lambda>x. x \<in> float)"
locally_compact_open_Int_closure: fixes S :: "'c set" assumes "locally compact S" obtains T :: "'c set" where "open T" and "S = T \<inter> closure S"
is_extensional: fixes f :: "'d \<times> 'c" assumes "\<not> DopxC.arr f" shows "\<Psi> f = S.null"
atomic_EphK: fixes X :: "ephkey" shows "atomic (EphK X)"
col_preserves_two_sides: fixes C :: "'p"   and D :: "'p"   and A :: "'p"   and B :: "'p"   and X :: "'p"   and Y :: "'p" assumes "C \<noteq> D"   and "Col A B C"   and "Col A B D"   and "A B TS X Y" shows "C D TS X Y"
m_nth: fixes ia :: "nat" assumes "ia < max_Var_floatariths fas" shows "ia < DIM('a) + length ramsch"
hypnat_diff_add_0: fixes n :: "nat star"   and m :: "nat star" shows "n - (n + m) = 0"
all_dnodes_def: shows "all_dnodes \<equiv> Set.insert s {v. \<exists>u. G_\<alpha> g (u, v) \<noteq> \<infinity>}"
weak_arrow_of_homs_axioms: shows "weak_arrow_of_homs (\<cdot>\<^sub>B) src\<^sub>B trg\<^sub>B (\<cdot>) src trg UP"
dfree: fixes f :: "'sf"   and f' :: "('sf + 'sz, 'sz) trm" assumes "SFunctions \<sigma> f = Some f'" shows "dfree f'"
assms_3: shows "g \<in> space M \<rightarrow> space M"
jw_Unroll: fixes \<alpha> :: "Syntax.tvar"   and \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and \<tau> :: "ty" assumes "atom \<alpha> \<sharp> \<Gamma>"   and "\<Gamma> \<turnstile>\<^sub>W e : Mu \<alpha> \<tau>" shows "\<Gamma> \<turnstile>\<^sub>W Unroll e : subst_type \<tau> (Mu \<alpha> \<tau>) \<alpha>"
sets_C: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "sets (C n m x) = sets (Pi\<^sub>M {0..<n + m} M)"
t_not_Fault: shows "t \<notin> Fault ` F"
vec_nth_cbox_1_eq: fixes u :: "('a, 1) vec"   and v :: "('a, 1) vec" shows "(\<lambda>x. x $ 1) ` cbox u v = cbox (u $ 1) (v $ 1)"
from_list_ga_def: fixes e :: "'a Heap"   and i :: "'b \<Rightarrow> 'a \<Rightarrow> 'a Heap"   and l :: "'b list" shows "from_list_ga e i l = e \<bind> from_list_ga_aux i l"
sfilter_scons_let: fixes p :: "'r \<rightarrow> tr"   and x :: "'r"   and xs :: "[:'r:]" assumes "p\<cdot>\<bottom> = \<bottom>" shows "sfilter\<cdot>p\<cdot>(x :# xs) = (let xs' = sfilter\<cdot>p\<cdot>xs in If p\<cdot>x then x :# xs' else xs')"
icache_val_def: fixes c :: "word20 \<times> word12"   and state :: "'b sparc_state" shows "icache_val c state \<equiv> icache (cache state) c"
arev_arev: fixes a :: "'b" shows "arev (arev a) = a"
smult_add_right: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" shows "smult a (p + q) = smult a p + smult a q"
execn_Normal_elim_cases_12: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>DynCom c,Normal s\<rangle> =n\<Rightarrow> t" obtains "\<Gamma>\<turnstile> \<langle>c s,Normal s\<rangle> =n\<Rightarrow> t"
poly_gcd_monic: fixes p :: "'c poly"   and q :: "'c poly" assumes "p \<noteq> 0 \<or> q \<noteq> 0" shows "monic (gcd p q)"
sheaf_of_rings_axioms: shows "sheaf_of_rings S is_open local.im_sheaf im_sheaf_morphisms b add_im_sheaf mult_im_sheaf zero_im_sheaf one_im_sheaf"
span_superset: fixes S :: "'b set" shows "S \<subseteq> local.span S"
acyclic_4_implies_2: fixes x :: "'a" assumes "symmetric x"   and "acyclic_4 x" shows "acyclic_2 x"
h6: shows "get_matrix (reduce_system p (qs, matrix, subsets, signs)) = M_mat (get_signs (reduce_system p (qs, matrix, subsets, signs))) (get_subsets (reduce_system p (qs, matrix, subsets, signs)))"
simps_2: fixes p :: "nat"   and a :: "floatarith"   and bs :: "floatarith list"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatariths_aformerr p (a # bs) vs = approx_floatarith p a vs \<bind> (\<lambda>a. approx_floatariths_aformerr p bs vs \<bind> (\<lambda>r. Some (a # r)))"
ey_def: shows "ey \<equiv> y - aform_val e (fst Y)"
univ_poly_is_ring: fixes K :: "('c multiset \<Rightarrow> 'a) set" assumes "subring K L" shows "ring (K [X]\<^bsub>L\<^esub>)"
cycle_remove_edge_2: fixes x :: "'a"   and s :: "'a"   and e :: "'a" assumes "cycle x"   and "point s"   and "point e"   and "e ; s\<^sup>T \<le> x" shows "start_points (x \<cdot> - (e ; s\<^sup>T)) \<le> s"
ltrm_cfs: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "monom (UP R) (f (deg R f)) (deg R f) n = (if n = deg R f then f (deg R f) else \<zero>\<^bsub>R\<^esub>)"
crypts_hash_1: fixes X :: "msg" shows "crypts {Hash X} \<subseteq> insert (Hash X) (crypts {X})"
moebius_similarity_inv: fixes a :: "complex"   and b :: "complex" assumes "a \<noteq> 0" shows "moebius_inv (moebius_similarity a b) = moebius_similarity (1 / a) (- b / a)"
left_adjoints_compose: fixes f :: "'a"   and f' :: "'a" assumes "B.is_left_adjoint f"   and "B.is_left_adjoint f'"   and "src f = trg f'" shows "B.is_left_adjoint (f \<star> f')"
simps_2: fixes s :: "nat \<Rightarrow> nat option"   and z :: "nat"   and i :: "nat"   and j :: "nat"   and x :: "nat" assumes "s \<in> \<R>" shows "prefixes s z i j (Suc x) = prefixes s z i j x @ [if x = 0 then j else if s (Partial_Recursive.list_encode (prefixes s z i j x @ [0])) \<noteq> s (Partial_Recursive.list_encode (prefixes s z i j x)) then 0 else if s (Partial_Recursive.list_encode (prefixes s z i j x @ [1])) \<noteq> s (Partial_Recursive.list_encode (prefixes s z i j x)) then 1 else z]"
sum_closed_free_Abelian_group: fixes I :: "'b set"   and x :: "'b \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and S :: "'c set" assumes "\<And>i. i \<in> I \<Longrightarrow> x i \<in> carrier (free_Abelian_group S)" shows "sum x I \<in> carrier (free_Abelian_group S)"
foldl_aux_def: shows "(tp', tss', len', maskL', maskR', a1_map', a2_map', done', done_length') = foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss_list"
preserves_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "trg\<^sub>C (F \<mu>) = F.map\<^sub>0 (trg\<^sub>B \<mu>)"
alphaGood_equivP: shows "equivP qGood alphaGood"
impl_inj: fixes implem :: "chan \<Rightarrow> msg"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg"   and x' :: "tag"   and A' :: "agent"   and B' :: "agent"   and M' :: "msg" assumes "semivalid_implem implem" shows "(implem (Chan x A B M) = implem (Chan x' A' B' M')) = (x = x' \<and> A = A' \<and> B = B' \<and> M = M')"
special_subgroup_genby_rev_mono: fixes a :: "'w set"   and b :: "'w set" assumes "a \<in> Pow S"   and "b \<in> Pow S"   and "order.greater_eq \<langle>b\<rangle> \<langle>a\<rangle>" shows "order.greater_eq b a"
compat_f: shows "\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<longrightarrow> gt (f y) (f x)"
minim_isMinim: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "B \<noteq> {}" shows "r.isMinim B (r.minim B)"
swapLeaves_id: fixes t :: "'a tree"   and a :: "'a" assumes "consistent t" shows "swapLeaves t (freq t a) a (freq t a) a = t"
d_gchainTr2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and l :: "nat"   and j :: "nat" assumes "0 < n"   and "d_gchain G n f"   and "l \<le> n"   and "j \<le> n"   and "l \<le> j" shows "f j \<subseteq> f l"
stream_rel_coinduct: fixes R :: "'a stream \<Rightarrow> 'b stream \<Rightarrow> bool"   and u :: "'a stream"   and v :: "'b stream"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R u v"   and "\<And>a u b v. R (a ## u) (b ## v) \<Longrightarrow> P a b \<and> R u v" shows "stream_all2 P u v"
simps_2: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" shows "higher_differentiable_on S f (Suc n) = ((\<forall>x\<in>S. f differentiable at x) \<and> (\<forall>v. higher_differentiable_on S (\<lambda>x. frechet_derivative f (at x) v) n))"
funpow_dist1_prop1: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "(f ^^ n) x = y"   and "0 < n" shows "(f ^^ funpow_dist1 f x y) x = y"
tHyp: shows "t \<in> U (X t\<^sub>0)"
sint_greater_eq: fixes w :: "'a word" shows "- (2 ^ (LENGTH('a) - Suc 0)) \<le> sint w"
base_repeats_or_shares: fixes B :: "base"   and Bs :: "base list"   and D :: "char list" assumes "B \<in> set Bs"   and "D = getbase B" shows "Repeats D \<in> set Bs \<or> Shares D \<in> set Bs"
liminf_subseq_lim: fixes u :: "nat \<Rightarrow> 'b" shows "\<exists>r. strict_mono r \<and> (u \<circ> r) \<longlonglongrightarrow> liminf u"
check_ipresIGWlsAll: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGWlsAll check check (errMOD MOD) MOD"
sc_r_distr: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and a :: "'f"   and m :: "'e"   and n :: "'e" assumes "R module M"   and "a \<in> carrier R"   and "m \<in> carrier M"   and "n \<in> carrier M" shows "a \<cdot>\<^sub>s\<^bsub>M\<^esub> (m \<plusminus>\<^bsub>M\<^esub> n) = a \<cdot>\<^sub>s\<^bsub>M\<^esub> m \<plusminus>\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> n"
sum_wt: fixes c :: "('test, 'atom, 'choice) cmd"   and s :: "'state" assumes "proper c" shows "sum (wt c s) {..<brn c} = 1"
right_total_rel_resource: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "right_unique A"   and "left_total A"   and "right_total B" shows "right_total (rel_resource A B)"
bf_not_def: fixes b :: "'b \<Rightarrow> bool" shows "bf_not b \<equiv> bf_ite b bf_False bf_True"
simps: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "top0"   and a5 :: "'b state \<times> (v sem_env, v) result" shows "evaluate_top a1 a2 a3 a4 a5 = ((\<exists>ck s1 s2 env d new_env. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = Tdec d \<and> a5 = (s2, Rval new_env) \<and> evaluate_dec ck [] env s1 d (s2, Rval new_env)) \<or> (\<exists>ck s1 s2 env d err. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = Tdec d \<and> a5 = (s2, Rerr err) \<and> evaluate_dec ck [] env s1 d (s2, Rerr err)) \<or> (\<exists>ck s1 s2 env ds mn specs new_env. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = Tmod mn specs ds \<and> a5 = (update_defined_mods (\<lambda>_. {[mn]} \<union> defined_mods s2) s2, Rval (make_sem_env (nsLift mn (sem_env.v new_env)) (nsLift mn (c new_env)))) \<and> [mn] \<notin> defined_mods s1 \<and> no_dup_types ds \<and> evaluate_decs ck [mn] env s1 ds (s2, Rval new_env)) \<or> (\<exists>ck s1 s2 env ds mn specs err. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = Tmod mn specs ds \<and> a5 = (update_defined_mods (\<lambda>_. {[mn]} \<union> defined_mods s2) s2, Rerr err) \<and> [mn] \<notin> defined_mods s1 \<and> no_dup_types ds \<and> evaluate_decs ck [mn] env s1 ds (s2, Rerr err)) \<or> (\<exists>ck s1 env ds mn specs. a1 = ck \<and> a2 = env \<and> a3 = s1 \<and> a4 = Tmod mn specs ds \<and> a5 = (s1, Rerr (Rabort Rtype_error)) \<and> \<not> no_dup_types ds) \<or> (\<exists>ck env s mn specs ds. a1 = ck \<and> a2 = env \<and> a3 = s \<and> a4 = Tmod mn specs ds \<and> a5 = (s, Rerr (Rabort Rtype_error)) \<and> [mn] \<in> defined_mods s))"
inject_4: fixes x41 :: "exp"   and x42 :: "exp"   and y41 :: "exp"   and y42 :: "exp" shows "(EApp x41 x42 = EApp y41 y42) = (x41 = y41 \<and> x42 = y42)"
mg: fixes r :: "('V Standard_Constant, 'b) labeled_graph \<times> ('V Standard_Constant, 'b) labeled_graph" assumes "maintained r G'" shows "maintained r (map_graph_fn G' h)"
distinct_1: fixes x1 :: "'a"   and x21 :: "'a"   and x22 :: "'b"   and x23 :: "('a, 'b) chain" shows "\<lbrace> x1 \<rbrace> \<noteq> \<lbrace> x21 \<rbrace> \<cdot> x22 \<cdot> x23"
assms: shows "s' = snd (fst (mul_instr_sub1 instr_name result s)) \<and> get_S (cpu_reg_val PSR s) = 0"
wp_loop_guard: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "healthy (wp body)"   and "sound P"   and "G s" shows "wp (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip) P s = wp (body ;; (\<mu>x. body ;; x \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)) P s"
allPh: fixes u :: "'a"   and y :: "'b" shows "PROP P (h (u, y))"
assms: shows "\<forall>v\<in>fmdom' f. fmlookup f v = fmlookup g v"
simps_2: fixes k :: "nat"   and x :: "'a" shows "Suc k \<times> x = k \<otimes> x"
pos_cons: fixes xs :: "real list"   and x :: "real" shows "xs \<noteq> [] \<longrightarrow> pos (x # xs) = (if 0 < x then pos xs else False)"
ln_float_interval: fixes p :: "nat"   and x :: "float interval"   and y :: "float interval" assumes "ln_float_interval p x = Some y" shows "ln ` set_of (real_interval x) \<subseteq> set_of (real_interval y)"
assms_3: shows "\<tt>\<tt> i \<le> \<tt>\<tt> q"
FinFunc_singleton: fixes z :: "'e"   and s :: "('d \<times> 'd) set" shows "FinFunc {(z, z)} s = {\<lambda>x. if x \<in> Field s then z else undefined}"
loop_forward: fixes \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and I :: "('b, 'sc, 'a) interp"   and id1 :: "'a" shows "\<nu> \<in> fml_sem I ([[$\<alpha> id1**]]Pc pid1) \<longrightarrow> \<nu> \<in> fml_sem I (Pc pid1 && [[$\<alpha> id1]][[$\<alpha> id1**]]Pc pid1)"
integrable_mult_left: fixes c :: "'a"   and M :: "'b measure"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a) \<Longrightarrow> integrable M f" shows "integrable M (\<lambda>x. f x * c)"
reindex_bij_witness: fixes S :: "'a set"   and i :: "'b \<Rightarrow> 'a"   and j :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" assumes "\<And>a. a \<in> S \<Longrightarrow> i (j a) = a"   and "\<And>a. a \<in> S \<Longrightarrow> j a \<in> T"   and "\<And>b. b \<in> T \<Longrightarrow> j (i b) = b"   and "\<And>b. b \<in> T \<Longrightarrow> i b \<in> S"   and "\<And>a. a \<in> S \<Longrightarrow> h (j a) = g a" shows "sum g S = sum h T"
select_first_def: fixes x :: "'a"   and p :: "'a \<Rightarrow> real"   and y :: "'a" shows "select_first x p y = (y \<in> I x \<and> (\<forall>y'\<in>I x - {y}. p y < p y'))"
simplex_like_no_pseudominimal_imp_singleton: fixes P :: "'a set" assumes "simplex_like P"   and "\<And>x. \<not> pseudominimal_in P x" shows "\<exists>p. P = {p}"
neg_0_le_iff_le: fixes a :: "'b" shows "((0::'b) \<le> - a) = (a \<le> (0::'b))"
Deq_def: fixes ic :: "state \<Rightarrow> 'a"   and q :: "state \<Rightarrow> 'a list"   and oc :: "state \<Rightarrow> 'a" shows "Deq ic q oc \<equiv> $q \<noteq> #[] \<and> oc$ = hd<$q> \<and> q$ = tl<$q> \<and> ic$ = $ic"
count_mset_set': fixes A :: "'k set"   and x :: "'k" shows "count (mset_set A) x = (if finite A \<and> x \<in> A then 1 else 0)"
sat_imp_ground_sat: fixes N :: "'f set" assumes "saturated N"   and "ground.Inf_from (\<G>_Fset N) \<subseteq> {\<iota>. \<exists>\<iota>'\<in>Inf_from N. \<G>_I \<iota>' \<noteq> None \<and> \<iota> \<in> the (\<G>_I \<iota>')} \<union> Red_I_G (\<G>_Fset N)" shows "ground.saturated (\<G>_Fset N)"
assms_3: shows "\<exists>rev. bidirected_digraph G rev"
poly_x_mult_y_code: fixes p :: "'a poly" shows "poly_x_mult_y p = (let cs = coeffs p in poly_of_list (map2 (\<lambda>i ai. monom ai i) [0..<length cs] cs))"
ldf5: shows "\<alpha> ! (i - length \<delta>) = \<gamma>' ! (j - length \<delta>)"
exp_Lambert_W': fixes x :: "real" assumes "x \<in> {- exp (- 1)..<0}" shows "exp (Lambert_W' x) = x / Lambert_W' x"
config_snoc: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s :: "'a list"   and rs :: "'a list"   and r :: "'a" shows "config\<^sub>p A s (rs @ [r]) = Partial_Cost_Model.Step A (config\<^sub>p A s rs) r"
aux_ineq_with_card_1: fixes A :: "nat set"   and g :: "nat \<Rightarrow> nat" assumes "finite A" shows "(\<Sum>k\<in>A. (- 1) ^ g k) \<le> int (card A)"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "Ik.wtE \<xi> \<equiv> \<forall>x. intT (GE.tpOfV x) (\<xi> x)"
isnpolyh_polynate_id: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "polynate p = p"