hcju/isabelleps huggingface dataset:
cSUP_subset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'d set"   and g :: "'d \<Rightarrow> 'c"   and B :: "'d set"   and f :: "'d \<Rightarrow> 'c" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (g ` B)"   and "A \<subseteq> B"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` B))"
prefixed_Imagep_imp_empty: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set" assumes "well_founded A less"   and "X \<subseteq> (less ``` X) \<inter> A" shows "X = {}"
Op_def: fixes delta :: "'c"   and inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binp :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option" shows "Op delta inp binp \<equiv> asTerm (qOp delta (pickInp inp) (pickBinp binp))"
power_minus: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "power.power one times (uminus a) n = times (power.power one times (uminus one) n) (power.power one times a n)"
a5: fixes x :: "'a" assumes "x \<in> S - F" shows "0 \<le> norm (f x)"
hyps: fixes A :: "(('a, 'c) vec, 'b) vec" shows "\<exists>P. invertible P \<and> Hermite_of_upt_row_i A k ass res = P ** A"
streams_shd: fixes s :: "'d stream"   and A :: "'d set" assumes "s \<in> streams A" shows "shd s \<in> A"
hyps: fixes kv :: "'a \<times> 'b" shows "set (update_by_pair kv xs) \<subseteq> insert kv (set xs)"
find_sub_False_sound: fixes xs :: "'b set list"   and ns :: "'b set list"   and rs :: "'b set list"   and w :: "'b set"   and ws :: "'b set list"   and ys :: "'b set list"   and z :: "'b set"   and zs :: "'b set list" assumes "find_sub_False xs ns = Some ((rs, w, ws), ys, z, zs)" shows "xs = rs @ w # ws \<and> ns = ys @ z # zs \<and> z \<subseteq> w"
select_convs_1: fixes ifaces :: "idecl list"   and classes :: "cdecl list"   and more :: "'a" shows "ifaces \<lparr>ifaces = ifaces, classes = classes, \<dots> = more\<rparr> = ifaces"
merge_emb_step_at: fixes p :: "dir list"   and d1 :: "dir"   and q :: "dir list"   and d2 :: "dir"   and t :: "('a, 'b) tm" shows "emb_step_at p d1 (emb_step_at (p @ [opp d1] @ q) d2 t) = emb_step_at p d1 t"
hyps_5: shows "atom \<alpha> \<sharp> \<Gamma>"
inv: shows "\<psi> \<circ> \<phi> = J.map"
Image_iff: fixes b :: "'g"   and r :: "('h \<times> 'g) set"   and A :: "'h set" shows "(b \<in> r `` A) = (\<exists>x\<in>A. (x, b) \<in> r)"
mult_right_dist_sup: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<squnion> y) * z = x * z \<squnion> y * z"
rep_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "\<F>C.rep (f \<otimes> g) = \<^bold>\<parallel>\<F>C.rep f \<^bold>\<otimes> \<F>C.rep g\<^bold>\<parallel>"
remove_def: fixes x :: "'b"   and A :: "'b set" shows "Set.remove (x::'b) (A::'b set) = A - {x}"
ardenR: fixes r :: "'a language"   and x :: "'a language"   and s :: "'a language" assumes "Plus r (Times x s) \<le> x" shows "Times r (Star s) \<le> x"
element_put_get: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and setter :: "('u \<Rightarrow> 'u) \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> ('n, 'l, 'm, 'p, 'r option) RElement_scheme"   and v :: "'u"   and h' :: "(_) heap"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "h \<turnstile> put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr setter v \<rightarrow>\<^sub>h h'"   and "\<And>x. getter (setter (\<lambda>_. v) x) = v" shows "h' \<turnstile> get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr getter \<rightarrow>\<^sub>r v"
inline_lift_spmf1: fixes p :: "'b \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'b) spmf"   and gpv :: "('a, 'e, 'f) gpv"   and s :: "'b" shows "inline (\<lambda>s c. lift_spmf (p s c)) gpv s = lift_spmf (map_spmf projl (inline1 (\<lambda>s c. lift_spmf (p s c)) gpv s))"
supp_fun_plus_subset: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "supp_fun (s + t) \<subseteq> supp_fun s \<union> supp_fun t"
main_bound: fixes u :: "real"   and n :: "nat" assumes "0 < u" shows "u * Mn.prob n {x. 2 / u \<le> \<bar>x\<bar>} \<le> Re (CLBINT t:{- u..u}. 1 - char (M n) t)"
pair: fixes A :: "'a trm"   and B :: "'a trm" assumes "NF A"   and "NF B" shows "NF (SimplyTyped.Pair A B)"
cong: fixes x :: "real" assumes "real_of_rat l \<le> x"   and "x \<le> real_of_rat r" shows "(poly (real_of_int_poly p) x = 0) = (poly (real_of_int_poly q) x = 0)"
usubstappp_loop_conv: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and \<alpha> :: "game" assumes "snd (usubstappp \<sigma> U \<alpha>**) \<noteq> undefg" shows "snd (usubstappp \<sigma> U \<alpha>) \<noteq> undefg \<and> snd (usubstappp \<sigma> (fst (usubstappp \<sigma> U \<alpha>)) \<alpha>) \<noteq> undefg"
supr_under: fixes r :: "('c \<times> 'c) set"   and a :: "'c" assumes "wo_rel r"   and "a \<in> Field r" shows "a = supr r (under r a)"
trace_to_steps: fixes c :: "nat \<Rightarrow> ('p \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and k :: "nat" assumes "trace c 0 k" shows "c 0 \<Rightarrow>\<^sub>d\<^sup>* c k"
wpt_def: fixes c :: "com"   and Q :: "state \<Rightarrow> bool" shows "wp\<^sub>t c Q = (\<lambda>s. wp c Q s \<and> c \<down> s)"
update_commute: fixes G :: "'b set"   and g :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and f :: "('b \<Rightarrow> 'c) \<Rightarrow> 'b \<Rightarrow> 'c"   and F :: "'b set"   and m :: "'b \<Rightarrow> 'c" assumes "\<forall>a m. a \<notin> G \<longrightarrow> g m a = m a"   and "\<forall>a m. a \<in> G \<longrightarrow> g (f m) a = g m a"   and "\<forall>a m. a \<notin> F \<longrightarrow> f m a = m a"   and "\<forall>a m. a \<in> F \<longrightarrow> f (g m) a = f m a"   and "G \<inter> F = {}" shows "f (g m) = g (f m)"
trans_nest: fixes a :: "'a set"   and c :: "'a set"   and g :: "'a set" assumes "a \<lless> c"   and "c \<lless> g" shows "a \<lless> g"
vertices_quad: fixes f :: "face"   and a :: "nat" assumes "|vertices f| = 4"   and "a \<in> \<V> f"   and "distinct (vertices f)" shows "\<V> f = {a, f \<bullet> a, f \<bullet> (f \<bullet> a), f \<bullet> (f \<bullet> (f \<bullet> a))}"
calc2: shows "CLBINT s=0..ereal x. f s n = complex_of_real (x ^ Suc n / real (Suc n))"
in_quotient_imp_subset: fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and X :: "'c set" assumes "equiv A r"   and "X \<in> A // r" shows "X \<subseteq> A"
Int_Inter_eq_1: fixes A :: "'c set"   and \<B> :: "'c set set" shows "A \<inter> \<Inter> \<B> = (if \<B> = {} then A else \<Inter> ((\<inter>) A ` \<B>))"
T_delete_amor: fixes n :: "nat"   and t :: "'a tree"   and x :: "'a"   and dl :: "nat" assumes "n = size t" shows "real (T_delete2 x (t, n, dl)) + \<Phi>\<^sub>d (local.delete2 x (t, n, dl)) - \<Phi>\<^sub>d (t, n, dl) \<le> (6 * e + 1) * real (height t) + 4 / cd + 4"
iMOD_0_Max: fixes r :: "nat" shows "Max [ r, mod 0 ] = r"
retraction_map_def: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "retraction_map X Y f \<equiv> \<exists>g. retraction_maps X Y f g"
degree_eq_iff: fixes p :: "'b mpoly"   and x :: "nat"   and k :: "nat" shows "(MPoly_Type.degree p x = k) = (if k = 0 then \<forall>m\<in>monomials p. lookup m x = 0 else (\<exists>m\<in>monomials p. lookup m x = k) \<and> (\<forall>m\<in>monomials p. lookup m x \<le> k))"
newton_sqrt_def: fixes n :: "nat" shows "newton_sqrt n = newton_sqrt_aux n n"
size_simp1: fixes t :: "'a"   and ts :: "'a list" assumes "t \<in> set ts" shows "size t < Suc (size_list size ts)"
the_g: shows "(THE g. \<forall>i. w $ i = (\<Sum>x\<in>UNIV. g x * X $ x $ i)) = g"
assms_3: shows "y \<in> set (fst (interp_ops ops))"
dropWhile_append2: fixes xs :: "'f list"   and P :: "'f \<Rightarrow> bool"   and ys :: "'f list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "dropWhile P (xs @ ys) = dropWhile P ys"
connected_Times: fixes S :: "'c set"   and T :: "'d set" assumes "connected S"   and "connected T" shows "connected (S \<times> T)"
Restr_incr2: fixes r :: "('c \<times> 'c) set"   and r' :: "('c \<times> 'c) set"   and A :: "'c set" assumes "r \<subseteq> r'" shows "Restr r A \<subseteq> Restr r' A"
det: assumes "length fs = n" shows "det (mat_of_rows n fs) = det (mat_of_rows n (map gso [0..<length fs]))"
atMost_Suc: fixes g :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "sum g {..Suc n} = sum g {..n} + g (Suc n)"
extend_leadsTo: fixes F :: "'a program"   and A :: "'a set"   and B :: "'a set" shows "(extend h F \<in> extend_set h A \<longmapsto> extend_set h B) = (F \<in> A \<longmapsto> B)"
get_put: fixes \<sigma> :: "'b" shows "put \<sigma> (get \<sigma>) = \<sigma>"
idijkstra_refines: fixes g :: "'G"   and v0 :: "'V" assumes "g.invar g" shows "RETURN (idijkstra g v0) \<le> cdijkstra g v0"
comm_monoid_add_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "class.comm_monoid_add plus zero \<equiv> class.ab_semigroup_add plus \<and> class.comm_monoid_add_axioms plus zero"
map_sel_1: fixes a :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "a \<noteq> []" shows "hd (map f a) = f (hd a)"
elims: fixes x :: "'b hp"   and y :: "'b multiset" assumes "mset_hp x = y" obtains xa :: "'b" and hs :: "'b hp list" where "x = Hp xa hs" and "y = {#xa#} + \<Sum>\<^sub># (mset (map mset_hp hs))"
C1_differentiable_on_components: fixes f :: "real \<Rightarrow> 'a"   and s :: "real set" assumes "\<And>i. i \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> i) C1_differentiable_on s" shows "f C1_differentiable_on s"
h_inc: fixes n :: "nat"   and x :: "(real, 'n) vec" shows "h n x \<le> h (Suc n) x"
pathfinish_pathstart_partcirclepath_simps_7: fixes z0 :: "complex"   and r :: "real"   and tt :: "real" shows "pathstart (part_circlepath z0 r pi tt) = z0 - complex_of_real r"
attract: shows "\<forall>q\<in>A. \<forall>x\<in>A. f q \<sim> q \<longrightarrow> x \<sqsubseteq> f q \<longrightarrow> x \<sqsubseteq> q"
conj_mono: fixes P1 :: "bool"   and Q1 :: "bool"   and P2 :: "bool"   and Q2 :: "bool" assumes "P1 \<longrightarrow> Q1"   and "P2 \<longrightarrow> Q2" shows "P1 \<and> P2 \<longrightarrow> Q1 \<and> Q2"
scaleQ_zero: fixes x :: "'e['f, 'g]" shows "(0::'e) *\<^sub>Q x = 0"
ccBindsExtra_cong: fixes heap2 :: "(Vars.var \<times> exp) list"   and cccexp1 :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and cccexp2 :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and heap1 :: "(Vars.var \<times> exp) list" assumes "\<And>e. e \<in> snd ` set heap2 \<Longrightarrow> cccexp1 e = cccexp2 e"   and "heap1 = heap2" shows "CoCallAnalysis.ccBindsExtra cccexp1 heap1 = CoCallAnalysis.ccBindsExtra cccexp2 heap2"
hcomplex_hIm_zero: shows "hIm 0 = 0"
rotate10: fixes A10 :: "fm"   and A1 :: "fm"   and A2 :: "fm"   and A3 :: "fm"   and A4 :: "fm"   and A5 :: "fm"   and A6 :: "fm"   and A7 :: "fm"   and A8 :: "fm"   and A9 :: "fm"   and H :: "fm set"   and B :: "fm" assumes "insert A10 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H))))))))) \<turnstile> B"
simps_1: fixes ts :: "ct list"   and ts' :: "t list" shows "produce (TopType ts) (Type ts') = TopType (ts @ to_ct_list ts')"
division_ofD_6: fixes s :: "'c set set"   and i :: "'c set" assumes "s division_of i" shows "\<Union> s = i"
distinct_7: fixes x2 :: "(val \<times> val) fset"   and x3 :: "val option" shows "val.Fun x2 \<noteq> Abs x3"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsC c" shows "\<xi>1 x = \<xi>2 x"
coarser_partitions_with_list_equivalence: fixes P :: "'b set list"   and new_el :: "'b" assumes "distinct P" shows "set (map set (coarser_partitions_with_list new_el P)) = coarser_partitions_with new_el (set P)"
le_bottom: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'c"   and B :: "'c"   and x :: "'c" assumes "lbv A r f T B" shows "(x \<sqsubseteq>\<^bsub>r\<^esub> B) = (x = B)"
isolate_variable_sparse_monom: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a"   and v :: "nat"   and i :: "nat" shows "isolate_variable_sparse (MPoly_Type.monom m a) v i = (if a = (0::'a) \<or> lookup m v \<noteq> i then 0 else MPoly_Type.monom (Poly_Mapping.update v 0 m) a)"
mmonitorable_exec_mmonitorable: fixes \<phi> :: "'b MFOTL.formula" assumes "mmonitorable_exec \<phi>" shows "mmonitorable \<phi>"
Zp_int_inc_rep: fixes n :: "int" shows "[n] \<cdot> \<one> = (\<lambda>m. p_residue m n)"
abs_eq: fixes xa :: "'c"   and x :: "('a, 'b, 'c) Tree" shows "Act\<^sub>\<alpha> xa (abs_Tree\<^sub>\<alpha> x) = abs_Tree\<^sub>\<alpha> (tAct xa x)"
cadjoint_exists: fixes G :: "'b \<Rightarrow> 'c" assumes "bounded_clinear G" shows "\<exists>F. is_cadjoint F G"
mps_4: assumes "nmu (dports m2)" shows "php m2"
finPremSet: fixes Ps :: "sequent list" shows "finite (set_of_prem Ps)"
gallery_CConsI: fixes w :: "'w set set"   and x :: "'w set set"   and xs :: "'w set set list" assumes "chamber w"   and "gallery (x # xs)"   and "w \<sim> x" shows "gallery (w # x # xs)"
e_type_comp: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and es :: "e list"   and e :: "e"   and t1s :: "t list"   and t3s :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> es @ [e] : t1s _> t3s" shows "\<exists>ts'. \<S>\<bullet>\<C> \<turnstile> es : t1s _> ts' \<and> \<S>\<bullet>\<C> \<turnstile> [e] : ts' _> t3s"
set_relI: fixes A :: "'a set"   and B :: "'b set"   and R :: "('a \<times> 'b) set" assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. (x, y) \<in> R"   and "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. (x, y) \<in> R" shows "(A, B) \<in> \<langle>R\<rangle>set_rel"
endq: shows "basepoint linePQ \<leadsto> direction linePQ = q"
Suc_j_less_nrows: shows "Suc j < nrows A"
ts''_def_1: shows "\<C> \<turnstile> [C ConstInt32 c] : ts _> ts''"
flift2_bottom_iff: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b lift" shows "(flift2 f\<cdot>x = \<bottom>) = (x = \<bottom>)"
simps_1: fixes x :: "('b, 'c) Combinators"   and l :: "'b set list" shows "smaller DenyAll x l = True"
iT_Plus_0: fixes I :: "nat set" shows "I \<oplus> 0 = I"
bv_to_int_utos: fixes w :: "bit list" shows "bv_to_int (utos w) = int (bv_to_nat w)"
assms_2: assumes "ide f"   and "ide g"   and "src f \<cong> g \<star> f"   and "f \<star> g \<cong> trg f" shows "T"
ide_a': fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and a :: "'a"   and a' :: "'a" assumes "yoneda_lemma_for_hom C S \<phi> a a'" shows "partial_magma.ide C a'"
uncountable_infinite: fixes A :: "'d set" assumes "uncountable A" shows "infinite A"
wqo_on_lists: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" assumes "wqo_on P A" shows "wqo_on (list_emb P) (lists A)"
p_def: fixes n :: "nat"   and x :: "real" shows "p n x = (if n = 0 then 1 / x else \<Sum>r<n. D (real r + x))"
minus_of_int: fixes x :: "'a"   and n :: "int" shows "f (x - of_int n) = f x"
rep_cod: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.rep (\<F>C.cod f) = \<^bold>\<parallel>\<F>C.COD f\<^bold>\<parallel>"
transp_le: shows "transp (\<preceq>\<^sub>t)"
assms: shows "(IF b THEN SKIP ELSE SKIP, s) \<Rightarrow> t"
1: shows "[\<^bold>\<forall>F. \<^bold>\<diamond>(\<lbrace>x\<^sup>P,F\<rbrace> \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p))) in v]"
Aenc_synth_eq1: fixes K :: "msg"   and H :: "msg set"   and X :: "msg" assumes "K \<notin> synth H" shows "(Aenc X K \<in> synth H) = (Aenc X K \<in> H)"
i2: fixes x :: "('a, 'n) vec"   and y :: "('a, 'n) vec"   and z :: "('a, 'n) vec" shows "inner (x + y) z = inner x z + inner y z"
subcocycle_bounded_by_birkhoffN: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and N :: "nat"   and n :: "nat"   and x :: "'a" assumes "subcocycle u"   and "2 * N < n"   and "0 < N" shows "u n x \<le> birkhoff_sum (\<lambda>x. u N x / real N) (n - 2 * N) x + (\<Sum>i<N. \<bar>u 1 ((T ^^ i) x)\<bar>) + 2 * (\<Sum>i<2 * N. \<bar>u 1 ((T ^^ (n - (2 * N - i))) x)\<bar>)"
real_0_le_add_iff: fixes x :: "real"   and y :: "real" shows "(0 \<le> x + y) = (- x \<le> y)"
\<gamma>': shows "\<gamma> = \<delta> @ \<gamma>' \<and> LeftDerivation \<alpha> (derivation_shift D (length \<delta>) 0) \<gamma>'"
minus_of_nat: fixes x :: "'a"   and n :: "nat" shows "f (x - of_nat n) = f x"
less_le_trans: fixes x :: "'a word"   and y :: "'a word"   and z :: "'a word" assumes "x <s y"   and "y \<le>s z" shows "x <s z"
eFreshAbs_simp1: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and us :: "'varSort"   and s :: "'sort"   and A :: "'d"   and ys :: "'varSort"   and y :: "'var" assumes "igWlsAbs MOD (us, s) A" shows "eFreshAbs MOD ys y (OK A) = igFreshAbs MOD ys y A"
map_drop_set_def: fixes A :: "'a set" shows "map_drop_set A = map_filter (\<lambda>a. a \<notin> A)"
B21: fixes x :: "'a" shows "(1::'a) + x \<le> x\<^sup>\<star>"
mmult_def: fixes m :: "int"   and x :: "int"   and y :: "int" shows "x **\<^bsub>m\<^esub> y = x * y mod m"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) (\<lambda>f. T (\<I>, f))"
polyhedral_cone_carrier: fixes A :: "'a mat"   and nr :: "nat" assumes "A \<in> carrier_mat nr nr" shows "nr.polyhedral_cone A \<subseteq> carrier_vec nr"
F2_f: shows "Fubx_get F2 f = Some fd2"
fps_of_poly_nth: fixes p :: "'a poly"   and n :: "nat" shows "fps_of_poly p $ n = coeff p n"
ex1_def: shows "ex1 \<equiv> \<lambda>F w. {x. F (\<nu>\<upsilon> x) dj w}"
has_integral_powr_to_inf: fixes e :: "real"   and a :: "real" assumes "e < - 1"   and "0 < a" shows "((\<lambda>x. x powr e) has_integral - (a powr (e + 1)) / (e + 1)) {a..}"
simps_15: fixes x4 :: "ty"   and x3 :: "qtname" shows "ArrayT x4 \<noteq> ClassT x3"
nat_term_compD4: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and u :: "'a"   and a :: "'a"   and v :: "'a"   and b :: "'a" assumes "nat_term_comp cmp"   and "fst (rep_nat_term u) = fst (rep_nat_term a)"   and "fst (rep_nat_term v) = fst (rep_nat_term b)"   and "snd (rep_nat_term u) = snd (rep_nat_term v)"   and "snd (rep_nat_term a) = snd (rep_nat_term b)"   and "cmp a b = Lt" shows "cmp u v = Lt"
le_st_iff: fixes F :: "'b st"   and G :: "'b st" shows "(F \<le> G) = (\<forall>x. fun F x \<le> fun G x)"
inf_prefix_length_leR: fixes xs :: "'b list"   and ys :: "'b list" shows "inf_prefix_length xs ys \<le> length ys"
assms_1: shows "\<not> hb (i, Store e1 mo i) (r, Expunge e2 mo2 r)"
assms_1: shows "nonneg_dirichlet_series (fds_ln l f)"
zero_less_power: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less zero a" shows "less zero (power.power one times a n)"
ll_on_open_it_def: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" shows "ll_on_open_it T f X \<equiv> ll_on_open T f X"
lookup_pair_sort_oalist_aux: fixes xs :: "('a \<times> 'e) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs" shows "lookup_pair ko (sort_oalist_aux ko xs) = lookup_raw xs"
simps_2: fixes x :: "brick list"   and ys :: "wall" shows "domain_wall (x * ys) = domain_block x"
lir_fps_def: fixes cs :: "'b list"   and fs :: "'b list"   and g :: "('b \<times> nat \<times> 'b) list" shows "lir_fps cs fs g = (if cs = [] \<or> length fs < length cs - 1 then None else let m = length fs + 1 - length cs; p = lir_fps_numerator m cs ((!) fs) (eval_polyexp g); q = lr_fps_denominator cs in Some ((ratfps_of_poly p + polyexp_ratfps g) * inverse (ratfps_of_poly q)))"
measure_space_inter: fixes s :: "'b set"   and t :: "'b set" assumes "s \<in> events"   and "t \<in> events"   and "prob t = prob (space p)" shows "prob (s \<inter> t) = prob s"
execute_update: fixes r :: "'a ref"   and v :: "'a"   and h :: "heap" shows "execute (r := v) h = Some ((), Ref.set r v h)"
assms_3: shows "consume (Trans s1 a1 ou1 s1') vl1 vl1'"
euler_witness_def: fixes a :: "int"   and p :: "nat" shows "euler_witness a p = [Jacobi a (int p) \<noteq> a ^ ((p - 1) div 2)] (mod int p)"
full_subcategory_axioms: shows "full_subcategory (\<cdot>) (\<lambda>f. \<F>C.ide f \<and> Diag (\<F>C.DOM f))"
image_linear_lessThan: fixes c :: "'c"   and b :: "'c"   and x :: "'c" assumes "c \<noteq> (0::'c)" shows "(\<lambda>x. c * x + b) ` {..<x} = (if (0::'c) < c then {..<c * x + b} else {c * x + b<..})"
cmp_components_are_iso: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.iso (\<Phi>\<^sub>G (f, g))"
proj_append_1: fixes l :: "'d"   and A' :: "('d strand_label \<times> 'e) list"   and B' :: "('d strand_label \<times> 'e) list" shows "proj l (A' @ B') = proj l A' @ proj l B'"
P1_adv_real_ideal_model_def: fixes n :: "nat"   and D :: "'d \<times> 'grp \<Rightarrow> bool spmf"   and M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'd spmf)"   and z :: "'e" shows "P1_adv_real_ideal_model n D M \<sigma> \<A> z = \<bar>spmf (P1_real_model n M \<sigma> z \<A> \<bind> D) True - spmf (P1_ideal_model n M \<sigma> z \<A> \<bind> D) True\<bar>"
dg_Obj_vsubset_Vset: shows "\<AA>\<lparr>Obj\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"