hcju/isabelleps huggingface dataset:
and_one_eq: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "and a one = modulo a (numeral.numeral one plus (num.Bit0 num.One))"
perp_in_perp_bis: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X PerpAt A B C D " shows "X B Perp C D \<or> A X Perp C D"
system_U_implies_vNM_utility: fixes outcomes :: "'b set"   and \<R> :: "('b pmf \<times> 'b pmf) set" assumes "finite outcomes"   and "outcomes \<noteq> {}"   and "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "continuous_vnm (\<P> outcomes) \<R>" shows "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
nth_root_nat_aux_le: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "k ^ m \<le> n"   and "0 < m" shows "nth_root_nat_aux m k (k ^ m) n ^ m \<le> n"
less_eq_def: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" shows "(x \<subseteq> y) = (x \<union> y = y)"
path_Exit_source_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFGExit sourcenode targetnode kind valid_edge Entry Exit"   and "CFG.path sourcenode targetnode valid_edge Exit as n'" shows "n' = Exit"
monom_deg_mult: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "g \<in> carrier P"   and "deg R g \<le> deg R f"   and "a \<in> carrier R" shows "deg R (g \<otimes>\<^bsub>P\<^esub> monom P a (deg R f - deg R g)) \<le> deg R f"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C\<^sup>o\<^sup>p) C'.\<a> C'.\<i> C'.src C'.trg (\<cdot>\<^sub>D) (\<star>\<^sub>D\<^sup>o\<^sup>p) D'.\<a> D'.\<i> D'.src D'.trg F (\<lambda>f. \<Phi> (snd f, fst f))"
codomains_comp: fixes g :: "'a1 \<times> 'a2"   and f :: "'a1 \<times> 'a2" assumes "A1xA2.seq g f" shows "A1xA2.codomains (g \<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2 f) = A1xA2.codomains g"
l2_inv3E: fixes x :: "l2_state" assumes "x \<in> l2_inv3" obtains "extr (bad x) (ik x) (chan x) \<subseteq> synth (analz (generators x)) \<and> chan x \<subseteq> faked_chan_msgs x \<union> chan_generators"
now: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A\<rbrakk> \<Longrightarrow> P x y" shows "af A P"
int_def: fixes n :: "nat" shows "int n = Abs_Integ (n, 0)"
abs_le_square_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sgn :: "'d \<Rightarrow> 'd"   and abs :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and x :: "'d"   and y :: "'d" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq (abs x) (abs y) = less_eq (power.power one times x 2) (power.power one times y 2)"
assms_3: shows "s1' = s1(V := \<lbrakk>e\<rbrakk>s1)"
in_set_clearjunk_iff_map_of_eq_Some: fixes a :: "'d"   and b :: "'e"   and xs :: "('d \<times> 'e) list" shows "((a, b) \<in> set (AList.clearjunk xs)) = (map_of xs a = Some b)"
f1: shows "\<forall>A. {\<Sum>a\<in>Aa. f a *\<^sub>C a |Aa f. finite Aa \<and> Aa \<subseteq> A} = cspan A"
map_bset_bempty: fixes f :: "'c \<Rightarrow> 'a" shows "map_bset f bempty = bempty"
IH: fixes i :: "nat"   and j :: "nat" assumes "i < j" shows "match test r i j = (\<Squnion> (match test ` rpd test j r)) i (j - 1)"
unique_connecting_path_unique: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "Tree G"   and "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w" shows "xs = v \<leadsto>\<^bsub>G\<^esub> w"
non_zero_vectors_def: shows "local.non_zero_vectors \<equiv> {x. x \<noteq> (0::'b)}"
sound_modes_no_read: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and i :: "nat" assumes "sound_mode_use (cms, mem)"   and "x \<in> (map snd cms ! i) GuarNoRead"   and "i \<sqsubset> length cms" shows "doesnt_read (fst (cms ! i)) x"
R_def: fixes h :: "'a"   and w :: "nat"   and e :: "nat" shows "Schnorr_\<Sigma>.R h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (a, e, z)))"
serial_strips_equivalent_to_serial_sas_plus_ii: fixes \<Pi> :: "('c, 'd) sas_plus_problem"   and \<pi> :: "('c \<times> 'd) strips_operator list" shows "concat (\<phi>\<^sub>P\<inverse> \<Pi> List_Supplement.embed \<pi>) = map (strips_op_to_sasp \<Pi>) \<pi>"
var_writable\<^sub>A: fixes mds\<^sub>C :: "Mode \<Rightarrow> 'Var\<^sub>C set"   and x :: "'Var\<^sub>A" assumes "\<not> var_asm_not_written mds\<^sub>C (var\<^sub>C_of x)" shows "\<not> var_asm_not_written (mds\<^sub>A_of mds\<^sub>C) x"
app_free_distinct: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a"   and name :: "name" shows "app u\<^sub>1 u\<^sub>2 \<noteq> free name"
unary_wt_sym_0_gt: fixes f :: "'s"   and g :: "'s" assumes "arity_sym f = 1"   and "wt_sym f = 0" shows "f >\<^sub>s g \<or> g = f"
asset_v_state: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset n, Crypt (SesK SK) (Num 0)) \<in> s" shows "(Asset n, SesKey SK) \<in> s \<and> Crypt (SesK SK) (Pwd n) \<in> used s"
hom_dvd: fixes p :: "int poly"   and q :: "int poly" assumes "p dvd q" shows "of_int_poly p dvd of_int_poly q"
rtranclp_trans: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "r\<^sup>*\<^sup>* x y"   and "r\<^sup>*\<^sup>* y z" shows "r\<^sup>*\<^sup>* x z"
nodeToDigit_list: fixes nd :: "('a, 'b) Node" shows "FingerTreeStruc.digitToList (FingerTreeStruc.nodeToDigit nd) = FingerTreeStruc.nodeToList nd"
mset_set_insert: fixes x :: "'b"   and E :: "'b set" assumes "x \<notin> E"   and "finite E" shows "mset_set (E \<union> {x}) = mset_set E + {#x#}"
clearQuantifiers_eval: fixes \<phi> :: "atom fm"   and xs :: "real list" shows "eval (clearQuantifiers \<phi>) xs = eval \<phi> xs"
ALength_\<tau>red1r_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and a :: "'addr expr1"   and xs :: "'addr val list"   and a' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gr uf P t h (a, xs) (a', xs')" shows "\<tau>red1gr uf P t h (a\<bullet>length, xs) (a'\<bullet>length, xs')"
a_def: shows "a = Polynomial.smult (unit_factor b__) e"
decode_plus: fixes base :: "nat"   and e :: "nat"   and M :: "hmultiset"   and N :: "hmultiset" assumes "2 \<le> base"   and "aligned\<^sub>d base e M" shows "decode base e (M + N) = decode base e M + decode base e N"
independent_card_le_aff_dim: fixes B :: "'c set"   and V :: "'c set" assumes "B \<subseteq> V"   and "\<not> affine_dependent B" shows "int (card B) \<le> aff_dim V + 1"
intro: fixes hom :: "'c \<Rightarrow> 'b" assumes "hom (1::'c) = (1::'b)" shows "one_hom hom"
space: fixes \<Omega> :: "'a set"   and M :: "'a set set" assumes "Dynkin_system \<Omega> M" shows "\<Omega> \<in> M"
pps_full_gb: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "pp_of_term ` Keys (fst ` set (full_gb bs)) \<subseteq> {0::'a}"
sum_eq: fixes n :: "nat" shows "(\<Sum>\<^sup>+ x\<in>A_n' n. ennreal (f n x)) = (\<Sum>\<^sup>+ x\<in>Y n. ennreal (g n x))"
IH_1: fixes ys :: "('a \<times> 'b) list"   and xs :: "'a list" assumes "ys = map (\<lambda>x. (x, p x)) xs"   and "inj_on p (set xs)"   and "filter (\<lambda>y. y < z) zs = sort_key p xs" shows "map_tree fst (treap_of (set ys)) = bst_of_list (filter (\<lambda>y. y < z) zs)"
lemma_starrel_refl: fixes x :: "nat \<Rightarrow> 'b" shows "x \<in> starrel `` {x}"
bot_eq_sup_iff: fixes x :: "'d multiset"   and y :: "'d multiset" shows "({#} = x \<union># y) = (x = {#} \<and> y = {#})"
map_poly_Im_of_real: fixes p :: "real poly" shows "map_poly Im (map_poly complex_of_real p) = 0"
inv_into_f_f: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d" assumes "inj_on f A"   and "x \<in> A" shows "inv_into A f (f x) = x"
InvGlobalIncomingInfoUpright_def: fixes c :: "('c, 'd) configuration" shows "InvGlobalIncomingInfoUpright c = (\<forall>k p q. upright (GlobalIncomingInfo c k p q))"
reaches1_reaches_iff2: fixes x :: "'a"   and y :: "'a" shows "(x \<rightarrow>\<^sup>+ y) = (\<exists>z. x \<rightarrow>* z \<and> z \<rightarrow> y)"
division_ofI: fixes s :: "'a set set"   and i :: "'a set" assumes "finite s"   and "\<And>K. K \<in> s \<Longrightarrow> K \<subseteq> i"   and "\<And>K. K \<in> s \<Longrightarrow> K \<noteq> {}"   and "\<And>K. K \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"   and "\<And>K1 K2. \<lbrakk>K1 \<in> s; K2 \<in> s; K1 \<noteq> K2\<rbrakk> \<Longrightarrow> interior K1 \<inter> interior K2 = {}"   and "\<Union> s = i" shows "s division_of i"
\<beta>_def: shows "\<beta> = Matrix.vec N (\<lambda>x. complex_of_real (if f x then 1 / sqrt (real M) else 0))"
map_vec_def: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b vec" shows "map_vec f v \<equiv> vec (dim_vec v) (\<lambda>i. f (v $ i))"
cInf_greaterThan: fixes x :: "'a" shows "Inf {x<..} = x"
e: fixes meets :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and \<I> :: "'a \<Rightarrow> bool"   and e :: "('a \<times> 'a) set"   and m :: "('a \<times> 'a) set"   and b :: "('a \<times> 'a) set"   and ov :: "('a \<times> 'a) set"   and d :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set"   and f :: "('a \<times> 'a) set"   and p :: "'a"   and q :: "'a" assumes "class.arelations meets \<I> e m b ov d s f" shows "((p, q) \<in> e) = (p = q)"
perfect_def: fixes m :: "nat" shows "perfect m \<equiv> 0 < m \<and> 2 * m = sigma m"
main: assumes "Suc (i_end - i_begin) \<noteq> 0 \<Longrightarrow> i_end + 0 = i_end"   and "Suc (i_end - i_begin) \<noteq> 0 \<Longrightarrow> l + 0 = l"   and "Suc (i_end - i_begin) + 0 = Suc (i_end - i_begin)" shows "step_3_c_inner_loop quot l i_end (Suc (i_end - i_begin)) (mat n n (\<lambda>(i, j). if (i, j) = (i_end, k) then if Suc (i_end - i_begin) = Suc (i_end - i_begin) then A $$ (i_end, k) else (0::'a) else if i_begin + Suc (i_end - i_begin) \<le> i \<and> i \<le> i_end \<and> k < j then A $$ (i, j) - quot * A $$ (l + i - i_end, j) else if (i, j) = (i_end - 0, Suc l - 0) \<and> 0 \<notin> {0, Suc (i_end - i_begin)} then quot else A $$ (i, j))) = mat n n (\<lambda>(i, j). if (i, j) = (i_end, k) then 0::'a else if i_begin \<le> i \<and> i \<le> i_end \<and> k < j then A $$ (i, j) - A $$ (i_end, k) / A $$ (l, k) * A $$ (l + i - i_end, j) else A $$ (i, j))"
plus_emeasure: fixes a :: "'b set"   and M :: "'b measure"   and b :: "'b set" assumes "a \<in> sets M"   and "b \<in> sets M"   and "a \<inter> b = {}" shows "emeasure M a + emeasure M b = emeasure M (a \<union> b)"
upd_list_Cons: fixes x :: "'c"   and y :: "'d"   and xs :: "('c \<times> 'd) list" assumes "sorted1 ((x, y) # xs)" shows "upd_list x y xs = (x, y) # xs"
cases: fixes x :: "'c btree" obtains "x = Leaf"   | ts :: "('c btree \<times> 'c) list" and t :: "'c btree" where "x = Node ts t"
induced_morphism_is_well_defined: fixes V :: "'c set"   and q :: "'d"   and V' :: "'c set"   and q' :: "'d" assumes "stfx.rel (V, q) (V', q')" shows "stx.class_of (f \<^sup>\<inverse> X V) (\<phi>\<^sub>f V q) = stx.class_of (f \<^sup>\<inverse> X V') (\<phi>\<^sub>f V' q')"
ExFalso: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"   and "\<not> \<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
adjoint_eval: fixes i :: "nat"   and A :: "'a mat"   and j :: "nat" assumes "i < dim_col A"   and "j < dim_row A" shows "adjoint A $$ (i, j) = conjugate (A $$ (j, i))"
axioms: fixes A :: "V"   and f :: "V" assumes "unop_bij A f" shows "nop_bij A (1\<^sub>\<nat>) f"
subtopology_eq_discrete_topology_sing: fixes X :: "'a topology"   and a :: "'a" shows "(X = discrete_topology {a}) = (topspace X = {a})"
list_3_cases: fixes xs :: "'c list" obtains "xs = []"   | x :: "'c" where "xs = [x__]"   | x :: "'c" and y :: "'c" and ys :: "'c list" where "xs = x # y # ys"
WT_oracle1: fixes s1 :: "'b" shows "\<I> \<turnstile>c oracle1 s1 \<surd>"
gis_done_impl_def: fixes v :: "'q"   and s :: "nat set list \<times> 'q list \<times> nat list \<times> ('q \<Rightarrow> node_state option) \<times> (nat \<times> 'q set) list" shows "gis_done_impl v s \<equiv> is_done_impl v (snd s)"
abs_eq: fixes xa :: "'c fset"   and x :: "'d" shows "[ xa |=> x ] = Abs_fmap (make_map xa x)"
finite_regular: fixes A :: "'a list set" assumes "finite A" shows "regular A"
zero_le_divide_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (divide a b) = (less_eq zero a \<and> less_eq zero b \<or> less_eq a zero \<and> less_eq b zero)"
homotopic_paths_imp_pathfinish: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q" shows "pathfinish p = pathfinish q"
neg_mod_conj: fixes b :: "int"   and a :: "int" assumes "b < 0" shows "a mod b \<le> 0 \<and> b < a mod b"
sel_2: fixes x1 :: "'c \<Rightarrow> char list \<Rightarrow> bool"   and x2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'c set" shows "\<K> (Kripke x1 x2) = x2"
invC: shows "invC = invocation_class (invmode statM e) s a statT"
a_rcosetsI: fixes H :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier R"   and "x \<in> carrier R" shows "H +> x \<in> a_rcosets H"
AbstrLevels_A1_A11: fixes i :: "AbstrLevelsID" assumes "sA1 \<in> AbstrLevel i" shows "sA11 \<notin> AbstrLevel i"
relcompE: fixes xz :: "'a \<times> 'b"   and r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" assumes "xz \<in> r O s" obtains x :: "'a" and y :: "'c" and z :: "'b" where "xz = (x, z)" and "(x, y) \<in> r" and "(y, z) \<in> s"
INF_greatest: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and A :: "'c::type set"   and u :: "'b::type"   and f :: "'c::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "\<And>i::'c::type. i \<in> (A::'c::type set) \<Longrightarrow> (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (u::'b::type) ((f::'c::type \<Rightarrow> 'b::type) i)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (u::'b::type) ((Inf::'b::type set \<Rightarrow> 'b::type) ((f::'c::type \<Rightarrow> 'b::type) ` (A::'c::type set)))"
subpath0_flow_to_path: fixes u :: "real"   and x :: "'a"   and t :: "real"   and t' :: "real" shows "subpath 0 u (rev.flow_to_path x t t') = rev.flow_to_path x t (t + u * (t' - t))"
pre_weaken_pre: fixes x :: "'d"   and P :: "('d, 'e, 'f) ann"   and P' :: "'d set" assumes "x \<in> pre P" shows "x \<in> pre (weaken_pre P P')"
ipresWlsAll_defs_1: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'e, 'k, 'l, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresWlsAll Delta h hA MOD \<equiv> FixSyn.ipresWls Delta h MOD \<and> FixSyn.ipresWlsAbs Delta hA MOD"
sum_SDG_parent_cdep_edge: fixes n' :: "'node SDG_node"   and m :: "'node"   and n :: "'node SDG_node" assumes "valid_SDG_node n'"   and "m = parent_node n'"   and "n = CFG_node m"   and "n \<noteq> n'" shows "n s\<longrightarrow>\<^bsub>cd\<^esub> n'"
ExI: fixes A :: "fm"   and i :: "name" assumes "se_fm A" shows "se_fm (SyntaxN.Ex i A)"
cartesian_product_margin2: fixes b :: "'a"   and B :: "'a set"   and A :: "'b set" assumes "b \<in> B" shows "{p \<in> A \<times> B. snd p = b} = A \<times> {b}"
reduce_basis_external_def: fixes \<alpha> :: "rat"   and fs :: "int vec list" shows "reduce_basis_external \<alpha> fs = (case fs of [] \<Rightarrow> [] | f # x \<Rightarrow> let rb = reduce_basis \<alpha>; fsi = map (map integer_of_int \<circ> list_of_vec) fs; n = dim_vec f; m = length fs in case external_lll_solver (map_prod integer_of_int integer_of_int (quotient_of \<alpha>)) fsi of (gsi, co) \<Rightarrow> let gs = map (vec_of_list \<circ> map int_of_integer) gsi in if \<not> (length gs = m \<and> (\<forall>gi\<in>set gs. dim_vec gi = n)) then Code.abort (STR ''error in external LLL invocation: dimensions of reduced basis do not fit\<newline>input to external solver: '' + String.implode (show fs) + STR ''\<newline>\<newline>'') (\<lambda>_. rb fs) else case co of None \<Rightarrow> if n = m \<and> integer_equivalent n fs gs then rb gs else Code.abort (STR ''error in external LLL invocation:\<newline>'' + (if n = m then STR ''reduced matrix does not span same lattice'' else STR ''no certificate only allowed for square matrices'')) (\<lambda>_. rb fs) | Some (u1i, u2i) \<Rightarrow> let u1 = mat_of_rows_list m (map (map int_of_integer) u1i); u2 = mat_of_rows_list m (map (map int_of_integer) u2i); gs = map (vec_of_list \<circ> map int_of_integer) gsi; Fs = mat_of_rows n fs; Gs = mat_of_rows n gs in if dim_row u1 = m \<and> dim_col u1 = m \<and> dim_row u2 = m \<and> dim_col u2 = m \<and> Fs = u1 * Gs \<and> Gs = u2 * Fs then rb gs else Code.abort (STR ''error in external lll invocation\<newline>f,g,u1,u2 are as follows\<newline>'' + String.implode (show Fs) + STR ''\<newline>\<newline>'' + String.implode (show Gs) + STR ''\<newline>\<newline>'' + String.implode (show u1) + STR ''\<newline>\<newline>'' + String.implode (show u2) + STR ''\<newline>\<newline>'') (\<lambda>_. rb fs))"
isSimplePath_def: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" shows "cf.isSimplePath u p v \<equiv> cf.isPath u p v \<and> distinct (cf.pathVertices u p)"
bin_split_def: fixes n :: "nat"   and k :: "int" shows "bin_split n k = (drop_bit n k, take_bit n k)"
bibd_point_occ_rep: fixes x :: "'a"   and bl :: "'a set" assumes "x \<in> bl"   and "bl \<in># \<B>" shows "remove1_mset bl \<B> rep x = \<Lambda> * (\<v> - 1) div (\<k> - 1) - 1"
tagged_division_of: fixes s :: "('b \<times> 'b set) set"   and i :: "'b set" shows "(s tagged_division_of i) = (finite s \<and> (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and> (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow> interior K1 \<inter> interior K2 = {}) \<and> \<Union> {K. \<exists>x. (x, K) \<in> s} = i)"
resumption_ord_refl: fixes r :: "('f, 'g, 'h) Resumption.resumption" shows "resumption_ord r r"
blocksReds: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "distinct Vs"   and "P \<turnstile> \<langle>e,(h, l(Vs [\<mapsto>] vs))\<rangle> \<rightarrow>* \<langle>e',(h', l')\<rangle>" shows "P \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l)\<rangle> \<rightarrow>* \<langle>blocks (Vs, Ts, map (the \<circ> l') Vs, e'),(h', override_on l' l (set Vs))\<rangle>"
sup1: fixes A :: "'d"   and k :: "'d"   and a :: "'d"   and b :: "'d" assumes "A \<equiv> sup k a" shows "sup A b \<equiv> sup k (sup a b)"
prm_set_unit_action: fixes a :: "'b"   and S :: "'b set"   and b :: "'b" assumes "a \<in> S"   and "b \<notin> S" shows "[a \<leftrightarrow> b] {$} S = S - {a} \<union> {b}"
additive_wp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "Transformers.additive t" shows "Transformers.additive (wp (Embed t))"
Nmul_normN: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "isnormNum x"   and "isnormNum y" shows "isnormNum (x *\<^sub>N y)"
prime_factorization_prod: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "finite A"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> zero" shows "factorial_semiring.prime_factorization zero times one normalize (comm_monoid_mult.prod times one f A) = (\<Sum>n\<in>A. factorial_semiring.prime_factorization zero times one normalize (f n))"
assms: fixes x :: "Preliminaries.var" assumes "x \<in> varsPB \<Phi>" shows "\<xi>1 x = \<xi>2 x"
comp_null_2: fixes f :: "'a" shows "Right.comp f Right.null = Right.null"
posprod: shows "\<forall>i w. 0 \<le> X w * indpre i w"