hcju/isabelleps huggingface dataset:
log3: fixes ck :: "bool"   and env :: "v sem_env"   and s1 :: "'b state"   and e1 :: "exp0"   and s2 :: "'b state"   and v1 :: "v"   and op0 :: "lop"   and e2 :: "exp0" assumes "evaluate ck env s1 e1 (s2, Rval v1) \<and> do_log op0 v1 e2 = None" shows "evaluate ck env s1 (Log op0 e1 e2) (s2, Rerr (Rabort Rtype_error))"
from_hma\<^sub>m_add: fixes a :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and b :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>m a + from_hma\<^sub>m b = from_hma\<^sub>m (a + b)"
zero_less_numeral: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero (numeral.numeral one plus n)"
interval_lebesgue_integral_uminus: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a" shows "LINT x=a..b|M. (- f x) = - interval_lebesgue_integral M a b f"
roots_of_rat_poly_main_code: fixes p :: "rat poly" shows "roots_of_rat_poly_main p = (let n = degree p in if n = 0 then [] else if n = 1 then [roots1 p] else if n = 2 then rat_roots2 p else case rational_root_test p of None \<Rightarrow> [] | Some x \<Rightarrow> x # roots_of_rat_poly_main (p div [:- x, 1:]))"
state_succ_def: fixes s :: "('a, 'b) fmap"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" shows "state_succ s a \<equiv> if fst a \<subseteq>\<^sub>f s then snd a ++ s else s"
arc_blocks_end_subset: fixes E :: "'b set"   and x :: "'a"   and y :: "'a" assumes "E \<subseteq> arcs G"   and "{x, y} \<in> arc_to_block ` E" shows "(x, y) \<in> arc_to_ends G ` E \<or> (y, x) \<in> arc_to_ends G ` E"
add_zerol: fixes x :: "'a" shows "(zero_class.zero::'a) + x = x"
vars_def: fixes e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "Aexp.vars e = {v. \<exists>\<sigma> val. e (\<sigma>(v := val)) \<noteq> e \<sigma>}"
assms: fixes x :: "'a" assumes "\<And>y. y \<in> set (coeffs f) \<Longrightarrow> x dvd y" shows "x dvd (1::'a)"
outstanding_non_volatile_refs_owned_or_read_only_axioms: shows "outstanding_non_volatile_refs_owned_or_read_only \<S> ts"
col_mult2: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and B :: "'c mat"   and nc :: "nat"   and j :: "nat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "j < nc" shows "col (A * B) j = A *\<^sub>v col B j"
ON_imp_Ord: fixes H :: "V set"   and x :: "V" assumes "H \<subseteq> ON"   and "x \<in> H" shows "Ord x"
sign_r_pos_nzero_left: fixes d :: "real"   and c :: "real"   and p :: "real poly" assumes "\<forall>x. d \<le> x \<and> x < c \<longrightarrow> poly p x \<noteq> 0"   and "d < c" shows "if even (order c p) = sign_r_pos p c then 0 < poly p d else poly p d < 0"
orthogonal_currentD_SAT: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and S :: "'a set"   and x :: "'a" assumes "orthogonal_current \<Gamma> f S"   and "x \<in> S" shows "x \<in> SAT \<Gamma> f"
not_outside_1: fixes x :: "'a"   and y :: "'a" assumes "of_weak_ranking (weak_ranking_aux carrier) x y" shows "x \<in> carrier"
prems_3: fixes r :: "'a" assumes "Some r \<in> snd ` set (xs @ [x])" shows "r \<noteq> head"
inj_on_contraD: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and y :: "'d" assumes "inj_on f A"   and "x \<noteq> y"   and "x \<in> A"   and "y \<in> A" shows "f x \<noteq> f y"
simps: fixes A :: "rat mat"   and b :: "rat vec" shows "mat_x_leq_vec A b = map (\<lambda>i. LEQ (matrix_to_lpolies A ! i) (b $ i)) [0..<dim_vec b]"
Pbij_extends_reflexive: fixes \<beta> :: "(Location \<times> Location) set" shows "Pbij_extends \<beta> \<beta>"
Psub: shows "P \<subseteq> A\<^sup>\<omega>"
REMOVE_INTERNAL_EQ_def: fixes a :: "'a"   and b :: "'a" shows "REMOVE_INTERNAL_EQ a b \<equiv> a = b"
bounded_linear_def: fixes f :: "'b \<Rightarrow> 'c" shows "bounded_linear f \<equiv> linear f \<and> bounded_linear_axioms f"
test_absorb_3: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.il_semiring inf sup bot less_eq less top"   and "\<exists>y. sup x y = top \<and> inf x y = bot \<and> inf y x = bot"   and "\<exists>ya. sup y ya = top \<and> inf y ya = bot \<and> inf ya y = bot" shows "inf x (sup x y) = x"
scaleR_2: fixes x :: "'a" shows "2 *\<^sub>R x = x + x"
raw_has_prod_def: fixes f :: "nat \<Rightarrow> 'e"   and M :: "nat"   and p :: "'e" shows "raw_has_prod f M p \<equiv> (\<lambda>n. \<Prod>i\<le>n. f (i + M)) \<longlonglongrightarrow> p \<and> p \<noteq> (0::'e)"
other_net_tree_ips_par_left: fixes U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and p\<^sub>1 :: "net_tree"   and p\<^sub>2 :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'd"   and \<sigma>' :: "nat \<Rightarrow> 'd" assumes "other U (net_tree_ips (p\<^sub>1 \<parallel> p\<^sub>2)) \<sigma> \<sigma>'"   and "\<And>\<xi>. U \<xi> \<xi>" shows "other U (net_tree_ips p\<^sub>1) \<sigma> \<sigma>'"
intro: fixes WAC :: "'a while_algo"   and WAA :: "'b while_algo"   and \<alpha> :: "'a \<Rightarrow> 'b" assumes "wa_refine WAC WAA \<alpha>"   and "wa_precise_refine_axioms WAC WAA \<alpha>" shows "wa_precise_refine WAC WAA \<alpha>"
epi_def: fixes G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" shows "epi G H = {f \<in> hom G H. f ` carrier G = carrier H}"
set_objectI: fixes A :: "'b set"   and U :: "'b set" assumes "A \<subseteq> U" shows "A \<in> Ob\<^bsub>set_cat U\<^esub>"
set_def: fixes a :: "'a array"   and x :: "'a list" shows "Array.set a x = arrays_update (\<lambda>h. h(TYPEREP('a) := (h TYPEREP('a))(addr_of_array a := map to_nat x)))"
ord_eq_le_trans: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" assumes "a = b"   and "b \<le>s c" shows "a \<le>s c"
cases: fixes x :: "('b \<times> nat) option \<times> 'b" obtains t :: "'b" where "x = (None, tb__)"   | t' :: "'b" and n :: "nat" and t :: "'b" where "x = (\<lfloor>(t', n)\<rfloor>, t)"
inv_char: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and y :: "'a" assumes "Group.monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "x \<otimes>\<^bsub>G\<^esub> y = \<one>\<^bsub>G\<^esub>"   and "y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> x = y"
sublist_altdef': fixes xs :: "'c list"   and ys :: "'c list" shows "sublist xs ys = (\<exists>ys'. suffix ys' ys \<and> prefix xs ys')"
unat_gt_0: fixes x :: "'b word" shows "(0 < unat x) = (x \<noteq> 0)"
card_surjective_functions_domain_permutation: fixes A :: "'c set"   and B :: "'d set" assumes "finite A"   and "finite B"   and "card B \<le> card A" shows "card ({f \<in> A \<rightarrow>\<^sub>E B. f ` A = B} // domain_permutation A B) = card A - 1 choose (card A - card B)"
CleanRepr_Proofs_axioms: shows "CleanRepr_Proofs var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc fls dsj proof prfOf encPf Pf"
assms_1: shows "(w, x) \<in> r \<or> w = x"
all: shows "\<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ B"
root_int_main_5: fixes n :: "int"   and p :: "nat"   and y :: "int"   and b :: "bool"   and x :: "int" assumes "0 \<le> n"   and "root_int_main p n = (y, b)"   and "0 < p"   and "0 \<le> x"   and "x ^ p = n" shows "y = x \<and> b"
n: fixes m :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> 'a" assumes "n \<le> m"   and "f \<in> Poly_Mapping.keys ((singular_subdivision p ^^ m) c)" shows "\<exists>V\<in>\<C>. f ` standard_simplex p \<subseteq> V"
pw_bind_inres: fixes M :: "'c nres"   and f :: "'c \<Rightarrow> 'b nres" shows "inres (M \<bind> f) = (\<lambda>x. nofail M \<longrightarrow> (\<exists>y. inres M y \<and> inres (f y) x))"
Seq1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and e2 :: "'addr expr1" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e;; e2,s\<rangle> -ta\<rightarrow> \<langle>e';; e2,s'\<rangle>"
2: shows "G_perm (binop x (\<ii> y)) = id"
div_left_ineq: fixes x :: "nat"   and y :: "nat"   and z :: "nat" assumes "x < y *n z" shows "x div z < y"
independent_length_eq_dimension: fixes n :: "nat"   and E :: "'a set"   and Us :: "'a list" assumes "dimension n K E"   and "local.independent K Us"   and "set Us \<subseteq> E" shows "(length Us = n) = (Span K Us = E)"
adjoint_transpose_right_1: fixes u :: "'a"   and v :: "'a" assumes "ide u"   and "ide v"   and "src v = trg e"   and "src u = trg e'" shows "e'e.trnr\<^sub>\<eta> v \<in> hom (v \<star> e) u \<rightarrow> hom v (u \<star> e')"
set_sequence_of: fixes enum :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "bij_betw enum {0..<card A} A"   and "f \<in> A \<rightarrow>\<^sub>E B" shows "set (sequence_of A enum f) \<subseteq> B"
Join_project_ensures: fixes F :: "'a program"   and G :: "'c program"   and C :: "'c set"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<squnion> G \<in> stable C"   and "F \<squnion> project h C G \<in> A ensures B" shows "extend h F \<squnion> G \<in> C \<inter> extend_set h A ensures extend_set h B"
steps_preserves_sconf: 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 es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and E :: "char list \<Rightarrow> ty option"   and Ts :: "ty list" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>"   and "P,E,hp s \<turnstile> es [:] Ts"   and "P,E \<turnstile> s \<surd>" shows "P,E \<turnstile> s' \<surd>"
floor_zero: shows "\<lfloor>0::'d\<rfloor> = 0"
arr_iff_in_hom: fixes f :: "'b \<times> 'b" shows "B.VV.arr f = B.VV.in_hom f (B.VV.dom f) (B.VV.cod f)"
distinct_11: fixes x1 :: "nat"   and x71 :: "expr"   and x72 :: "expr"   and x73 :: "expr" shows "Var x1 \<noteq> IF x71 THEN x72 ELSE x73"
inv_check_left_moving_in_middle_Bk_Oc: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and r :: "cell list"   and ires :: "cell list"   and s' :: "nat" assumes "inv_check_left_moving_in_middle (as, lm) (s, [], Oc # r) ires" shows "inv_check_left_moving_in_middle (as, lm) (s', [Bk], Oc # r) ires"
hm_is_it_def: fixes m :: "'a \<Rightarrow> 'b option"   and ht :: "('a, 'b) hashtable"   and m' :: "'a \<Rightarrow> 'b option"   and it :: "nat \<times> ('a \<times> 'b) list \<times> ('a, 'b) hashtable" shows "hm_is_it m ht m' it \<equiv> \<exists>\<^sub>Al l'. hm_is_it' l ht l' it * \<up> (map_of (concat l) = m \<and> map_of l' = m')"
assms_5: shows "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> I)"
second_mean_value_theorem_full: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real" assumes "f integrable_on {a..b}"   and "a \<le> b"   and "\<And>x y. \<lbrakk>a \<le> x; x \<le> y; y \<le> b\<rbrakk> \<Longrightarrow> g x \<le> g y" obtains c :: "real" where "c \<in> {a..b}" and "((\<lambda>x. g x * f x) has_integral g a * integral {a..c} f + g b * integral {c..b} f) {a..b}"
enabled_def: fixes name :: "char list"   and s :: "nat \<Rightarrow> nat option" shows "enabled name s \<equiv> case lookup_operator name of None \<Rightarrow> False | Some (x, pres, effs, xa) \<Rightarrow> s \<in> subsuming_states (map_of pres) \<and> s \<in> subsuming_states (map_of (implicit_pres effs))"
ide_in_hom: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d" assumes "partial_magma C" shows "partial_magma.ide C a = partial_magma.in_hom C a a a"
coview_slens_compl: fixes a :: "<'d, 'f> \<Longleftrightarrow> 'e" shows "\<C>\<^bsub>-\<^sub>L a\<^esub> = \<V>\<^bsub>a\<^esub>"
arrI: fixes f :: "'a2"   and a :: "'a2"   and b :: "'a2" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>2 b\<guillemotright>" shows "C2.arr f"
set_intros: fixes x2 :: "'b" shows "x2 \<in> set_option \<lfloor>x2\<rfloor>"
SIod_carrier: fixes D :: "'c Order"   and T :: "'c set" assumes "Order D"   and "T \<subseteq> carrier D" shows "carrier (SIod D T) = T"
add_le_divisor_imp_le_Suc_div: fixes x :: "nat"   and m :: "nat"   and n :: "nat"   and y :: "nat" assumes "x div m \<le> n"   and "y \<le> m" shows "(x + y) div m \<le> Suc n"
deg_of_def: fixes w :: "real" shows "deg_of w = 180 * w / pi"
Diff: fixes a :: "'c set"   and M :: "'c measure"   and b :: "'c set" assumes "a \<in> fmeasurable M"   and "b \<in> fmeasurable M" shows "a - b \<in> fmeasurable M"
index_one_mat_1: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "1\<^sub>m n $$ (i, j) = (if i = j then 1::'b else (0::'b))"
outstanding_volatile_writes_unowned_by_others_axioms: shows "outstanding_volatile_writes_unowned_by_others ts\<^sub>s\<^sub>b"
inc: shows "increment f x h = (*f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)"
unit_preserving_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "unit_preserving_hom hom \<equiv> comm_semiring_1_hom hom \<and> unit_preserving_hom_axioms hom"
A: fixes y :: "'a"   and k :: "nat" shows "\<bar>u y k - real k * l y\<bar> \<le> 2 * deltaG TYPE('a)"
distinct_3: fixes x31 :: "comb"   and x32 :: "comb" shows "K \<noteq> x31 \<bullet> x32"
apply_operations_def: fixes es :: "('msgid \<times> 'op) event list" shows "apply_operations es \<equiv> hb.apply_operations (node_deliver_messages es) initial_state"
atS_def: fixes p :: "'g"   and ls :: "'h set"   and s :: "('i, 'h, 'g, 'j, 'k) system_state" shows "atS p ls s = (\<exists>l\<in>ls. at p l s)"
insert_inc2: fixes a :: "'e"   and A :: "'e set" shows "a \<in> insert a A"
mset_bunion_filter: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and A :: "'a multiset" shows "{#a \<in># A. P a \<or> Q a#} = filter_mset P A \<union># filter_mset Q A"
cases: fixes x :: "('c, 'd) lit" obtains at :: "('c, 'd) atm" where "x = Pos at__"   | at :: "('c, 'd) atm" where "x = Neg at"
suminf_eq_SUP: fixes f :: "nat \<Rightarrow> 'a" shows "suminf f = (\<Squnion>n. sum f {..<n})"
colossless_gpv_catch_gpv: fixes gpv :: "('f, 'd, 'e) gpv" shows "colossless_gpv \<I>_full (catch_gpv gpv)"
set_mkIde: fixes A :: "'s set" assumes "A \<subseteq> S.Univ" shows "S.set (S.mkIde A) = A"
sinks_empty: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'c list" assumes "sinks I D u xs = {}" shows "ipurge_tr I D u xs = xs"
elims_2: fixes x :: "real_min"   and xa :: "real_min" assumes "x \<le> xa" obtains x :: "real" and y :: "real" where "xa__ = Rmin x" and "xa = Rmin y" and "x \<le> y"
sum_g: fixes x :: "'a" assumes "x \<in> carrier" shows "(\<Sum>i | i \<in> N \<and> x \<in> W i. g i x) = 1"
RID\<^sub>G_finite_invariant: fixes r :: "'r"   and s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "revision_step r s s'"   and "finite (RID\<^sub>G s)" shows "finite (RID\<^sub>G s')"
while_type: fixes \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and e :: "'BExp"   and t :: "'BExp set"   and P :: "'BExp set"   and \<S> :: "'Var set \<times> 'Var set"   and c :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<Gamma> \<turnstile>\<^sub>b e \<in> t "   and "P \<turnstile> t"   and "\<turnstile> \<Gamma>,\<S>,P +\<^sub>\<S> e {c} \<Gamma>,\<S>,P" shows "\<turnstile> \<Gamma>,\<S>,P {While e c} \<Gamma>,\<S>,P"
permutes_def: fixes p :: "'a \<Rightarrow> 'a"   and S :: "'a set" shows "(p permutes S) = ((\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y))"
not_le: fixes x :: "'a"   and y :: "'a" shows "(\<not> x \<preceq> y) = (y \<prec> x)"
Term_imp_wf_dbtm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "Term x" obtains t :: "dbtm" where "wf_dbtm t" and "x = \<lbrakk>quot_dbtm t\<rbrakk>e"
set_eq_subset: fixes A :: "'a set"   and B :: "'a set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
ts_process_rep: fixes T :: "'a list set" assumes "trace_set T" shows "Rep_process (ts_process T) = ({(xs, X). xs \<in> T \<and> (\<forall>x\<in>X. xs @ [x] \<notin> T)}, {})"
ide_char': fixes a :: "'a \<times> 'a" shows "CC.ide a = (CC.arr a \<and> (CC.dom a = a \<or> CC.cod a = a))"
minus_1_eq: fixes x :: "'a" shows "g x (- (1::'a)) = gn1 x"
in_chain_finite: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq A"   and "finite A"   and "A \<noteq> {}" shows "Sup A \<in> A"
univ_for_pr_tot_rec_lm: shows "c_graph univ_for_pr \<in> ce_sets"
security: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv"   and q :: "nat" assumes "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>"   and "interaction_bounded_by' isl \<A> q" shows "local.ind_cca.advantage \<A> \<le> PRF.advantage (reduction_prf \<A>) + UPF.advantage (reduction_upf \<A>) + real q / real (card prf_domain) * real q"
frag_cmul_diff_distrib: fixes a :: "int"   and b :: "int"   and c :: "'b \<Rightarrow>\<^sub>0 int" shows "frag_cmul (a - b) c = frag_cmul a c - frag_cmul b c"
symmetric: fixes x :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and y :: "'d state \<times> 'd state \<Rightarrow> Set('e)" shows "(x \<doteq> y) = (y \<doteq> x)"
Dp_Mp_eq: fixes f :: "int poly" shows "f = Mp f + Polynomial.smult m (Dp f)"
trans: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "c1 x y = Lt"   and "c1 y z = Lt" shows "c1 x z = Lt"
lower_or_eq_def: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and t :: "'b trm"   and s :: "'b trm" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms" shows "basic_superposition.lower_or_eq trm_ord t s = (t = s \<or> (t, s) \<in> trm_ord)"
assms_6: fixes u :: "'b"   and v :: "'b" shows "\<exists>w z. h ` cbox u v = cbox w z"
IA: shows "Iagree I I ({Inl x |x. Inl x \<in> SIGO ODE} \<union> {Inr (Inr x) |x. Inr x \<in> SIGO ODE})"
const_imply2: fixes P :: "'a \<Rightarrow> 'b"   and Q :: "'a \<Rightarrow> 'c" assumes "\<And>\<tau> \<tau>'. P \<tau> = P \<tau>' \<Longrightarrow> Q \<tau> = Q \<tau>'"   and "const P" shows "const Q"
map_ident: shows "map (\<lambda>x. x) = (\<lambda>xs. xs)"
simps_2: fixes a :: "'c \<times> ('d, 'e) fmap"   and as :: "('c \<times> ('d, 'e) fmap) list" shows "rem_effectless_act (a # as) = (if fmdom' (snd a) \<noteq> {} then a # rem_effectless_act as else rem_effectless_act as)"
P_no_deadend_v0_llength: fixes n :: "nat" assumes "enat (Suc n) < llength P" shows "\<not> deadend v"
new_element_get_tag_name_is_l_new_element_get_tag_name: shows "l_new_element_get_tag_name ShadowRootClass.type_wf get_tag_name get_tag_name_locs"
zero_not_one: shows "\<zero>\<^bsub>poly_ring R\<^esub> \<noteq> \<one>\<^bsub>poly_ring R\<^esub>"
has_subprob_density_equal_on_space: fixes M :: "'a measure"   and N :: "'a measure"   and f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "has_subprob_density M N f"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "has_subprob_density M N g"
simps: fixes e :: "'b"   and a :: "'c"   and t :: "('b, 'c) SkewBinomialTree"   and t' :: "('b, 'c) SkewBinomialTree" shows "skewlink e a t t' = (if a \<le> prio t \<and> a \<le> prio t' then Node e a (Suc (rank t)) [t, t'] else if prio t \<le> prio t' then Node (val t) (prio t) (Suc (rank t)) (Node e a 0 [] # t' # children t) else Node (val t') (prio t') (Suc (rank t')) (Node e a 0 [] # t # children t'))"
the_inverseD_2: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and g :: "'a" assumes "BinOpSetGroup G binop e"   and "g \<in> G" shows "binop g (BinOpSetGroup.the_inverse G binop e g) = e"
linear_scale: fixes f :: "'d \<Rightarrow> 'e"   and r :: "complex"   and b :: "'d" assumes "clinear f" shows "f (r *\<^sub>C b) = r *\<^sub>C f b"
B: fixes a :: "'a"   and b :: "'a" assumes "Pre sc1 (S a)"   and "sc2 (a, b)" shows "sc1 (S a, S b)"
valid_ownership_tl: fixes \<S> :: "nat \<Rightarrow> bool option"   and t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership \<S> (t # ts)" shows "valid_ownership \<S> ts"
bounded_unique_outside: fixes S :: "'a set"   and c :: "'a set" assumes "bounded S"   and "2 \<le> DIM('a)" shows "(c \<in> components (- S) \<and> \<not> bounded c) = (c = outside S)"
Lp_D_2: fixes p :: "real"   and f :: "'a \<Rightarrow> real"   and M :: "'a measure" assumes "0 < p"   and "f \<in> lspace M (ennreal p)" shows "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"
SeqStTerm_def: fixes v :: "hf"   and u :: "hf"   and x :: "hf"   and x' :: "hf"   and s :: "hf"   and k :: "hf" shows "SeqStTerm v u x x' s k \<equiv> is_Var v \<and> BuildSeq2 (\<lambda>y y'. (is_Ind y \<or> Ord y) \<and> y' = (if y = v then u else y)) (\<lambda>u u' v v' w w'. u = q_Eats v w \<and> u' = q_Eats v' w') s k x x'"
top_neq_numeral: fixes i :: "num" shows "\<top> \<noteq> numeral i"
prems_1: shows "Dag q (rep \<propto> low) (rep \<propto> high) (Node lrt q' rrt)"
div_write_new_val_def: fixes i :: "sparc_operation \<times> inst_operand list"   and result :: "word32"   and temp_V :: "word1" shows "div_write_new_val i result temp_V \<equiv> if fst i \<in> {arith_type UDIVcc, arith_type SDIVcc} then do psr_val \<leftarrow> gets (cpu_reg_val PSR); new_n_val \<leftarrow> gets (\<lambda>s. ucast (result >> 31)); new_z_val \<leftarrow> gets (\<lambda>s. if result = 0 then 1 else 0); new_v_val \<leftarrow> gets (\<lambda>s. temp_V); new_c_val \<leftarrow> gets (\<lambda>s. 0); new_psr_val \<leftarrow> gets (\<lambda>s. update_PSR_icc new_n_val new_z_val new_v_val new_c_val psr_val); write_cpu new_psr_val PSR; return () od else return ()"
positions_in_row: fixes w :: "('c \<times> bool list) list"   and i :: "nat" shows "positions_in_row w i = {p. p < length w \<and> snd (w ! p) ! i}"
assms_3: fixes \<sigma> :: "'\<sigma>"   and k :: "'u"   and it :: "'u set" assumes "c \<sigma>"   and "k \<in> it"   and "\<forall>j\<in>it. k \<le> j"   and "\<forall>j\<in>\<alpha> m - it. j \<le> k"   and "it \<subseteq> \<alpha> m"   and "I it \<sigma>" shows "I (it - {k}) (f k \<sigma>)"
pre_mat: shows "pre = mat N N (\<lambda>(i, j). if i = j \<and> i = 0 then 1 else 0)"
min_bot2: fixes x :: "'c list" shows "prefix_order.min x [] = []"