hcju/isabelleps huggingface dataset:
pfa_blit_rule: fixes si :: "nat"   and len :: "nat"   and sn :: "nat"   and di :: "nat"   and dn :: "nat"   and dc :: "nat"   and sc :: "nat"   and src :: "'b list"   and srci :: "'b array"   and dst :: "'b list"   and dsti :: "'b array" assumes "si + len \<le> sn"   and "di \<le> dn"   and "di + len \<le> dc" shows "<is_pfa sc src (srci, sn) * is_pfa dc dst (dsti, dn)> pfa_blit (srci, sn) si (dsti, dn) di len <\<lambda>_. is_pfa sc src (srci, sn) * is_pfa dc (take di dst @ take len (drop si src) @ drop (di + len) dst) (dsti, max (di + len) dn)>"
index_mat_multrow_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and mul :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "nat"   and a :: "'c" assumes "i < dim_row A"   and "j < dim_col A" shows "mat_multrow_gen mul k a A $$ (i, j) = (if k = i then mul a (A $$ (i, j)) else A $$ (i, j))"
comp_assoc: fixes h :: "'a"   and g :: "'a"   and f :: "'a" shows "(h \<cdot>\<^sub>B g) \<cdot>\<^sub>B f = h \<cdot>\<^sub>B g \<cdot>\<^sub>B f"
bv_mapzip_def: fixes f :: "bit \<Rightarrow> bit \<Rightarrow> bit"   and w1 :: "bit list"   and w2 :: "bit list" shows "bv_mapzip f w1 w2 = (let g = bv_extend (max (length w1) (length w2)) \<zero> in map2 f (g w1) (g w2))"
simps_1: fixes a :: "'d"   and b :: "'e"   and a' :: "'d"   and b' :: "'e" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
homeomorphic_convex_compact_sets: fixes S :: "'c set"   and T :: "'d set" assumes "convex S"   and "compact S"   and "convex T"   and "compact T"   and "aff_dim S = aff_dim T" shows "S homeomorphic T"
def_body: fixes x :: "'a"   and y :: "'b" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null"   and "y \<noteq> \<bottom>" shows "g x y \<noteq> \<bottom> \<and> g x y \<noteq> null"
word_next_unfold: fixes w :: "'b word" shows "word_next w = (if w = - 1 then - 1 else w + 1)"
merges_bounded_lemma: fixes step :: "nat \<Rightarrow> 'a \<Rightarrow> (nat \<times> 'a) list"   and n :: "nat"   and p :: "nat"   and ss :: "'a list"   and ts :: "'a list" assumes "SemilatAlg.mono r step n A"   and "bounded step n"   and "\<forall>(p', s')\<in>set (step p (ss ! p)). s' \<in> A"   and "ss \<in> list n A"   and "ts \<in> list n A"   and "p < n"   and "ss [\<sqsubseteq>\<^bsub>r\<^esub>] ts"   and "\<forall>p<n. stable r step ts p" shows "merges f (step p (ss ! p)) ss [\<sqsubseteq>\<^bsub>r\<^esub>] ts"
sigma_finite_density_unique: fixes f :: "'c \<Rightarrow> ennreal"   and M :: "'c measure"   and g :: "'c \<Rightarrow> ennreal" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "sigma_finite_measure (density M f)" shows "(density M f = density M g) = (AE x in M. f x = g x)"
return_node_determines_call_node: fixes n :: "'b" assumes "return_node n" shows "\<exists>!n'. \<exists>a a'. valid_edge a \<and> n' = sourcenode a \<and> valid_edge a' \<and> a' \<in> get_return_edges a \<and> n = targetnode a'"
funionCI: fixes c :: "'c"   and B :: "'c fset"   and A :: "'c fset" assumes "c |\<notin>| B \<Longrightarrow> c |\<in>| A" shows "c |\<in>| A |\<union>| B"
UTM_halt_lemma_pre: fixes tp :: "(action \<times> nat) list"   and rs :: "nat"   and args :: "nat list"   and i :: "nat"   and stp :: "nat"   and m :: "nat"   and k :: "nat" assumes "tm_wf0 tp"   and "0 < rs"   and "args \<noteq> []"   and "steps0 (Suc 0, Bk \<up> i, <args>) tp stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> k)" shows "\<exists>stp m n. steps0 (Suc 0, [], <code tp # args>) UTM_pre stp = (0, Bk \<up> m, Oc \<up> rs @ Bk \<up> n)"
abs_5: shows "fBall (frees (\<Lambda> t) |\<union>| fset_of_list \<Gamma>) (\<lambda>x. x \<le> s)"
recfun_aux_domI: fixes x :: "'a" assumes "local.dom x" shows "recfun_aux_dom x"
signed_take_bit_of_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and not :: "'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n zero = zero"
eq1: shows "(\<exists>(n, t)\<in>set (uset p). Ifm vs (x # bs) (msubst2 p (n *\<^sub>p (- 2)\<^sub>p) t)) = (\<exists>(n, t)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> (0::'a) \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (2::'a)) # bs) p)"
InitConf_HAStates2: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> InitConf A" shows "S \<in> HAStates A"
fresh_8: fixes a :: "name"   and x1 :: "('d, 'e, 'f) psi" shows "a \<sharp> !x1 = a \<sharp> x1"
prod_list_dvd_prod_list_subset: fixes A :: "'b list"   and B :: "'b list" assumes "distinct A"   and "distinct B"   and "set A \<subseteq> set B" shows "prod_list A dvd prod_list B"
distinct_207: fixes x81 :: "Syntax.term"   and x82 :: "Syntax.var"   and x83 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Syntax.Let x81 x82 x83 \<noteq> Unroll x14"
a: shows "\<forall>y\<in>S. dist x y \<le> a"
Arg2pi_eq: fixes y :: "complex" assumes "cmod y = 1" shows "1 - Arg2pi (cnj y) / (2 * pi) = Arg2pi y / (2 * pi) \<or> Arg2pi y = 0 \<and> Arg2pi (cnj y) = 0"
take_Suc_Cons: fixes n :: "nat"   and x :: "'c"   and xs :: "'c list" shows "take (Suc n) (x # xs) = x # take n xs"
compute_lookup_pp: fixes xs :: "('b, 'a) fmap"   and x :: "'b" shows "lookup (Pm_fmap xs) x = lookup0 xs x"
simple_function_borel_measurable: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "f \<in> borel_measurable M"   and "finite (f ` space M)" shows "simple_function M f"
card_of_Un_singl_ordLess_infinite1: fixes B :: "'a set"   and A :: "'b set"   and a :: "'b" assumes "infinite B"   and "|A| <o |B|" shows "|{a} \<union> A| <o |B|"
listrel_rtrancl_trans: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set"   and zs :: "'g list" assumes "(xs, ys) \<in> listrel (r\<^sup>*)"   and "(ys, zs) \<in> listrel (r\<^sup>*)" shows "(xs, zs) \<in> listrel (r\<^sup>*)"
P1_real_ideal_DDH_advantage_false: fixes \<sigma> :: "bool"   and M :: "'a \<times> 'a"   and z :: "'e"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'e \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'f \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'f) spmf) \<times> ('f \<Rightarrow> 'g spmf)"   and D :: "'g \<times> 'a \<Rightarrow> bool spmf" assumes "\<sigma> = False" shows "mal_def.adv_P1 M \<sigma> z (P1_S1, P1_S2) \<A> D = local.ddh.DDH_advantage (P1_DDH_mal_adv_\<sigma>_false M z \<A> D)"
mono_imp_derivation_infl: shows "\<forall>X x y. derivation X \<longrightarrow> x \<in> X \<longrightarrow> y \<in> X \<longrightarrow> x \<sqsubseteq> y \<longrightarrow> x \<sqsubseteq> f y"
le_of_nat_term_order_zero_min: fixes to :: "'b nat_term_order"   and t :: "'b" shows "le_of_nat_term_order to (0::'b) t"
mono: fixes c :: "'b"   and a :: "'b"   and d :: "'b"   and b :: "'b" assumes "c \<le> a"   and "d \<le> b" shows "sup c d \<le> sup a b"
add_neg_numeral_special_8: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "plus (uminus one) one = zero"
exec_gpv_fixp_induct: fixes P :: "(('d, 'e, 'f) gpv \<Rightarrow> 'g \<Rightarrow> ('d \<times> 'g) spmf) \<Rightarrow> bool"   and callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf" assumes "spmf.admissible (\<lambda>f. P (\<lambda>c s. f (c, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>exec_gpv. P exec_gpv \<Longrightarrow> P (\<lambda>c s. the_gpv c \<bind> case_generat (\<lambda>x. return_spmf (x, s)) (\<lambda>out c. callee s out \<bind> (\<lambda>(x, y). exec_gpv (c x) y)))" shows "P (exec_gpv callee)"
init_fin_silent_moves_RunningI: fixes r :: "('c, 'd, 'a, 'b, 'e, 'f) semantics"   and t :: "'d"   and \<tau>move :: "('c, 'd, 'a, 'b, 'e, 'f) \<tau>moves"   and x :: "'a"   and m :: "'b"   and x' :: "'a"   and m' :: "'b"   and final :: "'a \<Rightarrow> bool" assumes "(\<tau>trsys.silent_move (r t) \<tau>move)\<^sup>*\<^sup>* (x, m) (x', m')" shows "(\<tau>trsys.silent_move (multithreaded_base.init_fin final r t) (\<tau>multithreaded.init_fin_\<tau>move \<tau>move))\<^sup>*\<^sup>* ((Running, x), m) ((Running, x'), m')"
simps: fixes a :: "'c"   and A :: "('d, 'c) nba" shows "(a \<in> nodes A) = ((\<exists>p. a = p \<and> p \<in> initial A) \<or> (\<exists>p aa. a = snd aa \<and> p \<in> nodes A \<and> fst aa \<in> alphabet A \<and> snd aa \<in> transition A (fst aa) p))"
image_insert: fixes f :: "'c::type \<Rightarrow> 'b::type"   and a :: "'c::type"   and B :: "'c::type set" shows "(f::'c::type \<Rightarrow> 'b::type) ` insert (a::'c::type) (B::'c::type set) = insert (f a) (f ` B)"
inf_top_left: fixes inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and top :: "'d"   and x :: "'d" assumes "class.bounded_semilattice_inf_top inf less_eq less top" shows "inf top x = x"
has_integral_cmul: assumes "((\<lambda>x. c *\<^sub>R f x) has_integral c *\<^sub>R I) A" shows "((\<lambda>x. c *\<^sub>R f x /\<^sub>R c) has_integral c *\<^sub>R I /\<^sub>R c) A"
of_int_sum: 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 f :: "'d \<Rightarrow> int"   and A :: "'d set" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (sum f A) = comm_monoid_add.sum plus zero (\<lambda>x. ring_1.of_int one plus zero minus (f x)) A"
wf_prog_Object_mdecls: fixes G :: "prog" assumes "wf_prog G" shows "\<forall>m\<in>set Object_mdecls. accmodi m \<noteq> Package"
lift_gi_def: fixes f :: "'d \<Rightarrow> 'a"   and gi :: "bool \<times> 'd" shows "lift_gi f gi = (if fst gi then inv f (snd gi) else f (snd gi))"
seqsim_stutstep: fixes s :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "stutstep s n" shows "s |\<^sub>s Suc n \<approx> s |\<^sub>s n"
card_less_Suc: fixes M :: "nat set"   and i :: "nat" assumes "0 \<in> M" shows "Suc (card {k. Suc k \<in> M \<and> k < i}) = card {k \<in> M. k < Suc i}"
vreal_zero_closed: shows "0\<^sub>\<real> \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>"
exp_Ln: fixes z :: "complex" assumes "z \<noteq> 0" shows "exp (ln z) = z"
emeasure_0_AE: fixes M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "emeasure M (space M) = 0" shows "almost_everywhere M P"
borel_sigma_sets_subset: fixes A :: "'d set set" assumes "A \<subseteq> sets borel" shows "sigma_sets UNIV A \<subseteq> sets borel"
Left_Coset_Class_unit: fixes g :: "'a" assumes "g \<in> G" shows "g \<cdot>| K = Class g"
term_order_rel'_alt_def: fixes x :: "char list list"   and y :: "char list list" shows "term_order_rel' x y = (x < y)"
cil_cons_cases: fixes e :: "'d"   and w :: "'d list"   and w1 :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w2 :: "'d list" assumes "e # w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2" obtains w1' :: "'d list" where "w1 = e # w1'" and "w \<in> w1' \<otimes>\<^bsub>\<alpha>\<^esub> w2" and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w2) = {}"   | w2' :: "'d list" where "w2 = e # w2'" and "w \<in> w1 \<otimes>\<^bsub>\<alpha>\<^esub> w2'" and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> w1) = {}"
rPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' P') A\<^sub>P \<Psi>\<^sub>P"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* M"   and "A\<^sub>Q \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* P"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* M"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) M (\<lparr>\<nu>*xvec\<rparr>N \<prec>' (P' \<parallel> Q)) (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
dis_introR: fixes \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<psi> \<pi>l" shows "dis \<phi> \<psi> \<pi>l"
simps_1: fixes x1 :: "agent"   and y1 :: "agent" shows "(Agent x1 = Agent y1) = (x1 = y1)"
powsum_one: fixes x :: "'a"   and n :: "nat" shows "(1::'a) \<le> x\<^bsub>0\<^esub>\<^bsup>Suc n\<^esup>"
hd_in_awalk_verts_2: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G"   and "pre_digraph.apath G u p v" shows "u \<in> set (pre_digraph.awalk_verts G u p)"
cases: fixes x :: "'a list" obtains x :: "'a" and xs :: "'a list" where "xa__ = x # xs"   | "xa__ = []"
poly_as_sum_of_monoms: fixes p :: "'a poly" shows "(\<Sum>i\<le>degree p. monom (coeff p i) i) = p"
eq_last_state_imp_append_nempty_as: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and slist_1 :: "('a, 'b) fmap list"   and slist_2 :: "('a, 'b) fmap list" assumes "as \<noteq> []"   and "s \<in> valid_states PROB"   and "as \<in> valid_plans PROB"   and "slist_1 \<noteq> []"   and "slist_2 \<noteq> []"   and "slist_1 \<in> state_set (state_list s as)"   and "slist_2 \<in> state_set (state_list s as)"   and "length slist_1 \<noteq> length slist_2"   and "last slist_1 = last slist_2" shows "\<exists>as1 as2 as3. as1 @ as2 @ as3 = as \<and> exec_plan s (as1 @ as2) = exec_plan s as1 \<and> as2 \<noteq> []"
OrdDom_insf: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "OrdDom s"   and "Ord k" shows "OrdDom (insf s (succ k) y)"
Power_language: fixes M :: "'b nfa" assumes "nfa M" shows "dfa.language (nfa.Power_dfa M) = nfa.language M"
prems_6: shows "set (Fss ! i) \<inter> set newFs \<subseteq> {olfF}"
assms_1: shows "\<GG> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
cmod_eq_one: fixes a :: "complex" shows "(cmod a = 1) = (a * cnj a = 1)"
rel_state_lookup: shows "((=) ===> local.rel_state (=)) lookup_st lookup"
b: shows "r2 = (q # ps, c) \<and> r2 \<in> upRules"
substAbs_compose1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y1 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y2 :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y1"   and "good Y2" shows "A $[Y1 / y]_ys $[Y2 / y]_ys = A $[Y1 #[Y2 / y]_ys / y]_ys"
h: shows "MPoly_Type.degree p var = 1 \<or> MPoly_Type.degree p var = 2"
ofilter_def: fixes r :: "('d \<times> 'd) set"   and A :: "'d set" assumes "wo_rel r" shows "ofilter r A \<equiv> A \<subseteq> Field r \<and> (\<forall>a\<in>A. under r a \<subseteq> A)"
\<i>_def: shows "\<i> \<equiv> \<lambda>a. a"
2: assumes "j' \<noteq> j"   and "i' = i - 1" shows "A $$ (j, j') = (0::'a)"
hIm_hcomplex_of_hypreal: fixes z :: "real star" shows "hIm (hcomplex_of_hypreal z) = 0"
assms_3: shows "fmrel_on_fset (fmdom zm) f ym zm"
that: fixes c :: "real" assumes "\<And>x. \<lbrakk>x\<^sub>0 \<le> x; x \<le> x\<^sub>1\<rbrakk> \<Longrightarrow> f x \<le> 1 / 2 * (c * f_approx x)"   and "c' \<le> c"   and "0 < c" shows "thesis"
clean_paras: fixes gs :: "recf list"   and ffp :: "nat"   and xs :: "nat list"   and f :: "recf"   and anything :: "nat list" assumes "length gs \<le> ffp" shows "{\<lambda>nl. nl = map (\<lambda>i. rec_exec i xs) gs @ 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) - length gs) @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything} empty_boxes (length gs) {\<lambda>nl. nl = 0 \<up> max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> length gs @ xs @ anything}"
proper_dia_def: fixes p :: "('c, 'd) fm" shows "proper_dia p \<equiv> case p of Pro x \<Rightarrow> Map.empty x | Nom x \<Rightarrow> Map.empty x | \<^bold>\<not> x \<Rightarrow> Map.empty x | fm1 \<^bold>\<or> x \<Rightarrow> Map.empty x | \<^bold>\<diamond> p \<Rightarrow> if \<nexists>a. p = Nom a then Some p else None | \<^bold>@ b x \<Rightarrow> Map.empty x"
aff_dim_empty: fixes S :: "'a set" shows "(S = {}) = (aff_dim S = - 1)"
S_def: shows "S = range (\<lambda>x. ereal (sum a {0..x}))"
SeqSubstFormP_lemma: fixes m :: "name"   and v :: "tm"   and u :: "tm"   and x :: "tm"   and y :: "tm"   and s :: "tm"   and k :: "tm"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name" assumes "atom m \<sharp> (v, u, x, y, s, k, n, sm, sm', sn, sn')"   and "atom n \<sharp> (v, u, x, y, s, k, sm, sm', sn, sn')"   and "atom sm \<sharp> (v, u, x, y, s, k, sm', sn, sn')"   and "atom sm' \<sharp> (v, u, x, y, s, k, sn, sn')"   and "atom sn \<sharp> (v, u, x, y, s, k, sn')"   and "atom sn' \<sharp> (v, u, x, y, s, k)" shows "{SeqSubstFormP v u x y s k} \<turnstile> SubstAtomicP v u x y OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN k AND Var n IN k AND SeqSubstFormP v u (Var sm) (Var sm') s (Var m) AND SeqSubstFormP v u (Var sn) (Var sn') s (Var n) AND (x EQ Q_Disj (Var sm) (Var sn) AND y EQ Q_Disj (Var sm') (Var sn') OR x EQ Q_Neg (Var sm) AND y EQ Q_Neg (Var sm') OR x EQ Q_Ex (Var sm) AND y EQ Q_Ex (Var sm'))))))))"
heap_upds_ok_invariant: shows "invariant (\<Rightarrow>) heap_upds_ok_conf"
mono_post: fixes c :: "'b acom"   and c' :: "'b acom" assumes "c \<sqsubseteq> c'" shows "post c \<sqsubseteq> post c'"
rel_fundef_body_nth: fixes fd1 :: "('a, 'b, 'c, 'op) Std.instr fundef"   and fd2 :: "('a, 'b, 'c, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat" assumes "rel_fundef (\<lambda>x y. x = norm_instr y) fd1 fd2"   and "pc < length (body fd1)" shows "body fd1 ! pc = norm_instr (body fd2 ! pc)"
rep_eq: fixes x :: "'c"   and xa :: "'d" shows "lookup (monomial xa x) = (\<lambda>k'. xa when x = k')"
n2m_ctor_fold_G_F_def: fixes s1 :: "('e, 'f) G0 \<Rightarrow> 'g"   and s2 :: "('g, 'f) F0 \<Rightarrow> 'f" shows "n2m_ctor_fold_G_F s1 s2 = ctor_fold_F (s2 \<circ> map_pre_GF\<^sub>M id (n2m_ctor_fold_G s1 s2) id \<circ> BNF_Composition.id_bnf \<circ> BNF_Composition.id_bnf)"
generates_max_idealI_alt: fixes F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set"   and X :: "'c set" assumes "ideal F \<noteq> UNIV"   and "\<And>p. \<lbrakk>p \<in> P[X]; p \<notin> ideal F\<rbrakk> \<Longrightarrow> 1 \<in> ideal (insert p F)" shows "generates_max_ideal X F"
pseudo_proj_preimage: fixes g :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and N :: "'b measure"   and z :: "bool stream" assumes "g \<in> F n \<rightarrow>\<^sub>M N"   and "set_discriminating n g N" shows "pseudo_proj_True n -` g -` {g z} = pseudo_proj_True n -` pseudo_proj_True n ` g -` {g z}"
matches_ipassmt_iface_constrain_srcip_mexpr: fixes \<alpha> :: "action \<Rightarrow> ('a, 'b) tagged_packet_scheme \<Rightarrow> bool"   and ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface"   and a :: "action"   and p :: "('a, 'b) tagged_packet_scheme" shows "matches (common_matcher, \<alpha>) (ipassmt_iface_constrain_srcip_mexpr ipassmt ifce) a p = (case ipassmt ifce of None \<Rightarrow> match_iface ifce (p_iiface p) | Some ips \<Rightarrow> match_iface ifce (p_iiface p) \<and> p_src p \<in> ipcidr_union_set (set ips))"
dom_simp: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" assumes "VVV.arr f" shows "VVV.dom f = VxVxV.dom f"
add_le_cancel_left: fixes x :: "V"   and y :: "V"   and z :: "V" shows "(x + y \<le> x + z) = (y \<le> z)"
is_volatile_simps_1: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_volatile (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = volatile"
divideC_field_splits_simps_1_1: fixes a :: "'d"   and c :: "complex"   and b :: "'d" shows "(a = b /\<^sub>C c) = (if c = 0 then a = (0::'d) else c *\<^sub>C a = b)"
h4: shows "t mod cycleLength (nC i) mem schedule (nC i)"
init'_tableau_equiv: fixes t :: "(nat \<times> linear_poly) list"   and v :: "nat \<Rightarrow> 'a" assumes "\<triangle> t" shows "v \<Turnstile>\<^sub>t t = v \<Turnstile>\<^sub>t \<T> (init t)"
assms: shows "i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
l11_31_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and A :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet B A C"   and "D \<noteq> E"   and "F \<noteq> E" shows "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F"
butterfly_def_one_dim: fixes s :: "'b"   and t :: "'a" shows "butterfly s t = vector_to_cblinfun s o\<^sub>C\<^sub>L vector_to_cblinfun t*"
simps_1: fixes a :: "name"   and b :: "name" shows "a \<sharp> b = (a \<noteq> b)"
norm_int_preservation: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and i :: "nat"   and j :: "nat" assumes "dbm_int M n"   and "i \<le> n"   and "j \<le> n"   and "DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n i j \<noteq> \<infinity>" shows "get_const (DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n i j) \<in> \<int>"
span_singleton: fixes x :: "'a" shows "vector_space_over_itself.span {x} = range (\<lambda>k. k * x)"
Alist_inverse: fixes y :: "('a \<times> 'b) list" assumes "y \<in> {xs. (distinct \<circ> list.map fst) xs}" shows "impl_of (Alist y) = y"
find_index_eq_size_conv: fixes xs :: "'b list"   and n :: "nat"   and P :: "'b \<Rightarrow> bool" assumes "length xs = n" shows "(find_index P xs = n) = (\<forall>x\<in>set xs. \<not> P x)"
rep_eq: fixes x :: "\<o>" shows "d\<^sub>0 x = Some (eval\<o> x)"
c_prod_idl: fixes x :: "('a \<times> 'a set) set" shows "1\<^sub>\<pi> \<parallel> x = x"