hcju/isabelleps huggingface dataset:
sum_ereal: fixes f :: "'d \<Rightarrow> real"   and A :: "'d set" shows "(\<Sum>x\<in>A. ereal (f x)) = ereal (sum f A)"
one_reduced_homology_group: fixes p :: "int"   and X :: "'a topology" shows "\<one>\<^bsub>reduced_homology_group p X\<^esub> = \<one>\<^bsub>homology_group p X\<^esub>"
new_actions_for_fun: fixes w :: "nat"   and adal :: "'b \<times> addr_loc"   and w' :: "nat" assumes "w \<in> new_actions_for P E'' adal"   and "w' \<in> new_actions_for P E'' adal" shows "w = w'"
mods_with_vec: fixes v :: "nat list"   and w :: "nat list"   and b :: "nat list"   and a :: "nat" assumes "v <\<^sub>v w"   and "0 \<notin> set b"   and "length b = length w"   and "v \<bullet> b mod a = w \<bullet> b mod a" shows "(w -\<^sub>v v) \<bullet> b mod a = 0"
simps_2: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a" shows "f ^^ Suc n = f \<circ> f ^^ n"
apex_def: shows "\<eta>.dom.apex \<equiv> C.dom \<eta>.dom.leg0"
subsumable_axioms: shows "subsumable (\<le>\<cdot>)"
homotopic_loops_sym: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_loops s p q" shows "homotopic_loops s q p"
nth_default_eq_dflt_iff: fixes dflt :: "'b"   and xs :: "'b list"   and k :: "nat" shows "(nth_default dflt xs k = dflt) = (k < length xs \<longrightarrow> xs ! k = dflt)"
addition_assoc: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "assoc_bpp A f"   and "x \<in> addition_set f A"   and "y \<in> addition_set f A"   and "z \<in> addition_set f A" shows "x \<^sub>f+ y \<^sub>f+ z = x \<^sub>f+ (y \<^sub>f+ z)"
err_expecting_def: fixes msg :: "char list"   and ts :: "'b list" shows "err_expecting msg ts = Inl (''expecting '' @ msg @ '', but found: '' @ shows_quote (shows (take 30 ts)) [])"
filter_True: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. P x" shows "filter P xs = xs"
degree_primitive_part: fixes p :: "'b poly" shows "degree (primitive_part p) = degree p"
isin_set_tree: fixes t :: "('b \<times> 'c) tree"   and x :: "'b" assumes "Tree2.bst t" shows "isin t x = (x \<in> Tree2.set_tree t)"
nn_integral_snd: fixes f :: "'a \<times> ('i \<Rightarrow> 'a) \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (M i \<Otimes>\<^sub>M Pi\<^sub>M I M)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M i \<partial>Pi\<^sub>M I M = integral\<^sup>N (M i \<Otimes>\<^sub>M Pi\<^sub>M I M) f"
qSwapBinp_def: fixes xs :: "'f"   and x :: "'g"   and y :: "'g"   and qbinp :: "'c \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "qSwapBinp xs x y qbinp \<equiv> lift (qSwapAbs xs x y) qbinp"
defs_3: shows "\<S>\<bullet>\<C> \<turnstile> es : [] _> ts"
emb_restrict: fixes f :: "'e \<Rightarrow> 'a"   and D :: "'e set" shows "restrict (emb f D) D = restrict f D"
pt_set_nil: fixes Xs :: "'a set" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "[] \<bullet> Xs = Xs"
add_divide_eq_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "z \<noteq> zero" shows "plus x (divide y z) = divide (plus (times x z) y) z"
submonoid_axioms_def: fixes N :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" shows "submonoid_axioms N M composition unit \<equiv> N \<subseteq> M \<and> (\<forall>a b. a \<in> N \<longrightarrow> b \<in> N \<longrightarrow> composition a b \<in> N) \<and> unit \<in> N"
col_ABA: fixes A :: "Points"   and B :: "Points" shows "col A B A"
Actual_in_SDG_Use: fixes n :: "'node SDG_node"   and m :: "'node"   and x :: "nat"   and V :: "'var" assumes "valid_SDG_node n"   and "n = Actual_in (m, x)"   and "V \<in> ParamUses m ! x" shows "V \<in> Use\<^bsub>SDG\<^esub> n"
simps_5: fixes f :: "nat \<Rightarrow> atom \<Rightarrow> atom fm"   and \<phi> :: "atom fm"   and \<psi> :: "atom fm"   and var :: "nat" shows "liftmap f (Or \<phi> \<psi>) var = Or (liftmap f \<phi> var) (liftmap f \<psi> var)"
reorder_eq: fixes fps :: "(char list \<times> pat) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and \<Delta> :: "binding list"   and fs :: "(char list \<times> 'a) list" assumes "\<turnstile> fps [:] fTs \<Rightarrow> \<Delta>"   and "\<forall>(l, U)\<in>set fTs. \<exists>u. fs\<langle>l\<rangle>\<^sub>? = \<lfloor>u\<rfloor>" shows "\<forall>(l, p)\<in>set fps. fs\<langle>l\<rangle>\<^sub>? = map (\<lambda>(l, T). (l, the (fs\<langle>l\<rangle>\<^sub>?))) fTs\<langle>l\<rangle>\<^sub>?"
gs_pos: fixes g :: "'b \<Rightarrow> real" assumes "g \<in> set (map get_fun gs)" shows "filterlim g at_top F"
INFM_nat_add: fixes P :: "nat \<Rightarrow> bool"   and j :: "nat" shows "(\<exists>\<^sub>\<infinity>i. P i) = (\<exists>\<^sub>\<infinity>i. P (i + j))"
n_eq1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(!x \<cdot> y \<le> z \<and> t x \<cdot> y \<le> z) = (y \<le> z)"
expand: fixes vertex :: "'a vertex"   and vertex' :: "'a vertex" assumes "(vertex = SOURCE) = (vertex' = SOURCE)"   and "(vertex = vertex.SINK) = (vertex' = vertex.SINK)"   and "\<lbrakk>is_Inner vertex; is_Inner vertex'\<rbrakk> \<Longrightarrow> vertex.inner vertex = vertex.inner vertex'" shows "vertex = vertex'"
trivial_hom_iff: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h" shows "(h ` carrier G = {\<one>\<^bsub>H\<^esub>}) = (kernel G H h = carrier G)"
eventually_unless_until: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and Q :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "(\<lambda>s. (P \<W> Q) s \<and> (\<diamond>Q) s) = (P \<U> Q)"
assms_3: shows "f ` (P .\<^bold>\<le> x) = Pow A"
N_4: fixes vs :: "nat \<Rightarrow> nat" shows "\<exists>vs'. ((Suc 0, vs), 0, vs') \<in> (step N)\<^sup>* \<and> Q vs vs'"
foldli_conj: fixes list :: "'b list"   and P :: "'b \<Rightarrow> bool"   and b :: "bool" shows "foldli list id (\<lambda>kv \<sigma>. P kv) b = (b \<and> (\<forall>x\<in>set list. P x))"
map_FGl_def: fixes l1 :: "'au \<Rightarrow> 'bh"   and l2 :: "'av \<Rightarrow> 'bi"   and l3 :: "'bb \<Rightarrow> 'bn"   and co1 :: "'aw \<Rightarrow> 'bj"   and co2 :: "'ax \<Rightarrow> 'bk"   and co3 :: "'bc \<Rightarrow> 'bo"   and co4 :: "'bd \<Rightarrow> 'bp"   and contra1 :: "'bl \<Rightarrow> 'ay"   and contra2 :: "'bm \<Rightarrow> 'az"   and contra3 :: "'bq \<Rightarrow> 'be"   and contra4 :: "'br \<Rightarrow> 'bf" shows "map_FGl l1 l2 l3 co1 co2 co3 co4 contra1 contra2 contra3 contra4 = map_F (map_G l1 l2 co1 co2 contra1 contra2) l1 l3 co1 co3 co4 contra1 contra3 contra4"
subrelI: fixes r :: "('a \<times> 'b) set"   and s :: "('a \<times> 'b) set" assumes "\<And>x y. s2p r (x, y) \<Longrightarrow> s2p s (x, y)" shows "r \<subseteq> s"
a_loc: fixes x :: "'a"   and y :: "'a" shows "ad (x \<cdot> am_d y) = ad (x \<cdot> y)"
Sup_distl: fixes x :: "'a \<times> 'b"   and Y :: "('a \<times> 'b) set" shows "times_prod' x (Sup_prod Y) = Sup_prod (times_prod' x ` Y)"
factorize_polys_coprime: fixes ps :: "rat poly list"   and fs :: "rat poly list"   and data :: "(rat \<times> (nat \<times> nat) list) list"   and f :: "rat poly"   and g :: "rat poly" assumes "factorize_polys ps = (fs, data)"   and "f \<in> set fs"   and "g \<in> set fs"   and "f \<noteq> g" shows "comm_monoid_mult_class.coprime f g"
simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" shows "efficient_funpow f y x (Suc 0) = f x y"
I_nnf: fixes \<phi> :: "atom fm"   and xs :: "'b list" shows "DLO.I (DLO.nnf \<phi>) xs = DLO.I \<phi> xs"
chf_lm_1: fixes A :: "nat set" shows "zero_set (chf A) = A"
Cons_6: shows "set (u # Us) \<subseteq> carrier R"
remove_def: fixes x :: "'b"   and A :: "'b set" shows "Set.remove x A = A\<setminus>{x}"
mat_of_row_dim_2: fixes y :: "'b vec" shows "dim_col (mat_of_row y) = dim_vec y"
Can_implies_Arr: fixes t :: "'c local.term" assumes "Can t" shows "Arr t"
map_image_eq: fixes A :: "'a set"   and m :: "'a \<Rightarrow> 'b option"   and m' :: "'a \<Rightarrow> 'b option" assumes "A \<subseteq> dom m"   and "m \<subseteq>\<^sub>m m'" shows "m ` A = m' ` A"
abs_eq: fixes xa :: "'b ptrm"   and x :: "'b ptrm" shows "App (abs_trm xa) (abs_trm x) = abs_trm (PApp xa x)"
comp_arr_ide: fixes a :: "'a"   and f :: "'a" assumes "C.ide a"   and "C.seq f a" shows "f \<cdot> a = f"
irreflexive_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "irreflexive A r = irreflexive A r'"
zero_adds: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and a :: "'e" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus zero a"
Cons_le_Cons: fixes x :: "'b"   and xs :: "'b list"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list" shows "(x # xs [\<sqsubseteq>\<^bsub>r\<^esub>] y # ys) = (x \<sqsubseteq>\<^bsub>r\<^esub> y \<and> xs [\<sqsubseteq>\<^bsub>r\<^esub>] ys)"
is_rreq_def: fixes \<xi> :: "'b state_scheme" shows "is_rreq \<xi> \<equiv> case msg \<xi> of Rreq hops' rreqid' dip' dsn' dsk' oip' osn' sip' handled' \<Rightarrow> {\<xi>\<lparr>hops := hops', rreqid := rreqid', dip := dip', dsn := dsn', dsk := dsk', oip := oip', osn := osn', sip := sip', handled := handled'\<rparr>} | _ \<Rightarrow> {}"
deg_add: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P"   and "q \<in> carrier P" shows "deg R (p \<oplus>\<^bsub>P\<^esub> q) \<le> max (deg R p) (deg R q)"
conc_Un_distrib_1: fixes A :: "'b list set"   and B :: "'b list set"   and C :: "'b list set" shows "A @@ (B \<union> C) = A @@ B \<union> A @@ C"
imageI: fixes x :: "'g"   and A :: "'g set"   and f :: "'g \<Rightarrow> 'h" assumes "x \<in> A" shows "f x \<in> f ` A"
bounded_convex_hull: fixes s :: "'a set" assumes "bounded s" shows "bounded (convex hull s)"
subset_Pow_Union: fixes A :: "'g set set" shows "A \<subseteq> Pow (\<Union> A)"
count_greater_eq_one_iff: fixes M :: "'a multiset"   and x :: "'a" shows "(1 \<le> count M x) = (x \<in># M)"
charpoly_of_real: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "charpoly (HMA_Connect.map_matrix complex_of_real A) = map_poly complex_of_real (charpoly A)"
unrestricte_path: fixes g' :: "'b ugraph"   and E :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path (restrict_edges g' E) u p v" shows "path g' u p v"
zero_imp_mult_zero: fixes a :: "'b"   and b :: "'b" assumes "a = (0::'b) \<or> b = (0::'b)" shows "a * b = (0::'b)"
simps_2: fixes \<xi> :: "nat \<Rightarrow> 'a set" shows "\<xi> \<Turnstile>\<^sub>n false\<^sub>n = False"
norm_diff_ineq: fixes a :: "'a"   and b :: "'a" shows "norm a - norm b \<le> norm (a + b)"
distrib_sup_le: fixes 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 x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.lattice inf less_eq less sup" shows "less_eq (sup x (inf y z)) (inf (sup x y) (sup x z))"
hfref_to_ASSERT_conv: fixes P :: "'b \<Rightarrow> bool"   and a :: "'c \<Rightarrow> 'd Heap"   and b :: "'b \<Rightarrow> 'e nres"   and R :: "('b \<Rightarrow> 'c \<Rightarrow> assn) \<times> ('b \<Rightarrow> 'c \<Rightarrow> assn)"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> assn" assumes "NO_MATCH (\<lambda>_. True) P" shows "((a, b) \<in> [P]\<^sub>a R \<rightarrow> S) = ((a, \<lambda>x. ASSERT (P x) \<bind> (\<lambda>_. b x)) \<in> R \<rightarrow>\<^sub>a S)"
assms_3: shows "P \<subseteq> {s. init s \<in> P' s}"
assms: shows "z \<in> (\<Union>m\<in>\<int>. \<Union>n\<in>{0<..}. {Complex m (ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)}) \<union> (\<Union>m\<in>\<int>. \<Union>n\<in>{0<..}. {Complex m (- ln (real n + sqrt ((real n)\<^sup>2 - 1)) / pi)})"
HcompNml_Nml: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = t \<^bold>\<star> u"
cqt_further_2: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<not>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<equiv> (\<^bold>\<exists>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
subgroup_imp_group: fixes H :: "'a set" assumes "subgroup H G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
of_rat_val_constraint: fixes v :: "nat \<Rightarrow> rat"   and c :: "constraint" shows "of_rat_val v \<Turnstile>\<^sub>c c = v \<Turnstile>\<^sub>c c"
AtAwait: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and r :: "'e set"   and ae :: "('e, 'd, 'f) ann"   and b :: "'e set"   and e :: "('e, 'd, 'f) com" shows "atomicsR \<Gamma> \<Theta> (AnnRec r ae) (Await b e) (r \<inter> b, Await b e)"
induct: fixes P :: "('a, 'b) Combinators list \<Rightarrow> bool"   and a0 :: "('a, 'b) Combinators list" assumes "\<And>x. P x \<Longrightarrow> P (DenyAll # x)"   and "\<And>v va y z. \<lbrakk>first_bothNet (DenyAllFromTo v va) = first_bothNet y \<Longrightarrow> P (DenyAllFromTo v va \<oplus> y # z); first_bothNet (DenyAllFromTo v va) \<noteq> first_bothNet y \<Longrightarrow> P (y # z)\<rbrakk> \<Longrightarrow> P (DenyAllFromTo v va # y # z)"   and "\<And>v va vb y z. \<lbrakk>first_bothNet (AllowPortFromTo v va vb) = first_bothNet y \<Longrightarrow> P (AllowPortFromTo v va vb \<oplus> y # z); first_bothNet (AllowPortFromTo v va vb) \<noteq> first_bothNet y \<Longrightarrow> P (y # z)\<rbrakk> \<Longrightarrow> P (AllowPortFromTo v va vb # y # z)"   and "\<And>v va y z. \<lbrakk>first_bothNet (v \<oplus> va) = first_bothNet y \<Longrightarrow> P ((v \<oplus> va) \<oplus> y # z); first_bothNet (v \<oplus> va) \<noteq> first_bothNet y \<Longrightarrow> P (y # z)\<rbrakk> \<Longrightarrow> P (v \<oplus> va # y # z)"   and "P []"   and "\<And>vb vc. P [DenyAllFromTo vb vc]"   and "\<And>vb vc vd. P [AllowPortFromTo vb vc vd]"   and "\<And>vb vc. P [vb \<oplus> vc]" shows "P a0"
assn_ext: fixes P :: "assn"   and Q :: "assn" assumes "\<forall>h as. (pHeap h as \<Turnstile> P) = (pHeap h as \<Turnstile> Q)" shows "P = Q"
assms: shows "Ball (set \<pi>s) wf_plan_action"
Ql_def: shows "Ql = (\<lambda>C. (C, Old)) ` Q"
simps_2: fixes x21 :: "'c net_state"   and x22 :: "'c net_state"   and y21 :: "'c net_state"   and y22 :: "'c net_state" shows "(SubnetS x21 x22 = SubnetS y21 y22) = (x21 = y21 \<and> x22 = y22)"
And: fixes n :: "nat"   and p :: "pf"   and q :: "pf" shows "dfa_of_pf n (And p q) = and_dfa (dfa_of_pf n p) (dfa_of_pf n q)"
\<theta>: shows "\<theta> \<in> hom (relcycle_group p X S) H"
assms: shows "s \<in> reachable (pnet np p) I"
f_def: shows "f = (\<lambda>t. Im (g t - z) / Re (g t - z))"
rel_gpv''D: fixes A :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'m \<Rightarrow> 'n \<Rightarrow> bool"   and gpv :: "('i, 'k, 'm) gpv"   and gpv' :: "('j, 'l, 'n) gpv" assumes "rel_gpv'' A C R gpv gpv'" shows "rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv')"
lec_of_nsc: fixes c :: "'a ns_constraint"   and v :: "nat \<Rightarrow> 'a" assumes "is_leq_ns c" shows "v \<Turnstile>\<^sub>l\<^sub>e lec_of_nsc c = v \<Turnstile>\<^sub>n\<^sub>s c"
card_of_Times_infinite_simps_4: fixes A :: "'a set"   and B :: "'b set" assumes "infinite A"   and "B \<noteq> {}"   and "|B| \<le>o |A|" shows "|A| =o |B \<times> A|"
permute_Abs_lst: fixes p :: "perm"   and as :: "atom list"   and x :: "'e" shows "p \<bullet> ([as]lst. x) = [p \<bullet> as]lst. p \<bullet> x"
le_iff_add: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subseteq># b) = (\<exists>c. b = a + c)"
simps_2: fixes x :: "'r"   and xs :: "[:'r:]"   and f :: "'s \<rightarrow> 'r \<rightarrow> 's"   and z :: "'s" assumes "x \<noteq> \<bottom>"   and "xs \<noteq> \<bottom>" shows "sfoldl\<cdot>f\<cdot>z\<cdot>(x :# xs) = sfoldl\<cdot>f\<cdot>(f\<cdot>z\<cdot>x)\<cdot>xs"
nu_star_sub: fixes x :: "'a" shows "\<nu> x\<^sup>\<star> \<le> \<nu> (x\<^sup>\<star>)"
constrains_weaken: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B :: "'c set"   and B' :: "'c set" assumes "F \<in> A co A'"   and "B \<subseteq> A"   and "A' \<subseteq> B'" shows "F \<in> B co B'"
subfrmlsn_subset: fixes \<psi> :: "'a ltln"   and \<phi> :: "'a ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "subfrmlsn \<psi> \<subseteq> subfrmlsn \<phi>"
Union_image_single_mset: fixes m :: "'b multiset" shows "(\<Sum>x\<in>#m. x, {#}) = m"
a1: shows "t' - t \<in> existence_ivl0 (flow0 y t)"
weakly_reduced_def: fixes \<alpha> :: "'a"   and k :: "nat" shows "weakly_reduced \<alpha> k = (\<forall>i. Suc i < k \<longrightarrow> \<parallel>gso i\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>gso (Suc i)\<parallel>\<^sup>2)"
dual_comp_neg: fixes x :: "'a"   and y :: "'a" shows "x ^ o * y \<squnion> neg_assert x * \<top> = x ^ o * y"
up_defined: fixes x :: "'a" shows "up\<cdot>x \<noteq> \<bottom>"
sel_15: fixes x61 :: "'b mrexp2"   and x62 :: "bool" shows "nul (Star2 x61 x62) = True"
Nodes_var_pret: fixes pret :: "dag"   and levellista :: "ref list list"   and var :: "ref \<Rightarrow> nat"   and nb :: "nat"   and p :: "ref" assumes "wf_ll pret levellista var"   and "nb < length levellista"   and "p \<in> Nodes nb levellista" shows "var p < nb \<and> p \<in> set_of pret"
simps_10: fixes i :: "int"   and a :: "MIR.num" shows "bound0 (NDvd i a) = numbound0 a"
intros_16: fixes \<pi> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" shows "Tuple \<pi> \<sqsubset> OclSuper"
zero_in_syzygy_module: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set" shows "0 \<in> pmdl.syzygy_module B"
bounded_iff: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "semilattice_order f less_eq less" shows "less_eq a (f b c) = (less_eq a b \<and> less_eq a c)"
P: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> 'c) \<Rightarrow> 'c measure"   and M :: "nat \<Rightarrow> 'c measure"   and i :: "nat" assumes "Ionescu_Tulcea P M" shows "P i \<in> Pi\<^sub>M {0..<i} M \<rightarrow>\<^sub>M subprob_algebra (M i)"
sphere_cball: fixes z :: "'c"   and r :: "real" shows "sphere z r \<subseteq> cball z r"
intros_5: fixes xs :: "'b list"   and ys :: "'b list"   and a :: "'b" assumes "xs \<approx>2 ys" shows "a # xs \<approx>2 a # ys"
filter_pdevs_raw_nonzeros: fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" shows "{i. filter_pdevs_raw s f i \<noteq> (0::'a)} = {i. f i \<noteq> (0::'a)} \<inter> {x. s x (f x)}"
not_SN_onI: fixes f :: "nat \<Rightarrow> 'a"   and X :: "'a set"   and R :: "('a \<times> 'a) set" assumes "f 0 \<in> X"   and "\<forall>i. (f i, f (Suc i)) \<in> R" shows "\<not> SN_on R X"
lexord_lex: fixes x :: "'b list"   and y :: "'b list"   and r :: "('b \<times> 'b) set" shows "((x, y) \<in> lex r) = ((x, y) \<in> lexord r \<and> length x = length y)"
prems_1: shows "(name, rhs) \<in> set (r # rs)"
w: shows "P w u \<and> \<not> O w (\<midarrow> x)"
rep_eq: fixes x :: "'c fset"   and xa :: "('c, 'd) fmap" shows "fmlookup (fmrestrict_fset x xa) = map_restrict_set (fset x) (fmlookup xa)"
g_cong: fixes xs :: "'a list"   and ys :: "'a list" assumes "mset xs = mset ys" shows "g xs = g ys"
open_dist: fixes dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool"   and S :: "'d set" assumes "class.metric_space dist uniformity open" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
FH: fixes i :: "nat" assumes "i \<noteq> 0" shows "F i = ffp (H i)"
c: shows "Cinv g ;; Cinv f maps cod g to dom f"
concurrent_history_steps_simulates_store_buffer_step: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> 'f \<times> 'g \<times> 'h) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'i"   and ts' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> 'f \<times> 'g \<times> 'h) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "'i"   and ts\<^sub>h :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S>\<^sub>h :: "nat \<Rightarrow> bool option" assumes "(ts, m, \<S>) \<Rightarrow>\<^sub>s\<^sub>b (ts', m', \<S>')"   and "ts \<sim>\<^sub>h ts\<^sub>h " shows "\<exists>ts\<^sub>h' \<S>\<^sub>h'. (ts\<^sub>h, m, \<S>\<^sub>h) \<Rightarrow>\<^sub>s\<^sub>b\<^sub>h\<^sup>* (ts\<^sub>h', m', \<S>\<^sub>h') \<and> ts' \<sim>\<^sub>h ts\<^sub>h' "
cases: fixes r :: "parts_connection" obtains pc_iiface :: "char list" and pc_oiface :: "char list" and pc_proto :: "8 word" and pc_sport :: "16 word" and pc_dport :: "16 word" where "r = \<lparr>pc_iiface = pc_iiface, pc_oiface = pc_oiface, pc_proto = pc_proto, pc_sport = pc_sport, pc_dport = pc_dport\<rparr>"
eval_Disj_imp_Disj: fixes e :: "(name, hf) finfun"   and \<alpha> :: "fm"   and x :: "hf"   and y :: "hf" assumes "\<lbrakk>\<guillemotleft>\<alpha>\<guillemotright>\<rbrakk>e = q_Disj x y" shows "\<exists>A B. \<alpha> = A OR B \<and> \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e = x \<and> \<lbrakk>\<guillemotleft>B\<guillemotright>\<rbrakk>e = y"
ide_char: fixes f :: "cospan_shape.Arr" shows "partial_magma.ide cospan_shape.comp f = (f = cospan_shape.AA \<or> f = cospan_shape.BB \<or> f = cospan_shape.TT)"
hypnat_minus_zero: fixes z :: "nat star" shows "z - z = 0"
no_zero_height_derivsAux2: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and MM :: "Var list \<times> (int \<times> Instr) list \<times> int"   and lll :: "int"   and ss :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and h :: "(Addr \<times> Class \<times> (Field \<times> Val) list) list"   and v :: "Val" shows "((M, l, s, 0, ll, t) \<in> Step \<longrightarrow> False) \<and> ((MM, lll, ss, 0, h, v) \<in> Exec \<longrightarrow> False)"
sum_over_fun_eq: fixes A :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in># A \<Longrightarrow> f x = g x" shows "\<Sum>\<^sub># (image_mset f A) = \<Sum>\<^sub># (image_mset g A)"
C_restr_C_case: fixes r :: "C"   and f :: "C \<rightarrow> 'a" shows "C_case\<cdot>f|\<^bsub>C\<cdot>r\<^esub> = C_case\<cdot>(f|\<^bsub>r\<^esub>)"
insert_Nil: fixes x :: "'e" shows "List.insert x [] = [x]"