hcju/isabelleps huggingface dataset:
case_cong: fixes action :: "'a action"   and action' :: "'a action"   and f1 :: "'a \<Rightarrow> 'b"   and g1 :: "'a \<Rightarrow> 'b"   and f2 :: "'b"   and g2 :: "'b"   and f3 :: "'b"   and g3 :: "'b" assumes "action = action'"   and "\<And>x1. action' = NormalAction x1 \<Longrightarrow> f1 x1 = g1 x1"   and "action' = InitialThreadAction \<Longrightarrow> f2 = g2"   and "action' = ThreadFinishAction \<Longrightarrow> f3 = g3" shows "(case action of NormalAction x1 \<Rightarrow> f1 x1 | InitialThreadAction \<Rightarrow> f2 | ThreadFinishAction \<Rightarrow> f3) = (case action' of NormalAction x1 \<Rightarrow> g1 x1 | InitialThreadAction \<Rightarrow> g2 | ThreadFinishAction \<Rightarrow> g3)"
Least_equality: fixes P :: "'a word \<Rightarrow> bool"   and x :: "'a word" assumes "P x"   and "\<And>y. P y \<Longrightarrow> x \<le>s y" shows "signed.Least P = x"
NTExt: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTransExt \<eta>" shows "($$) \<eta> \<in> extensional (obj\<^bsub>NTCatDom \<eta>\<^esub>)"
wf: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "well_founded R" shows "wfP R"
qSwap_preserves_qGood: fixes zs :: "'varSort"   and x :: "'var"   and y :: "'var"   and X :: "('e, 'f, 'varSort, 'var, 'g) qTerm" shows "qGood (X #[[x \<and> y]]_zs) = qGood X"
empty_Shift: fixes Kl :: "'d list set"   and k :: "'d" assumes "[] \<in> Kl"   and "k \<in> BNF_Greatest_Fixpoint.Succ Kl []" shows "[] \<in> BNF_Greatest_Fixpoint.Shift Kl k"
kpower_inv: fixes I :: "'b \<Rightarrow> bool"   and F :: "'b \<Rightarrow> 'b set"   and n :: "nat" assumes "\<forall>s. I s \<longrightarrow> (\<forall>s'. s' \<in> F s \<longrightarrow> I s')" shows "\<forall>s. I s \<longrightarrow> (\<forall>s'. s' \<in> kpower F n s \<longrightarrow> I s')"
eval_simps_1: fixes b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide b"   and "ide c" shows "arr (hfsetcat.eval b c)"
open_diagonal_complement: shows "open {(x, y) |x y. x \<noteq> y}"
SpanI_lincomb_arb_len_coeffs: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "rs \<bullet>\<cdot> ms \<in> RSpan ms"
inorder_del: fixes t :: "'a tree"   and x :: "'a" assumes "Sorted_Less.sorted (inorder t)" shows "inorder (case del x t of None \<Rightarrow> t | Some t' \<Rightarrow> t') = del_list x (inorder t)"
meval_sched_length: fixes sched :: "nat list"   and gc :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)"   and gc' :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) list \<times> ('Var\<^sub>A \<Rightarrow> 'Val)" assumes "abs.meval_sched_abv gc sched gc'" shows "length (fst gc') = length (fst gc)"
measurable_sets: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and A :: "'e measure"   and S :: "'e set" assumes "f \<in> M \<rightarrow>\<^sub>M A"   and "S \<in> sets A" shows "f -` S \<inter> space M \<in> sets M"
admissible_def: fixes ts :: "(symbol \<times> character list) list" shows "admissible ts = (terminals ts \<in> \<L>\<^sub>P)"
numeral_plus_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus n) one = numeral.numeral one plus (n + num.One)"
dotprod_gt0D: fixes x :: "nat list"   and y :: "nat list" assumes "length x = length y"   and "0 < x \<bullet> y" shows "\<exists>i<length y. 0 < x ! i \<and> 0 < y ! i"
IH: shows "Brouwer_degree2 p (\<lambda>x i. if i = 0 then - x i else x i) = - 1"
subadditive_converges_bounded'': fixes u :: "nat \<Rightarrow> real"   and a :: "real" assumes "subadditive u"   and "\<And>n. 0 < n \<Longrightarrow> real n * a \<le> u n" shows "\<exists>l. (\<lambda>n. u n / real n) \<longlonglongrightarrow> l \<and> (\<forall>n>0. real n * l \<le> u n)"
add_missing_ret_unfoldings_def: fixes rs1 :: "'b rule list"   and rs2 :: "'b rule list" shows "add_missing_ret_unfoldings rs1 rs2 \<equiv> foldr (\<lambda>rf. (\<circ>) (add_match (MatchNot (get_match rf)))) (filter (\<lambda>r. get_action r = Return) rs1) id rs2"
Inv_preserves_Diag: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t"   and "monoidal_language.Diag C t" shows "monoidal_language.Diag C (monoidal_language.Inv C t)"
homomorphism_thm: fixes G :: "('b, 'c) monoid_scheme"   and h :: "'b \<Rightarrow> 'b" assumes "group_hom G G h" shows "G Mod kernel G (G\<lparr>carrier := h ` carrier G\<rparr>) h \<cong> G\<lparr>carrier := h ` carrier G\<rparr>"
distinct_9: fixes x21 :: "variable"   and x22 :: "typ"   and x3 :: "nat" shows "Fv x21 x22 \<noteq> Bv x3"
gen_mult: fixes x :: "'e"   and G :: "('e, 'f) monoid_scheme"   and gens :: "'e set"   and y :: "'e" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"   and "y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
filter_min_relE: fixes x :: "'a"   and xs :: "'a list" assumes "transp rel"   and "reflp rel"   and "x \<in> set xs" obtains y :: "'a" where "y \<in> set (local.filter_min xs)" and "rel y x"
assms: fixes \<tau> :: "'a"   and \<tau>' :: "'a" assumes "P \<tau> = P \<tau>'" shows "Q \<tau> = Q \<tau>'"
rel_nondetT_simps: fixes M :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'e"   and m' :: "'f" shows "rel_nondetT M (NondetT m) (NondetT m') = M m m'"
LIMSEQ_iff: fixes X :: "nat \<Rightarrow> 'c"   and L :: "'c" shows "X \<longlonglongrightarrow> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r)"
fps_euclidean_size_def: fixes f :: "'b fps" shows "euclidean_size f = (if f = 0 then 0 else 2 ^ subdegree f)"
eq2: shows "set (characterize_consistent_signs_at_roots (fss ! i) (removeAt i fss)) = consistent_sign_vectors_r (removeAt i fss) {x. poly (fss ! i) x = 0}"
eqD_17: fixes y :: "cmsg"   and N :: "ephkey" assumes "eq y (cEphK N)" shows "y = cEphK N"
prime_factors_characterization_int: fixes S :: "int set"   and f :: "int \<Rightarrow> nat"   and n :: "int" assumes "S = {p. 0 < f p}"   and "finite S"   and "\<forall>p\<in>S. prime p"   and "\<bar>n\<bar> = (\<Prod>p\<in>S. p ^ f p)" shows "prime_factors n = S"
neg_power_inverse: fixes G :: "('a, 'b) cyclic_group_scheme"   and g :: "'a"   and x :: "nat" assumes "Cyclic_Group.cyclic_group G"   and "g \<in> carrier G"   and "x < order G" shows "g [^]\<^bsub>G\<^esub> (order G - x) = inv\<^bsub>G\<^esub> (g [^]\<^bsub>G\<^esub> x)"
elims: fixes x :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool"   and xa :: "'at Literal set set"   and y :: "'at Literal set set" assumes "add_all_deducible_clauses x xa = y" obtains R :: "'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> 'at Literal set \<Rightarrow> bool" and S :: "'at Literal set set" where "x = R" and "xa = S" and "y = S \<union> all_deducible_clauses R S"
essentialPPA: shows "\<forall>c x. \<lfloor>PPA x\<rfloor>\<^sub>c \<longrightarrow> \<lfloor>PPA x\<rfloor>\<^sup>D"
m_comm: fixes x :: "'a list"   and y :: "'a list" assumes "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])" shows "x \<oplus>\<^bsub>K [X]\<^esub> y = y \<oplus>\<^bsub>K [X]\<^esub> x"
conc_mono: fixes A :: "'c list set"   and C :: "'c list set"   and B :: "'c list set"   and D :: "'c list set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A @@ B \<subseteq> C @@ D"
inv_into_injective: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "inv_into A f x = inv_into A f y"   and "x \<in> f ` A"   and "y \<in> f ` A" shows "x = y"
set_to_nat_at_empty: shows "set_to_nat {} = 0"
list_permutesI: fixes xs :: "('a \<times> 'a) list"   and A :: "'a set" assumes "set (map fst xs) \<subseteq> A"   and "set (map snd xs) = set (map fst xs)"   and "distinct (map fst xs)" shows "list_permutes xs A"
suf_pref_eq: fixes s :: "'b list"   and p :: "'b list" assumes "s \<le>s p"   and "p \<le>p s" shows "p = s"
abc_ex_path_unique: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "[[a b c]]" shows "\<exists>!Q. Q \<in> \<P> \<and> a \<in> Q \<and> b \<in> Q \<and> c \<in> Q"
i'_2: shows "\<tau> \<sigma> i + b + 1 \<le> \<tau> \<sigma> i'"
det_state_cover_finite: fixes M :: "('e, 'f, 'g) FSM"   and V :: "'e list set" assumes "is_det_state_cover M V"   and "finite (nodes M)" shows "finite V"
sel: fixes a :: "'c \<Rightarrow> 'e"   and b :: "'f \<Rightarrow> 'd"   and res :: "('e, 'f) resource" shows "run_resource (map_resource a b res) = map_spmf (map_prod b (map_resource a b)) \<circ> run_resource res \<circ> a"
prv_imp_cnj3L: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<chi> \<in> fmla" shows "bprv (imp (imp \<phi>1 \<chi>) (imp (cnj \<phi>1 \<phi>2) \<chi>))"
ee_is_fmset: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and bs :: "'c list" assumes "comm_monoid_cancel G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "essentially_equal G as bs = (fmset G as = fmset G bs)"
S_intersections_defs_2: fixes p :: "(real, 3) vec"   and q :: "(real, 3) vec" shows "S_intersection2 p q \<equiv> proj2_abs (S_intersection2_rep p q)"
qGoodEnv_imp_card_of_qTerm2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option" assumes "qGoodEnv rho" shows "|{X. \<exists>ys y. rho ys y = Some X}| <o |UNIV|"
card_M_le_weight_NVLi: fixes V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" assumes "matching V E M"   and "OSC L E" shows "card M \<le> weight {i \<in> L ` V. 1 < i} (N V L)"
fst_comp_apsnd: fixes f :: "'e \<Rightarrow> 'f" shows "fst \<circ> apsnd f = fst"
succeeding_states: fixes i :: "nat" assumes "smallest_accepting_rank\<^sub>R = Some i" shows "\<forall>\<^sub>\<infinity>n. \<forall>q. (Bex (configuration q n) token_succeeds \<longrightarrow> q \<in> \<S>\<^sub>R n) \<and> (q \<in> \<S>\<^sub>R n \<longrightarrow> Ball (configuration q n) token_succeeds)"
\<J>_subset_Suc_u: fixes k :: "nat"   and u :: "nat" shows "\<J> k u \<subseteq> \<J> k (Suc u)"
Keys_mat_to_polys: fixes ts :: "'c list"   and A :: "'d mat" shows "Keys (set (mat_to_polys ts A)) \<subseteq> set ts"
open_segment_furthest_le: fixes x :: "'b"   and a :: "'b"   and b :: "'b"   and y :: "'b" assumes "x \<in> open_segment a b" shows "norm (y - x) < norm (y - a) \<or> norm (y - x) < norm (y - b)"
netgmap_netmask_subnets: fixes sr :: "'f \<Rightarrow> 'd \<times> 'e"   and s1 :: "'f net_state"   and n1 :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'd"   and s2 :: "'f net_state"   and n2 :: "net_tree" assumes "netgmap sr s1 = netmask (net_tree_ips n1) (\<sigma>, snd (netgmap sr s1))"   and "netgmap sr s2 = netmask (net_tree_ips n2) (\<sigma>, snd (netgmap sr s2))" shows "fst (netgmap sr (SubnetS s1 s2)) = fst (netmask (net_tree_ips (n1 \<parallel> n2)) (\<sigma>, snd (netgmap sr (SubnetS s1 s2))))"
assms: shows "\<forall>x\<in>A. \<forall>y\<in>A. x \<noteq> y \<longrightarrow> incomparable subseq x y"
abupd_def: fixes f :: "abopt \<Rightarrow> abopt" shows "abupd f = map_prod f id"
retraction_refl: fixes S :: "'b set" shows "retraction S S (\<lambda>x. x)"
red_sat_eq_red_calc_sat: fixes N :: "'f set" shows "reduc_saturated N = reduc_calc.saturated N"
independent_length_le_dimension: fixes K :: "'a list set"   and n :: "nat"   and E :: "'a list set"   and Us :: "'a list list" assumes "subfield K (K [X])"   and "UP.dimension n K E"   and "UP.independent K Us"   and "set Us \<subseteq> E" shows "length Us \<le> n"
Aexp_eq: fixes cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and e :: "exp"   and a :: "Arity" shows "CoCallArityAnalysis.Aexp cccExp e\<cdot>a = fst (cccExp e\<cdot>a)"
br_run_conv: fixes \<alpha> :: "'a \<Rightarrow> 'b"   and I :: "'a \<Rightarrow> bool"   and rb :: "nat \<Rightarrow> 'b" assumes "R = br \<alpha> I" shows "b.is_run rb = (\<exists>ra. rb = \<alpha> \<circ> ra \<and> a.is_run ra)"
join_fun_def: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
assms_3: shows "A \<subseteq> U\<^sub>2"
lattice_of_eq_set: fixes fs :: "'a vec list"   and gs :: "'a vec list"   and n :: "nat" assumes "set fs = set gs"   and "set fs \<subseteq> carrier_vec n" shows "vec_module.lattice_of n fs = vec_module.lattice_of n gs"
get_disconnected_nodes_impl: shows "get_disconnected_nodes = CD.a_get_disconnected_nodes"
memberdecl_memberid_def: fixes m :: "memberdecl" shows "memberid m = (case m of fdecl (vn, f) \<Rightarrow> fid vn | mdecl (sig, m) \<Rightarrow> mid sig)"
nor2: shows "\<not> (\<exists>x>z. x < - b / (2 * a) \<and> poly [:c, b, a:] x = 0)"
tseytin_toatom_def: fixes F :: "'a formula" shows "tseytin_toatom F \<equiv> Atom \<circ> the \<circ> map_of (map (\<lambda>(a, b). (b, a)) (tseytin_assmt F))"
FG_advice_prop_entailment_1: fixes \<A> :: "'b ltln set"   and \<phi> :: "'b ltln"   and Y :: "'b ltln set" assumes "true\<^sub>n \<in> \<A>"   and "\<A> \<Turnstile>\<^sub>P \<phi>[Y]\<^sub>\<mu>" shows "{\<psi>. \<psi>[Y]\<^sub>\<mu> \<in> \<A>} \<Turnstile>\<^sub>P \<phi>"
hyps: assumes "\<forall>a\<in>set les. case a of (d, e, f) \<Rightarrow> \<exists>y'>q. \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0" shows "\<exists>y'>q. \<forall>a\<in>set les. case a of (d, e, f) \<Rightarrow> \<forall>x\<in>{q<..y'}. d * x\<^sup>2 + e * x + f < 0"
fold_mset_add_carrier: fixes a :: "'a"   and M :: "'a multiset" assumes "a \<in> carrier G"   and "set_mset M \<subseteq> carrier G" shows "fold_mset (\<oplus>) a M \<in> carrier G"
\<gamma>_def: shows "\<gamma> \<equiv> (common_matcher, in_doubt_allow)"
NRR_sender: fixes B' :: "agent"   and A :: "agent"   and n :: "msg"   and a :: "msg"   and l :: "msg"   and B :: "agent"   and X :: "msg"   and evs :: "event list" assumes "Says B' A \<lbrace>n, a, l, Crypt (priEK B) X\<rbrace> \<in> set evs"   and "evs \<in> zg" shows "B' \<in> {B, Spy}"
h'_def: shows "h' \<equiv> fst (bezout_coefficients g h)"
d_OUT_diff: fixes g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "\<And>y. g (x, y) \<le> f (x, y)"   and "d_OUT g x \<noteq> \<top>" shows "d_OUT (\<lambda>e. f e - g e) x = d_OUT f x - d_OUT g x"
exhaust: fixes y :: "'b extC" obtains "y = extC.Bot"   | x2 :: "'b" where "y = extC.Val x2"   | "y = extC.Top"
mem_Sigma_iff: fixes a :: "'g"   and b :: "'h"   and A :: "'g set"   and B :: "'g \<Rightarrow> 'h set" shows "((a, b) \<in> Sigma A B) = (a \<in> A \<and> b \<in> B a)"
Q_sets: fixes i :: "nat" shows "Q i \<in> sets M"
phi'_nonneg: fixes x :: "real" assumes "a < x"   and "x < b" shows "0 \<le> \<phi>' x"
dgrad_p_set_closed_tail: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and d :: "'d \<Rightarrow> nat"   and m :: "nat" assumes "gd_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<in> gd_term.dgrad_p_set pair_of_term d m" shows "ordered_term.tail term_of_pair ord_term p \<in> gd_term.dgrad_p_set pair_of_term d m"
borel_measurable_uminus: fixes g :: "'c \<Rightarrow> 'd"   and M :: "'c measure" assumes "g \<in> borel_measurable M" shows "(\<lambda>x. - g x) \<in> borel_measurable M"
dvd_lcmI1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b" shows "a dvd lcm b c"
aux_def: shows "aux = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)"
da: assumes "Q = (\<lambda>s. False)"   and "P = (\<lambda>s. False)" shows "thesis"
\<C>pub_unfold: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool" assumes "intruder_model arity Ana" shows "{f. public f} \<inter> {f. arity f = 0} = {f. arity f = 0 \<and> public f}"
of_int_complex_interval_def: fixes x :: "int" shows "of_int_complex_interval x = Complex_Interval (of_int_interval x) 0"
is_ground_subst_lit: fixes L :: "'a literal"   and \<sigma> :: "'s" assumes "is_ground_lit L" shows "L \<cdot>l \<sigma> = L"
bind_left_identity: fixes a :: "'f"   and f :: "'f \<Rightarrow> ['d| 'e]" shows "\<langle>a\<rangle> \<bind> f = f a"
evaluate_iff_2: fixes env :: "v sem_env"   and st :: "'a state"   and es :: "exp0 list"   and r' :: "'a state \<times> (v list, v) result" shows "evaluate_list True env st es r' = (r' = eval_list env es st)"
double_add_le_zero_iff_single_add_le_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (plus a a) zero = less_eq a zero"
two_comparisons_into_case_order_3: fixes x :: "'b"   and y :: "'b"   and P :: "'d"   and Q :: "'d"   and R :: "'d" shows "(if c1.le x y then if c1.le y x then P else Q else R) = (case c1 x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
complex_of_real_def: fixes r :: "real" shows "cor r = Complex r 0"
vimage_empty: fixes f :: "'a \<Rightarrow> 'b" shows "f -` {} = {}"
assms_4: shows "\<GG>' = \<HH> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG>"
assms: shows "[\<lparr>A!,x\<rparr> in v]"
less_le_trans: fixes x :: "'e word"   and y :: "'e word"   and z :: "'e word" assumes "x <s y"   and "y \<le>s z" shows "x <s z"
least: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.pcpo below" shows "\<exists>x. \<forall>y. below x y"
iprev_atLeast': fixes n :: "nat"   and t :: "nat" assumes "n \<le> t" shows "iprev (Suc t) {n..} = t"
last_message_hold_append: fixes xs :: "'b message_af list"   and ys :: "'b message_af list" shows "last_message_hold (xs @ ys) = last_message_hold xs @ last_message_hold_init ys (last_message xs)"
InfCard_def: fixes \<kappa> :: "V" shows "InfCard \<kappa> \<equiv> Card \<kappa> \<and> \<omega> \<le> \<kappa>"
reach_def: shows "reach \<equiv> \<lambda>T. {x. reachp T x}"
degree_numeral: fixes n :: "num" shows "degree (numeral n) = 0"
val_Zp_eq_frac_0: fixes p :: "int"   and a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int"   and d :: "nat \<Rightarrow> int" assumes "padic_integers p"   and "a \<in> nonzero (padic_int p)"   and "b \<in> nonzero (padic_int p)"   and "c \<in> nonzero (padic_int p)"   and "d \<in> nonzero (padic_int p)"   and "a \<otimes>\<^bsub>padic_int p\<^esub> d = b \<otimes>\<^bsub>padic_int p\<^esub> c" shows "padic_integers.val_Zp p a - padic_integers.val_Zp p b = padic_integers.val_Zp p c - padic_integers.val_Zp p d"
function_uminus: fixes f :: "'c \<Rightarrow> 'a" assumes "f \<in> carrier F" shows "\<ominus>\<^bsub>F\<^esub> f = function_uminus S R f"
aadd_minus_inv: fixes x :: "ant" shows "- x + x = 0"
subset_basis_imp_indep: fixes X :: "'a set"   and Y :: "'a set" assumes "basis X"   and "Y \<subseteq> X" shows "indep Y"
divisor_count_primorial: fixes x :: "real" shows "real (divisor_count (primorial x)) = 2 powr \<pi> x"
keys_powsum_mpoly_aux: fixes m :: "nat \<Rightarrow>\<^sub>0 nat"   and A :: "nat set"   and k :: "nat" assumes "m \<in> keys (powsum_mpoly_aux A k)" shows "keys m \<subseteq> A"
limit_along_weak_subseq: fixes u :: "nat \<Rightarrow> nat"   and v :: "nat \<Rightarrow> 'b"   and l :: "'b" assumes "(\<lambda>x. ereal (real (u x))) \<longlonglongrightarrow> \<infinity>"   and "v \<longlonglongrightarrow> l" shows "(\<lambda>n. v (u n)) \<longlonglongrightarrow> l"
case_1: fixes f1 :: "'c"   and f2 :: "'c"   and f3 :: "'d \<Rightarrow> 'c" shows "(case dSUCCEEDi of dSUCCEEDi \<Rightarrow> f1 | dFAILi \<Rightarrow> f2 | dRETURN x \<Rightarrow> f3 x) = f1"
Aab0': fixes a :: "nat"   and b :: "'cols" assumes "from_nat a \<noteq> b"   and "a < nrows A" shows "A $ from_nat a $ b = (0::'a)"
ritem_greatest_in_rset: fixes r :: "'b tree" assumes "r \<noteq> ET"   and "is_ord r" shows "\<forall>x. x \<in> set_of r \<longrightarrow> x \<noteq> rightmost_item r \<longrightarrow> x < rightmost_item r"
fst_cs_changed_by_recv_recording: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and cid :: "nat"   and p :: "'a"   and q :: "'a" assumes "c \<turnstile> ev \<mapsto> c'"   and "fst (cs c cid) \<noteq> fst (cs c' cid)"   and "channel cid = Some (p, q)" shows "snd (cs c cid) = Recording \<and> (\<exists>p q u u' m. ev = Recv cid q p u u' m)"
x: shows "x \<notin> Fvars (subst \<phi> (Var y) x)"
prefix_subst: fixes l :: "'c list"   and t :: "'c list"   and m :: "'c list" assumes "l @ t = m" shows "m - l = t"
csorted_list_of_set_def: fixes A :: "'b set" shows "csorted_list_of_set A = (if ID ccompare = None \<or> infinite A then undefined else linorder.sorted_list_of_set cless_eq A)"
append_butlast_last_id: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
assms: shows "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')"
Lp_D_2: fixes p :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "0 < p"   and "f \<in> space\<^sub>N (\<LL> (ennreal p) M)" shows "integrable M (\<lambda>x. \<bar>f x\<bar> powr p)"
simps_6: fixes x1 :: "'b" shows "p \<noteq> v x1"
flift1_def: shows "flift1 = (\<lambda>f. \<Lambda> x. case x of \<bottom> \<Rightarrow> \<bottom> | Def x \<Rightarrow> f x)"
t: shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set M"
wcode_fourtimes_case_first_correctness: fixes ires :: "cell list"   and rs :: "nat"   and m :: "nat"   and n :: "nat" shows "let P = \<lambda>(st, l, r). st = t_twice_len + 14; Q = \<lambda>(st, l, r). wcode_fourtimes_case_inv st ires rs (l, r); f = steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Oc # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main in \<exists>n. P (f n) \<and> Q (f n)"
set_integral_norm_bound: fixes M :: "'c measure"   and k :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "set_integrable M k f" shows "norm (set_lebesgue_integral M k f) \<le> \<integral>x\<in>k. norm (f x)\<partial>M"
simps_17: fixes x5 :: "nat \<times> char list list" shows "Ref x5 \<noteq> Null"
cc_subs_n: shows "set_mset CC \<subseteq> N"
a_rcos_module_imp: fixes H :: "'e set"   and G :: "('e, 'f) ring_scheme"   and x :: "'e"   and x' :: "'e" assumes "abelian_subgroup H G"   and "x \<in> carrier G"   and "x' \<in> H +>\<^bsub>G\<^esub> x" shows "x' \<oplus>\<^bsub>G\<^esub> \<ominus>\<^bsub>G\<^esub> x \<in> H"
m_inv_def: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a" shows "inv\<^bsub>G\<^esub> x = (THE y. 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>)"
dvd_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" shows "dvd.dvd times b a = (\<exists>k. a = times b k)"
fps_exp_eq_iff: fixes c :: "'a"   and d :: "'a" shows "(fps_exp c = fps_exp d) = (c = d)"
refl: fixes a :: "'a" shows "a \<sqsubseteq> a"
comm_monoid_def: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and z :: "'b::type" shows "comm_monoid (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (z::'b::type) \<equiv> abel_semigroup f \<and> comm_monoid_axioms f z"
antimono_iff_le_Suc: fixes f :: "nat \<Rightarrow> 'a" shows "decseq f = (\<forall>n. f (Suc n) \<le> f n)"
c2: shows " Lam [x].P'' \<longrightarrow>\<^sub>1 M3"
comp_null_2: fixes f :: "'a" shows "f \<cdot>\<^sub>C C.null = C.null"
join_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<squnion> y \<squnion> z = x \<squnion> (y \<squnion> z)"