hcju/isabelleps huggingface dataset:
assert_demonic_refinement: fixes p :: "'e \<Rightarrow> bool"   and r :: "'e \<Rightarrow> 'd"   and p' :: "'e \<Rightarrow> bool"   and r' :: "'e \<Rightarrow> 'd" shows "({. p .} \<circ> [: r :] \<le> {. p' .} \<circ> [: r' :]) = (p \<le> p' \<and> (\<forall>x. p x \<longrightarrow> r' x \<le> r x))"
strategy_extends_VVpstar: fixes v0 :: "'a"   and p :: "Player"   and w0 :: "'a"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v0 \<in> VV p**"   and "v0 \<rightarrow> w0"   and "winning_strategy p \<sigma> v0" shows "winning_strategy p \<sigma> w0"
atLeast_iff: fixes i :: "'b"   and b :: "'b"   and k :: "'b" shows "(i \<in> flat_ord.atLeast b k) = flat_ord b k i"
argmin_forall: fixes f :: "'a \<Rightarrow> 'c" assumes "xs \<noteq> []"   and "\<And>x. x \<in> set xs \<Longrightarrow> x \<in> set xs" shows "argmin f xs \<in> set xs"
hyps: fixes pending_write :: "bool"   and A :: "nat set" shows "acquired_reads pending_write (xs @ ys) A = acquired_reads pending_write xs A \<union> acquired_reads (pending_write \<or> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs \<noteq> {}) ys (acquired pending_write xs A)"
simps_11: fixes v :: "'a rexp" shows "nTimes rexp.One (Star v) = Star v"
Cons_3: shows "b # v \<preceq>\<^sub>F w"
ring_iso_memE_4: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme" assumes "h \<in> ring_iso R S" shows "h \<one>\<^bsub>R\<^esub> = \<one>\<^bsub>S\<^esub>"
Heq0: fixes Z :: "'a set"   and x :: "'a" assumes "Z \<in> \<C>"   and "x \<in> U - S"   and "x \<notin> Z" shows "H Z x = 0"
scale_left_distrib: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and x :: "'b" assumes "module scale" shows "scale (a + b) x = scale a x + scale b x"
simps: fixes a1 :: "'b list"   and a2 :: "'b list" shows "(a1 \<prec> a2) = (\<exists>xs ys. a1 = xs \<and> a2 = ys \<and> length xs < length ys \<and> take (length xs) ys = xs)"
O_simps_1: fixes \<gamma> :: "('b, 'c, 'd) trans \<Rightarrow> bool"   and g :: "('b, 'c, 'd) trans \<Rightarrow> 'a" shows "BD_Security.O \<gamma> g [] = []"
lfilter_empty_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "(lfilter P xs = LNil) = (\<forall>x\<in>lset xs. \<not> P x)"
dvd_0_left: fixes a :: "'a" assumes "(0::'a) dvd a" shows "a = (0::'a)"
add_divide_eq_if_simps_5: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and z :: "'a"   and b :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "minus (divide a z) b = (if z = zero then uminus b else divide (minus a (times b z)) z)"
min_arg_le_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "'b"   and n :: "'b" assumes "class.order less_eq less" shows "less_eq m (ord.min less_eq m n) = (ord.min less_eq m n = m)"
bounded_iff: fixes S :: "'b set" shows "bounded S = (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
tlookup_none: fixes h :: "'b \<Rightarrow> int"   and t :: "'b Tree"   and k :: "int" shows "sortedTree h t \<and> tlookup h k t = None \<longrightarrow> (\<forall>x\<in>setOf t. h x \<noteq> k)"
Field_def: fixes r :: "('b \<times> 'b) set" shows "Field r = Domain r \<union> Range r"
quasi_inverts_components_1: fixes a :: "'a" assumes "C.obj a" shows "D.isomorphic (map\<^sub>0 a \<star>\<^sub>D \<tau>\<^sub>0 a) (F.map\<^sub>0 a)"
r_zero: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "abelian_monoid G"   and "x \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> \<zero>\<^bsub>G\<^esub> = x"
nonneg_irreducible_mat_def: fixes A :: "'a mat" shows "nonneg_irreducible_mat A = (nonneg_mat A \<and> irreducible_mat A)"
Diff_eqvt: fixes p :: "perm"   and A :: "'b set"   and B :: "'b set" shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"
val_exp_ring: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and n :: "nat" assumes "valuation K v"   and "x \<in> carrier K"   and "x \<noteq> \<zero>" shows "int n *\<^sub>a v x = v (x^\<^bsup>K n\<^esup>)"
poly_deg_pp_def: fixes p :: "('x, nat) pp \<Rightarrow>\<^sub>0 'b" shows "poly_deg_pp p = (if p = 0 then 0 else max_list (map deg_pp (punit.keys_to_list p)))"
c: fixes a :: "nat"   and b :: "nat" shows "P a b = P b a"
poly_embed_EX: shows "\<exists>\<phi>. bij \<phi>"
kbo_code: fixes s :: "('f, 'a) Term.term"   and t :: "('f, 'a) Term.term" shows "kbo s t = (let wt = weight t; ws = weight s in if vars_term_ms (SCF t) \<subseteq># vars_term_ms (SCF s) \<and> wt \<le> ws then if wt < ws then (True, True) else case s of Var y \<Rightarrow> (False, case t of Var x \<Rightarrow> True | Fun g ts \<Rightarrow> ts = [] \<and> least g) | Fun f ss \<Rightarrow> case t of Var x \<Rightarrow> (True, True) | Fun g ts \<Rightarrow> let ff = (f, length ss); gg = (g, length ts) in if pr_strict ff gg then (True, True) else if pr_weak ff gg then lex_ext_unbounded kbo ss ts else (False, False) else (False, False))"
diff_add_assoc: fixes a :: "'a multiset"   and b :: "'a multiset"   and c :: "'a multiset" assumes "a \<subseteq># b" shows "c + b - a = c + (b - a)"
zariski_is_topological_space: shows "topological_space pi.spectrum pi.is_zariski_open"
inB: fixes x :: "nat"   and y :: "nat" assumes "P x y" shows "y \<in> B"
statement_63: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "q \<in> hyp2 \<union> S"   and "r \<in> hyp2 \<union> S"   and "is_K2_isometry J"   and "B\<^sub>\<real> cart2_pt p cart2_pt q cart2_pt r" shows "B\<^sub>\<real> cart2_pt (apply_cltn2 p J) cart2_pt (apply_cltn2 q J) cart2_pt (apply_cltn2 r J)"
monom_eq_1: shows "monom (1::'b) 0 = 1"
apsnd_def: fixes f :: "'d \<Rightarrow> 'e" shows "apsnd f = map_prod id f"
ufa_invar_def: fixes l :: "nat list" shows "ufa_invar l = (\<forall>i<length l. rep_of_dom (l, i) \<and> l ! i < length l)"
ccpo_Sup_least_inv': fixes C :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain (\<le>) C"   and "x < \<Squnion> C" obtains y :: "'b" where "y \<in> C" and "x < y"
c4: assumes "sqn rt ip = \<pi>\<^sub>2 r"   and "the (flag rt ip) = Aodv_Basic.inv" shows "P (rt(ip \<mapsto> addpre r (\<pi>\<^sub>7 (the (rt ip)))))"
word_set_set_same: fixes w :: "'b word"   and n :: "nat"   and x :: "bool"   and y :: "bool" shows "set_bit_class.set_bit (set_bit_class.set_bit w n x) n y = set_bit_class.set_bit w n y"
table_classes_Ext: shows "table_of Classes Ext = Some ExtCl"
seqE: fixes g :: "'b"   and f :: "'b" assumes "\<phi>.B.seq g f" obtains "\<phi>.B.arr f" and "\<phi>.B.arr g" and "\<phi>.B.dom g = \<phi>.B.cod f"
map\<^sub>0_simps_3: fixes a :: "'b" assumes "B.obj a" shows "trg\<^sub>C (\<tau>\<^sub>0 a) = \<tau>.G.map\<^sub>0 a"
frac_neg: fixes x :: "'b" shows "frac (- x) = (if x \<in> \<int> then 0::'b else (1::'b) - frac x)"
exec1_appendR: fixes P :: "instr list"   and c :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and c' :: "int \<times> (char list \<Rightarrow> int) \<times> int list"   and P' :: "instr list" assumes "P \<turnstile> c \<rightarrow> c'" shows "P @ P' \<turnstile> c \<rightarrow> c'"
simps_2: fixes A :: "'c set"   and S :: "(('b, 'c) fm list \<times> 'c) set"   and f :: "nat \<Rightarrow> ('b, 'c) fm list \<times> 'c"   and n :: "nat" shows "extend A S f (Suc n) = (if \<not> consistent A ({f n} \<union> extend A S f n) then extend A S f n else let used = A \<union> \<Union> (block_nominals ` ({f n} \<union> extend A S f n)) in {f n, witness (f n) used} \<union> extend A S f n)"
\<ww>_def: shows "\<ww> \<equiv> {cq.frac r s |r s. r \<in> I \<and> s \<in> R\<setminus>I}"
Jacobi_mult_right: fixes a :: "int"   and n :: "int"   and m :: "int" shows "Jacobi a (n * m) = Jacobi a n * Jacobi a m"
taylor_bigo_linear: fixes f :: "'a \<Rightarrow> 'a"   and x0 :: "'a"   and A :: "'a set" assumes "f field_differentiable at x0 within A" shows "(\<lambda>x. f x - f x0) \<in> O[at x0 within A](\<lambda>x. x - x0)"
set_diff_eq: fixes A :: "'e set"   and B :: "'e set" shows "A - B = {x \<in> A. x \<notin> B}"
iso_class_eqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VV.isomorphic f g" shows "VV.iso_class f = VV.iso_class g"
ctr_transfer_1: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "list_all2 R [] []"
AB_fv_disj: shows "fv\<^sub>s\<^sub>e\<^sub>t (set A) \<inter> fv\<^sub>s\<^sub>e\<^sub>t (set B) = {}"
stream_all2_shift1: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and xs1 :: "'b list"   and xs2 :: "'b stream"   and ys :: "'c stream" shows "stream_all2 P (xs1 @- xs2) ys = (\<exists>ys1 ys2. ys = ys1 @- ys2 \<and> list_all2 P xs1 ys1 \<and> stream_all2 P xs2 ys2)"
wf_design: shows "design (del_point p) (str_del_point_blocks p)"
invar_reachable: fixes I :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "is_invar I"   and "reachable s" shows "I s"
1: shows "set (transaction_fresh T) \<inter> bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T) = {}"
diag_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(diag x = diag y) = (x = y)"
LstSeqP_EQ: fixes H :: "fm set"   and s :: "tm"   and k :: "tm"   and y :: "tm"   and y' :: "tm" assumes "H \<turnstile> LstSeqP s k y"   and "H \<turnstile> HPair k y' IN s" shows "H \<turnstile> y EQ y'"
rec_power_def: shows "rec_power = Pr 1 (constn 1) (Cn 3 rec_mult [recf.id 3 0, recf.id 3 2])"
state_io_relD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "('a, 'b) state"   and s :: "'a"   and a :: "'b"   and s' :: "'a" assumes "state_io_rel P m"   and "run_state m s = (a, s')" shows "P s s'"
ide_in_iso_class: fixes f :: "'a \<times> 'a" assumes "VoV.ide f" shows "f \<in> VoV.iso_class f"
Sel_precondition: fixes k :: "nat"   and u :: "nat" shows "\<Z> k u \<subseteq> \<W> (\<P> k u) k"
simps_1: fixes uu :: "'a \<times> 'b \<times> 'a"   and uv :: "('a \<times> 'b \<times> 'a) list" shows "replace_by_UnPath [] uu uv = []"
splay_not_Leaf: fixes t :: "'b tree"   and a :: "'b" assumes "t \<noteq> \<langle>\<rangle>" shows "\<exists>l x r. splay a t = \<langle>l, x, r\<rangle>"
dlist_member_insert_eq: fixes y :: "'b"   and dl :: "'b dlist"   and x :: "'b" shows "member (DList.insert y dl) x = (x = y \<or> member dl x)"
dist_stereographic_finite: fixes M1 :: "riemann_sphere"   and m1 :: "complex"   and M2 :: "riemann_sphere"   and m2 :: "complex" assumes "stereographic M1 = of_complex m1"   and "stereographic M2 = of_complex m2" shows "dist_riemann_sphere' M1 M2 = 2 * cmod (m1 - m2) / (sqrt (1 + (cmod m1)\<^sup>2) * sqrt (1 + (cmod m2)\<^sup>2))"
tail_del_arc: fixes a :: "'b" shows "tail (del_arc a) = tail G"
x: shows "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> < (0::'a) \<and> \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> * x \<le> - Itm vs (x # bs) s \<or> (0::'a) < \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<and> - Itm vs (x # bs) s \<le> \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> * x"
homeomorphic_imp_surjective_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "homeomorphic_map X Y f" shows "f ` topspace X = topspace Y"
take_all: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "take n xs = xs"
free_hull_idem: fixes G :: "'b list set" shows "\<langle>\<langle>G\<rangle>\<^sub>F\<rangle>\<^sub>F = \<langle>G\<rangle>\<^sub>F"
at_right_to_top': fixes c :: "real" shows "at_right c = filtermap (\<lambda>x. c + inverse x) at_top"
card_fun: shows "CARD('a \<Rightarrow> 'b) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 \<or> CARD('b) = 1 then CARD('b) ^ CARD('a) else 0)"
extremum_uniqueI: fixes a :: "'a" assumes "top \<le> a" shows "a = top"
vars_monom_keys: fixes a :: "'b"   and m :: "nat \<Rightarrow>\<^sub>0 nat" assumes "a \<noteq> (0::'b)" shows "vars (MPoly_Type.monom m a) = keys m"
poly_mod_2_axioms: shows "poly_mod_2 (p ^ n)"
measure_semicontinuous_with_hausdist_explicit: fixes S :: "'a set"   and e :: "real" assumes "bounded S"   and "negligible (frontier S)"   and "0 < e" obtains d :: "real" where "0 < d" and "\<And>T. \<lbrakk>T \<in> lmeasurable; \<And>y. y \<in> T \<Longrightarrow> \<exists>x. x \<in> S \<and> dist x y < d\<rbrakk> \<Longrightarrow> Sigma_Algebra.measure lebesgue T < Sigma_Algebra.measure lebesgue S + e"
tail_lens_mwb: shows "mwb_lens tl\<^sub>L"
convex_epigraph_convex: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "convex S" shows "convex_on S f = convex (epigraph S f)"
route2match_def: fixes r :: "('a, 'b) routing_rule_scheme" shows "route2match r = \<lparr>iiface = ifaceAny, oiface = ifaceAny, src = (0, 0), dst = (pfxm_prefix (routing_match r), routing_prefix r), proto = ProtoAny, sports = (0, - 1), dports = (0, - 1)\<rparr>"
conforms_init_class_obj: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lenv"   and C :: "qtname"   and y :: "class" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "wf_prog G"   and "class G C = Some y"   and "\<not> inited C (globs s)" shows "(x, init_class_obj G C s)\<Colon>\<preceq>(G, L)"
assms: shows "t \<oplus> v \<prec>\<^sub>t t \<oplus> w"
LCons: fixes xs :: "'a llist"   and x :: "'a" assumes "lfinite xs"   and "P xs" shows "P (LCons x xs)"
divide_poly_main: fixes d :: "'a poly"   and lc :: "'a"   and r :: "'a poly"   and dr :: "nat"   and q :: "'a poly"   and n :: "nat"   and q' :: "'a poly" assumes "d \<noteq> 0"   and "lc = lead_coeff d"   and "degree (d * r) \<le> dr"   and "local.divide_poly_main lc q (d * r) d dr n = q'"   and "n = 1 + dr - degree d \<or> dr = 0 \<and> n = 0 \<and> d * r = 0" shows "q' = q + r"
igSubstIGAbsSTR_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igSubstIGAbsSTR MOD \<equiv> \<forall>ys y Y xs x X. (xs \<noteq> ys \<or> x \<noteq> y) \<and> igFresh MOD xs x Y \<longrightarrow> igSubstAbs MOD ys Y y (igAbs MOD xs x X) = igAbs MOD xs x (igSubst MOD ys Y y X)"
approx_def: shows "approx \<equiv> \<Sum>k<n. 2 * y ^ (2 * k + 1) / real (2 * k + 1)"
inject_2: fixes x2 :: "abort"   and y2 :: "abort" shows "(Rabort x2 = Rabort y2) = (x2 = y2)"
ps_def: shows "ps = sorted_list_of_set (prime_factors n)"
fdomain_vinsert: fixes a :: "V"   and b :: "V"   and r :: "V" shows "\<D>\<^sub>\<bullet> (vinsert [a, b]\<^sub>\<circ> r) = vinsert a (\<D>\<^sub>\<bullet> r)"
simps_17: fixes f1 :: "'a"   and f2 :: "qtname \<Rightarrow> 'a"   and f3 :: "qtname \<Rightarrow> 'a"   and f4 :: "ty \<Rightarrow> 'a"   and x2 :: "qtname" shows "(case IfaceT x2 of NullT \<Rightarrow> f1 | IfaceT x \<Rightarrow> f2 x | ClassT x \<Rightarrow> f3 x | ArrayT x \<Rightarrow> f4 x) = f2 x2"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" assumes "Labeling c s t f l"   and "Height_Bounded_Labeling_axioms c l" shows "Height_Bounded_Labeling c s t f l"
analz_Decrypt': fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "K \<in> symKeys"   and "Key K \<in> analz H" shows "X \<in> analz H"
multi_member_last: fixes x :: "'a" shows "x \<in># {#x#}"
simps_1: fixes red :: "bool list \<Rightarrow> bool"   and ds :: "bool list"   and T\<^sub>1 :: "tree"   and T\<^sub>2 :: "tree" shows "cutoff red ds (Branching T\<^sub>1 T\<^sub>2) = (if red ds then Leaf else Branching (cutoff red (ds @ [Left]) T\<^sub>1) (cutoff red (ds @ [Right]) T\<^sub>2))"
closed_Real_halfspace_Re_ge: fixes x :: "real" shows "closed (\<real> \<inter> {w. x \<le> Re w})"
distinct_3: fixes x1 :: "'f" shows "Node x1 \<noteq> NewExit"
mdqdb: shows "bound0 (MIR.disj (evaldjf (\<lambda>j. simpfm (subst0 (C j) (simpfm (minusinf (fst (chooset p)))))) [1..snd (snd (chooset p))]) (evaldjf (stage (fst (chooset p)) (snd (snd (chooset p)))) (fst (snd (chooset p)))))"
that_1: fixes i :: "'x" assumes "i \<in> Basis" shows "c \<bullet> i \<le> k \<bullet> i"
edge_begin_tup: fixes x :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort" shows "edge_begin x = fst (fst x)"
dictates_def: fixes i :: "indi"   and a :: "alt"   and b :: "alt" shows "i dictates a < b \<equiv> \<forall>P. P i a < P i b \<longrightarrow> F P a < F P b"
descartes_roots_test: fixes p :: "real poly"   and a :: "real"   and b :: "real" assumes "p \<noteq> 0"   and "a < b" shows "proots_count p {x. a < x \<and> x < b} \<le> descartes_roots_test a b p \<and> even (descartes_roots_test a b p - proots_count p {x. a < x \<and> x < b})"
\<omega>_sum_of_Cantor: fixes \<alpha>s :: "V list"   and ms :: "nat list" shows "\<omega>_sum (\<omega>_of_Cantor \<alpha>s ms) = Cantor_sum \<alpha>s ms"
translation_diff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "class.ab_group_add plus zero minus uminus" shows "plus a ` (s - t) = plus a ` s - plus a ` t"
coin_def: fixes n :: "nat"   and dc :: "nat option \<times> bool list"   and c :: "nat" assumes "dining_cryptographers_space n" shows "dining_cryptographers_space.coin n dc c = snd dc ! (c mod n)"
Guard: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and g :: "'f set"   and P :: "'f set"   and c :: "('f, 'e, 'g) com"   and Q :: "'f set"   and A :: "'f set"   and f :: "'g" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(g \<inter> P) Guard f g c Q,A"
integrate_insert_result: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message"   and s :: "('a, 'b option) insert_message list" assumes "consistent (M \<union> {Insert m})"   and "Insert m \<notin> M"   and "is_associated_string M s" shows "is_certified_associated_string (M \<union> {Insert m}) (integrate_insert m s (P m) (S m))"
bn_eqvt: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and bn :: "'b \<Rightarrow> atom set"   and p :: "perm"   and \<alpha> :: "'b" assumes "class.bn permute bn" shows "p \<bullet> bn \<alpha> = bn (permute p \<alpha>)"
that: fixes P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'"   and "(\<Psi>, P', Q') \<in> Rel" shows "thesis"
rep_eq: fixes x :: "'a lazy" shows "force x = Rep_lazy x"
cases: fixes x :: "('c \<Rightarrow> 'd) \<times> 'c list" obtains f :: "'c \<Rightarrow> 'd" where "xa__ = (fb__, [])"   | f :: "'c \<Rightarrow> 'd" and x :: "'c" and xs :: "'c list" where "xa__ = (f, x # xs)"
TESL_interp_stepwise_implies_coind_unfold: fixes master :: "clock"   and slave :: "clock"   and n :: "nat" shows "\<lbrakk> master implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = (\<lbrakk> master \<not>\<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<union> \<lbrakk> master \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> slave \<Up> n \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m) \<inter> \<lbrakk> master implies slave \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"
LIM_imp_LIM: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and a :: "'a"   and g :: "'a \<Rightarrow> 'c"   and m :: "'c" assumes "f \<midarrow>a\<rightarrow> l"   and "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)" shows "g \<midarrow>a\<rightarrow> m"
c: shows "common_reparam_exists (boundary circle_cube) (two_chain_vertical_boundary {rot_circle_cube})"
a_star_redu_OrR2_elim: fixes a :: "coname"   and M :: "trm"   and y :: "coname"   and R :: "trm" assumes "OrR2 <a>.M y \<longrightarrow>\<^sub>a* R" shows "\<exists>M'. R = OrR2 <a>.M' y \<and> M \<longrightarrow>\<^sub>a* M'"
psi_def: fixes i :: "nat"   and x :: "nat" shows "\<psi> i x \<equiv> Some (last (prefixes i x))"
in_ChainsD: fixes M :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "M \<in> Chains r"   and "x \<in> M"   and "y \<in> M" shows "(x, y) \<in> r \<or> (y, x) \<in> r"
IH_2: fixes env :: "('a, bool) mapping"   and t1 :: "'a ifex"   and t2 :: "'a ifex" shows "reduced (normif env s2 t1 t2) (Mapping.keys env)"
supp_transform: fixes a :: "'a"   and e :: "exp" shows "supp (transform a e) \<subseteq> supp e"
cdim_UNIV_basis_enum: shows "cdim UNIV = length canonical_basis"
progress_prefix_conv: fixes \<pi> :: "(char list \<times> 'b list) prefix"   and \<sigma> :: "(char list \<times> 'b list) trace"   and \<sigma>' :: "(char list \<times> 'b list) trace"   and \<phi> :: "'b MFOTL.formula" assumes "prefix_of \<pi> \<sigma>"   and "prefix_of \<pi> \<sigma>'" shows "Monitor.progress \<sigma> \<phi> (plen \<pi>) = Monitor.progress \<sigma>' \<phi> (plen \<pi>)"
C: fixes x :: "complex" assumes "x \<in> T" shows "cmod x \<le> C"
assms_2: fixes x :: "nat" assumes "k < x" shows "g x = \<zero>"
thm_relation_negation_1_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<nu>" shows "[\<lparr>F\<^sup>-,x\<^sup>P\<rparr> \<^bold>\<equiv> \<^bold>\<not>\<lparr>F,x\<^sup>P\<rparr> in v]"
assms_1: shows "\<forall>n\<le>m. psi n \<le> c * ln 2 * real n"
rule5'': fixes a :: "nat"   and b :: "nat" assumes "(a, b) \<in> \<E> g' \<and> (a, b) \<notin> \<E> g \<and> a \<in> set (removeNones vol) \<and> b \<in> set (removeNones vol)" shows "(a, b) \<in> \<E> f21"
min_gen_is_basis: fixes A :: "'c set" assumes "minimal A (\<lambda>S. S \<subseteq> carrier V \<and> gen_set S)" shows "basis A"
pfa_append_grow_def: shows "pfa_append_grow \<equiv> \<lambda>(a, n) x. pfa_capacity (a, n) \<bind> (\<lambda>l. (if l = n then array_grow a (l + 1) x else Array.upd n x a) \<bind> (\<lambda>a'. return (a', n + 1)))"
relcompp_relcomp_eq: fixes r :: "('a \<times> 'c) set"   and s :: "('c \<times> 'b) set" shows "(\<lambda>x xa. (x, xa) \<in> r) OO (\<lambda>x xa. (x, xa) \<in> s) = (\<lambda>x xa. (x, xa) \<in> r O s)"
assms: shows "(\<sigma>, ns) \<in> oreachable (opnet np p) S U"
r: fixes s :: "'a" assumes "s \<le> t"   and "0 < zcount va s" shows "False"
invariant_def: fixes A :: "'b set" shows "invariant A \<equiv> {F. Init F \<subseteq> A} \<inter> stable A"
smallest_ccdi_sets_Un: fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> smallest_ccdi_sets \<Omega> M"   and "B \<in> smallest_ccdi_sets \<Omega> M"   and "A \<inter> B = {}" shows "A \<union> B \<in> smallest_ccdi_sets \<Omega> M"
segment_unique1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec> b" shows "\<not> ord_equiv (Iod D (segment D b)) (Iod D (segment D a))"
mset_le_irrefl: fixes M :: "'k multiset"   and R :: "bool" assumes "M < M" shows "R"
intro: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "Network.push_precond c f l (u, v)" shows "push_effect_locale_axioms c f l u v"
lens_quotient_id: fixes X :: "'d \<Longrightarrow> 'e" assumes "weak_lens X" shows "X /\<^sub>L X = 1\<^sub>L"
the_f: shows "(THE f. \<forall>i. v $ i = (\<Sum>x\<in>UNIV. f x * Y $ x $ i)) = f"
assms_2: shows "\<forall>\<^sub>F x in at a. f x = g x"
blaschke_unit_disc_fix: fixes a :: "complex" assumes "cmod a < 1" shows "unit_disc_fix (blaschke a)"
assoc_int: fixes d :: "nat"   and i :: "int"   and r :: "char list"   and s :: "char list" shows "shows_prec d i r @ s = shows_prec d i (r @ s)"
p_res_ring_car: fixes k :: "nat" shows "carrier (residue_ring (p ^ k)) = {0..<p ^ k}"
equivP_def: fixes A :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "equivP A \<phi> \<equiv> reflP A \<phi> \<and> symP \<phi> \<and> transP \<phi>"
Keys_not_empty: fixes a :: "'a \<Rightarrow>\<^sub>0 'b"   and A :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "a \<in> A"   and "a \<noteq> 0" shows "Keys A \<noteq> {}"
is_extensional: fixes f :: "'c \<times> 'c" assumes "\<not> C.VV.arr f" shows "\<Phi> f = D.null"
simps_1: fixes S :: "nat \<Rightarrow> typ"   and n :: "nat" shows "$ S (FVar n) = mk_scheme (S n)"
countably_compact_eq_compact: fixes U :: "'b set" shows "countably_compact U = compact U"