hcju/isabelleps huggingface dataset:
adopt_node_pointers_preserved: fixes w :: "((_) heap, exception, unit) prog"   and parent :: "(_) object_ptr option"   and owner_document :: "(_) document_ptr"   and document_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> Shadow_DOM.adopt_node_locs parent owner_document document_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "object_ptr_kinds h = object_ptr_kinds h'"
Gateway_L6: fixes req :: "nat \<Rightarrow> reqType list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and a :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and d :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and k :: "nat"   and t :: "nat" assumes "Gateway req dt a stop lose d ack i vc"   and "\<forall>m\<le>k. req (t + m) \<noteq> [send]"   and "\<forall>j\<le>k. lose (t + j) = [False]"   and "ack t = [connection_ok]"   and "msg (Suc 0) req"   and "msg (Suc 0) stop"   and "msg (Suc 0) a"   and "ts lose" shows "\<forall>m\<le>k. ack (t + m) = [connection_ok]"
create_alt_def: shows "fresh_create = State (\<lambda>a. (next a, next a))"
S_transition_cases: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and P :: "'a"   and \<alpha>\<^sub>S :: "('c, 'b) S_action"   and P' :: "'a" assumes "nominal_ts satisfies transition"   and "nominal_ts.S_transition satisfies transition P \<langle>\<alpha>\<^sub>S,P'\<rangle>" obtains \<alpha> :: "'c" where "\<alpha>\<^sub>S = S_action.Act \<alpha>" and "transition P \<langle>\<alpha>,P'\<rangle>"   | \<phi> :: "'b" where "\<alpha>\<^sub>S = S_action.Pred \<phi>" and "P' = P" and "satisfies P \<phi>"
impl_def: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" shows "P \<longrightarrow> Q \<equiv> \<lambda>A. P A \<longrightarrow> Q A"
g0_1: shows "g0 \<in> borel_measurable Invariants"
swap_simps_2: fixes a :: "vrs"   and b :: "vrs" shows "[(a, b)] \<bullet> a = b"
slist_apply_msum_slist: fixes m :: "nat"   and xs :: "(nat, 'd) slist"   and ys :: "(nat, 'd) slist"   and i :: "nat" shows "slist_apply (msum_slist m xs ys) i = (if i < m then slist_apply xs i else slist_apply ys (i - m))"
g_def: shows "g = Cn 3 r_ifle [r_constn 2 2, Cn 3 r_nth [recf.Id 3 2, recf.Id 3 0], Cn 3 r_nth [recf.Id 3 2, recf.Id 3 0], recf.Id 3 1]"
in_set_subset: fixes t :: "('c, 'd) SkewBinomialTree"   and q :: "('c, 'd) SkewBinomialTree list" assumes "t \<in> set q" shows "tree_to_multiset t \<subseteq># queue_to_multiset q"
is_expansion_sin_ms_zero_exp: fixes b :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list"   and C :: "'a"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and Sin :: "'a"   and Cos :: "'a" assumes "basis_wf (b # basis)"   and "is_expansion (MS (MSLCons (C, 0) xs) f) (b # basis)"   and "((\<lambda>x. sin (eval C x)) expands_to Sin) basis"   and "((\<lambda>x. cos (eval C x)) expands_to Cos) basis" shows "is_expansion (MS (plus_ms_aux (scale_shift_ms_aux (Sin, 0) (cos_ms_aux' xs)) (scale_shift_ms_aux (Cos, 0) (sin_ms_aux' xs))) (\<lambda>x. sin (f x))) (b # basis)"
top_greatest: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and top :: "'b"   and a :: "'b" assumes "class.order_top less_eq less top" shows "less_eq a top"
BotL_canonical: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "\<bottom>, \<Gamma> \<Rightarrow> \<Delta>"
not_eq_order_implies_strict: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<noteq> b"   and "a \<preceq> b" shows "a \<prec> b"
is_singleton_def: fixes A :: "'g set" shows "is_singleton A = (\<exists>x. A = {x})"
tauPrefix_def: fixes P :: "('a, 'b, 'c) psi" shows "\<tau>.P \<equiv> THE P'. \<exists>x. x \<sharp> P \<and> P' = \<lparr>\<nu>x\<rparr>((nameTerm x\<lparr>\<lambda>*[] nameTerm x\<rparr>.\<zero>) \<parallel> nameTerm x\<langle>nameTerm x\<rangle>.P)"
hd_var: fixes x :: "nat"   and V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "x \<in> V" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Var x \<Rightarrow> marg_dens (V, V', \<Gamma>, \<delta>) x"
high_proc\<^sub>4_def: fixes s\<^sub>H :: "stmt" shows "high_proc\<^sub>4 s\<^sub>H \<equiv> \<forall>\<sigma> \<sigma>'. match \<sigma> \<Gamma>\<^sub>0 \<and> s\<^sub>H \<rhd> \<sigma> \<leadsto> Some \<sigma>' \<longrightarrow> the (\<sigma>' ''highOut'') = the (\<sigma> ''lowIn'') + the (\<sigma> ''highIn'')"
intros_5: shows "CK_nf (K \<circ> C)"
span_zero: shows "local.span {\<zero>\<^bsub>V\<^esub>} = {\<zero>\<^bsub>V\<^esub>}"
Lambert_W_eq_iff: fixes x :: "real"   and y :: "real" assumes "- exp (- 1) \<le> x"   and "- exp (- 1) \<le> y" shows "(Lambert_W x = Lambert_W y) = (x = y)"
simps: fixes a1 :: "'a tree"   and a2 :: "'a stream" shows "ipath a1 a2 = (\<exists>t steps t'. a1 = t \<and> a2 = steps \<and> tree.root t = shd steps \<and> t' |\<in>| cont t \<and> ipath t' (stl steps))"
simps: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" shows "rbtreeify_f n kvs = (if n = 0 then (Empty, kvs) else if n = 1 then case kvs of (k, v) # kvs' \<Rightarrow> (Branch R Empty k v Empty, kvs') else if n mod 2 = 0 then case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow> apfst (Branch B t1 k v) (rbtreeify_g (n div 2) kvs') else case rbtreeify_f (n div 2) kvs of (t1, (k, v) # kvs') \<Rightarrow> apfst (Branch B t1 k v) (rbtreeify_f (n div 2) kvs'))"
c1_2: fixes i :: "'a" assumes "i \<in> Basis" shows "continuous_on (S \<times> cbox a b) (\<lambda>(x, t). fx x t i)"
filterlim_mono: fixes f :: "'a \<Rightarrow> 'b"   and F2 :: "'b filter"   and F1 :: "'a filter"   and F2' :: "'b filter"   and F1' :: "'a filter" assumes "filterlim f F2 F1"   and "F2 \<le> F2'"   and "F1' \<le> F1" shows "filterlim f F2' F1'"
k: shows "k \<in> hom (free_Abelian_group T) (free_Abelian_group S)"
filter_for_uniflows_def: fixes E :: "('a \<times> 'a) list" shows "filter_for_uniflows E \<equiv> filter (\<lambda>e. (snd e, fst e) \<notin> set E) E"
subseq_mset_subseteq_mset: fixes xs :: "'b list"   and ys :: "'b list" assumes "subseq xs ys" shows "mset xs \<subseteq># mset ys"
diff_Suc_pos: fixes a :: "nat" assumes "0 < a - Suc 0" shows "0 < a"
norm_vec_def: fixes x :: "('a, 'b) Finite_Cartesian_Product.vec" shows "norm x = L2_set (\<lambda>i. norm (x $h i)) UNIV"
map_scale_def: fixes c :: "'b" shows "(\<cdot>) c = Poly_Mapping.map ((*) c)"
CONSTANT_case: fixes k :: "nat"   and l :: "nat list" shows "CONSTANT k l < ack k (sum_list l)"
rel_spmf_bind_reflI: fixes p :: "'d spmf"   and P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e spmf"   and g :: "'d \<Rightarrow> 'f spmf" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> rel_spmf P (f x) (g x)" shows "rel_spmf P (p \<bind> f) (p \<bind> g)"
S4bReturn_enabled: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $(S4 rmhist p \<and> ires!p \<noteq> #NotAResult) \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> $Enabled (<MemReturn rmCh ires p>_(m p))"
eval_primfuns_pos: fixes fs :: "(primfun \<times> real) list" shows "\<forall>\<^sub>F x in at_top. 0 < eval_primfuns fs x"
vpairs_eqE: fixes r :: "V" assumes "vpairs r = r" obtains "\<And>x. x \<in>\<^sub>\<circ> r \<Longrightarrow> \<exists>a b. x = \<langle>a, b\<rangle>"
hyps: assumes "set Us \<subseteq> carrier R"   and "\<And>Ks. \<lbrakk>set Ks \<subseteq> K; combine Ks Us = \<zero>\<rbrakk> \<Longrightarrow> set (take (length Us) Ks) \<subseteq> {\<zero>}" shows "local.independent K Us"
R_linear: fixes x :: "real"   and y :: "real" assumes "\<not> x \<le> y" shows "y \<le> x"
gso_scalar_zero: fixes k :: "nat"   and i :: "nat" assumes "k < length (map of_int_hom.vec_hom fs)"   and "i < k" shows "fs.gs.gso k \<bullet> map of_int_hom.vec_hom fs ! i = 0"
lcm_A: fixes a :: "'a"   and b :: "'a" shows "a * (a r\<rightarrow> b) = b * (b r\<rightarrow> a)"
Vinv3_event_pres: fixes r :: "nat"   and v_f :: "process \<Rightarrow> val option"   and d_f :: "process \<Rightarrow> val option" shows "{Vinv3 \<inter> Vinv1} v_round r v_f d_f {> Vinv3}"
s_inter_1: shows "finite (s \<inter> {0..1})"
intro: fixes R :: "('a, 'b) ring_scheme" assumes "ring_functions R"   and "cring R" shows "cring_functions R"
freshAlphaPerm: fixes p :: "(name \<times> name) list"   and Xs :: "name set"   and Ys :: "name set"   and xvec :: "name list" assumes "set p \<subseteq> Xs \<times> Ys"   and "Xs \<sharp>* xvec"   and "Ys \<sharp>* xvec" shows "xvec \<sharp>* p"
distinct_list_of_dlist: fixes dxs :: "'b set_dlist" assumes "ID CEQ('b) \<noteq> None" shows "distinct (list_of_dlist dxs)"
sorted_list_of_set_1: fixes A :: "'a word set" assumes "finite A" shows "set (signed.sorted_list_of_set A) = A"
subdag_neq: fixes t :: "dag"   and s :: "dag" assumes "subdag t s" shows "t \<noteq> s"
assms: shows "exact_seq (G1 # l, q)  \<longlongrightarrow>\<^bsub>g1\<^esub> G2  \<longlongrightarrow>\<^bsub>g2\<^esub> G3"
simps_2: fixes n :: "nat"   and s :: "dB"   and t :: "dB"   and k :: "nat" shows "liftn n (s \<degree> t) k = liftn n s k \<degree> liftn n t k"
i_expand_0: fixes f :: "nat \<Rightarrow> 'b message_af" shows "f \<odot> 0 = (\<lambda>n. NoMsg)"
ordLeq_ordLess_trans: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and r'' :: "('d \<times> 'd) set" assumes "r \<le>o r'"   and "r' <o r''" shows "r <o r''"
dk': fixes k :: "nat" assumes "k < n" shows "adjoint (M k) * M k - adjoint (M k) * P k * M k \<in> carrier_mat d d"
invertible_matrix_mult_right_rank: fixes Q :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'b) vec, 'c) vec" assumes "invertible Q" shows "rank (A ** Q) = rank A"
dagger_of_dagger_is_id: fixes M :: "complex Matrix.mat" shows "M\<^sup>\<dagger>\<^sup>\<dagger> = M"
Casts_Casts_eq_result: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and Ts :: "ty list"   and vs :: "val list"   and vs' :: "val list"   and ws' :: "val list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and Ts' :: "ty list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option" assumes "wf_C_prog P"   and "P \<turnstile> Ts Casts vs to vs' "   and "P \<turnstile> Ts Casts vs to ws' "   and "\<forall>T\<in>set Ts. is_type P T"   and "P,E \<turnstile> es [::] Ts'"   and "P \<turnstile> Ts' [\<le>] Ts"   and "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>map Val vs,(h, l)\<rangle>"   and "P,E \<turnstile> s \<surd>" shows "vs' = ws'"
2: fixes f :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "f \<in> F'"   and "t \<in> keys (q f * f)" shows "deg_pm t = poly_deg p"
convergent_bounded_family: fixes A :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>x. x \<in> A \<Longrightarrow> bounded (range (\<lambda>n. f n x))"   and "countable A" obtains k :: "nat \<Rightarrow> nat" where "strict_mono k" and "\<And>x. x \<in> A \<Longrightarrow> convergent (\<lambda>n. f (k n) x)"
mkarr_eqI: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term"   and g :: "'b local.term" assumes "free_monoidal_category C"   and "monoidal_language.Arr C f \<and> monoidal_language.Arr C g \<and> monoidal_language.Dom C f = monoidal_language.Dom C g \<and> monoidal_language.Cod C f = monoidal_language.Cod C g"   and "monoidal_language.Diagonalize C f = monoidal_language.Diagonalize C g" shows "free_monoidal_category.mkarr C f = free_monoidal_category.mkarr C g"
impl_safe: fixes loc1 :: "'loc"   and t :: "'t"   and s :: "'sum"   and loc2 :: "'loc" assumes "0 < zcount (c_imp c0 loc1) t"   and "s \<in>\<^sub>A path_summary loc1 loc2" shows "\<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc2) \<and> t' \<le> results_in t s"
Yield: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d"   and x :: "'c"   and s' :: "'d" assumes "g s = Yield x s'" shows "s \<in> productive_on g"
lb_mem: fixes h :: "nat \<Rightarrow> 'b"   and A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "nat" assumes "h \<in> mbs.BAD A P" shows "minimal_element.lexmin (\<lambda>x y. size x < size y) A (mbs.BAD A P) i \<in> ith (eq_upto (mbs.BAD A P) (minimal_element.lexmin (\<lambda>x y. size x < size y) A (mbs.BAD A P)) i) i"
ps1_sub: shows "snd ` set ps1 \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"
times_float_def: fixes a :: "('c, 'd) IEEE.float"   and b :: "('c, 'd) IEEE.float" shows "a * b = fmul To_nearest a b"
stk_esl_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and mxs :: "nat" shows "stk_esl P mxs \<equiv> upto_esl mxs (SemiType.esl P)"
left_MPair: fixes X :: "msg"   and Y :: "msg" shows "left (MPair X Y) = X"
select_convs_2: fixes ok :: "bool"   and more :: "'c" shows "alpha_d.more \<lparr>ok = ok, \<dots> = more\<rparr> = more"
nodeslist_not_Null: shows "nodeslist \<noteq> Null"
assoc: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd"   and c :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<odot> b \<odot> c = a \<odot> (b \<odot> c)"
l2product_pos_le: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S" shows "0 \<le> l2product S f f"
of_bits_of: fixes bits_of :: "'a \<Rightarrow> bool list"   and of_bits :: "bool list \<Rightarrow> 'a"   and a :: "'a" assumes "class.bit_representation bits_of of_bits" shows "of_bits (bits_of a) = a"
IH_2: fixes i :: "int"   and j :: "int" assumes "inorder t2 = [i..j]"   and "i \<le> j + 1" shows "fst ` leaves i t2 = {i..j + 1}"
unit_factor_normalize: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "a \<noteq> zero" shows "unit_factor (normalize a) = one"
coboundedI1: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c" shows "a \<cdot> b \<le> c"
topcontinuous_at_atin: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "topcontinuous_at X Y f x = (x \<in> topspace X \<and> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> limitin Y f (f x) (atin X x))"
splitAt_subset_ab: fixes a :: "'a list"   and b :: "'a list"   and ram :: "'a"   and vs :: "'a list" assumes "(a, b) = splitAt ram vs" shows "set a \<subseteq> set vs \<and> set b \<subseteq> set vs"
f0a: fixes w :: "complex" assumes "w \<in> M - {z}" shows "f w \<noteq> 0 \<and> f w \<noteq> a"
measurable_prob_algebraI: fixes N :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and M :: "'d measure" assumes "\<And>x. x \<in> space N \<Longrightarrow> prob_space (f x)"   and "f \<in> N \<rightarrow>\<^sub>M subprob_algebra M" shows "f \<in> N \<rightarrow>\<^sub>M prob_algebra M"
assms_1: shows "safe_distance_4r < s\<^sub>o - s\<^sub>e"
bij: fixes y :: "'a" assumes "y \<in> V" shows "bij (blinfun_apply (invf o\<^sub>L f' (g y + x0)))"
apply_swap_same: fixes c :: "'a"   and a :: "'a"   and b :: "'a" assumes "c \<noteq> a"   and "c \<noteq> b" shows "\<langle>a \<leftrightarrow> b\<rangle> \<langle>$\<rangle> c = c"
assert_def: fixes P :: "'a \<Rightarrow> bool"   and m :: "'m" shows "assert P m = bind m (\<lambda>x. if P x then return x else fail)"
chine_composite: shows "chine = \<langle>\<mu>.chine \<cdot> \<p>\<^sub>1[\<mu>.leg0, \<nu>.leg1] \<lbrakk>\<mu>.cod.leg0, \<nu>.cod.leg1\<rbrakk> \<nu>.chine \<cdot> \<p>\<^sub>0[\<mu>.leg0, \<nu>.leg1]\<rangle>"
fun_upd_finfun: fixes y :: "'d \<Rightarrow> 'e"   and a :: "'d"   and b :: "'e" shows "(y(a := b) \<in> finfun) = (y \<in> finfun)"
get_scdom_component_contains_get_dom_component: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and sc :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM_SC_DOM_Components.get_scdom_component ptr \<rightarrow>\<^sub>r sc"   and "ptr' \<in> set sc" obtains c :: "(_) object_ptr list" where "h \<turnstile> Shadow_DOM_DOM_Components.get_dom_component ptr' \<rightarrow>\<^sub>r c" and "set c \<subseteq> set sc"
eucl_of_list_inner: fixes i :: "'a"   and xs :: "real list" assumes "i \<in> Basis"   and "length xs = DIM('a)" shows "eucl_of_list xs \<bullet> i = xs ! index Basis_list i"
invertible_left_inverse2: fixes u :: "'a"   and v :: "'a" assumes "comm.multiplicative.invertible u"   and "u \<in> R"   and "v \<in> R" shows "comm.multiplicative.inverse u \<cdot> (u \<cdot> v) = v"
ap_spec_def: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "ap_spec ap = (\<forall>gs bs ps hs data. set (ap gs bs ps hs data) \<subseteq> set ps \<union> set hs \<times> (set gs \<union> set bs \<union> set hs) \<and> (\<forall>B d m. \<forall>h\<in>set hs. \<forall>g\<in>set gs \<union> set bs \<union> set hs. dickson_grading d \<longrightarrow> set gs \<union> set bs \<union> set hs \<subseteq> B \<longrightarrow> fst ` B \<subseteq> dgrad_p_set d m \<longrightarrow> set ps \<subseteq> set bs \<times> (set gs \<union> set bs) \<longrightarrow> unique_idx (gs @ bs @ hs) data \<longrightarrow> is_Groebner_basis (fst ` set gs) \<longrightarrow> h \<noteq> g \<longrightarrow> fst h \<noteq> 0 \<longrightarrow> fst g \<noteq> 0 \<longrightarrow> (\<forall>a b. (a, b) \<in>\<^sub>p set (ap gs bs ps hs data) \<longrightarrow> fst a \<noteq> 0 \<longrightarrow> fst b \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)) \<longrightarrow> (\<forall>a b. a \<in> set gs \<union> set bs \<longrightarrow> b \<in> set gs \<union> set bs \<longrightarrow> fst a \<noteq> 0 \<longrightarrow> fst b \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)) \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)) \<and> (\<forall>B d m h g. dickson_grading d \<longrightarrow> set gs \<union> set bs \<union> set hs \<subseteq> B \<longrightarrow> fst ` B \<subseteq> dgrad_p_set d m \<longrightarrow> set ps \<subseteq> set bs \<times> (set gs \<union> set bs) \<longrightarrow> (set gs \<union> set bs) \<inter> set hs = {} \<longrightarrow> unique_idx (gs @ bs @ hs) data \<longrightarrow> is_Groebner_basis (fst ` set gs) \<longrightarrow> h \<noteq> g \<longrightarrow> fst h \<noteq> 0 \<longrightarrow> fst g \<noteq> 0 \<longrightarrow> (h, g) \<in> set ps -\<^sub>p set (ap gs bs ps hs data) \<longrightarrow> (\<forall>a b. (a, b) \<in>\<^sub>p set (ap gs bs ps hs data) \<longrightarrow> (a, b) \<in>\<^sub>p set hs \<times> (set gs \<union> set bs \<union> set hs) \<longrightarrow> fst a \<noteq> 0 \<longrightarrow> fst b \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst a) (fst b)) \<longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst h) (fst g)))"
assms_1: shows "\<forall>\<^sub>F x in F. norm (f x) \<le> Re (g x)"
mIH: fixes val :: "bool" shows "val_ifex (ifex_ite (restrict_top i a val) (restrict_top t a val) (restrict_top e a val)) ass = bf_ite (val_ifex (restrict_top i a val)) (val_ifex (restrict_top t a val)) (val_ifex (restrict_top e a val)) ass"
inf_bot_left: fixes x :: "'b upred" shows "(false \<and> x) = false"
as_raw_invar_def: fixes s :: "'c array \<times> nat" shows "as_raw_invar s \<equiv> snd s \<le> array_length (fst s)"
w_def: fixes x :: "real" shows "w x = \<psi> x - \<phi> x"
proots_card_uball_eq: fixes r :: "real"   and p :: "complex poly"   and z0 :: "complex" assumes "0 < r" shows "card (proots_within p (ball z0 r)) = card (proots_within (p \<circ>\<^sub>p [:z0, complex_of_real r:]) (ball 0 1))"
assms: shows "(red F)\<^sup>*\<^sup>* (p - q) r"
finite_measure_compl: fixes M :: "'b measure"   and S :: "'b set" assumes "finite_measure M"   and "S \<in> sets M" shows "Sigma_Algebra.measure M (space M - S) = Sigma_Algebra.measure M (space M) - Sigma_Algebra.measure M S"
smult_vec_def: fixes a :: "'b"   and v :: "'b vec" shows "a \<cdot>\<^sub>v v \<equiv> vec (dim_vec v) (\<lambda>i. a * v $ i)"
lower_eq_zero_iff': fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(lower p v = 0) = (p = 0 \<or> v \<preceq>\<^sub>t tt p)"
rep_eq: fixes x :: "('c, 'b) comp_fun_commute" shows "set_fold_cfc x = Finite_Set.fold (comp_fun_commute_apply x)"
encode12_inv2: fixes i :: "nat"   and j :: "nat" assumes "i < d1"   and "j < d2" shows "encode2 (encode12 (i, j)) = j"
SET_CR4: fixes evs4 :: "event list"   and NCA :: "nat"   and KC1 :: "nat"   and i :: "nat"   and EKi :: "nat"   and C :: "agent"   and NC2 :: "nat"   and X :: "agent" assumes "evs4 \<in> set_cr"   and "Nonce NCA \<notin> used evs4"   and "KC1 \<in> symKeys"   and "Gets (CA i) (EXHcrypt KC1 EKi \<lbrace>Agent C, Nonce NC2\<rbrace> (Pan (pan X))) \<in> set evs4" shows "Says (CA i) C \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC2, Nonce NCA\<rbrace>, cert (CA i) (pubEK (CA i)) onlyEnc (priSK RCA), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace> # evs4 \<in> set_cr"
sorted: fixes f :: "'a list \<Rightarrow> 'a"   and xss :: "'a list list" shows "sorted (map f (sort1 f xss))"
multiset_of_em_conv_nth: fixes l :: "'b list"   and x :: "'b"   and m' :: "'b multiset" shows "(mset l = {#x#} + m') = (\<exists>i<length l. x = l ! i \<and> m' = mset l - {#x#})"
learn_bc_def: fixes \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" shows "learn_bc \<psi> U s \<equiv> environment \<psi> U s \<and> (\<forall>f\<in>U. \<exists>n\<^sub>0. \<forall>n\<ge>n\<^sub>0. \<psi> (the (s (f \<triangleright> n))) = f)"
if_SE_split_asm: fixes \<sigma> :: "'a"   and P :: "'a \<Rightarrow> bool"   and B\<^sub>1 :: "'a \<Rightarrow> (bool \<times> 'a) option"   and B\<^sub>2 :: "'a \<Rightarrow> (bool \<times> 'a) option" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2) = (P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>1) \<or> \<not> P \<sigma> \<and> (\<sigma> \<Turnstile> B\<^sub>2))"
DataRefinement_mono: fixes T :: "'e \<Rightarrow> 'f"   and S :: "'e \<Rightarrow> 'f"   and R :: "'f \<Rightarrow> 'g"   and R' :: "'e \<Rightarrow> 'h"   and S' :: "'h \<Rightarrow> 'g" assumes "T \<sqsubseteq> S"   and "mono R"   and "DataRefinement S R R' S'" shows "DataRefinement T R R' S'"
IH: fixes j :: "nat"   and i :: "nat" assumes "j = i + 2 ^ k"   and "i < min (length (drop (2 ^ k) xs)) (2 ^ (k + 1))" shows "braun_list (brauns (Suc k) (drop (2 ^ k) xs) ! i) (take_nths j (Suc k) xs)"
n': fixes n' :: "nat" assumes "n' \<in> {0<..<n}" shows "fds_nth g n' = (0::'a)"