hcju/isabelleps huggingface dataset:
poly_eval_times: fixes a :: "'d \<Rightarrow> 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_eval a (p * q) = poly_eval a p * poly_eval a q"
admissible_heap: fixes P :: "'a \<Rightarrow> heap \<Rightarrow> heap \<Rightarrow> 'b \<Rightarrow> bool" shows "heap.admissible (\<lambda>f. \<forall>x h h' r. effect (f x) h h' r \<longrightarrow> P x h h' r)"
NS_trans: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and u :: "('f, 'b) Term.term" assumes "NS s t"   and "NS t u" shows "NS s u"
pseudominimal_map: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "OrderingSetIso less_eq less less_eq' less' P f"   and "ordering.has_bottom less_eq P"   and "ordering.pseudominimal_in less_eq less P x" shows "ordering.pseudominimal_in less_eq' less' (f ` P) (f x)"
hoaret_sound': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>,{}\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
nf: shows "nf \<in> nnfis (np f) M"
freshAll_psubstAll: fixes P :: "('d, 'e, 'varSort, 'var, 'f) param"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and z :: "'var"   and zs :: "'varSort"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodPar P" shows "(good X \<longrightarrow> z \<in> varsOf P \<longrightarrow> (\<forall>rho\<in>envsOf P. fresh zs z X #[rho] = (\<forall>ys y. fresh ys y X \<or> freshImEnvAt zs z rho ys y))) \<and> (goodAbs A \<longrightarrow> z \<in> varsOf P \<longrightarrow> (\<forall>rho\<in>envsOf P. freshAbs zs z A $[rho] = (\<forall>ys y. freshAbs ys y A \<or> freshImEnvAt zs z rho ys y)))"
top_ccsubspace_def: shows "\<top> \<equiv> Abs_clinear_space UNIV"
that: fixes xs1 :: "'a list"   and xs2 :: "'a list" assumes "xs = xs1 @ xs2"   and "subseq xs1 ys"   and "subseq xs2 zs" shows "thesis"
Fun_eval: 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 "Fun (hfsetcat.eval b c) = (\<lambda>x\<^bold>\<in>local.set (HF'.prod (hfsetcat.exp b c) b). UP (happ (hfst (DOWN x)) (hsnd (DOWN x))))"
spec_ops_oid_unique: fixes op_list :: "('c \<times> 'd) list"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and op1 :: "'d"   and op2 :: "'d" assumes "spec_ops op_list deps"   and "(oid, op1) \<in> set op_list"   and "(oid, op2) \<in> set op_list" shows "op1 = op2"
ide_char: fixes a :: "'d" shows "Fop.Aop.ide a = Dop.ide a"
affine_hull_eq: fixes s :: "'a set" shows "(affine hull s = s) = affine s"
disjAll_def: fixes \<Delta>s :: "('i \<Rightarrow> 'j \<Rightarrow> 'k \<Rightarrow> 'l \<Rightarrow> bool) set"   and s :: "'i"   and vl :: "'j"   and s1 :: "'k"   and vl1 :: "'l" shows "disjAll \<Delta>s s vl s1 vl1 \<equiv> \<exists>\<Delta>\<in>\<Delta>s. \<Delta> s vl s1 vl1"
select_index_def: fixes n :: "nat"   and i :: "nat"   and j :: "nat" shows "select_index n i j \<equiv> i \<le> n - 1 \<and> j \<le> 2 ^ n - 1 \<and> 2 ^ (n - 1 - i) \<le> j mod 2 ^ (n - i)"
R24_3: shows "pmf (sds R24) d = 1 - pmf (sds R24) c"
in_hhom_def: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" shows "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>B b\<guillemotright> \<equiv> B.arr \<mu> \<and> src \<mu> = a \<and> trg \<mu> = b"
rep_def: fixes r :: "'b \<Rightarrow>\<^sub>0 'a" shows "rep r = (\<Sum>v\<in>keys r. lookup r v *s v)"
interleave_cont_conc: fixes a :: "'b list"   and b :: "'b list" shows "a @ b \<in> a \<otimes> b"
zero_flow_simp: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<notin> E" shows "f' (u, v) = (0::'a)"
not_None_eq: fixes x :: "'d option" shows "(x \<noteq> None) = (\<exists>y. x = \<lfloor>y\<rfloor>)"
Rep_hierauto_tuple: fixes HA :: "('b, 'c, 'a) hierauto" shows "Rep_hierauto HA = (HAInitValue HA, SAs HA, HAEvents HA, CompFun HA)"
inject_4: fixes x6 :: "'a fm"   and y6 :: "'a fm" shows "(Neg x6 = Neg y6) = (x6 = y6)"
assms_4: shows "h \<turnstile> create_document \<rightarrow>\<^sub>r result"
content_unit: shows "content (cbox (0::'c) One) = 1"
blinfun_of_cblinfun_zero: shows "blinfun_of_cblinfun 0 = 0"
normal_imp_commuting: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a"   and y :: "'a" assumes "A \<lhd> G"   and "B \<lhd> G"   and "A \<inter> B \<subseteq> {\<one>}"   and "x \<in> A"   and "y \<in> B" shows "x \<otimes> y = y \<otimes> x"
SeqCTermP_graph_aux_def: shows "SeqCTermP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>SeqCTermP_aux l s k sl m n sm sn vf t. x1 = (vf, s, k, t) \<and> x2 = LstSeqP s k t AND All2 l (SUCC k) (SyntaxN.Ex sl (HPair (Var l) (Var sl) IN s AND (Var sl EQ Zero OR (if vf then VarP (Var sl) else Fls) OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sn (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (Var sm) IN s AND HPair (Var n) (Var sn) IN s AND Var sl EQ Q_Eats (Var sm) (Var sn)))))))) \<and> atom l \<sharp> (s, k, sl, m, n, sm, sn) \<and> atom sl \<sharp> (s, m, n, sm, sn) \<and> atom m \<sharp> (s, n, sm, sn) \<and> atom n \<sharp> (s, sm, sn) \<and> atom sm \<sharp> (s, sn) \<and> atom sn \<sharp> s)"
blaschke_real_preserve_x_axis: fixes a :: "complex"   and z :: "complex_homo" assumes "is_real a"   and "cmod a < 1" shows "(moebius_pt (blaschke a) z \<in> circline_set x_axis) = (z \<in> circline_set x_axis)"
assms_7: shows "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> Xout i \<inter> Xout j = {}"
hyps_1: shows "g \<turnstile> m'-ms\<rightarrow>m"
b: shows "(inter_aform_plane optns (eucl_of_list_aform xi) (Sctn (eucl_of_list (normal si)) (pstn si)), inter_sctn_spec (Affine (eucl_of_list_aform xi)) (Sctn (eucl_of_list (normal si)) (pstn si))) \<in> \<langle>br Affine top\<rangle>nres_rel"
card_UNIV_expr'_def: shows "card_UNIV_class.card_UNIV = Phantom('a expr') 0"
H1: shows "\<not> (X = Y \<or> X = Z \<or> Y = Z)"
Max_wrt_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "Max_wrt R = Max_wrt_among R UNIV"
p'_1: shows "G.apath (head G a) p' (tail G a)"
observable_path_unique: fixes io :: "('d \<times> 'e) list"   and M :: "('d, 'e, 'f) FSM"   and q :: "'f"   and tr1 :: "'f list"   and tr2 :: "'f list" assumes "io \<in> LS M q"   and "observable M"   and "path M (io || tr1) q"   and "length io = length tr1"   and "path M (io || tr2) q"   and "length io = length tr2" shows "tr1 = tr2"
convex_on_subset: fixes t :: "'c set"   and f :: "'c \<Rightarrow> real"   and S :: "'c set" assumes "convex_on t f"   and "S \<subseteq> t" shows "convex_on S f"
B: fixes C :: "ennreal" assumes "C < \<infinity>" shows "\<exists>g\<in>{g \<in> space\<^sub>N (\<LL> q M). Norm (\<LL> q M) g \<le> 1}. C \<le> ennreal (LINT x|M. f x * g x)"
coreflexive_comp_top_inf_one: fixes x :: "'a" assumes "coreflexive x" shows "x * top \<sqinter> (1::'a) = x"
assms: shows "mcont Sup (\<le>) lSup (\<sqsubseteq>) (\<lambda>x. f x (eSuc x))"
main: fixes "is" :: "nat list" assumes "distinct is" shows "eliminate_entries_rec B I (map (\<lambda>i. (- A $$ (i, J), i)) (filter (\<lambda>i. i \<noteq> I) is)) = one_go B is"
encode1_def: fixes i :: "nat" shows "encode1 i = digit_decode dims1 (nths (digit_encode dims i) vars)"
HomFtorMapsTo: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>" shows "Hom\<^bsub>C\<^esub>[X,f] maps\<^bsub>SET\<^esub> Hom\<^bsub>C\<^esub> X dom\<^bsub>C\<^esub> f to Hom\<^bsub>C\<^esub> X cod\<^bsub>C\<^esub> f"
prod_as_plus: fixes X :: "'a \<Longrightarrow> 'c"   and Y :: "'b \<Longrightarrow> 'd" shows "X \<times> Y = X ;\<^sub>L fst\<^sub>L +\<^sub>L Y ;\<^sub>L snd\<^sub>L"
assms_3: fixes ss :: "'s"   and ce :: "'s multiset"   and ss' :: "'s"   and csp :: "'s multiset" assumes "c = add_mset ss ce"   and "ce' = add_mset ss' (csp + ce)"   and "((ss, ce), e, ss', add_mset s' (csp + ce)) \<in> gtrs" shows "P"
Suc_diff_eq_diff_pred: fixes n :: "nat"   and m :: "nat" assumes "0 < n" shows "Suc m - n = m - (n - 1)"
hom_1_iff: fixes x :: "'b" shows "(to_fract x = 1) = (x = (1::'b))"
f: shows "(f \<longlongrightarrow> f l) (at l within s)"
right_total_mod_ring_rel64: shows "right_total mod_ring_rel64"
init_done: fixes C :: "qtname"   and s :: "state"   and G :: "prog" assumes "initd C s" shows "G\<turnstile>s \<midarrow>Init C\<rightarrow> s"
f2: shows "\<forall>B Ba f fa. (B \<noteq> Ba \<or> bb fa f Ba \<in> Ba \<and> f (bb fa f Ba) \<noteq> fa (bb fa f Ba)) \<or> f ` B = fa ` Ba"
assms: shows "b_e_type_checker \<C> es (tn _> tm)"
conga_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a"   and E :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.CongA Bet Cong C B A F E D"
trg_src: fixes \<mu> :: "'c" assumes "C.arr \<mu>" shows "trg\<^sub>C (src\<^sub>C \<mu>) = src\<^sub>C \<mu>"
assms_1: shows "integrable M (\<lambda>x. f x * g x)"
simps_2: fixes final :: "'a \<Rightarrow> bool"   and s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and ct :: "'c conditional_action"   and cts :: "'c conditional_action list" shows "final_thread.cond_action_oks final s t (ct # cts) = (final_thread.cond_action_ok final s t ct \<and> final_thread.cond_action_oks final s t cts)"
intro: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option"   and \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> Sec option"   and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"   and "\<forall>x\<in>dom \<Gamma>'. \<Gamma>' x = Some High"   and "\<turnstile> \<Gamma>\<^sub>1 {c\<^sub>1} \<Gamma>'"   and "\<turnstile> \<Gamma>\<^sub>2 {c\<^sub>2} \<Gamma>'"   and "mds_consistent mds \<Gamma>\<^sub>1"   and "mds_consistent mds \<Gamma>\<^sub>2" shows "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>"
size_correct: fixes \<alpha> :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar :: "'a \<Rightarrow> bool"   and size :: "'a \<Rightarrow> nat"   and s :: "'a" assumes "map_size \<alpha> invar size"   and "invar s" shows "size s = card (dom (\<alpha> s))"
tT: shows "t \<in> {t \<in> T. tLength t = n}"
less_irrefl: fixes xa :: "'o"   and x :: "'a" shows "\<not> lt xa x x"
presOp_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'e, 'opSym, 'var, 'c, 'd, 'f) Recursion.model_scheme" shows "presOp h hA MOD \<equiv> \<forall>delta inp binp. wlsInp delta inp \<and> wlsBinp delta binp \<longrightarrow> h (Op delta inp binp) = gOp MOD delta inp (lift h inp) binp (lift hA binp)"
exhaust: fixes y :: "'a Literal" obtains x1 :: "'a" where "y = Pos x1"   | x2 :: "'a" where "y = Neg x2"
n1_is: shows "N1 = N \<union> {(C, L)}"
lstrict_prefix_def: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lstrict_prefix xs ys \<equiv> lprefix xs ys \<and> xs \<noteq> ys"
assms_2: shows "\<not> bounded {(f ^^ n) x |n. True}"
map_commutes: fixes Q :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "concat (map (\<lambda>n. if Q (f n) then [f n] else []) xs) = filter Q (map f xs)"
induct: fixes x :: "'a list"   and T :: "('a, 'b) TS_scheme"   and P :: "'a list \<Rightarrow> bool" assumes "x \<in> beh T"   and "P []"   and "\<And>s. s \<in> init T \<Longrightarrow> P [s]"   and "\<And>s b t. \<lbrakk>s # b \<in> beh T; P (s # b); (s, t) \<in> TS.trans T\<rbrakk> \<Longrightarrow> P (t # s # b)" shows "P x"
add_set_def: fixes A :: "rat set"   and B :: "rat set" shows "add_set A B = {w. \<exists>x\<in>A. \<exists>y\<in>B. w = x + y}"
bs_delete_def: fixes i :: "nat"   and s :: "integer" shows "bs_delete i s \<equiv> set_bit_class.set_bit s i False"
Macaulay_mat_def: fixes ps :: "(('x option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" shows "pp_pm.extended_ord.punit.Macaulay_mat ps = polys_to_mat (pp_pm.extended_ord.punit.Keys_to_list ps) ps"
roofed_greater: fixes S :: "'a set"   and G :: "('a, 'b) graph_scheme"   and B :: "'a set" shows "S \<subseteq> roofed_gen G B S"
h2: shows "MPoly_Type.degree p var = 2 \<longrightarrow> p = isolate_variable_sparse p var 0 + isolate_variable_sparse p var 1 * Var var + isolate_variable_sparse p var 2 * (Var var)\<^sup>2"
deg_le_sect_alt: fixes X :: "'b set"   and d :: "nat" shows "deg_le_sect X d = .[X] \<inter> {t. deg_pm t \<le> d}"
exhaust_sel: fixes BinomialTree :: "('c, 'd) BinomialTree" obtains "BinomialTree = Node (val BinomialTree) (prio BinomialTree) (rank BinomialTree) (children BinomialTree)"
fg_start_round_def: fixes rho :: "nat \<Rightarrow> ('f, 'g, 'h, 'i) config_scheme"   and p :: "'g"   and r :: "nat" shows "fg_start_round rho p r \<equiv> LEAST n. round (rho n) p = r"
simps_2: fixes m :: "n" shows "add (S m) = S \<circ> add m"
le_t_stop_iff: fixes x :: "real" assumes "a \<noteq> 0" shows "(x \<le> t_stop) = (if a < 0 then 0 \<le> p' x else p' x \<le> 0)"
gt_irrefl: fixes s :: "('s, 'v) tm" shows "\<not> s >\<^sub>t s"
abs_update_map_of_hcneq: fixes l :: "('a \<times> 'b) list list"   and x :: "'a"   and k :: "'a"   and v :: "'b" assumes "ht_hash l"   and "ht_distinct l"   and "1 < length l"   and "x \<noteq> k"   and "bounded_hashcode_nat (length l) x \<noteq> bounded_hashcode_nat (length l) k" shows "map_of (concat (abs_update k v l)) x = map_of (concat l) x"
onestep': fixes s :: "'s" shows "p \<le> wp body \<guillemotleft> \<N> G \<guillemotright> s"
compile_cn_correct': fixes f :: "recf"   and xs :: "nat list"   and gs :: "recf list"   and ffp :: "nat"   and far :: "nat"   and fap :: "abc_inst list"   and anything :: "nat list" assumes "\<And>anything r. rec_exec f (map (\<lambda>g. rec_exec g xs) gs) = rec_exec (Cn (length xs) f gs) xs \<Longrightarrow> {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ 0 \<up> (ffp - far) @ anything} fap {\<lambda>nl. nl = map (\<lambda>g. rec_exec g xs) gs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (ffp - Suc far) @ anything}"   and "rec_ci f = (fap, far, ffp)"   and "terminate f (map (\<lambda>g. rec_exec g xs) gs)"   and "\<forall>g\<in>set gs. terminate g xs \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ rec_exec g xs # 0 \<up> (xb - Suc xa) @ xc}))" shows "{\<lambda>nl. nl = xs @ 0 # 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs) @ anything} cn_merge_gs (map rec_ci gs) (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) [+] (mv_boxes 0 (Suc (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs)) (length xs) [+] (mv_boxes (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) 0 (length gs) [+] (fap [+] (mv_box far (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) [+] (empty_boxes (length gs) [+] (mv_box (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs)))) (length xs) [+] mv_boxes (Suc (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs)) 0 (length xs))))))) {\<lambda>nl. nl = xs @ rec_exec (Cn (length xs) f gs) xs # 0 \<up> (max (Suc (length xs)) (Max (insert ffp ((\<lambda>(aprog, p, n). n) ` rec_ci ` set gs))) + length gs) @ anything}"
is_OK_returns_result_E: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog" assumes "h \<turnstile> ok f" obtains x :: "'c" where "h \<turnstile> f \<rightarrow>\<^sub>r x"
x_pi_pj: shows "x mod p_i \<noteq> x mod p_j"
LT3: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<turnstile> F \<longrightarrow> G" shows "\<turnstile> F \<leadsto> G"
bind_rpv_def: fixes rpv :: "('k, 'j, 'h) rpv"   and f :: "'k \<Rightarrow> ('i, 'j, 'h) gpv" shows "rpv \<bind> f = (\<lambda>input. rpv input \<bind> f)"
splits_wrtD_2: fixes ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and hU :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set" assumes "splits_wrt (ps, qs) T F"   and "hU \<in> set ps" shows "cone hU \<subseteq> ideal F \<inter> P[X]"
cswitch_independent_of_state: shows "\<forall>n s t. current s = current t \<longrightarrow> current (cswitch n s) = current (cswitch n t)"
compositor_is_iso: fixes r :: "'a"   and s :: "'a" assumes "ide r"   and "ide s"   and "src r = trg s" shows "Span.iso (CMP r s)"
distinct_1: fixes x1 :: "'b" shows "top.Value x1 \<noteq> top.Top"
axioms_1: fixes R :: "('a, 'b) ring_scheme" assumes "domain R" shows "cring R"
Suc_0_not_prime_nat: shows "\<not> prime (Suc 0)"
fUnionM_return: fixes A :: "'a fset" shows "fUnionM ((\<lambda>x. return {|x|}) |`| A) = return A"
linear_simps_3: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "f (0::'c) = (0::'d)"
simps_2: fixes p :: "'d"   and args :: "'d \<Rightarrow> ('b, 'd) trm"   and \<sigma> :: "'e \<Rightarrow> ('b, 'c, 'd) formula" shows "PFsubst ($\<phi> p args) \<sigma> = $\<phi> p args"
B'_prop: shows "B' \<subseteq> B \<and> card B' = k"
summable_geometric: fixes c :: "'a" assumes "norm c < 1" shows "summable ((^) c)"
comparator_of_rep: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and equal :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat :: "'b \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "comparator_of (rep_nat x) (rep_nat y) = linorder.comparator_of less x y"
conc_star_star: fixes A :: "'b list set" shows "star A @@ star A = star A"
power2_diff: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "power.power one times (minus x y) 2 = minus (plus (power.power one times x 2) (power.power one times y 2)) (times (times (numeral.numeral one plus (num.Bit0 num.One)) x) y)"
his_ex: fixes t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid"   and n' :: "nat" assumes "\<not> Option.is_none (devBC t n nid n')" shows "\<exists>nid'. (n', nid') \<in> his t n nid"
h1: shows "steps0 (1, tp) (A |+| B) stp' = steps0 (1, tp) A stp'"
neg_cong: fixes m :: "int"   and x :: "int" assumes "residues m" shows "\<ominus>\<^bsub>residue_ring m\<^esub> (x mod m) = - x mod m"
rsp: shows "eq_onp (\<lambda>t. ord.is_rbt cless t \<or> ID ccompare = None) rbt.Empty rbt.Empty"
bdd_below: fixes x :: "real" shows "bdd_below {lim (\<lambda>k. f (d k) (r n)) |n. x < r n}"
dims_unit_vec: fixes n :: "nat"   and i :: "nat" shows "dims (unit_vec n i) = [n]"
MNotOut: shows "m \<noteq> <\<bottom>, outM b>"
Lim_transform: fixes g :: "'a \<Rightarrow> 'b"   and a :: "'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b" assumes "(g \<longlongrightarrow> a) F"   and "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'b)) F" shows "(f \<longlongrightarrow> a) F"
convergent_const: fixes c :: "'a" shows "convergent (\<lambda>n. c)"
assms_4: shows "YFtor C @@ A = YFtor C @@ B"
P: shows "{x \<in> space M. P x} \<in> sets M"
reduced_reduce: fixes env :: "('d, bool) mapping"   and t :: "'d ifex" shows "reduced (reduce env t) (Mapping.keys env)"
Sup_to_Inf: shows "Sup = \<partial> \<circ> Inf \<circ> (`) \<partial>"
P_def: shows "P \<equiv> exi yy (PPf (Var yy) (Var xx))"
satC_o: fixes \<xi> :: "Preliminaries.var \<Rightarrow> 'univ"   and \<rho> :: "Preliminaries.var \<Rightarrow> 'fsym trm"   and c :: "('fsym, 'psym) lit list" shows "satC (int \<xi> \<circ> \<rho>) c = satC \<xi> (substC \<rho> c)"
2: shows "inj_on (\<lambda>(is1, is2). is1 @ is2) ({is1. is1 \<lhd> ds1} \<times> {is2. is2 \<lhd> ds2})"
nat_diff_right_cancel_eq1: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m - k = n - k"   and "k < m" shows "m = n"
h3: assumes "\<exists>(a', b', c')\<in>set eq. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set eq. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f = 0) \<and> (\<forall>(d, e, f)\<in>set les. d * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'))\<^sup>2 + e * ((- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')) + f < 0)" shows "\<exists>x. (\<forall>(a, b, c)\<in>set eq. a * x\<^sup>2 + b * x + c = 0) \<and> (\<forall>(a, b, c)\<in>set les. a * x\<^sup>2 + b * x + c < 0)"
vcg_wp_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and mods :: "char list set"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT_mods \<pi> mods P c Q"   and "P s"   and "\<And>s'. \<lbrakk>modifies mods s' s; Q s s'\<rbrakk> \<Longrightarrow> Q' s'" shows "wp \<pi> c Q' s"
p_res_ring_zero: fixes k :: "nat" shows "\<zero>\<^bsub>residue_ring (p ^ k)\<^esub> = 0"
stutter_extend_simps_1: fixes G :: "('b, 'd) graph_rec_scheme" shows "g_V (stutter_extend G) = g_V G"
get_child_nodes\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr"   and uu :: "unit" shows "get_child_nodes\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr uu = get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr RShadowRoot.child_nodes"
v_1_AE: fixes cfg :: "'s cfg" assumes "cfg \<in> valid_cfg" shows "(v cfg = 1) = (AE \<omega> in T cfg. (HLD S1 suntil HLD S2) (state cfg ## \<omega>))"