hcju/isabelleps huggingface dataset:
consequence_relation_family_axioms: shows "consequence_relation_family Bot Q entails_q"
is_prefix_empty: fixes a :: "'a list" shows "is_prefix [] a"
cong_mod_left: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd"   and b :: "'d"   and a :: "'d"   and c :: "'d" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (modulo b a) c a = unique_euclidean_semiring.cong modulo b c a"
rsp: shows "eq_onp (\<lambda>s. s \<noteq> [] \<and> is_upper (hd s)) ''X'' ''X''"
check_poly_weak_mono_smart_def: fixes discrete :: "bool" shows "check_poly_weak_mono_smart discrete \<equiv> if discrete then check_poly_weak_mono_discrete else check_poly_weak_mono"
mkcan_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.mkcan C a b = free_monoidal_category.mkarr C (monoidal_language.Inv C (monoidal_language.red C (monoidal_language.Cod C (free_monoidal_category.rep C b))) \<^bold>\<cdot> monoidal_language.red C (monoidal_language.Dom C (free_monoidal_category.rep C a)))"
sublist_NIL: fixes l1 :: "'d list" shows "subseq l1 [] = (l1 = [])"
5: shows "(H * (d \<sqinter> - a) \<squnion> H * (d \<sqinter> - a)\<^sup>T) * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
matrix_matrix_mult_def: fixes m :: "(('a, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and m' :: "(('a, 'b) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $h i $h k * m' $h k $h j"
strict_order_iff: fixes b :: "'b"   and a :: "'b" shows "(b < a) = (a = sup a b \<and> a \<noteq> b)"
split_face_edges_f12_bet_vs: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 []"   and "(f12, f21) = split_face f ram1 ram2 []"   and "between (vertices f) ram1 ram2 = []" shows "\<E> f12 = {(ram2, ram1), (ram1, ram2)}"
objE: fixes a :: "'a" assumes "S.obj a" obtains "S.ide a" and "S.src a = a" and "S.trg a = a"
IH: fixes y :: "nat"   and i :: "nat"   and i' :: "nat" assumes "y < k"   and "obsp \<sigma> i \<noteq> obsp \<sigma>' i' \<or> cs\<^bsup>\<pi>\<^esup> i \<noteq> cs\<^bsup>\<pi>'\<^esup> i'"   and "is_kth_obs \<pi> y i"   and "is_kth_obs \<pi>' y i'" shows "\<exists>k k'. (\<sigma>', k') \<cc> (\<sigma>, k) \<and> \<pi> k \<in> dom local.att \<or> (\<sigma>, k) \<cc> (\<sigma>', k') \<and> \<pi>' k' \<in> dom local.att"
zhmset_pos_as_neg: fixes M :: "zhmultiset" shows "zhmset_of (hmset_pos M) = zhmset_of (hmset_neg M) + M"
transfer_FOREACHoci_nres: fixes iterate :: "('c dres \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c dres \<Rightarrow> 'c dres) \<Rightarrow> 'c dres \<Rightarrow> 'c dres"   and s :: "'d set"   and ordR :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and fi :: "'d \<Rightarrow> 'c \<Rightarrow> 'c dres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and I :: "'d set \<Rightarrow> 'c \<Rightarrow> bool" assumes "set_iterator_genord iterate s ordR"   and "\<And>x \<sigma>. nres_of (fi x \<sigma>) \<le> f x \<sigma>" shows "nres_of (iterate (case_dres False False c) (\<lambda>x s. s \<bind> fi x) (dRETURN \<sigma>)) \<le> FOREACH\<^sub>O\<^sub>C\<^bsup>ordR,I\<^esup> s c f \<sigma>"
intros_3: assumes "valid []"   and "3 \<le> 3" shows "valid (replicate 3 R @ [])"
not_matches_add_matchNot_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match (MatchNot m) rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
msg4_priEK_disj: fixes B :: "agent"   and KM1 :: "nat"   and K :: "nat"   and M :: "agent"   and NM2 :: "nat"   and merSK :: "nat"   and merEK :: "nat"   and Y :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>Crypt KM1 (sign K \<lbrace>Agent M, Nonce NM2, Key merSK, Key merEK\<rbrace>), Y\<rbrace> \<in> set evs"   and "evs \<in> set_mr" shows "(Key merSK \<in> analz (knows Spy evs) \<or> merSK \<notin> range priEK) \<and> (Key merEK \<in> analz (knows Spy evs) \<or> merEK \<notin> range priEK)"
restrict_PiE: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and S :: "'a \<Rightarrow> 'b set" shows "(restrict f I \<in> Pi\<^sub>E I S) = (f \<in> Pi I S)"
info_rel_def: fixes I :: "('e \<times> 'h) set"   and S :: "('f \<times> 'g) set" shows "\<langle>I, S\<rangle>info_rel = (I \<times>\<^sub>r S) O br snd top"
ccsubspace_leI: fixes A :: "'d ccsubspace"   and B :: "'d ccsubspace" assumes "space_as_set A \<subseteq> space_as_set B" shows "A \<le> B"
assms_2: shows "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r None"
Card_order_trans: fixes r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "Card_order r"   and "x \<noteq> y"   and "(x, y) \<in> r"   and "y \<noteq> z"   and "(y, z) \<in> r" shows "x \<noteq> z \<and> (x, z) \<in> r"
select_convs_2: fixes pid :: "pname"   and tid :: "tname"   and more :: "'a" shows "tid \<lparr>pid = pid, tid = tid, \<dots> = more\<rparr> = tid"
prod_eqI: fixes p :: "'b \<times> 'c"   and q :: "'b \<times> 'c" assumes "hamlet p = hamlet q"   and "time p = time q" shows "p = q"
fdomain_vempty: shows "\<D>\<^sub>\<bullet> []\<^sub>\<circ> = []\<^sub>\<circ>"
tj_stack_reach_hd_stack: fixes G :: "('c, 'd) graph_rec_scheme"   and s :: "'c tarjan_state"   and v :: "'c" assumes "Tarjan_invar G s"   and "v \<in> set (tj_stack s)" shows "(v, hd (stack s)) \<in> (g_E G)\<^sup>*"
absorb_iff2: fixes b :: "'a word"   and a :: "'a word" shows "(b \<le>s a) = (signed.min a b = b)"
continuous_map_eq_image_closure_subset: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "continuous_map X Y f = (\<forall>S. f ` (X closure_of S) \<subseteq> Y closure_of f ` S)"
mono_tllist_ord_case: fixes ord :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> ('d, 'b) tllist \<Rightarrow> ('d, 'b) tllist \<Rightarrow> 'e"   and bot :: "'b \<Rightarrow> 'e" assumes "\<And>x. monotone local.tllist_ord ord (\<lambda>xs. f x xs (TCons x xs))"   and "class.preorder ord (mk_less ord)"   and "\<And>x. ord (bot b) x" shows "monotone local.tllist_ord ord (\<lambda>xs. case xs of TNil b \<Rightarrow> bot b | TCons x xs' \<Rightarrow> f x xs' xs)"
trans_trancl: fixes r :: "('e \<times> 'e) set" shows "trans (r\<^sup>+)"
prod_algebra_mono: fixes I :: "'d set"   and E :: "'d \<Rightarrow> 'e measure"   and F :: "'d \<Rightarrow> 'e measure" assumes "\<And>i. i \<in> I \<Longrightarrow> space (E i) = space (F i)"   and "\<And>i. i \<in> I \<Longrightarrow> sets (E i) \<subseteq> sets (F i)" shows "prod_algebra I E \<subseteq> prod_algebra I F"
hen_1: shows "poly_mod.factorization_m (p ^ n) f (lead_coeff f, mset gs)"
assms_2: assumes "\<Phi>'"   and "\<Phi>" shows "(m, m') \<in> \<langle>R\<rangle>nres_rel"
less_divide_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and 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 sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less a (divide b c) = (if less zero c then less (times a c) b else if less c zero then less b (times a c) else less a zero)"
returns_heap_eq: fixes h :: "'x"   and f :: "('x, 'y, 'z) prog"   and h' :: "'x"   and h'' :: "'x" assumes "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h \<turnstile> f \<rightarrow>\<^sub>h h''" shows "h' = h''"
simps_1: fixes uu :: "('b, 'c) kripke"   and uv :: "'c" shows "(uu, uv \<Turnstile> \<^bold>\<bottom>) = False"
part_witness_interior_simps_3: shows "interior {part_witness.c} = {}"
P1: shows "Col A B X \<and> A B Perp C X"
assoc_coherence: fixes f :: "'b"   and g :: "'b"   and h :: "'b" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "F \<a>\<^sub>B[f, g, h] \<cdot>\<^sub>C \<Phi>\<^sub>F (f \<star>\<^sub>B g, h) \<cdot>\<^sub>C (\<Phi>\<^sub>F (f, g) \<star>\<^sub>C F h) = \<Phi>\<^sub>F (f, g \<star>\<^sub>B h) \<cdot>\<^sub>C (F f \<star>\<^sub>C \<Phi>\<^sub>F (g, h)) \<cdot>\<^sub>C \<a>\<^sub>C[F f, F g, F h]"
strict_prefixE': fixes xs :: "'b list"   and ys :: "'b list" assumes "xs <p ys" obtains z :: "'b" and zs :: "'b list" where "ys = xs \<cdot> z # zs"
assms_1: shows "s \<in> real_of_ereal ` {a<..<b}"
presSwapAbs_fromIMor: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<times> 'c"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<times> 'd"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "ipresCons h hA (fromMOD MOD)"   and "ipresSwapAbs hA (fromMOD MOD)" shows "presSwapAbs (fromIMorAbs hA) MOD"
ag_l_zero: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c" assumes "aGroup A"   and "x \<in> carrier A" shows "\<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> x = x"
admits_triangular_reduction_1x2: assumes "\<forall>A. A \<in> carrier_mat 1 2 \<longrightarrow> admits_triangular_reduction A" shows "\<forall>C. admits_triangular_reduction C"
hyps: fixes v :: "'v" assumes "is_trail v xs v'" shows "set xs \<subseteq> E"
wp_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and P :: "(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 "wp \<pi> c P s"   and "\<And>s. P s \<Longrightarrow> Q s" shows "wp \<pi> c Q s"
ex_cong: fixes Q :: "'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool" assumes "\<And>x. Q x \<Longrightarrow> P x = P' x" shows "(\<exists>x. Q x \<and> P x) = (\<exists>x. Q x \<and> P' x)"
simps: fixes \<delta>\<^sub>1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'b"   and \<delta>\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'c" shows "\<delta>\<^sub>1 \<times> \<delta>\<^sub>2 = (\<lambda>(q\<^sub>1, q\<^sub>2) \<nu>. (\<delta>\<^sub>1 q\<^sub>1 \<nu>, \<delta>\<^sub>2 q\<^sub>2 \<nu>))"
run_nondet_def: fixes nondetT :: "('e, 'd) nondetT" shows "run_nondet nondetT = (case nondetT of NondetT x \<Rightarrow> x)"
Di'_def: fixes a :: "nat list"   and b :: "nat list"   and i :: "nat"   and y :: "nat list" shows "hlde_ops.Di' a b i y = {dij a b i (j + length b - length y) - 1 |j. j < length y \<and> eij a b i (j + length b - length y) \<le> y ! j}"
distinct_5: fixes x21 :: "agent"   and x22 :: "msg"   and x31 :: "agent"   and x32 :: "msg" shows "Gets x21 x22 \<noteq> Notes x31 x32"
ShiftD: fixes kl :: "'b list"   and Kl :: "'b list set"   and k :: "'b" assumes "kl \<in> BNF_Greatest_Fixpoint.Shift Kl k" shows "k # kl \<in> Kl"
applicableBacktrack_def: fixes state :: "State"   and F0 :: "Literal list list" shows "applicableBacktrack state F0 \<equiv> \<exists>state'. appliedBacktrack state state' F0"
cdf_nonneg: fixes k :: "nat"   and x :: "real" shows "0 \<le> cdf (\<mu> k) x"
distinct_map: fixes f :: "'l \<Rightarrow> 'k"   and xs :: "'l list" shows "distinct (map f xs) = (distinct xs \<and> inj_on f (set xs))"
Cd_inj_on_Xd: fixes d :: "'b"   and X :: "'a set" shows "inj_on Xd (Singleton_for_d.Cd d X)"
add_mset_lt_lt_le: fixes a :: "'b"   and b :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" assumes "a < b"   and "A \<le> B" shows "add_mset a A < add_mset b B"
mergesort_by_rel_simps_1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "mergesort_by_rel R [] = []"
hop1: shows "prf'.game_0 \<A> = local.prf.game_0 (prf'_reduction \<A>)"
open_prod_elim: fixes S :: "('a \<times> 'b) set"   and x :: "'a \<times> 'b" assumes "open S"   and "x \<in> S" obtains A :: "'a set" and B :: "'b set" where "open A" and "open B" and "x \<in> A \<times> B" and "A \<times> B \<subseteq> S"
ex_map_conv: fixes ys :: "'l list"   and f :: "'k \<Rightarrow> 'l" shows "(\<exists>xs. ys = map f xs) = (\<forall>y\<in>set ys. \<exists>x. y = f x)"
monic_def: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "punit.monic p = local.punit.monom_mult ((1::'b) / lcf p) 0 p"
writes_def: fixes S :: "('a, 'b, 'c) prog set"   and setter :: "('a, 'b, 'd) prog"   and h :: "'a"   and h' :: "'a" shows "writes S setter h h' = (h \<turnstile> setter \<rightarrow>\<^sub>h h' \<longrightarrow> (\<exists>progs. set progs \<subseteq> S \<and> h \<turnstile> iterate_M progs \<rightarrow>\<^sub>h h'))"
simps_3: fixes x2 :: "'b" shows "\<lfloor>x2\<rfloor> \<noteq> \<bottom>"
iso_prop: fixes S :: "('a \<times> 'a set) set"   and R :: "('a \<times> 'a set) set" shows "mono (\<lambda>X. S \<union> R \<cdot> X)"
char_to_digit_simps_7: shows "char_to_digit CHR ''6'' = 6"
legs_form_cospan_1: shows "C.cospan cods.\<mu>.leg0 cods.\<nu>.leg1"
hyps_2: shows "Q'' \<longmapsto> \<tau> \<prec> Q'"
no_error: fixes s :: "'b sparc_state" assumes "good_context s" shows "snd (execute_instruction () s) = False"
insert_seq_butlast: fixes start :: "'a option"   and xs :: "('a \<times> 'a option) list" assumes "insert_seq start xs"   and "xs \<noteq> []"   and "xs \<noteq> [last xs]" shows "insert_seq start (butlast xs)"
contains_pred_def: fixes A :: "'c set"   and x :: "'c" shows "Predicate_Compile.contains_pred A x = (if x \<in> A then Predicate.single () else bot)"
op_dghm_dghm_comp: fixes \<GG> :: "V"   and \<FF> :: "V" shows "op_dghm (\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF>) = op_dghm \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M op_dghm \<FF>"
exp_eq_0_imp_not_bit: fixes n :: "nat"   and a :: "'a" assumes "(2::'a) ^ n = (0::'a)" shows "\<not> bit a n"
emeasure_HLD: fixes s :: "'s"   and X :: "'s set" shows "emeasure (T s) {\<omega> \<in> space (T s). HLD X \<omega>} = emeasure (measure_pmf (K s)) X"
eq': shows "f \<equiv> C (lfp (\<lambda>f. U (F (C f))))"
dist_float_triangle2: fixes x :: "float"   and y :: "float"   and z :: "float" shows "dist x y \<le> dist x z + dist y z"
trace_set_def: fixes T :: "'a list set" shows "trace_set T \<equiv> [] \<in> T \<and> (\<forall>xs x. xs @ [x] \<in> T \<longrightarrow> xs \<in> T)"
fvi_Suc: fixes x :: "nat"   and b :: "nat"   and \<phi> :: "'c MFOTL.formula" shows "(x \<in> MFOTL.fvi (Suc b) \<phi>) = (Suc x \<in> MFOTL.fvi b \<phi>)"
norm_blinfun_compose: fixes f :: "'d \<Rightarrow>\<^sub>L 'c"   and g :: "'b \<Rightarrow>\<^sub>L 'd" shows "norm (f o\<^sub>L g) \<le> norm f * norm g"
Poly_on_rev_starting_with_0: fixes as :: "'b list" assumes "hd as = (0::'b)" shows "Poly (rev (tl as)) = Poly (rev as)"
quasi_inverse_simps_3: fixes f :: "'b" assumes "D.equivalence_map f" shows "src\<^sub>D f\<^sup>~\<^sup>D = trg\<^sub>D f"
measurable_const_iff: fixes c :: "'b"   and A :: "'a measure"   and B :: "'b measure" shows "((\<lambda>x. c) \<in> A \<rightarrow>\<^sub>M B) = (space A = {} \<or> c \<in> space B)"
wtl_cert_def: fixes cert :: "'c list"   and f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and T :: "'c"   and B :: "'c"   and step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and pc :: "nat"   and s :: "'c" shows "wtl_cert cert f r T B step pc s = (if cert ! pc = B then wtl_inst cert f r T step pc s else if s \<sqsubseteq>\<^bsub>r\<^esub> cert ! pc then wtl_inst cert f r T step pc (cert ! pc) else T)"
ctran_in_comm: fixes s :: "'b"   and guar :: "('b \<times> 'b) set"   and Q :: "'b com option"   and xs :: "('b com option \<times> 'b) list"   and post :: "'b set"   and P :: "'b com option" assumes "(s, s) \<in> guar"   and "(Q, s) # xs \<in> comm (guar, post)" shows "(P, s) # (Q, s) # xs \<in> comm (guar, post)"
\<F>'_1: shows "\<F>' \<subseteq> balls ` carrier"
coeff_0_interp_factorization: fixes fctrs :: "'a \<times> ('a \<times> nat) list" shows "(coeff (interp_factorization fctrs) 0 = (0::'a)) = (fst fctrs = (0::'a) \<or> (0::'a) \<in> fst ` set (snd fctrs))"
res_carrier_eq: shows "carrier (residue_ring (int p)) = {0..int p - 1}"
model_def: fixes s :: "(nat \<times> form) list" shows "model s = (range ntou, \<lambda>p ms. let f = failing_path (deriv s) in \<forall>n m. \<not> contains f n (m, PAtom p (map uton ms)))"
simps: fixes g :: "'b poly"   and h :: "'b poly"   and F :: "'b poly fps"   and n :: "nat" assumes "fps_hensel_aux g h" shows "fps_hensel_aux.hensel_fpxs_aux F g h n = (if n = 0 then (g, h) else let U = F $ n - (\<Sum>(i, j)\<in>{(i, j). i < n \<and> j < n \<and> i + j = n}. fst (fps_hensel_aux.hensel_fpxs_aux F g h i) * snd (fps_hensel_aux.hensel_fpxs_aux F g h j)) in (U * snd (bezout_coefficients g h) + g * (U * fst (bezout_coefficients g h) div h), U * fst (bezout_coefficients g h) mod h))"
iptables_bigstep_rm_LogEmpty: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rm_LogEmpty rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
fps_nth_fps_const: fixes c :: "'a"   and n :: "nat" shows "fps_const c $ n = (if n = 0 then c else (0::'a))"
add_5: fixes z :: "'a"   and Z :: "'a multiset" assumes "add_mset y' Y' = add_mset z Z"   and "(x, z) \<in> ns"   and "(X, Z) \<in> multpw ns" shows "thesis"
zero_less_of_rat_iff: fixes r :: "rat" shows "((0::'a) < of_rat r) = (0 < r)"
exec_move_AAcc1: fixes pc :: "nat"   and e :: "'addr expr1"   and ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and xcp :: "'addr option" assumes "pc < length (compE2 e)" shows "exec_move ci P t (e\<lfloor>e'\<rceil>) h (stk, loc, pc, xcp) = exec_move ci P t e h (stk, loc, pc, xcp)"
mono_ndet_FD_right: fixes P :: "'b process"   and Q :: "'b process"   and S :: "'b process" assumes "P \<le> Q" shows "(S |-| P) \<le> Q"
v'_def: shows "beta_interp.v' \<equiv> \<lambda>i. if i \<le> n then THE c. c \<in> X \<and> v c = i else not_in_X"
Derivs_def: fixes xs :: "'b list"   and A :: "'b list set" shows "Derivs xs A = {ys. xs @ ys \<in> A}"
R_equiv_entailment: fixes lc\<^sub>1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc\<^sub>2 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set"   and \<Gamma>'' :: "'Var \<Rightarrow> 'BExp set option"   and P'' :: "'BExp set" assumes "lc\<^sub>1 \<R>\<^sup>u\<^bsub>\<Gamma>',\<S>',P'\<^esub> lc\<^sub>2"   and "\<Gamma>' =:\<^sub>P' \<Gamma>''"   and "P' \<turnstile> P''"   and "\<forall>mds. tyenv_wellformed mds \<Gamma>' \<S>' P' \<longrightarrow> tyenv_wellformed mds \<Gamma>'' \<S>' P''" shows "lc\<^sub>1 \<R>\<^sup>u\<^bsub>\<Gamma>'',\<S>',P''\<^esub> lc\<^sub>2"
list_ex_append: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'e list" shows "list_ex P (xs @ ys) = (list_ex P xs \<or> list_ex P ys)"
group_hom_imp_group_hom_image: fixes G :: "('b, 'c) monoid_scheme"   and h :: "'b \<Rightarrow> 'b" assumes "group_hom G G h" shows "group_hom G (G\<lparr>carrier := h ` carrier G\<rparr>) h"
compact_eq_bounded_closed: fixes S :: "'d set" shows "compact S = (bounded S \<and> closed S)"
equals0D: fixes A :: "'a set"   and a :: "'a" assumes "A = {}" shows "a \<notin> A"
factorization_m_smult: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset"   and d :: "int" assumes "factorization_m f (c, fs)" shows "factorization_m (smult d f) (c * d, fs)"
b_1: fixes xs :: "'e list"   and P :: "'e \<Rightarrow> bool" shows "Ball (set xs) P = list_all P xs"
monad_alt_optionT: fixes return :: "'a option \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a option \<Rightarrow> 'b) \<Rightarrow> 'b"   and alt :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "monad_alt return bind alt" shows "monad_alt (return_option return) (bind_option return bind) (alt_option alt)"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_inv (s, l, r) lm n ires = (if s = 0 then mopup_stop (s, l, r) lm n ires else if s \<le> 2 * n then if s mod 2 = 1 then mopup_bef_erase_a (s, l, r) lm n ires else mopup_bef_erase_b (s, l, r) lm n ires else if s = 2 * n + 1 then mopup_jump_over1 (s, l, r) lm n ires else if s = 2 * n + 2 then mopup_aft_erase_a (s, l, r) lm n ires else if s = 2 * n + 3 then mopup_aft_erase_b (s, l, r) lm n ires else if s = 2 * n + 4 then mopup_aft_erase_c (s, l, r) lm n ires else if s = 2 * n + 5 then mopup_left_moving (s, l, r) lm n ires else if s = 2 * n + 6 then mopup_jump_over2 (s, l, r) lm n ires else False)"
exist_C: shows "\<exists>C. C \<in> N \<and> D \<in> \<G>_F C"
psubst_ty_eqvt: fixes pi :: "(W.tvar \<times> W.tvar) list"   and \<theta> :: "(W.tvar \<times> ty) list"   and T :: "ty" shows "pi \<bullet> \<theta><T> = (pi \<bullet> \<theta>)<pi \<bullet> T>"
numeral_uexpr_rep_eq: fixes x :: "num"   and b :: "'d" shows "\<lbrakk>numeral x\<rbrakk>\<^sub>e b = numeral x"
is_terminal: fixes x' :: "'d"   and f :: "'c" assumes "arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x' x f" shows "\<exists>!g. x\<epsilon>.is_coext x' f g"
whole_ideal: fixes R :: "('d, 'e) Ring_scheme" assumes "Ring R" shows "ideal R (carrier R)"
Collect_conv_if2: fixes a :: "'c"   and P :: "'c \<Rightarrow> bool" shows "_Collect x (a = x \<and> P x) = (if P a then {a} else {})"
least_squares_approximation5': fixes S :: "'a set"   and v :: "'a" assumes "subspace S" shows "\<exists>!p. p \<in> S \<and> (\<forall>y\<in>S. norm (v - p) \<le> norm (v - y) \<and> v - p \<in> orthogonal_complement S)"
assertion_inf_comp_eq: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf x y = times x y"
permutation_bijective: fixes p :: "'e \<Rightarrow> 'e" assumes "permutation p" shows "bij p"
dg_Dom_vempty_if_Arr_vempty: assumes "\<CC>\<lparr>Arr\<rparr> = []\<^sub>\<circ>" shows "\<CC>\<lparr>Dom\<rparr> = []\<^sub>\<circ>"
arr_char: fixes f :: "'b \<times> 'b \<times> 'b" shows "D.VVV.arr f = (D.arr (fst f) \<and> D.VV.arr (snd f) \<and> src\<^sub>D (fst f) = trg\<^sub>D (fst (snd f)))"