hcju/isabelleps huggingface dataset:
subformulas\<^sub>\<nu>_subfrmlsn: fixes \<phi> :: "'a ltln" shows "subformulas\<^sub>\<nu> \<phi> \<subseteq> subfrmlsn \<phi>"
times_empty2: fixes A :: "V" shows "vtimes A 0 = 0"
mult_indicator_cong: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "indicator A x * f x = indicator A x * g x"
split_sum_mid_less: fixes i :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "i < n" shows "sum f {..<n} = sum f {..<i} + sum f {i..<n}"
Mk_\<mu>: fixes i :: "nat"   and j :: "nat" assumes "i < k"   and "j < k" shows "Mk $$ (i, j) = \<mu> i j"
isMinim_unique: fixes B :: "'a set"   and a :: "'a"   and a' :: "'a" assumes "local.isMinim B a"   and "local.isMinim B a'" shows "a = a'"
nb\<^sub>3: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set (map (strips_op_to_sasp \<Psi>) ops')" shows "map_of (sas_plus_operator.precondition_of op) \<subseteq>\<^sub>m s"
LeftDerivationFix_empty: fixes \<alpha> :: "symbol list"   and i :: "nat" assumes "is_sentence \<alpha>"   and "i < length \<alpha>" shows "LeftDerivationFix \<alpha> i [] i \<alpha>"
x'_y'_def_1: shows "x' \<equiv> min x y"
swapidseq_comp_add: fixes n :: "nat"   and p :: "'b \<Rightarrow> 'b"   and m :: "nat"   and q :: "'b \<Rightarrow> 'b" assumes "swapidseq n p"   and "swapidseq m q" shows "swapidseq (n + m) (p \<circ> q)"
assms: fixes f :: "three \<Rightarrow> three \<Rightarrow> three"   and xs :: "three list" assumes "associative f"   and "xs \<noteq> []" shows "candidate f xs = scanl1 f xs"
qequiv_refl: fixes a :: "'a['b, 'c]" shows "a \<cong>\<^sub>Q a"
dg_Par_is_arrI: fixes \<alpha> :: "V"   and S :: "V"   and A :: "V"   and B :: "V" assumes "arr_Par \<alpha> S"   and "S\<lparr>ArrDom\<rparr> = A"   and "S\<lparr>ArrCod\<rparr> = B" shows "S : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B"
cp0: fixes X :: "'b state \<times> 'b state \<Rightarrow> Set('a)"   and Y :: "'b state \<times> 'b state \<Rightarrow> 'a"   and \<tau> :: "'b state \<times> 'b state" shows "X->including\<^sub>S\<^sub>e\<^sub>t(Y) \<tau> = \<lambda>_. X \<tau>->including\<^sub>S\<^sub>e\<^sub>t(\<lambda>_. Y \<tau>) \<tau>"
read_only_reads_unowned: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "j < length ts\<^sub>s\<^sub>b"   and "i \<noteq> j"   and "ts\<^sub>s\<^sub>b ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts\<^sub>s\<^sub>b ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<O>\<^sub>i) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) = {}"
reduced_word_Cons_reduce: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero (a # as)) (a # as)" shows "monoid_add.reduced_word_for plus zero A (monoid_add.sum_list plus zero as) as"
if_eq: fixes n :: "nat"   and x' :: "real" shows "(if even n then (- 1) ^ (n div 2) * 1 / real (Suc n) else 0) * real (Suc n) * x' ^ n = (if even n then (- 1) ^ (n div 2) * x' ^ (2 * (n div 2)) else 0)"
fresh_left: fixes a :: "name"   and pi :: "(name \<times> name) list"   and x :: "'a" shows "a \<sharp> pi \<bullet> x = rev pi \<bullet> a \<sharp> x"
finite_2sat_iff: fixes cnf :: "lit uprod set" assumes "finite (vars_of_cnf cnf)"   and "is_2sat cnf" shows "satisfiable cnf = (\<forall>x\<in>vars_of_cnf cnf. \<not> ((Pos x, Neg x) \<in> (imp_graph cnf)\<^sup>* \<and> (Neg x, Pos x) \<in> (imp_graph cnf)\<^sup>*))"
list_succ_commute: fixes xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "set xs \<inter> set ys = {}" shows "list_succ xs (list_succ ys x) = list_succ ys (list_succ xs x)"
eventually_principal: fixes P :: "'b \<Rightarrow> bool"   and S :: "'b set" shows "eventually P (principal S) = (\<forall>x\<in>S. P x)"
cp0: fixes X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f X \<tau> = f (\<lambda>_. X \<tau>) \<tau>"
headconst_zero: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "(headconst p = 0\<^sub>N) = (p = 0\<^sub>p)"
INF_of_enat_nat_conv1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat"   and d :: "nat" assumes "finite A" shows "((INF x\<in>A. enat (f x)) = enat d) = (\<exists>x\<in>A. d = f x \<and> (\<forall>y\<in>A. f x \<le> f y))"
H2_def: fixes P :: "'c alpha_d_scheme \<times> 'c alpha_d_scheme \<Rightarrow> bool" shows "H2 P \<equiv> P ;; J"
token_run_P: fixes P :: "'b \<Rightarrow> bool"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "\<not> P (token_run x n)"   and "P (token_run x (Suc (n + m)))" shows "\<exists>m'\<le>m. \<not> P (token_run x (n + m')) \<and> P (token_run x (Suc (n + m')))"
u': shows "(u has_derivative blinfun_apply (- blinfun_scaleR_left (inverse (blinfun_apply (Ds x) (f x))) o\<^sub>L (Ds x o\<^sub>L flowderiv x 0) o\<^sub>L embed1_blinfun)) (at x)"
sets_m: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M" shows "sets (m N M y) = sets (stream_space (count_space UNIV))"
False: shows "\<not> length as \<le> card (reachable_s PROB s) - 1"
A3a: fixes x :: "i" shows "(\<box>x) \<^bold>\<cdot> x \<cong> x"
SDG_Def_parent_Def: fixes V :: "'var"   and n :: "'node SDG_node" assumes "V \<in> Def\<^bsub>SDG\<^esub> n" shows "V \<in> Def (parent_node n)"
cs_order_le: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and \<pi>' :: "nat \<Rightarrow> 'a"   and i :: "nat"   and i' :: "nat"   and j :: "nat"   and j' :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_path prob \<pi>'"   and "IFC_def.cs prob \<pi> i = IFC_def.cs prob \<pi>' i'"   and "IFC_def.cs prob \<pi> j = IFC_def.cs prob \<pi>' j'"   and "\<pi> i \<noteq> IFC_def.return prob"   and "i \<le> j" shows "i' \<le> j'"
inject_4: fixes x41 :: "'d"   and x42 :: "('b, 'c, 'd) Tree"   and y41 :: "'d"   and y42 :: "('b, 'c, 'd) Tree" shows "(tAct x41 x42 = tAct y41 y42) = (x41 = y41 \<and> x42 = y42)"
one_dim_iso_of_one: shows "one_dim_iso (1::'b) = (1::'a)"
dist_les_6: fixes nat' :: "nat" shows "ValTT \<notsqsubseteq> ValN\<cdot>nat'"
fcb1: shows "(as \<inter> supp x) \<sharp>* f (as \<inter> supp x) x c"
simps_1: fixes \<nu> :: "'a set" shows "af_letter true\<^sub>n \<nu> = true\<^sub>n"
remdups_adj_Nil_iff: fixes xs :: "'c list" shows "(remdups_adj xs = []) = (xs = [])"
Entry_empty: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val" shows "Def Entry = {} \<and> Use Entry = {}"
is_invI_1: fixes G :: "('a, 'b) monoid_scheme"   and a :: "'a"   and b :: "'a" assumes "Group.comm_monoid G"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "a \<otimes>\<^bsub>G\<^esub> b = \<one>\<^bsub>G\<^esub>" shows "inv\<^bsub>G\<^esub> b = a"
fv_sub: shows "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V"
equiv_uimp_le_alt: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and u :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "equiv_utrans t u" shows "le_utrans u t"
power_up_neg_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(power_up prec b n < 0) = (b < 0 \<and> odd n)"
deleted_ids_exist: fixes ref :: "'c"   and ops :: "('c \<times> ('c, 'd) list_op) list" shows "(ref \<in> set (deleted_ids ops)) = (\<exists>i. (i, Delete ref) \<in> set ops)"
w1_cast: shows "sint (SCAST(32 \<rightarrow> 64) w1) = sint w1"
add_eq_inf_sup: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "plus a b = plus (sup a b) (inf a b)"
final_def: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event llist" assumes "cocomputation channel trans send recv init"   and "distributed_system.cotrace channel trans send recv init t" shows "cocomputation.final channel trans send recv init t = distributed_system.cos channel trans send recv init t (cocomputation.final_i channel trans send recv init t)"
bivl: shows "(f (real_of_float l) + f (real_of_float u) - real_of_float (a * (l + u))) / 2 \<in>\<^sub>r bivl"
HcompNml_assoc: fixes t :: "'a E.term"   and u :: "'a E.term"   and v :: "'a E.term" assumes "E.Nml t"   and "E.Nml u"   and "E.Nml v"   and "E.Src t = E.Trg u"   and "E.Src u = E.Trg v" shows "(t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v"
basis_expansion_cart_basis': fixes x :: "('a, 'b) vec" shows "(\<Sum>i\<in>UNIV. x $ i *s cart_basis' $ i) = x"
infsetsum_Diff: fixes f :: "'d \<Rightarrow> 'e"   and B :: "'d set"   and A :: "'d set" assumes "f abs_summable_on B"   and "A \<subseteq> B" shows "infsetsum f (B - A) = infsetsum f B - infsetsum f A"
assms_2: fixes a :: "'a" shows "r a x \<or> r x a"
P2_inv_g_rewrite: fixes s :: "nat"   and u1' :: "nat" assumes "s < order \<G>" shows "inv \<^bold>g [^] (u1' + (order \<G> - s)) = \<^bold>g [^] s \<otimes> inv (\<^bold>g [^] u1')"
Range_Un_eq: fixes A :: "('e \<times> 'd) set"   and B :: "('e \<times> 'd) set" shows "Range (A \<union> B) = Range A \<union> Range B"
ipresIGOp_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('b, 'c, 'i, 'j, 'f, 'k, 'g, 'h, 'l) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.ipresIGOp Delta h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresOp Delta h hA MOD"
connected_component_of_eq_empty: fixes X :: "'c topology"   and x :: "'c" shows "(connected_component_of_set X x = {}) = (x \<notin> topspace X)"
prefix_same_cases: fixes xs\<^sub>1 :: "'c list"   and ys :: "'c list"   and xs\<^sub>2 :: "'c list" assumes "prefix xs\<^sub>1 ys"   and "prefix xs\<^sub>2 ys" shows "prefix xs\<^sub>1 xs\<^sub>2 \<or> prefix xs\<^sub>2 xs\<^sub>1"
simps_2: fixes \<phi> :: "'fmla"   and t1 :: "'trm"   and x1 :: "'var"   and txs :: "('trm \<times> 'var) list" shows "rawpsubst \<phi> ((t1, x1) # txs) = rawpsubst (subst \<phi> t1 x1) txs"
sub_measure_space: fixes M :: "'b measure"   and F :: "'b measure" assumes "subalgebra M F" shows "measure_space (space M) (sets F) (emeasure M)"
times_list_def: fixes xs :: "'a list"   and ys :: "'a list" shows "xs * ys \<equiv> xs @ ys"
WHILEIT_def: shows "aWHILEIT \<equiv> \<lambda>I b f. REC\<^sub>T (WHILEI_body abind areturn I b f)"
assms_2: shows "A, n \<turnstile> (p # ps, a) # branch"
homeomorphism_cont2: fixes S :: "'b set"   and T :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "homeomorphism S T f g" shows "continuous_on T g"
assms_4: shows "ht_distinct (l[bounded_hashcode_nat (length l) k := fst (ls_delete k (l ! bounded_hashcode_nat (length l) k))])"
Rep_assn: fixes x :: "assn" shows "Rep_assn x \<in> Collect proper"
Ga0b0: shows "geodesic_segment_between G a0 b0"
leg1_composite_2: shows "HH\<mu>H\<nu>\<pi>\<rho>.dom.leg1 = \<mu>.leg1 \<cdot> \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1\<^sub>1"
iprev_nth_imirror_inext_nth_conv: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "imirror I \<leftarrow> n = mirror_elem (I \<rightarrow> n) I"
type_less_left_simps_3: fixes \<rho> :: "'a basic_type"   and \<sigma> :: "'a OCL_Types.type" shows "(\<rho>[?] < \<sigma>) = (\<exists>\<upsilon>. \<sigma> = OclSuper \<or> \<sigma> = \<upsilon>[?] \<and> \<rho> < \<upsilon>)"
poly_add_zero_2: fixes K :: "'a set"   and p :: "'a list" assumes "subring K R"   and "polynomial K p" shows "poly_add [] p = p"
that: fixes Z :: "'a set" assumes "Z \<in> sets M"   and "Z \<subseteq> W"   and "emeasure M Z < \<infinity>"   and "C < emeasure M Z" shows "thesis"
mod_add_eq_mod_conv: fixes m :: "nat"   and x :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < m" shows "((x + a) mod m = b mod m) = (x mod m = (m + b mod m - a mod m) mod m)"
is_volatile_Read\<^sub>s\<^sub>b_simps_4: fixes A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_volatile_Read\<^sub>s\<^sub>b (Ghost\<^sub>s\<^sub>b A L R W) = False"
inconsistent_head: fixes S :: "('a, 'b) fm set"   and p :: "('a, 'b) fm" assumes "consistent S"   and "maximal S"   and "p \<notin> S" obtains S' :: "('a, 'b) fm list" where "set S' \<subseteq> S" and "p # S' \<turnstile> \<^bold>\<bottom>"
pure_assn_eq_conv: fixes P :: "bool"   and Q :: "bool" shows "(\<up> P = \<up> Q) = (P = Q)"
mb_eq: shows "mb = (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows)"
wakeup: fixes t :: "'t"   and x :: "'x"   and w :: "wake_up_status" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" shows "wset s t \<noteq> \<lfloor>PostWS w\<rfloor>"
iDdomAux: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "p \<noteq> []"   and "x \<notin> dom (Cp (list2FWpolicy p))"   and "x \<in> dom (Cp (list2FWpolicy (insertDenies p)))" shows "Cp (list2FWpolicy (insertDenies p)) x = \<lfloor>deny ()\<rfloor>"
cr_real_eq: shows "pcr_real = (\<lambda>x y. Real.cauchy x \<and> Real.Real x = y)"
simps_5: fixes one' :: "'c \<Rightarrow> 'c"   and carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "one_update one' \<lparr>carrier = carrier, mult = mult, one = one, \<dots> = more\<rparr> = \<lparr>carrier = carrier, mult = mult, one = one' one, \<dots> = more\<rparr>"
uniformly_continuous_imp_continuous: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "uniformly_continuous_on s f" shows "continuous_on s f"
linf_norm_zero_vec: fixes n :: "nat" shows "\<parallel>0\<^sub>v n\<parallel>\<^sub>\<infinity> = (0::'b)"
mat_impl_def: shows "mat_impl \<equiv> map_fun Rep_mat_impl Abs_mat (\<lambda>(nr, nc, m). (nr, nc, mk_mat nr nc (\<lambda>(i, j). m !! i !! j)))"
Y_simp: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<phi> :: "'a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'b"   and f :: "'a" assumes "yoneda_functor C S \<phi>"   and "partial_magma.arr C f" shows "yoneda_functor.Y C S \<phi> f = (\<lambda>g. hom_functor.map C S \<phi> (g, f))"
1: shows "(f \<sqinter> - q) * (f \<sqinter> q)\<^sup>T = bot"
inj_on_subset: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set"   and B :: "'g set" assumes "inj_on f A"   and "B \<subseteq> A" shows "inj_on f B"
insert_DiffM: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "x, M - (x, {#}) = M"
greaterThanAtMost_empty_iff2: fixes k :: "'t"   and l :: "'t" shows "({} = ord_term_lin.greaterThanAtMost k l) = (\<not> k \<prec>\<^sub>t l)"
unit_imp_dvd: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times b a"
pinf_7: fixes t :: "'a" shows "\<exists>z. \<forall>x>z. (t < x) = True"
snd_pseudo_mod_smult_invar_right: fixes x :: "'b"   and lc :: "'b"   and q :: "'b poly"   and r :: "'b poly"   and d :: "'b poly"   and dr :: "nat"   and n :: "nat"   and q' :: "'b poly" shows "snd (pseudo_divmod_main (x * lc) q r (Polynomial.smult x d) dr n) = snd (pseudo_divmod_main lc q' (Polynomial.smult (x ^ n) r) d dr n)"
Scan_empty: fixes k :: "nat"   and I :: "item set" shows "Scan {} k I = I"
increase_def: fixes f :: "('c \<Rightarrow> 'd list) \<times> (('d list \<Rightarrow> 'd list) \<Rightarrow> 'c \<Rightarrow> 'c)"   and val :: "'d" shows "increase f val \<equiv> snd f ((#) val)"
alw_sdrop: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and n :: "nat" assumes "alw \<phi> xs" shows "alw \<phi> (sdrop n xs)"
assms: assumes "degree p \<noteq> 0" shows "a * lead_coeff p \<noteq> (0::'a)"
Ln_times_simple: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0"   and "- pi < Im (Ln w) + Im (Ln z)"   and "Im (Ln w) + Im (Ln z) \<le> pi" shows "Ln (w * z) = Ln w + Ln z"
ct_prefix_code_3: fixes x :: "ct"   and xs :: "ct list"   and y :: "ct"   and ys :: "ct list" shows "ct_prefix (x # xs) (y # ys) = (ct_eq x y \<and> ct_prefix xs ys)"
intensional_simps_5: fixes P :: "'b \<Rightarrow> bool" shows "\<not> P = P \<equiv> #False"
has_vector_derivative_circlepath: fixes z :: "complex"   and r :: "real"   and x :: "real"   and X :: "real set" shows "(circlepath z r has_vector_derivative complex_of_real (2 * pi) * \<i> * complex_of_real r * exp (2 * complex_of_real pi * \<i> * complex_of_real x)) (at x within X)"
ereal_of_enat_simps_2: shows "ereal_of_enat \<infinity> = \<infinity>"
derived_in_carrier: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "derived G H \<subseteq> carrier G"
IH: assumes "xs \<in> paths (substitute (f_nxt f T x) T (nxt t x \<otimes>\<otimes> f x))"   and "\<And>xa. (xa \<in> T) = (xa \<in> T') \<or> f_nxt f T x xa = TTree.empty" shows "xs \<in> paths (substitute (f_nxt f T x) T' (nxt t x \<otimes>\<otimes> f x))"
as_good_as_def: fixes x :: "'b"   and B :: "'b set"   and P :: "('b \<times> 'b) set" shows "as_good_as x B P = {y \<in> B. x \<approx>[P] y}"
awalkI_apath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "G.apath u p v" shows "G.awalk u p v"
Runit_in_runit: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and t :: "'b local.term" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "t \<in> f" shows "\<^bold>\<r>\<^bold>[t\<^bold>] \<in> free_monoidal_category.runit\<^sub>F\<^sub>M\<^sub>C C f"
Lp_space_def: fixes p :: "ennreal"   and M :: "'b measure" shows "\<LL> p M = (if p = 0 then quasinorm_of (1, \<lambda>f. if f \<in> borel_measurable M then 0 else \<infinity>) else if p < \<infinity> then quasinorm_of (if p < 1 then 2 powr (1 / enn2real p - 1) else 1, \<lambda>f. if f \<in> borel_measurable M \<and> integrable M (\<lambda>x. \<bar>f x\<bar> powr enn2real p) then ennreal ((LINT x|M. \<bar>f x\<bar> powr enn2real p) powr (1 / enn2real p)) else \<infinity>) else quasinorm_of (1, \<lambda>f. if f \<in> borel_measurable M then e2ennreal (esssup M (\<lambda>x. ereal \<bar>f x\<bar>)) else \<infinity>))"
assms_2: shows "ik\<^sub>e\<^sub>s\<^sub>t A \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t"
pred_of_rel_def: fixes s :: "('d \<times> 'e) set" shows "pred_of_rel s \<equiv> \<lambda>a b. (a, b) \<in> s"
nths_Cons: fixes x :: "'a"   and l :: "'a list"   and A :: "nat set" shows "nths (x # l) A = (if 0 \<in> A then [x] else []) @ nths l {j. Suc j \<in> A}"
solution_uminus_left: fixes D :: "nat"   and x :: "'a"   and y :: "'a" assumes "pell D" shows "pell.solution D (- x, y) = pell.solution D (x, y)"
T2: shows "S.UP (\<r>\<^sup>-\<^sup>1[g] \<cdot> \<l>[g]) = (S.cmp\<^sub>U\<^sub>P (g, src g) \<cdot>\<^sub>S (S.UP g \<star>\<^sub>S UP.unit (src g))) \<cdot>\<^sub>S (S.inv (UP.unit (trg g)) \<star>\<^sub>S S.UP g) \<cdot>\<^sub>S S.inv (S.cmp\<^sub>U\<^sub>P (trg g, g))"
invariant_reachable: fixes F :: "'b program" shows "F \<in> invariant (reachable F)"
path_graph_cong: fixes g\<^sub>1 :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b"   and g\<^sub>2 :: "'b ugraph" assumes "path g\<^sub>1 u p v"   and "set p \<subseteq> edges g\<^sub>1 \<Longrightarrow> set p \<subseteq> edges g\<^sub>2" shows "path g\<^sub>2 u p v"
content_eq_0_interior: fixes a :: "'c"   and b :: "'c" shows "(content (cbox a b) = 0) = (interior (cbox a b) = {})"
that: fixes c :: "'a" assumes "p = [:c:]"   and "c dvd (1::'a)" shows "thesis"
simps_3: fixes carrier :: "'c set"   and top :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and iop :: "'c \<Rightarrow> 'c"   and one :: "'c"   and more :: "'d" shows "\<one>\<^bsub>\<lparr>carrier = carrier, Group.top = top, iop = iop, one = one, \<dots> = more\<rparr>\<^esub> = one"
finGSi: shows "finite (\<GG> ! (i + 1))"
oldDefs_correct: fixes g :: "'g"   and n :: "'node" shows "defs g n = var g ` defs' g n"
simps_37: fixes f1 :: "agr_key \<Rightarrow> 'a"   and f2 :: "enc_key \<Rightarrow> 'a"   and f3 :: "msg \<Rightarrow> 'a"   and f4 :: "enc_key \<Rightarrow> msg \<Rightarrow> 'a"   and f5 :: "msg \<Rightarrow> msg \<Rightarrow> 'a"   and f6 :: "nat \<Rightarrow> msg \<Rightarrow> 'a"   and x1 :: "agr_key" shows "(case AgrKey x1 of AgrKey x \<Rightarrow> f1 x | EncKey x \<Rightarrow> f2 x | Hash x \<Rightarrow> f3 x | Crypt x xa \<Rightarrow> f4 x xa | \<lbrace>x, xa\<rbrace> \<Rightarrow> f5 x xa | \<langle>x, xa\<rangle> \<Rightarrow> f6 x xa) = f1 x1"
cyclic_group_axioms: shows "Finite_And_Cyclic_Groups.cyclic_group H h"
lookup_zero: fixes k :: "'c" shows "poly_mapping.lookup 0 k = (0::'b)"
iprod_def: fixes xs :: "'a list"   and ys :: "'a list" shows "\<langle>xs,ys\<rangle> = sum_list (map2 (*) xs ys)"
ag_pOp_add_r: fixes A :: "('d, 'e) aGroup_scheme"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "a = b" shows "a \<plusminus>\<^bsub>A\<^esub> c = b \<plusminus>\<^bsub>A\<^esub> c"
r_neg2: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier H"   and "y \<in> carrier H" shows "x \<oplus>\<^bsub>H\<^esub> (\<ominus>\<^bsub>H\<^esub> x \<oplus>\<^bsub>H\<^esub> y) = y"
assms_1: shows "dim_col A\<^sup>T = dim_vec x"
set_1: fixes x1 :: "'b" shows "set_trm (Var x1) = {x1}"
C13: fixes x :: "'a" shows "(x\<^sup>\<star>)\<^sup>\<star> = x\<^sup>\<star>"
cod_simps_2: shows "trg.cod.leg0 = trg.leg0"
arc_edge: fixes e :: "'a"   and v :: "'a"   and g :: "'a"   and r :: "'a"   and t :: "'a"   and w :: "'a" assumes "e \<le> v * - v\<^sup>T \<sqinter> g"   and "vector v"   and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"   and "t \<le> g"   and "r\<^sup>T * g\<^sup>\<star> \<le> r\<^sup>T * w\<^sup>\<star>"   and "w * v \<le> v"   and "injective w"   and "arc e" shows "arc (prim_E w v e)"