hcju/isabelleps huggingface dataset:
inner_node_def: fixes n :: "'b LDCFG_node" shows "CFGExit_wf.inner_node n \<equiv> CFGExit_wf.valid_node n \<and> n \<noteq> NewEntry \<and> n \<noteq> NewExit"
collapse_1: fixes "term" :: "('a, 'b) Term.term" assumes "is_Var term" shows "Var (the_Var term) = term"
wrap_unwrap: shows "fix\<cdot>(wrap oo unwrap oo body) = fix\<cdot>body"
can_lock_devreserp: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t' :: "'t"   and ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and L :: "('l + 't + 't) set" assumes "s \<in> wf_state"   and "s -t'\<triangleright>ta'\<rightarrow> s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s'\<rangle> L \<wrong>" shows "\<exists>L'\<subseteq>L. t \<turnstile> \<langle>x, shr s\<rangle> L' \<wrong>"
CD: fixes K :: "nat"   and X :: "freemsg" shows "CRYPT K (DECRYPT K X) \<sim> X"
assms_1: fixes x :: "'a"   and y :: "'a" shows "h (x + y) = h x + h y"
one_neq_neg_one: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_char_0 one times plus zero minus uminus" shows "one \<noteq> uminus one"
taut: shows "w \<le> 0x80000000 \<or> 0x80000000 < w"
cases: fixes a :: "trm" assumes "valuep a" obtains T :: "POPLmarkRecord.type" and t :: "trm" where "a = (\<lambda>:Tb__. tb__)"   | T :: "POPLmarkRecord.type" and t :: "trm" where "a = (\<lambda><:T. t)"   | fs :: "(char list \<times> trm) list" where "a = Rcd fs" and "\<forall>(l, t)\<in>set fs. valuep t"
upt_Suc_append: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "[i..<Suc j] = [i..<j] @ [j]"
Convm_def: shows "Convm = Conv (id_matrix nr (output_size' m)) m"
assms: fixes F :: "'a set" assumes "finite F"   and "F \<subseteq> S" shows "(\<Sum>x\<in>F. norm (f x)) \<le> B"
limit_Append_path_nonelem_split: fixes p :: "(symbol \<times> character list) list"   and T :: "(symbol \<times> character list) set"   and k :: "nat"   and u :: "nat" assumes "p \<in> limit (Append T k) (\<P> k u)"   and "p \<notin> \<P> k u" shows "\<exists>q ts. p = q @ ts \<and> q \<in> \<P> k u \<and> charslength q = k \<and> admissible (q @ ts) \<and> (\<forall>t\<in>set ts. t \<in> T) \<and> (\<forall>t\<in>set (butlast ts). chars_of_token t = [])"
OUT: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a" assumes "antiparallel_edges \<Delta>"   and "edge \<Delta> x y" shows "antiparallel_edges.edg \<Delta> (antiparallel_edges.Vertex x) (antiparallel_edges.Edge x y)"
assms: shows "(u, v) \<in> E\<^sup>*"
insert_DiffM: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "add_mset x (M - {#x#}) = M"
lists_length_eq_eq_empty_iff: fixes A :: "'a set"   and n :: "nat" shows "({xs. set xs \<subseteq> A \<and> length xs = n} = {}) = (A = {} \<and> 0 < n)"
f12: shows "\<not> (\<exists>(a', b', c')\<in>set d. a' \<noteq> 0 \<and> - b'\<^sup>2 + 4 * a' * c' \<le> 0 \<and> (\<forall>(d, e, f)\<in>set a. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f = 0) \<and> (\<forall>(d, e, f)\<in>set b. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f < 0) \<and> (\<forall>(d, e, f)\<in>set c. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<le> 0) \<and> (\<forall>(d, e, f)\<in>set d. \<exists>y'>(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a'). \<forall>x\<in>{(- b' + - 1 * sqrt (b'\<^sup>2 - 4 * a' * c')) / (2 * a')<..y'}. d * x\<^sup>2 + e * x + f \<noteq> 0))"
counit_simps_2: shows "S.dom \<xi>' = S.UP f \<star>\<^sub>S S.UP g"
power_inductr: fixes z :: "'a ppath set"   and y :: "'a ppath set"   and x :: "'a ppath set"   and n :: "nat" assumes "z \<union> pp_prod y x \<subseteq> y" shows "pp_prod z (ppath_dioid.power x n) \<subseteq> y"
Min_le: fixes A :: "'b set"   and x :: "'b" assumes "finite A"   and "x \<in> A" shows "Min A \<le> x"
mult_poly_by_monom_def: shows "mult_poly_by_monom = (\<lambda>ys. (`#) (\<lambda>xs. (fst xs + fst ys, snd ys * snd xs)))"
getNeg_NegPos_map_simp2: fixes X :: "'e \<Rightarrow> 'd"   and src :: "'e negation_type list" shows "getNeg (NegPos_map X src) = map X (getNeg src)"
nothing_on_v2: shows "\<forall>c. \<parallel>len v2 ts c\<parallel> = 0 \<or> restrict v2 (clm ts) c = bot \<and> restrict v2 (res ts) c = bot"
alg_dvd_triv_left_iff: fixes x :: "'b"   and y :: "'b" shows "(x alg_dvd x * y) = (x = (0::'b) \<or> algebraic_int y)"
moebius_ocircline_circline: fixes M :: "moebius"   and H :: "ocircline" shows "moebius_ocircline M H = of_circline (moebius_circline M (of_ocircline H)) \<or> moebius_ocircline M H = opposite_ocircline (of_circline (moebius_circline M (of_ocircline H)))"
nat_of_natural_of_nat_inverse: fixes n :: "nat" shows "nat_of_natural (natural_of_nat n) = n"
irreducibleD: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.irreducible times one zero p"   and "p = times a b" shows "dvd.dvd times a one \<or> dvd.dvd times b one"
hyps: fixes x2a :: "('f, 'b) Term.term" assumes "x2a \<in> set ts" shows "weight (x2a \<cdot> \<sigma>) = weight x2a + (\<Sum>x\<in>#vars_term_ms (SCF x2a). weight (\<sigma> x) - w0)"
pext': shows "primitive_extractor (disc, sel) m = (spts, rst)"
reduced_word_Cons_reduce_contra: fixes A :: "'a set"   and as :: "'a list"   and a :: "'a" assumes "\<not> reduced_word A as" shows "\<not> reduced_word A (a # as)"
str_del_point_blocks_def: fixes p :: "'a" shows "kgdd.str_del_point_blocks p \<equiv> filter_mset ((\<notin>) p) (str_del_point_blocks x)"
one_le_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq one (numeral.numeral one plus n)"
prob_space_normal_density: fixes \<sigma> :: "real"   and \<mu> :: "real" assumes "0 < \<sigma>" shows "prob_space (density lborel (\<lambda>x. ennreal (normal_density \<mu> \<sigma> x)))"
lgrid_empty: fixes p :: "(nat \<times> int) list"   and ds :: "nat set" shows "lgrid p ds (level p) = {}"
nat_le_add_iff1: fixes j :: "nat"   and i :: "nat"   and u :: "nat"   and m :: "nat"   and n :: "nat" assumes "j \<le> i" shows "(i * u + m \<le> j * u + n) = ((i - j) * u + m \<le> n)"
induct: fixes P :: "('a, 'b) aformula \<Rightarrow> bool"   and a0 :: "('a, 'b) aformula" assumes "\<And>\<phi> \<psi>. \<lbrakk>P \<phi>; P \<psi>\<rbrakk> \<Longrightarrow> P (FAnd \<phi> \<psi>)"   and "\<And>v. P (FBool v)"   and "\<And>v. P (FBase v)"   and "\<And>v. P (FNot v)"   and "\<And>v va. P (FOr v va)"   and "\<And>v va. P (FEx v va)"   and "\<And>v va. P (FAll v va)" shows "P a0"
E_def: shows "E \<equiv> some_third_chamber C D (C \<inter> D)"
assms_1: shows "X \<in>\<^sub>\<circ> Vset \<alpha>"
east_west_on_equator_2: shows "proj2_incident west equator"
rtrancl_is_path: fixes u :: "'e"   and v :: "'e"   and E :: "('e \<times> 'e) set" assumes "(u, v) \<in> E\<^sup>*" obtains l :: "'e list" where "path E u l v"
intros: fixes x :: "Predicate_Compile_Tests.char" assumes "x = D \<or> x = E" shows "is_D_or_E x"
arg_max_set_def: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "arg_max_set f S = {x. is_arg_max f (\<lambda>x. x \<in> S) x}"
conj_def: shows "(\<^bold>&) \<equiv> \<lambda>x y. \<^bold>\<not>(x \<^bold>\<rightarrow> \<^bold>\<not>y)"
wordinterval_lowest_none_empty: fixes r :: "'b wordinterval" shows "(wordinterval_lowest_element r = None) = wordinterval_empty r"
induct: fixes P :: "'a poly \<Rightarrow> 'a poly \<Rightarrow> bool"   and a0 :: "'a poly"   and a1 :: "'a poly" assumes "\<And>c c'. P (c)\<^sub>p (c')\<^sub>p"   and "\<And>c c' n' p'. \<lbrakk>P (c)\<^sub>p c'; P (c)\<^sub>p p'\<rbrakk> \<Longrightarrow> P (c)\<^sub>p (CN c' n' p')"   and "\<And>c n p c'. \<lbrakk>P c (c')\<^sub>p; P p (c')\<^sub>p\<rbrakk> \<Longrightarrow> P (CN c n p) (c')\<^sub>p"   and "\<And>c n p c' n' p'. \<lbrakk>P c (CN c' n' p'); P p (CN c' n' p'); P (CN c n p) c'; P (CN c n p) p'; P (CN c n p) c'; P (CN c n p) p'\<rbrakk> \<Longrightarrow> P (CN c n p) (CN c' n' p')"   and "\<And>v b. P (poly.Bound v) b"   and "\<And>v va b. P (Add v va) b"   and "\<And>v va b. P (Sub v va) b"   and "\<And>v va b. P (Mul v va) b"   and "\<And>v b. P (Neg v) b"   and "\<And>v va b. P (Pw v va) b"   and "\<And>v va vb vc. P (CN v va vb) (poly.Bound vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Add vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Sub vc vd)"   and "\<And>v va vb vc vd. P (CN v va vb) (Mul vc vd)"   and "\<And>v va vb vc. P (CN v va vb) (Neg vc)"   and "\<And>v va vb vc vd. P (CN v va vb) (Pw vc vd)"   and "\<And>a v. P a (poly.Bound v)"   and "\<And>a v va. P a (Add v va)"   and "\<And>a v va. P a (Sub v va)"   and "\<And>a v va. P a (Mul v va)"   and "\<And>a v. P a (Neg v)"   and "\<And>a v va. P a (Pw v va)"   and "\<And>vc v va vb. P (poly.Bound vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Add vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Sub vc vd) (CN v va vb)"   and "\<And>vc vd v va vb. P (Mul vc vd) (CN v va vb)"   and "\<And>vc v va vb. P (Neg vc) (CN v va vb)"   and "\<And>vc vd v va vb. P (Pw vc vd) (CN v va vb)" shows "P a0 a1"
simps_1: fixes v :: "'vertex"   and p :: "('form, 'var) antecedent" shows "tree.root (edge_tree v p) = (adjacentTo v p, v, p)"
less_sets_singleton2: fixes A :: "'a set"   and a :: "'a" shows "(A \<lless> {a}) = (\<forall>x\<in>A. x < a)"
traces_failures: fixes xs :: "'b list"   and P :: "'b process" assumes "xs \<in> traces P" shows "(xs, {}) \<in> failures P"
wf_subst_properties: fixes \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" shows "(\<theta> \<in> wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_set) = wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>"
integrate_insert_final_step: fixes pm :: "'a extended"   and sm :: "'a extended"   and l :: "nat" assumes "invariant pm sm"   and "idx s pm = Inr l"   and "idx s sm = Inr (Suc l)" shows "is_certified_associated_string' (Inr (take l s @ to_woot_char m # drop l s))"
set_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "local.set a = some_img ` hom unity a"
comp_arr_point: fixes f :: "'s"   and x :: "'s" assumes "S.arr f"   and "\<guillemotleft>x : S.unity \<rightarrow>\<^sub>S S.dom f\<guillemotright>" shows "f \<cdot>\<^sub>S x = S.mkPoint (S.cod f) (S.Fun f (S.img x))"
a_trees: fixes i :: "nat" shows "\<forall>t\<in>set (a i). t \<in> trees"
totient_Suc_0: shows "totient (Suc 0) = Suc 0"
ring_is_ag: fixes R :: "('e, 'f) Ring_scheme" assumes "Ring R" shows "aGroup R"
P1DQ1: shows "(P1, D, Q1) = diagonalize_2x2 A"
simps_1: fixes s :: "'b stream" shows "sdrop 0 s = s"
span_zero: fixes S :: "'b set" shows "(0::'b) \<in> ideal S"
RedCondT: fixes E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   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 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" shows "P,E \<turnstile> \<langle>if (true) e\<^sub>1 else e\<^sub>2,s\<rangle> \<rightarrow> \<langle>e\<^sub>1,s\<rangle>"
assms_4: fixes i :: "'c" assumes "i \<in> A - B" shows "g i = \<one>"
naturality: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "\<tau>\<^sub>1 (B.cod \<mu>) \<cdot>\<^sub>C (G \<mu> \<star>\<^sub>C \<tau>\<^sub>0 (src\<^sub>B \<mu>)) = (\<tau>\<^sub>0 (trg\<^sub>B \<mu>) \<star>\<^sub>C F \<mu>) \<cdot>\<^sub>C \<tau>\<^sub>1 (B.dom \<mu>)"
ipt_tcp_flags_NoMatch_def: shows "ipt_tcp_flags_NoMatch \<equiv> TCP_Flags {} {TCP_SYN}"
simps_3: fixes v :: "('fun, 'var) Term.term" shows "tfr\<^sub>s\<^sub>t\<^sub>p (send\<langle>v\<rangle>\<^sub>s\<^sub>t) = True"
elims_3: fixes x :: "bool list"   and xa :: "tree" assumes "\<not> branch x xa" obtains d :: "bool" and ds :: "bool list" and l :: "tree" and r :: "tree" where "x = d # ds" and "xa = Branching l r" and "\<not> (if d then branch ds l else branch ds r)"   | v :: "bool" and va :: "bool list" where "x = v__ # va__" and "xa = Leaf"   | v :: "tree" and va :: "tree" where "x = []" and "xa = Branching v va"
sums_le: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and s :: "'a"   and t :: "'a" assumes "\<And>n. f n \<le> g n"   and "f sums s"   and "g sums t" shows "s \<le> t"
FORKs_is_FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m: shows "FORKs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N (replicate N 0) = fold (\<lambda>i P. P|||FORK i) [1..<N] (FORK 0)"
impl_times: fixes a :: "'a N"   and b :: "'a N" shows "(a l\<rightarrow> b) * a = (b l\<rightarrow> a) * b"
BitM_plus_one: fixes n :: "num" shows "Num.BitM n + num.One = num.Bit0 n"
add: fixes u :: "'a \<Rightarrow> ennreal"   and v :: "'a \<Rightarrow> ennreal" assumes "simple_function M u"   and "P u"   and "simple_function M v"   and "\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0"   and "P v" shows "P (\<lambda>x. v x + u x)"
Delay_L1: fixes t :: "nat"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and y :: "nat \<Rightarrow> aType list"   and d :: "nat"   and x :: "nat \<Rightarrow> aType list"   and i2 :: "nat \<Rightarrow> ECall_Info list"   and t2 :: "nat" assumes "\<forall>t1<t. i1 t1 = []"   and "Delay y i1 d x i2"   and "t2 < t + d" shows "i2 t2 = []"
gallery_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and Cs :: "'b set list" assumes "ChamberComplexFolding X f"   and "SimplicialComplex.maxsimpchain X Cs" shows "SimplicialComplex.maxsimpchain X (f \<Turnstile> Cs)"
gcd_eq_1_imp_coprime: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "gcd a b = one" shows "algebraic_semidom.coprime times one a b"
calc_spread_def: fixes k :: "'a"   and ps :: "(real, 'a) vec list" shows "calc_spread k ps = (case ps of [] \<Rightarrow> 0 | a # list \<Rightarrow> let ks = map (\<lambda>p. p $ k) (tl (a # list)) in fold max ks (hd (a # list) $ k) - fold min ks (hd (a # list) $ k))"
dd: shows "dim ((+) (- a) ` S) = dim ((+) (- b) ` T)"
coprime_n_idealsTr0: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "ideal R A"   and "ideal R B"   and "ideal R C"   and "coprime_ideals R A C"   and "coprime_ideals R B C" shows "coprime_ideals R (A \<diamondsuit>\<^sub>r B) C"
every_eqcl_has_reg: fixes A :: "'b list set"   and X :: "'b list set" assumes "finite (UNIV // \<approx>A)"   and "X \<in> UNIV // \<approx>A" shows "\<exists>r. X = lang r"
f_the_inv_into_f_bij_betw: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and x :: "'d" assumes "bij_betw f A B"   and "bij_betw f A B \<Longrightarrow> x \<in> B" shows "f (the_inv_into A f x) = x"
U_3: fixes x :: "'a" assumes "x \<in> U" shows "returns_to p2 x"
gStates_finite: fixes prog :: "program"   and g :: "gState" shows "finite ((gState_progress_rel prog)\<^sup>* `` {g})"
assms_3: shows "(b, S) \<notin> tvsT \<tau>"
schur_upper_triangular_def: fixes A :: "'b mat"   and es :: "'b list" shows "schur_upper_triangular A es = (case schur_decomposition A es of (B, x, xa) \<Rightarrow> B)"
Rep_formula_inverse: fixes x :: "('b, 'c, 'd, 'e) formula" shows "Abs_formula (Rep_formula x) = x"
invariant_factor_decomposition_Zn: obtains ns :: "nat list" where "DirProds (\<lambda>n. Z (ns ! n)) {..<length ns} \<cong> G" and "successively (dvd) ns" and "length ns \<le> card gen"
3: shows "S' = u # U \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>"
integral_pmf: fixes X :: "'b set"   and M :: "'b pmf" shows "integral\<^sup>L (count_space X) (pmf M) = measure_pmf.prob M X"
biessentially_surjective_on_objects: fixes a' :: "'b" assumes "C.obj a'" shows "\<exists>a. B.obj a \<and> C.equivalent_objects (F.map\<^sub>0 a) a'"
assms_6: shows "ptr \<notin> set |h \<turnstile> get_dom_component (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node_ptr)|\<^sub>r"
c'_insertion: shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 0) = c'"
assms_4: shows "f \<star> a \<cong> g \<star> a"
sym_trans_comp_subset: fixes r :: "('b \<times> 'b) set" assumes "sym r"   and "trans r" shows "r\<inverse> O r \<subseteq> r"
l8_14_2_1b_bis: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and X :: "'p" assumes "A B Perp C D"   and "Col X A B"   and "Col X C D" shows "X PerpAt A B C D "
len_left: fixes v :: "view"   and ts :: "traffic"   and c :: "cars" shows "left (ext v) \<le> left (len v ts c)"
add_top: fixes x :: "'k" assumes "(0::'k) \<le> x" shows "x + \<top> = \<top>"
assms_1: shows "A \<turnstile> \<langle>l, [D]\<^bsub>v,n\<^esub>\<rangle> \<leadsto> \<langle>l', Z\<rangle>"
simps_2: fixes f :: "char list"   and ts :: "fterm list" shows "ground\<^sub>t (Fun f ts) = ground\<^sub>t\<^sub>s ts"
length_append_singleton: fixes xs :: "'a list"   and x :: "'a" shows "|xs @ [x]| = Suc |xs|"
monotone_lzip2: fixes xs :: "'d llist" shows "monotone lprefix lprefix (lzip xs)"
invariant_next_subseqs: fixes base :: "'b"   and elements :: "'a list"   and n :: "nat"   and state :: "'a \<times> 'a iarray \<times> (nat \<times> 'b) list"   and out :: "'b list"   and state' :: "'a \<times> 'a iarray \<times> (nat \<times> 'b) list" assumes "invariant base elements n state"   and "local.next_subseqs state = (out, state')" shows "invariant base elements (Suc n) state'"
det3_nonneg_segments_of_aformI: fixes e :: "nat \<Rightarrow> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and seg :: "(real \<times> real) \<times> real \<times> real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "length (half_segments_of_aform X) \<noteq> 1"   and "seg \<in> set (segments_of_aform X)" shows "0 \<le> det3 (fst seg) (snd seg) (aform_val e X)"
nonEmptySet: fixes A :: "'a list" shows "A \<noteq> [] \<longrightarrow> (\<exists>a. a \<in> set A)"
takeWhile_append2: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool"   and ys :: "'c list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "takeWhile P (xs @ ys) = xs @ takeWhile P ys"
closed_UN_well_related: fixes XX :: "'c set set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<forall>X\<in>XX. well_related_set X less_eq \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. \<not> less_eq x y \<longrightarrow> y \<in> X)" shows "well_related_set (\<Union> XX) less_eq"
add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "n x \<oplus> (n y \<oplus> n z) = n x \<oplus> n y \<oplus> n z"
index_lfp: fixes c :: "com"   and f :: "'a acom \<Rightarrow> 'a acom" shows "Collecting.lfp c f \<in> {C. strip C = c}"
assms_2: shows "p \<in> dterms \<Gamma> (p1 \<oplus> p2)"
dirichlet_prod_assoc_aux1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b"   and h :: "nat \<Rightarrow> 'b" assumes "0 < n" shows "dirichlet_prod f (dirichlet_prod g h) n = (\<Sum>(a, b, c)\<in>{(a, b, c). a * b * c = n}. f a * g b * h c)"
of_nat_less_two_power: fixes n :: "nat" shows "of_nat n < (2::'c) ^ n"
K: shows "\<forall>n>K. f $$ - int n = (0::'a)"
smult_distrib_left: fixes r :: "'r"   and m :: "'m"   and n :: "'m" assumes "r \<in> R"   and "m \<in> M"   and "n \<in> M" shows "r \<cdot> (m + n) = r \<cdot> m + r \<cdot> n"
finter_funion_distrib2: fixes B :: "'a fset"   and C :: "'a fset"   and A :: "'a fset" shows "B |\<union>| C |\<inter>| A = B |\<inter>| A |\<union>| (C |\<inter>| A)"
Suc_natural_minus_one: fixes n :: "natural" shows "Code_Numeral.Suc n - 1 = n"
assms: fixes i :: "'a" assumes "i \<in> Basis" shows "continuous F (\<lambda>x. blinfun_apply (f x) i)"
right_total_def: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "right_total R = (\<forall>y. \<exists>x. R x y)"
assms_1: shows "(p, e) \<in> prob_proj PROB vs"
mem_affine: fixes S :: "'b set"   and x :: "'b"   and y :: "'b"   and u :: "real"   and v :: "real" assumes "affine S"   and "x \<in> S"   and "y \<in> S"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> S"
ts_ok_mono: fixes P :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool"   and ts :: "'f \<Rightarrow> ('g \<times> 'i \<Rightarrow>f nat) option"   and m :: "'h"   and Q :: "'f \<Rightarrow> 'g \<Rightarrow> 'h \<Rightarrow> bool" assumes "ts_ok P ts m"   and "\<And>t x. P t x m \<Longrightarrow> Q t x m" shows "ts_ok Q ts m"
arrE: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" obtains "f \<noteq> Null" and "Dom f \<in> {A. |A| <o \<AA>}" and "Cod f \<in> {A. |A| <o \<AA>}" and "Map f \<in> extensional (Dom f) \<inter> (Dom f \<rightarrow> Cod f)"
sel_4: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list" shows "args (Fun x21 x22) = x22"
wt_Hd_Sym: fixes f :: "'s" shows "wt (Hd (Sym f)) = wt_sym f + the_enat (enat \<delta> * arity_sym f)"
simps_3: fixes p :: "real mpoly"   and L :: "real list" shows "aEval (Leq p) L = (insertion (nth_default 0 L) p \<le> 0)"
CFG_wf_axioms_def: fixes \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list"   and Entry :: "'f \<Rightarrow> 'g"   and defs :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set"   and "uses" :: "'f \<Rightarrow> 'g \<Rightarrow> 'i set" shows "CFG_wf_axioms \<alpha>n invar inEdges' Entry defs uses \<equiv> \<forall>g. \<forall>m\<in>set (\<alpha>n g). \<forall>v\<in>uses g m. CFG_base.defAss' \<alpha>n invar inEdges' Entry defs g m v"
icd_imp_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and n :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_icdi prob n \<pi> k" shows "IFC_def.is_cdi prob n \<pi> k"
homf_preserves_cod: fixes f :: "'a"   and A :: "'o" assumes "f \<in> Ar" shows "Hom(A,_) \<^bsub>\<o>\<^esub> (Cod f) = Cod\<^bsub>Set\<^esub> Hom(A,_) \<^bsub>\<a>\<^esub> f"
proj2_Col_def: fixes p :: "proj2"   and q :: "proj2"   and r :: "proj2" shows "proj2_Col p q r \<equiv> \<exists>i j k. i *\<^sub>R proj2_rep p + j *\<^sub>R proj2_rep q + k *\<^sub>R proj2_rep r = 0 \<and> (i \<noteq> 0 \<or> j \<noteq> 0 \<or> k \<noteq> 0)"
TESL_interp_stepwise_weakly_precedes_coind_unfold: fixes K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and n :: "nat" shows "\<lbrakk> K\<^sub>1 weakly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> n\<^esup> = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y \<rbrakk>\<^sub>p\<^sub>r\<^sub>i\<^sub>m \<inter> \<lbrakk> K\<^sub>1 weakly precedes K\<^sub>2 \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L\<^bsup>\<ge> Suc n\<^esup>"