hcju/isabelleps huggingface dataset:
exec_elim_cases_6: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and f :: "'f"   and g :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Guard f g c,s\<rangle> \<Rightarrow> t" obtains s :: "'e" where "sb__ = Normal sc__" and "sc__ \<in> g" and "\<Gamma>\<turnstile> \<langle>c,Normal sc__\<rangle> \<Rightarrow> t"   | s :: "'e" where "sb__ = Normal sd__" and "t = Fault f__" and "sd__ \<notin> g"   | f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"
represents_nth_root_pos_real: fixes n :: "nat"   and p :: "int poly"   and x :: "real" assumes "n \<noteq> 0"   and "p represents x"   and "0 < x" shows "poly_nth_root n p represents root n x"
productive_imp_INTERP: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {}" shows "ground_resolution_with_selection.INTERP S N \<TTurnstile> C"
ord_eq_le_subst: fixes a :: "'k"   and f :: "'l \<Rightarrow> 'k"   and b :: "'l"   and c :: "'l" assumes "a = f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
round_numeral: fixes n :: "num" shows "round (numeral n) = numeral n"
focus_def: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus X p = (\<Sum>t\<in>keys p. monomial (monomial (lookup p t) (except t X)) (except t (- X)))"
revis_closure: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A" shows "(K \<div> .\<not> \<phi>) \<oplus> \<phi> = Cn ((K \<div> .\<not> \<phi>) \<oplus> \<phi>)"
ind_hyp_xs: assumes "linorder_class.sorted xs"   and "distinct xs" shows "Sorted_Less.sorted xs"
arctan_upper_33_def: shows "arctan_upper_33 \<equiv> \<lambda>x. (64 * x ^ 4 + 735 * x\<^sup>2 + 945) * x / (15 * (15 * x ^ 4 + 70 * x\<^sup>2 + 63))"
limproc_is_thelub: fixes S :: "nat \<Rightarrow> 'a process" assumes "chain S" shows "Lub S = lim_proc (range S)"
ordertype_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "ordertype A r \<equiv> ZFC_in_HOL.set (ordermap A r ` A)"
base_spset: shows "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV (\<sharp>\<cdot>) vs \<and> R_scalar_mult.R_lin_independent UNIV (\<sharp>\<cdot>) vs"
csa1: shows "set (characterize_consistent_signs_at_roots p qs1) \<subseteq> set signs1"
HoHV_def: fixes \<mu> :: "'a \<times> 'a \<times> 'a" shows "B.HoHV \<mu> \<equiv> if B.VVV.arr \<mu> then (fst \<mu> \<star>\<^sub>B fst (snd \<mu>)) \<star>\<^sub>B snd (snd \<mu>) else B.null"
same_program_def: fixes c :: "('b com option list \<times> 'b) list"   and clist :: "('b com option \<times> 'b) list list" shows "same_program c clist \<equiv> \<forall>j<length c. fst (c ! j) = map (\<lambda>x. fst (x ! j)) clist"
isTrivial_the_trivial: fixes g :: "'g"   and v :: "'val"   and vs :: "'val list"   and v' :: "'val" assumes "uninst_code.ssa.phi phis g v = Some vs"   and "uninst_code.ssa.isTrivialPhi phis g v v'" shows "the_trivial v vs = Some v'"
borel_measurable_continuous_Pair: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "continuous_on UNIV (\<lambda>x. H (fst x) (snd x))" shows "(\<lambda>x. H (f x) (g x)) \<in> borel_measurable M"
l1: fixes m :: "'a set"   and x :: "'a"   and y :: "'a" assumes "m open"   and "x \<in> X"   and "y \<in> X"   and "x \<le> y"   and "x \<in> m" shows "y \<in> m"
fac_triv: fixes p :: "'b list"   and x :: "'b list"   and q :: "'b list" assumes "p \<cdot> x \<cdot> q = x" shows "p = \<epsilon>"
divisor_hd_coeff1_neq0: fixes \<phi> :: "atom fm"   and a :: "atom" assumes "qfree \<phi>"   and "a \<in> atoms \<phi>"   and "divisor a \<noteq> 0" shows "divisor (hd_coeff1 (zlcms (map hd_coeff (atoms\<^sub>0 \<phi>))) a) \<noteq> 0"
filterlim_at_within_If: fixes f :: "'a \<Rightarrow> 'c"   and G :: "'c filter"   and x :: "'a"   and A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and g :: "'a \<Rightarrow> 'c" assumes "filterlim f G (at x within A \<inter> {x. P x})"   and "filterlim g G (at x within A \<inter> {x. \<not> P x})" shows "LIM x at x within A. if P x then f x else g x :> G"
domain_permutation_def: fixes A :: "'c set"   and B :: "'d set" shows "domain_permutation A B = {(f, f'). (f, f') \<in> (A \<rightarrow>\<^sub>E B) \<times> (A \<rightarrow>\<^sub>E B) \<and> (\<exists>p. p permutes A \<and> (\<forall>x\<in>A. f x = f' (p x)))}"
fs_carrier: shows "set (map of_int_hom.vec_hom fs) \<subseteq> Rn"
Act_eq_iff_perm: fixes \<alpha>1 :: "'c"   and x1 :: "('a, 'b, 'c) formula"   and \<alpha>2 :: "'c"   and x2 :: "('a, 'b, 'c) formula" shows "(Formula.Act \<alpha>1 x1 = Formula.Act \<alpha>2 x2) = (\<exists>p. supp x1 - bn \<alpha>1 = supp x2 - bn \<alpha>2 \<and> (supp x1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> x1 = x2 \<and> supp \<alpha>1 - bn \<alpha>1 = supp \<alpha>2 - bn \<alpha>2 \<and> (supp \<alpha>1 - bn \<alpha>1) \<sharp>* p \<and> p \<bullet> \<alpha>1 = \<alpha>2)"
span_induct_alt: fixes x :: "'b"   and S :: "'b set"   and h :: "'b \<Rightarrow> bool" assumes "x \<in> ideal S"   and "h (0::'b)"   and "\<And>c x y. \<lbrakk>x \<in> S; h y\<rbrakk> \<Longrightarrow> h (c * x + y)" shows "h x"
AbGroup_inner_dirsum_subset_proj_eq_0: fixes Gs :: "'a set list"   and X :: "'a set" assumes "Gs \<noteq> []"   and "\<forall>G\<in>set Gs. AbGroup G"   and "add_independentS Gs"   and "X \<subseteq> (\<Oplus>G\<leftarrow>Gs. G)"   and "\<forall>i<length Gs. (\<lambda>a. if a \<in> (\<Oplus>G\<leftarrow>Gs. G) then \<Oplus>Gs\<leftarrow> a ! i else (0::'a)) ` X = 0" shows "X = 0"
max2_equals1: fixes r :: "('d \<times> 'd) set"   and a :: "'d"   and b :: "'d" assumes "wo_rel r"   and "a \<in> Field r"   and "b \<in> Field r" shows "(max2 r a b = a) = ((b, a) \<in> r)"
ctr_1: fixes xs :: "('a, 'b) tllist" assumes "is_TNil xs" shows "llist_of_tllist xs = LNil"
conservation_const_pointwise: fixes u :: "nat" assumes "u \<in> V - {s, t}" shows "(\<Sum>v\<in>E `` {u}. f (u, v)) = (\<Sum>v\<in>E\<inverse> `` {u}. f (v, u))"
set_sort: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "class.linorder less_eq less" shows "set (linorder.sort_key less_eq f xs) = set xs"
Minimal_SolutionsI': fixes x :: "nat list"   and y :: "nat list"   and a :: "nat list"   and b :: "nat list" assumes "(x, y) \<in> Linear_Diophantine_Equations.Solutions a b"   and "nonzero x"   and "\<not> (\<exists>(u, v)\<in>Linear_Diophantine_Equations.Minimal_Solutions a b. u @ v <\<^sub>v x @ y)" shows "(x, y) \<in> Linear_Diophantine_Equations.Minimal_Solutions a b"
frechet_derivative_componentwise: fixes f :: "'b \<Rightarrow> real"   and a :: "'b"   and v :: "'b" assumes "f differentiable at a" shows "frechet_derivative f (at a) v = (\<Sum>i\<in>Basis. v \<bullet> i * frechet_derivative f (at a) i)"
open_Collect_neq: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "open {x. f x \<noteq> g x}"
peek_st_def: fixes P :: "st \<Rightarrow> 'b assn" shows "peek_st P = (\<lambda>Y s. P (snd s) Y s)"
round_up: fixes x :: "real"   and prec :: "int" shows "x \<le> round_up prec x"
fps_shift_fps_X_power: fixes n :: "nat"   and m :: "nat" assumes "n \<le> m" shows "fps_shift n (fps_X ^ m) = fps_X ^ (m - n)"
v12: shows "M.sumlist (map (\<lambda>i. g i \<cdot>\<^sub>v gs ! i) [0..<k]) + g k \<cdot>\<^sub>v gs ! k \<in> carrier_vec n"
linear_iff: fixes f :: "'d \<Rightarrow> 'e" shows "linear f = ((\<forall>x y. f (x + y) = f x + f y) \<and> (\<forall>c x. f (c *\<^sub>R x) = c *\<^sub>R f x))"
Rf_Cd_mono: fixes d :: "'d" shows "mono (Rf (Cd d))"
final_threads_def: fixes s :: "('f, 'e) locks \<times> (('e \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option) \<times> 'g) \<times> ('e \<Rightarrow> 'h wait_set_status option) \<times> 'e set" shows "final_threads s \<equiv> {t. final_thread s t}"
card_image: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" assumes "inj_on f A" shows "card (f ` A) = card A"
base: fixes x :: "nat"   and y :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat" assumes "(x, y) \<in> rel" shows "(x, y) \<in> Clos rel n"
splitAt_ram: fixes ram :: "'c"   and vs :: "'c list" assumes "ram \<in> set vs" shows "vs = fst (splitAt ram vs) @ ram # snd (splitAt ram vs)"
rtrancl_imp_seq: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, y) \<in> r\<^sup>*" shows "\<exists>f n. f 0 = x \<and> f n = y \<and> (\<forall>i<n. (f i, f (Suc i)) \<in> r)"
th2: shows "s @ t setinterleaves ((ta, u), A) \<longrightarrow> s @ t = u"
bounded_by_mono_alt: fixes b :: "real"   and Q :: "'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real" assumes "bounded_by b Q"   and "P \<tturnstile> Q" shows "bounded_by b P"
map_add_find_left: fixes n :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and m :: "'b \<Rightarrow> 'a option" assumes "n k = None" shows "(m ++ n) k = m k"
seq_Sup_distl: fixes P :: "('b, 'd) urel"   and A :: "('d, 'c) urel set" shows "P ;; \<Or>A = \<Or>((;;) P ` A)"
op_list_tl_def: shows "op_list_tl \<equiv> tl"
prems: shows "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
Unit_Right: fixes P :: "'a hrel" assumes "P is \<H>" shows "P ;; \<I>\<I> = P"
8: shows "minarc (v * - v\<^sup>T \<sqinter> g) \<le> v"
res_def: fixes key_gen :: "'a spmf" shows "key.res key_gen = RES (key.key_oracle key_gen \<oplus>\<^sub>O key.key_oracle key_gen) None"
closure_of_restrict: fixes X :: "'c topology"   and S :: "'c set" shows "X closure_of S = X closure_of (topspace X \<inter> S)"
consts_list_comb: fixes f :: "'a"   and xs :: "'a list" shows "consts (list_comb f xs) = consts f |\<union>| ffUnion (fset_of_list (map consts xs))"
pointwise_def: fixes f :: "'b set \<Rightarrow> 'c set" shows "pointwise f = (\<forall>X. f X = \<Union> {f {x} |x. x \<in> X})"
log_minus_eq_powr: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "0 < b"   and "b \<noteq> 1"   and "0 < x" shows "log b x - y = log b (x * b powr - y)"
lsu_bot: shows "bot_pres \<nu>\<^sup>\<natural>"
finite_emeasure_space: shows "emeasure M (space M) \<noteq> top"
psubstAbs_preserves_good: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set" assumes "goodEnv rho"   and "goodAbs A" shows "goodAbs A $[rho]"
maximalD1: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set" assumes "x \<in> maximal f S" shows "x \<in> S"
bounded_iff: fixes b :: "'a DBMEntry"   and c :: "'a DBMEntry"   and a :: "'a DBMEntry" shows "(linordered_monoid.max b c \<preceq> a) = (b \<preceq> a \<and> c \<preceq> a)"
rel_spmf_reflI: fixes p :: "'d spmf"   and P :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>x. x \<in> set_spmf p \<Longrightarrow> P x x" shows "rel_spmf P p p"
assoc: fixes a :: "'a word"   and b :: "'a word"   and c :: "'a word" shows "signed.min (signed.min a b) c = signed.min a (signed.min b c)"
almost_full_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "almost_full_on P A = (\<forall>f\<in>SEQ A. good P f)"
comp_trancl: fixes R :: "('b \<times> 'b) set"   and S :: "('b \<times> 'b) set" assumes "R O S \<subseteq> S" shows "R O S\<^sup>+ \<subseteq> S\<^sup>+"
span_induct_alt: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and S :: "'b set"   and h :: "'b \<Rightarrow> bool" assumes "module scale"   and "x \<in> module.span scale S"   and "h (0::'b)"   and "\<And>c x y. \<lbrakk>x \<in> S; h y\<rbrakk> \<Longrightarrow> h (scale c x + y)" shows "h x"
echelon_form_echelon_form_of: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec" assumes "is_bezout_ext bezout" shows "echelon_form (echelon_form_of A bezout)"
p'_def: shows "p' = (case convert_poly var p (xs' @ x # xs) of Some p' \<Rightarrow> p')"
wf_syscls_is_class_xcpt: fixes C :: "String.literal"   and P :: "'a prog" assumes "C \<in> sys_xcpts"   and "wf_syscls P" shows "is_class P C"
globs_lupd: fixes vn :: "lname"   and v :: "val"   and s :: "st" shows "globs (lupd(vn\<mapsto>v) s) = globs s"
SeqQuoteP_Mem_imp_All2: fixes i :: "name"   and i' :: "name"   and \<alpha> :: "fm"   and Fi :: "name \<Rightarrow> tm"   and j :: "name"   and j' :: "name"   and pi :: "perm"   and s :: "name"   and k :: "name" assumes "insert (QuoteP (Var i) (Var i')) (quote_all p Vs) \<turnstile> \<alpha> IMP PfP (ssubst \<lfloor>\<alpha>\<rfloor>(insert i Vs) (insert i Vs) Fi)"   and "supp \<alpha> - {atom i} \<subseteq> atom ` Vs"   and "j \<in> Vs"   and "p \<bullet> j = j'"   and "pi = (atom i \<rightleftharpoons> atom i') + p"   and "Fi = make_F (insert i Vs) pi"   and "atom i \<sharp> (j, j', s, k, p)"   and "atom i' \<sharp> (i, p, \<alpha>)"   and "atom j \<sharp> (j', s, k, \<alpha>)"   and "atom j' \<sharp> (s, k, \<alpha>)"   and "atom s \<sharp> (k, \<alpha>)"   and "atom k \<sharp> (\<alpha>, p)" shows "insert (SeqQuoteP (Var j) (Var j') (Var s) (Var k)) (quote_all p (Vs - {j})) \<turnstile> All2 i (Var j) \<alpha> IMP PfP (ssubst \<lfloor>All2 i (Var j) \<alpha>\<rfloor>Vs Vs F)"
n\<^sub>2_def: shows "n\<^sub>2 = Suc (Max (fail_t \<union> \<Union> {succeed_t j |j. j < i}))"
p_max_eq: shows "ego2.p_max = ego.q \<delta> - v\<^sub>e\<^sup>2 / a\<^sub>e / 2"
RPD_safe_fv_regex: fixes mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and ms :: "mregex" assumes "safe_regex Past Strict (from_mregex mr \<phi>s)"   and "ms \<in> RPD mr" shows "fv_regex (from_mregex ms \<phi>s) = fv_regex (from_mregex mr \<phi>s)"
assms_13: fixes \<phi> :: "'a formula"   and \<psi> :: "'a formula"   and I :: "\<I>" assumes "fv \<phi> \<subseteq> fv \<psi>"   and "safe_formula \<phi>"   and "P \<phi>"   and "P \<psi>" shows "P (Since \<phi> I \<psi>)"
ax1: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<box>F \<longrightarrow> F"
finite_mult_not_eq_zero_leftI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite {a. f a \<noteq> (0::'b)}" shows "finite {a. g a * f a \<noteq> (0::'b)}"
summable_norm_sin: fixes x :: "'a" shows "summable (\<lambda>n. norm (sin_coeff n *\<^sub>R x ^ n))"
mirror_l: fixes p :: "fm" assumes "iszlfm p \<and> d_\<beta> p 1" shows "iszlfm (mirror p) \<and> d_\<beta> (mirror p) 1"
tick_count_fsuc: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and n :: "nat" assumes "dilating f sub r" shows "tick_count r c (f (Suc n)) = tick_count r c (f n) + card {k. k = f (Suc n) \<and> hamlet (Rep_run r k c)}"
eq: fixes x :: "real" shows "f x * ennreal \<bar>c\<bar> / ennreal \<bar>c\<bar> = f x"
assms_3: shows "ContractsWithBilateralSubstitutesAndIRC Xd Xh (Pd'(d := Pd'')) Ch'"
distinct_card: fixes xs :: "'q list" assumes "distinct xs" shows "card (set xs) = length xs"
hb_consistent_append_porder: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a"   and y :: "'a" assumes "hb_consistent (xs @ ys)"   and "x \<in> set xs"   and "y \<in> set ys" shows "\<not> y \<prec> x"
assms_2: shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t'"
HCP_game_def: fixes A :: "'index \<Rightarrow> bool \<Rightarrow> bool \<Rightarrow> (bool \<times> 'index \<times> bool \<times> bool \<Rightarrow> bool spmf) \<Rightarrow> 'range \<Rightarrow> bool spmf" shows "local.etp.HCP_game A = (\<lambda>\<sigma> b\<^sub>\<sigma> D. I \<bind> (\<lambda>(\<alpha>, \<tau>). local.etp.S \<alpha> \<bind> (\<lambda>x. A \<alpha> \<sigma> b\<^sub>\<sigma> D x \<bind> (\<lambda>b'. let b = B \<alpha> (F\<^sub>i\<^sub>n\<^sub>v \<alpha> \<tau> x) in return_spmf (b = b')))))"
isProperPrefixPrefixToLevel: fixes level :: "nat"   and t :: "('b \<times> bool) list" assumes "level < currentLevel t" shows "\<exists>s. prefixToLevel level t @ s = t \<and> s \<noteq> [] \<and> marked (hd s)"
Ints_mult: 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"   and a :: "'b"   and b :: "'b" assumes "class.ring_1 one times plus zero minus uminus"   and "a \<in> ring_1.Ints one plus zero minus"   and "b \<in> ring_1.Ints one plus zero minus" shows "times a b \<in> ring_1.Ints one plus zero minus"
ctr_simps_mem_Collect_eq: fixes a :: "'e"   and P :: "'e \<Rightarrow> bool" shows "(a \<in> Collect P) = P a"
Proj_sandwich: fixes A :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b"   and S :: "'a ccsubspace" assumes "isometry A" shows "sandwich A *\<^sub>V Proj S = Proj (A *\<^sub>S S)"
fps_binomial_ODE_unique': fixes a :: "'a fps"   and c :: "'a" shows "(fps_deriv a = fps_const c * a / (1 + fps_X) \<and> fps_nth a 0 = (1::'a)) = (a = fps_binomial c)"
distr_bij_count_space: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and B :: "'e set" assumes "bij_betw f A B" shows "distr (count_space A) (count_space B) f = count_space B"
indicator_simps_2: fixes x :: "'k"   and S :: "'k set" assumes "x \<notin> S" shows "indicator S x = (0::'l)"
pathsConverge'_def: fixes g :: "'g"   and x :: "'node"   and xs :: "'node list"   and y :: "'node"   and ys :: "'node list"   and z :: "'node" shows "pathsConverge' g x xs y ys z \<equiv> g \<turnstile> x-xs\<rightarrow>z \<and> g \<turnstile> y-ys\<rightarrow>z \<and> 1 < length xs \<and> 1 < length ys \<and> x \<noteq> y \<and> set (butlast xs) \<inter> set (butlast ys) = {}"
store_instr_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and t1 :: "'b sparc_state"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> (fst instr = load_store_type STB \<or> fst instr = load_store_type STH \<or> fst instr = load_store_type ST \<or> fst instr = load_store_type STA \<or> fst instr = load_store_type STD) \<and> get_S (cpu_reg_val PSR s1) = 0 \<and> get_S (cpu_reg_val PSR s2) = 0 \<and> t1 = snd (fst (store_instr instr s1)) \<and> t2 = snd (fst (store_instr instr s2))" shows "low_equal t1 t2"
simps_7: fixes x :: "'sz"   and y :: "'sz"   and \<theta> :: "('sf, 'sz) trm" shows "TUrename x y (Differential \<theta>) = Differential (TUrename x y \<theta>)"
add_0_right_vecset: fixes A :: "'b vec set"   and n :: "nat" assumes "A \<subseteq> carrier_vec n" shows "A + {0\<^sub>v n} = A"
h4: shows "\<forall>u v. (u, v) \<in> Partitions x \<and> strict_prefix u x \<and> u \<in> A\<star> \<and> v @ z \<in> A\<star> \<longrightarrow> length u \<le> length u_max"
TP_epxr_Sound: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list"   and e :: "Expr"   and n :: "nat"   and l :: "int"   and code :: "(int \<times> Instr) list"   and code1 :: "(int \<times> Instr) list"   and l1 :: "int"   and C :: "Class"   and m :: "Method"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and MI :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list" assumes "(\<Sigma>, e, n) \<in> TP_expr"   and "Sig_good \<Sigma>"   and "(l, code, e, code1, l1) \<in> compileExpr"   and "MST\<down>(C, m) = Some (T, MI, Anno)"   and "Segment C m l l1 code1" shows "deriv G C m l (Cachera n)"
prems_4: shows "trace (complex_of_real c \<cdot>\<^sub>m \<rho>) \<le> 1"
sel_1: fixes x21 :: "'g"   and x22 :: "'g list" shows "hd (x21 # x22) = x21"
sdescending_sset_scons: fixes a :: "'b"   and w :: "'b stream"   and b :: "'b" assumes "sdescending (a ## w)"   and "b \<in> sset w" shows "b \<le> a"
lookup_def: fixes xs :: "'y" shows "oay.lookup xs = lookup_raw (list_of_oalisty xs)"
max_Suc1: fixes n :: "nat"   and m :: "nat" shows "max (Suc n) m = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max n m'))"
span_base: fixes a :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "('a \<Rightarrow>\<^sub>0 'b) set" assumes "a \<in> S" shows "a \<in> phull S"
invertible_make_mat: fixes B :: "'a mat"   and k :: "nat"   and n :: "nat" assumes "invertible_mat B"   and "B \<in> carrier_mat 2 2"   and "k < n"   and "k \<noteq> 0" shows "invertible_mat (make_mat n k B)"
period_rootE: fixes u :: "'b list"   and t :: "'b list" assumes "u \<le>p t\<^sup>\<omega>" obtains n :: "nat" and r :: "'b list" where "r <p t" and "t \<^sup>@ n \<cdot> r = u"
none: shows "kronecker_factorization_rat_main df dp bnd p = None"
poly_const_pow: fixes a :: "'a"   and b :: "nat" shows "[:a:] ^ b = [:a ^ b:]"
mult_zero_left: fixes a :: "'c" shows "bot \<sqinter> a = bot"
simps: fixes f :: "Val \<Rightarrow> Val \<Rightarrow> bool"   and e1 :: "Expr"   and e2 :: "Expr"   and s :: "Var \<Rightarrow> Val" shows "evalB (compB f e1 e2) s = f (evalE e1 s) (evalE e2 s)"
factorization_m_lead_coeff: fixes f :: "int poly"   and c :: "int"   and fs :: "int poly multiset" assumes "pl.factorization_m f (c, fs)" shows "lead_coeff (pl.Mp f) = pl.M c"
real_of_of_int: fixes z :: "int" shows "real_of_rat (rat_of_int z) = real_of (of_int z)"
option2set_def: fixes n :: "'c option" shows "option2set n \<equiv> case n of None \<Rightarrow> {} | Some s \<Rightarrow> {s}"
subsumed_by_one: fixes subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "wf_sub_rel subs" shows "\<forall>v\<in>subsumees subs. \<exists>!v'. (v, v') \<in> subs"
veqpoll_vinsert: fixes A :: "V"   and B :: "V"   and a :: "V"   and b :: "V" assumes "A \<approx>\<^sub>\<circ> B"   and "a \<notin>\<^sub>\<circ> A"   and "b \<notin>\<^sub>\<circ> B" shows "vinsert a A \<approx>\<^sub>\<circ> vinsert b B"
G_def: shows "G \<equiv> \<lambda>t. C + K * integral {t..0} g"
def_hashmap_size_option_def: shows "def_hashmap_size = (\<lambda>_. def_hashmap_size TYPE('a) + 1)"
reflexive: fixes p :: "nat \<times> 'c"   and A :: "('d, 'c) nba"   and w :: "'d stream"   and V :: "(nat \<times> 'c) set" shows "p \<in> greachable A w V p"
lists_hd: fixes us :: "'b list"   and Q :: "'b set" assumes "us \<noteq> \<epsilon>"   and "us \<in> lists Q" shows "hd us \<in> Q"
stream_minus: fixes t :: "'a tree"   and t' :: "'a tree" shows "stream (t - t') = stream t - stream t'"
irrat_sqrt_nonsquare: fixes n :: "nat" assumes "\<not> is_square n" shows "sqrt (real n) \<notin> \<rat>"
rtlI: shows "ta_lang TA1 \<in> regular_languages A"