hcju/isabelleps huggingface dataset:
list_all_take: fixes P :: "'a \<Rightarrow> bool"   and n :: "nat"   and s :: "'a list" shows "list_all P (take n s) = (\<forall>i. i < n \<and> i < length s \<longrightarrow> P (s ! i))"
roots_of_complex_main_2: fixes p :: "complex poly" assumes "p \<noteq> 0"   and "degree p \<le> 2 \<or> set (coeffs p) \<subseteq> \<rat>" shows "distinct (roots_of_complex_main p)"
execn_block: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat"   and t :: "'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and u :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Normal t"   and "\<Gamma>\<turnstile> \<langle>c s t,Normal (return s t)\<rangle> =n\<Rightarrow> u" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> =n\<Rightarrow> u"
fmrelD: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('e, 'c) fmap"   and n :: "('e, 'd) fmap"   and x :: "'e" assumes "fmrel P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
lnil: fixes l :: "'a llist" assumes "l = LNil" shows "P l"
pos_mult_2: fixes c :: "'a"   and a :: "'a" assumes "(0::'a) < c" shows "(c * a < (0::'a)) = (a < (0::'a))"
O\<alpha>\<gamma>: shows "Ord (\<alpha> \<up> \<gamma>)"
rep_eq: fixes x :: "'a vec" shows "($) x = snd (Rep_vec x)"
Hintikka_model: fixes H :: "('a, 'b) fm set"   and p :: "('a, 'b) fm" assumes "Hintikka H" shows "(p \<in> H \<longrightarrow> \<lbrakk>\<^bold>#, \<^bold>\<dagger>, \<lambda>P ts. \<^bold>\<ddagger> P ts \<in> H\<rbrakk> p) \<and> ((\<^bold>\<not> p) \<in> H \<longrightarrow> \<not> \<lbrakk>\<^bold>#, \<^bold>\<dagger>, \<lambda>P ts. \<^bold>\<ddagger> P ts \<in> H\<rbrakk> p)"
hiding_game_ind_cpa_def: shows "abstract_com.hiding_game_ind_cpa \<equiv> \<lambda>uu. case uu of (\<A>1, \<A>2) \<Rightarrow> TRY key_gen \<bind> (\<lambda>(ck, vk). \<A>1 vk \<bind> (\<lambda>((m0, m1), \<sigma>). assert_spmf (valid_msg m0 \<and> valid_msg m1) \<bind> (\<lambda>_. coin_spmf \<bind> (\<lambda>b. commit ck (if b then m0 else m1) \<bind> (\<lambda>(c, d). \<A>2 c \<sigma> \<bind> (\<lambda>b'. return_spmf (b' = b))))))) ELSE coin_spmf"
lemma1: fixes r :: "lam"   and r' :: "lam"   and s :: "lam"   and s' :: "lam" assumes "r \<rightarrow>\<^sub>s r'"   and "s \<rightarrow>\<^sub>s s'" shows "r \<degree> s \<rightarrow>\<^sub>s r' \<degree> s'"
invariant_next_subseqs: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and base :: "'d"   and elements :: "'c list"   and n :: "nat"   and state :: "'c \<times> 'c iarray \<times> (nat \<times> 'd) list"   and out :: "'d list"   and state' :: "'c \<times> 'c iarray \<times> (nat \<times> 'd) list" assumes "my_subseqs.invariant f base elements n state"   and "my_subseqs.next_subseqs f state = (out, state')" shows "my_subseqs.invariant f base elements (Suc n) state'"
length_sums: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'b list" assumes "class.monoid_add plus zero" shows "length (monoid_add.sums plus zero xs) = Suc (length xs)"
field_lbound_gt_zero: fixes d1 :: "'c"   and d2 :: "'c" assumes "(0::'c) < d1"   and "(0::'c) < d2" shows "\<exists>e>0::'c. e < d1 \<and> e < d2"
ring_hom_memE_3: fixes h :: "'c \<Rightarrow> 'd"   and R :: "('c, 'e) ring_scheme"   and S :: "('d, 'f) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "h \<in> ring_hom R S"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "h (x \<oplus>\<^bsub>R\<^esub> y) = h x \<oplus>\<^bsub>S\<^esub> h y"
ate: shows "approx_tse prec x s c k f vs = Some ivl"
assms: shows "\<xi>\<^sup>2 + b * \<xi> + c = 0"
gtree_def: fixes g :: "'c growth" shows "gtree g = (case g of Stay t \<Rightarrow> t | Sprout l p r \<Rightarrow> Branch2 l p r)"
cs: shows "sign [:c, b, a:] x \<noteq> sign [:c, b, a:] y \<and> x < y \<longrightarrow> (\<exists>c\<in>root_list [:c, b, a:]. x \<le> c \<and> c \<le> y)"
nz_1: fixes s :: "complex" assumes "s0 < ereal (Re s)" shows "eval_fds f s \<noteq> 0"
X_distr: fixes a :: "real" assumes "0 \<le> a" shows "emeasure M {x \<in> space M. X x \<le> a} = 1 - ennreal (exp (- a * l))"
simply_connected_eq_contractible_circlemap: fixes S :: "'a set" shows "simply_connected S = (path_connected S \<and> (\<forall>f. continuous_on (sphere 0 1) f \<and> f ` sphere 0 1 \<subseteq> S \<longrightarrow> (\<exists>a. homotopic_with_canon (\<lambda>h. True) (sphere 0 1) S f (\<lambda>x. a))))"
assms_6: shows "f2 ` S \<subseteq> T"
erf_remainder_integral_bigo: fixes n :: "nat" shows "erf_remainder_integral n \<in> O(\<lambda>x. exp (- x\<^sup>2) / x ^ (2 * n + 1))"
simps_1: fixes f :: "'a" shows "Cod \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>C.cod f\<^bold>\<rangle>"
inline1_inline_conv_inline2: fixes callee1 :: "'f \<Rightarrow> 'j \<Rightarrow> ('i \<times> 'f, 'g, 'h) gpv"   and callee2 :: "'e \<Rightarrow> 'k \<Rightarrow> ('l \<times> 'e, 'j, 'i) gpv"   and gpv :: "('d, 'k, 'l) gpv"   and s2 :: "'e"   and s1 :: "'f" shows "inline1 callee1 (inline callee2 gpv s2) s1 = map_spmf (map_sum (\<lambda>(x, s2, s1). ((x, s2), s1)) (\<lambda>(x, rpv'', rpv', rpv). (x, rpv'', \<lambda>r1. rpv' r1 \<bind> (\<lambda>(r2, s2). inline callee2 (rpv r2) s2)))) (inline2 callee1 callee2 gpv s2 s1)"
ExFalso: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"   and "\<not> \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P c Q,A"
coprime_absorb_left: fixes x :: "'a"   and y :: "'a" assumes "x dvd y" shows "coprime x y = is_unit x"
s1: shows "s1 = (ls1, (ts1, m1), ws1, is1)"
crename_OrR1_aux: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and e :: "coname" assumes "R[a\<turnstile>c>b] = OrR1 <c>.M e" shows "a = e \<and> a = b \<or> a \<noteq> e"
snd_eucl_of_list_prod: fixes xs :: "real list" shows "snd (eucl_of_list xs) = eucl_of_list (drop DIM('b) xs)"
assms_1: fixes n :: "nat" shows "f (Suc n) \<equiv> h n"
subalgebra_def: fixes K :: "'c set"   and V :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subalgebra K V R \<equiv> subgroup V (add_monoid R) \<and> subalgebra_axioms K V R"
p\<^sub>1_simps: shows "trg fg\<^sub>0h\<^sub>1.p\<^sub>1 = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)"
takeWhile_idem: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "takeWhile P (takeWhile P xs) = takeWhile P xs"
max_absorb1: fixes y :: "'b"   and x :: "'b" assumes "y \<le> x" shows "max x y = x"
nat_pow_Suc: fixes x :: "int"   and n :: "nat" shows "x [^]\<^bsub>R\<^esub> Suc n = x [^]\<^bsub>R\<^esub> n \<otimes>\<^bsub>R\<^esub> x"
range_enumerate: fixes S :: "nat set" assumes "infinite S" shows "range (wellorder_class.enumerate S) = S"
append_eq_conv_conj: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs @ ys = zs) = (xs = take (length xs) zs \<and> ys = drop (length xs) zs)"
preserves_section_retraction: fixes e :: "'a"   and m :: "'a" assumes "A.ide (e \<cdot>\<^sub>A m)" shows "B.ide (G e \<cdot>\<^sub>B G m)"
extended_predist_nonneg: fixes x :: "'a Gromov_completion"   and y :: "'a Gromov_completion" shows "0 \<le> extended_predist x y"
op_ivl_rep_of_set_def: fixes X :: "'c set" shows "op_ivl_rep_of_set X = (let X = X in Inf_spec X \<bind> (\<lambda>i. Sup_spec X \<bind> (\<lambda>s. RETURN (inf i s, s))))"
cong_exp_iff_simps_12: 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'c \<times> 'c"   and divmod_step :: "num \<Rightarrow> 'c \<times> 'c \<Rightarrow> 'c \<times> 'c"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit1 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 q))) = False"
periodic_orbit_imp_flow0_regular: fixes x :: "'a"   and t :: "real" assumes "periodic_orbit x" shows "f (flow0 x t) \<noteq> (0::'a)"
map_comp_empty_1: fixes m :: "'f \<Rightarrow> 'e option" shows "m \<circ>\<^sub>m \<emptyset> = \<emptyset>"
univ_ce_is_not_comp2: shows "\<not> total_recursive (chf univ_ce)"
domain_eq: shows "Domainp (pcr_fls (=)) = (\<lambda>f. \<forall>\<^sub>\<infinity>n. f (- int n) = (0::'a))"
continuous_on_compose_fracI: fixes f :: "real \<Rightarrow> real" assumes "continuous_on {0..1} f"   and "f 0 = f 1" shows "continuous_on UNIV (\<lambda>x. f (frac x))"
iso_imp_finite: fixes H :: "('e, 'f) monoid_scheme" assumes "H \<cong> H"   and "Group.group H" shows "finite_group H"
finalStateCharacterizationLemma: fixes state :: "State"   and decisionVars :: "nat set" assumes "InvariantUniq (getM state)"   and "InvariantReasonClauses (getF state) (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "\<not> applicableDecide state decisionVars"   and "\<not> applicableConflict state"   and "\<not> applicableExplain state"   and "\<not> applicableBackjump state" shows "getConflictFlag state = False \<and> \<not> formulaFalse (getF state) (elements (getM state)) \<and> decisionVars \<subseteq> vars (elements (getM state)) \<or> getConflictFlag state = True \<and> getC state = []"
\<rho>_cong: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and R :: "(MIR.num \<times> int) set" assumes "iszlfm p (a # bs)"   and "(\<lambda>(b, k). (Inum (a # bs) b, k)) ` R = (\<lambda>(b, k). (Inum (a # bs) b, k)) ` set (\<rho> p)" shows "(\<exists>(e, c)\<in>R. \<exists>j\<in>{1..c * \<delta> p}. Ifm (a # bs) (\<sigma> p c (Add e (C j)))) = (\<exists>(e, c)\<in>set (\<rho> p). \<exists>j\<in>{1..c * \<delta> p}. Ifm (a # bs) (\<sigma> p c (Add e (C j))))"
is_generalisation_of_def: fixes t' :: "transition"   and t :: "transition"   and i :: "nat"   and r :: "nat" shows "Store_Reuse.is_generalisation_of t' t i r = (t' = remove_guard_add_update t i r \<and> i < Arity t \<and> (\<exists>v. Eq (V (I i)) (L v) \<in> set (Guards t)) \<and> r \<notin> set (map fst (Updates t)))"
comp_opt_p_def: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "comp_opt_p p q = (if p = q then Some Eq else if ord_strict_p p q then Some Lt else if ord_strict_p q p then Some Gt else None)"
hermitean_transpose: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "hermitean A = (mat_transpose A = mat_cnj A)"
member_le_sum_list: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "x \<in> set xs" shows "less_eq x (monoid_add.sum_list plus zero xs)"
B_greater_zero: fixes x :: "int" assumes "x \<in> B" shows "0 < x"
simps_3: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln"   and M :: "'b ltln set" shows "\<phi> M\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2 = \<phi>[M]\<^sub>\<Sigma>\<^sub>2 M\<^sub>n \<psi>[M]\<^sub>\<Sigma>\<^sub>2"
evalRR_one: fixes x :: "'a" assumes "x \<in> carrier R" shows "eval R R id x \<one>\<^bsub>UP R\<^esub> = \<one>"
preserves_cod: fixes f :: "'a" assumes "S.arr f" shows "S.cod (L f) = L (S.cod f)"
OrL_sim: fixes F :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula"   and G :: "'b formula" assumes "F \<triangleright> \<Gamma> \<turnstile> H"   and "G \<triangleright> \<Gamma> \<turnstile> H" shows "F \<^bold>\<or> G \<triangleright> \<Gamma> \<turnstile> H"
pequiv_DC: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and d :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<simeq> c"   and "b \<simeq> d" shows "(a \<Sqinter> b) \<simeq> (c \<Sqinter> d)"
that: fixes x :: "int" assumes "Finite_And_Cyclic_Groups.cyclic_group (Z n) x" shows "thesis"
worklist_tree_coll_def: fixes succs :: "'c \<Rightarrow> 'c list"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and ws :: "'c list" shows "worklist_tree_coll succs P f ws = worklist_tree_coll_aux succs P f ws empty"
eucl_fold_mono: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and xs :: "'a list" assumes "\<And>w x y z. \<lbrakk>w \<le> x; y \<le> z\<rbrakk> \<Longrightarrow> f w y \<le> f x z"   and "x \<le> y" shows "fold f xs x \<le> fold f xs y"
impl_test: fixes b1 :: "'b"   and b2 :: "'b" shows "impl_test ifex_of b1 b2 = (\<forall>s. val b1 s \<longrightarrow> val b2 s)"
components_intermediate_subset: fixes s :: "'b set"   and u :: "'b set"   and t :: "'b set" assumes "s \<in> components u"   and "s \<subseteq> t"   and "t \<subseteq> u" shows "s \<in> components t"
leg_simps_3: shows "C.arr \<epsilon>.dom.leg1"
tiny_dg_Obj_in_Vset: shows "\<BB>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
sub_size: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm" assumes "sub s t" shows "size s \<le> size t"
prod_ge_1: 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 A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq one (f x)" shows "less_eq one (comm_monoid_mult.prod times one f A)"
pth_5: fixes c :: "real"   and d :: "real"   and x :: "'a" shows "c *\<^sub>R d *\<^sub>R x \<equiv> (c * d) *\<^sub>R x"
nhds_def: fixes "open" :: "'c set \<Rightarrow> bool"   and a :: "'c" assumes "class.topological_space open" shows "topological_space.nhds open a = Inf (principal ` {S. open S \<and> a \<in> S})"
iso_in_homE: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.iso_in_hom f a b" obtains "VV.in_hom f a b" and "VV.iso f"
Can_implies_Arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Can C t" shows "monoidal_language.Arr C t"
cases: fixes a :: "('s, 'v) tm" assumes "wary_fo a" obtains s :: "('s, 'v) tm" where "a = s" and "is_Hd s \<or> is_Sym (head s)" and "enat (num_args s) = arity_hd (head s)" and "Ball (set (args s)) wary_fo"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and p :: "float poly"   and e :: "float interval" shows "tm_lower_order_of_normed prec ord I a (TaylorModel p e) = (let x = split_by_degree ord p in case case x of (x, y) \<Rightarrow> (x, map_poly interval_of y) of (l, r) \<Rightarrow> TaylorModel l (round_interval prec (e + compute_bound_poly prec r I a)))"
csmall_def: fixes P :: "'prog"   and \<sigma> :: "'state" shows "csmall P \<sigma> \<equiv> {(\<sigma>'', coll). \<sigma>'' \<in> small P \<sigma> \<and> collect P \<sigma> \<sigma>'' = coll}"
Jordan_Brouwer_separation: fixes S :: "'a set"   and a :: "'a"   and r :: "real" assumes "S homeomorphic sphere a r"   and "0 < r" shows "\<not> connected (- S)"
mat_gt_arc_trans: fixes A :: "'a mat"   and B :: "'a mat"   and C :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "mat_gt_arc A B"   and "mat_gt_arc B C"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "mat_gt_arc A C"
lookup_mult_scalar_monomial_right_plus: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'t"   and c :: "'c"   and t :: "'a" shows "lookup (p \<odot> monomial c v) (t \<oplus> v) = lookup p t * c"
simps_3: fixes \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "propos (\<phi> and \<psi>) = propos \<phi> \<union> propos \<psi>"
subformulas\<^sub>\<mu>_subset: fixes \<psi> :: "'b ltln"   and \<phi> :: "'b ltln" assumes "\<psi> \<in> subfrmlsn \<phi>" shows "subformulas\<^sub>\<mu> \<psi> \<subseteq> subformulas\<^sub>\<mu> \<phi>"
less_divide_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" 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)"
chain1: shows "Complete_Partial_Order.chain (ord_spmf (=)) (fst ` Y)"
scale_shift_ms_aux_MSLCons: fixes x :: "'a \<times> real"   and y :: "'a \<times> real"   and F :: "('a \<times> real) msllist" shows "scale_shift_ms_aux x (MSLCons y F) = MSLCons (fst x * fst y, snd x + snd y) (scale_shift_ms_aux x F)"
REMOVE_INTERNAL_def: fixes x :: "'g" shows "REMOVE_INTERNAL x \<equiv> x"
op_fmap_contains_key_def: shows "op_fmap_contains_key \<equiv> \<lambda>k m. k \<in># dom_m m"
A: shows "\<not> (\<exists>v\<in>D ` set ys. \<exists>u\<in>sources_aux I D (sources_aux I D U [z]) zs. (v, u) \<in> I)"
finite_tableaus: fixes V :: "nat set"   and t0 :: "(nat \<times> linear_poly) list" assumes "finite V" shows "finite {t. tvars t = V \<and> \<triangle> t \<and> (\<forall>v. v \<Turnstile>\<^sub>t t = v \<Turnstile>\<^sub>t t0)}"
cinner_sup_onorm: fixes A :: "'a \<Rightarrow> 'b" assumes "bounded_linear A" shows "onorm A = (SUP (\<psi>, \<phi>). cmod (\<psi> \<bullet>\<^sub>C A \<phi>) / (norm \<psi> * norm \<phi>))"
cindex_polyE_0_2: fixes a :: "real"   and b :: "real"   and q :: "real poly" shows "cindex_polyE a b q 0 = 0"
v_def: shows "v = (\<lambda>n. root n (u n))"
RPCFail_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rst :: "state \<Rightarrow> PrIds \<Rightarrow> rpcState"   and p :: "PrIds" shows "RPCFail send rcv rst p \<equiv> \<not> $Calling rcv p \<and> Return send p #RPCFailure \<and> (rst!p)$ = #rpcA \<and> unchanged (caller rcv!p)"
valid_bind_cong: fixes f :: "'b \<Rightarrow> ('a \<times> 'b) option"   and \<sigma> :: "'b"   and g :: "'b \<Rightarrow> ('a \<times> 'b) option"   and M :: "'a \<Rightarrow> 'b \<Rightarrow> (bool \<times> 'b) option" assumes "f \<sigma> = g \<sigma>" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E f M) = (\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E g M)"
not_eq_newpkt_rreq: fixes hops :: "nat"   and rreqid :: "nat"   and dip :: "nat"   and dsn :: "nat"   and dsk :: "k"   and oip :: "nat"   and osn :: "nat"   and sip :: "nat" shows "\<not> eq_newpkt (Rreq hops rreqid dip dsn dsk oip osn sip)"
fixp_induct: fixes P :: "(Core.term \<Rightarrow> Core.term option) \<Rightarrow> bool" assumes "option.admissible P"   and "P Map.empty"   and "\<And>t. P t \<Longrightarrow> P (\<lambda>ta. case ta of Ct literal typ \<Rightarrow> Some (Ct literal typ) | Fv variable typ \<Rightarrow> Some (Fv variable typ) | Bv nat \<Rightarrow> Some (Bv nat) | Abs T body \<Rightarrow> map_option (Abs T) (t body) | Ct literal typ $ u \<Rightarrow> case t (Ct literal typ) of None \<Rightarrow> None | Some (Ct literal typ) \<Rightarrow> map_option (($) (Ct literal typ)) (t u) | Some (Fv variable typ) \<Rightarrow> map_option (($) (Fv variable typ)) (t u) | Some (Bv nat) \<Rightarrow> map_option (($) (Bv nat)) (t u) | Some (Abs T body) \<Rightarrow> t (subst_bv u body) | Some (term1 $ term2) \<Rightarrow> map_option (($) (term1 $ term2)) (t u) | Fv variable typ $ u \<Rightarrow> case t (Fv variable typ) of None \<Rightarrow> None | Some (Ct literal typ) \<Rightarrow> map_option (($) (Ct literal typ)) (t u) | Some (Fv variable typ) \<Rightarrow> map_option (($) (Fv variable typ)) (t u) | Some (Bv nat) \<Rightarrow> map_option (($) (Bv nat)) (t u) | Some (Abs T body) \<Rightarrow> t (subst_bv u body) | Some (term1 $ term2) \<Rightarrow> map_option (($) (term1 $ term2)) (t u) | Bv nat $ u \<Rightarrow> case t (Bv nat) of None \<Rightarrow> None | Some (Ct literal typ) \<Rightarrow> map_option (($) (Ct literal typ)) (t u) | Some (Fv variable typ) \<Rightarrow> map_option (($) (Fv variable typ)) (t u) | Some (Bv nat) \<Rightarrow> map_option (($) (Bv nat)) (t u) | Some (Abs T body) \<Rightarrow> t (subst_bv u body) | Some (term1 $ term2) \<Rightarrow> map_option (($) (term1 $ term2)) (t u) | Abs T body $ u \<Rightarrow> t (subst_bv u body) | term1 $ term2 $ u \<Rightarrow> case t (term1 $ term2) of None \<Rightarrow> None | Some (Ct literal typ) \<Rightarrow> map_option (($) (Ct literal typ)) (t u) | Some (Fv variable typ) \<Rightarrow> map_option (($) (Fv variable typ)) (t u) | Some (Bv nat) \<Rightarrow> map_option (($) (Bv nat)) (t u) | Some (Abs T body) \<Rightarrow> t (subst_bv u body) | Some (term1 $ term2) \<Rightarrow> map_option (($) (term1 $ term2)) (t u))" shows "P beta_norm"
iff_simps_6: fixes p :: "fm" assumes "p \<noteq> NOT T" shows "Ferrack.iff T p = p"
lex_comp'_EqD: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rep_nat_pp :: "'b \<Rightarrow> (nat, nat) pp"   and abs_nat_pp :: "(nat, nat) pp \<Rightarrow> 'b"   and lex_comp' :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and deg' :: "'b \<Rightarrow> nat"   and x :: "'b"   and y :: "'b" assumes "class.nat_pp_compare plus zero less_eq less rep_nat_pp abs_nat_pp lex_comp' deg'"   and "lex_comp' x y = Eq" shows "x = y"
condensation_nodes_def: fixes g :: "'g"   and P :: "'val set" shows "condensation_nodes g P \<equiv> scc_of (induced_phi_graph g P) ` P"
scd: shows "pseudo_proj_True (Suc n) -` {pseudo_proj_False n w} \<subseteq> pseudo_proj_True n -` {w}"
extr_insert_dy_fake_msg: fixes M :: "msg"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" assumes "M \<in> dy_fake_msg bad IK CH" shows "extr bad (insert M IK) CH \<subseteq> dy_fake_msg bad IK CH"
l_zero: fixes x :: "'c" assumes "x \<in> carrier V" shows "\<zero>\<^bsub>V\<^esub> \<oplus>\<^bsub>V\<^esub> x = x"
prems: shows "\<Gamma>\<turnstile>sequence Seq ((x # xs) @ ys) \<down> Normal s"
fmupd_lookup: fixes a :: "'d"   and b :: "'c"   and m :: "('d, 'c) fmap"   and a' :: "'d" shows "m(a $$:= b) $$ a' = (if a = a' then Some b else m $$ a')"
not_poss4: shows "squash (rpoly q x1) \<noteq> 0"
snoc_1: assumes "\<exists>a\<in>set xs. P a" shows "\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>z\<in>set zs. \<not> P z)"
lprefix_antisym: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "ys \<sqsubseteq> xs" shows "xs = ys"
simps_1: fixes uu :: "'c" shows "chainlen \<lbrace> uu \<rbrace> = 0"
R_O_Id: fixes R :: "('a \<times> 'b) set" shows "R ; Id = R"
qtable2: shows "qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) (ne + length aux) \<psi>) rel2"