hcju/isabelleps huggingface dataset:
berlekamp_mat_def: fixes u :: "'a mod_ring poly" shows "local.berlekamp_mat u = (let n = degree u; mul_p = power_poly_f_mod u [:0, 1:] CARD('a); xks = power_polys mul_p u 1 n in mat_of_rows_list n (map (\<lambda>cs. let coeffs_cs = coeffs cs; k = n - length (coeffs cs) in coeffs cs @ replicate k 0) xks))"
converse_relcomp: fixes r :: "('b \<times> 'c) set"   and s :: "('c \<times> 'a) set" shows "(r O s)\<inverse> = s\<inverse> O r\<inverse>"
pcr_fin_string_def: shows "pcr_fin_string \<equiv> list_all2 (=) OO cr_fin_string"
V: shows "V = vec (db + dc) \<langle>X\<rangle>"
IH1': shows "quicksort_post R h lo (p - Suc 0) xs' xs''"
no_rep_length: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "is_trail v ps v'" shows "length ps = card (set ps)"
good_item_simps_41: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and delta :: "'c" assumes "goodInp inp"   and "goodBinp binpa" shows "skel (Op delta inp binpa) = Branch (skelInp inp) (skelBinp binpa)"
monom_altdef: fixes c :: "'b"   and n :: "nat" shows "monom c n = smult c ([:0::'b, 1::'b:] ^ n)"
ereal_of_enat_simps_1: fixes n :: "nat" shows "ereal_of_enat (enat n) = ereal (real n)"
size_splay_max: fixes t :: "'b tree" shows "size (splay_max t) = size t"
f: shows "C.iso f \<and> arr f \<and> arr (C.inv f)"
right_totalE: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d" assumes "right_total A" obtains x :: "'c" where "A x y"
memo_list_trancl: fixes r :: "('b \<times> 'b) list"   and a :: "'b" shows "set (memo_list_trancl r a) = {b. (a, b) \<in> (set r)\<^sup>+}"
size_queue_sum_list: fixes bq :: "('c, 'd) BinomialTree list" shows "size (queue_to_multiset bq) = sum_list (map (size \<circ> tree_to_multiset) bq)"
pmod_def: fixes p :: "'a list list"   and q :: "'a list list" shows "UP.pmod p q = (if q = [] then p else snd (UP.long_division p q))"
assms: shows "\<exists>n. w !! n \<noteq> x !! n"
Rep_uint16'_def: shows "Rep_uint16' = Rep_uint16"
abs_eq: fixes xa :: "perm"   and x :: "('a, 'b, 'c) Tree" shows "xa \<bullet> abs_Tree\<^sub>\<alpha> x = abs_Tree\<^sub>\<alpha> (xa \<bullet> x)"
funcset_compose: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and C :: "'c set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "g \<circ> f \<down> A \<in> A \<rightarrow> C"
success_heapI: fixes f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap" shows "success (Heap_Monad.heap f) h"
simps: fixes a1 :: "string_ternop + collection_ternop"   and a2 :: "call_kind"   and a3 :: "'a OCL_Types.type"   and a4 :: "'b OCL_Types.type"   and a5 :: "'a OCL_Types.type"   and a6 :: "'a OCL_Types.type" shows "ternop_type a1 a2 a3 a4 a5 a6 = ((\<exists>op \<tau> \<sigma> \<rho> \<upsilon>. a1 = Inl op \<and> a2 = DotCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> a6 = \<upsilon> \<and> string_ternop_type op \<tau> \<sigma> \<rho> \<upsilon>) \<or> (\<exists>op \<tau> \<sigma> \<rho> \<upsilon>. a1 = Inr op \<and> a2 = ArrowCall \<and> a3 = \<tau> \<and> a4 = \<sigma> \<and> a5 = \<rho> \<and> a6 = \<upsilon> \<and> collection_ternop_type op \<tau> \<sigma> \<rho> \<upsilon>))"
scale_zero_left: fixes x :: "'b" shows "0 *\<^sub>R x = (0::'b)"
image: fixes a :: "'b"   and b :: "'b" assumes "(a, b) \<in> s" shows "(f a, f b) \<in> r"
power_add: fixes a :: "'b list"   and m :: "nat"   and n :: "nat" shows "a \<^sup>@ (m + n) = a \<^sup>@ m \<cdot> a \<^sup>@ n"
leading_coeff_0_iff: fixes p :: "'b poly" shows "(Polynomial.lead_coeff p = (0::'b)) = (p = 0)"
infprod_in_sets: fixes E :: "nat \<Rightarrow> 'b set"   and M :: "nat \<Rightarrow> 'b measure" assumes "\<And>i. E i \<in> sets (M i)" shows "Pi UNIV E \<in> sets (Pi\<^sub>M UNIV M)"
p: shows "p = (\<Sum>b\<in>set bs. q b *s b)"
Sup_upper2: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and u :: "'a"   and A :: "'a set"   and v :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq v u" shows "less_eq v (Sup A)"
between_exchange4: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A C D" shows "Bet A B D"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b llist" shows "prepend (x # xs) ys = LCons x (prepend xs ys)"
transcendental_cos: fixes z :: "complex" assumes "algebraic z"   and "z \<noteq> 0" shows "\<not> algebraic (cos z)"
IndP_Q_Ind: fixes H :: "fm set"   and x :: "tm" assumes "H \<turnstile> OrdP x" shows "H \<turnstile> IndP (Q_Ind x)"
IH: fixes n :: "nat" assumes "wf_formula n \<phi>" shows "lang n (rexp_of n \<phi>) = lang n (rexp.Inter (rexp_of_alt n \<phi>) (ENC n (FOV \<phi>)))"
card_Suc_eq: fixes A :: "'c set"   and k :: "nat" shows "(card A = Suc k) = (\<exists>b B. A = Set.insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {}))"
simps_5: fixes t :: "'c local.term" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = D.\<ll> \<lbrace>t\<rbrace>"
line_integral_on_path: shows "one_chain_line_integral F {j} \<gamma> = one_chain_line_integral F {j} \<H> + one_chain_line_integral F {j} (two_chain_vertical_boundary two_chain)"
perms_def: shows "perms = {f. bij f \<and> finite {x. f x \<noteq> x}}"
convex_less_in_riS: fixes f :: "'b \<Rightarrow> ereal"   and S :: "'b set"   and a :: "ereal" assumes "Lower_Semicontinuous.convex_on UNIV f"   and "convex S"   and "rel_interior S \<subseteq> domain f"   and "\<exists>x\<in>closure S. f x < a" shows "\<exists>x\<in>rel_interior S. f x < a"
exhaust: fixes y :: "('a, 'b) atomic" obtains x1 :: "'a" where "y = Fo x1"   | x2 :: "'a" where "y = Z x2"   | x31 :: "'a" and x32 :: "'a" where "y = Less x31 x32"   | x41 :: "'a" and x42 :: "'b" where "y = In x41 x42"
malicious_sec_P2: fixes \<G> :: "'d cyclic_group"   and x1 :: "'d"   and x0 :: "'d"   and \<sigma> :: "bool"   and z :: "'e"   and \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('d \<times> 'd \<times> 'd \<times> 'd \<times> 'd) \<times> 'f) spmf) \<times> ('d \<times> 'd \<times> 'd \<times> 'd \<times> 'd \<Rightarrow> 'f \<Rightarrow> ((('d \<times> 'd \<times> 'd) \<times> nat) \<times> 'f) spmf) \<times> ('d \<times> 'd \<Rightarrow> 'd \<times> 'd \<Rightarrow> 'f \<Rightarrow> 'g spmf)" assumes "ot \<G>"   and "x1 \<in> carrier \<G>"   and "x0 \<in> carrier \<G>" shows "malicious_base.perfect_sec_P2 funct_OT_12 (ot_base.P2_real_model \<G>) (x0, x1) \<sigma> z (ot_base.P2_S1 \<G>, ot_base.P2_S2 \<G>) \<A>"
get_owner_document_is_l_get_owner_document: shows "l_get_owner_document get_owner_document"
lift_prime_elem_poly: fixes c :: "'a" assumes "prime_elem c" shows "prime_elem [:c:]"
hyps: shows "fst (Gauss_Jordan_upt_k_det_P A k) * det A = det (snd (Gauss_Jordan_upt_k_det_P A k))"
reduced_word_for_length: 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 a as" shows "length as = monoid_add.word_length plus zero A a"
flip_altsublist_chain_lists: fixes ss :: "'w list"   and xss :: "'w list list"   and ts :: "'w list" assumes "ss \<in> lists S"   and "flip_altsublist_chain (ss # xss @ [ts])" shows "ts \<in> lists S"
simps_1: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and vertices :: "'e fset"   and nodeOf :: "'e \<Rightarrow> ('b, 'd) graph_node"   and edges :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) set"   and vidx :: "'e \<Rightarrow> nat"   and inst :: "'e \<Rightarrow> 'c"   and v :: "'e"   and p :: "('b, 'a) antecedent"   and pth :: "(('e \<times> ('b, 'a) out_port) \<times> 'e \<times> ('b, 'a) antecedent) list" assumes "Tasked_Proof_Graph freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions vertices nodeOf edges vidx inst" shows "tree.root (Tasked_Proof_Graph.tree freshenLC renameLCs subst anyP antecedent vertices nodeOf edges vidx inst v p pth) = (Tasked_Proof_Graph.hyps_along freshenLC renameLCs subst antecedent vertices nodeOf vidx inst ((Tasked_Proof_Graph.adjacentTo edges v p, v, p) # pth) \<turnstile> Instantiation.labelAtIn nodeOf labelsIn freshenLC renameLCs subst vidx inst v p, case Tasked_Proof_Graph.adjacentTo edges v p of (v', p') \<Rightarrow> Tasked_Proof_Graph.toNatRule nodeOf v' p')"
favorites_altdef: fixes i :: "'agent" shows "favorites R i = Max_wrt_among (R i) alts"
gcd_mult_distrib': 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 c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times (normalize c) (gcd a b) = gcd (times c a) (times c b)"
nonzero_mult_in_car: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> nonzero Zp"   and "b \<in> nonzero Zp" shows "a \<otimes> b \<in> carrier Zp"
quasi_inversesI: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.isomorphic (Span.src f) (g \<circ> f)"   and "Span.isomorphic (f \<circ> g) (Span.trg f)" shows "Span.quasi_inverses f g"
monomial_prod_sum: fixes a :: "'d \<Rightarrow> 'b"   and I :: "'d set"   and c :: "'d \<Rightarrow> 'c" shows "monomial (prod c I) (sum a I) = (\<Prod>i\<in>I. monomial (c i) (a i))"
valid_cert_concatI: fixes xs :: "pratt list list" assumes "\<forall>x\<in>set xs. valid_cert x" shows "valid_cert (concat xs)"
iprev_nth_card_iMin: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}"   and "card I \<le> Suc n" shows "I \<leftarrow> n = iMin I"
approx_tm_num_Vars: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list"   and t :: "taylor_model" assumes "approx_tm prec ord I a f env = Some t" shows "max_Var_floatarith f \<le> length env"
igSwapInpIPresIGWlsInp_def: fixes MOD :: "('c, 'd, 'e, 'sort, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSwapInpIPresIGWlsInp MOD \<equiv> \<forall>zs z1 z2 delta inp. igWlsInp MOD delta inp \<longrightarrow> igWlsInp MOD delta (igSwapInp MOD zs z1 z2 inp)"
assms_3: shows "f \<in> mor\<^bsub>C\<^esub>"
initial_def: fixes v0 :: "'v"   and D :: "'v set" shows "initial v0 D \<equiv> ([{v0}], D, E \<inter> {v0} \<times> UNIV)"
elims_part: fixes \<Gamma>\<^sub>1 :: "'b constr list"   and n\<^sub>1 :: "nat"   and \<Psi>\<^sub>1 :: "'b TESL_atomic list"   and \<Phi>\<^sub>1 :: "'b TESL_atomic list"   and \<Gamma>\<^sub>2 :: "'b constr list"   and n\<^sub>2 :: "nat"   and \<Psi>\<^sub>2 :: "'b TESL_atomic list"   and \<Phi>\<^sub>2 :: "'b TESL_atomic list" assumes "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2" shows "\<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow> \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2"
row_to_poly_vec_plus: fixes ts :: "'d list"   and n :: "nat"   and f1 :: "nat \<Rightarrow> 'e"   and f2 :: "nat \<Rightarrow> 'e" assumes "distinct ts"   and "length ts = n" shows "row_to_poly ts (vec n (f1 + f2)) = row_to_poly ts (vec n f1) + row_to_poly ts (vec n f2)"
assms_1: fixes c :: "'a"   and var :: "Vars.var" shows "P c (Var var)"
l_new_element_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_element_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_new_element type_wf \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_new_element_get_child_nodes_axioms get_child_nodes get_child_nodes_locs"
less_antisym: fixes n :: "nat"   and m :: "nat" assumes "\<not> n < m"   and "n < Suc m" shows "m = n"
mcont_max_enat2: fixes x :: "enat" shows "mcont Sup (\<le>) Sup (\<le>) (max x)"
incrIndexList_help7: fixes ls :: "nat list"   and m :: "nat"   and nmax :: "nat" assumes "incrIndexList ls m nmax" shows "ls \<noteq> []"
no_abs_cases: fixes t :: "'c" assumes "no_abs t" obtains name :: "name" where "t = free name__"   | name :: "name" where "t = const name"   | t\<^sub>1 :: "'c" and t\<^sub>2 :: "'c" where "t = app t\<^sub>1 t\<^sub>2" and "no_abs t\<^sub>1" and "no_abs t\<^sub>2"
vrange_vcomp_vsubset: fixes r :: "V"   and B :: "V"   and s :: "V" assumes "\<R>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> B" shows "\<R>\<^sub>\<circ> (r \<circ>\<^sub>\<circ> s) \<subseteq>\<^sub>\<circ> B"
inv_inv: fixes x :: "cltn2" assumes "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> (inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x) = x"
0: shows "\<forall>n\<in>num. prv (subst (cnj fls (eql (Var x) (Var x))) n x)"
passive_subset_def: fixes M :: "('f \<times> 'l) set" shows "passive_subset M = {CL \<in> M. snd CL \<noteq> active}"
exec_gpv_bind: fixes callee :: "'e \<Rightarrow> 'f \<Rightarrow> ('g \<times> 'e) spmf"   and c :: "('h, 'f, 'g) gpv"   and f :: "'h \<Rightarrow> ('d, 'f, 'g) gpv"   and s :: "'e" shows "exec_gpv callee (c \<bind> f) s = exec_gpv callee c s \<bind> (\<lambda>xa. case xa of (x, s') \<Rightarrow> exec_gpv callee (f x) s')"
iprev_cut_le_conv: fixes n :: "nat"   and t :: "nat"   and I :: "nat set" assumes "n \<le> t" shows "iprev n (I \<down>\<le> t) = iprev n I"
unfoldE_1: fixes P :: "pi"   and Q :: "pi"   and s :: "(name \<times> name) list" assumes "P \<simeq>\<^sup>s Q" shows "P[
] \<leadsto><weakBisim> Q[
]"
simps_1: fixes t1 :: "('b \<times> nat) tree"   and b :: "'b"   and lvb :: "nat"   and t2 :: "('b \<times> nat) tree"   and a :: "'b"   and lva :: "nat"   and t3 :: "('b \<times> nat) tree" shows "skew \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle> = (if lva = lvb then \<langle>t1, (b, lvb), \<langle>t2, (a, lva), t3\<rangle>\<rangle> else \<langle>\<langle>t1, (b, lvb), t2\<rangle>, (a, lva), t3\<rangle>)"
deadlock_free_v2_is_right: fixes P :: "'b process" shows "deadlock_free_v2 P = (\<forall>s\<in>T P. tickFree s \<longrightarrow> (s, UNIV) \<notin> F P)"
i_shrink_def: fixes f :: "nat \<Rightarrow> 'd message_af"   and k :: "nat" shows "f \<div> k \<equiv> i_aggregate f k last_message"
simps: fixes v :: "nat \<Rightarrow> 'a"   and rel :: "le_rel"   and l :: "linear_poly"   and r :: "'a" shows "v \<Turnstile>\<^sub>l\<^sub>e Le_Constraint rel l r = rel_of rel l \<lbrace> v \<rbrace> r"
satisfies_gen_restrict_RESTR: fixes r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "satisfies_gen (\<lambda>k P n. restrict k P \<and> r k P n) \<AA> \<phi> = satisfies_gen r \<AA> (RESTR \<phi>)"
l: shows "unwind_cont (last \<Delta>s) {last \<Delta>s, \<Delta>e}"
linear_inj_imp_surj: fixes f :: "'b \<Rightarrow> 'b" assumes "linear (*s) (*s) f"   and "inj f" shows "surj f"
span_minimal: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set"   and T :: "'c set" assumes "module scale"   and "S \<subseteq> T"   and "module.subspace scale T" shows "module.span scale S \<subseteq> T"
domintros_14: fixes v :: "(char list \<times> char list \<times> exp0) list"   and va :: "exp0" shows "is_value_dom (Letrec v va)"
eq: shows "adjoint (2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N) = 2 \<cdot>\<^sub>m proj_psi - 1\<^sub>m N"
adjoint_equivalence_axioms: shows "adjoint_equivalence (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G F \<epsilon>'.map \<eta>'.map"
simps_1: fixes size :: "'a \<Rightarrow> nat"   and frees :: "'a \<Rightarrow> name fset"   and subst :: "'a \<Rightarrow> (name, 'a) fmap \<Rightarrow> 'a"   and "consts" :: "'a \<Rightarrow> name fset"   and app :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unapp :: "'a \<Rightarrow> ('a \<times> 'a) option"   and const :: "name \<Rightarrow> 'a"   and unconst :: "'a \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'a"   and unfree :: "'a \<Rightarrow> name option"   and abs_pred :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f [] = f"
llength_ldrop: fixes n :: "enat"   and xs :: "'a llist" shows "llength (ldrop n xs) = (if n = \<infinity> then 0 else llength xs - n)"
exhaust: fixes y :: "('a, 'b) tm" obtains x1 :: "('a, 'b) hd" where "y = Hd x1"   | x21 :: "('a, 'b) tm" and x22 :: "('a, 'b) tm" where "y = App x21 x22"
sim_det_def_def: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c spmf"   and S1 :: "'a \<Rightarrow> 'd \<Rightarrow> 'c spmf"   and R2 :: "'a \<Rightarrow> 'b \<Rightarrow> 'e spmf"   and S2 :: "'b \<Rightarrow> 'f \<Rightarrow> 'e spmf"   and funct :: "'a \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf" shows "sim_det_def R1 S1 R2 S2 funct \<equiv> ((\<forall>m1 m2. lossless_spmf (R1 m1 m2)) \<and> (\<forall>m1 out1. lossless_spmf (S1 m1 out1))) \<and> (\<forall>m1 m2. lossless_spmf (R2 m1 m2)) \<and> (\<forall>m2 out2. lossless_spmf (S2 m2 out2)) \<and> (\<forall>m1 m2. lossless_spmf (funct m1 m2))"
preserves_cod: fixes f :: "'a" assumes "A.arr f" shows "B.cod (F f) = F (A.cod f)"
Red1CASFail: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v'' :: "'addr val"   and v :: "'addr val"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and v' :: "'addr val"   and l :: "'addr val list" assumes "heap_read h a (CField D F) v''"   and "v \<noteq> v''" shows "uf,P,t \<turnstile>1 \<langle>addr a\<bullet>compareAndSwap(D\<bullet>F, Val v, Val v'),(h, l)\<rangle> -\<lbrace>ReadMem a (CField D F) v''\<rbrace>\<rightarrow> \<langle>false,(h, l)\<rangle>"
hh: shows "hermitean H \<and> H \<noteq> mat_zero"
sf: shows "sets (fct_gen_subalgebra M borel (geom_proc n)) = {geom_proc n -` B |B. B \<in> sets borel}"
exists_n_dvd_n: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n" shows "\<exists>k. k dvd n \<and> P k"
act_sup_distl: fixes x :: "'a"   and p :: "'b"   and q :: "'b" shows "\<alpha> x (p \<squnion> q) = \<alpha> x p \<squnion> \<alpha> x q"
ccSup_le_iff: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and A :: "'a set"   and b :: "'a" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
lformula_lderiv0: fixes a :: "'a"   and x :: "'x" assumes "lformula0 a" shows "lformula (lderiv0 x a)"
simps: fixes A :: "'d mat" shows "square_mat A = (dim_col A = dim_row A)"
cases: fixes r :: "('d, 'f, 'e, 'g) galcon" obtains orderA :: "('d, 'e) gorder_scheme" and orderB :: "('f, 'g) gorder_scheme" and lower :: "'d \<Rightarrow> 'f" and upper :: "'f \<Rightarrow> 'd" where "r = \<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper\<rparr>"
mono: fixes nxs :: "enat \<times> 'a llist" shows "mono_llist (\<lambda>ltake. case nxs of (n, xs) \<Rightarrow> F (curry ltake) n xs)"
uex: shows "(\<lambda>t. (t, u t)) ` cball x d0 \<subseteq> Sigma X existence_ivl0"
foldr_option_Cons_SomeE: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a option"   and x :: "'b"   and xs :: "'b list"   and e :: "'a"   and y :: "'a" assumes "foldr_option f (x # xs) e = Some y" obtains y' :: "'a" where "foldr_option f xs e = Some y'" and "f x y' = Some y"
f2: shows "f |` (- dom h) = (f ++ h) |` (- dom h)"
tObsC_abs_conv: fixes u :: "('a, 'es, 'as) BEState"   and v :: "('a, 'es, 'as) BEState"   and t :: "('a, 'es, 'as) BEState Trace" shows "((u, v) \<in> tObsC_abs t) = (\<exists>t'. t' \<in> SPR.jkbpC \<and> tObsC t' = tObsC t \<and> u = tFirst t' \<and> v = tLast t')"
get_tag_name_ptr_in_heap: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr" assumes "h \<turnstile> ok get_tag_name element_ptr" shows "element_ptr |\<in>| element_ptr_kinds h"
monoidal_category_axioms: shows "monoidal_category (\<cdot>\<^sub>D) T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>1"
double_bind_assoc: fixes g :: "'a \<Rightarrow> 'b measure"   and N :: "'a measure"   and N' :: "'b measure"   and f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and M' :: "'d measure"   and h :: "'c \<Rightarrow> 'd \<Rightarrow> 'a" assumes "g \<in> N \<rightarrow>\<^sub>M subprob_algebra N'"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra M'"   and "(\<lambda>(x, y). h x y) \<in> M \<Otimes>\<^sub>M M' \<rightarrow>\<^sub>M N" shows "M \<bind> (\<lambda>x. f x \<bind> (\<lambda>y. g (h x y))) = M \<bind> (\<lambda>x. f x \<bind> (\<lambda>y. return N (h x y))) \<bind> g"
fold_union: fixes x :: "'b fset" shows "Finite_Set.fold (\<lambda>x. (\<union>) {x}) {} (fset x) = fset x"
enum_sum_def: shows "enum = map Inl enum @ map Inr enum"
mstep_output_iff: fixes \<phi> :: "Formula.formula"   and \<pi> :: "(String.literal \<times> event_data list) prefix"   and R :: "event_data list set"   and st :: "('a, 'b) mstate"   and tdb :: "(String.literal \<times> event_data list) set \<times> nat"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and v :: "event_data option list"   and i :: "nat" assumes "wf_mstate \<phi> \<pi> R st"   and "last_ts \<pi> \<le> snd tdb"   and "prefix_of (psnoc \<pi> tdb) \<sigma>"   and "mem_restr R v" shows "((i, v) \<in> flatten_verdicts (fst (mstep (map_prod mk_db id tdb) st))) = (Monitor.progress \<sigma> Map.empty \<phi> (plen \<pi>) \<le> i \<and> i < Monitor.progress \<sigma> Map.empty \<phi> (Suc (plen \<pi>)) \<and> wf_tuple (Formula.nfv \<phi>) (fv \<phi>) v \<and> Formula.sat \<sigma> Map.empty (map the v) i \<phi>)"
UniversalObject_def: shows "\<^bold>a\<^sub>V \<equiv> \<^bold>\<iota>x. TAO_8_Definitions.Universal (x\<^sup>P)"