hcju/isabelleps huggingface dataset:
bezout_coefficients: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and Gcd :: "'b set \<Rightarrow> 'b"   and Lcm :: "'b set \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.euclidean_ring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm uminus"   and "euclidean_ring_gcd.bezout_coefficients divide minus zero times one modulo unit_factor normalize a b = (x, y)" shows "plus (times x a) (times y b) = gcd a b"
right_unit: fixes a :: "'a" assumes "a \<in> M" shows "a \<cdot> \<one> = a"
raise_trap_def: fixes t :: "Trap" shows "raise_trap t \<equiv> do modify (add_trap_set t); return () od"
dimension_vector: shows "finite_dimensional_vector_space.dimension cart_basis = CARD('c)"
insertI: fixes A :: "'f set"   and a :: "'f" assumes "finite A" shows "finite (insert a A)"
llength_lconcat_lfinite_conv_sum: fixes xss :: "'a llist llist" assumes "lfinite xss" shows "llength (lconcat xss) = (\<Sum>i | enat i < llength xss. llength (lnth xss i))"
oalist_inv_raw_sort_oalist: fixes xs :: "('a \<times> 'c) list" shows "cmp2.oalist_inv_raw (cmp2.sort_oalist xs)"
nopaid: fixes s :: "'a list \<times> 'b"   and n :: "'a" shows "map_pmf (\<lambda>x. snd (fst x)) (snd A s n) = return_pmf []"
traces: shows "{t \<in> T. tLength t = n} = {t \<in> T'. tLength t = n}"
normal_invE_1: fixes G :: "('a, 'b) monoid_scheme"   and N :: "'a set" assumes "Group.group G"   and "N \<lhd> G" shows "subgroup N G"
Abs_inj_on: shows "inj_on word_of_nat (unats LENGTH('b))"
mult_monom: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'b"   and y :: "nat \<Rightarrow>\<^sub>0 nat"   and b :: "'b" shows "MPoly_Type.monom x a * MPoly_Type.monom y b = MPoly_Type.monom (x + y) (a * b)"
induct: fixes P :: "'b nmultiset \<Rightarrow> bool"   and nmultiset :: "'b nmultiset" assumes "\<And>x. P (Elem x)"   and "\<And>x. (\<And>xa. xa \<in># x \<Longrightarrow> P xa) \<Longrightarrow> P (MSet x)" shows "P nmultiset"
rep_eq: fixes x :: "'a cfg" shows "state x = fst (rep_cfg x)"
not_eq_order_implies_strict: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "border a b" shows "strict_border a b"
simps_1: fixes h1 :: "'b heap"   and h2 :: "'b heap"   and hs :: "'b heap list" shows "pass\<^sub>1 (h1 # h2 # hs) = merge h1 h2 # pass\<^sub>1 hs"
2: fixes a :: "'t \<Rightarrow>\<^sub>0 'b" assumes "a \<in> A"   and "a \<noteq> 0"   and "lt a adds\<^sub>t v" shows "rword (spp_of a) (spp_of q)"
simps_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "'a tree"   and m :: "'a"   and r :: "'a tree" assumes "class.linorder less_eq less" shows "linorder.heap less_eq \<langle>l, m, r\<rangle> = (Ball (set_tree l \<union> set_tree r) (less_eq m) \<and> linorder.heap less_eq l \<and> linorder.heap less_eq r)"
tiny_dg_Arr_in_Vset: shows "\<AA>\<lparr>Arr\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
matching_i_edges_as_sets: fixes i :: "nat"   and V :: "'b set"   and E :: "('b \<times> 'b) set"   and M :: "('b \<times> 'b) set"   and L :: "'b \<Rightarrow> nat" shows "edge_as_set ` matching_i i V E M L = {e1. \<exists>(u, v)\<in>matching_i i V E M L. edge_as_set (u, v) = e1}"
sorted_msort: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" assumes "transp f"   and "\<And>x y. f x y \<or> f y x" shows "sorted_wrt f (msort f xs)"
cut_ge_Diff_conv: fixes I :: "'a set"   and t :: "'a" shows "I \<down>\<ge> t = I - {..<t}"
add_to_nat_def: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "plus x y = mod_type.from_nat Abs (mod_type.to_nat Rep x + mod_type.to_nat Rep y)"
strictly_preserves_unity: shows "E.map \<F>C.unity = \<I>\<^sub>D"
nn_cond_exp_intg: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable G"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * nn_cond_exp M G g x \<partial>M = \<integral>\<^sup>+ x. f x * g x \<partial>M"
get_owner_document_pure: fixes get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_get_owner_document get_owner_document" shows "pure (get_owner_document ptr) h"
zero_less_abs_iff: fixes abs :: "'d \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less zero (abs a) = (a \<noteq> zero)"
inf_le2: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> y \<le> y"
star_inf: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "\<And>x y z. \<lbrakk>r x y; r x z\<rbrakk> \<Longrightarrow> y = z"   and "r\<^sup>*\<^sup>* x y"   and "Inf.inf r x" shows "Inf.inf r y"
poly_deg_subst_pp_le: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and f :: "'b \<Rightarrow> ('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" assumes "\<And>x. x \<in> keys t \<Longrightarrow> poly_deg (f x) \<le> 1" shows "poly_deg (subst_pp f t) \<le> deg_pm t"
simps_6: shows "c\<^sub>c \<noteq> b\<^sub>c"
Sup_ne_def: fixes X :: "'b set" shows "Sup_ne X = FOLD_bin_ne sup X"
perm_swap_eq: fixes a :: "atom"   and b :: "atom"   and p :: "perm" shows "((a \<rightleftharpoons> b) \<bullet> p = p) = (p \<bullet> (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b))"
trms\<^sub>e\<^sub>s\<^sub>tD: fixes t :: "('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list" assumes "t \<in> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set A)" shows "t \<in> trms\<^sub>s\<^sub>t (to_st A)"
distinct_take_drop: fixes vs :: "'a list"   and i :: "nat" assumes "distinct vs"   and "i < length vs" shows "distinct (take i vs @ drop (Suc i) vs)"
kolmogorov_0_1_law: fixes A :: "nat \<Rightarrow> ('b \<Rightarrow> 'a) set set"   and X :: "('b \<Rightarrow> 'a) set" assumes "\<And>i. sigma_algebra (space (Pi\<^sub>M UNIV (\<lambda>_. M))) (A i)"   and "P.P.indep_sets A UNIV"   and "X \<in> P.P.tail_events A" shows "P.P.prob X = 0 \<or> P.P.prob X = 1"
not_span_independent_insert: fixes S :: "('a, 'b) vec set"   and v :: "('a, 'b) vec" assumes "independent S"   and "v \<notin> span S" shows "independent (insert v S)"
gState2HASH_eq: fixes x :: "'b gState_scheme"   and y :: "'b gState_scheme" assumes "gState2HASH x = gState2HASH y" shows "x = y"
old_document: shows "h \<turnstile> get_owner_document (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) \<rightarrow>\<^sub>r old_document"
currentD_OUT: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and b :: "'a" assumes "current \<Gamma> f"   and "b \<in> B \<Gamma>" shows "d_OUT f b = 0"
linear_order_on_def: fixes A :: "'d set"   and r :: "('d \<times> 'd) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> total_on A r"
index_of_nth: fixes xs :: "'c list"   and i :: "nat" assumes "distinct xs"   and "i < length xs" shows "index_of (xs ! i) xs = i"
init_lvars_def2: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and mode :: "inv_mode"   and a' :: "val"   and pvs :: "val list"   and x :: "abopt"   and s :: "st" shows "init_lvars G C sig mode a' pvs (x, s) = set_lvars (\<lambda>k. case k of EName (VNam v) \<Rightarrow> [pars (mthd (the (methd G C sig))) [\<mapsto>] pvs] v | EName Res \<Rightarrow> None | This \<Rightarrow> if mode = Static then None else Some a') (if mode = Static then x else np a' x, s)"
l8_14_2_1a: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B C D"
get_ancestors_si_ptrs_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_ancestors_si ptr \<rightarrow>\<^sub>r ancestors"   and "ptr' \<in> set ancestors" shows "ptr' |\<in>| object_ptr_kinds h"
attractive_axioms_def: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "attractive_axioms A less_eq \<equiv> semiattractive A (\<lambda>x y. less_eq y x)"
P_def: shows "P = (\<lambda>p. p \<in> local.\<P> \<longrightarrow> p \<succeq>[\<R>] return_pmf B)"
tendsto_add_filterlim_at_infinity: fixes f :: "'a \<Rightarrow> 'b"   and c :: "'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "(f \<longlongrightarrow> c) F"   and "filterlim g at_infinity F" shows "LIM x F. f x + g x :> at_infinity"
get_plane_def: fixes X :: "'d set" shows "get_plane X = SPEC (\<lambda>sctn. X = plane_of sctn)"
empty_ordering: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" shows "\<exists>f. TernaryOrdering.ordering f ord {}"
qs_def: shows "qs = (map 0 ps)[i := p]"
adjuster_in_span: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us \<in> local.span (set us)"
restrict_codomain_to_image: shows "ChamberComplexMorphism X (f \<turnstile> X) f"
mem_correct_rbt_mapping: shows "mem_correct (\<lambda>k. State_Monad.get \<bind> (\<lambda>m. State_Monad.return (Mapping.lookup m k))) (\<lambda>k v. State_Monad.get \<bind> (\<lambda>m. State_Monad.set (Mapping.update k v m))) (\<lambda>_. True)"
count_replicate_mset: fixes n :: "nat"   and x :: "'k"   and y :: "'k" shows "count (replicate_mset n x) y = (if y = x then n else 0)"
Cod_comp: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.arr f"   and "Maps.arr g"   and "Maps.Dom g = Maps.Cod f" shows "Maps.Cod (g \<odot> f) = Maps.Cod g"
winning_path_supergame: fixes p :: "Player"   and P :: "'a llist"   and G' :: "('a, 'c) ParityGame_scheme" assumes "winning_path p P"   and "ParityGame G'"   and "VV p** \<subseteq> (if p** = Even then V0\<^bsub>G'\<^esub> else V\<^bsub>G'\<^esub> - V0\<^bsub>G'\<^esub>)"   and "\<omega> = \<omega>\<^bsub>G'\<^esub>" shows "ParityGame.winning_path G' p P"
A_to_prodag_rHom: fixes A :: "('d, 'e) Ring_scheme"   and I :: "'f set"   and B :: "'f \<Rightarrow> ('g, 'h) Ring_scheme"   and S :: "'f \<Rightarrow> 'd \<Rightarrow> 'g" assumes "Ring A"   and "\<forall>k\<in>I. Ring (B k)"   and "\<forall>k\<in>I. S k \<in> rHom A (B k)" shows "A_to_prodag A I S B \<in> rHom A (r\<Pi>\<^bsub>I\<^esub> B)"
Quotient3_Gromov_completion: shows "Quotient3 Gromov_completion_rel abs_Gromov_completion rep_Gromov_completion"
in_homE: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and b :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] b\<guillemotright>" obtains "A_B.arr f" and "A_B.dom f = a" and "A_B.cod f = b"
in_keys_plusI2: fixes t :: "'d"   and q :: "'d \<Rightarrow>\<^sub>0 'e"   and p :: "'d \<Rightarrow>\<^sub>0 'e" assumes "t \<in> keys q"   and "t \<notin> keys p" shows "t \<in> keys (p + q)"
simps_9: fixes i :: "String.literal"   and x :: "String.literal"   and p :: "fol fm" shows "free_in i (Exists x p) = (i \<noteq> x \<and> free_in i p)"
Gram_Schmidt_column_k_efficient_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "Gram_Schmidt_column_k_efficient A k = (\<chi>a b. (if b = mod_type_class.from_nat k then column b A - (\<Sum>x\<leftarrow>map (\<lambda>n. column (mod_type_class.from_nat n) A) [0..<mod_type_class.to_nat b]. (column b A \<bullet> x / (x \<bullet> x)) *\<^sub>R x) else column b A) $ a)"
assms_1: shows "check_single \<C> e (Type t1s) = ctm"
smult_pCons: fixes a :: "'a"   and b :: "'a"   and p :: "'a poly" shows "smult a (pCons b p) = pCons (a * b) (smult a p)"
simps_1: fixes x :: "vrs"   and Y :: "tyvrs"   and T2 :: "ty" shows "Var x[Y \<mapsto>\<^sub>\<tau> T2] = Var x"
frameImpEmpty: fixes \<Psi> :: "'b"   and \<phi> :: "'c" shows "(\<langle>\<epsilon>, \<Psi>\<rangle>) \<turnstile>\<^sub>F \<phi> = \<Psi> \<turnstile> \<phi>"
lessThan_Int_lessThan: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.linorder less_eq less" shows "ord.greaterThan less a \<inter> ord.greaterThan less b = ord.greaterThan less (ord.max less_eq a b)"
diff: fixes x :: "real"   and y :: "real"   and xa :: "'c" shows "(x - y) *\<^sub>R xa = x *\<^sub>R xa - y *\<^sub>R xa"
preserves_reflects_arr: fixes f :: "'a1" shows "A2_B.arr (\<tau> f) = A1.arr f"
sortedP_Cons: fixes xa :: "real \<times> real"   and x :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "ccw.sortedP xa (x # xs)" obtains "\<forall>xb\<in>set xs. ccw xa x xb" and "ccw.sortedP xa xs"
assms_3: shows "(r, Expunge e2 i r) \<in> set (node_deliver_messages xs)"
left_secure: shows "component_secure_prot_stateful n (Pi n) Sec attack"
dim_R_def: fixes M :: "'m set" shows "dim_R M = (if \<exists>ms. set ms \<subseteq> M \<and> RSpan ms = M then LEAST n. \<exists>ms. length ms = n \<and> set ms \<subseteq> M \<and> RSpan ms = M else 0)"
valid_ReqsACS: shows "valid_reqs (get_ACS M)"
scaleC_eq_iff: fixes b :: "'d"   and u :: "complex"   and a :: "'d" shows "(b + u *\<^sub>C a = a + u *\<^sub>C b) = (a = b \<or> u = 1)"
assms: shows "a \<in> parcs (list_digraph G_list)"
DPR: fixes x :: "'i list" shows "Ex (list_all2 R x) = list_all (arith_ops_record.DP ops) x"
hyps: shows "TypeExn (mk_id mn_ cn_) \<notin> defined_types s_"
unfold2': fixes E :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> 'res) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state"   and self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3)"   and POST' :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0) \<Rightarrow> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and BODY :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0) \<Rightarrow> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> 'res" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self \<and> (case a1 of (a1, a2, a3) \<Rightarrow> \<lambda>\<tau>. \<tau> \<Turnstile> \<upsilon> a1 \<and> \<tau> \<Turnstile> \<upsilon> a2 \<and> \<tau> \<Turnstile> \<upsilon> a3) \<tau>"   and "\<tau> \<Turnstile> case a1 of (a, b, c) \<Rightarrow> PRE self a b c"   and "\<tau> \<Turnstile> POST' self a1"   and "\<And>res. (case a1 of (a, b, c) \<Rightarrow> POST self a b c) res = (POST' self a1 and (res \<triangleq> BODY self a1))" shows "(\<tau> \<Turnstile> E (case a1 of (a, b, c) \<Rightarrow> f self a b c)) = (\<tau> \<Turnstile> E (BODY self a1))"
polymul_norm: fixes p :: "poly"   and q :: "poly" assumes "isnpoly p"   and "isnpoly q" shows "isnpoly (p *\<^sub>p q)"
u_1: fixes i :: "'a" assumes "i \<in> I"   and "a \<le> i"   and "i < b" shows "(f \<longlongrightarrow> u i) (at_right i)"
distinct_append: fixes xs :: "'c list"   and ys :: "'c list" shows "distinct (xs @ ys) = (distinct xs \<and> distinct ys \<and> set xs \<inter> set ys = {})"
union_disjoint: fixes A :: "'b set"   and B :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "prod g (A \<union> B) = prod g A * prod g B"
simps_2: fixes T'' :: "ty"   and Ts'' :: "ty list"   and T' :: "ty"   and Ts' :: "ty list" shows "conformable (T'' # Ts'') (T' # Ts') = ((T'' = T' \<or> (\<exists>C. T'' = NT \<and> T' = Class C)) \<and> conformable Ts'' Ts')"
phi_k: shows "\<phi> k = j # a @ [k] \<odot> 0\<^sup>\<infinity>"
sset_flat: fixes s :: "'a list stream" assumes "\<forall>xs\<in>i.R s. xs \<noteq> []" shows "i.R (flat s) = \<Union> (set ` i.R s)"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "S.src f = S.null"
apos_neq_minf: fixes a :: "ant" assumes "0 \<le> a" shows "a \<noteq> - \<infinity>"
bezout_coefficients_mod_int: fixes F :: "'a mod_ring poly"   and f :: "int poly"   and G :: "'a mod_ring poly"   and g :: "int poly"   and A :: "'a mod_ring poly"   and B :: "'a mod_ring poly"   and a :: "int poly"   and b :: "int poly" assumes "F = of_int_poly f"   and "G = of_int_poly g"   and "coprime_m f g"   and "bezout_coefficients F G = (A, B)"   and "a = to_int_poly A"   and "b = to_int_poly B" shows "f * a + g * b =m 1"
totient_code_naive: fixes n :: "nat" shows "totient n = totient_naive n 0 n"
tendsto_add_ereal_general: fixes x :: "ereal"   and y :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and F :: "'b filter"   and g :: "'b \<Rightarrow> ereal" assumes "\<not> (x = \<infinity> \<and> y = - \<infinity> \<or> x = - \<infinity> \<and> y = \<infinity>)"   and "(f \<longlongrightarrow> x) F"   and "(g \<longlongrightarrow> y) F" shows "((\<lambda>x. f x + g x) \<longlongrightarrow> x + y) F"
mult_coprime: fixes a :: "nat"   and b :: "nat" assumes "coprime a b" shows "moebius_mu (a * b) = moebius_mu a * moebius_mu b"
real_scalar_mult: 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 cnj :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and real_of :: "'a \<Rightarrow> real"   and c :: "real"   and a :: "'a" assumes "class.real_of_extended divide inverse times one plus zero minus uminus cnj scaleR real_of" shows "real_of (scaleR c a) = c * real_of a"
lipschitz_on_constant: fixes U :: "'a set"   and c :: "'b" shows "0-lipschitz_on U (\<lambda>x. c)"
transfer_imp: fixes p :: "bool"   and P :: "nat \<Rightarrow> bool"   and q :: "bool"   and Q :: "nat \<Rightarrow> bool" assumes "p \<equiv> eventually P \<U>"   and "q \<equiv> eventually Q \<U>" shows "p \<longrightarrow> q \<equiv> \<forall>\<^sub>F n in \<U>. P n \<longrightarrow> Q n"
ide_char: fixes f :: "'d \<times> 'c" shows "DopxC.ide f = (Dop.ide (fst f) \<and> C.ide (snd f))"
that: fixes k :: "nat" assumes "k \<le> l"   and "\<pi> k = n" shows "thesis"
mask_shift: fixes x :: "'b word"   and y :: "nat" shows "x AND NOT (mask y) >> y = x >> y"
aux_lemma: fixes T :: "'c ring"   and f :: "'a \<Rightarrow> 'd"   and S :: "'d ring"   and g :: "'a \<Rightarrow> 'c" assumes "ring T"   and "f \<in> ring_hom R S"   and "g \<in> ring_hom R T"   and "f ` carrier R \<subseteq> f ` a_kernel R T g" shows "(\<lambda>a. (f a, g a)) ` carrier R = f ` carrier R \<times> g ` carrier R"
cgderiv_def: fixes f :: "'b \<Rightarrow> complex"   and x :: "'b"   and D :: "'b" shows "cGDERIV f x :> D = (f has_derivative cinner D) (at x)"
assms: shows "\<R>\<^sub>\<circ> r = 0"
simple_syntactic_and_def: fixes P :: "'c \<Rightarrow> bool" shows "simple_syntactic_and P \<equiv> \<forall>t u. P (app t u) = (P t \<and> P u)"
Word_compareTr2: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme" assumes "Worder D"   and "Worder E"   and "ord_equiv D E"   and "\<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b))" shows "False"
cong_simp: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and h :: "'d \<Rightarrow> 'e" assumes "A = B"   and "\<And>x. x \<in> B =simp=> g x = h x" shows "sum g A = sum h B"
sublist_Cons_right: fixes xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "sublist xs (y # ys) = (prefix xs (y # ys) \<or> sublist xs ys)"
grun_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and V :: "(nat \<times> 'a) set" shows "grun A w V \<equiv> transition_system.run (\<lambda>x _. x) (\<lambda>u (k, p). w !! k \<in> alphabet A \<and> u \<in> {Suc k} \<times> transition A (w !! k) p \<inter> V)"
spr_repMC_S5n: fixes X :: "(('a, 'es, 'as) BEState \<times> ('a, 'es, 'as) BEState) set" shows "S5n (spr_repMC X)"
Sup_pres_Sup_closed: fixes f :: "'c \<Rightarrow> 'd" assumes "Sup_pres f" shows "Sup_closed_set (range f)"
complete_lattice_criterion1: fixes L :: "('c, 'd) gorder_scheme" assumes "partial_order L"   and "\<exists>g. greatest L g (carrier L)"   and "\<And>A. \<lbrakk>A \<subseteq> carrier L; A \<noteq> {}\<rbrakk> \<Longrightarrow> \<exists>i. is_glb L i A" shows "complete_lattice L"
Union_components: fixes u :: "'c set" shows "\<Union> (components u) = u"
not_Iic_eq_Icc: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h' :: "'a"   and l :: "'a"   and h :: "'a" assumes "class.no_bot less_eq less" shows "ord.atMost less_eq h' \<noteq> ord.atLeastAtMost less_eq l h"
filtrated_prob_space_axioms_def: fixes M :: "'c measure"   and F :: "'d \<Rightarrow> 'c measure" shows "filtrated_prob_space_axioms M F \<equiv> Filtration.filtration M F"
lists_eq_set: fixes A :: "'g set" shows "lists A = {xs. set xs \<subseteq> A}"
While_le: fixes Inv :: "'a"   and b :: "bexp"   and c :: "'a acom"   and P :: "'a"   and w :: "'a acom" shows "({Inv} WHILE b DO c {P} \<le> w) = (\<exists>Inv' c' P'. w = {Inv'} WHILE b DO c' {P'} \<and> c \<le> c' \<and> Inv \<le> Inv' \<and> P \<le> P')"
unit_in_vhom: shows "\<guillemotleft>\<phi> : src f \<Rightarrow> g' \<star> f\<guillemotright>"
ipresIGWlsAll_def: fixes h :: "'u \<Rightarrow> 'v"   and hA :: "'w \<Rightarrow> 'x"   and MOD :: "('y, 'z, 'aa, 'ab, 'ac, 'ad, 'u, 'w, 'ae) model_scheme"   and MOD' :: "('af, 'ag, 'aa, 'ab, 'ah, 'ai, 'v, 'x, 'aj) model_scheme" shows "ipresIGWlsAll h hA MOD MOD' \<equiv> ipresIGWls h MOD MOD' \<and> ipresIGWlsAbs hA MOD MOD'"
mat0I_def: fixes ze :: "'c"   and nr :: "nat"   and nc :: "nat" shows "mat0I ze nr nc = replicate nc (vec0I ze nr)"
\<tau>_ide: fixes b :: "'c" assumes "Cop.ide b" shows "\<tau> b = S.mkArr (Hom.set (b, a)) (F.SET b) (\<lambda>x. F.FUN (\<psi> (b, a) x) (\<tau>.FUN a (\<phi> (a, a) a)))"