hcju/isabelleps huggingface dataset:
path_sub_events: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and Q :: "'b set" assumes "MinkowskiPrimitive \<E> \<P>"   and "Q \<in> \<P>" shows "Q \<subseteq> \<E>"
Diff_empty: fixes A :: "'b set" shows "A\<setminus>{} = A"
Phase0Read_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk" shows "Phase0Read s s' p d = (phase s p = 0 \<and> blocksRead s' = (blocksRead s)(p := (blocksRead s p)(d := blocksRead s p d \<union> {\<lparr>block = disk s d p, proc = p\<rparr>})) \<and> inpt s' = inpt s \<and> outpt s' = outpt s \<and> disk s' = disk s \<and> phase s' = phase s \<and> dblock s' = dblock s \<and> disksWritten s' = disksWritten s)"
commute: fixes a :: "'b"   and b :: "'b" shows "max a b = max b a"
f_expand_Cons: fixes k :: "nat"   and x :: "'b message_af"   and xs :: "'b message_af list" assumes "0 < k" shows "(x # xs) \<odot> k = x # NoMsg\<^bsup>k - Suc 0\<^esup> @ xs \<odot> k"
list_all_length: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" shows "list_all P xs = (\<forall>n<length xs. P (xs ! n))"
simps_1: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "sorted_wrt P [] = True"
eq_fold': fixes A :: "'a set" shows "\<Sqinter> A = the (Finite_Set.fold (\<lambda>x y. Some (case y of None \<Rightarrow> x | Some z \<Rightarrow> x \<sqinter> z)) None A)"
add_assocD_var1: fixes y :: "'a"   and z :: "'a"   and x :: "'a" assumes "D y z \<and> D x (y \<oplus> z)" shows "D x y \<and> D (x \<oplus> y) z"
fold_insert: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'d set"   and x :: "'d"   and z :: "'e" assumes "comp_fun_commute f"   and "finite A"   and "x \<notin> A" shows "Finite_Set.fold f z (insert x A) = f x (Finite_Set.fold f z A)"
MkIde_Dom: fixes a :: "(bool, 'arr list) concrete_category.arr" assumes "arr a" shows "MkIde (Dom a) = local.dom a"
sp_fundefs_get: fixes F :: "'a \<Rightarrow> ('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef option"   and f :: "'a"   and fd :: "('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef" assumes "sp_fundefs F"   and "F f = Some fd" shows "sp_fundef F fd (body fd) = Ok [None]"
hom_char_matrix: fixes A :: "'a mat"   and n :: "nat"   and x :: "'a" assumes "A \<in> carrier_mat n n" shows "char_matrix (mat\<^sub>h A) (hom x) = mat\<^sub>h (char_matrix A x)"
unconst_rel: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and t :: "'c"   and u :: "'d" assumes "term_struct_rel_strong P"   and "P t u" shows "unconst t = unconst u"
list2policy_def: fixes l :: "('c \<Rightarrow> 'd decision option) list" shows "list2policy l = foldr (\<lambda>x y. x \<Oplus> y) l \<emptyset>"
sgn_mult: fixes x :: "'a"   and y :: "'a" shows "sgn (x * y) = sgn x * sgn y"
coprime_exp: fixes d :: "nat"   and a :: "nat"   and n :: "nat" assumes "coprime d a" shows "coprime d (a ^ n)"
set_return_pmf: fixes x :: "'e" shows "set_pmf (return_pmf x) = {x}"
monom_altdef: fixes c :: "'b"   and n :: "nat" shows "monom c n = Polynomial.smult c ([:0::'b, 1::'b:] ^ n)"
length_ge_zero: fixes r :: "real_int" shows "0 \<le> \<parallel>r\<parallel>"
conjugE1: fixes u :: "'b list"   and v :: "'b list" assumes "u \<sim> v" obtains r :: "'b list" where "u \<cdot> r = r \<cdot> v"
Cons: fixes a :: "'a"   and A :: "'a set"   and l :: "'a list" assumes "a \<in> A"   and "l \<in> lists A" shows "a # l \<in> lists A"
llist_exhaust: fixes xs :: "'c llist" obtains ys :: "'c list" where "xs = llist_of ys"   | f :: "nat \<Rightarrow> 'c" where "xs = inf_llist f"
vec_upd_def: fixes s :: "('b, 'c) vec"   and i :: "'c"   and a :: "'b" shows "HS_Preliminaries.vec_upd s i a = (\<chi>j. ((($) s)(i := a)) j)"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'c list" shows "take n (drop m xs) = drop m (take (n + m) xs)"
ordLeq_csum1: fixes p1 :: "('c \<times> 'c) set"   and p2 :: "('d \<times> 'd) set" assumes "Card_order p1" shows "p1 \<le>o p1 +c p2"
merge_guards_simps_4: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "merge_guards (Seq c\<^sub>1 c\<^sub>2) = Seq (merge_guards c\<^sub>1) (merge_guards c\<^sub>2)"
wellformed_items_Predict: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and I :: "item set"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "LocalLexing.wellformed_items \<RR> Doc I" shows "LocalLexing.wellformed_items \<RR> Doc (LocalLexing.Predict \<RR> k I)"
rtrancl_trancl_absorb: fixes R :: "('a \<times> 'a) set" shows "(R\<^sup>*)\<^sup>+ = R\<^sup>*"
map_poly_0: fixes f :: "'b \<Rightarrow> 'a" shows "map_poly f 0 = 0"
isolated_verts_def: shows "isolated_verts \<equiv> {v \<in> verts G. out_arcs G v = {}}"
tinres_nonempty_2: fixes r' :: "cell list" assumes "tinres [] r'"   and "r' \<noteq> []" shows "hd r' = Bk"
Min_in: fixes A :: "'a Word.word set" assumes "finite A"   and "A \<noteq> {}" shows "signed.Min A \<in> A"
frac_1_eq: fixes x :: "'b" shows "frac (x + (1::'b)) = frac x"
proj_zero: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and k :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.proj_poly term_of_pair k 0 = 0"
brw_inner_invar_initial: fixes q :: "'d"   and W :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and Q :: "'d \<Rightarrow> 'e tree option"   and rcm :: "('d, 'e) ta_rule \<Rightarrow> nat option" assumes "q \<in> W" shows "brw_inner_invar {r \<in> \<delta>. q \<in> set (rhsq r)} q (Q, W - {q}, rcm) {r \<in> \<delta>. q \<in> set (rhsq r)} (Q, W - {q}, rcm)"
umf_pair: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and qs :: "'c list"   and init :: "'c list" assumes "List_Factoring.pairwise A"   and "\<And>is s q. \<forall>((free, paid), uu_)\<in>set_pmf (snd A (s, is) q). paid = []"   and "set qs \<subseteq> set init"   and "distinct init"   and "\<And>x. x < length qs \<Longrightarrow> finite (set_pmf (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) (take x qs)))" shows "T\<^sub>p_on_rand' A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs = (\<Sum>(x, y)\<in>{(x, y). x \<in> set init \<and> y \<in> set init \<and> x < y}. T\<^sub>p_on_rand' A (fst A (Lxy init {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy init {x, y}, is))) (Lxy qs {x, y}))"
assms_2: 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 child) \<rightarrow>\<^sub>r owner_document"
insertAssertionSimps: fixes A\<^sub>F :: "name list"   and \<Psi> :: "'b"   and \<Psi>\<^sub>F :: "'b" assumes "A\<^sub>F \<sharp>* \<Psi>" shows "insertAssertion (\<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>) \<Psi> = \<langle>A\<^sub>F, \<Psi> \<otimes> \<Psi>\<^sub>F\<rangle>"
Abs_uint8_inverse: fixes y :: "8 word" assumes "y \<in> UNIV" shows "Rep_uint8 (Abs_uint8 y) = y"
reaches1_reaches_iff1: fixes x :: "'a"   and y :: "'a" shows "G'.reaches1 x y = (\<exists>z. E' x z \<and> G'.reaches z y)"
factorial_condition_one: fixes G :: "('b, 'c) monoid_scheme" shows "(divisor_chain_condition_monoid G \<and> primeness_condition_monoid G) = factorial_monoid G"
porder_on_linorder_on_tranclp_porder_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and B :: "'b set"   and s :: "('b \<times> 'b) set" assumes "partial_order_on A r"   and "linear_order_on B s"   and "order_consistent r s"   and "B \<subseteq> A" shows "partial_order_on A ((r \<union> s)\<^sup>+)"
ordinal_of_nat_plus_omega: fixes n :: "nat" shows "ordinal_of_nat n + \<omega> = \<omega>"
lift_None: fixes h :: "'b \<Rightarrow> 'a"   and inp :: "'c \<Rightarrow> 'b option"   and i :: "'c" shows "(lift h inp i = None) = (inp i = None)"
nonInterferenceI: fixes \<Gamma> :: "char list \<Rightarrow> secLevel option"   and c :: "com" assumes "\<And>s1 s2 s1' s2'. \<lbrakk>\<Gamma> \<turnstile> s1 \<approx>\<^sub>L s2; \<langle>c,s1\<rangle> \<rightarrow>* \<langle>Skip,s1'\<rangle>; \<langle>c,s2\<rangle> \<rightarrow>* \<langle>Skip,s2'\<rangle>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> s1' \<approx>\<^sub>L s2'" shows "nonInterference \<Gamma> c"
adm_edges_inv_disj: fixes f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat" shows "adm_edges f l \<inter> (adm_edges f l)\<inverse> = {}"
power_mult_distrib: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.comm_monoid_mult times one" shows "power.power one times (times a b) n = times (power.power one times a n) (power.power one times b n)"
list_slice_aux_nth: fixes m :: "nat"   and n :: "nat"   and xs :: "'b list"   and k :: "nat" assumes "m < n" shows "list_slice_aux xs k n ! m = xs \<up> (m * k) \<down> k"
conseq_ir_valid: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and Q' :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "ir_valid P c c' Q"   and "\<And>s s'. P s s' \<Longrightarrow> P' s s'"   and "\<And>s s'. Q' s s' \<Longrightarrow> Q s s'" shows "ir_valid P' c c' Q'"
minus_apply: fixes A :: "'b \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'a"   and x :: "'b" shows "(A - B) x = A x - B x"
some_geodesic_endpoints_1: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" shows "x \<in> {x--S--y}"
ord0_precD: fixes x :: "ord0"   and y :: "ord0" assumes "(x, y) \<in> ord0_prec" shows "\<exists>f n. x = f n \<and> y = ord0_Lim f"
literalElFormulaCharacterization: fixes literal :: "Literal"   and formula :: "Literal list list" shows "literal el formula = (\<exists>clause. clause el formula \<and> literal el clause)"
zero_order_5: fixes n :: "'b multiset" shows "(\<not> {#} \<subset># n) = (n = {#})"
is_empty_unfold_2: fixes A :: "'b set" shows "set_eq {} A = Set.is_empty A"
assms_1: shows "[a] @ u =\<^sub>F u\<^sub>1 @ [a] @ u\<^sub>2"
arrE: fixes f :: "('a2 \<Rightarrow> 'b, 'a2 \<Rightarrow> 'b) concrete_category.arr" assumes "A2_B.arr f" obtains "f \<noteq> A2_B.null" and "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (A2_B.Dom f) (A2_B.Cod f) (A2_B.Map f)"
representsI: fixes p :: "int poly"   and x :: "'a" assumes "ipoly p x = (0::'a)"   and "p \<noteq> 0" shows "p represents x"
\<delta>\<^sub>L_reach: fixes \<Sigma> :: "'b list"   and \<delta> :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and q\<^sub>0 :: "'a" assumes "finite (reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0)" shows "\<delta>\<^sub>L \<Sigma> \<delta> q\<^sub>0 = reach\<^sub>t (set \<Sigma>) \<delta> q\<^sub>0"
cfg_eq: fixes \<Gamma> :: "(String.literal \<Rightarrow> unit option) \<times> (String.literal \<Rightarrow> unit option)"   and c :: "cmd"   and a :: "action"   and c' :: "cmd" assumes "ty_cmd (\<Gamma>, False) c" shows "cfg' c a c' = cfg c (Inl a) c'"
lincomb_Cons: fixes r :: "'f"   and rs :: "'f list"   and m :: "'v"   and ms :: "'v list" shows "(r # rs) \<bullet>\<cdot> (m # ms) = r \<cdot> m + rs \<bullet>\<cdot> ms"
distinct_21: fixes name1 :: "name"   and name2 :: "name"   and pi :: "pi"   and name1' :: "name"   and name2' :: "name"   and pi' :: "pi" shows "name1{name2}.pi \<noteq> name1'<name2'>.pi'"
simps_1: shows "max_Var_floatariths [] = 0"
absorb_iff2: fixes a :: "'g multiset"   and b :: "'g multiset" shows "(a \<subseteq># b) = (a \<union># b = b)"
1: shows "h (inv a) = inv\<^bsub>S\<^esub> h a"
grid_disjunct': fixes p :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and p' :: "(nat \<times> int) list"   and x :: "(nat \<times> int) list"   and ds' :: "nat set" assumes "p \<in> grid b ds"   and "p' \<in> grid b ds"   and "x \<in> grid p ds'"   and "p \<noteq> p'"   and "ds \<inter> ds' = {}" shows "x \<notin> grid p' ds'"
r0: shows "r (Suc k) (a $ 0) ^ Suc k = a $ 0"
simps_17: fixes x61 :: "msg"   and x62 :: "msg"   and x1 :: "agent" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Agent x1"
assms: shows "P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'"
pset_pnorm_alt: fixes r :: "'a rexp" shows "pset (pnorm_alt r) = pnorm_alt ` pset r"
concurrentD2: fixes hb_weak :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and hb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and s1 :: "'c"   and s2 :: "'c" assumes "happens_before hb_weak hb"   and "happens_before.concurrent hb s1 s2" shows "\<not> hb s2 s1"
is_WAIT_def: fixes thread :: "thread" shows "is_WAIT thread = (\<exists>x2. thread = WAIT x2)"
le: fixes y :: "'b"   and x :: "real" assumes "y \<in> A"   and "a \<le> x" shows "norm (f y x) \<le> g x"
span_is_submodule: fixes S :: "'c set" assumes "S \<subseteq> carrier M" shows "LinearCombinations.submodule R (local.span S) M"
assms_1: shows "I \<circ> K \<circ> I = I"
valid_conj_lift: fixes P :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool"   and p :: "'d process_name"   and Q :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool"   and P' :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool"   and Q' :: "(('a, 'b, 'c) response, char list, 'd process_name, 'd process_name \<times> ('a, 'd, 'b, 'c) request_op, ('a, 'd, 'b, 'c) Model.local_state) system_state \<Rightarrow> bool" assumes "\<lbrace>P\<rbrace> p \<lbrace>Q\<rbrace>"   and "\<lbrace>P'\<rbrace> p \<lbrace>Q'\<rbrace>" shows "\<lbrace>\<lambda>s. P s \<and> P' s\<rbrace> p \<lbrace>\<lambda>s. Q s \<and> Q' s\<rbrace>"
prems_1: shows "ro_ifex (IF xv xb1 xb2)"
h: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and x :: "'a" assumes "Measure1 m h" shows "m x \<le> h"
cos_ge_minus_one: fixes x :: "real" shows "- 1 \<le> cos x"
nfinal_notin_def: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and T :: "('e, 'f) xstate set" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>T = (\<forall>t. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t \<longrightarrow> t \<notin> T)"
fixespointwise_insert: fixes f :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and a :: "'b" assumes "fixespointwise f A"   and "f ` insert a A = insert a A" shows "fixespointwise f (insert a A)"
fold_mset_fun_left_comm: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'d"   and s :: "'e"   and M :: "'d multiset" assumes "comp_fun_commute f" shows "f x (fold_mset f s M) = fold_mset f (f x s) M"
ldropn_ltl: fixes n :: "nat"   and xs :: "'c llist" shows "ldropn n (ltl xs) = ldropn (Suc n) xs"
Var_subtermeq: fixes t :: "('a, 'b) Term.term"   and v :: "'b" assumes "t \<sqsubseteq> Var v" shows "t = Var v"
R2: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and X :: "'b set"   and Y :: "('b, 'c) ipurge_rec" assumes "\<lparr>Pol = I, Map = D, Doms = U, List = x # xs, ListOp = Some ys, Set = X\<rparr> \<in> ipurge_fail_aux_t_set Y"   and "\<exists>u\<in>U. (u, D x) \<in> I" shows "\<lparr>Pol = I, Map = D, Doms = U, List = ipurge_tr I D (D x) xs, ListOp = Some ys, Set = ipurge_ref I D (D x) xs X\<rparr> \<in> ipurge_fail_aux_t_set Y"
le: shows "ccw' (fst (half_segments_of_aform X ! n)) (snd (half_segments_of_aform X ! n)) (2 *\<^sub>R fst X - aform_val e X)"
\<alpha>\<beta>2\<gamma>_lt_\<alpha>\<beta>1\<gamma>': shows "zhmset_of \<alpha>2 + zhmset_of \<beta>2 * zhmset_of \<gamma> < zhmset_of \<alpha>1 + zhmset_of \<beta>1 * zhmset_of \<gamma>"
polydivide_def: fixes s :: "poly"   and p :: "poly" shows "polydivide s p = polydivide_aux (head p) (Reflected_Multivariate_Polynomial.degree p) p 0 s"
hyps: fixes a :: "int" shows "length [a..a + int n] = Suc n"
disj: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> C \<inter> sigma_sets \<Omega> M"   and "disjoint_family A" shows "\<Union> (range A) \<in> C"
s'_2: shows "learn_bc \<phi> (U\<^sub>0 \<union> V\<^sub>0) s'"
octo_cnj_mult_conv_norm: fixes x :: "octo" shows "Octonions.cnj x * x = (octo_of_real (norm x))\<^sup>2"
case_nat_unfold: fixes x :: "'c"   and f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "case n of 0 \<Rightarrow> x | Suc x \<Rightarrow> f x \<equiv> if n = 0 then x else f (n - 1)"
enum_bit0_def: shows "enum_class.enum = map (Abs_bit0' \<circ> int) [0..<CARD('a bit0)]"
better_T_Let: fixes \<Gamma> :: "(name \<times> ty) list"   and t :: "trm"   and T :: "ty"   and p :: "pat"   and \<Delta> :: "(name \<times> ty) list"   and u :: "trm"   and U :: "ty" assumes "\<Gamma> \<turnstile> t : T"   and "\<turnstile> p : T \<Rightarrow> \<Delta>"   and "\<Delta> @ \<Gamma> \<turnstile> u : U" shows "\<Gamma> \<turnstile> (LET p = t IN u) : U"
singleton_orbits: shows "\<Union> {N \<in> orbits. card N = 1} = fixed_points"
insertCI: fixes a :: "'a"   and B :: "'a set"   and b :: "'a" assumes "a \<notin> B \<Longrightarrow> a = b" shows "a \<in> insert b B"
nth_prime_partition'': fixes x :: "real" assumes "2 \<le> x" shows "x \<in> {real (nth_prime (nat \<lfloor>primes_pi x\<rfloor> - 1))..<real (nth_prime (nat \<lfloor>primes_pi x\<rfloor>))}"
that: fixes r :: "'a"   and h' :: "heap" assumes "execute f h = Some (r, h')" shows "thesis"
cong: fixes coef_sym :: "'a \<Rightarrow> nat \<Rightarrow> hmultiset" shows "tkbo_coefs.coef coef_sym \<equiv> tkbo_coefs.coef coef_sym"
trans_less: fixes cmp :: "'b \<Rightarrow> 'b \<Rightarrow> comp"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "comparator cmp"   and "cmp a b = Less"   and "cmp b c = Less" shows "cmp a c = Less"
lhr_fps_aux: fixes m :: "nat"   and c :: "nat \<Rightarrow> 'b"   and f :: "nat \<Rightarrow> 'b"   and N :: "nat" assumes "\<And>n. m \<le> n \<Longrightarrow> (\<Sum>k\<le>N. c k * f (n + k)) = (0::'b)"   and "c N \<noteq> (0::'b)" shows "Abs_fps f = fps_of_poly (Poly (map (\<lambda>k. \<Sum>i\<le>min N k. c (N - i) * f (k - i)) [0..<N + m])) / fps_of_poly (Poly (map (\<lambda>k. c (N - k)) [0..<Suc N]))"
map_conv_bind_option: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d option" shows "map_option f x = x \<bind> Some \<circ> f"
inject_1: fixes a :: "'a"   and a' :: "'a" shows "(Inl a = Inl a') = (a = a')"
simps_3: fixes m :: "bool"   and a :: "'b" shows "final (Atom (m, a)) = m"
ftoh: shows "(red G)\<^sup>*\<^sup>* f h"
F_def: shows "F \<equiv> \<lambda>f1 a. if a \<in> A2 then if a \<in> A1 then f1 a else bb else undefined"
ball_Un: fixes A :: "'c set"   and B :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>A \<union> B. P x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x))"
sop_distr1: fixes f :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and z :: "'d"   and i :: "'d \<Rightarrow> 'd"   and s :: "'a \<Rightarrow> 'd \<Rightarrow> 'd"   and A :: "'d set" assumes "commute_bpp f (aug_pm_set z i (s_set R s A))"   and "assoc_bpp (aug_pm_set z i (s_set R s A)) f"   and "ipp_cond1 (s_set R s A) i"   and "ipp_cond2 z (s_set R s A) i f"   and "ipp_cond3 z i"   and "inv_ipp z i f (s_set R s A)"   and "zeroA z i f (s_set R s A) z"   and "sop_distr1 R s f i (s_set R s A) z"   and "sop_distr2 R s f i (s_set R s A) z"   and "sop_assoc R s (aug_pm_set z i (s_set R s A))"   and "sop_inv R s i (s_set R s A)"   and "sop_one R s (aug_pm_set z i (s_set R s A))"   and "sop_z R s z" shows "\<forall>a\<in>carrier R. \<forall>b\<in>carrier R. \<forall>x\<in>addition_set f (aug_pm_set z i (s_set R s A)). (a \<plusminus> b) \<^sub>s\<cdot> x = a \<^sub>s\<cdot> x \<^sub>f+ b \<^sub>s\<cdot> x"
fbrelation_vintersection_vdomain: fixes s :: "V" assumes "\<D>\<^sub>\<bullet> r \<inter>\<^sub>\<circ> \<D>\<^sub>\<bullet> s = []\<^sub>\<circ>" shows "r \<inter>\<^sub>\<circ> s = []\<^sub>\<circ>"
real_of_ints_4: fixes w :: "int"   and z :: "int" shows "of_int (w * z) = of_int w * of_int z"
ord_of_eq_0_iff: fixes n :: "nat" shows "(n\<^sub>\<nat> = 0) = (n = 0)"
prod_triangle: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'd"   and ff :: "('c \<Rightarrow> 'd) \<Rightarrow> 'h"   and B :: "'h set"   and h1 :: "'h \<Rightarrow> 'f"   and h2 :: "'f \<Rightarrow> 'h" assumes "I \<noteq> {}"   and "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "\<forall>j\<in>I. g j \<in> aHom S (A j)"   and "ff \<in> carrier (a\<Pi>\<^bsub>I\<^esub> A) \<rightarrow> B"   and "bij_to ff (carrier (a\<Pi>\<^bsub>I\<^esub> A)) B"   and "h1 \<in> aHom (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) S"   and "\<forall>j\<in>I. compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) (g j) h1 = ProjInd I A ff j"   and "h2 \<in> aHom S (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff)"   and "\<forall>j\<in>I. compos S (ProjInd I A ff j) h2 = g j" shows "compos (Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff) h2 h1 = aI\<^bsub>Ag_ind (a\<Pi>\<^bsub>I\<^esub> A) ff\<^esub>"
RPO_S_subst: fixes s :: "('f, 'b) Term.term"   and t :: "('f, 'b) Term.term"   and \<sigma> :: "'b \<Rightarrow> ('f, 'b) Term.term" assumes "(s, t) \<in> RPO_S" shows "(s \<cdot> \<sigma>, t \<cdot> \<sigma>) \<in> RPO_S"
assms: shows "1 \<le> length xs \<and> length xs \<le> 4"
term_pair: fixes v :: "'a" shows "fst (to_pair_unit v) = v"
Fst: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<lbrace>X, Y\<rbrace> \<in> items H" shows "X \<in> items H"
size_0: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat" assumes "class.euclidean_semiring divide plus minus zero times one modulo euclidean_size" shows "euclidean_size zero = 0"
distinct_1: shows "dSUCCEEDi \<noteq> dFAILi"
Int_UNIV: fixes A :: "'b set"   and B :: "'b set" shows "(A \<inter> B = UNIV) = (A = UNIV \<and> B = UNIV)"
l2_inv6D: fixes x :: "l2_state"   and R :: "fid_t" assumes "in_progress (progress x R) xsk"   and "x \<in> l2_inv6" shows "\<exists>X N. guessed_frame R xsk = Some (Exp X (NonceF (R $ N)))"
less_complex_lex_def: fixes x :: "complex"   and y :: "complex" shows "(x <\<^sub>\<complex> y) = (Re x < Re y \<or> Re x = Re y \<and> Im x < Im y)"
var_rename_fv_set_disjoint': fixes M :: "('a, 'b \<times> nat) Term.term set"   and N :: "('c, 'b \<times> nat) Term.term set" assumes "finite M" shows "fv\<^sub>s\<^sub>e\<^sub>t M \<inter> fv\<^sub>s\<^sub>e\<^sub>t (N \<cdot>\<^sub>s\<^sub>e\<^sub>t var_rename (max_var_set (fv\<^sub>s\<^sub>e\<^sub>t M))) = {}"
inv_is_inverse: fixes f :: "'a" assumes "local.iso f" shows "inverse_arrows f (local.inv f)"
two_equivalences_in_bicategory_axioms: shows "two_equivalences_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg e\<^sub>0 d\<^sub>0 \<eta>\<^sub>0 \<epsilon>\<^sub>0 e\<^sub>1 d\<^sub>1 \<eta>\<^sub>1 \<epsilon>\<^sub>1"
axioms_2: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "commutative_monoid M composition unit" shows "commutative_monoid_axioms M composition"