hcju/isabelleps huggingface dataset:
wqo_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "wqo_on P A = (transp_on P A \<and> almost_full_on P A)"
simps_3: fixes f :: "'c"   and u :: "'c local.term" shows "Diag (\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> u) = (C.arr f \<and> Diag u \<and> u \<noteq> \<^bold>\<I>)"
simps_18: fixes x1 :: "'b"   and x6 :: "'b local.term" shows "\<^bold>\<langle>x1\<^bold>\<rangle> \<noteq> \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[x6\<^bold>]"
filterlim_tan_at_right: shows "filterlim tan at_bot (at_right (- (pi / 2)))"
embed_measure_count_space: fixes f :: "'k \<Rightarrow> 'l"   and A :: "'k set" assumes "inj f" shows "embed_measure (count_space A) f = count_space (f ` A)"
fresh_delete: fixes a :: "atom"   and \<Gamma> :: "('c \<times> 'd) list"   and v :: "'c" assumes "a \<sharp> \<Gamma>" shows "a \<sharp> delete v \<Gamma>"
frequently_rev_mp: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter"   and Q :: "'b \<Rightarrow> bool" assumes "frequently P F"   and "\<forall>\<^sub>F x in F. P x \<longrightarrow> Q x" shows "frequently Q F"
nmult_smult: fixes a :: "'a"   and f :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "f \<in> carrier P" shows "n_mult (a \<odot>\<^bsub>P\<^esub> f) = a \<odot>\<^bsub>P\<^esub> n_mult f"
type': shows "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"
HPhase0Read_HInv4d_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase0Read s s' p d"   and "HInv4d s q"   and "p \<noteq> q" shows "HInv4d s' q"
scalar_prod_smult_right: fixes w :: "'a vec"   and v :: "'a vec"   and k :: "'a" assumes "dim_vec w = dim_vec v" shows "w \<bullet> (k \<cdot>\<^sub>v v) = k * (w \<bullet> v)"
isomorphic_implies_ide_2: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f g" shows "Span.ide g"
inj_comm_ring_hom_axioms: shows "inj_comm_ring_hom poly_lift"
eheight_def: fixes t :: "'a tree" shows "eheight t = (if t = \<langle>\<rangle> then 0 else 2 ^ (height t - 1))"
a: shows "alw (\<lambda>xs. \<phi> xs \<longrightarrow> nxt \<phi> xs) xs"
iteration_subdenest: fixes x :: "'a"   and y :: "'a" shows "x\<^sup>\<infinity> \<cdot> y\<^sup>\<infinity> \<le> (x + y)\<^sup>\<infinity>"
z_in_S: shows "z \<in> {xa. xa < (as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i) in fst (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)) \<and> count_list (take (index (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i))) ((as @ [x] @ bs @ [x] @ cs) ! (length (as @ [x] @ bs @ [x]) + i))) (snd (TSdet init h (as @ [x] @ bs @ [x] @ cs) (length (as @ [x] @ bs @ [x]) + i)))) xa \<le> 1}"
i_expand_nth_if: fixes k :: "nat"   and f :: "nat \<Rightarrow> 'b message_af"   and n :: "nat" assumes "0 < k" shows "(f \<odot> k) n = (if n mod k = 0 then f (n div k) else NoMsg)"
open_closed_segment: fixes u :: "'b"   and w :: "'b"   and z :: "'b" assumes "u \<in> open_segment w z" shows "u \<in> closed_segment w z"
uninst_chooseNext_def: fixes u :: "'c"   and p :: "('a \<times> 'b, 'b list) mapping"   and g :: "('a, 'd, 'e, 'b) gen_ssa_cfg_wf" shows "uninst_chooseNext u p g \<equiv> Max (uninst_trivial_phis (\<lambda>_. p) g)"
sorted_wrt1: fixes P :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g" shows "sorted_wrt P [x] = True"
edgeSet_finite: fixes s :: "(integer \<times> edge list) iarray" shows "finite (edgeSet s)"
Pi_anti_mono: fixes A' :: "'b set"   and A :: "'b set"   and B :: "'b \<Rightarrow> 'c set" assumes "A' \<subseteq> A" shows "Pi A B \<subseteq> Pi A' B"
ins_list_split_right: fixes ts :: "('a btree \<times> 'a) list"   and x :: "'a"   and ls :: "('a btree \<times> 'a) list"   and sub :: "'a btree"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "split ts x = (ls, (sub, sep) # rs)"   and "sorted_less (inorder (Node ts t))"   and "sep \<noteq> x" shows "ins_list x (inorder_list ((sub, sep) # rs) @ inorder t) = ins_list x (inorder sub) @ sep # inorder_list rs @ inorder t"
fold_mset_parametric: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f1 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f2 :: "'e \<Rightarrow> 'g \<Rightarrow> 'g" assumes "(A ===> B ===> B) f1 f2"   and "comp_fun_commute f1"   and "comp_fun_commute f2" shows "(B ===> rel_mset A ===> B) (fold_mset f1) (fold_mset f2)"
neg_0_equal_iff_equal: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "class.group_add minus plus zero uminus" shows "(zero = uminus a) = (zero = a)"
stream_eq_Stream_iff: fixes s :: "'a stream"   and x :: "'a"   and t :: "'a stream" shows "(s = x ## t) = (shd s = x \<and> stl s = t)"
simps_1: fixes \<rho> :: "val set list" shows "T TNat \<rho> = {v. \<exists>n. v = VNat n}"
pbdom_in_hom: fixes f :: "'a"   and g :: "'a" assumes "cospan f g" shows "\<guillemotleft>f \<down>\<down> g : f \<down>\<down> g \<rightarrow> f \<down>\<down> g\<guillemotright>"
Throw_\<tau>red1t_xt: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (throw e, xs) (throw e', xs')"
simps_1: fixes s :: "'c" shows "last_state (s, []) = s"
gf_G_eq_gf_F: fixes z :: "real"   and x :: "'s"   and y :: "'s" assumes "norm z < 1" shows "gf_G x y z = gf_F x y z * gf_G y y z"
cases: fixes x :: "('h, 'i, 'j) generat" obtains uu_ :: "'h" where "x = Pure uu_"   | out :: "'i" and c :: "'j" where "x = IO out c"
eq_\<I>_gpv_left_gpv_cong: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "('c, 'd) \<I>"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'c, 'd) gpv"   and \<I>' :: "('e, 'f) \<I>" assumes "eq_\<I>_gpv A \<I> gpv gpv'" shows "eq_\<I>_gpv A (\<I> \<oplus>\<^sub>\<I> \<I>') (left_gpv gpv) (left_gpv gpv')"
Sup_sup_dual: fixes f :: "'b \<Rightarrow> 'c" assumes "Sup_dual f" shows "sup_dual f"
pos_le_divideR_eq: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a \<le> b /\<^sub>R c) = (c *\<^sub>R a \<le> b)"
bounded: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "bounded_bilinear prod" shows "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
dim_eq_card: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set"   and V :: "'b set" assumes "vector_space scale"   and "module.span scale B = module.span scale V"   and "\<not> module.dependent scale B" shows "vector_space.dim scale V = card B"
k: shows "x \<in> Sup_upto_llist Xs (enat k)"
div_mult_mod_eq: fixes a :: "'a"   and b :: "'a" shows "a div b * b + a mod b = a"
simps_1: fixes cc1 :: "('d, 'c) cconstraint"   and cc2 :: "('d, 'c) cconstraint"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'d \<Rightarrow> nat" shows "abstr (AND cc1 cc2) M v = And (abstr cc1 M v) (abstr cc2 M v)"
fds_converges_deriv: fixes f :: "'a fds"   and s :: "'a" assumes "conv_abscissa f < ereal (s \<bullet> (1::'a))" shows "fds_converges (fds_deriv f) s"
aneq_natneq: fixes n :: "nat"   and m :: "nat" shows "(an n \<noteq> an m) = (n \<noteq> m)"
map_mono_prefix: fixes xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'e" assumes "prefix xs ys" shows "prefix (map f xs) (map f ys)"
divide_nonpos_nonpos: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less_eq y zero" shows "less_eq zero (divide x y)"
strand_fv_card_rm_eq_le: fixes S :: "('b, 'a) strand_step list"   and S' :: "('b, 'a) strand_step list"   and a :: "poscheckvariant"   and t :: "('b, 'a) Term.term"   and t' :: "('b, 'a) Term.term" shows "card (fv\<^sub>s\<^sub>t (S @ S')) \<le> card (fv\<^sub>s\<^sub>t (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S'))"
mult_scalar_zero_right: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "p \<odot> 0 = 0"
notin_fset: fixes x :: "'g"   and S :: "'g fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
gfTC: shows "(g \<circ> f) ` T \<subseteq> C"
default_rel_internal: fixes d :: "'b"   and X :: "('a \<times> 'b) set" shows "default_rel d X = insert (None, d) ((\<lambda>(x, y). (Some x, y)) ` X)"
ext_inject: fixes TDomain :: "'a"   and TExprSet :: "('a, 'b) Expression set"   and TCodomain :: "'a"   and more :: "'c"   and TDomain' :: "'a"   and TExprSet' :: "('a, 'b) Expression set"   and TCodomain' :: "'a"   and more' :: "'c" shows "(\<lparr>TDomain = TDomain, TExprSet = TExprSet, TCodomain = TCodomain, \<dots> = more\<rparr> = \<lparr>TDomain = TDomain', TExprSet = TExprSet', TCodomain = TCodomain', \<dots> = more'\<rparr>) = (TDomain = TDomain' \<and> TExprSet = TExprSet' \<and> TCodomain = TCodomain' \<and> more = more')"
assms: shows "\<exists>x'. (x, x') \<in> R \<and> x' \<in> X"
frontier_convex_hull_eq_rel_frontier: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "frontier (convex hull S) = (if card S \<le> DIM('a) then convex hull S else rel_frontier (convex hull S))"
sol_eq_l: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "x \<in> carrier G"   and "a \<cdot> x = b" shows "x = \<rho> a \<cdot> b"
map_cong0: fixes x :: "'a list"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>z. z \<in> set x \<Longrightarrow> f z = g z" shows "map f x = map g x"
summable_Integral: fixes x :: "real" assumes "\<bar>x\<bar> < 1" shows "summable (\<lambda>n. (- 1) ^ n * x ^ (2 * n))"
alw_InvGlobalIncomingInfoUpright: fixes s :: "('c, 'd) configuration stream" assumes "spec s" shows "alw (holds InvGlobalIncomingInfoUpright) s"
fls_integral_const: fixes x :: "'a" shows "fls_integral (fls_const x) = fls_const x * fls_X"
simps: fixes c :: "nat" shows "stat c = lo c (Pi 1)"
path_connectedin_euclidean: fixes S :: "'a set" shows "path_connectedin euclidean S = path_connected S"
AE_mp: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b measure"   and Q :: "'b \<Rightarrow> bool" assumes "almost_everywhere M P"   and "AE x in M. P x \<longrightarrow> Q x" shows "almost_everywhere M Q"
infinite_Icc_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.dense_linorder less_eq less" shows "infinite (ord.atLeastAtMost less_eq a b) = less a b"
assms_2: shows "{c..d} \<subseteq> {a..b}"
bi_unique_OO: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A"   and "bi_unique B" shows "bi_unique (A OO B)"
simps_3: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" shows "(P,h \<turnstile> v :\<le> Integer) = (P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Integer\<rfloor>)"
cong: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and "uses" :: "'a \<Rightarrow> ('b, 'c set) mapping"   and phis :: "'a \<Rightarrow> ('b \<times> 'c, 'c list) mapping" shows "CFG_SSA_Transformed_notriv_linorder_code.substAll_efficient \<alpha>n uses phis \<equiv> CFG_SSA_Transformed_notriv_linorder_code.substAll_efficient \<alpha>n uses phis"
lit_semantics_reducible_to_subset_if: fixes C' :: "'b literal set"   and C :: "'b literal set"   and \<A> :: "'b \<Rightarrow> bool" assumes "C' \<subseteq> C"   and "\<forall>L\<in>C'. \<not> lit_semantics \<A> L" shows "clause_semantics \<A> C = clause_semantics \<A> (C - C')"
card_eq_1_iff: fixes A :: "'a set" shows "(card A = Suc 0) = (\<exists>x. A = {x})"
assms_2: shows "B \<Longrightarrow>\<^sub>A A"
trans_tm_inject: fixes e :: "name list"   and t :: "tm"   and u :: "tm" shows "(trans_tm e t = trans_tm e u) = (t = u)"
is_tiny_dghmI: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"   and "tiny_digraph \<alpha> \<AA>"   and "tiny_digraph \<alpha> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>i\<^sub>n\<^sub>y\<^bsub>\<alpha>\<^esub> \<BB>"
cblinfun_of_mat_inverse: fixes M :: "complex mat" assumes "M \<in> carrier_mat (length canonical_basis) (length canonical_basis)" shows "mat_of_cblinfun (cblinfun_of_mat M) = M"
dests_iD_invalidate: fixes dests :: "nat \<Rightarrow> nat option"   and ip :: "nat"   and rsn :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dests ip = Some rsn"   and "ip \<in> kD rt" shows "ip \<in> iD (invalidate rt dests)"
induct: fixes x1 :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and x2 :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat"   and P :: "('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> ('a literal multiset \<times> nat) multiset \<times> nat \<Rightarrow> bool" assumes "x1 \<leadsto>\<^sub>w x2"   and "\<And>A C N i Pa Q n. \<lbrakk>Neg A \<in># C; Pos A \<in># C\<rbrakk> \<Longrightarrow> P (N + {#(C, i)#}, Pa, Q, n) (N, Pa, Q, n)"   and "\<And>D Pa Q C N i n. \<lbrakk>D \<in># image_mset fst (Pa + Q); subsumes D C\<rbrakk> \<Longrightarrow> P (N + {#(C, i)#}, Pa, Q, n) (N, Pa, Q, n)"   and "\<And>D N C Pa Q n. \<lbrakk>D \<in># image_mset fst N; C \<in># image_mset fst Pa; strictly_subsumes D C\<rbrakk> \<Longrightarrow> P (N, Pa, Q, n) (N, {#(E, k) \<in># Pa. E \<noteq> C#}, Q, n)"   and "\<And>D N C Pa Q i n. \<lbrakk>D \<in># image_mset fst N; strictly_subsumes D C\<rbrakk> \<Longrightarrow> P (N, Pa, Q + {#(C, i)#}, n) (N, Pa, Q, n)"   and "\<And>D L' Pa Q L \<sigma> C N i n. \<lbrakk>D + {#L'#} \<in># image_mset fst (Pa + Q); - L = L' \<cdot>l \<sigma>; D \<cdot> \<sigma> \<subseteq># C\<rbrakk> \<Longrightarrow> P (N + {#(C + {#L#}, i)#}, Pa, Q, n) (N + {#(C, i)#}, Pa, Q, n)"   and "\<And>D L' N L \<sigma> C Pa i Q n. \<lbrakk>D + {#L'#} \<in># image_mset fst N; - L = L' \<cdot>l \<sigma>; D \<cdot> \<sigma> \<subseteq># C; \<forall>j. (C + {#L#}, j) \<in># Pa \<longrightarrow> j \<le> i\<rbrakk> \<Longrightarrow> P (N, Pa + {#(C + {#L#}, i)#}, Q, n) (N, Pa + {#(C, i)#}, Q, n)"   and "\<And>D L' N L \<sigma> C Pa Q i n. \<lbrakk>D + {#L'#} \<in># image_mset fst N; - L = L' \<cdot>l \<sigma>; D \<cdot> \<sigma> \<subseteq># C\<rbrakk> \<Longrightarrow> P (N, Pa, Q + {#(C + {#L#}, i)#}, n) (N, Pa + {#(C, i)#}, Q, n)"   and "\<And>N C i Pa Q n. P (N + {#(C, i)#}, Pa, Q, n) (N, Pa + {#(C, i)#}, Q, n)"   and "\<And>Pa C i N n Q. \<lbrakk>\<forall>(D, j)\<in>#Pa. weight (C, i) \<le> weight (D, j); N = mset_set ((\<lambda>D. (D, n)) ` concls_of (inference_system.inferences_between (ord_FO_\<Gamma> S) (set_mset (image_mset fst Q)) C))\<rbrakk> \<Longrightarrow> P ({#}, Pa + {#(C, i)#}, Q, n) (N, {#(D, j) \<in># Pa. D \<noteq> C#}, Q + {#(C, i)#}, Suc n)" shows "P x1 x2"
2: shows "(\<lambda>x. interpret_floatarith (deriv_rec f 0) ((map real_of_float xs)[0 := x])) = (\<lambda>x. interpret_floatarith f ((map real_of_float xs)[0 := x]))"
associatedD2: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" shows "dvd.dvd times b a"
S_fairness_bad_R: fixes S :: "agent"   and R :: "agent"   and K :: "nat"   and m :: "nat"   and AO :: "nat"   and cleartext :: "nat"   and q :: "nat"   and S2TTP :: "msg"   and evs :: "event list"   and hs :: "msg" assumes "Says S R \<lbrace>Agent S, Agent TTP, Crypt K (Number m), Number AO, Number cleartext, Nonce q, S2TTP\<rbrace> \<in> set evs"   and "S2TTP = Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace>"   and "Key K \<in> analz (knows Spy evs)"   and "evs \<in> certified_mail"   and "S \<noteq> Spy" shows "R \<in> bad \<and> Gets S (Crypt (priSK TTP) S2TTP) \<in> set evs"
fs_def: shows "fs = snd (sym_preproc (map fst bs) (pdata_pairs_to_list ps))"
ssubseqs_lists: fixes as :: "'b list"   and A :: "'b set"   and bs :: "'b list" assumes "as \<in> lists A"   and "bs \<in> ssubseqs as" shows "bs \<in> lists A"
orthogonal_extension_aux: fixes T :: "'a set"   and S :: "'a set" assumes "finite T"   and "finite S"   and "pairwise orthogonal S" shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
valid_in_valid: fixes p :: "fm"   and U :: "nat set" assumes "card (props p) \<le> card U"   and "valid_in U p" shows "valid p"
prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "moebius_mu (p ^ k) = (if k = 1 then - (1::'a) else (0::'a))"
U_parOf_def: shows "U_parOf \<equiv> length \<circ> parOf"
b_r2_q2: shows "b = m * q2 + r2"
max_def: fixes a :: "nat \<Rightarrow> nat"   and b :: "nat \<Rightarrow> nat" shows "preorder.max a b = (if a \<lesssim> b then b else a)"
f_2: shows "f \<in> UNIV \<rightarrow> I"
committed_subset_actions_def: fixes J :: "nat \<Rightarrow> ('a, 'b) justifying_execution" shows "committed_subset_actions J = (\<forall>n. committed (J n) \<subseteq> actions (justifying_exec (J n)))"
RM_1: fixes \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and v :: "i" assumes "\<And>v. [\<phi> \<^bold>\<rightarrow> \<psi> in v]" shows "[\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<psi> in v]"
M_def: shows "M \<equiv> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l \<A>))"
nomatchD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and r :: "'c rule list"   and s :: "state"   and t :: "state"   and m :: "'c match_expr"   and a :: "action" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>r, s\<rangle> \<Rightarrow> t"   and "r = [Rule m a]"   and "s = Undecided"   and "\<not> matches \<gamma> m p" shows "t = Undecided"
RecvMarker_given_channel: fixes ev :: "('p, 's, 'm) event"   and cid :: "nat"   and p :: "'p"   and q :: "'p"   and c :: "('p, 's, 'm) configuration" assumes "isRecvMarker ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "can_occur ev c" shows "ev = RecvMarker cid q p"
Auth_Snd: fixes x :: "rat"   and y :: "rat"   and H :: "msg set" assumes "Auth_Data x y \<in> parts H" shows "Pri_AgrK y \<in> parts H"
ide_prod: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "ide (local.prod a b)"
that: assumes "wf_net_tree p1"   and "wf_net_tree p2" shows "thesis"
hyps_3: shows "(P, P') \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*"
Ball_def: fixes A :: "'d set"   and P :: "'d \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
uniformity_fpxs_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
same_messages_if_no_occurrence: fixes c :: "('p, 's, 'm) configuration"   and ev :: "('p, 's, 'm) event"   and c' :: "('p, 's, 'm) configuration"   and p :: "'p"   and q :: "'p"   and cid :: "nat" assumes "c \<turnstile> ev \<mapsto> c'"   and "occurs_on ev \<noteq> p"   and "occurs_on ev \<noteq> q"   and "channel cid = Some (p, q)" shows "msgs c cid = msgs c' cid \<and> cs c cid = cs c' cid"
piecewise_continuous_on_insertE: fixes a :: "'c"   and b :: "'c"   and i :: "'c"   and I :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "piecewise_continuous_on a b (insert i I) f"   and "i \<in> {a..b}" obtains g :: "'c \<Rightarrow> 'd" and h :: "'c \<Rightarrow> 'd" where "piecewise_continuous_on a i I g" and "piecewise_continuous_on i b I h" and "\<And>x. \<lbrakk>a \<le> x; x < i\<rbrakk> \<Longrightarrow> g x = f x" and "\<And>x. \<lbrakk>i < x; x \<le> b\<rbrakk> \<Longrightarrow> h x = f x"
step: fixes s :: "'a"   and s' :: "'a" assumes "C s"   and "I (Some s)"   and "B s = Some s'" shows "I (Some s')"
create_gba_from_nodes_def: fixes \<phi> :: "'a ltlr"   and qs :: "'a node set" shows "create_gba_from_nodes \<phi> qs \<equiv> \<lparr>g_V = qs, g_E = {(q, q'). q \<in> qs \<and> q' \<in> qs \<and> name q \<in> incoming q'}, g_V0 = {q \<in> qs. expand_init \<in> incoming q}, gbg_F = {{q \<in> qs. \<mu> U\<^sub>r \<eta> \<in> old q \<longrightarrow> \<eta> \<in> old q} |\<mu> \<eta>. \<mu> U\<^sub>r \<eta> \<in> subfrmlsr \<phi>}, gba_L = \<lambda>q l. q \<in> qs \<and> {p. prop\<^sub>r(p) \<in> old q} \<subseteq> l \<and> {p. nprop\<^sub>r(p) \<in> old q} \<inter> l = {}\<rparr>"
plus_p64_def: fixes p :: "uint64"   and x :: "uint64"   and y :: "uint64" shows "plus_p64 p x y \<equiv> let z = x + y in if p \<le> z then z - p else z"
DERIV_imp_deriv: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and x :: "'a" assumes "(f has_field_derivative f') (at x)" shows "deriv f x = f'"
span_sum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and S :: "'b set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> ideal S" shows "sum f A \<in> ideal S"
SourcesA6_L0: shows "Sources level0 sA6 = {}"
Diff_Int2: fixes A :: "'h set"   and C :: "'h set"   and B :: "'h set" shows "A \<inter> C - B \<inter> C = A \<inter> C - B"
dec_interp_Inr: fixes i :: "nat"   and FO :: "nat set"   and n :: "nat"   and x :: "('a \<times> bool list) list" assumes "i \<notin> FO"   and "i < n" shows "\<exists>P. dec_interp n FO x ! i = Inr P"
list_all_rev: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list" shows "list_all P (rev xs) = list_all P xs"
map_tree_ap_tree_pure_tree: fixes f :: "'c \<Rightarrow> 'b"   and u :: "'c tree" shows "pure f \<diamondop> u = map_tree f u"
dims_def: fixes A :: "'a tensor" shows "dims A = fst (Rep_tensor A)"
PiE_cong: fixes I :: "'e set"   and A :: "'e \<Rightarrow> 'f set"   and B :: "'e \<Rightarrow> 'f set" assumes "\<And>i. i \<in> I \<Longrightarrow> A i = B i" shows "Pi\<^sub>E I A = Pi\<^sub>E I B"
polynomial_pow_not_zero: fixes p :: "'a list list"   and n :: "nat" assumes "p \<in> carrier (poly_ring (poly_ring R))"   and "p \<noteq> []" shows "p [^]\<^bsub>poly_ring (poly_ring R)\<^esub> n \<noteq> []"
adjacent_Nil: fixes w :: "nat \<times> nat" shows "adjacent w []"
upto_enum_red: fixes n :: "'b"   and m :: "'b" shows "[n .e. m] = map toEnum [fromEnum n..<Suc (fromEnum m)]"
Ana_const': fixes c :: "'fun"   and T :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term list" assumes "arity c = 0" shows "Ana (Fun c T) = ([], [])"
adj_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow> v" shows "u \<in> verts G"
isomorphic_implies_hpar_3: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and f' :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.isomorphic f f'" shows "Span.src f = Span.src f'"