hcju/isabelleps huggingface dataset:
select_correct_factor: fixes init :: "'a"   and x :: "real"   and polys :: "(int poly \<times> root_info) list"   and q :: "int poly"   and ri :: "root_info"   and l :: "rat"   and r :: "rat" assumes "converges_to (\<lambda>i. bnd_get ((bnd_update ^^ i) init)) x"   and "local.select_correct_factor init polys = ((q, ri), l, r)"   and "\<And>q ri. (q, ri) \<in> set polys \<Longrightarrow> root_info_cond ri q"   and "\<And>q ri. (q, ri) \<in> set polys \<Longrightarrow> q \<noteq> 0"   and "\<exists>q. q \<in> fst ` set polys \<and> poly (real_of_int_poly q) x = 0"   and "distinct (map fst polys)"   and "\<And>x. \<exists>q. q \<in> fst ` set polys \<and> poly (real_of_int_poly q) x = 0 \<Longrightarrow> \<exists>!q. q \<in> fst ` set polys \<and> poly (real_of_int_poly q) x = 0" shows "(\<exists>!x. root_cond (q, l, r) x) \<and> (q, ri) \<in> set polys \<and> x = real_of_1 (q, l, r)"
UNIV1_def: shows "UNIV1 \<equiv> UNIV - {[]}"
disc_2: fixes R :: "('f, 'g, 'h) Resumption.resumption set" assumes "\<not> (\<forall>r\<in>R. Resumption.resumption.is_Done r)" shows "\<not> Resumption.resumption.is_Done (resumption_lub R)"
sum_SDG_edge_SDG_edge: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and n' :: "'node SDG_node" assumes "sum_SDG_edge n Vopt popt False n'" shows "SDG_edge n Vopt popt n'"
splus_assoc: fixes s :: "'a"   and t :: "'a"   and v :: "'t" shows "(s + t) \<oplus> v = s \<oplus> (t \<oplus> v)"
convex_Affine: fixes X :: "'b \<times> 'b pdevs" shows "convex (Affine X)"
word_lsb_def: fixes a :: "'c word" shows "lsb a = odd (uint a)"
totally_bounded_def: fixes S :: "'a set" shows "totally_bounded S = (\<forall>E. eventually E uniformity \<longrightarrow> (\<exists>X. finite X \<and> (\<forall>s\<in>S. \<exists>x\<in>X. E (x, s))))"
that_2: fixes i :: "'a" shows "f i \<in> topspace (Y i) \<rightarrow> X"
elims_3: fixes wtFsym :: "'f \<Rightarrow> bool"   and wtPsym :: "'g \<Rightarrow> bool"   and arOf :: "'f \<Rightarrow> 'e list"   and resOf :: "'f \<Rightarrow> 'e"   and x :: "'f trm" assumes "Signature TYPE('e) wtFsym wtPsym"   and "\<not> Signature.wt wtFsym arOf resOf x" obtains f :: "'f" and Tl :: "'f trm list" where "x = Fn f Tl" and "\<not> (wtFsym f \<and> list_all (Signature.wt wtFsym arOf resOf) Tl \<and> arOf f = map (Signature.tpOf resOf) Tl)"
stake_inter_snth: fixes n :: "nat"   and M :: "'b stream measure"   and x :: "'b stream" assumes "Suc 0 \<le> n" shows "{w \<in> space M. stake n w = stake n x} = (\<Inter>i\<in>{0..n - 1}. {w \<in> space M. w !! i = x !! i})"
LSCategory_Op: fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "LSCategory (Op C)"
lstrict_prefix_lfinite1: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "lstrict_prefix xs ys" shows "lfinite xs"
V_def: shows "V \<equiv> {root n (norm (x ^ n)) |n. 0 < n}"
trans: fixes a :: "nat list"   and b :: "nat list"   and c :: "nat list" assumes "a \<le>\<^sub>v b"   and "b \<le>\<^sub>v c" shows "a \<le>\<^sub>v c"
count_diff: fixes M :: "'k multiset"   and N :: "'k multiset"   and a :: "'k" shows "count (M - N) a = count M a - count N a"
lin_def: fixes x :: "'b" shows "lin x = {a \<cdot> x |a. True}"
iso_lunit: fixes f :: "'a" assumes "Left_a.ide f" shows "Left_a.iso (Left_a.lunit f)"
False: shows "\<not> f contour_integrable_on part_circlepath c r a b"
less_le: fixes x :: "'loc \<times> 't"   and y :: "'loc \<times> 't" shows "(x<\<^sub>py) = (x\<le>\<^sub>py \<and> x \<noteq> y)"
br_dsq_ss: fixes Q :: "'d set"   and W :: "'d set"   and \<delta> :: "('d, 'e) ta_rule set"   and q :: "'d" assumes "(Q, W) \<in> br_invar \<delta>"   and "W \<noteq> {}"   and "q \<in> W" shows "br_dsq \<delta> q (Q, W) \<subseteq> b_accessible \<delta>"
rel_Grp: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i" shows "rel_option (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_option x \<subseteq> A} (map_option f)"
index_of_const_lm_1: fixes n :: "nat" shows "nat_to_pr (index_of_const n) 0 = n"
seqI: fixes f :: "'b"   and g :: "'b" assumes "D.arr f"   and "D.arr g"   and "D.dom g = D.cod f" shows "D.seq g f"
S12: shows "pr_gr upb = g_step (pr_gr (c_pair (c_pair n x) (upb1 + upb2))) (c_fst (c_pair (c_pair n x) (upb1 + upb2)))"
del_point_order: fixes p :: "'a" assumes "p \<in> \<V>" shows "int (card (del_point p)) = \<v> - 1"
simps_1: fixes bs :: "('c \<times> ('d, 'c) Term.term) list" shows "unify [] bs = Some bs"
spr_simAbs_def: shows "spr_simAbs \<equiv> \<lambda>ss. rel_ext (\<lambda>uu_. \<exists>s. uu_ = (toSet ss, s) \<and> s \<in> toSet ss)"
asset_seskey_nonce: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and SK :: "nat option \<times> nat set \<times> nat set" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset n, SesKey SK) \<in> s" shows "\<exists>S. fst SK = Some S \<and> (Asset n, Crypt (Auth_ShaKey n) (PriKey S)) \<in> s"
pred_cong: fixes x :: "'c dual"   and ya :: "'c dual"   and P :: "'c \<Rightarrow> bool"   and Pa :: "'c \<Rightarrow> bool" assumes "x = ya"   and "\<And>z. z \<in> set_dual ya \<Longrightarrow> P z = Pa z" shows "pred_dual P x = pred_dual Pa ya"
assoc_in_vhom\<^sub>A\<^sub>W\<^sub>C: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "associative_weak_composition V H \<a>"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "H f g \<noteq> partial_magma.null V"   and "H g h \<noteq> partial_magma.null V" shows "partial_magma.in_hom V (\<a> f g h) (H (H f g) h) (H f (H g h))"
disj: shows "\<aa> \<subseteq> x \<or> \<bb> \<subseteq> x"
f1: shows "ad (ad (x \<cdot> \<nabla>x)) = ad (ad (x \<cdot> \<nabla>x)) + \<nabla>x"
collapse_1: fixes sum :: "'a + 'b" assumes "isl sum" shows "Inl (projl sum) = sum"
mset_map_single_rightE: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b multiset"   and y :: "'a" assumes "f `# P = {#y#}" obtains x :: "'b" where "P = {#x#}" and "f x = y"
hyps: fixes p :: "'a poly" assumes "degree p < degree pa__"   and "p \<noteq> 0" shows "card {x. poly p x = (0::'a)} \<le> degree p"
map_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and v :: "'e pmf" shows "map_pmf g (map_pmf f v) = map_pmf (g \<circ> f) v"
102: fixes x :: "'a" shows "x \<squnion> x \<sqinter> bot = x"
card_suc_ge_insert: fixes x :: "'a"   and A :: "'a set" shows "card (insert x A) \<le> card A + 1"
set_rw: shows "{..degree u - 1} = {..<degree u}"
on_triple_def: fixes R :: "('b \<times> 'c) set" shows "on_triple R \<equiv> {((l, s, t), l', s', t'). l = l' \<and> (s, s') \<in> R \<and> (t, t') \<in> R}"
irreflexive_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "irreflexive B ((\<sqsubset>) \<restriction> A)"
inter_empty: shows "subsumers subs \<inter> subsumees subs = {}"
valid_cfg_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set" assumes "Markov_Decision_Process_Invariant K S" shows "Markov_Decision_Process_Invariant.valid_cfg K S = \<Union> (Markov_Decision_Process.cfg_on K ` S)"
dg_tiny_digraph_if_ge_Limit: fixes \<beta> :: "V" assumes "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "tiny_digraph \<beta> \<BB>"
simps: fixes a :: "'d"   and v :: "'c"   and ts :: "('d, 'c) bintree list"   and q :: "('c, 'd) pq" shows "bt_augment (Node a v ts) q = PQ.push v a (bts_augment ts q)"
assms: fixes x :: "'a"   and y :: "'b" shows "mf x y \<equiv> ASSERT (P x y) \<bind> (\<lambda>_. RETURN (f x y))"
single_valued_def: fixes r :: "('a \<times> 'b) set" shows "single_valued r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
normf: fixes x :: "'a" assumes "x \<in> S" shows "norm (f x) \<le> g x"
open_iff: fixes m :: "'a llist set" shows "(m open) = (m \<in> topo (\<Union>s\<in>A\<^sup>\<clubsuit>. {suff A s}))"
test_preserve2: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "p \<cdot> q \<cdot> !p = (0::'a)"
avl_split_max: fixes t :: "('c \<times> nat) tree" assumes "avl t"   and "t \<noteq> \<langle>\<rangle>" shows "avl (fst (split_max t)) \<and> height t \<in> {height (fst (split_max t)), height (fst (split_max t)) + 1}"
get_disconnected_document_def: shows "get_disconnected_document = (\<lambda>node_ptr. Heap_Error_Monad.bind (check_in_heap (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)) (\<lambda>_. Heap_Error_Monad.bind document_ptr_kinds_M (\<lambda>ptrs. Heap_Error_Monad.bind (filter_M (\<lambda>document_ptr. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disconnected_nodes. return (node_ptr \<in> set disconnected_nodes))) ptrs) (case_list (error HierarchyRequestError) (\<lambda>document_ptr. case_list (return document_ptr) (\<lambda>aa lista. error HierarchyRequestError))))))"
star_star_absorb: fixes y :: "'a"   and x :: "'a" shows "y\<^sup>\<star> * (y\<^sup>\<star> * x)\<^sup>\<star> * y\<^sup>\<star> = (y\<^sup>\<star> * x)\<^sup>\<star> * y\<^sup>\<star>"
has_integral_restrict: fixes S :: "'c set"   and T :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and i :: "'d" assumes "S \<subseteq> T" shows "((\<lambda>x. if x \<in> S then f x else (0::'d)) has_integral i) T = (f has_integral i) S"
map_bset_eq_bempty_iff: fixes f :: "'c \<Rightarrow> 'a"   and X :: "'c set['b]" shows "(map_bset f X = bempty) = (X = bempty)"
eq_snd_iff: fixes b :: "'b"   and p :: "'c \<times> 'b" shows "(b = snd p) = (\<exists>a. p = (a, b))"
Var_apps_eq_Var_apps_conv: fixes m :: "nat"   and rs :: "dB list"   and n :: "nat"   and ss :: "dB list" shows "(Var m \<degree>\<degree> rs = Var n \<degree>\<degree> ss) = (m = n \<and> rs = ss)"
index_map_mat_2: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b Matrix.mat" shows "dim_row (map_mat f A) = dim_row A"
"case": fixes f :: "'b \<Rightarrow> 'b stream \<Rightarrow> 'a"   and x1 :: "'b"   and x2 :: "'b stream" shows "(case x1 ## x2 of x ## xa \<Rightarrow> f x xa) = f x1 x2"
of_int_less_iff: fixes 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and w :: "int"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (ring_1.of_int one plus zero minus w) (ring_1.of_int one plus zero minus z) = (w < z)"
neq_Leaf_iff: fixes t :: "'c tree" shows "(t \<noteq> \<langle>\<rangle>) = (\<exists>l a r. t = \<langle>l, a, r\<rangle>)"
Un_commute: fixes A :: "'k set"   and B :: "'k set" shows "A \<union> B = B \<union> A"
C_immediate_prefix_containment: fixes vs :: "'e list"   and xs :: "'e list"   and M2 :: "('e, 'f, 'g) FSM"   and M1 :: "('e, 'f, 'h) FSM"   and \<Omega> :: "('e, 'f) ATC set"   and V :: "'e list set"   and m :: "nat"   and i :: "nat" assumes "vs @ xs \<in> C M2 M1 \<Omega> V m (Suc (Suc i))"   and "xs \<noteq> []" shows "vs @ butlast xs \<in> C M2 M1 \<Omega> V m (Suc i) - RM M2 M1 \<Omega> V m (Suc i)"
simps_25: fixes vc :: "'b rexp"   and v :: "'b rexp"   and va :: "'b rexp" shows "le_rexp (Star vc) (Times v va) = True"
order_refl: fixes less_eq :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and less :: "'r \<Rightarrow> 'r \<Rightarrow> bool"   and x :: "'r" assumes "class.preorder less_eq less" shows "less_eq x x"
mor_select: fixes i :: "'d IIT"   and s1' :: "('d, 'e, 'f) F1 \<Rightarrow> 'e"   and s2' :: "('d, 'e, 'f) F2 \<Rightarrow> 'f"   and f :: "(((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> 'e"   and g :: "(((bd_type_F1 + (bd_type_F1, bd_type_F1, bd_type_F1) F1) + bd_type_F2 + (bd_type_F2, bd_type_F2, bd_type_F2) F2) set \<Rightarrow> 'd + bool) \<Rightarrow> 'f"   and dummy :: "'d" assumes "i \<in> II"   and "mor (fst (fst (Rep_IIT i))) (snd (fst (Rep_IIT i))) (fst (snd (Rep_IIT i))) (snd (snd (Rep_IIT i))) UNIV UNIV s1' s2' f g" shows "mor (car_init1 dummy) (car_init2 dummy) (str_init1 dummy) (str_init2 dummy) UNIV UNIV s1' s2' (f \<circ> (\<lambda>h. h i)) (g \<circ> (\<lambda>h. h i))"
expand_ord_def: fixes \<phi> :: "'f ltlr" shows "expand_ord \<phi> \<equiv> inv_image (finite_psupset (old_next_limit \<phi>) <*lex*> less_than) (\<lambda>(n, ns). (old_next_pair ` ns, size_set (new n)))"
eqA: shows "{w \<in> space N. \<not> w !! n} = {w \<in> space N. \<not> w !! n} \<union> {w \<in> space N. w !! n} - {w \<in> space N. w !! n}"
coeff_def: shows "Abstract_Linear_Poly.coeff \<equiv> map_fun Rep_linear_poly id fun_coeff"
split_vals_e_conv_app: fixes xs :: "e list"   and as :: "v list"   and bs :: "e list" assumes "split_vals_e xs = (as, bs)" shows "xs = ($$* as) @ bs"
incl_in_def: fixes a :: "'s"   and b :: "'s" shows "incl_in a b = (ide a \<and> ide b \<and> local.set a \<subseteq> local.set b)"
y\<phi>: shows "y \<notin> Fvars \<phi>"
assms_1: fixes x :: "nat \<Rightarrow> 'a" shows "A x \<in> carrier_mat n n"
geom_proc_filt_info': fixes f :: "bool stream \<Rightarrow> 'b"   and n :: "nat"   and w :: "bool stream" assumes "f \<in> borel_measurable (G n)" shows "f w = f (pseudo_proj_False n w)"
memop_empty_rels_safe_free_flowing_implies_safe_delayed: fixes \<O>s :: "nat set list"   and i :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and m :: "nat \<Rightarrow> nat"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R>s :: "(nat \<Rightarrow> bool option) list" assumes "\<O>s,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"   and "\<forall>\<R>\<in>set \<R>s. \<R> = Map.empty"   and "length \<O>s = length \<R>s"   and "\<forall>a. (\<forall>i<length \<O>s. a \<notin> \<O>s ! i) \<longrightarrow> a \<in> dom \<S>"   and "\<O>s ! i = \<O>" shows "\<O>s,\<R>s,i\<turnstile> (is, \<theta>, m, \<D>, \<O>, \<S>) \<surd>"
reflpE: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e" assumes "reflp r" obtains "r x x"
rcostOp: fixes I :: "'a set"   and a :: "'a"   and b :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "rcostOp R I (a \<uplus>\<^bsub>R\<^esub> I) (b \<uplus>\<^bsub>R\<^esub> I) = a \<cdot>\<^sub>r b \<uplus>\<^bsub>R\<^esub> I"
maxim_in: fixes A :: "'b set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field s" shows "s.maxim A \<in> A"
unique_sxalloc: fixes G :: "prog"   and s :: "state"   and s' :: "state"   and s'' :: "state" assumes "G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'"   and "G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s''" shows "s'' = s'"
backjumpLevelLastLast: fixes l :: "Literal"   and C :: "Literal list"   and M :: "(Literal \<times> bool) list"   and ll :: "Literal" assumes "isUIP l C M"   and "uniq (elements M)"   and "clauseFalse C (elements M)"   and "isLastAssertedLiteral (opposite ll) (removeAll (opposite l) (oppositeLiteralList C)) (elements M)" shows "isBackjumpLevel (elementLevel (opposite ll) M) l C M"
le_approx_implies_le_ref: fixes P :: "'d process"   and Q :: "'d process" assumes "P \<sqsubseteq> Q" shows "P \<le> Q"
sprefix_stake_same: fixes n :: "nat"   and xs :: "'c stream" shows "sprefix (stake n xs) xs"
cases: fixes a :: "'a multiset" assumes "local.well_base a" obtains M :: "'a multiset" where "a = M" and "\<forall>n. count M n < base"
list_of_oalist_ntm_inverse: fixes x :: "('a, 'b) oalist_ntm" shows "Abs_oalist_ntm (list_of_oalist_ntm x) = x"
While_backwards_frontier_ir_valid': fixes P :: "nat \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and k :: "nat"   and b :: "bexp"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c' :: "com"   and t :: "char list \<Rightarrow> int"   and t' :: "char list \<Rightarrow> int" assumes "\<And>n. \<forall>t t'. P (k + Suc n) t t' \<longrightarrow> (\<exists>s. P (k + n) s t' \<and> bval b s \<and> (c, s) \<Rightarrow> t)"   and "\<forall>t t'. Q t t' \<longrightarrow> (\<exists>s s'. (\<exists>n. P (k + n) s s') \<and> (WHILE b DO c, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t')"   and "Q t t'" shows "\<exists>s s'. P k s s' \<and> (WHILE b DO c, s) \<Rightarrow> t \<and> (c', s') \<Rightarrow> t'"
CallRedsParams: fixes es :: "expr list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 E :: "char list \<Rightarrow> ty option"   and v :: "val"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>es,s\<rangle> [\<rightarrow>]* \<langle>es',s'\<rangle>" shows "P,E \<turnstile> \<langle>Call (Val v) Copt M es,s\<rangle> \<rightarrow>* \<langle>Call (Val v) Copt M es',s'\<rangle>"
Deq_enabledE: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "\<turnstile> Enabled \<langle>Deq ic q oc\<rangle>_(ic, q, oc) \<longrightarrow> $q \<noteq> #[]"
permutes_natset_le: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set" assumes "p permutes S"   and "\<forall>i\<in>S. p i \<le> i" shows "p = id"
state_succ_def: fixes s :: "('c, 'd) fmap"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap" shows "state_succ s a \<equiv> if fst a \<subseteq>\<^sub>f s then snd a ++ s else s"
enn2ereal_inverse: fixes x :: "ennreal" shows "e2ennreal' (enn2ereal x) = x"
set_map0_2: fixes f1 :: "'l \<Rightarrow> 'p"   and f2 :: "'m \<Rightarrow> 'o"   and f3 :: "'n \<Rightarrow> 'q" shows "F'set1 \<circ> F'map f2 f3 f1 = (`) f2 \<circ> F'set1"
dterms_pinduct: fixes a0 :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and P :: "('a \<Rightarrow> ('b, 'c, 'a, 'd) seqp) \<Rightarrow> ('b, 'c, 'a, 'd) seqp \<Rightarrow> bool"   and a1 :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed a0"   and "\<And>\<Gamma> l g p. dterms_dom (\<Gamma>, {l}\<langle>g\<rangle> p) \<Longrightarrow> P \<Gamma> ({l}\<langle>g\<rangle> p)"   and "\<And>\<Gamma> l u p. dterms_dom (\<Gamma>, {l}\<lbrakk>u\<rbrakk> p) \<Longrightarrow> P \<Gamma> ({l}\<lbrakk>u\<rbrakk> p)"   and "\<And>\<Gamma> p1 p2. \<lbrakk>dterms_dom (\<Gamma>, p1 \<oplus> p2); P \<Gamma> p1; P \<Gamma> p2\<rbrakk> \<Longrightarrow> P \<Gamma> (p1 \<oplus> p2)"   and "\<And>\<Gamma> l s\<^sub>i\<^sub>p s\<^sub>m\<^sub>s\<^sub>g p q. dterms_dom (\<Gamma>, {l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q) \<Longrightarrow> P \<Gamma> ({l}unicast(s\<^sub>i\<^sub>p, s\<^sub>m\<^sub>s\<^sub>g) . p \<triangleright> q)"   and "\<And>\<Gamma> l s\<^sub>m\<^sub>s\<^sub>g p. dterms_dom (\<Gamma>, {l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p) \<Longrightarrow> P \<Gamma> ({l}broadcast(s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>\<Gamma> l s\<^sub>i\<^sub>p\<^sub>s s\<^sub>m\<^sub>s\<^sub>g p. dterms_dom (\<Gamma>, {l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p) \<Longrightarrow> P \<Gamma> ({l}groupcast(s\<^sub>i\<^sub>p\<^sub>s, s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>\<Gamma> l s\<^sub>m\<^sub>s\<^sub>g p. dterms_dom (\<Gamma>, {l}send(s\<^sub>m\<^sub>s\<^sub>g) . p) \<Longrightarrow> P \<Gamma> ({l}send(s\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>\<Gamma> l s\<^sub>d\<^sub>a\<^sub>t\<^sub>a p. dterms_dom (\<Gamma>, {l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p) \<Longrightarrow> P \<Gamma> ({l}deliver(s\<^sub>d\<^sub>a\<^sub>t\<^sub>a) . p)"   and "\<And>\<Gamma> l u\<^sub>m\<^sub>s\<^sub>g p. dterms_dom (\<Gamma>, {l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p) \<Longrightarrow> P \<Gamma> ({l}receive(u\<^sub>m\<^sub>s\<^sub>g) . p)"   and "\<And>\<Gamma> pn. \<lbrakk>dterms_dom (\<Gamma>, call(pn)); P \<Gamma> (\<Gamma> pn)\<rbrakk> \<Longrightarrow> P \<Gamma> (call(pn))" shows "P a0 a1"
dickson_lessD2: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict"   and "gd_powerprod.dickson_less ord_strict d m s t" shows "d t \<le> m"
simps_9: fixes x4 :: "nat" shows "REL x4 \<noteq> INIT"
lookup_update: fixes k1 :: "'b"   and v :: "'a"   and xs :: "('b, 'a) alist"   and k2 :: "'b" shows "lookup (update k1 v xs) k2 = (if k1 = k2 then Some v else lookup xs k2)"
greaterThanAtMost_def: fixes l :: "'b word"   and u :: "'b word" shows "signed.greaterThanAtMost l u \<equiv> signed.greaterThan l \<inter> signed.atMost u"
fup_Aexp_edom: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and e :: "exp"   and a :: "Arity\<^sub>\<bottom>" assumes "EdomArityAnalysis Aexp" shows "edom (fup\<cdot>(Aexp e)\<cdot>a) \<subseteq> fv e"
stable_pair_on_CH: fixes Xd :: "'d \<Rightarrow> 'e"   and Xh :: "'d \<Rightarrow> 'f"   and Pd :: "'e \<Rightarrow> ('d \<times> 'd) set"   and Ch :: "'f \<Rightarrow> 'd set \<Rightarrow> 'd set"   and ds :: "'e set"   and XD_XH :: "'d set \<times> 'd set" assumes "Contracts Xd Xh Pd Ch"   and "Contracts.stable_pair_on Pd Ch ds XD_XH" shows "fst XD_XH \<inter> snd XD_XH = Contracts.CH Ch (snd XD_XH)"
zero_min: fixes t :: "'a \<Rightarrow>\<^sub>0 nat" shows "0 \<preceq> t"
inj_on: fixes G :: "'b set"   and T :: "'b \<Rightarrow> 'c" assumes "GroupIso G T" shows "inj_on T G"
flow_attainability_def: fixes \<Delta> :: "('a, 'b) network_scheme" shows "flow_attainability \<Delta> \<equiv> countable_network \<Delta> \<and> flow_attainability_axioms \<Delta>"
assms_1: shows "(E, E') \<in> unif\<^sup>!"
stable_on_CH: fixes ds :: "'d set"   and X :: "'x set" assumes "stable_on ds X" shows "CH X = X"
strong_local_confluence: fixes r :: "'r"   and s\<^sub>1 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and s\<^sub>2 :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r' :: "'r"   and s\<^sub>2' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "revision_step r s\<^sub>1 s\<^sub>2"   and "revision_step r' s\<^sub>1 s\<^sub>2'"   and "reachable s\<^sub>1"   and "infinite UNIV"   and "infinite UNIV" shows "\<exists>s\<^sub>3 s\<^sub>3'. s\<^sub>3 \<approx> s\<^sub>3' \<and> (revision_step r' s\<^sub>2 s\<^sub>3 \<or> s\<^sub>2 = s\<^sub>3) \<and> (revision_step r s\<^sub>2' s\<^sub>3' \<or> s\<^sub>2' = s\<^sub>3')"
gSwapPresGWls_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'e, 'f) Recursion.model_scheme" shows "gSwapPresGWls MOD \<equiv> \<forall>zs z1 z2 s X' X. wls s X' \<and> gWls MOD s X \<longrightarrow> gWls MOD s (gSwap MOD zs z1 z2 X' X)"
fls_nth_vanishes_below_natE: fixes f :: "'a fls" obtains N :: "nat" where "\<forall>n>N. fls_nth f (- int n) = (0::'a)"
unique_solution: fixes s :: "'a"   and U :: "real set"   and t :: "real"   and Y\<^sub>1 :: "real \<Rightarrow> 'a"   and Y\<^sub>2 :: "real \<Rightarrow> 'a" assumes "s \<in> S"   and "t\<^sub>0 \<in> U"   and "t \<in> U"   and "is_interval U"   and "U \<subseteq> ex_ivl s"   and " D Y\<^sub>1 = (\<lambda>t. f t (Y\<^sub>1 t)) on U"   and "Y\<^sub>1 t\<^sub>0 = s"   and "Y\<^sub>1 \<in> U \<rightarrow> S"   and " D Y\<^sub>2 = (\<lambda>t. f t (Y\<^sub>2 t)) on U"   and "Y\<^sub>2 t\<^sub>0 = s"   and "Y\<^sub>2 \<in> U \<rightarrow> S" shows "Y\<^sub>1 t = Y\<^sub>2 t"