hcju/isabelleps huggingface dataset:
lift_Use_High: fixes V :: "'f"   and H :: "'f set"   and E :: "'g"   and Use :: "'g \<Rightarrow> 'f set"   and X :: "'g"   and L :: "'f set" assumes "V \<in> H" shows "(Node E, V) \<in> lift_Use_set Use E X H L"
subset_decode_imp_le: fixes m :: "nat"   and n :: "nat" assumes "set_decode m \<subseteq> set_decode n" shows "m \<le> n"
contI: fixes orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g"   and luba :: "'f set \<Rightarrow> 'f"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>Complete_Partial_Order.chain orda Y; Y \<noteq> {}\<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)" shows "cont luba orda lubb ordb f"
type_definition_rat: shows "type_definition Rep_rat Abs_rat {c. \<exists>x. ratrel x x \<and> c = Collect (ratrel x)}"
simps_1: fixes c :: "'a"   and s :: "'a FOL_Fitting.term" shows "subc c s FF = FF"
simps: fixes xs :: "('a \<times> 'b) list"   and ko :: "'a key_order" shows "sort_oalist_ko (xs, ko) = (sort_oalist_ko' ko xs, ko)"
solution_uminus_left: fixes x :: "'b"   and y :: "'b" shows "solution (- x, y) = solution (x, y)"
impl_Impl: fixes implem :: "chan \<Rightarrow> msg"   and x :: "tag"   and A :: "agent"   and B :: "agent"   and M :: "msg" assumes "semivalid_implem implem" shows "implem (Chan x A B M) \<notin> payload"
\<II>\<nn>\<ll>_invertible: fixes op :: "'op"   and xs :: "'dyn list"   and opinl :: "'opinl" assumes "\<II>\<nn>\<ll> op xs = Some opinl" shows "\<DD>\<ee>\<II>\<nn>\<ll> opinl = op"
mk_disjoint_finsert: fixes a :: "'a"   and A :: "'a fset" assumes "a |\<in>| A" shows "\<exists>B. A = finsert a B \<and> a |\<notin>| B"
subgroup_indicator_le_ord: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> carrier G"   and "subgroup H G" shows "subgroup_indicator G H a \<le> ord a"
set_delete_aux: fixes xs :: "('c \<times> 'd) list"   and k :: "'c" assumes "distinct (map fst xs)" shows "set (AList.delete_aux k xs) = set xs - {k} \<times> UNIV"
comm_ring_hom_axioms: fixes p :: "'b poly" shows "comm_ring_hom (\<lambda>q. q \<circ>\<^sub>p p)"
nprv_zer_suc_2contrE01: fixes t :: "'trm"   and F :: "'fmla set"   and t1 :: "'trm"   and \<phi> :: "'fmla" assumes "eql t zer \<in> F"   and "eql t (suc t1) \<in> F"   and "finite F"   and "F \<subseteq> fmla"   and "t \<in> atrm"   and "t1 \<in> atrm"   and "\<phi> \<in> fmla" shows "nprv F \<phi>"
split_int_nz: fixes t :: "MIR.num" assumes "nozerocoeff t" shows "nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))"
lfp_def: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and f :: "'b::type \<Rightarrow> 'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)" shows "complete_lattice.lfp (Inf::'b::type set \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (f::'b::type \<Rightarrow> 'b::type) = Inf {u::'b::type. less_eq (f u) u}"
that: fixes c :: "'x" assumes "c \<notin> A" shows "thesis"
rbt_insert_is_rbt: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and t :: "('a, 'b) RBT_Impl.rbt"   and k :: "'a"   and v :: "'b" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (ord.rbt_insert less k v t)"
cltn2_compose_is_K2_isometry: fixes H :: "cltn2"   and J :: "cltn2" assumes "is_K2_isometry H"   and "is_K2_isometry J" shows "is_K2_isometry (cltn2_compose H J)"
sum_strict_mono_ex1: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "\<forall>x\<in>A. f x \<le> g x"   and "\<exists>a\<in>A. f a < g a" shows "sum f A < sum g A"
dbm_positive: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'d \<Rightarrow> nat"   and c :: "'d"   and n :: "nat"   and u :: "'d \<Rightarrow> 'c" assumes "M 0 (v c) \<le> \<one>"   and "v c \<le> n"   and "u \<turnstile>\<^bsub>v,n\<^esub> M" shows "(0::'c) \<le> u c"
find_lambda: shows "\<lbrace>\<lambda>s. nu s \<in> {0<..lambda + mu} \<and> tortoise s = seq mu \<and> m s = mu\<rbrace> find_lambda \<lbrace>\<lambda>s. nu s \<in> {0<..lambda + mu} \<and> l s = lambda \<and> m s = mu\<rbrace>"
P1: shows "A B Perp Q C \<and> A B OS D Q"
ord_embed_alt: fixes f :: "'a \<Rightarrow> 'b" shows "ord_embed f = (mono f \<and> (\<forall>x y. f x \<le> f y \<longrightarrow> x \<le> y))"
WHILET_eq_WHILE_tproof: fixes V :: "('a \<times> 'a) set"   and I :: "'a \<Rightarrow> bool"   and s0 :: "'a"   and b :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a nres" assumes "wf V"   and "I s0"   and "\<And>s. \<lbrakk>I s; b s\<rbrakk> \<Longrightarrow> f s \<le> SPEC (\<lambda>s'. I s' \<and> (s', s) \<in> V)" shows "WHILE\<^sub>T b f s0 = WHILE b f s0"
ids_axioms: shows "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4"
induct: fixes x1 :: "'node SDG_node set"   and x2 :: "'edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind"   and x3 :: "'node list"   and x4 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and x5 :: "'edge"   and x6 :: "'node list"   and x7 :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and P :: "'node SDG_node set \<Rightarrow> ('edge \<Rightarrow> ('var, 'val, 'ret, 'pname) edge_kind) \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> 'edge \<Rightarrow> 'node list \<Rightarrow> (('var \<Rightarrow> 'val option) \<times> 'ret) list \<Rightarrow> bool" assumes "x1,x2 \<turnstile> (x3,x4) -x5\<rightarrow> (x6,x7)"   and "\<And>f a s s' ms S ms'. \<lbrakk>pred (f a) s; transfer (f a) s = s'; valid_edge a; intra_kind (kind a); \<forall>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; hd ms \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; length s' = length s; length ms = length s; hd ms = sourcenode a; ms' = targetnode a # tl ms\<rbrakk> \<Longrightarrow> P S f ms s a ms' s'"   and "\<And>f a s s' Q r p fs a' ms S ms'. \<lbrakk>pred (f a) s; transfer (f a) s = s'; valid_edge a; kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; valid_edge a'; a' \<in> get_return_edges a; \<forall>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; hd ms \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; length ms = length s; length s' = Suc (length s); hd ms = sourcenode a; ms' = targetnode a # targetnode a' # tl ms\<rbrakk> \<Longrightarrow> P S f ms s a ms' s'"   and "\<And>f a s s' Q p f' ms S ms'. \<lbrakk>pred (f a) s; transfer (f a) s = s'; valid_edge a; kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f'; \<forall>m\<in>set (tl ms). \<exists>m'. call_of_return_node m m' \<and> m' \<in> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>; length ms = length s; length s = Suc (length s'); s' \<noteq> []; hd ms = sourcenode a; hd (tl ms) = targetnode a; ms' = tl ms\<rbrakk> \<Longrightarrow> P S f ms s a ms' s'" shows "P x1 x2 x3 x4 x5 x6 x7"
IH: shows "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<Rightarrow>* \<Delta> \<oplus> Modal M Ms, m) \<in> derivable (ext R R2 M1 M2) \<longrightarrow> (\<forall>r'\<in>R'. rightPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable (ext R R2 M1 M2))"
list_all2_conv_all_nth: fixes P :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and xs :: "'g list"   and ys :: "'h list" shows "list_all2 P xs ys = (length xs = length ys \<and> (\<forall>i<length xs. P (xs ! i) (ys ! i)))"
split_sum_first_elt_less: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "n < m" shows "sum f {n..<m} = f n + sum f {Suc n..<m}"
lamvar: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and S :: "stack_elem list" assumes "map_of \<Gamma> x = Some e"   and "isVal e"   and "I (\<Gamma>, Var x, S)" shows "P (\<Gamma>, Var x, S) ((x, e) # delete x \<Gamma>, e, S)"
OrR: fixes \<Gamma> :: "'a formula multiset"   and F :: "'a formula"   and G :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> F, G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<or> G, \<Delta>"
exec_final_notin_to_execn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and T :: "('e, 'f) xstate set"   and n :: "nat" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>T" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>T"
digit_to_char_simps_10: shows "digit_to_char 9 = CHR ''9''"
language: fixes p :: "'a"   and A :: "('b, 'a) nba"   and w :: "'b stream"   and r :: "'a stream" assumes "p \<in> initial A"   and "run A (w ||| r) p"   and "infs (accepting A) (p ## r)" shows "w \<in> language A"
sorted_sorted_wrt: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq = sorted_wrt less_eq"
wffp_induct2: fixes sl :: "('St, 'U, 'C) state list"   and P :: "('St, 'U, 'C) state list \<Rightarrow> bool" assumes "wffp sl"   and "\<And>s. P [s]"   and "\<And>s sl. \<lbrakk>hd sl \<in> \<delta> s; wffp sl; P sl\<rbrakk> \<Longrightarrow> P (s # sl)" shows "P sl"
lengthOneImpliesOnlyElement: fixes l :: "'b list"   and a :: "'b" assumes "length l = 1"   and "a \<in> set l" shows "\<forall>a'. a' \<in> set l \<longrightarrow> a' = a"
elims: fixes x :: "'b extT"   and xa :: "'b extT"   and y :: "'b extT" assumes "x + xa = y" obtains x :: "'b" and y :: "'b" where "xa__ = extT.Val x" and "xa = extT.Val y" and "ya__ = extT.Val (x + y)"   | uu_ :: "'b" where "xa__ = extT.Val uu_" and "xa = extT.Top" and "ya__ = extT.Top"   | uv_ :: "'b extT" where "xa__ = extT.Top" and "xa = uv_" and "ya__ = extT.Top"
continuous_on_id: fixes s :: "'c set" shows "continuous_on s (\<lambda>x. x)"
eligible_iff: fixes \<sigma> :: "'s"   and As :: "'a list"   and DA :: "'a literal multiset" shows "eligible S \<sigma> As DA = (As = [] \<or> length As = 1 \<and> maximal_wrt (hd As \<cdot>a \<sigma>) (DA \<cdot> \<sigma>))"
prems_2: shows "list_all3 (\<lambda>P j xs. i \<le> j \<and> list_all2 P [i..<j] xs) Ps js buf"
\<theta>_1: shows "subst_domain \<theta> = \<V> \<and> ground (subst_range \<theta>)"
nullhomotopic_through_contractible_space: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and Z :: "'d topology"   and g :: "'c \<Rightarrow> 'd" assumes "continuous_map X Y f"   and "continuous_map Y Z g"   and "contractible_space Y" obtains c :: "'d" where "homotopic_with (\<lambda>h. True) X Z (g \<circ> f) (\<lambda>x. c)"
openin_contains_ball: fixes T :: "'c set"   and S :: "'c set" shows "openin (top_of_set T) S = (S \<subseteq> T \<and> (\<forall>x\<in>S. \<exists>e>0. ball x e \<inter> T \<subseteq> S))"
list_all2_trans: fixes P1 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and P2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and P3 :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and as :: "'e list"   and bs :: "'f list"   and cs :: "'g list" assumes "\<And>a b c. \<lbrakk>P1 a b; P2 b c\<rbrakk> \<Longrightarrow> P3 a c"   and "list_all2 P1 as bs"   and "list_all2 P2 bs cs" shows "list_all2 P3 as cs"
intro: fixes P :: "'a \<Rightarrow> bool" assumes "\<forall>i. \<exists>j>i. P j" shows "infinitely_many1 P"
le_supI: fixes sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and a :: "'b::type"   and x :: "'b::type"   and b :: "'b::type" assumes "class.semilattice_sup (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (a::'b::type) (x::'b::type)"   and "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (b::'b::type) (x::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (a::'b::type) (b::'b::type)) (x::'b::type)"
hom_graph_finite_superset: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and x :: "'a formula"   and a :: "'b"   and T :: "'a set" assumes "hom_graph f S x a"   and "finite T"   and "S \<subseteq> T" shows "hom_graph f T x a"
distinct_49: fixes x51 :: "poly"   and x52 :: "poly"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "Mul x51 x52 \<noteq> CN x81 x82 x83"
iv: shows "queue_to_multiset (insertList ts (local.insert (val a) (prio a) q)) = queue_to_multiset ts + queue_to_multiset (local.insert (val a) (prio a) q)"
hom_power: fixes x :: "'a poly"   and n :: "nat" shows "coeff (x ^ n) 0 = coeff x 0 ^ n"
simps: fixes h :: "'b \<Rightarrow> 'a"   and data :: "'b"   and subtrees :: "'b rose_tree list" shows "hash_source_tree h (Tree (data, subtrees)) = Tree\<^sub>h (Content (h data, map (hash_source_tree h) subtrees))"
arbitrary_def: fixes \<U> :: "'c set set" shows "arbitrary \<U> \<equiv> True"
count_filter_mset: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b multiset"   and a :: "'b" shows "count (filter_mset P M) a = (if P a then count M a else 0)"
tail_edge_succ: fixes a :: "'b" shows "tail G (edge_succ M a) = tail G a"
bl_of_nth_nth_le: fixes n :: "nat"   and xs :: "'b list" assumes "n \<le> length xs" shows "bl_of_nth n ((!) (rev xs)) = drop (length xs - n) xs"
assms: fixes T' :: "('index, 'bindex) tree" assumes "\<And>T. (T, T') \<in> treeLess \<Longrightarrow> phi T" shows "phi T'"
Domainp_forall_transfer: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "right_total A" shows "rel_fun (rel_fun A (=)) (=) (transfer_bforall (Domainp A)) transfer_forall"
chamberD_maximal: fixes x :: "'b set"   and z :: "'b set" assumes "codomain.chamber x"   and "z \<in> Y"   and "order.greater_eq z x" shows "z = x"
size_arg_lt: fixes s :: "('a, 'b) tm" assumes "is_App s" shows "size (arg s) < size s"
rel_pmf_joinI: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf pmf"   and q :: "'c pmf pmf" assumes "rel_pmf (rel_pmf P) p q" shows "rel_pmf P (join_pmf p) (join_pmf q)"
lfilter_LCons: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c llist" shows "lfilter P (LCons x xs) = (if P x then LCons x (lfilter P xs) else lfilter P xs)"
assms_2: shows "x\<^sup>2 = 1 + int D * y\<^sup>2"
row_def: fixes m :: "'a list list"   and i :: "nat" shows "Matrix_Legacy.row m i \<equiv> map (\<lambda>w. w ! i) m"
nodes_it_def: fixes g :: "'G" shows "nodes_it g \<equiv> it_to_it (nodes_list_it g)"
simps_21: fixes v :: "'b rexp"   and vc :: "'b rexp"   and vb :: "'b" shows "le_rexp (Times v vc) (Atom vb) = False"
proj_stoch_preimage: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and n :: "nat"   and w :: "bool stream" assumes "borel_adapt_stoch_proc F X" shows "proj_stoch_proc X n -` {proj_stoch_proc X n w} = (\<Inter>i\<in>{m. m \<le> n}. X i -` {X i w})"
set_zip: fixes xs :: "'b list"   and ys :: "'c list" shows "set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
dvd_mult_div_cancel: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a b" shows "times a (divide b a) = b"
d: fixes l :: "rat" shows "local.initial_lower_bound l = (if l \<le> 1 then l else rat_of_int (root_rat_floor n l))"
prod: shows "inv_D ** cA ** D = (\<chi>i j. cis (- arg (y $h i)) * cA $h i $h j * cis (arg (y $h j)))"
sink_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and q :: "'d" shows "semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 q \<equiv> q\<^sub>0 \<noteq> q \<and> (\<forall>\<nu>\<in>\<Sigma>. \<delta> q \<nu> = q)"
out_arcs_del_arc_iff: fixes a :: "'b"   and u :: "'a" shows "out_arcs (G.del_arc a) u = (if tail G a = u then out_arcs G u - {a} else out_arcs G u)"
subst_gt: fixes i :: "nat"   and j :: "nat"   and u :: "dB" assumes "i < j" shows "Var j[u/i] = Var (j - 1)"
S_uminus: fixes a :: "'a permutation" assumes "a \<in> (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})" shows "- a = a"
Op: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" assumes "wlsInp delta inp"   and "wlsBinp delta binp" shows "wls (stOf delta) (Op delta inp binp)"
quantifier_change_order_twice: fixes m :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" assumes "\<forall>i<m. \<forall>j<n. \<exists>no. \<forall>n\<ge>no. P n i j" shows "\<exists>no. \<forall>i<m. \<forall>j<n. \<forall>n\<ge>no. P n i j"
scaleR_pdevs_def: fixes r :: "real"   and x :: "'b pdevs" shows "scaleR_pdevs r x = unop_pdevs ((*\<^sub>R) r) x"
\<R>_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option"   and \<S> :: "'Var set \<times> 'Var set"   and P :: "'BExp set" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains "(case \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>"   | "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>"
invar1_def: shows "invar1 \<equiv> \<lambda>(c, n, ts). c \<le> n \<and> (\<forall>t k. t < N \<and> has_ticket (ts t) k \<longrightarrow> c \<le> k \<and> k < n \<and> (\<forall>t' k'. t' < N \<and> has_ticket (ts t') k' \<and> t \<noteq> t' \<longrightarrow> k \<noteq> k') \<and> (\<forall>k. k \<noteq> c \<longrightarrow> \<not> locks_ticket (ts t) k))"
special_subgroup_int_S: fixes S :: "'b set"   and T :: "'b set" assumes "PreCoxeterSystemWithDeletion S"   and "T \<in> Pow S" shows "\<langle>T\<rangle> \<inter> S = T"
lm093: fixes x :: "'c"   and Y :: "'d list"   and F :: "('c \<times> 'd) set list" shows "set (concat (map (\<lambda>f. map (\<lambda>y. f \<union> {(x, y)}) (filter (\<lambda>y. y \<notin> Range f) Y)) F)) = (\<Union>f\<in>set F. {f \<union> {(x, y)} |y. y \<in> set Y - Range f})"
mkArr_in_hom: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>"   and "B \<subseteq> Collect (category.terminal S)"   and "|B| <o \<AA>"   and "F \<in> A \<rightarrow> B" shows "partial_magma.in_hom S (set_category_given_img.mkArr S img \<AA> A B F) (set_category_given_img.mkIde S img \<AA> A) (set_category_given_img.mkIde S img \<AA> B)"
leadsTo_weaken_R: fixes F :: "'c program"   and A :: "'c set"   and A' :: "'c set"   and B' :: "'c set" assumes "F \<in> A \<longmapsto> A'"   and "A' \<subseteq> B'" shows "F \<in> A \<longmapsto> B'"
h: shows "map (\<lambda>j. inverse (real (p choose j)) * coeff (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:]) (p - j)) [0..<p + 1] = map (\<lambda>j. inverse (real (p choose j)) * nth_default 0 (map (\<lambda>j. nth_default 0 (coeffs (reciprocal_poly p P \<circ>\<^sub>p [:1, 1:])) (p - j)) [0..<p + 1]) j) [0..<p + 1]"
is_wt_instance_of_condD'': fixes s :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term"   and x :: "('fun, 'atom) Term.term \<times> nat" assumes "wf\<^sub>t\<^sub>r\<^sub>m s"   and "is_wt_instance_of_cond \<Gamma> t s"   and "t = Var x" shows "\<exists>y. s = Var y \<and> \<Gamma> (Var y) = \<Gamma> (Var x)"
return_restrict_space: fixes \<Omega> :: "'c set"   and M :: "'c measure"   and x :: "'c" assumes "\<Omega> \<in> sets M" shows "return (restrict_space M \<Omega>) x = restrict_space (return M x) \<Omega>"
apply_operations_never_fails: fixes xs :: "('a \<times> ('a, 'b) operation) event list"   and i :: "nat" assumes "xs prefix of i" shows "apply_operations xs \<noteq> None"
collapse_1: fixes pr_operation' :: "pr_operation'" assumes "is_RELABEL' pr_operation'" shows "pr_operation' = RELABEL'"
comp_simp: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>\<^sub>SC.seq g f" shows "\<F>\<^sub>SC.comp g f = g \<cdot> f"
dist_below_tr_4: shows "FF \<notsqsubseteq> TT"
subst_type_term_split_levels: fixes instT :: "((variable \<times> String.literal set) \<times> typ) list"   and insts :: "((variable \<times> typ) \<times> Core.term) list"   and t :: "Core.term" shows "subst_type_term instT insts t = subst_term insts (subst_typ' instT t)"
pure_def: fixes R :: "('d \<times> 'b) set" shows "pure R \<equiv> \<lambda>a c. \<up> ((c, a) \<in> R)"
add_left_imp_eq: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.cancel_semigroup_add plus"   and "plus a b = plus a c" shows "b = c"
in_gamma_option_iff: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and x :: "'c"   and r :: "'d \<Rightarrow> 'c set"   and u :: "'d option" assumes "Gamma_semilattice num' plus' \<gamma>" shows "(x \<in> \<gamma>_option r u) = (\<exists>u'. u = Some u' \<and> x \<in> r u')"
i_set_cont_minp_i_set_cont_min_eq: shows "i_set_cont_minp = (\<lambda>x. x \<in> i_set_cont_min)"
semiring_simprules_1: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
core_eq_open: fixes a :: "'a set" assumes "core a = a" shows "a open"
compat_cons: fixes gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list"   and x :: "'a" assumes "ext gt ys xs" shows "ext gt (x # ys) (x # xs)"
get_root_node_si_pure: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap" shows "pure (get_root_node_si ptr) h"
know_composition2: fixes m :: "KS"   and P :: "specID"   and Q :: "specID"   and PQ :: "specID" assumes "m \<notin> specKeysSecrets P"   and "m \<notin> specKeysSecrets Q"   and "know Q m"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionKS PQ" shows "know PQ m"
galleryD_chamber: fixes xs :: "'a set list"   and x :: "'a set" assumes "gallery xs"   and "x \<in> set xs" shows "chamber x"