hcju/isabelleps huggingface dataset:
idom_ops_def: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "idom_ops ops R \<equiv> ring_ops ops R"
analz_def: shows "analz \<equiv> \<lambda>H. {x. analzp (\<lambda>x. x \<in> H) x}"
is_empty_ivl_def: shows "is_empty_ivl \<equiv> map_fun rep_ivl id is_empty_rep"
assms: shows "(phiArg g)\<^sup>+\<^sup>+ \<phi> n"
const_implies_eq: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" assumes "\<phi> \<sim>\<^sub>C \<psi>" shows "\<phi> \<sim> \<psi>"
less_asym': fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and P :: "bool" assumes "class.preorder less_eq less"   and "less a b"   and "less b a" shows "P"
renaming_preserves_subsumption: fixes C :: "'at Literal set"   and D :: "'at Literal set"   and Sel :: "'at set" assumes "subsumes C D" shows "subsumes (rename_clause Sel C) (rename_clause Sel D)"
S_Max_Lt: shows "S_Max_Lt = {u c' + d |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> M (v c) (v c') = Lt d} \<union> {d |d. M (v c) 0 = Lt d}"
oreceived_rreq_rrep_nsqn_fresh_inv: fixes i :: "nat" shows "opaodv i \<Turnstile> (otherwith quality_increases {i} (orecvmsg msg_fresh), other quality_increases {i} \<rightarrow>) onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<sigma>, l). l \<in> {PAodv-:1} \<longrightarrow> msg_fresh \<sigma> (msg (\<sigma> i)))"
power_decreasing: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "nat"   and N :: "nat"   and a :: "'a" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "n \<le> N"   and "less_eq zero a"   and "less_eq a one" shows "less_eq (power.power one times a N) (power.power one times a n)"
row_def: fixes m :: "'b list list"   and i :: "nat" shows "row m i \<equiv> map (\<lambda>w. w ! i) m"
invariant_def: fixes P :: "'a \<Rightarrow> bool"   and succs :: "'a \<Rightarrow> 'a list" shows "invariant P succs \<equiv> \<forall>g g'. g' \<in> set (succs g) \<longrightarrow> P g \<longrightarrow> P g'"
gr_2: shows "\<forall>w\<in>cball z gr - {z}. g w = gp w * (w - z) powr of_int gn \<and> gp w \<noteq> 0"
constant_when_zero: fixes v :: "real \<Rightarrow> (real, 'a) vec"   and t0 :: "real"   and i :: "'a"   and x0 :: "real"   and f :: "real \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and T :: "real set"   and S :: "(real, 'a) vec set"   and t :: "real" assumes "v t0 $ i = x0"   and "(v solves_ode f) T S"   and "\<And>s x. s \<in> T \<Longrightarrow> f s x $ i = 0"   and "t0 \<in> T"   and "t \<in> T"   and "convex T" shows "v t $ i = x0"
elims_2: fixes cenum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and x :: "'a list \<Rightarrow> bool"   and xa :: "nat" assumes "all_n_lists cenum_all x xa" obtains P :: "'a list \<Rightarrow> bool" and n :: "nat" where "x = P" and "xa = n" and "if n = 0 then P [] else cenum_all (\<lambda>x. all_n_lists cenum_all (\<lambda>xs. P (x # xs)) (n - 1))"
pairwise_subset: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "pairwise P S"   and "T \<subseteq> S" shows "pairwise P T"
add_zeror: fixes x :: "'a" shows "x + (zero_class.zero::'a) = x"
closed_except_const: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name"   and S :: "name fset" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.closed_except frees (const name) S"
map_key_plus: fixes f :: "'a \<Rightarrow> 'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c" assumes "inj f" shows "Poly_Mapping.map_key f (p + q) = Poly_Mapping.map_key f p + Poly_Mapping.map_key f q"
span_eq_iff: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and s :: "'b set" assumes "module scale" shows "(module.span scale s = s) = module.subspace scale s"
\<Sigma>_def: fixes f :: "'d \<Rightarrow> 'c"   and j :: "int"   and l :: "int" shows "\<Sigma> f j l = (if j < l then sum (f \<circ> of_int) {j..<l} else if l < j then - sum (f \<circ> of_int) {l..<j} else (0::'c))"
rel_symp: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "symp R" shows "symp (llist_all2 R)"
mantissa_eq_zero_iff: fixes x :: "float" shows "(mantissa x = 0) = (x = 0)"
qconv_id: fixes x :: "'c['d, 'e]" shows "qconv id\<^sub>C x = x"
is_natural_2: fixes f :: "'b" assumes "arr f" shows "\<rr>'.map (cod f) \<cdot> local.map f = \<rr>'.map f"
eqvt: fixes x1 :: "pi"   and x2 :: "residual"   and pi :: "(name \<times> name) list" assumes "x1 \<longmapsto> x2" shows "pi \<bullet> x1 \<longmapsto> pi \<bullet> x2"
restr_un: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and m :: "'b \<Rightarrow> bool" shows "(r \<union> s | m) = (r | m) \<union> (s | m)"
pair_in_swap_image: fixes y :: "'b"   and x :: "'c"   and A :: "('c \<times> 'b) set" shows "((y, x) \<in> prod.swap ` A) = ((x, y) \<in> A)"
lm094: fixes Y :: "'c set"   and f :: "('d \<times> 'c) set"   and x :: "'d" assumes "finite Y" shows "set (map (\<lambda>y. f +< {(x, y)}) (sorted_list_of_set (Y - Range f))) = {f +< {(x, y)} |y. y \<in> Y - Range f}"
length_list_of_fset: fixes s :: "'b fset" shows "length (list_of_fset s) = size s"
simps_5: fixes n :: "nat"   and r :: "'b rexp"   and s :: "'b rexp" shows "lang n (Plus r s) = lang n r \<union> lang n s"
Ioc_subset_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" shows "({a<..b} \<subseteq> {c<..d}) = (b \<le> a \<or> c \<le> a \<and> b \<le> d)"
order_refl: fixes x :: "'c word" shows "x \<le>s x"
ltake_LNil: fixes n :: "enat" shows "ltake n []\<^sub>l = []\<^sub>l"
B: fixes n :: "nat" shows "Sigma_Algebra.measure M (\<Union> (S ` {..n})) \<le> B"
f_def: shows "f = (\<lambda>n. if n = 0 then 0 else 1)"
multiset_cases: fixes M :: "'b multiset" obtains "M = \<Empt>"   | x :: "'b" and N :: "'b multiset" where "M = add_mset x N"
simps_9: fixes x3 :: "nat"   and x2 :: "nat" shows "Greater x3 \<noteq> Intv x2"
simps_2: fixes s :: "'a"   and d :: "bool"   and \<alpha> :: "'b"   and t :: "'a"   and ss :: "(bool \<times> 'b \<times> 'a) list" shows "lst_conv (s, (d, \<alpha>, t) # ss) = lst_conv (t, ss)"
AAcc_\<tau>red1r_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and i :: "'addr expr1"   and xs :: "'addr val list"   and i' :: "'addr expr1"   and xs' :: "'addr val list"   and a :: "'addr val" assumes "\<tau>red1gr uf P t h (i, xs) (i', xs')" shows "\<tau>red1gr uf P t h (Val a\<lfloor>i\<rceil>, xs) (Val a\<lfloor>i'\<rceil>, xs')"
HomFtorFtor': fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "FunctorM (HomP\<^bsub>C\<^esub>[X,\<midarrow>])"
H_while_inv: fixes p :: "('a \<times> 'a) set"   and i :: "('a \<times> 'a) set"   and r :: "('a \<times> 'a) set"   and q :: "('a \<times> 'a) set"   and x :: "('a \<times> 'a) set" assumes "rel_kat.t_op p \<subseteq> rel_kat.t_op i"   and "rel_kat.t_op i ; rel_at r \<subseteq> rel_kat.t_op q"   and "rel_kat.H (rel_kat.t_op i ; rel_kat.t_op r) x i" shows "rel_kat.H p (rel_kat.while_inv r i x) q"
chamber_system_def: shows "folding.\<C> \<equiv> Collect folding.chamber"
take_map: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map f xs \<down> n = map f (xs \<down> n)"
sel_3: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "BinomialHeapStruc.rank (BinomialHeapStruc.Node x1 x2 x3 x4) = x3"
forest_components_equivalence: fixes x :: "'a" assumes "injective x" shows "equivalence (forest_components x)"
apply_guards_fold: fixes G :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "apply_guards G s = (gval (fold gAnd G (Bc True)) s = true)"
List_def: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and ps :: "ref list" shows "List p h ps = Path p h Null ps"
row_carrier_vec: fixes i :: "nat"   and nr :: "nat"   and A :: "'a mat"   and nc :: "nat" assumes "i < nr"   and "A \<in> carrier_mat nr nc" shows "row A i \<in> carrier_vec nc"
Union_Int_subset: fixes A :: "'b set set"   and B :: "'b set set" shows "order.greater_eq (\<Union> A \<inter> \<Union> B) (\<Union> (A \<inter> B))"
Obj_Trg: fixes t :: "'a local.term" assumes "Arr t" shows "Obj (Trg t)"
strict_dom_shortestPath_order: fixes n :: "'node"   and m :: "'node"   and g :: "'g" assumes "n \<noteq> m \<and> dominates g n m"   and "m \<in> set (\<alpha>n g)"   and "invar g" shows "shortestPath g n < shortestPath g m"
nat_le_prod_with_same: fixes m :: "nat"   and n :: "nat" assumes "m \<noteq> 0" shows "n \<le> m * n"
intro: fixes \<alpha> :: "V" assumes "Limit \<alpha>"   and "\<omega> \<in>\<^sub>\<circ> \<alpha>" shows "\<Z> \<alpha>"
bounded_by_update: fixes xs :: "real list"   and vs :: "float interval option list"   and i :: "nat"   and ivl :: "float interval" assumes "bounded_by xs vs"   and "xs ! i \<in>\<^sub>r ivl" shows "bounded_by xs (vs[i := Some ivl])"
corthogonalD: fixes vs :: "'c vec list"   and i :: "nat"   and j :: "nat" assumes "corthogonal vs"   and "i < length vs"   and "j < length vs" shows "(inner_prod (vs ! j) (vs ! i) = (0::'c)) = (i \<noteq> j)"
reflect_factorization: fixes p :: "'a poly"   and fctrs :: "'a \<times> ('a \<times> nat) list" assumes "constant_term p \<noteq> (0::'a)"   and "is_factorization_of fctrs p" shows "is_alt_factorization_of fctrs (reflect_poly p)"
zero: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "A *v 0 = 0"
poss1: shows "squash (rpoly q x1) = - 1 \<and> squash (rpoly q x2) = 1 \<longrightarrow> (\<exists>r>x1. r \<le> x2 \<and> (\<exists>q\<in>set qs. rpoly q r = 0))"
data_refinement_choice1: fixes S1 :: "'a \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'c"   and D' :: "'a \<Rightarrow> 'd"   and S2 :: "'d \<Rightarrow> 'c"   and S2' :: "'d \<Rightarrow> 'c" assumes "DataRefinement S1 D D' S2"   and "DataRefinement S1 D D' S2'" shows "DataRefinement S1 D D' (S2 \<sqinter> S2')"
CAS1Red3: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and D :: "String.literal"   and F :: "String.literal"   and v' :: "'addr val" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e),s\<rangle> -ta\<rightarrow> \<langle>Val v\<bullet>compareAndSwap(D\<bullet>F, Val v', e'),s'\<rangle>"
psubstAbs_cong: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option" assumes "goodAbs A"   and "goodEnv rho"   and "goodEnv rho'"   and "\<And>ys y. freshAbs ys y A \<or> rho ys y = rho' ys y" shows "A $[rho] = A $[rho']"
config_induct: fixes P :: "'b list \<Rightarrow> bool"   and s0 :: "'b list"   and A :: "('b list \<Rightarrow> 'c) \<times> ('b list \<times> 'c \<Rightarrow> 'b \<Rightarrow> (nat \<times> nat list) \<times> 'c)"   and qs :: "'b list" assumes "P s0"   and "\<And>s q a. P s \<Longrightarrow> P (step s q a)" shows "P (fst (config A s0 qs))"
intro: fixes R :: "('c, 'd) ring_scheme" assumes "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> z \<otimes>\<^bsub>R\<^esub> (x \<oplus>\<^bsub>R\<^esub> y) = z \<otimes>\<^bsub>R\<^esub> x \<oplus>\<^bsub>R\<^esub> z \<otimes>\<^bsub>R\<^esub> y" shows "ring_axioms R"
set_child_nodes_get_disconnected_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" assumes "l_set_child_nodes_get_disconnected_nodes type_wf set_child_nodes set_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs" shows "\<forall>w\<in>set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
left_total_alt_def: fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "left_total R = ((=) \<le> R OO R\<inverse>\<inverse>)"
map_def: fixes f :: "'j" shows "A'.F.map f = (if J.arr f then C.dom a' else C.null)"
gcd_mult_unit1: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a one" shows "gcd (times b a) c = gcd b c"
wf_bs_polyadd: fixes bs :: "'b list"   and p :: "'b poly"   and q :: "'b poly" shows "wf_bs bs p \<and> wf_bs bs q \<longrightarrow> wf_bs bs (p +\<^sub>p q)"
simps_6: fixes f1 :: "bool \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> 'a"   and x2 :: "nat" shows "(case Indet x2 of Det x \<Rightarrow> f1 x | Indet x \<Rightarrow> f2 x) = f2 x2"
sl_cs_3: fixes cs :: "'a list" assumes "cs \<in> set css" shows "distinct (some_list cs)"
msg_nonempty2: fixes a :: "nat \<Rightarrow> 'a list"   and t :: "nat" assumes "msg (Suc 0) a"   and "a t \<noteq> []" shows "length (a t) = Suc 0"
poly_of_pm_of_poly: fixes x :: "'b"   and p :: "'a poly" shows "poly_of_pm x (pm_of_poly x p) = p"
quot_Eq: fixes x :: "tm"   and y :: "tm" shows "\<guillemotleft>x EQ y\<guillemotright> = Q_Eq \<guillemotleft>x\<guillemotright> \<guillemotleft>y\<guillemotright>"
sel_3: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "('a, 'b) BsSkewBinomialHeapStruc.BsSkewBinomialTree list" shows "BsSkewBinomialHeapStruc.un_Element3 (BsSkewBinomialHeapStruc.Element x1 x2 x3) = x3"
ide_def: fixes a :: "'a" shows "V.ide a \<equiv> V a a \<noteq> V.null \<and> (\<forall>f. (V f a \<noteq> V.null \<longrightarrow> V f a = f) \<and> (V a f \<noteq> V.null \<longrightarrow> V a f = f))"
fun_le_inv: fixes t1 :: "(val \<times> val) list"   and t2 :: "(val \<times> val) list" assumes "t1 \<lesssim> t2" obtains "\<forall>v1 v2. (v1, v2) \<in> set t1 \<longrightarrow> (\<exists>v3 v4. (v3, v4) \<in> set t2 \<and> v1 \<sqsubseteq> v3 \<and> v3 \<sqsubseteq> v1 \<and> v2 \<sqsubseteq> v4 \<and> v4 \<sqsubseteq> v2)"
tree_chop_ap_tree: fixes f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "tree_chop (f \<diamondop> x) = tree_chop f \<diamondop> tree_chop x"
le_inf_iff: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c"   and z :: "'b \<times> 'c" shows "less_eq_prod x (inf_prod y z) = (less_eq_prod x y \<and> less_eq_prod x z)"
fds_left_inverse: fixes f :: "'b fds" assumes "fds_nth f 1 \<noteq> (0::'b)" shows "inverse f * f = 1"
ide_def: fixes a :: "'d" shows "Dop.ide a \<equiv> Dop.comp a a \<noteq> Dop.null \<and> (\<forall>f. (Dop.comp f a \<noteq> Dop.null \<longrightarrow> Dop.comp f a = f) \<and> (Dop.comp a f \<noteq> Dop.null \<longrightarrow> Dop.comp a f = f))"
Jacobi_0_right: fixes a :: "int" shows "Jacobi a 0 = 0"
A2: shows "[\<lambda>w. \<forall>x xa. ((P x m\<and> \<box> (\<lambda>v. \<forall>xb. (x xb m\<rightarrow> xa xb) v)) m\<rightarrow> P xa) w]"
rev_eq_Cons_iff: fixes xs :: "'f list"   and y :: "'f"   and ys :: "'f list" shows "(rev xs = y # ys) = (xs = rev ys @ [y])"
simps_4: fixes x :: "'b"   and xs :: "['b]"   and y :: "'b"   and ys :: "['b]" shows "eq\<cdot>(x : xs)\<cdot>(y : ys) = (eq\<cdot>x\<cdot>y andalso eq\<cdot>xs\<cdot>ys)"
bisim1FAccNull: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and xs :: "'addr val list" shows "P,e\<bullet>F{D},h \<turnstile> (Throw (addr_of_sys_xcpt NullPointer), xs) \<leftrightarrow> ([Null], xs, length (compE2 e), \<lfloor>addr_of_sys_xcpt NullPointer\<rfloor>)"
src_trg: fixes \<mu> :: "'b" assumes "B.arr \<mu>" shows "src\<^sub>B (trg\<^sub>B \<mu>) = trg\<^sub>B \<mu>"
assms: shows "monoid_congruence R (\<cdot>) \<one> Ring_Congruence"
Abs_scene_inverse: fixes y :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "y \<in> {F. overrider F}" shows "Rep_scene (Abs_scene y) = y"
eq_fps_sin: fixes a :: "'a fps"   and c :: "'a" assumes "fps_nth a 0 = (0::'a)"   and "fps_nth a 1 = c"   and "fps_deriv (fps_deriv a) = - (fps_const c * fps_const c * a)" shows "fps_sin c = a"
always_eventually: fixes P :: "'b \<Rightarrow> bool"   and F :: "'b filter" assumes "\<forall>x. P x" shows "eventually P F"
c_def: shows "c \<equiv> b + d *\<^sub>R One"
confs_Cons2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and xs :: "val list"   and y :: "ty"   and ys :: "ty list" shows "(P,h \<turnstile> xs [:\<le>] y # ys) = (\<exists>z zs. xs = z # zs \<and> P,h \<turnstile> z :\<le> y \<and> P,h \<turnstile> zs [:\<le>] ys)"
is_universal': fixes p0' :: "'c"   and p1' :: "'c" assumes "D.is_rendered_commutative_by p0' p1'" shows "\<exists>!h. \<guillemotleft>h : C.dom p0' \<rightarrow> C.dom p0\<guillemotright> \<and> p0 \<cdot> h = p0' \<and> p1 \<cdot> h = p1'"
sel_1: fixes x1 :: "int"   and x2 :: "int" shows "a (X x1 x2) = x1"
infin_imp_iNextWeak_iNextStrong_eq_iNext: fixes I :: "nat set"   and t0 :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "infinite I"   and "t0 \<in> I" shows "iNextWeak t0 I P = iNext t0 I P \<and> iNextStrong t0 I P = iNext t0 I P"
P_narrow: fixes C1 :: "'a acom"   and C2 :: "'a acom" assumes "P C1"   and "P C2" shows "P (C1 \<triangle> C2)"
hyps: fixes \<O> :: "nat set"   and \<S>' :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option"   and \<R> :: "nat \<Rightarrow> bool option" assumes "\<forall>a\<in>\<O> \<union> all_acquired sb. \<S>' a = \<S> a"   and "sharing_consistent \<S> \<O> sb" shows "release sb (dom \<S>') \<R> = release sb (dom \<S>) \<R>"
plus_dim1: fixes A :: "'m tensor"   and B :: "'m tensor" assumes "dims A = dims B" shows "dims (A + B) = dims A"
kmp3_def: fixes s :: "'b list"   and t :: "'b list" shows "kmp3 s t \<equiv> if s = [] then RETURN (Some 0) else kmp2 s t"
sub_rel_of: fixes g :: "('d \<times> nat, 'e) rgraph_scheme"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "wf_sub_rel_of g subs" shows "sub_rel_of g subs"
augment_solvable_seq: fixes H :: "'a set" assumes "subgroup H G"   and "solvable_seq G (derived G H)" shows "solvable_seq G H"
not_outside_1: fixes x :: "'alt"   and y :: "'alt" assumes "Pareto R x y" shows "x \<in> alts"
is_reducible_lit_imp_is_reducible: fixes L :: "'a literal"   and C' :: "'a literal list"   and Ds :: "'a literal list list"   and C :: "'a literal list" assumes "L \<in> set C'"   and "is_reducible_lit Ds (C @ remove1 L C') L" shows "reduce Ds C C' \<noteq> C'"
ll2f_lem1: fixes t :: "'b llist"   and i :: "nat"   and x :: "'b" assumes "t !! Suc i = Some x" shows "\<exists>y. t !! i = Some y"
prems: shows "n' \<in> sum_SDG_slice1 n''"
real_transfer_operator_diff: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "AE x in M. real_transfer_operator (\<lambda>x. f x - g x) x = real_transfer_operator f x - real_transfer_operator g x"
connectedI: fixes xs :: "'a list"   and v :: "'a"   and w :: "'a" assumes "walk xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<rightarrow>\<^sup>* w"
face_cycle_succ_inD: fixes x :: "'b"   and y :: "'b" assumes "G.face_cycle_succ x \<in> G.face_cycle_set y" shows "x \<in> G.face_cycle_set y"
unique_wall: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and f' :: "'b \<Rightarrow> 'b"   and g' :: "'b \<Rightarrow> 'b"   and C' :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "OpposedThinChamberComplexFoldings X f' g' C'"   and "A \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "A \<in> f' \<turnstile> ChamberComplex.chamber_system X"   and "B \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "B \<in> g' \<turnstile> ChamberComplex.chamber_system X"   and "A \<sim> B" shows "{f \<turnstile> ChamberComplex.chamber_system X, g \<turnstile> ChamberComplex.chamber_system X} = {f' \<turnstile> ChamberComplex.chamber_system X, g' \<turnstile> ChamberComplex.chamber_system X}"
Abs_bit1'_def: fixes x :: "int" shows "Abs_bit1' x = Abs_bit1 (x mod int CARD('b bit1))"
mon_ww_ileq: fixes w :: "('a, 'b) label list list"   and w' :: "('a, 'b) label list list"   and fg :: "('d, 'a, 'e, 'c, 'f) flowgraph_rec_scheme" assumes "w \<preceq> w'" shows "mon_ww fg w \<subseteq> mon_ww fg w'"
ml_def: shows "ml = Max {return_time {x \<in> S. s x = 0} x / 2, return_time {x \<in> S. s x = 0} x - e, d}"
singleton_eqI: fixes A :: "'a pred"   and x :: "'a"   and default :: "unit \<Rightarrow> 'a" assumes "\<exists>!x. eval A x"   and "eval A x" shows "singleton default A = x"
soundI: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and G :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" assumes "\<And>I. \<lbrakk>is_interp I; \<And>i. \<lbrakk>0 \<le> i; i < length SG\<rbrakk> \<Longrightarrow> seq_sem I (SG ! i) = UNIV\<rbrakk> \<Longrightarrow> seq_sem I G = UNIV" shows "sound (SG, G)"
search_r: shows "search r x = has_overlap (Tree2.set_tree r) x"
open_mapping_thm2: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and U :: "complex set" assumes "f holomorphic_on S"   and "open S"   and "open U"   and "U \<subseteq> S"   and "\<And>X. \<lbrakk>open X; X \<subseteq> S; X \<noteq> {}\<rbrakk> \<Longrightarrow> \<not> f constant_on X" shows "open (f ` U)"
simps: fixes m :: "nat"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list" shows "wprepare_add_one2 m lm (l, r) = (\<exists>rn. l = Bk # Bk # <rev lm> @ Bk # Bk # Oc \<up> Suc m \<and> (r = [] \<or> tl r = Bk \<up> rn))"
order_linear': fixes a :: "'a"   and b :: "'a" shows "order a [:b, 1::'a:] = (if b = - a then 1 else 0)"
lt_halfspace_def: fixes sctn :: "'b sctn"   and x :: "'b" shows "lt_halfspace sctn x = (x \<bullet> normal sctn < pstn sctn)"