hcju/isabelleps huggingface dataset:
Brouwer_degree2: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" assumes "x \<in> carrier (reduced_homology_group (int p) (nsphere p))" shows "hom_induced (int p) (nsphere p) {} (nsphere p) {} f x = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> Brouwer_degree2 p f"
expand_impl_def: shows "expand_impl \<equiv> REC\<^sub>T (\<lambda>x (xa, xb). if lss.isEmpty (new_impl xa) then if Gen_Set.gen_bex (\<lambda>x. foldli (id x)) xb (\<lambda>xc. lss.equal (old_impl xc) (old_impl xa) \<and> lss.equal (next_impl xc) (next_impl xa)) then RETURN (name_impl xa, upd_incoming_impl xa xb) else x (\<lparr>name_impl = Suc (name_impl xa), incoming_impl = map2set_insert rbt_insert (name_impl xa) rbt.Empty, new_impl = next_impl xa, old_impl = lss.empty (), next_impl = lss.empty ()\<rparr>, xa # xb) else RETURN (the (lss.sel (new_impl xa) (\<lambda>_. True))) \<bind> (\<lambda>xc. let y' = xa\<lparr>new_impl := lss.delete xc (new_impl xa)\<rparr> in case xc of true\<^sub>r \<Rightarrow> x (y'\<lparr>old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y')\<rparr>, xb) | false\<^sub>r \<Rightarrow> RETURN (name_impl y', xb) | prop\<^sub>r(xd) \<Rightarrow> if lss.memb nprop\<^sub>r(xd) (old_impl y') then RETURN (name_impl y', xb) else x (y'\<lparr>old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y')\<rparr>, xb) | nprop\<^sub>r(xd) \<Rightarrow> if lss.memb prop\<^sub>r(xd) (old_impl y') then RETURN (name_impl y', xb) else x (y'\<lparr>old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y')\<rparr>, xb) | xd and\<^sub>r xe \<Rightarrow> x (y'\<lparr>new_impl := lss.ins xd (lss.ins xe (new_impl y')), old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y'), next_impl := next_impl y'\<rparr>, xb) | xd or\<^sub>r xe \<Rightarrow> x (y'\<lparr>new_impl := lss.ins xd (new_impl y'), old_impl := lss.ins xc (old_impl y'), next_impl := lss.union_dj (lss.empty ()) (next_impl y')\<rparr>, xb) \<bind> (\<lambda>(xf, xg). x (y'\<lparr>name_impl := xf, new_impl := lss.union (lss.ins_dj xe (lss.empty ())) (new_impl y'), old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y')\<rparr>, xg)) | X\<^sub>r xd \<Rightarrow> x (y'\<lparr>new_impl := new_impl y', old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y'), next_impl := lss.ins xd (next_impl y')\<rparr>, xb) | xd U\<^sub>r xe \<Rightarrow> x (y'\<lparr>new_impl := lss.ins xd (new_impl y'), old_impl := lss.ins xc (old_impl y'), next_impl := lss.union (lss.ins_dj xc (lss.empty ())) (next_impl y')\<rparr>, xb) \<bind> (\<lambda>(xf, xg). x (y'\<lparr>name_impl := xf, new_impl := lss.union (lss.ins_dj xe (lss.empty ())) (new_impl y'), old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y')\<rparr>, xg)) | xd R\<^sub>r xe \<Rightarrow> x (y'\<lparr>new_impl := lss.ins xe (new_impl y'), old_impl := lss.ins xc (old_impl y'), next_impl := lss.union (lss.ins_dj xc (lss.empty ())) (next_impl y')\<rparr>, xb) \<bind> (\<lambda>(xf, xg). x (y'\<lparr>name_impl := xf, new_impl := lss.union (lss.ins xd (lss.ins_dj xe (lss.empty ()))) (new_impl y'), old_impl := lss.union (lss.ins_dj xc (lss.empty ())) (old_impl y')\<rparr>, xg))))"
conth: fixes s :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and t :: "'b set"   and k :: "'b \<Rightarrow> 'a" assumes "Retracts s h t k" shows "continuous_on s h"
mult_rel_def: fixes X :: "('a set \<times> 'b) set"   and Y :: "('a set \<times> 'b) set" shows "mult_rel X Y \<equiv> let x = SOME x. x \<in> X; y = SOME y. y \<in> Y; w = get_lower_bound (fst x) (fst y) in \<lfloor>w, \<cdot>\<^bsub>w\<^esub> (\<rho> (fst x) w (snd x)) (\<rho> (fst y) w (snd y))\<rfloor>"
map_o_corec: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> bool"   and ga :: "'c \<Rightarrow> 'e"   and gb :: "'c \<Rightarrow> bool"   and gc :: "'c \<Rightarrow> 'e llist"   and gd :: "'c \<Rightarrow> 'c" shows "lmap f \<circ> corec_llist g ga gb gc gd = corec_llist g (f \<circ> ga) gb (lmap f \<circ> gc) gd"
OK_OK_conv: fixes P :: "'c prog"   and T :: "ty"   and T' :: "ty" shows "P \<turnstile> OK T \<le>\<^sub>\<top> OK T' = P \<turnstile> T \<le> T'"
1: shows "prv (exi x (neg (cnj fls (eql (Var x) (Var x)))))"
greaterThan_def: fixes less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and l :: "'c" shows "ord.greaterThan less l \<equiv> {x. less l x}"
dom_inv: fixes f :: "'d" assumes "D.iso f" shows "D.dom (D.inv f) = D.cod f"
sqn_update_same: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and ip :: "nat"   and v :: "nat \<times> k \<times> f \<times> nat \<times> nat" shows "sqn (rt(ip \<mapsto> v)) ip = \<pi>\<^sub>2 v"
xs_x_def: shows "xs #\<^sub>\<circ> x = vfsequence_of_vlist (y # ys)"
simps: fixes t0 :: "'t"   and queue :: "'t list"   and n0 :: "nat"   and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "round_robin_reschedule t0 queue n0 s = (let t = hd queue; queue' = tl queue in if t = t0 then None else case round_robin_step n0 (t # queue', n0) s t of None \<Rightarrow> round_robin_reschedule t0 (queue' @ [t]) n0 s | \<lfloor>ttaxm\<sigma>\<rfloor> \<Rightarrow> \<lfloor>ttaxm\<sigma>\<rfloor>)"
Units_l_cancel: fixes x :: "'a \<times> 'a"   and y :: "'a \<times> 'a"   and z :: "'a \<times> 'a" assumes "x \<in> Units (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "y \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)"   and "z \<in> carrier (subgroup_generated G A \<times>\<times> subgroup_generated G B)" shows "(x \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> y = x \<otimes>\<^bsub>subgroup_generated G A \<times>\<times> subgroup_generated G B\<^esub> z) = (y = z)"
transpose_transpose: fixes A :: "(('d, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec" shows "Finite_Cartesian_Product.transpose (Finite_Cartesian_Product.transpose A) = A"
A: fixes Y :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "P Y \<tau> = P (\<lambda>_. Y \<tau>) \<tau>"
varFIX: fixes b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and s :: "Var \<Rightarrow> Val"   and t :: "Var \<Rightarrow> Val" assumes "(b, \<Phi>, s, t) \<in> var" shows "PhiWhile b \<Phi> (s, t)"
pfun_graph_inv: fixes f :: "('a, 'b) pfun" shows "graph_pfun (pfun_graph f) = f"
mod_exp_def: fixes b :: "'a"   and e :: "nat"   and m :: "'a" shows "mod_exp b e m = b ^ e mod m"
closure_act_3: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<A>(\<^bold>\<forall>\<alpha>. \<^bold>\<A>\<phi> \<alpha> \<^bold>\<equiv> \<phi> \<alpha>) in v]"
simps: fixes A :: "'a set"   and f :: "'a \<Rightarrow> nat"   and l :: "nat"   and up :: "nat"   and left :: "nat" shows "steps A f l up left = (if A = {} \<or> infinite A then 0 else let m = Max A in if f m < l then steps (A - {m}) f l up left else if l < f m then up + steps A f (l + 1) up left else left + steps (A - {m}) f l up left)"
elims: fixes x :: "'b tree"   and y :: "nat" assumes "imbal x = y" obtains "x = \<langle>\<rangle>" and "y = 0"   | l :: "'b tree" and uu_ :: "'b" and r :: "'b tree" where "x = \<langle>l, uu_, r\<rangle>" and "y = nat \<bar>int (size l) - int (size r)\<bar> - 1"
AinequalityTr0: fixes x :: "ant"   and m :: "nat" assumes "x \<noteq> - \<infinity>" shows "\<exists>L. \<forall>N>L. an m < x + an N"
bl_or_aux_bin: shows "map2 (\<or>) (bin_to_bl_aux n a []) (bin_to_bl_aux n b []) = bin_to_bl_aux n (a OR b) (map2 (\<or>) [] [])"
assms: shows "x \<rightarrow>\<^sup>*\<^bsub>H\<^esub> y"
invertible: fixes u :: "'a" assumes "u \<in> A" shows "maxI.additive.invertible u"
simps_3: fixes n :: "num" shows "Num.BitM (num.Bit1 n) = num.Bit1 (num.Bit0 n)"
inner_prod_def: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "x \<bullet> y = fst x \<bullet> fst y + snd x \<bullet> snd y"
complete_lattice_axioms_def: fixes L :: "('a, 'b) gorder_scheme" shows "complete_lattice_axioms L \<equiv> (\<forall>A\<subseteq>carrier L. \<exists>s. is_lub L s A) \<and> (\<forall>A\<subseteq>carrier L. \<exists>i. is_glb L i A)"
P_Q: fixes s :: "'a" assumes "PROP P" shows "PROP Q s"
fmfilter_fmmap: fixes P :: "'c \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'd"   and m :: "('c, 'e) fmap" shows "fmfilter P (fmmap f m) = fmmap f (fmfilter P m)"
And_def: fixes \<phi> :: "'b formula"   and \<psi> :: "'b formula" shows "And \<phi> \<psi> = Neg (Or (Neg \<phi>) (Neg \<psi>))"
heap_is_wellformed_disconnected_nodes_distinct: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes" shows "distinct disc_nodes"
matrix_vector_all_zero_from_index2: fixes i :: "'b"   and A :: "(('c, 'd) vec, 'b) vec"   and k :: "'d" shows "(\<forall>m>i. A $ m $ k = (0::'c)) = vector_all_zero_from_index (mod_type_class.to_nat i + 1, vec_to_iarray (column k A))"
diagonal_to_Smith_PQ': fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "(('b, 'd) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and S :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec"   and Q :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" assumes "isDiagonal A"   and "is_bezout_ext bezout"   and "(P, S, Q) = diagonal_to_Smith_PQ A bezout" shows "S = P ** A ** Q \<and> invertible P \<and> invertible Q \<and> Smith_normal_form S"
varnum_le_iff: fixes X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 'b"   and n :: "nat" shows "(varnum X t \<le> n) = (keys t \<subseteq> X \<union> {x. elem_index x < n})"
finite_supp_B: shows "finite (supp {p \<bullet> x |p. supp P \<sharp>* p})"
basic_just_sp_def: fixes G :: "('c, 'd) pre_digraph"   and dist :: "'c \<Rightarrow> ereal"   and c :: "'d \<Rightarrow> real"   and s :: "'c"   and num :: "'c \<Rightarrow> enat" shows "basic_just_sp G dist c s num \<equiv> basic_sp G dist c s \<and> basic_just_sp_axioms G dist c s num"
simp_regular: fixes x :: "'b regular" shows "in_p_image (Rep_regular x)"
degree_prod_sum_monic: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c poly" assumes "finite S"   and "0 \<notin> (degree \<circ> f) ` S"   and "\<And>a. a \<in> S \<Longrightarrow> monic (f a)" shows "degree (prod f S) = sum (degree \<circ> f) S \<and> coeff (prod f S) (sum (degree \<circ> f) S) = (1::'c)"
midpoint_existence: fixes A :: "'p"   and B :: "'p" shows "\<exists>X. X Midpoint A B"
abs_o_rep: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b" assumes "Quotient3 R Abs Rep" shows "Abs \<circ> Rep = id"
mem_simps_8: fixes b :: "'a"   and B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "(b \<in> \<Union> (B ` A)) = (\<exists>x\<in>A. b \<in> B x)"
inject: fixes x2 :: "'c"   and y2 :: "'c" shows "(Num_arc_delta x2 = Num_arc_delta y2) = (x2 = y2)"
pref_take_conv: fixes r :: "'b list"   and w :: "'b list" shows "(take \<^bold>|r\<^bold>| w = r) = (r \<le>p w)"
Cons: fixes x :: "'g"   and y :: "'h"   and r :: "('g \<times> 'h) set"   and xs :: "'g list"   and ys :: "'h list" assumes "(x, y) \<in> r"   and "(xs, ys) \<in> listrel r" shows "(x # xs, y # ys) \<in> listrel r"
iso_pullback_ide: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.cospan \<mu> \<nu>"   and "C.ide \<mu>" shows "C.iso \<p>\<^sub>0[\<mu>, \<nu>]"
simps_2: fixes f :: "'a \<Rightarrow> 'a" shows "atom_com (Basic f) = True"
lexordp_eq_refl: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list" shows "ord.lexordp_eq less xs xs"
fmdomE: fixes x :: "'c"   and m :: "('c, 'd) fmap" assumes "x |\<in>| fmdom m" obtains y :: "'d" where "fmlookup m x = Some y"
that: fixes a :: "'edge" assumes "valid_edge a"   and "inner_node (sourcenode a)"   and "targetnode a = (_Exit_)" shows "thesis"
bounded_lattice_top_ow_axioms: shows "bounded_lattice_top_ow U\<^sub>1 (\<sqinter>\<^sub>o\<^sub>w\<^sub>.\<^sub>1) (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>1) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>1) (\<squnion>\<^sub>o\<^sub>w\<^sub>.\<^sub>1) top\<^sub>1"
prod_coprime_right: 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 :: "'c set"   and a :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>i. i \<in> A \<Longrightarrow> algebraic_semidom.coprime times one a (f i)" shows "algebraic_semidom.coprime times one a (comm_monoid_mult.prod times one f A)"
hyps_1: fixes xs :: "'a list"   and ys :: "'a list" assumes "l = length xs + length ys"   and "length xs = card {a \<in> A. a < length xs + length ys}"   and "length ys = card {a \<in> - A. a < length xs + length ys}" shows "nths (weave A xs ys) A = xs \<and> nths (weave A xs ys) (- A) = ys"
dom_llp: fixes x :: "'a"   and y :: "'a" shows "(x \<le> ad (ad y) \<cdot> x) = (ad (ad x) \<le> ad (ad y))"
univ_poly_is_monoid: shows "Group.monoid (K [X])"
assms_5: shows "m1 \<le>\<^sub>L 1\<^sub>m d1"
zero_less_Fract_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(0 < Fract a b) = (0 < a)"
inf_greatest: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e"   and z :: "'e" assumes "class.semilattice_inf inf less_eq less"   and "less_eq x y"   and "less_eq x z" shows "less_eq x (inf y z)"
not_numeral_le_neg_one: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less_eq (numeral.numeral one plus m) (uminus one)"
elims: fixes x :: "'c llist"   and y :: "('c, 'c llist) step" assumes "lstream x = y" obtains "x = LNil" and "y = Done"   | xa :: "'c" and xs :: "'c llist" where "x = LCons xa xs" and "y = Yield xa xs"
degree_mult_eq2: fixes p :: "'a poly"   and q :: "'a poly" assumes "lead_coeff p * lead_coeff q \<noteq> (0::'a)" shows "degree (p * q) = degree p + degree q"
distinct_6: fixes x4 :: "('a, 'b, 'c) expr"   and x1 :: "('a, 'b, 'c) val" shows "Ref x4 \<noteq> VE x1"
assms_2: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
repc_funcs: fixes I :: "interp"   and f :: "char"   and d :: "real" shows "Funcs (repc I f d) = Funcs I"
set_case_simps_3: fixes s :: "'a set"   and c_empty :: "'b"   and c_sing :: "'a \<Rightarrow> 'b"   and c_else :: "'b" assumes "1 < card s" shows "set_case s c_empty c_sing c_else = c_else"
up_injective: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.order less_eq less"   and "{y. less_eq x y} = {ya. less_eq y ya}" shows "x = y"
scale_left_imp_eq: fixes a :: "'a"   and x :: "'a"   and y :: "'a" assumes "a \<noteq> (0::'a)"   and "a * x = a * y" shows "x = y"
R_def: shows "R = (\<Sum>(p, i)\<in>{(p, i). prime p \<and> 1 < i \<and> p ^ i \<le> n}. ln (real p) * real (n div p ^ i))"
compositionality: fixes f :: "'d \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'e"   and h :: "'f \<Rightarrow> 'h"   and i :: "'i \<Rightarrow> 'g"   and "fun" :: "'h \<Rightarrow> 'i" shows "map_fun f g (map_fun h i fun) = map_fun (h \<circ> f) (g \<circ> i) fun"
env_delete_restr: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" shows "env_delete x m = m f|` (- {x})"
distinct_factorize_complex_main: fixes p :: "complex poly"   and fctrs :: "complex \<times> (complex \<times> nat) list" assumes "factorize_complex_main p = Some fctrs" shows "distinct (map fst (snd fctrs))"
abstract_primitive_preserves_nodisc_nedgated: fixes disc' :: "'b common_primitive \<Rightarrow> bool"   and neg :: "bool"   and m :: "'b common_primitive match_expr"   and disc :: "'b common_primitive negation_type \<Rightarrow> bool" assumes "\<not> has_disc_negated disc' neg m"   and "\<forall>str. \<not> disc' (Extra str)" shows "\<not> has_disc_negated disc' neg (abstract_primitive disc m)"
dghm_op_dghm_op_dghm: 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>" shows "op_dghm (op_dghm \<FF>) = \<FF>"
list_encode_0: fixes xs :: "nat list" assumes "Partial_Recursive.list_encode xs = 0" shows "xs = []"
simps_1: shows "sklvl1.domain Init = {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}"
ortho_involution: fixes x :: "'a" shows "- (- x) = x"
top_ccsubspace_code: shows "\<top> = (let n = length canonical_basis in SPAN (unit_vecs n))"
that: fixes a :: "'a"   and b :: "'b" assumes "p = (a, b)"   and "P a b" shows "thesis"
these_def: fixes A :: "'c option set" shows "Option.these A = the ` {x \<in> A. x \<noteq> \<bottom>}"
RedTI: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and ttas :: "('t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "rtrancl3p redT s ttas s'" shows "s -\<triangleright>ttas\<rightarrow>* s'"
reachable_weaken_TT: fixes s :: "'d"   and A :: "('d, 'e) automaton"   and I :: "'e \<Rightarrow> bool" assumes "s \<in> reachable A I" shows "s \<in> reachable A TT"
left_inverse: fixes a :: "'a" shows "- a + a = (0::'a)"
per': fixes x :: "real" shows "f (- (x + (- a - - b))) = f (- x)"
simps_1: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "insert_sort_by cmp [] = []"
find_cld_name_eq: fixes P :: "cld list"   and ctx :: "ctx"   and dcl :: "char list"   and ctxcld :: "ctx \<times> cld" assumes "find_cld_f P ctx (fqn_def dcl) = Some ctxcld" shows "\<exists>cl fds mds. (ctx, cld_def dcl cl fds mds) = ctxcld"
sin_le_zero': fixes x :: "real" assumes "- pi \<le> x"   and "x \<le> 0" shows "sin x \<le> 0"
id_take_nth_drop: fixes i :: "nat"   and xs :: "'e list" assumes "i < length xs" shows "xs = take i xs @ xs ! i # drop (Suc i) xs"
simps_35: fixes x61 :: "msg"   and x62 :: "msg"   and x3 :: "nat" shows "\<lbrace>x61, x62\<rbrace> \<noteq> Nonce x3"
nsubcls_D_C: shows "(D, C) \<notin> (subcls1 G)\<^sup>+"
density_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "density M f = measure_of (space M) (sets M) (\<lambda>A. set_nn_integral M A f)"
minus_one_cong_solve: fixes n :: "nat"   and x :: "nat"   and y :: "nat" assumes "1 < n"   and "[x = n - 1] (mod n)"   and "[x * y = 1] (mod n)"   and "coprime x n"   and "coprime y n" shows "[y = n - 1] (mod n)"
cipher_queriesE: fixes x :: "'cipher"   and ose :: "'b \<Rightarrow> ('ekey \<times> 'dkey \<times> 'cipher list) option" assumes "x \<in> cipher_queries ose" obtains n :: "'b" and ek :: "'ekey" and dk :: "'dkey" and ciphers :: "'cipher list" where "ose n = Some (ek, dk, ciphers)" and "x \<in> set ciphers"
to_vec_matrix_scalar_mult: fixes x :: "'c"   and A :: "('c, 'd) sq_matrix" shows "to_vec (x *\<^sub>S A) = x *k to_vec A"
tree0Eq: fixes y :: "'b"   and subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b" shows "((0, y) \<in> tree subs gamma) = (y = gamma)"
memory_write_low_equal_none2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and bm :: "word4"   and data :: "word32" assumes "low_equal s1 s2"   and "memory_write 10 addr bm data s2 = None" shows "memory_write 10 addr bm data s1 = None"
hyps: fixes tp :: "(action \<times> nat) list" assumes "tp = concat (take as (tms_of ap))"   and "as \<le> length ap" shows "start_of (layout_of ap) as = Suc (length tp div 2)"
Red1Cast: fixes s :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and U :: "ty"   and P :: "'addr J1_prog"   and T :: "ty"   and uf :: "bool"   and t :: "'thread_id" assumes "typeof\<^bsub>hp s\<^esub> v = \<lfloor>U\<rfloor>"   and "P \<turnstile> U \<le> T" shows "uf,P,t \<turnstile>1 \<langle>Cast T (Val v),s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val v,s\<rangle>"
forward_ix_drop_one: fixes o :: "'t"   and os :: "'t list"   and s :: "'s"   and t :: "'s"   and n :: "nat" shows "forward_ix (o # os) s t (n + 1) = forward_ix os s t n"
set_filter_op_list_mem_vs: fixes v :: "'c"   and vs :: "'c list"   and ops :: "'b list" shows "set (filter (\<lambda>op. ListMem v vs) ops) = {op \<in> set ops. v \<in> set vs}"
OrderingSetMap_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less_eq' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "OrderingSetMap less_eq less less_eq' less' P f \<equiv> ordering less_eq less \<and> ordering less_eq' less' \<and> OrderingSetMap_axioms less_eq less_eq' P f"
Ord_in_Ord: fixes k :: "V"   and m :: "V" assumes "Ord k"   and "m \<in>\<^sub>\<circ> k" shows "Ord m"
distinct_set_conv_list: fixes xs :: "'c list"   and g :: "'c \<Rightarrow> 'd" assumes "distinct xs" shows "sum g (set xs) = sum_list (map g xs)"
Class_unit_normal_subgroup: shows "L.Class \<one> = L"
Sup_empty: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
finite_lattice_complete_bot_least: fixes x :: "'a" shows "bot \<le> x"
awalk_conv: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and p :: "'d list"   and v :: "'c" assumes "wf_digraph G" shows "pre_digraph.awalk G u p v = (set (pre_digraph.awalk_verts G u p) \<subseteq> verts G \<and> set p \<subseteq> arcs G \<and> hd (pre_digraph.awalk_verts G u p) = u \<and> last (pre_digraph.awalk_verts G u p) = v \<and> pre_digraph.cas G u p v)"
imp_recovery2: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" assumes "A \<turnstile> p .\<longrightarrow>. q"   and "A \<turnstile> (q .\<longrightarrow>. p) .\<longrightarrow>. p" shows "A \<turnstile> q"
trans_opar_comp: fixes s :: "((nat \<Rightarrow> 'c) \<times> 'd, 'f seq_action) automaton"   and i :: "nat"   and t :: "('e, 'f seq_action) automaton" shows "automaton.trans (s \<langle>\<langle>\<^bsub>i\<^esub> t) = oparp_sos i (automaton.trans s) (automaton.trans t)"
eqr: shows "flow0 (flow0 y t) ` {0..t' - t} = flow0 y ` {t..t'}"
EM_remainder'_combine: fixes a :: "real"   and c :: "real"   and f :: "real \<Rightarrow> 'a"   and b :: "real"   and n :: "nat" assumes "continuous_on {a..c} f"   and "a \<le> b"   and "b \<le> c" shows "EM_remainder' n f a b + EM_remainder' n f b c = EM_remainder' n f a c"
hyps_2: shows "M' = (\<lambda>i' j'. if i' = i \<and> j' = j then Le d else if i' = j \<and> j' = i then Le (- d) else M i' j')"