hcju/isabelleps huggingface dataset:
formula_signs_induct: fixes P :: "formula \<Rightarrow> bool"   and A :: "formula" assumes "\<forall>p vs. P (FAtom Pos p vs)"   and "\<forall>p vs. P (FAtom Neg p vs)"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (FConj Pos A B)"   and "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P (FConj Neg A B)"   and "\<And>A. P A \<Longrightarrow> P (FAll Pos A)"   and "\<And>A. P A \<Longrightarrow> P (FAll Neg A)" shows "P A"
smult_monom: fixes a :: "'c"   and b :: "'c"   and n :: "nat" shows "smult a (monom b n) = monom (a * b) n"
deg_diff_by_const': fixes g :: "nat \<Rightarrow> 'a"   and a :: "'a"   and h :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier (UP R)"   and "a \<in> carrier R"   and "h = g \<ominus>\<^bsub>UP R\<^esub> monom (UP R) a 0" shows "deg R g = deg R h"
quasisym_not_less: fixes cmp :: "'b comparator"   and a :: "'b"   and b :: "'b" assumes "compare cmp a b \<noteq> Less"   and "compare cmp b a \<noteq> Less" shows "compare cmp a b = Equiv"
function_ring_defs_5: shows "\<zero>\<^bsub>function_ring (carrier R) R\<^esub> = function_zero (carrier R) R"
simps_2: fixes ik :: "msg set"   and secret :: "msg set"   and progress :: "fid_t \<Rightarrow> Runs.var set option"   and signalsInit :: "signal \<Rightarrow> nat"   and signalsResp :: "signal \<Rightarrow> nat"   and signalsInit2 :: "signal \<Rightarrow> nat"   and signalsResp2 :: "signal \<Rightarrow> nat"   and chan :: "chan set"   and bad :: "agent set"   and more :: "'a" shows "l2_state.bad \<lparr>ik = ik, secret = secret, progress = progress, signalsInit = signalsInit, signalsResp = signalsResp, signalsInit2 = signalsInit2, signalsResp2 = signalsResp2, chan = chan, bad = bad, \<dots> = more\<rparr> = bad"
that: assumes "z \<in> inside (path_image c1 \<union> path_image c2)"   and "winding_number (c1 +++ reversepath c2) z = d" shows "thesis"
trans_rtrancl: fixes r :: "('b \<times> 'b) set" shows "trans (r\<^sup>*)"
appr_rel_def: shows "appr_rel = appr_rell O \<langle>lv_rel\<rangle>set_rel"
le_add_same_cancel2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
w1: fixes oif :: "char list" shows "(OF_priority_match OF_match_fields_safe oft p = Action [Forward oif]) = (simple_linux_router_nol12 rt fw p = Some (p\<lparr>p_oiface := oif\<rparr>))"
hom_mult: fixes x :: "'c poly"   and y :: "'c poly" shows "map_poly to_fract (x * y) = map_poly to_fract x * map_poly to_fract y"
exhaust: fixes y :: "'d ifex" obtains "y = Trueif"   | "y = Falseif"   | x31 :: "'d" and x32 :: "'d ifex" and x33 :: "'d ifex" where "y = IF x31 x32 x33"
len_lt_eq: shows "length (pivot_positions (gauss_jordan_single (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))\<^sup>T)) \<le> dim_col (reduce_mat_cols (M_mat_R signs subsets) (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))"
at_ds7: fixes pi :: "('a \<times> 'a) list" assumes "at TYPE('a)" shows " (rev pi @ pi) \<triangleq> [] "
InterE: fixes A :: "'a"   and C :: "'a set set"   and X :: "'a set" assumes "A \<in> \<Inter> C" obtains "X \<notin> C"   | "A \<in> X"
adjacent_append_atomics_reusables: fixes w :: "nat \<times> nat"   and as :: "'b list"   and t :: "'b tree" assumes "0 < fst w"   and "fst w \<le> snd w"   and "valid as t"   and "l t \<le> fst w"   and "r t \<le> snd w" shows "adjacent w (atomics as (max (fst w) (Suc (r t))) (snd w) @ reusables t w)"
powers_distl: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> qstar y = (\<Squnion>i. x \<cdot> y ^ i)"
e: fixes q :: "real \<Rightarrow> complex" assumes "path q"   and "pathfinish q = pathstart q"   and "\<forall>t\<in>{0..1}. cmod (q t - h t) < e" shows "homotopic_loops (- {z}) h q"
fps_binomial_1: shows "fps_binomial (1::'a) = 1 + fps_X"
unit_factor_field_def: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.field divide inverse times one plus zero minus uminus" shows "field.unit_factor_field x = x"
perm_1: fixes pi :: "(name \<times> name) list"   and x1 :: "name" shows "pi \<bullet> lPar x1 = lPar (pi \<bullet> x1)"
in_remove_fset: fixes y :: "'b"   and S :: "'b fset"   and x :: "'b" shows "(x |\<in>| remove_fset y S) = (x |\<in>| S \<and> x \<noteq> y)"
elims_2: fixes x :: "'d list set"   and xa :: "('d, 'e, 'f) FSM"   and xb :: "('d, 'e, 'f) FSM"   and xc :: "('d \<times> 'e) list"   and xd :: "('d \<times> 'e) list" assumes "minimal_sequence_to_failure_extending x xa xb xc xd" obtains V :: "'d list set" and M1 :: "('d, 'e, 'f) FSM" and M2 :: "('d, 'e, 'f) FSM" and v' :: "('d \<times> 'e) list" and io :: "('d \<times> 'e) list" where "x = V" and "xa = M1" and "xb = M2" and "xc = v'" and "xd = io" and "v' \<in> L\<^sub>i\<^sub>n M1 V \<and> sequence_to_failure M1 M2 (v' @ io) \<and> (\<nexists>io'. \<exists>w'\<in>L\<^sub>i\<^sub>n M1 V. sequence_to_failure M1 M2 (w' @ io') \<and> length io' < length io)"
lt_minus_lessI: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p - q \<noteq> 0"   and "lpp q = lpp p"   and "lcf q = lcf p" shows "lpp (p - q) \<prec> lpp p"
on_graph_comp: fixes G :: "('g, 'e) labeled_graph"   and f :: "'h \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'h" shows "on_graph G (f \<circ> g) = on_graph G g O on_graph (map_graph_fn G g) f"
P'eqP: shows "P' \<equiv>\<^sub>e P"
subst_atm_mset_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and AA :: "'b multiset"   and \<sigma> :: "'c" shows "substitution_ops.subst_atm_mset subst_atm AA \<sigma> = {#subst_atm A \<sigma>. A \<in># AA#}"
homeomorphic_subspace_Euclidean_space_dim: fixes S :: "'a set" assumes "subspace S" shows "top_of_set S homeomorphic_space Euclidean_space (dim S)"
dfree_Times_simps: fixes a :: "('a, 'b) trm"   and b :: "('a, 'b) trm" shows "dfree (Times a b) = (dfree a \<and> dfree b)"
inject: fixes x1 :: "bool"   and x2 :: "'b \<Rightarrow> 'b trie option"   and y1 :: "bool"   and y2 :: "'b \<Rightarrow> 'b trie option" shows "(Nd x1 x2 = Nd y1 y2) = (x1 = y1 \<and> x2 = y2)"
Lang_term: fixes w :: "'c list"   and G :: "('d \<times> ('d, 'c) RHS) list"   and S :: "'d" assumes "w \<in> Lang G S" shows "\<forall>x\<in>set w. \<exists>N. (N, Leaf x) \<in> set G"
swapAbs_substAbs: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var"   and ys :: "'varSort"   and y :: "'var" assumes "goodAbs A"   and "good Y" shows "A $[Y / y]_ys $[z1 \<and> z2]_zs = A $[z1 \<and> z2]_zs $[Y #[z1 \<and> z2]_zs / y @ys[z1 \<and> z2]_zs]_ys"
comp: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" shows "map_option f \<circ> map_option g = map_option (f \<circ> g)"
collect_clkt_def: fixes S :: "('f \<times> ('d, 'e) cconstraint \<times> 'g \<times> 'd list \<times> 'f) set" shows "collect_clkt S = \<Union> {collect_clock_pairs (fst (snd t)) |t. t \<in> S}"
cod_comp: fixes g :: "'a"   and f :: "'a" assumes "B.seq g f" shows "B.cod (g \<cdot>\<^sub>B f) = B.cod g"
cscalar_prod_is_scalar_prod: fixes x :: "'a vec"   and y :: "'a vec" shows "x \<bullet>c y = x \<bullet> y"
fstI: fixes x :: "'c \<times> 'd"   and y :: "'c"   and z :: "'d" assumes "x = (y, z)" shows "fst x = y"
axioms_1: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and index :: "int"   and u_block_size :: "int" assumes "k_\<Lambda>_PBD point_set block_collection index u_block_size" shows "pairwise_balance point_set block_collection index"
poly_eq_fold_coeffs: fixes p :: "'a poly" shows "poly p = fold_coeffs (\<lambda>a f x. a + x * f x) p (\<lambda>x. 0::'a)"
intro: fixes cl :: "'b potype"   and f :: "'b \<Rightarrow> 'b" assumes "(cl, f) \<in> CLF_set" shows "CLF cl f"
Pd_singletons_for_ds_linear: fixes X :: "'a set"   and ds :: "'b set"   and d :: "'b" assumes "inj_on Xd X" shows "Linear_order (Pd_singletons_for_ds X ds d)"
invI_2: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "x \<otimes> y = \<one>" shows "x = inv y"
pure_filter_def: fixes x :: "'d" shows "pure_filter x = principal {x}"
fun_valuate_def: fixes lp :: "nat \<Rightarrow> rat"   and val :: "nat \<Rightarrow> 'a" shows "fun_valuate lp val = (\<Sum>x\<in>{v. lp v \<noteq> 0}. lp x *R val x)"
rel_Grp: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i" shows "rel_pmf (BNF_Def.Grp A f) = BNF_Def.Grp {x. set_pmf x \<subseteq> A} (map_pmf f)"
coherent_Assoc_Ide: fixes a :: "'c local.term"   and b :: "'c local.term"   and c :: "'c local.term" assumes "Ide a"   and "Ide b"   and "Ide c" shows "coherent \<^bold>\<a>\<^bold>[a, b, c\<^bold>]"
1: fixes h :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "h \<in> hom_components p" shows "P h (q0 h)"
HPhase1or2ReadThen_HInv5_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv5_inner s q"   and "HInv4a s p"   and "p \<noteq> q" shows "HInv5_inner s' q"
word_arith_nat_mod: fixes a :: "'b word"   and b :: "'b word" shows "a mod b = word_of_nat (unat a mod unat b)"
\<psi>: shows "ChamberComplexIsomorphism A (CoxeterComplex.TheComplex S) \<psi>"
bv_chop_append: fixes w1 :: "bit list"   and w2 :: "bit list" shows "bv_chop (w1 @ w2) (length w2) = (w1, w2)"
assms_2: shows "cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
pfun_of_inverse: fixes x :: "('a, 'b) ffun" shows "ffun.Abs_pfun (pfun_of x) = x"
perm_simps_7: fixes p :: "perm"   and term1 :: "Syntax.term"   and term2 :: "Syntax.term" shows "p \<bullet> Syntax.Pair term1 term2 = Syntax.Pair (p \<bullet> term1) (p \<bullet> term2)"
sw_involutive: fixes xs :: "'c"   and x :: "'b"   and y :: "'b"   and zs :: "'c"   and z :: "'b" shows "z @zs[x \<and> y]_xs @zs[x \<and> y]_xs = z"
genby_0_closed: fixes S :: "'a set" shows "(0::'a) \<in> \<langle>S\<rangle>"
typ_of1_decr': fixes t :: "Core.term"   and T :: "typ"   and Ts :: "typ list"   and \<tau> :: "typ" assumes "\<not> loose_bvar1 t 0"   and "typ_of1 (T # Ts) t = Some \<tau>" shows "typ_of1 Ts (decr 0 t) = Some \<tau>"
cases: fixes x :: "'b formula" obtains "x = \<bottom>"   | "x = \<^bold>\<not> \<bottom>"   | uu_ :: "'b" where "x = Atom uu_"   | uv_ :: "'b" where "x = \<^bold>\<not> (Atom uv_)"   | va :: "'b formula" where "x = \<^bold>\<not> (\<^bold>\<not> va__)"   | va :: "'b formula" and vb :: "'b formula" where "x = \<^bold>\<not> (vaa__ \<^bold>\<and> vb__)"   | va :: "'b formula" and vb :: "'b formula" where "x = \<^bold>\<not> (vab__ \<^bold>\<or> vba__)"   | va :: "'b formula" and vb :: "'b formula" where "x = \<^bold>\<not> (vac__ \<^bold>\<rightarrow> vb)"   | v :: "'b formula" and va :: "'b formula" where "x = v__ \<^bold>\<and> vad__"   | v :: "'b formula" and va :: "'b formula" where "x = vc__ \<^bold>\<or> vae__"   | v :: "'b formula" and va :: "'b formula" where "x = v \<^bold>\<rightarrow> va"
lead_coeff_1: shows "lead_coeff 1 = (1::'b)"
simps_2: fixes v :: "'a"   and va :: "'a"   and l :: "'a set list" shows "in_list (DenyAllFromTo v va) l = (bothNet (DenyAllFromTo v va) \<in> set l)"
other_falsified: fixes C\<^sub>1' :: "fterm literal set"   and B :: "bool list"   and d :: "bool"   and l :: "fterm literal"   and lo :: "fterm literal" assumes "ground\<^sub>l\<^sub>s C\<^sub>1' \<and> falsifies\<^sub>g (B @ [d]) C\<^sub>1'"   and "l \<in> C\<^sub>1'"   and "nat_of_fatom (get_atom l) = length B"   and "lo \<in> C\<^sub>1'"   and "lo \<noteq> l" shows "falsifies\<^sub>l B lo"
\<sigma>_ineqs_fv_dom: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X \<subseteq> subst_domain \<sigma>"
theorem10: fixes g :: "'c \<rightarrow> 'd set"   and r :: "'c \<rightarrow> 'c" shows "(\<mu> f. \<Lambda> x. g\<cdot>x \<union> f\<cdot>(r\<cdot>x)) = (\<Lambda> x. \<Union>i. g\<cdot>(r\<^bsup>i\<^esup>\<cdot>x))"
XC_in_sets: fixes c :: "nat list" shows "XC c \<in> events"
iso_iff_group_isomorphisms: fixes G :: "('c, 'd) monoid_scheme"   and f :: "'c \<Rightarrow> 'e"   and H :: "('e, 'f) monoid_scheme" assumes "Group.group G" shows "(f \<in> Group.iso G H) = (\<exists>g. group_isomorphisms G H f g)"
simps: fixes fstate :: "(d \<times> d list \<times> ((nat \<times> char list) \<times> contour \<Rightarrow> d option) \<times> contour) discr" shows "\<F>\<cdot>fstate = (case undiscr fstate of (DI int, ba) \<Rightarrow> \<bottom> | (DC (Lambda lab vs c, \<beta>), as, ve, b) \<Rightarrow> if length vs = length as then let \<beta>' = \<beta>(lab \<mapsto> b); ve' = ve(map (\<lambda>v. (v, b)) vs [\<mapsto>] as) in \<C>\<cdot>(Discr (c, \<beta>', ve', b)) else \<bottom> | (DP (Plus c), [], bb) \<Rightarrow> \<bottom> | (DP (Plus c), [DI a1], bb) \<Rightarrow> \<bottom> | (DP (Plus c), [DI a1, DI a2], bb) \<Rightarrow> \<bottom> | (DP (Plus c), [DI a1, DI a2, cnt], ve, b) \<Rightarrow> if isProc cnt then let b' = nb b c; \<beta> = [c \<mapsto> b] in \<F>\<cdot>(Discr (cnt, [DI (a1 + a2)], ve, b')) \<union> {((c, \<beta>), cnt)} else \<bottom> | (DP (Plus c), DI a1 # DI a2 # cnt # aa # lista, bb) \<Rightarrow> \<bottom> | (DP (Plus c), DI a1 # _ # lista, bb) \<Rightarrow> \<bottom> | (DP (Plus c), _ # list, bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [], bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [DI v], bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [DI v, contt], bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), [DI v, contt, contf], ve, b) \<Rightarrow> if isProc contt \<and> isProc contf then if v \<noteq> 0 then let b' = nb b ct; \<beta> = [ct \<mapsto> b] in \<F>\<cdot>(Discr (contt, [], ve, b')) \<union> {((ct, \<beta>), contt)} else let b' = nb b cf; \<beta> = [cf \<mapsto> b] in \<F>\<cdot>(Discr (contf, [], ve, b')) \<union> {((cf, \<beta>), contf)} else \<bottom> | (DP (prim.If ct cf), DI v # contt # contf # ab # lista, bb) \<Rightarrow> \<bottom> | (DP (prim.If ct cf), _ # list, bb) \<Rightarrow> \<bottom> | (Stop, [], bb) \<Rightarrow> \<bottom> | (Stop, [DI i], x, xa) \<Rightarrow> {} | (Stop, DI i # a # lista, bb) \<Rightarrow> \<bottom> | (Stop, _ # list, bb) \<Rightarrow> \<bottom>)"
neq_0_iff': fixes xs :: "nat list"   and n :: "nat" shows "(xs \<noteq> zeroes n) = (length xs \<noteq> n \<or> (\<exists>x\<in>set xs. 0 < x))"
simps: fixes i :: "nat" shows "gs1.gso i = map of_int_hom.vec_hom fs ! i + gs.M.sumlist (map (\<lambda>j. - gs1.\<mu> i j \<cdot>\<^sub>v gs1.gso j) [0..<i])"
prems: shows "(x, y) \<in> r ^^ Suc n"
br_sv: fixes \<alpha> :: "'a \<Rightarrow> 'c"   and I :: "'a \<Rightarrow> bool" shows "single_valued (br \<alpha> I)"
Lt_Lt_dbm_lt_D: fixes a :: "'d"   and b :: "'d" assumes "Lt a \<prec> Lt b" shows "a < b"
dg_Par_incl_Par_is_arr: fixes A :: "V"   and B :: "V" assumes "A \<in>\<^sub>\<circ> Vset \<alpha>"   and "B \<in>\<^sub>\<circ> Vset \<alpha>"   and "A \<subseteq>\<^sub>\<circ> B" shows "incl_Rel A B : A \<mapsto>\<^bsub>dg_Par \<alpha>\<^esub> B"
test_iso: fixes p :: "'a"   and q :: "'a" assumes "p \<le> q" shows "\<iota> p \<le> \<iota> q"
s_UnrollRoll: fixes v :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and vb :: "mode" assumes "value v" shows "\<lless>\<pi>, Unroll (Roll v)\<ggreater> vb\<rightarrow> \<lless>\<pi>, v\<ggreater>"
star_class_defs_11: shows "abs \<equiv> *f* abs"
disjoint_intervals_def: fixes A :: "'b word \<times> 'b word"   and B :: "'b word \<times> 'b word" shows "disjoint_intervals A B \<equiv> disjoint (interval_of A) (interval_of B)"
stock_portfolio_def: fixes Mkt :: "('b, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real" shows "stock_portfolio Mkt p = (portfolio p \<and> support_set p \<subseteq> stocks Mkt)"
fmfilter_alt_defs_4: fixes A :: "'a set" shows "fmrestrict_set A = fmfilter (\<lambda>a. a \<in> A)"
start_addrs_def: fixes empty_heap :: "'b"   and allocate :: "'b \<Rightarrow> htype \<Rightarrow> ('b \<times> 'a) set" shows "heap_base.start_addrs empty_heap allocate = fst (snd (heap_base.start_heap_data empty_heap allocate))"
not_below2not_eq: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.po below"   and "\<not> below x y" shows "x \<noteq> y"
nodes_it_is_iterator: fixes g :: "'G" assumes "invar g" shows "set_iterator (nodes_it g) (nodes (\<alpha> g))"
assms: shows "order.greater_eq (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) (set ss)"
surjec_def: fixes F :: "('h, 'i) aGroup_scheme"   and G :: "('j, 'k) aGroup_scheme"   and f :: "'h \<Rightarrow> 'j" shows "surjec\<^bsub>F,G\<^esub> f = (f \<in> aHom F G \<and> surj_to f (carrier F) (carrier G))"
funset_eq_UN_fun_upd_I: fixes F :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) set"   and a :: "'a"   and A :: "'a set"   and d :: "'b"   and G :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b set" assumes "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f(a := d) \<in> F A"   and "\<And>f. f \<in> F (insert a A) \<Longrightarrow> f a \<in> G (f(a := d))"   and "\<And>f x. \<lbrakk>f \<in> F A; x \<in> G f\<rbrakk> \<Longrightarrow> f(a := x) \<in> F (insert a A)" shows "F (insert a A) = (\<Union>f\<in>F A. fun_upd f a ` G f)"
i_append_Nil: fixes f :: "nat \<Rightarrow> 'd" shows "[] \<frown> f = f"
diff_list_set: fixes xs :: "'a list"   and ys :: "'a list" shows "set (diff_list xs ys) = set xs - set ys"
close_eqvt: fixes pi :: "(W.tvar \<times> W.tvar) list"   and T :: "ty"   and \<Gamma> :: "(W.var \<times> tyS) list" shows "pi \<bullet> close \<Gamma> T = close (pi \<bullet> \<Gamma>) (pi \<bullet> T)"
inf: shows "infinite {n. 0 \<le> birkhoff_sum f n x}"
big_trans: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)"   and "g \<in> O[F](h)" shows "f \<in> O[F](h)"
PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m_1: fixes N :: "nat"   and ps :: "nat list" assumes "DiningPhilosophers N"   and "0 < length ps" shows "DiningPhilosophers.PHILs\<^sub>n\<^sub>o\<^sub>r\<^sub>m N 1 ps = DiningPhilosophers.LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m N (ps ! 0)"
bdd_all_bdd_lookup: fixes ws :: "bool list"   and bdd :: "'a bdd"   and P :: "'a \<Rightarrow> bool" assumes "bddh (length ws) bdd"   and "bdd_all P bdd" shows "P (bdd_lookup bdd ws)"
R27_1: shows "pmf (sds R27) a = 1 / 2"
glb_unique: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b"   and B :: "'b set" assumes "antisymp_on P A"   and "x \<in> A"   and "y \<in> A"   and "glb P B x"   and "glb P B y" shows "x = y"
sum_norm_bound: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and S :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and K :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> K" shows "norm (comm_monoid_add.sum plus zero f S) \<le> real (card S) * K"
closedin_retract: fixes S :: "'a set"   and T :: "'a set" assumes "S retract_of T" shows "closedin (top_of_set T) S"
abs_triangle_ineq2: fixes abs :: "'a \<Rightarrow> 'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (minus (abs a) (abs b)) (abs (minus a b))"
poly_div_minus_right: fixes x :: "'b poly"   and y :: "'b poly" shows "x div - y = - (x div y)"
prod_succs_def: fixes A :: "nat bdd list \<times> bool list"   and B :: "nat bdd list \<times> bool list" shows "prod_succs A B = (\<lambda>(i, j). add_leaves (bdd_binop Pair (fst A ! i) (fst B ! j)) [])"
unreachable: fixes \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" shows "\<C> \<turnstile> [Unreachable] : ts _> ts'"
axioms_1: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F_g :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F_g" shows "standard_lifting Inf_F Bot_G Inf_G entails_G Red_I_G Red_F_G Bot_F \<G>_F \<G>_I"
borel_measurable_iff_greater: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure" shows "(f \<in> borel_measurable M) = (\<forall>a. {w \<in> space M. a < f w} \<in> sets M)"
hoare_triple_while: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat" assumes "grover_state_sig n f dims R K" shows "state_sig.hoare_partial dims (adjoint (grover_state_sig.exM0 n dims R K) * grover_state_sig.P' n f dims R K * grover_state_sig.exM0 n dims R K + adjoint (grover_state_sig.exM1 n dims R K) * grover_state_sig.Q n f dims R K * grover_state_sig.exM1 n dims R K) (state_sig.While_P dims (grover_state_sig.vars2 n) (grover_state_sig.M0 R K) (grover_state_sig.M1 R K) (grover_state_sig.D n f dims K)) (grover_state_sig.P' n f dims R K)"
wf\<^sub>s\<^sub>t\<^sub>s'_update\<^sub>s\<^sub>t_eq: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" assumes "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"   and "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "wf\<^sub>s\<^sub>t\<^sub>s' (update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)) (\<A> @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
proves_ascend_abs_rule': fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and x :: "variable"   and \<tau>' :: "typ"   and \<Gamma> :: "Core.term set"   and s :: "Core.term"   and t :: "Core.term" assumes "wf_theory \<Theta>"   and "(x, \<tau>') \<notin> FV \<Gamma>"   and "(x, \<tau>') \<notin> fv (mk_eq (Abs \<tau>' s) (Abs \<tau>' t))"   and "typ_ok \<Theta> \<tau>'"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq (subst_bv (Fv x \<tau>') s) (subst_bv (Fv x \<tau>') t)"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (Abs \<tau>' s) (Abs \<tau>' t)"
assms_2: shows "Domain a \<inter> X \<noteq> {}"
mult_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "pn.eq_m (f * pn.Mp g) (f * g)"