hcju/isabelleps huggingface dataset:
valid_ConsE: fixes trn :: "('state, 'act, 'out) trans"   and tr :: "('state, 'act, 'out) trans list" assumes "valid (trn # tr)" obtains "tr = []" and "validTrans trn"   | "validTrans trn" and "tgtOf trn = srcOf (hd tr)" and "valid tr"
elims: fixes x :: "'a list"   and y :: "'a list" assumes "vwalk_to_vpath x = y" obtains "x = []" and "y = []"   | xa :: "'a" and xs :: "'a list" where "x = xa # xs" and "y = (if xa \<in> set xs then vwalk_to_vpath (dropWhile (\<lambda>y. y \<noteq> xa) xs) else xa # vwalk_to_vpath xs)"
mod_2: fixes x :: "nat" shows "is_even x \<or> x mod 2 = Suc 0"
tail_arev: fixes a :: "'a \<times> 'a" assumes "a \<in> arcs (with_proj G)" shows "tail (with_proj G) (edge_rev M a) = head (with_proj G) a"
decisive_def: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and C :: "'a set"   and x :: "'b"   and y :: "'b" shows "decisive scf A Is C x y \<equiv> C \<subseteq> Is \<and> (\<forall>P. profile A Is P \<and> (\<forall>i\<in>C. x \<^bsub>(P i)\<^esub>\<prec> y) \<longrightarrow> x \<^bsub>(scf P)\<^esub>\<prec> y)"
Trans: fixes H :: "fmla set"   and x :: "trm"   and y :: "trm"   and z :: "trm" assumes "H \<turnstile> x EQ y"   and "H \<turnstile> y EQ z" shows "H \<turnstile> x EQ z"
assms_1: shows "g \<in> F n \<rightarrow>\<^sub>M N"
pseudo_ordered_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "pseudo_ordered_set B (\<lambda>x y. y \<sqsubseteq> x)"
A_2: shows "fst gh' \<in> G \<and> snd gh' \<in> H \<and> x = fst gh' + snd gh'"
intros_2: fixes G :: "('c, 'd) ParityGame_scheme"   and v :: "'c"   and p :: "Player"   and W :: "'c set" assumes "ParityGame G"   and "v \<in> (if p = Even then V0\<^bsub>G\<^esub> else V\<^bsub>G\<^esub> - V0\<^bsub>G\<^esub>)"   and "\<exists>w. v \<rightarrow>\<^bsub>G\<^esub> w \<and> w \<in> ParityGame.attractor_inductive G p W" shows "v \<in> ParityGame.attractor_inductive G p W"
sort_oalist_def: fixes x :: "'o"   and xs :: "('a \<times> 'c) list" shows "sort_oalist x xs = foldr (update_by_pair x) xs []"
lcm_list_least: 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 bs :: "'b list"   and a :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "\<And>b. b \<in> set bs \<Longrightarrow> dvd.dvd times b a" shows "dvd.dvd times (semiring_gcd.Lcm_fin one zero lcm (set bs)) a"
nvo: shows "non_volatile_owned_or_read_only pending_write \<S> \<O> sb"
measure_def: fixes r :: "('a \<times> 'a) set"   and P :: "('b \<times> ('a \<times> 'b) list) \<times> 'b \<times> ('a \<times> 'b) list" shows "Decreasing_Diagrams.measure r P = r|labels (fst P)| + r|labels (snd P)|"
Tarski_2D_axioms: shows "Tarski_2D Bet Cong"
bi_total_alt_def: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
listE_nth_in: fixes xs :: "'b list"   and n :: "nat"   and A :: "'b set"   and i :: "nat" assumes "xs \<in> list n A"   and "i < n" shows "xs ! i \<in> A"
uminus_in_float_intervalI: fixes a :: "real"   and A :: "float interval" assumes "a \<in>\<^sub>r A" shows "- a \<in>\<^sub>r - A"
assms_2: assumes "\<And>y. y \<in>\<^sub>\<circ> A \<Longrightarrow> a \<in>\<^sub>\<circ> y" shows "P"
poly_mult_coeff: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "local.coeff (poly_mult p1 p2) = (\<lambda>i. \<Oplus>k\<in>{..i}. local.coeff p1 k \<otimes> local.coeff p2 (i - k))"
final: shows "abc_final (abc_steps_l (0, lm) p (Suc n)) p"
initmissing_oreachable_netmask: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and onp :: "nat \<Rightarrow> ((nat \<Rightarrow> 'e) \<times> 'f, 'd seq_action) automaton"   and sr :: "'c \<Rightarrow> 'e \<times> 'f"   and s :: "'c net_state"   and n :: "net_tree"   and U :: "(nat \<Rightarrow> 'e) \<Rightarrow> (nat \<Rightarrow> 'e) \<Rightarrow> bool" assumes "openproc np onp sr"   and "openproc.initmissing np sr (netgmap sr s) \<in> oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"   and "net_ips s = net_tree_ips n" shows "netgmap sr s \<in> netmask (net_tree_ips n) ` oreachable (oclosed (opnet onp n)) (\<lambda>_ _ _. True) U"
M_def: shows "M = (\<lambda>v. {(x, ecard Zs) |x Zs. Zs = {zs. length zs = b \<and> Formula.sat \<sigma> V (zs @ v) i \<phi> \<and> Formula.eval_trm (zs @ v) f = x} \<and> Zs \<noteq> {}})"
simps_5: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltl"   and \<psi> :: "'a ltl" shows "w \<Turnstile> \<phi> and \<psi> = (w \<Turnstile> \<phi> \<and> w \<Turnstile> \<psi>)"
compat_list: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and y :: "'a"   and x :: "'a"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and xs' :: "'a list" assumes "ext_compat_list_strong ext"   and "y \<noteq> x"   and "gt y x" shows "ext gt (xs @ y # xs') (xs @ x # xs')"
trans_L_Prec: fixes l1 :: "label"   and l2 :: "label"   and l3 :: "label" assumes "l1 \<sqsubset>l l2"   and "l2 \<sqsubset>l l3" shows "l1 \<sqsubset>l l3"
sat_slice_strong: fixes S :: "'b list set"   and \<phi> :: "'b formula"   and E :: "(char list \<times> 'b list) set"   and v :: "'b list"   and \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and i :: "nat" assumes "relevant_events \<phi> S \<subseteq> E"   and "v \<in> S" shows "sat \<sigma> v i \<phi> = sat (map_\<Gamma> (\<lambda>D. D \<inter> E) \<sigma>) v i \<phi>"
disjoint_witness_None: fixes s1 :: "('a, unit) HashMap.hashmap"   and s2 :: "('a, unit) HashMap.hashmap" assumes "hs.invar s1"   and "hs.invar s2" shows "(hh_set_xx.g_disjoint_witness s1 s2 = None) = (hs.\<alpha> s1 \<inter> hs.\<alpha> s2 = {})"
f2: shows "\<langle>|u\<rangle>| \<in> carrier_mat 1 (dim_vec v)"
open_non_path_component: fixes S :: "'a set"   and x :: "'a" assumes "open S" shows "open (S - path_component_set S x)"
f_Exec_one: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e"   and c :: "'d" shows "f_Exec_Comp trans_fun [m] c = trans_fun m c"
adjoint_pair_antipar_1: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.adjoint_pair f g" shows "Span.ide f"
mcalc_iff: fixes \<sigma> :: "'tp"   and c :: "('a, 'b) lit list" shows "(\<sigma> \<turnstile> c) = (infTp \<sigma> \<or> (\<forall>x\<in>nvC c. TE.tpOfV x \<noteq> \<sigma>))"
in_countE: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" obtains n :: "nat" where "count M x = Suc n"
bigomega_f: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and f :: "nat \<Rightarrow> real"   and integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and A :: "real" assumes "akra_bazzi_lower x\<^sub>0 x\<^sub>1 k as bs ts f integrable integral g g'" obtains a :: "real" where "A \<le> a" and "f \<in> \<Omega>(\<lambda>x. real x powr akra_bazzi_exponent as bs * (1 + integral (\<lambda>u. g' u / u powr (akra_bazzi_exponent as bs + 1)) a (real x)))"
util_def: fixes carrier :: "'a set"   and relation :: "('a \<times> 'a) set"   and u :: "'a \<Rightarrow> real"   and x :: "'a"   and y :: "'a" assumes "ordinal_utility carrier relation u"   and "x \<in> carrier"   and "y \<in> carrier" shows "x \<succeq>[relation] y = (u y \<le> u x)"
divide_inverse: fixes a :: "'b"   and b :: "'b" shows "a / b = a * inverse b"
cod_simp: fixes f :: "'c \<times> 'c \<times> 'c" assumes "C.VVV.arr f" shows "C.VVV.cod f = C.VxVxV.cod f"
t_count_def: fixes x :: "'b"   and xt :: "'b bintree" shows "t_count x xt \<equiv> count (t_multiset xt) x"
ccw'_mirror_point: fixes x0 :: "real \<times> real"   and a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" assumes "ccw' (mirror_point x0 a) (mirror_point x0 b) (mirror_point x0 c)" shows "ccw' a b c"
nn_integral_restrict_space: fixes \<Omega> :: "'b set"   and M :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "\<Omega> \<inter> space M \<in> sets M" shows "integral\<^sup>N (restrict_space M \<Omega>) f = \<integral>\<^sup>+ x. f x * indicator \<Omega> x \<partial>M"
assms_3: fixes i :: "'c" assumes "i \<in> domain P" shows "dist ((P)\<^sub>F i) ((Q)\<^sub>F i) < e"
finfun_const_def: shows "finfun_const \<equiv> map_fun id Abs_finfun (\<lambda>b x. b)"
mkVarChannelProc_def: fixes v :: "procVarDecl"   and g :: "'b gState_scheme"   and p :: "pState" shows "mkVarChannelProc v g p = (let v' = case v of ProcVarDeclNum lb hb name siz init \<Rightarrow> VarDeclNum lb hb name siz init | ProcVarDeclChan name siz \<Rightarrow> VarDeclChan name siz None; (k, v, cs) = toVariable g p v' in mkVarChannel v' (apsnd \<circ> pState.vars_update) g p)"
compute_pdevs_apply: fixes x :: "(nat, 'd) slist"   and i :: "nat" shows "pdevs_apply (Pdevs x) i = slist_apply x i"
acyclicI: fixes r :: "('b \<times> 'b) set" assumes "\<forall>x. (x, x) \<notin> r\<^sup>+" shows "acyclic r"
cases: fixes a1 :: "'v"   and a2 :: "'v"   and a3 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" assumes "path a1 a2 a3" obtains v :: "'v" where "a1 = v" and "a2 = v" and "a3 = []"   | e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort" and v' :: "'v" and pth :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" where "a1 = edge_begin e" and "a2 = v'" and "a3 = e # pth" and "e \<in> edges" and "path (edge_end e) v' pth"
finite_G: shows "finite (carrier (G\<lparr>carrier := P\<rparr>))"
poly_rel_def: fixes ops :: "'b arith_ops_record"   and R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b list"   and x' :: "'c poly" assumes "ring_ops ops R" shows "ring_ops.poly_rel R x x' = list_all2 R x (coeffs x')"
cod_simps_3: shows "\<pi>.cod.leg1 = \<pi>.leg1"
map_upd_eqD1: fixes m :: "'f \<Rightarrow> 'g option"   and a :: "'f"   and x :: "'g"   and n :: "'f \<Rightarrow> 'g option"   and y :: "'g" assumes "m(a \<mapsto> x) = n(a \<mapsto> y)" shows "x = y"
hd_remdups_adj: fixes xs :: "'c list" shows "hd (remdups_adj xs) = hd xs"
smult_0_left: fixes p :: "'c poly" shows "Polynomial.smult (0::'c) p = 0"
add_le_cancel_right: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" shows "(a + c \<subseteq># b + c) = (a \<subseteq># b)"
eval_lowerFm_helper: fixes i :: "nat"   and F :: "atom fm"   and init :: "real list"   and xs :: "real list"   and x :: "real" assumes "freeIn i F"   and "length init = i" shows "eval (lowerFm i 1 F) (init @ xs) = eval F (init @ [x] @ xs)"
set_of_def: fixes S :: "('ma, 'mt) AlgState" shows "KBPAlg.set_of S \<equiv> {x. k_isNode x \<and> k_memb x S}"
b_App: fixes x :: "name"   and e\<^sub>1 :: "trm"   and e\<^sub>2 :: "trm"   and e' :: "trm"   and e :: "trm"   and e\<^sub>2' :: "trm" assumes "x \<sharp> (e\<^sub>1, e\<^sub>2, e')"   and "e\<^sub>1 \<Down> Lam [x].e"   and "e\<^sub>2 \<Down> e\<^sub>2'"   and "e[x::=e\<^sub>2'] \<Down> e'" shows "App e\<^sub>1 e\<^sub>2 \<Down> e'"
restrict0_const_diff_fun_space: fixes c :: "real" shows "restrict0 manifold_eucl.dest.carrier (\<lambda>x. c) \<in> manifold_eucl.dest.diff_fun_space"
skip_doesnt_access: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and x :: "'Var" assumes "lc \<in> loc_reach \<langle>Skip \<otimes> annos, mds, mem\<rangle>"   and "lc = \<langle>c', mds', mem'\<rangle>" shows "doesnt_read_or_modify c' x \<and> doesnt_modify c' x"
Image_singleton: fixes r :: "('e \<times> 'd) set"   and a :: "'e" shows "r `` {a} = {b. (a, b) \<in> r}"
orthonormal_system_def: fixes S :: "'a set"   and w :: "'b \<Rightarrow> 'a \<Rightarrow> real" shows "orthonormal_system S w \<equiv> \<forall>m n. l2product S (w m) (w n) = (if m = n then 1 else 0)"
unique_rootD_6: fixes plr :: "int poly \<times> rat \<times> rat"   and y :: "real" assumes "\<exists>!x. root_cond plr x" shows "(y = real_of_1 plr) = root_cond plr y"
add_nonpos_nonpos: fixes a :: "'c multiset"   and b :: "'c multiset" assumes "a \<subseteq># {#}"   and "b \<subseteq># {#}" shows "a + b \<subseteq># {#}"
sdpos: fixes x :: "'a" assumes "x \<in> U" shows "0 < setdist {x} S + setdist {x} T"
objI_trg: fixes a :: "'a" assumes "arr a"   and "trg a = a" shows "obj a"
i_drop_i_take_1: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'd" shows "f \<Up> n \<Down> Suc 0 = [f n]"
wf_idecl_all: fixes p :: "prog" assumes "p = tprg" shows "Ball (set Ifaces) (wf_idecl p)"
And_int_preservation: fixes n :: "nat"   and M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'b DBMEntry" assumes "dbm_int M1 n"   and "dbm_int M2 n" shows "dbm_int (And M1 M2) n"
Un_closed_segment: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<in> closed_segment a c" shows "closed_segment a b \<union> closed_segment b c = closed_segment a c"
assms_2: fixes b :: "'a" assumes "b \<noteq> (0::'a)"   and "a * b = (0::'a)" shows "thesis"
path_component_of_def: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "path_component_of X x y \<equiv> \<exists>g. pathin X g \<and> g 0 = x \<and> g 1 = y"
assert_True: fixes f :: "unit \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'c) set \<times> bool" shows "assert True >>= f = f ()"
measurable_snth_count_space: fixes n :: "nat" shows "Measurable.pred (bernoulli_stream p) (\<lambda>w. w !! n)"
uminus_pdevs_def: shows "uminus_pdevs = unop_pdevs uminus"
exec_impl_execs: fixes s :: "state"   and c :: "com"   and s' :: "state"   and cs :: "com list" assumes "s -c\<rightarrow> s'" shows "(c # cs, s) \<rightarrow>\<^sup>* (cs, s')"
aezfun_set_def: shows "aezfun_set = {f. finite (supp f)}"
PUadmitFO_def: fixes \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm"   and \<theta> :: "('c + 'b, 'e, 'd) hp"   and U :: "('d + 'd) set" shows "PUadmitFO \<sigma> \<theta> U = ((\<Union>i\<in>{i. Inl (Inr i) \<in> SIGP \<theta>}. FVT (\<sigma> i)) \<inter> U = {})"
3: shows "w =\<^sub>F w\<^sub>1 @ w\<^sub>2"
iso_in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>B b\<guillemotright>"   and "B.iso f" shows "B.iso_in_hom f a b"
transfer_rule_of_bool: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R (0::'a) (0::'b)"   and "R (1::'a) (1::'b)" shows "((=) ===> R) of_bool of_bool"
pathVertices_fwd_simps_1: fixes s :: "nat" shows "pathVertices_fwd s [] = [s]"
rfd_rbb_dual: fixes R :: "('d \<times> 'c) set" shows "\<partial> \<circ> fd\<^sub>\<R> R = bb\<^sub>\<R> (\<smile> R) \<circ> \<partial>"
subspace_dual_space: fixes S :: "'c set" assumes "subspace S" shows "subspace (dual_space S)"
pruned_def: fixes \<alpha>n :: "'c \<Rightarrow> 'd list"   and defs :: "'c \<Rightarrow> 'd \<Rightarrow> 'e set"   and "uses" :: "'c \<Rightarrow> 'd \<Rightarrow> 'e set"   and phis :: "'c \<Rightarrow> 'd \<times> 'e \<Rightarrow> 'e list option"   and g :: "'c" shows "CFG_SSA_wf_base.pruned \<alpha>n defs uses phis g = (\<forall>n\<in>set (\<alpha>n g). \<forall>val. val \<in> CFG_SSA_base.phiDefs phis g n \<longrightarrow> CFG_SSA_wf_base.liveVal \<alpha>n defs uses phis g val)"
iffs: fixes getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and more :: "'a"   and getF' :: "Literal list list"   and getM' :: "(Literal \<times> bool) list"   and more' :: "'a" shows "(\<lparr>getF = getF, getM = getM, \<dots> = more\<rparr> = \<lparr>getF = getF', getM = getM', \<dots> = more'\<rparr>) = (getF = getF' \<and> getM = getM' \<and> more = more')"
take_Suc: fixes n :: "nat"   and xs :: "'b list" shows "take (Suc n) xs = (if xs = [] then [] else hd xs # take n (tl xs))"
simps: fixes xss :: "'c llist llist" shows "lconcat xss = (case xss of LNil \<Rightarrow> LNil | LCons xs xss' \<Rightarrow> lappend xs (lconcat xss'))"
Hilbert_fun_cone_decomp: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and z :: "nat" assumes "cone_decomp T ps"   and "valid_decomp X ps"   and "hom_decomp ps" shows "Hilbert_fun T z = (\<Sum>hU\<in>set ps. Hilbert_fun (cone hU) z)"
is_equality_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "is_equality R = (R = (=))"
wellformed2_sorted: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "all_in_list p l"   and "distinct p"   and "allNetsDistinct p"   and "singleCombinators p" shows "wellformed_policy2Pr (FWNormalisationCore.sort p l)"
igWlsBinp_def: fixes MOD :: "('b, 'c, 'varSort, 'sort, 'd, 'e, 'f, 'g, 'h) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'g option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
finite_set_random_bst: fixes A :: "'a set" assumes "finite A" shows "finite (set_pmf (random_bst A))"
rel_return_pmf: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'c" shows "rel_pmf R (return_pmf x1) (return_pmf x2) = R x1 x2"
signed_of_int: fixes n :: "int" shows "signed (word_of_int n) = of_int (signed_take_bit (LENGTH('c) - Suc 0) n)"
w: fixes n :: "real" assumes "N \<le> n" shows "AE x in M. norm (s n x) \<le> w x"
call_return_node_edge: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and a :: "'a"   and a' :: "'a" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "valid_edge a"   and "a' \<in> get_return_edges a" shows "\<exists>a''. valid_edge a'' \<and> sourcenode a'' = sourcenode a \<and> targetnode a'' = targetnode a' \<and> kind a'' = (\<lambda>cf. False)\<^sub>\<surd>"
hcomp_assoc_isomorphic: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and g :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and h :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.ide g"   and "Span.ide h"   and "Span.src f = Span.trg g"   and "Span.src g = Span.trg h" shows "Span.isomorphic ((f \<circ> g) \<circ> h) (f \<circ> g \<circ> h)"
subclsS_subcls1: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> C \<prec>\<^sub>S D" shows "P \<turnstile> C \<prec>\<^sup>1 D"
sum_SDG_edge_valid_SDG_node_2: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and b :: "bool"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.sum_SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n Vopt popt b n'" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n'"
slp_index_def: fixes vs :: "'a list"   and i :: "nat" shows "slp_index vs i = length vs - Suc i"
mult_smult_left: fixes a :: "'a"   and p :: "'a poly"   and q :: "'a poly" shows "smult a p * q = smult a (p * q)"
comp_apply: fixes f :: "'r \<Rightarrow> 'q"   and g :: "'s \<Rightarrow> 'r"   and x :: "'s" shows "(f \<circ> g) x = f (g x)"
elims: fixes x :: "'a array"   and xa :: "nat"   and y :: "(nat \<times> 'a) list" assumes "assoc_list_of_array_code x xa = y" obtains a :: "'a array" and n :: "nat" where "x = a" and "xa = n" and "y = (if array_length a \<le> n then [] else (n, array_get a n) # assoc_list_of_array_code a (n + 1))"
limit_elem: fixes x :: "'a"   and f :: "'a set \<Rightarrow> 'a set"   and n :: "nat"   and X :: "'a set" assumes "x \<in> funpower f n X" shows "x \<in> limit f X"
opnet_tau2: fixes \<sigma> :: "nat \<Rightarrow> 'c"   and t :: "'d net_state"   and \<sigma>' :: "nat \<Rightarrow> 'c"   and t' :: "'d net_state"   and T :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set"   and s :: "'d net_state"   and S :: "(((nat \<Rightarrow> 'c) \<times> 'd net_state) \<times> 'e node_action \<times> (nat \<Rightarrow> 'c) \<times> 'd net_state) set" assumes "((\<sigma>, t), \<tau>, \<sigma>', t') \<in> T" shows "((\<sigma>, SubnetS s t), \<tau>, \<sigma>', SubnetS s t') \<in> opnet_sos S T"
simps_10: fixes P :: "pi" shows "hnf (!P) = False"
vs2: shows "vs2 \<equiv> getFrN (map snd txs) (t2 # map fst txs) [] (length txs)"
setReason_def: fixes literal :: "Literal"   and clause :: "nat"   and state :: "State" shows "setReason literal clause state = state\<lparr>getReason := getReason state(literal \<mapsto> clause)\<rparr>"
r_into_trancl': fixes p :: "'b \<times> 'b"   and r :: "('b \<times> 'b) set" assumes "p \<in> r" shows "p \<in> r\<^sup>+"
poly_weak_mono_all_def: fixes p :: "('c monom \<times> 'd) list" shows "poly_weak_mono_all p \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. \<beta> x \<le> \<alpha> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> eval_poly \<beta> p \<le> eval_poly \<alpha> p"
mat_mult2_11: fixes U :: "'b mat"   and V :: "'b mat"   and A :: "'b mat" assumes "U \<in> carrier_mat 2 2"   and "V \<in> carrier_mat 2 2"   and "A = U * V" shows "A $$ (1, 1) = U $$ (1, 0) * V $$ (0, 1) + U $$ (1, 1) * V $$ (1, 1)"
dense_complement_openin_affine_hull: fixes T :: "'a set"   and S :: "'a set" assumes "aff_dim T < aff_dim S"   and "openin (top_of_set (affine hull S)) S" shows "closure (S - T) = closure S"
split_beta_bezout: fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and a :: "'a"   and b :: "'a" assumes "is_bezout_ext bezout" shows "bezout a b = (fst (bezout a b), fst (snd (bezout a b)), fst (snd (snd (bezout a b))), fst (snd (snd (snd (bezout a b)))), snd (snd (snd (snd (bezout a b)))))"
map_ide_simp: fixes a :: "'a" assumes "B.ide a" shows "B.\<rr>'.map a = B.inv (B.\<rr> a)"
resumption_corec: fixes f :: "'j \<Rightarrow> ('f, 'g, 'h, 'i, 'j) resumption_corec"   and e :: "'j" shows "resumption_corec f e = (case f e of Terminal_corec a \<Rightarrow> Terminal a | Linear_corec b e' \<Rightarrow> Linear b (resumption_corec f e') | Branch_corec c es \<Rightarrow> Branch c (\<lambda>d. resumption_corec f (es d)) | Resumption_corec r \<Rightarrow> r)"
simps_219: fixes x15 :: "expr"   and x121 :: "expr"   and x122 :: "expr" shows "throw x15 \<noteq> x121;; x122"