hcju/isabelleps huggingface dataset:
closure_invariant_termination: fixes r :: "'b rexp"   and s :: "'b rexp" assumes "local.wf n r"   and "local.wf n s"   and "closure_invariant ([([], init r, init s)], [], {(post (init r), post (init s))}) = None" shows "False"
assms_3: fixes P''' :: "('a, 'b, 'c) psi"   and P'' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'''"   and "\<Psi> \<rhd> P''' \<longmapsto> \<alpha> \<prec> P''"   and "\<Psi> \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P'" shows "Prop \<alpha> P'"
inv_all_diff_evD: fixes A :: "'a mat"   and i :: "nat"   and j :: "nat" assumes "local.inv_all local.diff_ev A"   and "i < j"   and "j < n"   and "A $$ (i, i) \<noteq> A $$ (j, j)" shows "A $$ (i, j) = (0::'a)"
simple_fw_iff_generalized_fw: fixes fw :: "'b simple_rule list"   and p :: "('b, 'c) simple_packet_scheme"   and a :: "simple_action" shows "(simple_fw fw p = simple_action_to_decision a) = (\<exists>r. generalized_sfw (map simple_rule_dtor fw) p = Some (r, a))"
poly_sum_list: fixes ps :: "'b poly list"   and x :: "'b" shows "poly (sum_list ps) x = (\<Sum>p\<leftarrow>ps. poly p x)"
inv_check_left_moving_on_leftmost_Bk_Bk: fixes as :: "nat"   and lm :: "nat list"   and s :: "nat"   and r :: "cell list"   and ires :: "cell list"   and s' :: "nat" assumes "inv_on_left_moving_in_middle_B (as, lm) (s, [], Bk # r) ires" shows "inv_check_left_moving_on_leftmost (as, lm) (s', [], Bk # Bk # r) ires"
rtc_isPath: fixes u :: "nat"   and v :: "nat" assumes "(u, v) \<in> cf.E\<^sup>*" shows "\<exists>p. cf.isPath u p v"
ordinal_exp_1: fixes x :: "ordinal" shows "x ** oSuc 0 = x"
norm_Pair1: fixes x :: "'e" shows "norm (0::'d, x) = norm x"
h2d: shows "set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2)) \<subseteq> set (signs_smash signs1 signs2)"
simps: fixes uu :: "int poly"   and l :: "rat"   and uv :: "rat" shows "rai_lb (uu, l, uv) = l"
split: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> nat \<Rightarrow> 'a"   and tm :: "'b tm" shows "P (case tm of ??.Time_Monad.tm.TM x xa \<Rightarrow> f x xa) = (\<forall>x1 x2. tm = ??.Time_Monad.tm.TM x1 x2 \<longrightarrow> P (f x1 x2))"
l_one: fixes x :: "'a list" assumes "x \<in> carrier (K [X])" shows "\<zero>\<^bsub>K [X]\<^esub> \<oplus>\<^bsub>K [X]\<^esub> x = x"
simps_5: fixes d :: "procVarDecl"   and uy :: "program"   and g :: "gState\<^sub>I"   and l :: "pState" shows "evalEffect (EEDecl d) uy g l = mkVarChannelProc d g l"
a: shows "\<not> \<pi> ` x \<notin># \<B>'"
sublens_refl: fixes X :: "'d \<Longrightarrow> 'e" shows "X \<subseteq>\<^sub>L X"
out546__suma: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and E :: "'p"   and D :: "'p"   and F :: "'p" assumes "A \<noteq> B"   and "B \<noteq> C"   and "E Out D F" shows "A B C D E F SumA A B C"
g: fixes x :: "'b" shows "(0::'a) \<le> g x"
space_count_space: fixes \<Omega> :: "'a set" shows "space (count_space \<Omega>) = \<Omega>"
map\<^sub>1_simps_3: fixes f :: "'a" assumes "C.ide f" shows "trg\<^sub>D (\<l>\<^sub>D\<^sup>-\<^sup>1[F f] \<cdot>\<^sub>D \<r>\<^sub>D[F f]) = F.map\<^sub>0 (trg\<^sub>C f)"
subst_fm_Conj: fixes A :: "fm"   and B :: "fm"   and i :: "name"   and x :: "tm" shows "(A AND B)(i::=x) = A(i::=x) AND B(i::=x)"
seqr_pre_transfer': fixes P :: "('b, 'd) urel"   and q :: "'d upred"   and R :: "('d, 'c) urel" shows "(P \<and> \<lceil>q\<rceil>\<^sub>>) ;; R = P ;; (\<lceil>q\<rceil>\<^sub>< \<and> R)"
chi_expr: fixes m :: "nat" shows "\<chi> m = \<tau> / complex_of_real (sqrt (real n)) * (\<Sum>k = 1..n. cnj (\<chi> k) * unity_root n (- int m * int k))"
simps_22: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and v :: "bool" shows "local.less_gexp_aux (Nor g1 g2) (Bc v) = False"
singleton_conv: fixes a :: "'b" shows "{x. x = a} = \<eta> a"
union_filter_mset_complement: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and M :: "'a multiset" assumes "\<forall>x. P x = (\<not> Q x)" shows "filter_mset P M + filter_mset Q M = M"
n_roots_def: shows "n_roots = (\<lambda>p. card (Roots p))"
RG_proj_mult_leftdelta': fixes g :: "'g"   and r :: "'a"   and s :: "'a"   and g' :: "'g" assumes "g \<in> H" shows "Supgroup.RG_proj (r \<delta>\<delta> g * (s \<delta>\<delta> g')) = r \<delta>\<delta> g * Supgroup.RG_proj (s \<delta>\<delta> g')"
in_listsI: fixes xs :: "'b list"   and A :: "'b set" assumes "\<forall>x\<in>set xs. x \<in> A" shows "xs \<in> lists A"
lm047: fixes a :: "('c \<times> 'd set) set" assumes "runiq a" shows "{(fst pair, Y) |Y pair. Y \<in> finestpart (snd pair) \<and> pair \<in> a} = {(x, Y) |Y x. Y \<in> finestpart (a ,, x) \<and> x \<in> Domain a}"
list_def: shows "list = map fst ((0, 0) # (2, 1) # xs_ys)"
ide_hcomp\<^sub>W\<^sub>C: fixes f :: "'a"   and g :: "'a" assumes "ide f"   and "ide g"   and "g \<star> f \<noteq> null" shows "ide (g \<star> f)"
coinduct: fixes X :: "'b llist \<Rightarrow> bool"   and x :: "'b llist"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "X x"   and "\<And>x. X x \<Longrightarrow> (\<exists>xa. x = LCons xa LNil) \<or> (\<exists>xa zs xs. x = LCons xa (prepend zs xs) \<and> chain R (llist_of (xa # zs @ [lhd xs])) \<and> (X xs \<or> chain' R xs))" shows "chain' R x"
simps_5: fixes I :: "interp"   and x :: "variable"   and \<phi> :: "fml" shows "fml_sem I (Exists x \<phi>) = {\<omega>. \<exists>r. repv \<omega> x r \<in> fml_sem I \<phi>}"
a_kill: fixes G :: "nat set \<times> nat set set" assumes "finite (short_cycles G k)"   and "short_cycles G k \<noteq> {}"   and "P (G -- choose_v G k) k" shows "P G k"
pseudofunctor_axioms: shows "pseudofunctor (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D G' \<Phi>\<^sub>G'"
size_6: fixes x2 :: "nat" shows "size (TVar x2) = 0"
interleavings_set: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "interleavings xs ys zs" shows "set xs \<union> set ys = set zs"
elims: fixes x :: "arctic"   and xa :: "arctic"   and y :: "arctic" assumes "x + xa = y" obtains y :: "arctic" where "xa__ = MinInfty" and "xa = yb__" and "ya__ = yb__"   | v :: "int" where "xa__ = Num_arc v" and "xa = MinInfty" and "ya__ = Num_arc v"   | x :: "int" and y :: "int" where "xa__ = Num_arc x" and "xa = Num_arc y" and "ya__ = Num_arc (max x y)"
mod_0_imp_diff_Suc_div_conv: fixes n :: "nat"   and r :: "nat"   and m :: "nat" assumes "(n - r) mod m = 0" shows "(n - Suc r) div m = (n - r) div m - Suc 0"
hyps: assumes "t j \<le> T"   and "max_stepsize j \<le> root p (\<bar>r\<bar> * L / B / (exp (L * (T - t 0) + 1) - 1))" shows "dist (x t j) (grid_function (discrete_evolution \<psi>) x0 t j) \<le> B / L * (exp (L * (t j - t 0) + 1) - 1) * max_stepsize j ^ p"
iso_lunit: fixes f :: "'a" assumes "B.ide f" shows "B.iso \<l>[f]"
pseudo_proj_True_stake: fixes p :: "real"   and M :: "bool stream measure"   and n :: "nat"   and w :: "bool stream" assumes "infinite_coin_toss_space p M" shows "stake n (pseudo_proj_True n w) = stake n w"
2: shows "length xs = length (tuple_decode (pdec1 k) (pdec2 k)) - 1"
E''_def: shows "E'' \<equiv> lappend (llist_of E') E"
hyps_1: fixes \<pi> :: "nat \<Rightarrow> 'n"   and n :: "nat"   and n' :: "nat" assumes "xs = cs\<^bsup>\<pi>\<^esup> n"   and "is_path \<pi>"   and "\<pi> n \<noteq> local.return"   and "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> n'" shows "n = n'"
inf_closed: fixes x :: "'b"   and y :: "'b" shows "- x \<sqinter> - y = - - (- x \<sqinter> - y)"
delete_edge_preserve_subgraph: fixes G :: "('c, 'd) graph"   and H :: "('c, 'd) graph"   and a :: "'c"   and w :: "'d"   and b :: "'c" assumes "valid_graph G"   and "subgraph H G" shows "subgraph (delete_edge a w b H) G"
coeff_add: fixes p :: "'a mpoly"   and q :: "'a mpoly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (p + q) m = MPoly_Type.coeff p m + MPoly_Type.coeff q m"
simps: fixes A_pos :: "nat set list"   and L_pos :: "'b option list set list" shows "dominate_True A_pos L_pos = (case find_sub_True A_pos of None \<Rightarrow> None | Some split \<Rightarrow> Some (split, proj_list_5 L_pos split))"
not_less_iff_gr_or_eq: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.linorder less_eq less" shows "(\<not> less x y) = (less y x \<or> x = y)"
has_products_def: fixes I :: "'a set" shows "has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J (\<cdot>) D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. has_as_product J D a))"
affine_hull_0_dependent: fixes S :: "'a set" assumes "(0::'a) \<in> affine hull S" shows "dependent S"
states_Plus: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme" shows "states (Plus fsm1 fsm2) = states fsm1 + states fsm2"
filter_union_mset: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" shows "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
Gcd_fin_greatest: fixes A :: "'a set"   and a :: "'a" assumes "finite A"   and "\<And>b. b \<in> A \<Longrightarrow> a dvd b" shows "a dvd Gcd\<^sub>f\<^sub>i\<^sub>n A"
mono_mult1: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "r \<subseteq> r'" shows "mult1 r \<subseteq> mult1 r'"
CondThrow\<^sub>1: fixes P :: "nat exp prog"   and e :: "nat exp"   and s\<^sub>0 :: "heap \<times> val list \<times> sheap"   and e' :: "nat exp"   and s\<^sub>1 :: "heap \<times> val list \<times> sheap"   and e\<^sub>1 :: "nat exp"   and e\<^sub>2 :: "nat exp" assumes "P \<turnstile>\<^sub>1 \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
prems: shows "s \<in> timpl_closure (Fun f T) TI"
simps_2: fixes \<S> :: "nat \<Rightarrow> bool option"   and x :: "'b memref"   and xs :: "'b memref list" shows "non_volatile_writes_unshared \<S> (x # xs) = (case x of Write\<^sub>s\<^sub>b volatile a sop v A L R W \<Rightarrow> if volatile then non_volatile_writes_unshared (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) xs else a \<notin> dom \<S> \<and> non_volatile_writes_unshared \<S> xs | Ghost\<^sub>s\<^sub>b A L R W \<Rightarrow> non_volatile_writes_unshared (\<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L) xs | _ \<Rightarrow> non_volatile_writes_unshared \<S> xs)"
hyps_3: fixes i :: "nat" shows "integral\<^sup>N (join M) (F i) = \<integral>\<^sup>+ M'. integral\<^sup>N M' (F i) \<partial>M"
earlyLateTauChain: fixes P :: "pi"   and Q :: "pi" assumes "(P, Q) \<in> {(P, P') |P P'. P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'}\<^sup>*" shows "P \<Longrightarrow>\<^sub>\<tau>\<^sub>l Q"
ACI_deriv: fixes r :: "'a rexp"   and s :: "'a rexp"   and a :: "'a" assumes "ACI r s" shows "ACI (deriv a r) (deriv a s)"
adjacent_half_chamber_system_image_fg: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set"   and C :: "'b set"   and D :: "'b set" assumes "OpposedThinChamberComplexFoldings X f g C0"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> g \<turnstile> ChamberComplex.chamber_system X"   and "C \<sim> D" shows "f ` D = C"
uniformity_complex_homo_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
image_prod_mset_multiplicity: fixes f :: "'c \<Rightarrow> 'b"   and M :: "'c multiset" shows "\<Prod>\<^sub># (image_mset f M) = (\<Prod>x\<in>set_mset M. f x ^ count M x)"
ipresFreshAbs_def: fixes hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('d, 'e, 'varSort, 'f, 'g, 'var, 'h, 'c, 'i) Iteration.model_scheme" shows "ipresFreshAbs hA MOD \<equiv> \<forall>ys y us s A. wlsAbs (us, s) A \<longrightarrow> freshAbs ys y A \<longrightarrow> igFreshAbs MOD ys y (hA A)"
Rmap_endomorph_is_RModuleEnd: fixes T :: "'m \<Rightarrow> 'm" assumes "GroupEnd M T"   and "\<And>r m. \<lbrakk>r \<in> R; m \<in> M\<rbrakk> \<Longrightarrow> T (r \<cdot> m) = r \<cdot> T m" shows "RModuleEnd R (\<cdot>) M T"
pow_conjug: fixes t :: "'b list"   and i :: "nat"   and r :: "'b list"   and u :: "'b list"   and k :: "nat"   and s :: "'b list" assumes "t \<^sup>@ i \<cdot> r \<cdot> u = t \<^sup>@ k"   and "r \<cdot> s = t" shows "u \<cdot> t \<^sup>@ i \<cdot> r = (s \<cdot> r) \<^sup>@ k"
reach_def: fixes A :: "('d, 'c) nba"   and w :: "'d stream"   and i :: "nat" shows "reach A w i \<equiv> {target r p |r p. path A r p \<and> p \<in> initial A \<and> map fst r = stake i w}"
prems: shows "wf'\<^sub>s\<^sub>s\<^sub>t V ((x # S) @ S')"
SkipHigh: fixes G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "G \<rhd> Skip : HighSec"
monom_mult_dist_right: fixes c :: "'d"   and t :: "'c"   and p :: "'c \<Rightarrow>\<^sub>0 'd"   and q :: "'c \<Rightarrow>\<^sub>0 'd" shows "punit.monom_mult c t (p + q) = punit.monom_mult c t p + punit.monom_mult c t q"
simps_3: fixes f :: "'d \<rightarrow> 'c \<rightarrow> 'b"   and ys :: "['c]" shows "zipWith\<cdot>f\<cdot>[]\<cdot>ys = []"
lemma5_1: fixes r :: "dB"   and r' :: "dB" assumes "r \<rightarrow>\<^sub>l r'" shows "WN r"
assms_2: fixes i :: "nat" shows "r (seq (Suc i)) (seq i)"
root_imp_reducible_poly: fixes p :: "'b poly"   and x :: "'b" assumes "poly p x = (0::'b)"   and "1 < degree p" shows "\<not> irreducible p"
aux_if: fixes clist :: "('b com option \<times> 'b) list list"   and xs :: "'b com option list"   and s :: "'b"   and ys :: "('b com option list \<times> 'b) list" assumes "length clist = length xs \<and> (\<forall>i<length xs. (xs ! i, s) # clist ! i \<in> cptn) \<and> (xs, s) # ys \<propto> map (\<lambda>i. (fst i, s) # snd i) (zip xs clist)" shows "(xs, s) # ys \<in> par_cptn"
remove: fixes A :: "'a set"   and x :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<in> A" shows "sum g A = g x + sum g (A - {x})"
comp_ide_arr: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and f :: "'b" assumes "partial_magma C"   and "partial_magma.ide C b"   and "partial_magma.arr C (C b f)" shows "C b f = f"
word_bw_lcs_1: fixes y :: "'a word"   and x :: "'a word"   and z :: "'a word" shows "y AND x AND z = x AND y AND z"
simps_3: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.rank_invar (t # t' # bq) = (SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t' \<and> SkewBinomialHeapStruc.rank_invar (t' # bq))"
mat_mult_assoc: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and nr :: "nat"   and n1 :: "nat"   and m1 :: "'a list list"   and n2 :: "nat"   and m2 :: "'a list list"   and nc :: "nat"   and m3 :: "'a list list" assumes "class.semiring_0 plus zero times"   and "mat nr n1 m1"   and "mat n1 n2 m2"   and "mat n2 nc m3" shows "mat_multI zero plus times nr (mat_multI zero plus times nr m1 m2) m3 = mat_multI zero plus times nr m1 (mat_multI zero plus times n1 m2 m3)"
gcd_poly_decompose: fixes p :: "'a poly"   and q :: "'a poly" shows "gcd p q = Polynomial.smult (gcd (content p) (content q)) (gcd (primitive_part p) (primitive_part q))"
add_increasing: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "{#} \<subseteq># a"   and "b \<subseteq># c" shows "b \<subseteq># a + c"
square_ccsqrt: fixes x :: "complex" shows "(ccsqrt x)\<^sup>2 = x"
preserves_reflects_arr: fixes f :: "'a" shows "arr (cod \<iota> \<otimes> f) = arr f"
composition_closed: fixes a :: "'b"   and b :: "'b" assumes "a \<in> \<FF> z"   and "b \<in> \<FF> z" shows "\<cdot>\<^bsub>z\<^esub> a b \<in> \<FF> z"
assms_3: shows "unat j * unat k < 2 ^ LENGTH('a)"
indRelLTPO_to_TRel_2: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q" shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> False"
list_emb_set: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "list_emb P xs ys"   and "x \<in> set xs" obtains y :: "'a" where "y \<in> set ys" and "P x y"
elims_3: fixes x :: "classes1"   and xa :: "classes1" assumes "\<not> subclass1_fun x xa" obtains \<C> :: "classes1" where "x = Object" and "xa = \<C>__"   | \<C> :: "classes1" where "x = Person" and "xa = \<C>'__" and "\<C>'__ \<noteq> Object"   | \<C> :: "classes1" where "x = Employee" and "xa = \<C>''__" and "\<not> (\<C>''__ = Object \<or> \<C>''__ = Person)"   | \<C> :: "classes1" where "x = Customer" and "xa = \<C>'''__" and "\<not> (\<C>'''__ = Object \<or> \<C>'''__ = Person)"   | \<C> :: "classes1" where "x = Project" and "xa = \<C>''''__" and "\<C>''''__ \<noteq> Object"   | \<C> :: "classes1" where "x = Task" and "xa = \<C>'''''__" and "\<C>'''''__ \<noteq> Object"   | \<C> :: "classes1" where "x = Sprint" and "xa = \<C>" and "\<C> \<noteq> Object"
CD_on_closed: fixes X :: "'a set"   and ds :: "'b set" assumes "inj_on Xd X"   and "X \<subseteq> (\<Union>d\<in>ds. Field (Pd d))" shows "CD_on ds X = X"
finsum_empty: fixes f :: "'e \<Rightarrow> 'c" shows "finsum M f {} = \<zero>\<^bsub>M\<^esub>"
request_first: fixes x :: "nat"   and y :: "nat"   and h :: "nat list"   and "is" :: "nat list" assumes "x \<noteq> y" shows "Partial_Cost_Model.Step (rTS h) ([x, y], is) x = ([x, y], x # is)"
assms_4: shows "InvariantVarsM (getM state) F0 Vbl"
fst_inf: fixes x :: "'a \<times> 'b"   and y :: "'a \<times> 'b" shows "fst (inf x y) = inf (fst x) (fst y)"
bij_betw_ball_uball: fixes r :: "real"   and z0 :: "complex" assumes "0 < r" shows "bij_betw (\<lambda>x. complex_of_real r * x + z0) (ball 0 1) (ball z0 r)"
simps_1: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and p :: "'d spmf" shows "trace_callee_aux callee p [] = p"
proots_count_pCons_1_iff: fixes a :: "'b"   and s :: "'b set" shows "proots_count [:- a, 1::'b:] s = (if a \<in> s then 1 else 0)"
subst_mset_subst_compose: fixes \<sigma> :: "'e \<Rightarrow> ('c, 'f) Term.term"   and \<tau> :: "'f \<Rightarrow> ('c, 'd) Term.term"   and M :: "(('c, 'e) Term.term \<times> ('c, 'e) Term.term) multiset" shows "subst_mset (\<sigma> \<circ>\<^sub>s \<tau>) M = subst_mset \<tau> (subst_mset \<sigma> M)"
left_diff_distrib': fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and b :: "'d"   and c :: "'d"   and a :: "'d" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "times (minus b c) a = minus (times b a) (times c a)"
P_def: shows "P \<equiv> \<lambda>n. prime_factors n \<subseteq> {p}"
psi_residues_compare: fixes n :: "nat" shows "psi_odd n \<le> psi_even n"
insertion_prod: fixes f :: "nat \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a mpoly"   and A :: "'b set" shows "insertion f (prod g A) = (\<Prod>x\<in>A. insertion f (g x))"
collapse: fixes sctn :: "'a sctn" shows "Sctn (normal sctn) (pstn sctn) = sctn"
strategy_winning_by_Even_def: fixes A :: "'c llist set"   and N :: "enat"   and \<sigma>\<^sub>e :: "'c list \<Rightarrow> 'c"   and p :: "'c list" assumes "GSgame A N" shows "GSgame.strategy_winning_by_Even A N \<sigma>\<^sub>e p \<equiv> \<forall>\<sigma>\<^sub>o. GSgame.induced_play N (joint_strategy \<sigma>\<^sub>e \<sigma>\<^sub>o) p \<in> A"
funsignature_abbrev_def: fixes f :: "'a"   and S :: "('b, 'a) Signature"   and A :: "'b"   and B :: "'b" shows "f \<in> Sig S : A \<rightarrow> B \<equiv> f \<in> Fn\<^bsub>S\<^esub> \<and> A \<in> Ty\<^bsub>S\<^esub> \<and> B \<in> Ty\<^bsub>S\<^esub> \<and> sDom\<^bsub>S\<^esub> f = A \<and> sCod\<^bsub>S\<^esub> f = B \<and> Signature S"
map_id: fixes t :: "'a \<Rightarrow> 'b" shows "id \<circ> t = t"
PiE_eq_empty_iff: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd set" shows "(Pi\<^sub>E I F = {}) = (\<exists>i\<in>I. F i = {})"
simps: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec" shows "create_optimal_solutions A b c = (case simplex (x_times_c_geq_y_times_b c b # mat_leqb_eqc A b c @ from_index_geq0_vector (dim_vec c) (0\<^sub>v (dim_vec b))) of Inl x \<Rightarrow> Inl x | Inr x \<Rightarrow> Inr x)"
maps_arr_to_Univ: fixes f :: "'c" assumes "C.arr f" shows "\<phi> (C.dom f, C.cod f) f \<in> S.Univ"
nat_pow_def: fixes G :: "('b, 'c) monoid_scheme"   and a :: "'b"   and n :: "nat" shows "a [^]\<^bsub>G\<^esub> n = rec_nat \<one>\<^bsub>G\<^esub> (\<lambda>u b. b \<otimes>\<^bsub>G\<^esub> a) n"
ord_pmf_increaseI: fixes p :: "'b spmf"   and q :: "'b spmf"   and R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. spmf p x \<le> spmf q x"   and "\<And>x. x \<in> set_spmf p \<Longrightarrow> R x x" shows "ord_spmf R p q"
l'upd: shows "(l'(V := l V)) V' = \<lfloor>v'\<rfloor>"
stut_all: fixes F :: "'a \<Rightarrow> (nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>x. STUTINV F x" shows "STUTINV (\<forall>x. F x)"
insertI: fixes A :: "'d set"   and a :: "'d" assumes "finite A" shows "finite (insert a A)"
maxsimpchain_append_reduce1: fixes xs :: "'a set list"   and ys :: "'a set list" assumes "maxsimpchain (xs @ ys)" shows "maxsimpchain xs"
finite_dom_ahm_\<alpha>: fixes hm :: "('a, 'b) hashmap" shows "finite (dom (ahm_\<alpha> hm))"
sum_choose_two_inter_num: fixes b1 :: "'a set" assumes "b1 \<in># \<B>" shows "int (\<Sum>b2\<in>#remove1_mset b1 \<B>. nat (b1 |\<inter>| b2) choose 2) = \<Lambda> * (\<Lambda> - 1) div 2 * (\<v> - 1)"
add_le_less_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d"   and c :: "'d"   and d :: "'d" assumes "class.ordered_cancel_ab_semigroup_add plus minus less_eq less"   and "less_eq a b"   and "less c d" shows "less (plus a c) (plus b d)"
assms_4: shows "\<forall>i j. bounded m r\<^sub>0 i \<longrightarrow> bounded m r\<^sub>0 j \<longrightarrow> i < j \<longrightarrow> arr j \<le> arr i"
sd_chain_Some_propagate: fixes n :: "nat"   and m :: "nat"   and q :: "'alt pmf" assumes "n \<le> m"   and "sd_chain m = Some q" shows "\<exists>q'. sd_chain n = Some q'"
antisym: fixes a :: "'a"   and b :: "'a" assumes "b \<succeq> a"   and "a \<succeq> b" shows "a = b"
Min_in: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "linorder.Min less_eq A \<in> A"
less_oLimitE: fixes x :: "ordinal"   and f :: "nat \<Rightarrow> ordinal" assumes "x < oLimit f" obtains n :: "nat" where "x < f n"
arsinh_real_aux: fixes x :: "real" shows "0 < x + sqrt (x\<^sup>2 + 1)"
map_split_alt: fixes f :: "'c \<Rightarrow> 'a \<times> 'b"   and xs :: "'c list" shows "map_split f xs = (map (fst \<circ> f) xs, map (snd \<circ> f) xs)"
s2_2: shows "successively f (remove1 a (y # zs))"
obj_def: fixes a :: "'d" shows "D'.obj a \<equiv> D'.arr a \<and> D'.src a = a"
surjective: fixes r :: "('a, 'b, 'c) graph_scheme" shows "r = \<lparr>nodes = nodes r, edges = edges r, \<dots> = more r\<rparr>"
prod_defs_2: fixes f :: "nat \<Rightarrow> 'b"   and p :: "'b" shows "f has_prod p \<equiv> raw_has_prod f 0 p \<or> (\<exists>i q. p = (0::'b) \<and> f i = (0::'b) \<and> raw_has_prod f (Suc i) q)"
real_sqrt_sum_squares_eq_cancel2: fixes x :: "real"   and y :: "real" assumes "sqrt (x\<^sup>2 + y\<^sup>2) = y" shows "x = 0"
yun_monic_factorization_def: fixes Gcd :: "'c poly \<Rightarrow> 'c poly \<Rightarrow> 'c poly"   and p :: "'c poly" shows "yun_gcd.yun_monic_factorization Gcd p = (let pp = pderiv p; u = Gcd p pp; b0 = p div u; c0 = pp div u in filter (\<lambda>(a, i). a \<noteq> 1) (yun_gcd.yun_factorization_main Gcd b0 c0 0 []))"