hcju/isabelleps huggingface dataset:
lasso_of_prpl_def: fixes prpl :: "'a list \<times> 'a list" shows "lasso_of_prpl prpl \<equiv> case prpl of (pr, pl) \<Rightarrow> \<lparr>lasso_reach = pr, lasso_va = hd pl, lasso_cysfx = tl pl\<rparr>"
prems_2: shows "P,h \<turnstile> l (:\<le>)\<^sub>w E"
1: fixes k :: "nat" assumes "k \<le> 2" shows "integrable M (\<lambda>x. x ^ k)"
comp_ide_arr: fixes b :: "('a, 'a set) Maps.arr"   and f :: "('a, 'a set) Maps.arr" assumes "Maps.ide b"   and "Maps.seq b f" shows "b \<odot> f = f"
in_upd_image: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and A :: "nat set"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "A \<subseteq> {..<n}"   and "i < n" shows "(upd i \<in> upd ` A) = (i \<in> A)"
that: shows "\<forall>x\<in>set xs. \<forall>y\<in>set xs. g x = g y \<longrightarrow> f x = f y"
Iagree_Prog: fixes I :: "('d, 'e, 'f) interp"   and J :: "('d, 'e, 'f) interp"   and V :: "('d + 'e + 'f) set"   and a :: "'f" assumes "Iagree I J V"   and "Inr (Inr a) \<in> V" shows "Programs I a = Programs J a"
comp_null_1: fixes f :: "'b" shows "J J.null f = J.null"
Init: fixes s :: "'b"   and F :: "'b program" assumes "s \<in> Init F" shows "reachablep F s"
Sup: fixes X :: "'a set"   and a :: "'a" assumes "X \<subseteq> carrier D"   and "minimum_elem D (upper_bounds D X) a" shows "Algebra2.Sup D X = a"
fresh_psubst_E2: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and ys :: "'varSort"   and y :: "'var"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho"   and "rho ys y = Some Y"   and "fresh zs z X #[rho]" shows "fresh ys y X \<or> fresh zs z Y"
offset_addition: fixes k :: "nat"   and x :: "'a"   and xs :: "'a list"   and n :: "nat"   and m :: "nat" assumes "offset k x xs \<noteq> None" shows "offset (n + m) x xs = Some (the (offset n x xs) + m)"
oinvariant_weakenE: fixes A :: "('a \<times> 'b, 'c) automaton"   and PS :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and PU :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and Q :: "'a \<times> 'b \<Rightarrow> bool"   and QS :: "'a \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and QU :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "A \<Turnstile> (PS, PU \<rightarrow>) P"   and "\<And>s. P s \<Longrightarrow> Q s"   and "\<And>s s' a. QS s s' a \<Longrightarrow> PS s s' a"   and "\<And>s s'. QU s s' \<Longrightarrow> PU s s'" shows "A \<Turnstile> (QS, QU \<rightarrow>) Q"
right_idem: fixes f :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and a :: "'b::type"   and b :: "'b::type" assumes "semilattice (f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(f::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (f (a::'b::type) (b::'b::type)) b = f a b"
cauchy_add: fixes X :: "nat \<Rightarrow> rat"   and Y :: "nat \<Rightarrow> rat" assumes "Real.cauchy X"   and "Real.cauchy Y" shows "Real.cauchy (\<lambda>n. X n + Y n)"
G_def: shows "G = (\<lambda>t. SOME g. crit t g) ` T"
program_equalityI: fixes F :: "'b program"   and G :: "'b program" assumes "Init F = Init G"   and "Acts F = Acts G"   and "AllowedActs F = AllowedActs G" shows "F = G"
collect_interrupt_subset: fixes Ts :: "'b set"   and Ts' :: "'b set"   and ia :: "'b interrupt_action" assumes "Ts \<subseteq> Ts'" shows "collect_interrupt ia Ts \<subseteq> collect_interrupt ia Ts'"
setdist_def: fixes s :: "'a set"   and t :: "'a set" shows "setdist s t \<equiv> if s = {} \<or> t = {} then 0 else Inf {dist x y |x y. x \<in> s \<and> y \<in> t}"
at_ds5': fixes a :: "'a"   and b :: "'a" assumes "at TYPE('a)" shows " [(a, b), (b, a)] \<triangleq> [] "
l2_trans_def: shows "l2_trans \<equiv> \<Union>m M KE Rb Ra A B K. (l2_step1 Ra A B \<union> l2_step2 Rb A B KE \<union> l2_step3 Ra A B m \<union> l2_dy_fake_chan M \<union> l2_dy_fake_msg m \<union> l2_lkr_others A \<union> l2_lkr_after A \<union> l2_skr Ra K)\<^sup>="
lsl_sup_closed_var: fixes x :: "'b"   and y :: "'b" shows "\<nu> (\<nu> x \<squnion> \<nu> y) = \<nu> x \<squnion> \<nu> y"
trms_of_deg_leq_iter: fixes f :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "f \<in> carrier P" shows "trms_of_deg_leq (Suc k) f = trms_of_deg_leq k f \<oplus>\<^bsub>P\<^esub> monom P (f (Suc k)) (Suc k)"
Abs_formula_inverse: fixes y :: "'a set set" assumes "y \<in> fba" shows "Rep_formula (Abs_formula y) = y"
leftderives_refl: fixes u :: "symbol list" shows "leftderives u u"
monoid_axioms: shows "monoid (\<parallel>) 1\<^sub>\<pi>"
smaller_fermat4: fixes a :: "int"   and b :: "int"   and c :: "int" assumes "a ^ 4 + b ^ 4 = c\<^sup>2"   and "a * b * c \<noteq> 0"   and "odd a"   and "coprime a b" shows "\<exists>p q r. p ^ 4 + q ^ 4 = r\<^sup>2 \<and> p * q * r \<noteq> 0 \<and> odd p \<and> coprime p q \<and> r\<^sup>2 < c\<^sup>2"
If_cases: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "finite A" shows "(\<Sum>x\<in>A. if P x then h x else g x) = sum h (A \<inter> {x. P x}) + sum g (A \<inter> - {x. P x})"
map_iterator_foldli_conv: fixes iti :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and m :: "'e \<Rightarrow> 'f option" shows "map_iterator iti m = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> iti = foldli l0)"
sgn_zero_iff: fixes x :: "'b" shows "(sgn x = (0::'b)) = (x = (0::'b))"
rel_simps_45: shows "\<not> (1::'a) \<le> (0::'a)"
bind_returns_result_I: fixes h :: "'aq"   and f :: "('aq, 'ar, 'as) prog"   and x :: "'as"   and h' :: "'aq"   and g :: "'as \<Rightarrow> ('aq, 'ar, 'at) prog"   and y :: "'at" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> Heap_Error_Monad.bind f g \<rightarrow>\<^sub>r y"
g_orbitalD: fixes s' :: "'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" assumes "s' \<in> g_orbital f G U S t\<^sub>0 s" obtains X :: "real \<Rightarrow> 'b" and t :: "real" where "X \<in> Sols f U S t\<^sub>0 s" and "X t = s'" and "t \<in> U s" and "\<P> X (down (U s) t) \<subseteq> {s. G s}"
is_SNF_def: fixes A :: "'b mat"   and R :: "'b mat \<times> 'b mat \<times> 'b mat" shows "is_SNF A R = (case R of (P, S, Q) \<Rightarrow> P \<in> carrier_mat (dim_row A) (dim_row A) \<and> Q \<in> carrier_mat (dim_col A) (dim_col A) \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S \<and> S = P * A * Q)"
empty_N_if_Nil_in_P_or_Q: fixes P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and N :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "[] \<in> fst ` set (P @ Q)" shows "wstate_of_dstate (N, P, Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate ([], P, Q, n)"
Exe1S: fixes v :: "i"   and F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>" shows "[\<lparr>F,x\<rparr> in v] = (\<exists>r o\<^sub>1. Some r = Semantics.d\<^sub>1 F \<and> Some o\<^sub>1 = Semantics.d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> Semantics.ex1 r v)"
diff_diff_eq2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a - (b - c) = a + c - b"
weakening: fixes \<Gamma>\<^sub>1 :: "(name \<times> ty) list"   and e :: "trm"   and T :: "ty"   and \<Gamma>\<^sub>2 :: "(name \<times> ty) list" assumes "\<Gamma>\<^sub>1 \<turnstile> e : T"   and "valid \<Gamma>\<^sub>2"   and "\<Gamma>\<^sub>1 \<subseteq> \<Gamma>\<^sub>2" shows "\<Gamma>\<^sub>2 \<turnstile> e : T"
rep_eq: fixes x :: "'c pmf" shows "pmf x = (\<lambda>xa. measure_pmf.prob x {xa})"
a_set_attribute_def: fixes ptr :: "(_) element_ptr"   and k :: "char list"   and v :: "char list option" shows "a_set_attribute ptr k v = Heap_Error_Monad.bind (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs) (\<lambda>m. put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr attrs_update (if v = None then fmdrop k m else fmupd k (the v) m))"
DBM_reset_sound'': fixes v :: "'c \<Rightarrow> nat"   and n :: "nat"   and c :: "'c"   and u :: "'c \<Rightarrow> 'd"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d"   and u'' :: "'c \<Rightarrow> 'd" assumes "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "v c \<le> n"   and "u \<turnstile>\<^bsub>v,n\<^esub> reset M n (v c) d"   and "u'' \<turnstile>\<^bsub>v,n\<^esub> M" obtains d' :: "'d" where "u(c := d') \<turnstile>\<^bsub>v,n\<^esub> M"
cont_def: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'g" shows "cont luba orda lubb ordb f = (\<forall>Y. Complete_Partial_Order.chain orda Y \<longrightarrow> Y \<noteq> {} \<longrightarrow> f (luba Y) = lubb (f ` Y))"
Domain_mono: fixes r :: "('e \<times> 'f) set"   and s :: "('e \<times> 'f) set" assumes "r \<subseteq> s" shows "Domain r \<subseteq> Domain s"
assms: fixes x :: "'a"   and ys :: "'b list" shows "set (f (g x) ys) = set (g x) \<union> set ys"
prefix_proj_1: fixes A :: "('e strand_label \<times> 'f) list"   and B :: "('e strand_label \<times> 'f) list" assumes "prefix A B" shows "prefix (unlabel A) (unlabel B)"
assms_2: fixes i :: "nat" assumes "i \<le> n" shows "f $ i = (0::'a)"
poly_rel_of_int_poly: fixes ff_ops :: "'a arith_ops_record"   and R :: "'a \<Rightarrow> 'b mod_ring \<Rightarrow> bool"   and p :: "int"   and f' :: "'a list"   and f :: "int poly"   and f'' :: "'b mod_ring poly" assumes "mod_ring_gen ff_ops R p"   and "f' = of_int_poly_i ff_ops (poly_mod.Mp p f)"   and "f'' = of_int_poly (poly_mod.Mp p f)" shows "ring_ops.poly_rel R f' f''"
ltl_llist_of_tllist: fixes xs :: "('a, 'b) tllist" shows "ltl (llist_of_tllist xs) = llist_of_tllist (ttl xs)"
map_in_bicategory_axioms: shows "map_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg (tab\<^sub>0 r)"
map_getminE: fixes m :: "'b"   and k :: "'v"   and p :: "nat" assumes "Q_getmin m = (k, p)"   and "Q_invar m"   and "Q_lookup m \<noteq> Map.empty" obtains "Q_lookup m k = Some p" and "\<forall>k' p'. Q_lookup m k' = Some p' \<longrightarrow> p \<le> p'"
rcos_mult_step2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "H <#> (x <# H) #> y = H <#> (H #> x) #> y"
mod_mult2_eq: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "m mod (n * q) = n * (m div n mod q) + m mod n"
a2: shows "r \<rightarrow>\<^sub>\<eta> s"
that: fixes Ks :: "'a list" assumes "length Ks = length Us"   and "combine Ks Us = \<zero>"   and "set Ks \<subseteq> K"   and "set Ks \<noteq> {\<zero>}" shows "thesis"
check_unsat_terminates: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"   and "\<U> s" shows "PivotUpdateMinVars.check_dom eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s"
s2: shows "(- 1 * a < 0 \<or> - 1 * a = 0 \<and> (0 < - 1 * b \<or> - 1 * b = 0 \<and> - 1 * c < 0)) = (0 < a \<or> a = 0 \<and> (b < 0 \<or> b = 0 \<and> 0 < c))"
mem_carrier: fixes x :: "'a" assumes "x \<in> H" shows "x \<in> carrier G"
select_convs_2: fixes Pol :: "('d \<times> 'd) set"   and Map :: "'c \<Rightarrow> 'd"   and Dom :: "'d"   and In :: "'c list"   and Out :: "'c list"   and more :: "'e" shows "Map \<lparr>Pol = Pol, Map = Map, Dom = Dom, In = In, Out = Out, \<dots> = more\<rparr> = Map"
bounded_byE: fixes xs :: "real list"   and vs :: "float interval option list"   and i :: "nat" assumes "bounded_by xs vs"   and "i < length vs" shows "case vs ! i of None \<Rightarrow> True | Some ivl \<Rightarrow> xs ! i \<in>\<^sub>r ivl"
scene_equiv_def: fixes S\<^sub>1 :: "'a"   and S\<^sub>2 :: "'a"   and X :: "'a scene" shows "S\<^sub>1 \<approx>\<^sub>S S\<^sub>2 on X = (S\<^sub>1 \<oplus>\<^sub>S S\<^sub>2 on X = S\<^sub>1)"
c_cex: shows "\<not> INTERP N \<Turnstile> C"
sum_pos2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and I :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "finite I"   and "i \<in> I"   and "less zero (f i)"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq zero (f i)" shows "less zero (comm_monoid_add.sum plus zero f I)"
isOK_forallM_index_aux: fixes P :: "'c \<Rightarrow> nat \<Rightarrow> 'd + unit"   and n :: "nat"   and xs :: "'c list" shows "isOK (forallM_index_aux P n xs) = (\<forall>i<length xs. isOK (P (xs ! i) (i + n)))"
arccos_minus_abs: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "arccos (- x) = pi - arccos x"
fold_set_fold: fixes y :: "'c set"   and xs :: "'c list" shows "Finite_Set.fold Set.remove y (set xs) = fold Set.remove xs y"
proj2_eq_sqn: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option" assumes "dip \<in> kD rt" shows "\<pi>\<^sub>2 (the (rt dip)) = sqn rt dip"
dvd_prod_mset: fixes x :: "'a"   and A :: "'a multiset" assumes "x \<in># A" shows "x dvd \<Prod>\<^sub># A"
of_nat_nat: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and z :: "int" assumes "class.ring_1 one times plus zero minus uminus"   and "0 \<le> z" shows "semiring_1.of_nat one plus zero (nat z) = ring_1.of_int one plus zero minus z"
induced_map_unique: fixes r\<^sub>0 :: "'a"   and \<omega> :: "'a"   and w :: "'a"   and w' :: "'a" assumes "\<sigma>.is_induced_by_cell w r\<^sub>0 \<omega>"   and "\<sigma>.is_induced_by_cell w' r\<^sub>0 \<omega>" shows "w \<cong> w'"
finitely_branching: fixes v :: "'v" assumes "v \<in> reachable" shows "finite (E `` {v})"
WTrtCAS: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'addr expr"   and U :: "ty"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal"   and e2 :: "'addr expr"   and T2 :: "ty"   and e3 :: "'addr expr"   and T3 :: "ty" assumes "P,E,h \<turnstile> e1 : U"   and "class_type_of' U = \<lfloor>C\<rfloor>"   and "P \<turnstile> C has F:T (fm) in D"   and "volatile fm"   and "P,E,h \<turnstile> e2 : T2"   and "P \<turnstile> T2 \<le> T"   and "P,E,h \<turnstile> e3 : T3"   and "P \<turnstile> T3 \<le> T" shows "P,E,h \<turnstile> e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3) : Boolean"
im_bottom: 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" assumes "OrderingSetMap less_eq less less_eq' less' P f"   and "ordering.has_bottom less_eq P" shows "f (ordering.bottom less_eq P) = ordering.bottom less_eq' (f ` P)"
that_4: fixes f :: "Value \<rightarrow> Value"   and g :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" assumes "x = Fn\<cdot>f"   and "y = CFn\<cdot>g" shows "thesis"
Infinitesimal_zero: shows "0 \<in> Infinitesimal"
idealI: fixes R :: "('a, 'b) ring_scheme"   and I :: "'a set" assumes "ring R"   and "subgroup I (add_monoid R)"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>R\<^esub> a \<in> I"   and "\<And>a x. \<lbrakk>a \<in> I; x \<in> carrier R\<rbrakk> \<Longrightarrow> a \<otimes>\<^bsub>R\<^esub> x \<in> I" shows "ideal I R"
elims: fixes x :: "'a iarray"   and xa :: "'a"   and y :: "'a iarray" assumes "iarray_append x xa = y" obtains xs :: "'a list" and xb :: "'a" where "x = IArray xs" and "xa = xb" and "y = IArray (xs @ [xb])"
of_int_def: shows "of_int \<equiv> map_fun Rep_Integ id (\<lambda>(i, j). of_nat i - of_nat j)"
drop_zero_ms_aux: fixes C :: "'a"   and e :: "real"   and xs :: "('a \<times> real) msllist"   and f :: "real \<Rightarrow> real"   and basis :: "(real \<Rightarrow> real) list" assumes "\<forall>\<^sub>F x in at_top. eval C x = 0"   and "is_expansion_aux (MSLCons (C, e) xs) f basis" shows "is_expansion_aux xs f basis"
lemmA: fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and i :: "nat"   and A :: "formula" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "EV (contains f (i, A))" shows "\<exists>n nAs. \<not> SATAxiom (sequent (f n)) \<and> nforms (f n) = (i, A) # nAs \<and> f (Suc n) \<in> subs (f n)"
i: shows "\<forall>u. u \<in># mset_set (cl_ecl C) \<longrightarrow> (mset_lit (subst_lit u \<sigma>), mset_lit (subst_lit u \<eta>)) \<in> mult trm_ord \<or> mset_lit (subst_lit u \<sigma>) = mset_lit (subst_lit u \<eta>)"
diag_mat_vec_nth_simps_1: fixes i :: "'b"   and j :: "'b"   and f :: "'b \<Rightarrow> 'c" assumes "i = j" shows "diag_mat f $ i $ j = f i"
wordinterval_subset_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_subset r1 r2 \<equiv> wordinterval_empty (wordinterval_setminus r1 r2)"
sequentially_even_odd: fixes P :: "nat \<Rightarrow> bool" assumes "\<forall>\<^sub>F n in sequentially. P (2 * n)"   and "\<forall>\<^sub>F n in sequentially. P (2 * n + 1)" shows "eventually P sequentially"
summarize_pdevs_listE: fixes e :: "nat \<Rightarrow> real"   and X :: "real pdevs list"   and d :: "nat"   and p :: "nat"   and I :: "nat \<Rightarrow> real list \<Rightarrow> bool" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "degrees X \<le> d" obtains e' :: "nat \<Rightarrow> real" where "pdevs_vals e X = pdevs_vals e' (summarize_pdevs_list p I d X)" and "\<And>i. i < d \<Longrightarrow> e i = e' i" and "e' \<in> UNIV \<rightarrow> {- 1..1}"
force_def: fixes xs :: "('a, 'b) tllist" shows "force xs = (case xs of TNil b \<Rightarrow> Inr b | TCons x ys \<Rightarrow> Inl (x, ys))"
eSwap_nchotomy: fixes eX :: "'c withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" shows "(\<exists>s X. eX = OK X \<and> igWls MOD s X) \<or> eSwap MOD zs z1 z2 eX = ERR"
union_is_single: fixes M :: "'a multiset"   and N :: "'a multiset"   and a :: "'a" shows "(M + N = {#a#}) = (M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#})"
GuardFault: fixes s :: "'a"   and g :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and f :: "'c"   and c :: "('a, 'b, 'c) com" assumes "s \<notin> g" shows "\<Gamma>\<turnstile>Guard f g c \<down> Normal s"
intro: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.bounded_lattice inf less_eq less sup bot top"   and "class.complemented_lattice_axioms minus uminus inf sup bot top" shows "class.complemented_lattice minus uminus inf less_eq less sup bot top"
image_Un: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and B :: "'d set" shows "f ` (A \<union> B) = f ` A \<union> f ` B"
comp_arr_dom: fixes f :: "'a"   and a :: "'a" assumes "L'.C.arr f"   and "L'.C.dom f = a" shows "local.Left a' f a = f"
aka_loopi_def: fixes x :: "'a"   and i :: "'a" shows "loop x inv i  = x\<^sup>\<star>"
fBall_cong: fixes A :: "'a fset"   and B :: "'a fset"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "A = B"   and "\<And>x. x |\<in>| B \<Longrightarrow> P x = Q x" shows "fBall A P = fBall B Q"
intros_2: fixes a :: "aexp"   and x :: "char list"   and l :: "nat" assumes "sec a \<le> sec x"   and "l \<le> sec x" shows "l \<turnstile> x ::= a"
L4_Boiler: fixes x :: "nat \<Rightarrow> bit list"   and s :: "nat \<Rightarrow> nat list"   and y :: "nat \<Rightarrow> nat list" assumes "SteamBoiler x s y"   and "ts x" shows "hd (s 0) \<le> 800"
taylor_term_def: fixes c :: "'a"   and p :: "nat \<Rightarrow> 'a"   and i :: "nat" shows "taylor_term c p i = taylor_expansion R c p i \<odot>\<^bsub>UP R\<^esub> X_poly_minus R c [^]\<^bsub>UP R\<^esub> i"
sum_outgoing_extend: fixes u :: "nat"   and g :: "nat \<times> nat \<Rightarrow> 'a" assumes "\<And>v. \<lbrakk>v \<in> V; (u, v) \<notin> E\<rbrakk> \<Longrightarrow> g (u, v) = (0::'a)" shows "sum g (outgoing u) = (\<Sum>v\<in>V. g (u, v))"
nprv_ptrmE_uni: fixes \<sigma> :: "'fmla"   and n :: "nat"   and F :: "'fmla set"   and t1 :: "'trm"   and t2 :: "'trm"   and \<psi> :: "'fmla" assumes "\<sigma> \<in> ptrm n"   and "nprv F (subst \<sigma> t1 out)"   and "nprv F (subst \<sigma> t2 out)"   and "nprv (insert (eql t1 t2) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "nprv F \<psi>"
label_def: fixes c :: "cmd"   and n :: "nat" shows "label c n \<equiv> THE c'. labels c n c'"
scale_one: fixes x :: "'a" shows "1 *\<^sub>R x = x"
cases: fixes a1 :: "'a"   and a2 :: "(bool \<times> 'b \<times> 'a) list"   and ars :: "('a \<times> 'b \<times> 'a) set" assumes "(a1, a2) \<in> conv ars" obtains a :: "'a" where "a1 = a__" and "a2 = []"   | a :: "'a" and \<alpha> :: "'b" and b :: "'a" and ss :: "(bool \<times> 'b \<times> 'a) list" where "a1 = aa__" and "a2 = (True, \<alpha>__, b__) # ss__" and "(aa__, \<alpha>__, b__) \<in> ars" and "(b__, ss__) \<in> conv ars"   | b :: "'a" and \<alpha> :: "'b" and a :: "'a" and ss :: "(bool \<times> 'b \<times> 'a) list" where "a1 = a" and "a2 = (False, \<alpha>, b) # ss" and "(b, \<alpha>, a) \<in> ars" and "(b, ss) \<in> conv ars"
2: shows "(\<lambda>a. Suc (index as a)) ` set as = Suc ` index as ` set as"
changes_poly_pos_inf_pders: fixes p :: "real poly" shows "changes_poly_pos_inf (pders p) = 0"
subtopology_discrete_topology: fixes U :: "'c set"   and S :: "'c set" shows "subtopology (discrete_topology U) S = discrete_topology (U \<inter> S)"
not_inf: fixes x :: "'a"   and i :: "nat" assumes "x \<in> space M" shows "U i x < top"
Abs_nat_int_inject: fixes x :: "nat set"   and y :: "nat set" assumes "x \<in> {S. \<exists>m n. {m..n} = S}"   and "y \<in> {S. \<exists>m n. {m..n} = S}" shows "(Abs_nat_int x = Abs_nat_int y) = (x = y)"
induct: fixes x :: "'c"   and A :: "('c, 'd) automaton"   and I :: "'d \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> reachable A I"   and "\<And>s. s \<in> init A \<Longrightarrow> P s"   and "\<And>s a s'. \<lbrakk>s \<in> reachable A I; P s; (s, a, s') \<in> automaton.trans A; I a\<rbrakk> \<Longrightarrow> P s'" shows "P x"
list_all2_map2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and as :: "'c list"   and f :: "'e \<Rightarrow> 'd"   and bs :: "'e list" shows "list_all2 P as (map f bs) = list_all2 (\<lambda>x y. P x (f y)) as bs"
mult_divisor_le_mod_ge_imp_ge: fixes m :: "nat"   and k :: "nat"   and n :: "nat"   and r :: "nat" assumes "m * k \<le> n"   and "r \<le> n mod m" shows "m * k + r \<le> n"
gen_subdiv_1: shows "common_boundary_sudivision_exists (one_chain1 - ps1) (one_chain2 - ps2)"