hcju/isabelleps huggingface dataset:
recvmsg_simps_7: fixes P :: "'a \<Rightarrow> bool"   and m :: "'a" shows "recvmsg P (receive m) = P m"
simps_8: fixes \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula" shows "Formula.future_bounded (formula.And \<phi> \<psi>) = (Formula.future_bounded \<phi> \<and> Formula.future_bounded \<psi>)"
reduced_GB_Polys: fixes X :: "'x set"   and F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite X"   and "F \<subseteq> P[X]" shows "i.punit.reduced_GB F \<subseteq> P[X]"
proves_eq_transitive_pre: 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 s :: "Core.term"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> s"   and "term_ok \<Theta> t"   and "term_ok \<Theta> u"   and "typ_of s = typ_of t"   and "typ_of t = typ_of u" shows "\<Theta>,{} \<turnstile> mk_eq s t \<longmapsto> mk_eq t u \<longmapsto> mk_eq s u"
eq: shows "real_of_int m = real_of_int (mantissa f) * 2 powr real_of_int (exponent f - e)"
esing_eq_up_iff: fixes x :: "'b"   and a :: "'a"   and y :: "'b"   and a' :: "'a" shows "((esing x\<cdot>(up\<cdot>a)) y = up\<cdot>a') = (x = y \<and> a = a')"
simps: fixes a1 :: "('fun, 'var) Term.term set"   and a2 :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and a3 :: "('fun, 'var) extstrand_step list" shows "sem\<^sub>e\<^sub>s\<^sub>t_c a1 a2 a3 = ((\<exists>M\<^sub>0 \<I>. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = []) \<or> (\<exists>M\<^sub>0 \<I> S t. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)] \<and> sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S \<and> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I>) \<or> (\<exists>M\<^sub>0 \<I> S t. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step (receive\<langle>t\<rangle>\<^sub>s\<^sub>t)] \<and> sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S) \<or> (\<exists>M\<^sub>0 \<I> S t t' a. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t] \<and> sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S \<and> Unifier \<I> t t') \<or> (\<exists>M\<^sub>0 \<I> S X F. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Step \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t] \<and> sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S \<and> ineq_model \<I> X F) \<or> (\<exists>M\<^sub>0 \<I> S t K M. a1 = M\<^sub>0 \<and> a2 = \<I> \<and> a3 = S @ [Decomp t] \<and> sem\<^sub>e\<^sub>s\<^sub>t_c M\<^sub>0 \<I> S \<and> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c t \<cdot> \<I> \<and> Ana t = (K, M) \<and> (\<forall>x. x \<in> set K \<longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile>\<^sub>c x \<cdot> \<I>)))"
eq_iff: fixes x :: "'a::order"   and y :: "'a::order" shows "((x::'a::order) = (y::'a::order)) = (x \<le> y \<and> y \<le> x)"
map_value_image_filter_correct_aux: fixes \<alpha>1 :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and invar1 :: "'a \<Rightarrow> bool"   and \<alpha>2 :: "'d \<Rightarrow> 'b \<Rightarrow> 'e option"   and invar2 :: "'d \<Rightarrow> bool"   and map_value_image_filter :: "('b \<Rightarrow> 'c \<Rightarrow> 'e option) \<Rightarrow> 'a \<Rightarrow> 'd"   and m :: "'a"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e option" assumes "map_value_image_filter \<alpha>1 invar1 \<alpha>2 invar2 map_value_image_filter"   and "invar1 m" shows "invar2 (map_value_image_filter f m) \<and> \<alpha>2 (map_value_image_filter f m) = (\<lambda>k. \<alpha>1 m k \<bind> f k)"
K4: shows "(D A B' LtA D B' A) = (D B' Lt D A)"
dghm_cn_comp_components_1: fixes \<GG> :: "V"   and \<FF> :: "V" shows "(\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ObjMap\<rparr> = \<GG>\<lparr>ObjMap\<rparr> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr>"
assms_1: shows "\<forall>n\<ge>m. d n < e n"
subdegree_c: fixes i :: "nat" assumes "i \<le> N" shows "rat_of_int (fls_subdegree (c i)) = fpxs_val (coeff p i) * rat_of_nat b"
q_inter: fixes Q :: "'a set"   and Q' :: "'a set" assumes "i2.proj.quorum Q"   and "i2.proj.quorum Q'"   and "Q \<inter> I\<^sub>2 \<noteq> {}"   and "Q' \<inter> I\<^sub>2 \<noteq> {}" shows "Q \<inter> Q' \<inter> I\<^sub>2 \<noteq> {}"
subdigraphE: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>" obtains "digraph \<alpha> \<BB>" and "digraph \<alpha> \<CC>" and "\<And>a. a \<in>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr> \<Longrightarrow> a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "\<And>a b f. f : a \<mapsto>\<^bsub>\<BB>\<^esub> b \<Longrightarrow> f : a \<mapsto>\<^bsub>\<CC>\<^esub> b"
p_A_def: fixes n :: "nat" shows "p_A n = int (length (paid_A ! n))"
padic_int_is_cring: fixes p :: "int" assumes "prime p" shows "cring (padic_int p)"
set_sort_on: fixes r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" assumes "Linear_order r" shows "set (sort_key_on r f xs) = set xs"
senior_states_cases_subseteq: fixes p :: "'b"   and n :: "nat"   and q :: "'b" obtains "senior_states p n \<subseteq> senior_states q n"   | "senior_states q n \<subseteq> senior_states p n"
intro: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "H \<lhd> G"   and "weak_group_morphism_axioms f H G" shows "weak_group_morphism f H G"
cobounded1: fixes a :: "'a"   and b :: "'a" shows "a \<le> max a b"
compl_sup_top: fixes x :: "'a" shows "- x \<squnion> x = \<top>"
has_derivative_proj: fixes f :: "'a \<Rightarrow> real \<Rightarrow> real"   and f' :: "'a \<Rightarrow> real \<Rightarrow> real"   and F :: "real filter"   and j :: "'a" assumes "((\<lambda>x. \<chi>i. f i x) has_derivative (\<lambda>h. \<chi>i. f' i h)) F" shows "(f j has_derivative f' j) F"
assms_1: shows "InvariantNoDecisionsWhenConflict F M (currentLevel M)"
simps_3: fixes e :: "name list"   and A :: "fm"   and B :: "fm" shows "trans_fm e (A OR B) = DBDisj (trans_fm e A) (trans_fm e B)"
T'_altdef: fixes a :: "(int, 'n) vec" shows "T' a = (\<lambda>x. x + of_int_vec a) -` T a"
a_def: shows "a = (\<lambda>k m. 1 / of_nat k * (\<Sum>d | d dvd gcd m k. moebius_mu d * of_nat (k div d) * of_nat d))"
max_intros_2: fixes f :: "'a \<Rightarrow> 'a" shows "maximal (wp (Apply f))"
sortnMT: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "p \<noteq> []" shows "FWNormalisationCore.sort p l \<noteq> []"
subst_typ_combine: fixes fresh_idns :: "variable list"   and insts :: "((variable \<times> String.literal set) \<times> typ) list"   and \<tau> :: "typ" assumes "length fresh_idns = length insts"   and "distinct fresh_idns"   and "distinct (map fst insts)"   and "\<forall>idn\<in>set fresh_idns. idn \<notin> fst ` (tvsT \<tau> \<union> tvsT_Set (snd ` set insts) \<union> fst ` set insts)" shows "subst_typ insts \<tau> = subst_typ (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (subst_typ (zip (map fst insts) (map2 Tv fresh_idns (map snd (map fst insts)))) \<tau>)"
prems_2: shows "length (h1 # t1) = length l2"
set_def: fixes s' :: "'c" shows "local.set s' = State (\<lambda>_. ((), s'))"
subst_eql: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and t :: "'b"   and t1 :: "'b"   and t2 :: "'b"   and x :: "'a" assumes "Syntax_with_Connectives var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi"   and "t \<in> trm"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x \<in> var" shows "subst (eql t1 t2) t x = eql (substT t1 t x) (substT t2 t x)"
map_times_neg1_eq_map_uminus: fixes rs :: "'a list" shows "map ((*) (- (1::'a))) rs = map uminus rs"
cong_iff_lin: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and m :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment" shows "unique_euclidean_semiring.cong modulo a b m = (\<exists>k. b = plus a (times m k))"
ord_strict_pE: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and ord :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_strict :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ord_term :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and ord_term_strict :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and q :: "'c \<Rightarrow>\<^sub>0 'f" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "ordered_term.ord_strict_p ord_term_strict p q" obtains v :: "'c" where "lookup p v = (0::'f)" and "lookup q v \<noteq> (0::'f)" and "\<And>u. ord_term_strict v u \<Longrightarrow> lookup p u = lookup q u"
Diagonalize_Comp_Arr_Dom: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<lfloor>t \<^bold>\<cdot> Dom t\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor>"
finite_atLeastLessThan_int: fixes l :: "int"   and u :: "int" shows "finite {l..<u}"
Abs_asTerm_asAbs_qAbs: fixes qX :: "('a, 'b, 'varSort, 'var, 'c) qTerm"   and xs :: "'varSort"   and x :: "'var" assumes "qGood qX" shows "Abs xs x (asTerm qX) = asAbs (qAbs xs x qX)"
ls_def: shows "ls = map (\<lambda>x. (x, p x)) (filter (\<lambda>y. y < z) zs)"
bin_to_bl_aux_cat: fixes nv :: "nat"   and nw :: "nat"   and w :: "int"   and v :: "int"   and bs :: "bool list" shows "bin_to_bl_aux (nv + nw) (concat_bit nw w v) bs = bin_to_bl_aux nv v (bin_to_bl_aux nw w bs)"
int_clset_def: fixes S :: "'a eclause set" shows "int_clset S = same_values (\<lambda>x. trm_rep x S)"
inv_one: shows "inv\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = \<one>\<^bsub>H\<^esub>"
assms_7: shows "R' = Ax \<union> R1 \<union> R2 \<union> R3"
irreducible_connect_field: fixes f :: "'a poly" shows "irreducible\<^sub>d f = irreducible f"
row_space_row_echelon: fixes A :: "'c mat" shows "row_space (row_echelon A) = row_space A"
is_App_apps: fixes s :: "('c, 'd) tm"   and ss :: "('c, 'd) tm list" shows "is_App (apps s ss) = (args (apps s ss) \<noteq> [])"
continuous_on_topological: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
assms_2: fixes i :: "nat" assumes "i < length T" shows "(x \<in> fv (T ! i)) = (x \<in> fv (S ! i))"
i_cut_Int_conv_2: fixes I :: "'b set"   and t :: "'b" shows "I \<down>< t = I \<inter> {..<t}"
simps_2: fixes l :: "label" shows "(Processed \<sqsubset>l l) = (l = New)"
it_to_it_map_rev_linord_correct: fixes it :: "(('a \<times> 'b) list \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'b \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list) \<Rightarrow> ('a \<times> 'b) list \<Rightarrow> ('a \<times> 'b) list"   and S :: "'a \<Rightarrow> 'b option" assumes "map_iterator_rev_linord it S" shows "map_iterator_rev_linord (it_to_it it) S"
poly_det_cong: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b poly mat"   and k :: "'b" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "\<And>i j. \<lbrakk>i < n; j < n\<rbrakk> \<Longrightarrow> poly (B $$ (i, j)) k = A $$ (i, j)" shows "poly (det B) k = det A"
allocatedToBuyerMeansSold: fixes a :: "(integer \<times> 'c set) set"   and N :: "integer set"   and \<Omega> :: "'c set" assumes "a \<in> allocationsUniverse"   and "Domain a \<subseteq> N - {seller}"   and "\<Union> (Range a) \<subseteq> \<Omega>" shows "a \<in> soldAllocations'' N \<Omega>"
assms_2: shows "continuous_on {real_of_int a..} f"
ode_slp_impl_def: fixes ode_ops :: "'c \<times> ('a \<times> 'd \<times> 'e \<times> 'f \<times> 'g) \<times> 'h \<times> 'i" shows "ode_slp_impl ode_ops = (case ode_ops of (xa, (x, xb, xc, xd, xe), xf, xg) \<Rightarrow> x)"
fun_upd_triv: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c" shows "f(x := f x) = f"
Heap_ordI: fixes x :: "'a Heap"   and y :: "'a Heap" assumes "\<And>h. execute x h = None \<or> execute x h = execute y h" shows "Heap_ord x y"
circ_sup_upper_bound: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "x \<le> z\<^sup>\<circ>"   and "y \<le> z\<^sup>\<circ>" shows "x \<squnion> y \<le> z\<^sup>\<circ>"
assms_4: shows "(ir, Delete is e) \<in> set (node_deliver_messages xs)"
compl_bot_eq: fixes minus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and uminus :: "'g \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "uminus bot = top"
make_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "Array.make n f = Heap_Monad.heap (Array.alloc (map f [0..<n]))"
simps_1: fixes x :: "'a" shows "lookup_pair [] x = (0::'c)"
le_trans_equiv_rsp_right: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and u :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and v :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "equiv_trans t u" shows "le_trans v t = le_trans v u"
finite_set_strict_sorted: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.linorder less_eq less"   and "finite A" obtains l :: "'b list" where "linorder.strict_sorted less l" and "set l = A" and "length l = card A"
length_lesss_le: fixes R :: "('b \<times> 'b) set"   and x :: "'b"   and xs :: "'b list" shows "length (lesss R x xs) \<le> length xs"
dis_log_break': fixes \<G> :: "'a cyclic_group"   and y :: "nat"   and m' :: "nat"   and m :: "nat"   and d :: "nat"   and d' :: "nat" assumes "pedersen \<G>"   and "y < order \<G>"   and "\<not> m' < m"   and "m \<noteq> m'"   and "m' < order \<G>"   and "\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> d \<otimes>\<^bsub>\<G>\<^esub> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) [^]\<^bsub>\<G>\<^esub> m = \<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> d' \<otimes>\<^bsub>\<G>\<^esub> (\<^bold>g\<^bsub>\<G>\<^esub> [^]\<^bsub>\<G>\<^esub> y) [^]\<^bsub>\<G>\<^esub> m'" shows "y = nat ((int d - int d') * Number_Theory_Aux.inverse (m' - m) (order \<G>) mod int (order \<G>))"
assms: fixes e :: "real" assumes "0 < e" shows "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
bottom_edge_explicit: shows "(\<lambda>x. twoC (x, 0)) = (\<lambda>x. (g2 a + x *\<^sub>R (g1 a - g2 a), a))"
ord_le_eq_subst: fixes a :: "'k"   and b :: "'k"   and f :: "'k \<Rightarrow> 'l"   and c :: "'l" assumes "a \<le> b"   and "f b = c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "f a \<le> c"
non_zero_int: fixes n :: "int" assumes "n \<noteq> 0" shows "0 < n \<or> n < 0"
qtname_declclass_simp: fixes q :: "qtname" shows "declclass q = q"
discrim_congruent: shows "(\<lambda>U. {freediscrim U}) respects msgrel"
digitToTree_inv_2: fixes d :: "('a, 'b) Digit" assumes "FingerTreeStruc.is_measured_digit d" shows "FingerTreeStruc.is_measured_ftree (FingerTreeStruc.digitToTree d)"
ring_homomorphism_axioms: shows "ring_homomorphism \<eta> R (+) (\<cdot>) \<zero> \<one> R' (+') (\<cdot>') \<zero>' \<one>'"
bij_betw_trans: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and C :: "'d set" assumes "bij_betw f A B"   and "bij_betw g B C" shows "bij_betw (g \<circ> f) A C"
s1_def: shows "s1 = State (do st1 u c) u c"
fds_shift_inverse: fixes a :: "'b"   and f :: "'b fds" shows "fds_shift a (inverse f) = inverse (fds_shift a f)"
dist_commute: fixes x :: "'c"   and y :: "'c" shows "dist x y = dist y x"
S_view: shows "SD \<V> Tr\<^bsub>(induceES SES)\<^esub>"
not_eq_extremum: fixes a :: "'a multiset" shows "(a \<noteq> {#}) = ({#} \<subset># a)"
span_li_not_depend_2: fixes S :: "'a vec set"   and N :: "'a vec set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule class_ring N V" shows "module.lin_dep class_ring (vs N) S = lin_dep S"
leg_simps_1: shows "C.arr HHfgh.cod.leg0"
abc_before_final: fixes lm :: "nat list"   and p :: "abc_inst list"   and n :: "nat" assumes "abc_final (abc_steps_l (0, lm) p n) p"   and "p \<noteq> []" shows "\<exists>n'. abc_notfinal (abc_steps_l (0, lm) p n') p \<and> abc_final (abc_steps_l (0, lm) p (Suc n')) p"
tendsto_zero_divide_iff: fixes c :: "'e"   and a :: "nat \<Rightarrow> 'e" assumes "c \<noteq> (0::'e)" shows "(\<lambda>n. a n / c) \<longlonglongrightarrow> (0::'e) = a \<longlonglongrightarrow> (0::'e)"
connected_set_of: fixes X :: "'b interval" shows "connected (set_of X)"
Hoare_plus_halt: fixes P :: "cell list \<times> cell list \<Rightarrow> bool"   and A :: "(action \<times> nat) list"   and Q :: "cell list \<times> cell list \<Rightarrow> bool"   and B :: "(action \<times> nat) list"   and S :: "cell list \<times> cell list \<Rightarrow> bool" assumes "{P} A {Q}"   and "{Q} B {S}"   and "tm_wf0 A" shows "{P} A |+| B {S}"
m'_2: shows "indexed_eval_aux (Ps @ [P]) i m' \<noteq> \<zero>"
doesnt_read_or_modify_vars_def: fixes c :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and X :: "var\<^sub>C set" shows "conc.doesnt_read_or_modify_vars c X = (\<forall>mds mem c' mds' mem'. eval_abv\<^sub>C \<langle>c, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C \<longrightarrow> (\<forall>x\<in>X. \<forall>v. eval_abv\<^sub>C \<langle>c, mds, mem(x := v)\<rangle>\<^sub>C \<langle>c', mds', mem'(x := v)\<rangle>\<^sub>C))"
dims_nths_one_lt_n: fixes l :: "nat" assumes "l < n" shows "nths dims {l} = [2]"
iso_digraph_is_iso_dghm: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V" assumes "\<AA> \<approx>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<exists>\<FF>. \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>"
map_member_alt_def: fixes x :: "'a"   and y :: "'b"   and A :: "'a \<Rightarrow> 'b option" shows "((x, y) \<in>\<^sub>m A) = (x \<in> dom A \<and> A(x)\<^sub>m = y)"
that: fixes x :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> ok g x" shows "thesis"
axioms_1: fixes H :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subcring H R" shows "subring H R"
typ_of_beta_redex_arg: fixes T :: "typ"   and s :: "Core.term"   and t :: "Core.term" assumes "typ_of (Abs T s $ t) \<noteq> None" shows "typ_of t = Some T"
sort'_def: fixes p :: "'b set list"   and l :: "('b, 'c) Combinators list" shows "sort' p l = FWNormalisationCore.sort l p"
simps_1: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and n :: "nat"   and v :: "'d \<Rightarrow> nat"   and d :: "'c" shows "reset' M n [] v d = M"
normalfilters_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "pseudo_hoop_algebra.normalfilters times imp_l less_eq imp_r = {F \<in> pseudo_hoop_algebra.filters times less_eq. \<forall>a b. (imp_l a b \<in> F) = (imp_r a b \<in> F)}"
suc_greatest_not_zero: shows "(GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'rows) \<noteq> (0::'rows)"
SAMEQUOT_mono: fixes A :: "'b list set"   and B :: "'b list set"   and a :: "'b" assumes "A \<subseteq> B" shows "SAMEQUOT a A \<subseteq> SAMEQUOT a B"
assms_2: fixes v :: "'a" assumes "v \<in> winning_region p" shows "winning_strategy p \<sigma>' v"
essential_classD2: fixes C :: "'s option set"   and x :: "'s option"   and y :: "'s option" assumes "K2.essential_class C"   and "x \<in> C"   and "(x, y) \<in> K2.acc" shows "y \<in> C"
T_Cons: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmarkRecord.type"   and fs :: "(char list \<times> trm) list"   and fTs :: "(char list \<times> POPLmarkRecord.type) list"   and l :: "char list" assumes "\<Gamma> \<turnstile> t : T"   and "\<Gamma> \<turnstile> fs [:] fTs"   and "fs\<langle>l\<rangle>\<^sub>? = \<bottom>" shows "\<Gamma> \<turnstile> (l, t) \<Colon> fs [:] (l, T) \<Colon> fTs"
ins_mset: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and q :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.tree_invar t"   and "BinomialHeapStruc.queue_invar q" shows "BinomialHeapStruc.queue_to_multiset (BinomialHeapStruc.ins t q) = BinomialHeapStruc.tree_to_multiset t + BinomialHeapStruc.queue_to_multiset q"
substn_subst_n: fixes t :: "Core.term"   and n :: "nat"   and s :: "Core.term" shows "subst_bv1 t n s = subst_bv2 t n (incr_bv n 0 s)"
nabla_unfold_eq: fixes x :: "'a" shows "\<nabla> x = d (x \<cdot> \<nabla> x)"
induct: fixes P :: "name list \<Rightarrow> Term_Class.term \<Rightarrow> bool"   and a0 :: "name list"   and a1 :: "Term_Class.term" assumes "\<And>uu_ name. P uu_ (Const name)"   and "\<And>uv_ name. P uv_ (Free name)"   and "\<And>\<Gamma> n. P \<Gamma> (Bound n)"   and "\<And>\<Gamma> t. (\<And>x. P (x # \<Gamma>) t) \<Longrightarrow> P \<Gamma> (\<Lambda> t)"   and "\<And>\<Gamma> t\<^sub>1 t\<^sub>2. \<lbrakk>P \<Gamma> t\<^sub>1; \<And>x. P \<Gamma> t\<^sub>2\<rbrakk> \<Longrightarrow> P \<Gamma> (t\<^sub>1 $ t\<^sub>2)" shows "P a0 a1"
inner2s_def: fixes x :: "'b list"   and b :: "'b list"   and c :: "'b list" shows "inner2s x b c = (inner_lv_rel x b, inner_lv_rel x c)"
exhaust: fixes y :: "'a tree" obtains "y = Node"   | x21 :: "'a" and x22 :: "'a tree list" where "y = Fork x21 x22"
spr_simInit_def: fixes envInit :: "'d list"   and envObs :: "'b \<Rightarrow> 'd \<Rightarrow> 'c" shows "SPRViewSingle.spr_simInit envInit envObs \<equiv> \<lambda>a iobs. fromList (concat (map (\<lambda>s. if envObs a s = iobs then [s] else []) envInit))"
comp_assoc_assoc'_1: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "D.ide a"   and "D.ide b"   and "D.ide c" shows "\<a>\<^sub>D[a, b, c] \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[a, b, c] = a \<otimes>\<^sub>D b \<otimes>\<^sub>D c"
of_int_divide_in_Ints_iff: fixes b :: "int"   and a :: "int" assumes "b \<noteq> 0" shows "(of_int a / of_int b \<in> \<int>) = (b dvd a)"
true_inf_2: fixes P :: "'b upred" shows "P \<squnion> true = P"
IncrEnvironment_axioms: shows "IncrEnvironment jkbp jview envObs jviewInit jviewIncr"
ovalid_is_triple: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "ovalid P f Q = triple_judgement P f (ovalid_property Q (\<lambda>s f. f s))"
connected_inV_iff: fixes u :: "nat"   and v :: "nat" assumes "local.connected u v" shows "(v \<in> V) = (u \<in> V)"