hcju/isabelleps huggingface dataset:
freeword_funlift_Abs_freeword_singleton: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b \<times> bool" shows "freeword_funlift f (Abs_freeword [x]) = apply_sign f x"
comp_arr_ide: fixes a :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.ide a"   and "\<F>C.seq f a" shows "f \<cdot> a = f"
sp_in_extends_w_subs: fixes a :: "('d \<times> nat) edge"   and g :: "('d \<times> nat, 'e) rgraph_scheme"   and g' :: "('d \<times> nat, 'e) rgraph_scheme"   and v1 :: "'d \<times> nat"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set" assumes "ArcExt.extends g a g'"   and "SubRel.subpath g v1 es v2 subs" shows "SubRel.subpath g' v1 es v2 subs"
uniform_limitD: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and S :: "'c set"   and l :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and e :: "real" assumes "uniform_limit S f l F"   and "0 < e" shows "\<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e"
InitSeqReds: fixes C :: "char list"   and b :: "bool"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and v' :: "val"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool" assumes "P \<turnstile> \<langle>INIT C ([C],b) \<leftarrow> unit,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v',s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e,s\<^sub>1,icheck P C e\<rangle> \<rightarrow>* \<langle>e\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>"   and "\<not> sub_RI e" shows "P \<turnstile> \<langle>INIT C ([C],b) \<leftarrow> e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e\<^sub>2,s\<^sub>2,b\<^sub>2\<rangle>"
bet_out__bet: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and P :: "'a"   and C :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A P C"   and "Tarski_neutral_dimensionless.Out Bet P A B" shows "Bet B P C"
assms: shows "P \<Longrightarrow>\<^sub>l \<tau> \<prec> P'"
coprime_fib_Suc: fixes n :: "nat" shows "coprime (fib n) (fib (n + 1))"
simps: fixes a1 :: "'a set"   and a2 :: "'a \<Rightarrow> nat"   and a3 :: "'a \<Rightarrow> Regions.intv"   and a4 :: "('a \<times> 'a) set" shows "Regions.valid_region a1 a2 a3 a4 = (\<exists>X\<^sub>0 X I r k. a1 = X \<and> a2 = k \<and> a3 = I \<and> a4 = r \<and> X\<^sub>0 = {x \<in> X. \<exists>d. I x = Regions.intv.Intv d} \<and> refl_on X\<^sub>0 r \<and> Relation.trans r \<and> total_on X\<^sub>0 r \<and> (\<forall>x\<in>X. Regions.valid_intv (k x) (I x)))"
some1_equality: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a" assumes "\<exists>!x. P x"   and "P a" shows "(SOME x. P x) = a"
IteratorCallNE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and \<tau> :: "'a OCL_Types.type"   and iter :: "iterator"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and call :: "'a call_expr" assumes "(\<Gamma>, \<tau>) \<turnstile>\<^sub>C Iterator iter its its_ty body \<Rrightarrow> call" obtains \<sigma> :: "'a OCL_Types.type" and body\<^sub>2 :: "'a expr" where "its_ty = Some \<sigma>__" and "call = Iterator iter its (Some \<sigma>__) body\<^sub>2__" and "\<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, \<sigma>__)) its) \<turnstile> body \<Rrightarrow> body\<^sub>2__"   | \<sigma> :: "'a OCL_Types.type" and body\<^sub>2 :: "'a expr" where "its_ty = None" and "call = Iterator iter its (Some \<sigma>) body\<^sub>2" and "element_type \<tau> \<sigma>" and "\<Gamma> ++\<^sub>f fmap_of_list (map (\<lambda>it. (it, \<sigma>)) its) \<turnstile> body \<Rrightarrow> body\<^sub>2"
hd_append2: fixes xs :: "'c list"   and ys :: "'c list" assumes "xs \<noteq> []" shows "hd (xs @ ys) = hd xs"
H_def: shows "H \<equiv> {g \<in> carrier G. M1 #> g = M1}"
arr_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "VV.arr f = (V.arr (fst f) \<and> V.arr (snd f) \<and> src (fst f) = trg (snd f))"
assms_3: shows "infs (degen cs) (w ||| k ## sscan (count cs) w k)"
alpha1: shows "(qinp, qinp') \<in> alphaInp"
sel_2: fixes x1 :: "'m"   and x2 :: "'l" shows "snd (x1, x2) = x2"
delete_edges_set_nodes: fixes G :: "'b list_graph"   and E :: "('b \<times> 'b) list" shows "set (nodesL (FiniteListGraph.delete_edges G E)) = set (nodesL G)"
closed_Un: fixes S :: "'b set"   and T :: "'b set" assumes "closed S"   and "closed T" shows "closed (S \<union> T)"
oppOrdering_simps_4: shows "oppOrdering\<cdot>\<bottom> = \<bottom>"
sum_list_update: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and xs :: "'a list"   and x :: "'a" assumes "class.ordered_cancel_comm_monoid_diff plus zero less_eq less minus"   and "k < length xs" shows "monoid_add.sum_list plus zero (xs[k := x]) = minus (plus (monoid_add.sum_list plus zero xs) x) (xs ! k)"
\<mu>_small_row_refl: fixes i :: "nat"   and fs :: "int vec list" shows "\<mu>_small_row i fs i"
grey_protects_white_def: fixes g :: "'g"   and w :: "'g" shows "g grey_protects_white w = (\<lambda>s. grey g s \<and> (g has_white_path_to w) s)"
add_le_left: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "Ord \<alpha>"   and "Ord \<beta>" shows "\<beta> \<le> \<alpha> + \<beta>"
assms_2: fixes x :: "nat" assumes "x < n" shows "coprime x n = odd x"
atoms_of_cnf_def: fixes c :: "'b literal set set" shows "atoms_of_cnf c = atoms_of_lit ` \<Union> c"
times_elements: fixes v :: "'a list"   and a :: "'a" shows "\<forall>i<length v. local.times a v ! i = a * v ! i"
denote_positive_trace_dim: fixes S :: "com"   and \<rho> :: "complex mat" assumes "well_com S"   and "\<rho> \<in> carrier_mat d d"   and "partial_density_operator \<rho>" shows "positive (denote S \<rho>) \<and> trace (denote S \<rho>) \<le> trace \<rho> \<and> denote S \<rho> \<in> carrier_mat d d"
merkle_interface_axioms: shows "merkle_interface hash_transaction blinding_of_transaction merge_transaction"
elims_2: fixes x :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xa :: "'a tree" assumes "bst_wrt x xa" obtains P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "x = P__" and "xa = \<langle>\<rangle>"   | P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and l :: "'a tree" and a :: "'a" and r :: "'a tree" where "x = P" and "xa = \<langle>l, a, r\<rangle>" and "(\<forall>x\<in>set_tree l. P x a) \<and> Ball (set_tree r) (P a) \<and> bst_wrt P l \<and> bst_wrt P r"
zero_less_one: fixes one :: "'g"   and zero :: "'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
finite: shows "finite {x. jump f x \<noteq> 0 \<and> a < x \<and> x < c}"
H_seq_swap: fixes p :: "'a"   and x :: "'a"   and r :: "'a"   and y :: "'a"   and q :: "'a" assumes "H p x r"   and "H r y q" shows "H p (x \<cdot> y) q"
scount_eq_0I: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "alw (\<lambda>xs. \<not> P xs) \<omega>" shows "scount P \<omega> = 0"
butlast_conv_take: fixes xs :: "'c list" shows "butlast xs = take (degree xs) xs"
assms: shows "wf_net_tree (p1 \<parallel> p2)"
fmlookup_restrict_set: fixes A :: "'f set"   and m :: "'f \<rightharpoonup>\<^sub>f 'e"   and x :: "'f" shows "fmlookup (fmrestrict_set A m) x = (if x \<in> A then fmlookup m x else None)"
assms: shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! q)"
S2: shows "fn_to_set q = {x. \<exists>y. q x y = 0}"
corres: fixes v :: "'a"   and w :: "'b"   and u' :: "'a" shows "((v, w, u') \<in> G.E) = ((u', w, v) \<in> G.E)"
Un_insert_right: fixes A :: "'x set"   and a :: "'x"   and B :: "'x set" shows "A \<union> insert a B = insert a (A \<union> B)"
conjugate_square_eq_0_1: fixes x :: "'b" shows "(sq_norm_conjugate x = (0::'b)) = (x = (0::'b))"
image_ring_def: fixes f :: "'b \<Rightarrow> 'a"   and R :: "('b, 'c) ring_scheme" shows "image_ring f R \<equiv> monoid.extend (image_group f R) \<lparr>zero = f \<zero>\<^bsub>R\<^esub>, add = \<lambda>a b. f (inv_into (carrier R) f a \<oplus>\<^bsub>R\<^esub> inv_into (carrier R) f b)\<rparr>"
sigma_finite_iff_density_finite: fixes f :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable N" shows "sigma_finite_measure (density N f) = (AE x in N. f x \<noteq> \<infinity>)"
simps: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b llist" shows "chain R a = ((\<exists>x. a = LCons x LNil) \<or> (\<exists>xs x. a = LCons x xs \<and> chain R xs \<and> R x (lhd xs)))"
pullback_sums_map_middle2: fixes F :: "'a \<Rightarrow> 'c"   and xs :: "'a list"   and ds :: "'c list"   and d :: "'c"   and e :: "'c"   and es :: "'c list" assumes "map F (sums xs) = ds @ [d, e] @ es" shows "\<exists>as a bs. xs = as @ [a] @ bs \<and> map F (sums as) = ds @ [d] \<and> d = F (sum_list as) \<and> e = F (sum_list (as @ [a]))"
unique_connecting_path: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a" assumes "Tree G"   and "v \<in> V\<^bsub>G\<^esub>"   and "w \<in> V\<^bsub>G\<^esub>" shows "\<exists>!xs. v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
card_inj: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "f \<in> A \<rightarrow> B"   and "inj_on f A"   and "finite B" shows "card A \<le> card B"
Sup_atom: fixes x :: "'a" assumes "x \<noteq> \<bottom>"   and "\<forall>Y. (\<exists>y\<in>Y. x \<le> y) = (x \<le> \<Squnion> Y)" shows "atom x"
sharing_consis_def: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "sharing_consis \<S> ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> sharing_consistent \<S> \<O> sb"
compl: fixes \<Omega> :: "'b set"   and M :: "'b set set"   and A :: "'b set" assumes "Dynkin_system \<Omega> M"   and "A \<in> M" shows "\<Omega> - A \<in> M"
abs_eq: shows "Y = abs_var (PMF (pmf_of_set {(True, undefined), (False, undefined)}))"
of_safe_unsafe_match_eq: fixes m :: "of_match_field set"   and p :: "(32, 'b) simple_packet_ext_scheme" assumes "all_prerequisites m" shows "OF_match_fields m p = Some (OF_match_fields_unsafe m p)"
sum_outgoing_alt: fixes g :: "nat \<times> nat \<Rightarrow> 'capacity" assumes "\<forall>e. (0::'capacity) \<le> g e \<and> g e \<le> cf e" shows "\<forall>v\<in>cf.V. sum g (cf.outgoing v) = (\<Sum>u\<in>cf.V. g (v, u))"
assms_1: shows "functor (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F"
simps: fixes a1 :: "trm"   and a2 :: "trm" shows "a1 \<longmapsto> a2 = ((\<exists>x v\<^sub>2 T\<^sub>11 t\<^sub>12. a1 = (\<lambda>x:T\<^sub>11. t\<^sub>12) \<cdot> v\<^sub>2 \<and> a2 = t\<^sub>12[x \<mapsto> v\<^sub>2] \<and> x \<sharp> v\<^sub>2 \<and> val v\<^sub>2) \<or> (\<exists>t t' u. a1 = t \<cdot> u \<and> a2 = t' \<cdot> u \<and> t \<longmapsto> t') \<or> (\<exists>v t t'. a1 = v \<cdot> t \<and> a2 = v \<cdot> t' \<and> val v \<and> t \<longmapsto> t') \<or> (\<exists>X T\<^sub>11 T\<^sub>2 t\<^sub>12. a1 = (\<lambda>X<:T\<^sub>11. t\<^sub>12) \<cdot>\<^sub>\<tau> T\<^sub>2 \<and> a2 = t\<^sub>12[X \<mapsto>\<^sub>\<tau> T\<^sub>2] \<and> X \<sharp> (T\<^sub>11, T\<^sub>2)) \<or> (\<exists>t t' T. a1 = t \<cdot>\<^sub>\<tau> T \<and> a2 = t' \<cdot>\<^sub>\<tau> T \<and> t \<longmapsto> t'))"
sharing_consistent_preservation: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "sharing_consistent \<S> \<O> sb"   and "all_acquired sb \<inter> dom \<S> - dom \<S>' = {}"   and "all_unshared sb \<inter> dom \<S>' - dom \<S> = {}" shows "sharing_consistent \<S>' \<O> sb"
fmrel_on_fsetD: fixes x :: "'a"   and S :: "'a fset"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and m :: "('a, 'b) fmap"   and n :: "('a, 'c) fmap" assumes "x |\<in>| S"   and "fmrel_on_fset S P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
Ord: fixes x :: "tm" assumes "\<exists>i. x = ORD_OF i" shows "coding_tm x"
inj_map_strong: fixes x :: "'d pmf"   and xa :: "'d pmf"   and f :: "'d \<Rightarrow> 'e"   and fa :: "'d \<Rightarrow> 'e" assumes "\<And>z za. \<lbrakk>z \<in> set_pmf x; za \<in> set_pmf xa; f z = fa za\<rbrakk> \<Longrightarrow> z = za"   and "map_pmf f x = map_pmf fa xa" shows "x = xa"
1: assumes "\<lbrakk>M; \<A>\<rbrakk>\<^sub>c' \<I>" shows "\<lbrakk>M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>; \<A>\<rbrakk>\<^sub>c \<I>"
list_set_rel_range: fixes R :: "('b \<times> 'a) set" shows "Range (\<langle>R\<rangle>list_set_rel) = {S. finite S \<and> S \<subseteq> Range R}"
sat_primalI: fixes A :: "'a mat"   and x :: "'a vec"   and b :: "'a vec" assumes "[A*\<^sub>vx]\<le>b" shows "x \<in> sat_primal A b"
simps_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vwelldefined (Vabs cs \<Gamma>) = (pred_fmap id (fmmap vwelldefined \<Gamma>) \<and> list_all (\<lambda>(uu_, t). consts t |\<subseteq>| all_consts) cs)"
ext_hd_or_tl_axioms: shows "ext_hd_or_tl lexext"
not_aligned_mod_nz: fixes a :: "'b word"   and n :: "nat" assumes "\<not> is_aligned a n" shows "a mod 2 ^ n \<noteq> 0"
bi_unique_rel: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique R" shows "bi_unique (list_all2 R)"
simps_1: fixes a :: "'c" shows "rsuml (Inl (Inl a)) = Inl a"
Cond: fixes P :: "'d set"   and b :: "'d set"   and P\<^sub>1 :: "'d set"   and P\<^sub>2 :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c\<^sub>1 :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and c\<^sub>2 :: "('d, 'e, 'f) com" assumes "P \<subseteq> {s. (s \<in> b \<longrightarrow> s \<in> P\<^sub>1) \<and> (s \<notin> b \<longrightarrow> s \<in> P\<^sub>2)}"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>1 c\<^sub>1 Q,A"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P\<^sub>2 c\<^sub>2 Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Cond b c\<^sub>1 c\<^sub>2 Q,A"
simps: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "'b pmf"   and q :: "'c pmf" shows "rel_pmf R p q = (\<exists>pq. (\<forall>x xa. (x, xa) \<in> set_pmf pq \<longrightarrow> R x xa) \<and> map_pmf fst pq = p \<and> map_pmf snd pq = q)"
get_signs_def: fixes data :: "rat mat \<times> nat list list \<times> rat list list" shows "get_signs data = snd (snd data)"
dg_length: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "digraph \<alpha> \<CC>" shows "vcard \<CC> = 4\<^sub>\<nat>"
extensional0_scaleR: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and c :: "real" assumes "extensional0 S f" shows "extensional0 S (c *\<^sub>R f)"
shiftr_shiftr: fixes a :: "'b word"   and b :: "nat"   and c :: "nat" shows "a >> b >> c = a >> b + c"
add_uminus_conv_diff: fixes minus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and plus :: "'q \<Rightarrow> 'q \<Rightarrow> 'q"   and zero :: "'q"   and uminus :: "'q \<Rightarrow> 'q"   and a :: "'q"   and b :: "'q" assumes "class.group_add minus plus zero uminus" shows "plus a (uminus b) = minus a b"
kD_None: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<notin> kD rt" shows "rt dip = None"
is_Equality_def: fixes stateful_strand_step :: "('d, 'e) stateful_strand_step" shows "stateful_strand_step.is_Equality stateful_strand_step = (\<exists>x31 x32 x33. stateful_strand_step = \<langle>x31: x32 \<doteq> x33\<rangle>)"
id_decomp: fixes a :: "Vertex"   and b :: "Vertex"   and c :: "Vertex"   and d :: "Vertex"   and r :: "Vertex \<Rightarrow> Vertex" assumes "distinct [a, b, c, d]"   and "\<forall>x\<in>{a, b, c, d}. r x = x" shows "r = id"
TC_CASES1_NEQ: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and z :: "'b" assumes "R\<^sup>+\<^sup>+ x z" shows "R x z \<or> (\<exists>y. x \<noteq> y \<and> y \<noteq> z \<and> R x y \<and> R\<^sup>+\<^sup>+ y z)"
frag_cmul_zero: fixes x :: "'c \<Rightarrow>\<^sub>0 int" shows "frag_cmul 0 x = 0"
assms_2: shows "max_Var_floatarith fa \<le> DIM('a)"
rbb_rfd_dual: fixes R :: "('c \<times> 'd) set" shows "\<partial> \<circ> bb\<^sub>\<R> R = fd\<^sub>\<R> (\<smile> R) \<circ> \<partial>"
X_def: shows "X = (\<lambda>i\<in>\<^sub>\<circ>I. if i \<in>\<^sub>\<circ> J \<Rightarrow> x\<lparr>i\<rparr> | otherwise \<Rightarrow> SOME x. x \<in>\<^sub>\<circ> A i)"
le_divide_eq_numeral1_1: fixes a :: "'c"   and b :: "'c"   and w :: "num" shows "(a \<le> b / numeral w) = (a * numeral w \<le> b)"
eq_diff0: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "(x \<ominus>\<^bsub>R\<^esub> y = \<zero>\<^bsub>R\<^esub>) = (x = y)"
map_def: fixes f :: "'e \<Rightarrow> 'd"   and dxs :: "'e dlist" shows "Dlist.map f dxs = Dlist (remdups (list.map f (list_of_dlist dxs)))"
wf_plan_sound_system: fixes \<pi>s :: "plan_action list" assumes "Ball (set \<pi>s) wf_plan_action" shows "sound_system (set (map resolve_instantiate \<pi>s)) I (wm_to_state I) pddl_opr_to_act"
intro: fixes create_document :: "((_) heap, exception, (_) document_ptr) prog" assumes "create_document = create_document" shows "l_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M create_document"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset" assumes "incidence_system point_set block_collection"   and "simple_incidence_system_axioms block_collection" shows "simple_incidence_system point_set block_collection"
c: shows "steps0 (Suc 0 + off, l, r) (A @ shift B off) n = (s'' + off, l'', r'')"
map_prod_def: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd" shows "map_prod f g = (\<lambda>(x, y). (f x, g y))"
inv_def: fixes f :: "'b" shows "B.inv f = (SOME g. B.inverse_arrows f g)"
targetnodes_def: 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 xs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.targetnodes targetnode xs \<equiv> map targetnode xs"
u_NotexpiredSK_NotexpiredAK: fixes Ts :: "nat"   and evs :: "event list"   and Ta :: "nat" assumes "\<not> expiredSK Ts evs"   and "servKlife + Ts \<le> authKlife + Ta" shows "\<not> expiredAK Ta evs"
card_of_leq: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set" assumes "replete_set_category S"   and "A \<subseteq> Collect (category.terminal S)" shows "|A| <o cardSuc (cmax |UNIV| natLeq)"
in_pair_Un: fixes p :: "'e \<times> 'e"   and A :: "('e \<times> 'e) set"   and B :: "('e \<times> 'e) set" shows "(p \<in>\<^sub>p A \<union> B) = (p \<in>\<^sub>p A \<or> p \<in>\<^sub>p B)"
continuous_within_tendsto_compose: fixes a :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'d \<Rightarrow> 'b"   and F :: "'d filter" assumes "continuous (at a within s) f"   and "\<forall>\<^sub>F n in F. x n \<in> s"   and "(x \<longlongrightarrow> a) F" shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
cyclic_group_alt: shows "cyclic_group G = (\<exists>x. subgroup_generated G {x} = G)"
differing_vars_neg: fixes x :: "'Var"   and mem1 :: "'Var \<Rightarrow> 'Val"   and mem2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat" assumes "x \<notin> differing_vars_lists mem1 mem2 mems i" shows "fst (mems ! i) x = mem1 x \<and> snd (mems ! i) x = mem2 x"
dg: fixes k :: "nat" assumes "k < l" shows "g k \<in> carrier_mat m m"
frpar2_def: fixes p :: "fm" shows "frpar2 p = simpfm (qelim (prep p) ferrack2)"
nth_upt: fixes i :: "nat"   and k :: "nat"   and j :: "nat" assumes "i + k < j" shows "[i..<j] ! k = i + k"
subring_iff: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "subring H R = ring (R\<lparr>carrier := H\<rparr>)"
Ds_mono: fixes A :: "char list set option"   and A' :: "char list set option"   and es :: "expr list" assumes "A \<sqsubseteq> A'"   and "\<D>s es A" shows "\<D>s es A'"
h2a: shows "\<forall>i. MPoly_Type.degree (Const (1 - real (i mod 2))) var = 0"
Ax: fixes Ax :: "('a, 'b) Axioms"   and A :: "('a, 'b) Language" assumes "Axioms Ax"   and "A \<in> aAxioms Ax" shows "A \<in> Axioms.Theory Ax"
sdescending_snth_antimono: fixes w :: "'b stream" assumes "sdescending w" shows "antimono ((!!) w)"
gval_fold_cons: fixes g :: "'a gexp"   and gs :: "'a gexp list"   and s :: "'a \<Rightarrow> value option" shows "gval (fold gAnd (g # gs) (Bc True)) s = gval g s \<and>? gval (fold gAnd gs (Bc True)) s"
assms_3: shows "p \<in> dgrad_p_set d m"
proper_def: shows "proper \<equiv> map_fun eval\<kappa> id ((\<noteq>) None)"
list_rel_sv: fixes R :: "('d \<times> 'e) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>list_rel)"
finfun_All_update: fixes f :: "('a, bool) finfun"   and a :: "'a"   and b :: "bool" shows "finfun_All (finfun_update f a b) = (b \<and> finfun_All_except [a] f)"
zip_map1: fixes f :: "'e \<Rightarrow> 'c"   and xs :: "'e list"   and ys :: "'d list" shows "zip (map f xs) ys = map2 (\<lambda>x. Pair (f x)) xs ys"
IH: assumes "\<And>i_begina i_enda. (i_begina, i_enda) \<in> set blocks \<Longrightarrow> (i_begina, i_enda) \<in> set ((i_begin, i_end) # blocks)" shows "similar_mat (step_3_c x l k blocks B) B"
SAsCarAudioSystem: shows "SAs CarAudioSystem = {CDMode_CTRL, TunerMode_CTRL, AudioPlayer_CTRL, CDPlayer_CTRL, Root_CTRL}"
simps_1: fixes x :: "'b"   and hsl :: "'b hp list"   and hsr :: "'b hp list" shows "size_hps (Hp x hsl # hsr) = size_hps hsl + size_hps hsr + 1"
unitary: fixes n :: "nat"   and A :: "complex mat" assumes "gate n A" shows "unitary A"
sigma_algebra_axioms_def: fixes M :: "'b set set" shows "sigma_algebra_axioms M \<equiv> \<forall>A. range A \<subseteq> M \<longrightarrow> \<Union> (range A) \<in> M"
assms_1: shows "f \<in> \<omega>[F](\<lambda>x. 1)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_val\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs"   and "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "l_set_val_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs"