hcju/isabelleps huggingface dataset:
fst_map2_val_pair_subset: fixes xs :: "('a \<times> 'e) list"   and ys :: "('a \<times> 'f) list"   and g :: "('a \<times> 'e) list \<Rightarrow> ('a \<times> 'g) list"   and h :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'g) list"   and f :: "'a \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> 'g" assumes "oalist_inv_raw xs"   and "oalist_inv_raw ys"   and "map2_val_compat g"   and "map2_val_compat h" shows "fst ` set (map2_val_pair f g h xs ys) \<subseteq> fst ` set xs \<union> fst ` set ys"
lex_ext_unbounded_stri_imp_nstri: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "fst (lex_ext_unbounded f xs ys)" shows "snd (lex_ext_unbounded f xs ys)"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> int vec \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "int vec" assumes "\<And>i l acc. (\<And>x xa. \<lbrakk>\<not> i \<le> l; x = \<mu>' l l \<cdot>\<^sub>v acc - \<mu>' i l \<cdot>\<^sub>v gso_int l l; xa = map_vec (\<lambda>k. k div \<mu>' (l - 1) (l - 1)) x\<rbrakk> \<Longrightarrow> P i (l + 1) xa) \<Longrightarrow> P i l acc" shows "P a0 a1 a2"
UNIF1_subst_domain_subset: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "subst_domain \<sigma> \<subseteq> vars_mset E"
SUP_eq_iff: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and I :: "'d set"   and c :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> less_eq c (f i)" shows "(Sup (f ` I) = c) = (\<forall>i\<in>I. f i = c)"
analz_subset_parts: fixes H :: "msg set" shows "analz H \<subseteq> parts H"
subspace_def: fixes Sa :: "'a set" assumes "Sa \<subseteq> S" shows "subspace Sa = ((0::'a) \<in> Sa \<and> (\<forall>x\<in>Sa. \<forall>y\<in>Sa. x + y \<in> Sa) \<and> (\<forall>c. \<forall>x\<in>Sa. c *\<^sub>R x \<in> Sa))"
distinct_1: fixes x1 :: "'b" shows "bot.Value x1 \<noteq> bot.Bot"
iwlsFSwSTR_iterSTR_termFSwImorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "iwlsFSwSTR MOD" shows "termFSwImorph (iterSTR MOD) (iterAbsSTR MOD) MOD"
elims: fixes x :: "'e"   and xa :: "('f, 'g) epsym"   and y :: "epol" assumes "pol x xa = y" obtains uu_ :: "'e" and p :: "'f" where "x = uu_" and "xa = Oldp p" and "y = Cext"   | uv_ :: "'e" and \<sigma> :: "'g" where "x = uv_" and "xa = Guard \<sigma>" and "y = Fext"
polysizes_tame: fixes g' :: "graph"   and n :: "nat"   and v :: "nat"   and f :: "face"   and g :: "graph"   and p :: "nat" assumes "g' \<in> set (generatePolygon n v f g)"   and "Invariants.inv g"   and "f \<in> set (nonFinals g)"   and "v \<in> \<V> f"   and "3 \<le> n"   and "n < 4 + p"   and "n \<notin> set (polysizes p g)" shows "notame7 g'"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_7: fixes l :: "'d strand_label"   and X :: "'f list"   and F :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and G :: "(('e, 'f) Term.term \<times> ('e, 'f) Term.term) list"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) # A) = (l, \<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
not_final_Normal_step: fixes c :: "('a, 'b, 'c) com"   and s :: "'a"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option" assumes "\<not> final (c, Normal s)" shows "\<exists>c' s'. \<Gamma>\<turnstile> (c, Normal s) \<rightarrow> (c', s')"
vimage_UN: fixes f :: "'a \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "f -` \<Union> (B ` A) = (\<Union>x\<in>A. f -` B x)"
finter_absorb: fixes A :: "'c fset" shows "A |\<inter>| A = A"
s5: fixes u :: "'a"   and v :: "'a" assumes "u \<in> {x. f x = (0::'b)}"   and "v \<in> {x. f x = (0::'b)}" shows "u + v \<in> {x. f x = (0::'b)}"
cases: fixes r :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and a1 :: "'k"   and a2 :: "'k list"   and a3 :: "'k" assumes "rtrancl_path r a1 a2 a3" obtains x :: "'k" where "a1 = xb__" and "a2 = []" and "a3 = xb__"   | x :: "'k" and y :: "'k" and ys :: "'k list" and z :: "'k" where "a1 = x" and "a2 = y # ys" and "a3 = z" and "r x y" and "rtrancl_path r y ys z"
node_net_state: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and s :: "'b net_state"   and i :: "nat"   and T :: "((nat \<Rightarrow> 'a) \<times> 'b, 'c seq_action) automaton"   and R\<^sub>i :: "nat set"   and S :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> 'c node_action \<Rightarrow> bool"   and U :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "(\<sigma>, s) \<in> oreachable (\<langle>i : T : R\<^sub>i\<rangle>\<^sub>o) S U" shows "\<exists>\<zeta> R. s = NodeS i \<zeta> R"
l_insert_before_wf_def: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog" shows "l_insert_before_wf heap_is_wellformed type_wf known_ptr known_ptrs insert_before get_child_nodes \<equiv> l_known_ptrs known_ptr known_ptrs \<and> l_insert_before_wf_axioms heap_is_wellformed type_wf known_ptrs insert_before get_child_nodes"
subst_term_combine: fixes fresh_idns :: "variable list"   and insts :: "((variable \<times> typ) \<times> Core.term) list"   and t :: "Core.term" 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 ` (fv t \<union> \<Union> (fv ` snd ` set insts) \<union> fst ` set insts)" shows "subst_term insts t = subst_term (zip (zip fresh_idns (map snd (map fst insts))) (map snd insts)) (subst_term (zip (map fst insts) (map2 Fv fresh_idns (map snd (map fst insts)))) t)"
ceuclidean_representation: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and CBasis :: "'a set"   and x :: "'a" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis" shows "comm_monoid_add.sum plus zero (\<lambda>b. scaleC (cinner b x) b) CBasis = x"
\<delta>: shows "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"
unwind_dec: fixes \<Delta>s :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set"   and "next" :: "('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) \<Rightarrow> ('state \<Rightarrow> 'value list \<Rightarrow> 'state \<Rightarrow> 'value list \<Rightarrow> bool) set" assumes "\<And>\<Delta>. \<Delta> \<in> \<Delta>s \<Longrightarrow> next \<Delta> \<subseteq> \<Delta>s \<and> unwind_to \<Delta> (next \<Delta>)" shows "unwind (disjAll \<Delta>s)"
mat_of_rows_carrier_1: fixes n :: "nat"   and vs :: "'b vec list" shows "mat_of_rows n vs \<in> carrier_mat (length vs) n"
nth_mem: fixes n :: "nat"   and xs :: "'e list" assumes "n < length xs" shows "xs ! n \<in> set xs"
multiset_cases: fixes M :: "'a multiset" obtains "M = \<Empt>"   | x :: "'a" and N :: "'a multiset" where "M = add_mset x N"
simps_1: fixes v :: "'a"   and t :: "'a ifex"   and e :: "'a ifex" shows "ifex_no_twice (IF v t e) = (v \<notin> ifex_var_set t \<union> ifex_var_set e \<and> ifex_no_twice t \<and> ifex_no_twice e)"
affine_affine_hull: fixes s :: "'c set" shows "affine (affine hull s)"
assms_17: shows "0 < getBackjumpLevel state \<longrightarrow> getF state \<noteq> [] \<and> last (getF state) = getC state"
harry_meas_step_aux: fixes xs :: "bool list" assumes "True \<in> set xs" shows "harry_meas xs = Suc (harry_meas (harry_step xs))"
float_divl_lower_bound: fixes x :: "float"   and y :: "float"   and prec :: "nat" assumes "0 \<le> x"   and "0 \<le> y" shows "0 \<le> real_of_float (float_divl prec x y)"
hom_sum: fixes f :: "'c \<Rightarrow> 'b poly poly"   and X :: "'c set"   and x :: "'b"   and y :: "'b" shows "poly2 (sum f X) x y = (\<Sum>x\<in>X. poly2 (f x) x y)"
reduced_GB_nonzero_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "0 \<notin> reduced_GB F"
S5: shows "mod7 (c_fst (c_pair 5 (c_pair (sch_to_nat sch1) (sch_to_nat sch2)))) = 5"
has_vector_derivative_linepath_within: fixes a :: "'b"   and b :: "'b"   and x :: "real"   and S :: "real set" shows "(linepath a b has_vector_derivative b - a) (at x within S)"
reachable_def: fixes xs :: "'rep set" shows "KBPAlg.reachable xs \<equiv> {(x, y). y \<in> set (k_succs x)}\<^sup>* `` xs"
list_rel_simp_4: fixes x :: "'b"   and l :: "'b list"   and x' :: "'c"   and l' :: "'c list"   and R :: "('b \<times> 'c) set" shows "((x # l, x' # l') \<in> \<langle>R\<rangle>list_rel) = ((x, x') \<in> R \<and> (l, l') \<in> \<langle>R\<rangle>list_rel)"
list_eq_def: fixes xs :: "'b list"   and ys :: "'b list" shows "(xs \<approx> ys) = (set xs = set ys)"
coprime_imp_gcd_eq_1: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "algebraic_semidom.coprime times one a b" shows "gcd a b = one"
adopt_node_def: fixes document :: "(_) document_ptr"   and node :: "(_) node_ptr" shows "adopt_node document node = Heap_Error_Monad.bind (get_owner_document (cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r node)) (\<lambda>old_document. Heap_Error_Monad.bind (get_parent node) (\<lambda>parent_opt. Heap_Error_Monad.bind (case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> remove_child parent node) (\<lambda>_. if document \<noteq> old_document then Heap_Error_Monad.bind (get_disconnected_nodes old_document) (\<lambda>old_disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes old_document (remove1 node old_disc_nodes)) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document) (\<lambda>disc_nodes. set_disconnected_nodes document (node # disc_nodes)))) else return ())))"
dns3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.domain_near_semiring plus times domain_op less_eq less" shows "domain_op (plus x y) = plus (domain_op x) (domain_op y)"
prems_1: shows "Subobjs\<^sub>R P (hd (C' # Cs')) ((C' # Cs') @ [D])"
tab_simps_4: shows "local.dom \<tau> = t\<^sub>1"
input2state_cons: fixes x1 :: "nat"   and ia :: "value list"   and a :: "value" assumes "0 < x1"   and "x1 < length ia" shows "input2state (a # ia) $ x1 = input2state ia $ (x1 - 1)"
set_mkIde: fixes A :: "(('c + ('j \<Rightarrow> 'c)) set, 'c + ('j \<Rightarrow> 'c) \<Rightarrow> 'c + ('j \<Rightarrow> 'c)) concrete_category.arr set" assumes "A \<subseteq> S.Univ" shows "S.set (S.mkIde A) = A"
SP_pre_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "SP_pre M l A = (\<lambda>s0 r. \<exists>s l1 n. A s0 s \<and> (M, l, s, n, l1, r) \<in> Step)"
inter_left_comm: fixes X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "X Inter A B C D" shows "X Inter B A C D"
elims_3: fixes x :: "heap \<times> nat set" assumes "\<not> one_assn_raw x" obtains h :: "heap" and as :: "nat set" where "x = (h, as)" and "as \<noteq> {}"
Int_absorb: fixes A :: "'b set" shows "A \<inter> A = A"
sn: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in># X"   and "y \<in># Y"   and "z \<in># Z"   and "(x, y) \<in> s"   and "(y, z) \<in> ns" shows "(x, z) \<in> s"
empty_def: shows "Tree_Set.empty = \<langle>\<rangle>"
dghm_cn_comp_ArrMap_vdomain: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<D>\<^sub>\<circ> ((\<GG> \<^sub>D\<^sub>G\<^sub>H\<^sub>M\<circ> \<FF>)\<lparr>ArrMap\<rparr>) = \<AA>\<lparr>Arr\<rparr>"
assms_3: shows "x \<in> set (minimize_wrt P xs)"
l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent DocumentClass.known_ptr DocumentClass.type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document"
simps_2: fixes r :: "'a rexp" shows "verund [r] = r"
until_always: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(INF n. (SUP i\<in>{i. i < n}. - p i) \<squnion> p n) \<le> p n"
vP_intro: fixes k :: "'k2"   and v :: "'v2" shows "(\<exists>k' v'. m1.\<alpha> m k' = Some v' \<and> f (k', v') = Some (k, v)) = vP k v"
lift_is_hom: fixes f :: "'e \<Rightarrow> 'a"   and gens :: "'e set" assumes "f \<in> gens \<rightarrow> carrier G" shows "G.lift f \<in> hom \<F>\<^bsub>gens\<^esub> G"
assms_4: shows "finite {v. f v \<noteq> (0::'a)}"
algebraic_diff_cancel_left: fixes x :: "'a"   and y :: "'a" assumes "algebraic x" shows "algebraic (x - y) = algebraic y"
suminf_cong: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "\<And>n. f n = g n" shows "suminf f = suminf g"
bounded_clinear_prod_right: fixes prod :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "bounded_cbilinear prod" shows "bounded_clinear (bounded_cbilinear.prod_right prod)"
abs_RED: fixes \<tau> :: "ty"   and x :: "name"   and t :: "trm"   and \<sigma> :: "ty" assumes "\<forall>s\<in>RED \<tau>. t[x::=s] \<in> RED \<sigma>" shows "\<Lambda> x . t \<in> RED (\<tau> \<rightarrow> \<sigma>)"
mertens_remainder_aux_bound: fixes x :: "real" shows "(\<Sum>(p, i)\<in>{(p, i). prime p \<and> 1 < i \<and> real (p ^ i) \<le> x}. ln (real p) / real (p ^ i)) < 9 / pi\<^sup>2"
rep_eq: fixes x :: "'b \<Rightarrow> 'd"   and xa :: "('c, 'd) uexpr" shows "\<lbrakk>x \<dagger> xa\<rbrakk>\<^sub>e = (\<lambda>b. \<lbrakk>xa\<rbrakk>\<^sub>e (x b))"
monom_mult_in_pmdl: fixes pair_of_term :: "'b \<Rightarrow> 'c \<times> 'd"   and term_of_pair :: "'c \<times> 'd \<Rightarrow> 'b"   and b :: "'b \<Rightarrow>\<^sub>0 'e"   and B :: "('b \<Rightarrow>\<^sub>0 'e) set"   and c :: "'e"   and t :: "'c" assumes "term_powerprod pair_of_term term_of_pair"   and "b \<in> B" shows "term_powerprod.monom_mult pair_of_term term_of_pair c t b \<in> module.span (term_powerprod.mult_scalar pair_of_term term_of_pair) B"
exec_mbindFSave_failure: fixes ioprog :: "'f \<Rightarrow> 'e \<Rightarrow> ('d \<times> 'e) option"   and a :: "'f"   and \<sigma> :: "'e"   and S :: "'f list"   and M :: "'d list \<Rightarrow> 'e \<Rightarrow> (bool \<times> 'e) option" assumes "ioprog a \<sigma> = None" shows "(\<sigma> \<Turnstile> bind\<^sub>S\<^sub>E (mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e (a # S) ioprog) M) = (\<sigma> \<Turnstile> M [])"
integral_scaleR_right: fixes M :: "'c measure"   and c :: "real"   and f :: "'c \<Rightarrow> 'b" shows "LINT x|M. c *\<^sub>R f x = c *\<^sub>R integral\<^sup>L M f"
the_flag_Some: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and ip :: "nat"   and x :: "f" assumes "the (flag rt ip) = x"   and "ip \<in> kD rt" shows "flag rt ip = Some x"
fls_right_inverse_base_factor: fixes y :: "'a"   and f :: "'a fls" assumes "y \<noteq> (0::'a)" shows "fls_right_inverse (fls_base_factor f) y = fls_base_factor (fls_right_inverse f y)"
merge_rt_F\<^sub>m_def: fixes ha :: "'c \<Rightarrow> 'e"   and ma :: "'c \<Rightarrow> 'c \<Rightarrow> 'c option"   and hr :: "'d \<Rightarrow> 'f"   and mr :: "'d \<Rightarrow> 'd \<Rightarrow> 'd option" shows "merge_rt_F\<^sub>m ha ma hr mr \<equiv> merge_blindable (map_prod ha (map hr)) (merge_prod ma (merge_list mr))"
exe_antisym_imp_antisym: fixes l :: "('a \<times> 'a) list" assumes "exe_antisym l" shows "antisym (set l)"
emeasure_ge_1_iff: fixes M :: "'b measure"   and A :: "'b set" assumes "prob_space M" shows "(1 \<le> emeasure M A) = (emeasure M A = 1)"
eq0: fixes r :: "'a" assumes "r \<in> carrier R" shows "f r = g r"
simps_2: fixes n :: "nat"   and x :: "'a"   and y :: "'a"   and ys :: "'a list" shows "offset n x (y # ys) = (if y = x then Some n else offset (Suc n) x ys)"
in_ivl_def: shows "in_ivl \<equiv> map_fun id (map_fun rep_ivl id) (\<lambda>i (l, h). l \<le> Fin i \<and> Fin i \<le> h)"
B: shows "\<r>[f] \<star> src f = (f \<star> \<i>[src f]) \<cdot> \<a>[f, src f, src f]"
HEndPhase1_HInv4b_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase1 s s' p" shows "HInv4b s' p"
gamma1_y_const: shows "\<forall>x. gamma1 x \<bullet> j = a"
assms_2: shows "\<forall>\<^sub>F x in F. f1 x = f2 x"
hyps: fixes xs :: "'a list"   and ys :: "'a list" assumes "length (xs @ ys) + length xs < length (xsa__ @ ysa__) + length xsa__"   and "xs @ ys = ys @ xs" shows "\<exists>m n zs. concat (replicate m zs) = xs \<and> concat (replicate n zs) = ys"
prime_nat_iff: fixes n :: "nat" shows "normalization_semidom_class.prime n = (1 < n \<and> (\<forall>m. m dvd n \<longrightarrow> m = 1 \<or> m = n))"
rename_welltyped: fixes u :: "expr"   and U :: "Unary_PCF.type"   and x :: "char list"   and T :: "Unary_PCF.type"   and y :: "char list" assumes "u ::: U" shows "subst (x, T) \<langle>(y, T)\<rangle> u ::: U"
0: fixes u :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Src t = Trg u" shows "Nml (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) \<and> Dom (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Dom t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Dom u \<and> Cod (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Cod t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> Cod u \<and> Src (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Src u \<and> Trg (t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) = Trg t"
choose_iapath_def: fixes G :: "'b pair_pre_digraph"   and u :: "'b"   and v :: "'b" shows "pair_pre_digraph.choose_iapath G u v \<equiv> let chosen_path = \<lambda>u v. SOME p. pre_digraph.gen_iapath (with_proj G) (verts3 (with_proj G)) u p v in if pair_pre_digraph.direct_arc (u, v) = (u, v) then chosen_path u v else rev_path (chosen_path v u)"
le_of_comp_of_ords_gen: fixes lt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y. lt x y \<Longrightarrow> le x y" shows "le_of_comp (comp_of_ords le lt) = le"
simps_2: fixes uv :: "const" shows "plus_const Any uv = Any"
SPEC_rule_conj_leofI1: fixes m :: "'a nres"   and \<Phi> :: "'a \<Rightarrow> bool"   and \<Psi> :: "'a \<Rightarrow> bool" assumes "m \<le> SPEC \<Phi>"   and "m \<le>\<^sub>n SPEC \<Psi>" shows "m \<le> SPEC (\<lambda>s. \<Phi> s \<and> \<Psi> s)"
exp_add_commuting: fixes x :: "'d"   and y :: "'d" assumes "x * y = y * x" shows "exp (x + y) = exp x * exp y"
bernoulli'_def: fixes n :: "nat" shows "bernoulli' n = (if n = 1 then 1 / 2 else bernoulli n)"
holomorphic_Dirichlet_L: fixes n :: "nat"   and \<chi> :: "nat \<Rightarrow> complex"   and A :: "complex set" assumes "1 < n"   and "\<chi> \<noteq> principal_dchar n \<and> dcharacter n \<chi> \<or> \<chi> = principal_dchar n \<and> 1 \<notin> A" shows "Dirichlet_L n \<chi> holomorphic_on A"
dsum_zero_func: fixes I :: "'h set"   and A :: "'h \<Rightarrow> ('i, 'j) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)" shows "prod_zero I A \<in> carr_dsumag I A"
a_4: shows "list_all is_Update (unlabel (transaction_updates T))"
f_succ_alt: fixes \<delta> :: "('a, 'b) ta_rule set" shows "f_succ \<delta> = {(q, q'). \<exists>l qs. q \<rightarrow> l qs \<in> \<delta> \<and> q' \<in> set qs}"
sigma_sets_sigma_sets_eq: fixes M :: "'c set set"   and S :: "'c set" assumes "M \<subseteq> Pow S" shows "sigma_sets S (sigma_sets S M) = sigma_sets S M"
singletonI: fixes a :: "'e" shows "a \<in> {a}"
intro: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.monoid G"   and "Group.comm_monoid_axioms G" shows "Group.comm_monoid G"
fdomain_vintersection_vsubset: fixes A :: "V"   and B :: "V" shows "\<D>\<^sub>\<bullet> (A \<inter>\<^sub>\<circ> B) \<subseteq>\<^sub>\<circ> \<D>\<^sub>\<bullet> A \<inter>\<^sub>\<circ> \<D>\<^sub>\<bullet> B"
a: shows "a = p\<^sup>2 - q\<^sup>2"
productF_simps_2: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM" assumes "productF A B FAIL AB" shows "inputs AB = inputs A"
find_ce_impl_def: shows "find_ce_impl \<equiv> stat_start_nres \<bind> (\<lambda>_. let os = goinitial_impl in FOREACH\<^sub>C\<^bsup>\<lambda>it os. fgl_outer_invar it (goGS_\<alpha> os)\<^esup> V0 go_is_no_brk_impl (\<lambda>v0 s0. if \<not> go_is_done_impl v0 s0 then let s = (None, ginitial_impl v0 s0) in WHILE\<^sub>T\<^bsup>\<lambda>(brk, s). fgl_invar v0 (oGS_\<alpha> (goD_impl s0)) (brk, snd (gGS_\<alpha> s))\<^esup> (\<lambda>(brk, s). brk = None \<and> \<not> gpath_is_empty_impl s) (\<lambda>(l, s). gselect_edge_impl s \<bind> (\<lambda>(vo, s). case vo of None \<Rightarrow> gpop_impl s \<bind> (\<lambda>s. RETURN (None, s)) | Some v \<Rightarrow> if gis_on_stack_impl v s then gcollapse_impl v s \<bind> (\<lambda>s. last_is_acc_impl s \<bind> (\<lambda>b. if b then ce_impl s else RETURN (None, s))) else if \<not> gis_done_impl v s then RETURN (None, gpush_impl v s) else RETURN (None, s))) s \<bind> (\<lambda>(brk, s). RETURN (gto_outer_impl brk s)) else RETURN s0) os \<bind> (\<lambda>os. stat_stop_nres \<bind> (\<lambda>_. RETURN (goBrk_impl os))))"