hcju/isabelleps huggingface dataset:
axioms_2: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   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"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and remove_child_locs :: "(_) object_ptr \<Rightarrow> (_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and remove :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool" assumes "l_remove_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs set_child_nodes set_child_nodes_locs get_parent get_parent_locs get_owner_document get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs remove_child remove_child_locs remove type_wf known_ptr known_ptrs" shows "l_get_owner_document get_owner_document"
sym_order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "sympartp (\<sim>) x y"   and "z \<sim> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sim> x"
ideD_3: fixes a :: "'a arrow_of_spans_data" assumes "V.ide a" shows "V.cod a = a"
sqrt_int_def: fixes x :: "int" shows "sqrt_int x = root_int 2 x"
\<Delta>'''_bounded_countable_network: shows "bounded_countable_network \<Delta>'''"
ie: shows "L\<lbrakk>Vx : A \<turnstile> e : B\<rbrakk>\<^bsub>CI T\<^esub> \<rightarrow> IMor [A,e,B]\<^bsub>T\<^esub>"
Compl_partition2: fixes A :: "'b set" shows "- A \<union> A = UNIV"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ptr_in_heap: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'k option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "l_get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_lemmas type_wf"   and "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter" shows "ptr |\<in>| shadow_root_ptr_kinds h"
kripke_rels_worlds: fixes w :: "'e"   and w' :: "'e"   and M :: "('f, 'g, 'e) KripkeStructure"   and a :: "'f" assumes "(w, w') \<in> relations M a"   and "kripke M" shows "w \<in> worlds M \<and> w' \<in> worlds M"
unfold_le_opt: fixes o\<^sub>1 :: "'b option"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and o\<^sub>2 :: "'b option" shows "(o\<^sub>1 \<sqsubseteq>\<^bsub>Opt.le r\<^esub> o\<^sub>2) = (case o\<^sub>2 of None \<Rightarrow> o\<^sub>1 = None | Some y \<Rightarrow> case o\<^sub>1 of None \<Rightarrow> True | Some x \<Rightarrow> x \<sqsubseteq>\<^bsub>r\<^esub> y)"
vsubstEnv_updEnv: fixes ys :: "'varSort"   and y1 :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "rho [x \<leftarrow> X]_xs &[y1 // y]_ys = rho &[y1 // y]_ys [x \<leftarrow> X #[y1 // y]_ys]_xs"
Zp_int_inc_zero': fixes x :: "nat \<Rightarrow> int" shows "[0] \<cdot> x = \<zero>"
linear_span_pOp_closed: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and A :: "'f set"   and H :: "'e set"   and a :: "'e"   and b :: "'e" assumes "R module M"   and "ideal R A"   and "H \<subseteq> carrier M"   and "a \<in> linear_span R M A H"   and "b \<in> linear_span R M A H" shows "a \<plusminus>\<^bsub>M\<^esub> b \<in> linear_span R M A H"
vals_merge_distinct: fixes t :: "('d, 'c) bintree"   and r :: "('d, 'c) bintree" assumes "distinct (bt_dfs val t)"   and "distinct (bt_dfs val r)"   and "set (bt_dfs val t) \<inter> set (bt_dfs val r) = {}" shows "distinct (bt_dfs val (merge t r))"
prepend_total: fixes f :: "nat \<Rightarrow> nat option"   and vs :: "nat list" assumes "total1 f" shows "total1 (vs \<odot> f)"
eq_add_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and x :: "'d"   and y :: "'d" assumes "class.ab_group_add plus zero minus uminus" shows "(x = plus x y) = (y = zero)"
gtD: fixes l :: "nat"   and n :: "nat" assumes "l < n" shows "\<exists>m. n = Suc m \<and> l \<le> m"
rel_cases: fixes R :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and a :: "('n, 'p, 'q, 'r, 's, 't) U"   and b :: "('o, 'p, 'q, 'r, 's, 't) U" assumes "rell_U R a b" obtains xa :: "(('n, 'p, 'q, 'r, 's, 't) U, 'n, 'p, 'q, 'r, 's, 't) G" and ya :: "(('o, 'p, 'q, 'r, 's, 't) U, 'o, 'p, 'q, 'r, 's, 't) G" where "a = C_U xa" and "b = C_U ya" and "rell_G (rell_U R) R xa ya"
weakCongSimE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q'" obtains P' :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<Longrightarrow>\<^sub>\<tau> P'" and "(\<Psi>, P', Q') \<in> Rel"
restrict_and_extend0_aezfun_is_aezfun: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "f \<in> aezfun_set" shows "f \<down> A \<in> aezfun_set"
lcd: shows "l cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
prems: shows "\<forall>l. wf\<^sub>l\<^sub>s\<^sub>t V (proj l (A @ [a]))"
psimps_5: fixes v :: "'b"   and M :: "'b trm" assumes "unify_dom (Var v, M)" shows "unify (Var v) M = (if Var v \<prec> M then None else Some [(v, M)])"
mono_read_FD: fixes P :: "'b \<Rightarrow> 'c process"   and Q :: "'b \<Rightarrow> 'c process"   and c :: "'b \<Rightarrow> 'c" assumes "\<And>x. P x \<le> Q x" shows "read c UNIV P \<le> read c UNIV Q"
class_of_1_in: fixes U :: "'a set" assumes "U \<in> I" shows "\<one>\<^bsub>U\<^esub> \<in> \<FF> U"
Rep_fs_set_inverse: fixes x :: "'b fs_set" shows "Abs_fs_set (Rep_fs_set x) = x"
sameDom_sym: fixes inp :: "'a \<Rightarrow> 'b option"   and inp' :: "'a \<Rightarrow> 'c option" shows "sameDom inp inp' = sameDom inp' inp"
Power_dfa_def: fixes M :: "'b nfa" assumes "nfa M" shows "nfa.Power_dfa M = \<lparr>dfa.states = {HF (nfa.epsclo M q) |q. q \<in> Pow (nfa.states M)}, init = HF (nfa.epsclo M (nfa.init M)), final = {HF (nfa.epsclo M Q) |Q. Q \<subseteq> nfa.states M \<and> Q \<inter> nfa.final M \<noteq> {}}, nxt = \<lambda>Q x. HF (\<Union>q\<in>nfa.epsclo M (hfset Q). nfa.epsclo M (nfa.nxt M q x))\<rparr>"
sk: shows "\<forall>x\<^sub>0 x\<^sub>1. \<not> list_emb P (xs @ x\<^sub>0) x\<^sub>1 \<or> sk\<^sub>0 x\<^sub>0 x\<^sub>1 @ sk\<^sub>1 x\<^sub>0 x\<^sub>1 = x\<^sub>1 \<and> list_emb P xs (sk\<^sub>0 x\<^sub>0 x\<^sub>1) \<and> list_emb P x\<^sub>0 (sk\<^sub>1 x\<^sub>0 x\<^sub>1)"
ctor_rel_induct: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and IR :: "'c G \<Rightarrow> 'd G \<Rightarrow> bool" assumes "\<And>y xa. rel_pre_G R IR y xa \<Longrightarrow> IR (ctor_G y) (ctor_G xa)" shows "rel_G R \<le> IR"
minGraphProps7: fixes g :: "graph"   and f :: "face"   and v :: "nat" assumes "minGraphProps g"   and "f \<in> \<F> g"   and "v \<in> \<V> f" shows "f \<in> set (facesAt g v)"
zero_le_even_power': fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq zero (power.power one times a (2 * n))"
maintained_def: fixes R :: "('b, 'c) labeled_graph \<times> ('b, 'c) labeled_graph"   and G :: "('b, 'd) labeled_graph" shows "maintained R G \<equiv> \<forall>f. graph_homomorphism (fst R) G f \<longrightarrow> extensible R G f"
assm': fixes i :: "'c" assumes "i \<in> I" shows "subgroup (S i) (G\<lparr>carrier := J\<rparr>)"
fun_eq_onD: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and a :: "'b" assumes "fun_eq_on f g A"   and "a \<in> A" shows "f a = g a"
fold_maybe_or_rev: fixes l :: "trilean list"   and b :: "trilean" shows "fold (\<or>?) l b = fold (\<or>?) (rev l) b"
simps_3: fixes n :: "nat"   and m :: "nat"   and ms :: "nat list" shows "omega_sum_aux (Suc n) (m # ms) = \<omega> \<up> ord_of_nat n * ord_of_nat m + omega_sum_aux n ms"
rel_option_None1: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d option" shows "rel_option P None x = (x = None)"
OO_eq: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "R OO (=) = R"
test_bit_cong: fixes xa :: "'b"   and y :: "'b"   and x :: "nat" assumes "xa = y" shows "xa !! x = y !! x"
lm080: shows "allocationsUniverse \<subseteq> runiqs \<inter> Pow (UNIV \<times> (UNIV - {{}}))"
vimage_mono: fixes A :: "'a set"   and B :: "'a set"   and f :: "'b \<Rightarrow> 'a" assumes "A \<subseteq> B" shows "f -` A \<subseteq> f -` B"
rsplit0_cs: fixes t :: "MIR.num"   and x :: "real"   and bs :: "real list" shows "\<forall>(p, n, s)\<in>set (rsplit0 t). (Ifm (x # bs) p \<longrightarrow> Inum (x # bs) t = Inum (x # bs) (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p"
sscan_smap: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and xs :: "'e stream"   and a :: "'c" shows "sscan f (smap g xs) a = sscan (f \<circ> g) xs a"
soundnessForSAT: fixes F0 :: "Literal list list"   and decisionVars :: "nat set"   and state0 :: "State"   and state :: "State" assumes "vars F0 \<subseteq> decisionVars"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "\<not> formulaFalse F0 (elements (getM state))"   and "decisionVars \<subseteq> vars (elements (getM state))" shows "model (elements (getM state)) F0"
is_serial_sol_then_valid_plan_2: fixes ops :: "(nat, nat) sas_plus_operator list" assumes "is_serial_solution_for_problem abs_prob ops" shows "\<exists>s'\<in>G. path_to I ((map (fst \<circ> the \<circ> lookup_action) \<circ> rem_condless_ops I) ops) s'"
e: fixes q :: "real \<Rightarrow> complex" assumes "path q"   and "pathstart q = pathstart h"   and "pathfinish q = pathfinish h"   and "\<forall>t\<in>{0..1}. cmod (q t - h t) < e" shows "homotopic_paths (- {z}) h q"
Par1B: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<longmapsto>\<^sub>l a\<guillemotleft>x\<guillemotright> \<prec> P' \<parallel> Q"
inv_matching_def: fixes C :: "'b set"   and F :: "'a set set"   and M :: "'a set set" shows "inv_matching C F M = (matching M \<and> M \<subseteq> E \<and> card C \<le> 2 * card M \<and> (\<forall>e\<in>M. \<forall>f\<in>F. e \<inter> f = {}))"
1578: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (x \<sqinter> (y \<squnion> z)) \<sqinter> (x \<sqinter> y) = bot"
simps: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a1 :: "'e"   and a2 :: "'e" shows "r\<^sup>+\<^sup>+ a1 a2 = ((\<exists>a b. a1 = a \<and> a2 = b \<and> r a b) \<or> (\<exists>a b c. a1 = a \<and> a2 = c \<and> r\<^sup>+\<^sup>+ a b \<and> r b c))"
strict_implies_order: fixes a :: "'b word"   and b :: "'b word" assumes "a <s b" shows "a \<le>s b"
assms_1: fixes B :: "'a set" assumes "B \<in># \<B>" shows "K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}"
scale_right_diff_distrib: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'b"   and x :: "'c"   and y :: "'c" assumes "module scale" shows "scale a (x - y) = scale a x - scale a y"
assms: fixes f :: "('a \<times> 'b) set" assumes "graph_homomorphism A G f" shows "extensible (A, B) G f"
language_state_for_inputs_map_fst: fixes io :: "('e \<times> 'f) list"   and M :: "('e, 'f, 'g) FSM"   and q :: "'g"   and T :: "'e list set" assumes "io \<in> LS M q"   and "map fst io \<in> T" shows "io \<in> LS\<^sub>i\<^sub>n M q T"
simps_31: fixes \<C> :: "t_context"   and ts :: "checker_type" shows "check_single \<C> Current_memory ts = (if memory \<C> \<noteq> None then type_update ts [] (Type [T_i32]) else Bot)"
solution_upd_but1: fixes ap :: "nat \<Rightarrow> 'b"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and p :: "nat"   and a :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and n :: "nat"   and c :: "nat \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'b" assumes "ap = A p"   and "\<forall>i j. i \<noteq> p \<longrightarrow> a i j = A i j"   and "p < n" shows "solution (\<lambda>i. if i = p then ap else (\<lambda>j. a i j - c i * ap j)) n x = solution A n x"
relation_rules_4: fixes Q :: "'b \<Rightarrow> bool"   and x :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "\<not> Q x \<Longrightarrow> P x" shows "(P \<or> Q) x"
transfer: shows "(\<lambda>f g. if g = 0 then f else let n = ratfps_subdegree g; h = ratfps_shift n g in ratfps_of_poly (ratfps_cutoff n (f * inverse h)) * h) = (mod)"
SCp_swap_applies_3: fixes \<Gamma> :: "'b formula multiset"   and G :: "'b formula"   and H :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> G, H, \<Delta>" shows "\<Gamma> \<Rightarrow> H, G, \<Delta>"
measure_zero_top: fixes M :: "'d measure"   and A :: "'d set" assumes "emeasure M A = top" shows "Sigma_Algebra.measure M A = 0"
einterpret_euler_err_fas: fixes CX :: "floatarith list"   and vs :: "real list"   and X0 :: "'d list"   and h :: "floatarith" assumes "einterpret CX vs \<in> Csafe"   and "length X0 = DIM('c)"   and "length CX = DIM('c)"   and "DIM('c) \<le> length vs" shows "einterpret (euler_err_fas X0 h CX) vs = ((interpret_floatarith h vs)\<^sup>2 / 2) *\<^sub>R blinfun_apply (ode_d 0 (einterpret CX vs) (ode (einterpret CX vs))) (ode (einterpret CX vs))"
sup_aci_3: fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_sup sup less_eq less" shows "sup x (sup y z) = sup y (sup x z)"
elementMemElements: fixes x :: "'b \<times> bool"   and M :: "('b \<times> bool) list" assumes "x \<in> set M" shows "element x \<in> set (elements M)"
fps_X_pow_binomial_coeffs: fixes p :: "nat"   and N :: "nat"   and k :: "nat" assumes "prime p" shows "(1 + fps_X ^ p) ^ N $ k = int (if p dvd k then N choose k div p else 0)"
zhmset_of_prod_list: fixes Ms :: "hmultiset list" shows "zhmset_of (prod_list Ms) = prod_list (map zhmset_of Ms)"
auto_ll_on_open_axioms: shows "auto_ll_on_open ode Csafe"
new_shadow_root_no_child_nodes: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^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 new_shadow_root_ptr) \<rightarrow>\<^sub>r []"
simps: fixes a1 :: "lam"   and a2 :: "lam" shows " a1 \<longrightarrow>\<^sub>1\<^sup>* a2 = ((\<exists>M. a1 = M \<and> a2 = M) \<or> (\<exists>M1 M2 M3. a1 = M1 \<and> a2 = M3 \<and>  M1 \<longrightarrow>\<^sub>1\<^sup>* M2 \<and>  M2 \<longrightarrow>\<^sub>1 M3))"
rel2p_def: fixes R :: "('g \<times> 'h) set"   and x :: "'g"   and y :: "'h" shows "rel2p R x y \<equiv> (x, y) \<in> R"
assms_3: shows "apply_operations (es @ [Deliver (i, Delete n)]) = Some xs2"
P3_def: shows "P3 \<equiv> \<lambda>tp. \<exists>k. tp = (Bk \<up> k, <1>)"
hom_into_singleton_iff: fixes G :: "('b, 'c) monoid_scheme"   and h :: "'b \<Rightarrow> 'd"   and a :: "'d" assumes "Group.group G" shows "(h \<in> hom G (singleton_group a)) = (h \<in> carrier G \<rightarrow> {a})"
ord_class_def: shows "OFCLASS('a, ord_class) \<equiv> OFCLASS('a, type_class)"
normalize_idem_imp_is_unit_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = a" shows "dvd.dvd times a one = (a = one)"
winding_number_linepath_pos_lt: fixes b :: "complex"   and a :: "complex"   and z :: "complex" assumes "0 < Im ((b - a) * cnj (b - z))" shows "0 < Re (winding_number (linepath a b) z)"
coalesce_eq_OK_ub_D: fixes A :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b err"   and xs :: "'b list"   and n :: "nat"   and ys :: "'b list"   and zs :: "'b list"   and us :: "'b list" assumes "semilat (err A, Err.le r, lift2 f)"   and "xs \<in> list n A"   and "ys \<in> list n A"   and "coalesce (xs [\<squnion>\<^bsub>f\<^esub>] ys) = OK zs \<and> xs [\<sqsubseteq>\<^bsub>r\<^esub>] us \<and> ys [\<sqsubseteq>\<^bsub>r\<^esub>] us \<and> us \<in> list n A" shows "zs [\<sqsubseteq>\<^bsub>r\<^esub>] us"
path_connected_space_euclidean_subtopology: fixes S :: "'a set" shows "path_connected_space (top_of_set S) = path_connected S"
isLb_def: fixes R :: "'b set"   and S :: "'b set"   and x :: "'b" shows "isLb R S x = (x <=* S \<and> x \<in> R)"
sams2_suma2__conga123: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and G :: "'p"   and H :: "'p"   and I :: "'p" assumes "SAMS A B C D E F"   and "SAMS A' B' C' D E F"   and "A B C D E F SumA G H I"   and "A' B' C' D E F SumA G H I" shows "A B C CongA A' B' C'"
restrict_nested: fixes A :: "nat set"   and B :: "nat set"   and x :: "'b option list" shows "restrict A (restrict B x) = restrict (A \<inter> B) x"
vpra_induct: fixes xs :: "'edge list"   and ys :: "'edge list"   and P :: "'edge list \<Rightarrow> 'edge list \<Rightarrow> bool" assumes "valid_path_rev_aux xs ys"   and "\<And>cs. P cs []"   and "\<And>cs a as. \<lbrakk>intra_kind (kind a); valid_path_rev_aux cs as; P cs as\<rbrakk> \<Longrightarrow> P cs (as @ [a])"   and "\<And>cs a as Q p f. \<lbrakk>kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f; valid_path_rev_aux (a # cs) as; P (a # cs) as\<rbrakk> \<Longrightarrow> P cs (as @ [a])"   and "\<And>cs a as Q r p fs. \<lbrakk>kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; cs = []; valid_path_rev_aux [] as; P [] as\<rbrakk> \<Longrightarrow> P cs (as @ [a])"   and "\<And>cs a as Q r p fs c' cs'. \<lbrakk>kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs; cs = c' # cs'; valid_path_rev_aux cs' as; c' \<in> get_return_edges a; P cs' as\<rbrakk> \<Longrightarrow> P cs (as @ [a])" shows "P xs ys"
bliftM2_def: fixes f :: "'a \<rightarrow> 'b \<rightarrow> 'c" shows "bliftM2 f \<equiv> \<Lambda> x y. unbox\<cdot>x >>= (\<Lambda> x'. unbox\<cdot>y >>= (\<Lambda> y'. box\<cdot>(f\<cdot>x'\<cdot>y')))"
genby_set_lconjby_set_lconjby_closed: fixes g :: "'g"   and x :: "'g"   and A :: "'g set" assumes "g \<in> G"   and "x \<in> normal_closure A" shows "lconjby g x \<in> normal_closure A"
disjoint_iff_not_equal: fixes A :: "'m set"   and B :: "'m set" shows "(A \<inter> B = {}) = (\<forall>x\<in>A. \<forall>y\<in>B. x \<noteq> y)"
amor_insert: fixes t :: "'b tree"   and x :: "'b" assumes "bst t" shows "real (T_insert x t) + \<Phi> (Splay_Tree.insert x t) - \<Phi> t \<le> 4 * \<phi> t + 3"
image_insert: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and B :: "'b set" shows "f ` insert a B = insert (f a) (f ` B)"
eq2: fixes A :: "'a set" assumes "A \<in> sets M" shows "(T' \<circ> T) -` A \<inter> space M = A"
continuous_map_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f \<equiv> (\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>U. openin Y U \<longrightarrow> openin X {x \<in> topspace X. f x \<in> U})"
rel_fun_def_butlast: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'g \<Rightarrow> bool"   and T :: "'f \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'f"   and g :: "'d \<Rightarrow> 'g \<Rightarrow> 'h" shows "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> rel_fun S T (f x) (g y))"
hd_nonzero_imp_sign_changes_const_aux: fixes x\<^sub>0 :: "real"   and ps' :: "real poly list" assumes "poly (hd ps) x\<^sub>0 \<noteq> 0"   and "ps' \<in> set (split_sign_changes ps x\<^sub>0)" shows "\<forall>\<^sub>F x in at x\<^sub>0. sign_changes ps' x = sign_changes ps' x\<^sub>0"
assms_4: shows "(max 0 b + 1) / (1 - a) \<le> real n0"
succ_rel_insert_Some: fixes list :: "'b list"   and oid :: "'b"   and ref :: "'b" assumes "distinct list" shows "succ_rel (insert_spec list (oid, Some ref)) = insert_alt (succ_rel list) (oid, ref)"
list_nth_append1: fixes i :: "nat"   and x :: "'a list"   and b :: "'a"   and y :: "'a list" assumes "i < length x" shows "(b # x) ! i = (b # x @ y) ! i"
ucast_NOT_down: fixes x :: "'b word" assumes "is_down UCAST('b \<rightarrow> 'c)" shows "UCAST('b \<rightarrow> 'c) (NOT x) = NOT (UCAST('b \<rightarrow> 'c) x)"
lift_Use_node: fixes V :: "'g"   and Use :: "'h \<Rightarrow> 'g set"   and n :: "'h"   and E :: "'h"   and X :: "'h"   and H :: "'g set"   and L :: "'g set" assumes "V \<in> Use n" shows "(Node n, V) \<in> lift_Use_set Use E X H L"
argmax_non_empty_iff: fixes X :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite X"   and "X \<noteq> {}" shows "argmax f X \<noteq> {}"
isRefV_def: fixes v :: "Value" shows "isRefV v = (case v of objV C a \<Rightarrow> True | arrV T a \<Rightarrow> True | nullV \<Rightarrow> True | _ \<Rightarrow> False)"
ns_def: shows "ns \<equiv> {(x, y). nss x y True}"
tolerance_def: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "tolerance A equiv \<equiv> symmetric A equiv \<and> reflexive A equiv"
tauCases': fixes P :: "pi"   and Rs :: "residual" assumes "\<tau>.P \<longmapsto> Rs" obtains Pa :: "pi" where "\<tau>.P = \<tau>.Pa" and "Rs = \<tau> \<prec> Pa"