hcju/isabelleps huggingface dataset:
divide_fps_const: fixes f :: "'a fps"   and c :: "'a" shows "f / fps_const c = fps_const (inverse c) * f"
expandTrans_4: fixes R :: "pi"   and P :: "pi"   and Q :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "(R, expandSet P Q) \<in> sumComposeSet"   and "hnf P"   and "hnf Q" shows "P \<parallel> Q \<longmapsto> a<\<nu>x> \<prec> P' = R \<longmapsto> a<\<nu>x> \<prec> P'"
lift_ltl_transformer_def: fixes f :: "'b ltl \<Rightarrow> 'c \<Rightarrow> 'b ltl" shows "lift_ltl_transformer f \<equiv> (\<forall>\<phi> \<psi> \<nu>. \<phi> \<equiv>\<^sub>P \<psi> \<longrightarrow> f \<phi> \<nu> \<equiv>\<^sub>P f \<psi> \<nu>) \<and> (\<forall>\<phi> \<nu>. nested_propos (f \<phi> \<nu>) \<subseteq> nested_propos \<phi>)"
intro: fixes C1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category C1"   and "category C2" shows "product_category C1 C2"
that: fixes ns\<^sub>1 :: "'node list"   and ns\<^sub>2 :: "'node list" assumes "g \<turnstile> n-ns\<^sub>1\<rightarrow>x"   and "g \<turnstile> x-ns\<^sub>2\<rightarrow>m"   and "ns = butlast ns\<^sub>1 @ ns\<^sub>2" shows "thesis"
aff_dim_hyperplane: fixes a :: "'a"   and r :: "real" assumes "a \<noteq> (0::'a)" shows "aff_dim {x. a \<bullet> x = r} = int (DIM('a) - 1)"
in_path_image_part_circlepath: fixes w :: "complex"   and z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" assumes "w \<in> path_image (part_circlepath z r s t)"   and "s \<le> t"   and "0 \<le> r" shows "cmod (w - z) = r"
unsigned_not_eq: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and "and" :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and or :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xor :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and mask :: "nat \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and not :: "'b \<Rightarrow> 'b"   and w :: "'c word" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "semiring_1.unsigned one plus zero (NOT w) = take_bit LENGTH('c) (not (semiring_1.unsigned one plus zero w))"
update_butlast_eq0: fixes n :: "nat"   and ba :: "nat"   and aa :: "nat"   and lm1 :: "nat list"   and lm2 :: "nat list"   and lm3 :: "nat list"   and lm4 :: "nat list" assumes "Suc n \<le> ba - aa"   and "aa < ba"   and "length lm1 = aa"   and "length lm2 = Suc n"   and "length lm3 = ba - Suc (aa + n)" shows "(lm1 @ 0 \<up> n @ last lm2 # lm3 @ butlast lm2 @ 0 # lm4)[ba + n := last lm2, aa + n := 0] = lm1 @ 0 # 0 \<up> n @ lm3 @ lm2 @ lm4"
concurrent_append_expunge_independent: fixes i :: "'a"   and e1 :: "'b"   and r :: "'a"   and e2 :: "'b"   and mo :: "'a"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat" assumes "\<not> hb (i, Append i e1) (r, Expunge e2 mo r)"   and "\<not> hb (r, Expunge e2 mo r) (i, Append i e1)"   and "xs prefix of j"   and "(i, Append i e1) \<in> set (node_deliver_messages xs)"   and "(r, Expunge e2 mo r) \<in> set (node_deliver_messages xs)" shows "i \<noteq> mo"
start: shows "(1 + fps_X) ^ (n + 1) = (1 + fps_X) * (1 + fps_X) ^ n"
downset_directed_downset: shows "ord.directed (\<lambda>x y. y \<le> x) \<circ> ord.downset_set (\<lambda>x y. y \<le> x) = ord.directed (\<lambda>x y. y \<le> x)"
InitConf_HAInitStates: fixes A :: "('a, 'b, 'c) hierauto" shows "InitConf A \<subseteq> HAInitStates A"
is_unit_power_iff: fixes a :: "'a"   and n :: "nat" shows "is_unit (a ^ n) = (is_unit a \<or> n = 0)"
hd_dropWhile: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "dropWhile P xs \<noteq> []" shows "\<not> P (hd (dropWhile P xs))"
not_eventuallyD: fixes P :: "'e \<Rightarrow> bool"   and F :: "'e filter" assumes "\<not> eventually P F" shows "\<exists>x. \<not> P x"
inject_2: fixes x21 :: "'a"   and x22 :: "'a FOL_Fitting.term list"   and y21 :: "'a"   and y22 :: "'a FOL_Fitting.term list" shows "(App x21 x22 = App y21 y22) = (x21 = y21 \<and> x22 = y22)"
sequentially_imp_eventually_within: fixes s :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> (\<forall>\<^sub>F n in sequentially. P (f n))" shows "eventually P (at a within s)"
assms_2: shows "A \<in> absOfS P (us, s)"
less_eq_imp_production_subseteq_Interp: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" assumes "C \<le> D" shows "production C \<subseteq> Interp D"
assms_3: fixes dir :: "('i, 'a) Direction"   and x\<^sub>i :: "nat"   and I :: "'i list" assumes "dir = Positive \<or> dir = Negative"   and "\<not> \<U> s"   and "min_lvar_not_in_bounds s = Some x\<^sub>i"   and "\<lhd>\<^sub>l\<^sub>b (lt dir) (\<langle>\<V> s\<rangle> x\<^sub>i) (LB dir s x\<^sub>i)"   and "min_rvar_incdec dir s x\<^sub>i = Inl I" shows "P (set_unsat I s)"
assms_1: shows "R \<subseteq> A \<times> A"
sum_list_zero: fixes xs :: "'a list" assumes "set xs \<subseteq> {0::'a}" shows "sum_list xs = (0::'a)"
ode_slp_def: shows "ode_slp = slp_of_fas ode_e"
val_reduce: fixes s :: "'d \<Rightarrow> bool"   and env :: "('d, bool) mapping"   and t :: "'d ifex" assumes "agree s env" shows "val_ifex (reduce env t) s = val_ifex t s"
Nom_7: shows "M, g \<Turnstile>\<^sub>\<Theta> (ps, a) # branch"
image_constant: fixes x :: "'e"   and A :: "'e set"   and c :: "'f" assumes "x \<in> A" shows "(\<lambda>x. c) ` A = {c}"
locally_ringed_space_def: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" shows "locally_ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<equiv> ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<and> locally_ringed_space_axioms is_open \<FF> \<rho> add_str mult_str zero_str one_str"
freshSubstChain: fixes a :: "name"   and P :: "pi"   and \<sigma> :: "(name \<times> name) list" assumes "a \<sharp> P"   and "a \<sharp> \<sigma>" shows "a \<sharp> P[<\<sigma>>]"
SeqQuoteP_Eats: fixes s :: "name"   and k :: "name"   and s1 :: "tm"   and s2 :: "tm"   and k1 :: "tm"   and k2 :: "tm"   and t1 :: "tm"   and t2 :: "tm"   and u1 :: "tm"   and u2 :: "tm" assumes "atom s \<sharp> (k, s1, s2, k1, k2, t1, t2, u1, u2)"   and "atom k \<sharp> (s1, s2, k1, k2, t1, t2, u1, u2)" shows "{SeqQuoteP t1 u1 s1 k1, SeqQuoteP t2 u2 s2 k2} \<turnstile> SyntaxN.Ex s (SyntaxN.Ex k (SeqQuoteP (Eats t1 t2) (Q_Eats u1 u2) (Var s) (Var k)))"
abort_def: fixes uu :: "String.literal"   and f :: "unit \<Rightarrow> 'd" shows "Code.abort uu f = f ()"
of_nat_less_top: fixes i :: "nat" shows "of_nat i < \<top>"
iwlsFSwSb_termFSwSbImorph_iff: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'b, 'c, 'd) model_scheme"   and h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c" assumes "iwlsFSwSb MOD" shows "termFSwSbImorph h hA MOD = ipresCons h hA MOD"
count_prime_factorization_nonprime: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<not> normalization_semidom.prime zero times one normalize p" shows "count (factorial_semiring.prime_factorization zero times one normalize x) p = 0"
rtrancl_on_refl: fixes a :: "'c"   and F :: "'c set"   and r :: "('c \<times> 'c) set" assumes "a \<in> F" shows "(a, a) \<in> rtrancl_on F r"
section_is_mono: fixes g :: "'c" assumes "C.section g" shows "C.mono g"
cases: fixes a1 :: "pi"   and a2 :: "pi set" assumes "(a1, a2) \<in> sumComposeSet" obtains "a1 = \<zero>" and "a2 = {}"   | Q :: "pi" and S :: "pi set" and P :: "pi" where "a1 = P \<oplus> Q" and "a2 = S" and "Q \<in> S" and "(P, S - {Q}) \<in> sumComposeSet"
Skip_simp: fixes \<pi> :: "char list \<Rightarrow> com option"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" shows "\<pi>: (SKIP, s) \<Rightarrow> t = (t = s)"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>C\<^sub>L 'c" shows "(*\<^sub>V) (sandwich x) = (\<lambda>B. x o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L x*)"
of_nat_mono: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and i :: "nat"   and j :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "i \<le> j" shows "less_eq (semiring_1.of_nat one plus zero i) (semiring_1.of_nat one plus zero j)"
endpoints_in_S_perp_foot_are_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2"   and c :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "c \<in> hyp2"   and "a \<noteq> b"   and "proj2_incident a l"   and "proj2_incident b l" shows "are_endpoints_in_S (endpoint_in_S a b) (endpoint_in_S b a) a (perp_foot c l)"
uminus_mult_right_mat: fixes A :: "'a mat"   and B :: "'a mat" assumes "dim_col A = dim_row B" shows "A * - B = - (A * B)"
Trivial_Aexp_simp: fixes e :: "exp"   and n :: "Arity" shows "Aexp_syn n e = (\<lambda>x. up\<cdot>0) f|` fv e"
simps_3: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and b :: "bool" shows "P \<turnstile> typeof\<^bsub>h\<^esub> (Bool b) = \<lfloor>Boolean\<rfloor>"
rel_prime: shows "\<forall>q. List.member qs q \<longrightarrow> coprime p q"
setmonotone_implies_chain_funpower: fixes f :: "'b set \<Rightarrow> 'b set"   and I :: "'b set" assumes "setmonotone f" shows "chain (\<lambda>n. funpower f n I)"
simps: fixes b :: "'b" shows "the_Inr (Inr b) = b"
list_of_oalist_OAlist_ntm: fixes xs :: "('a \<times> 'b) list \<times> 'a nat_term_order" shows "list_of_oalist_ntm (OAlist_ntm xs) = sort_oalist_ko_ntm xs"
assms: shows "ptr |\<in>| shadow_root_ptr_kinds h"
SPRAY_nonevent: fixes x :: "'a" assumes "x \<notin> \<E>" shows "SPRAY x = {}"
sg2: shows "if activation t = [] then nGet i t = [] \<and> nSend i t = [] else nGet i t = activation t \<and> nSend i t = nReturn i t"
ltl_prop_equiv_subst_S: fixes S :: "'a ltln set"   and \<phi> :: "'a ltln"   and m :: "'a ltln \<Rightarrow> 'a ltln option" shows "S \<Turnstile>\<^sub>P subst \<phi> m = (S - dom m \<union> {uu_. \<exists>\<chi> \<chi>'. uu_ = \<chi> \<and> \<chi> \<in> dom m \<and> m \<chi> = Some \<chi>' \<and> S \<Turnstile>\<^sub>P \<chi>'}) \<Turnstile>\<^sub>P \<phi>"
seq: 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) P"
indRelR_modulo_pred_impl_indRelRPO_modulo_pred: fixes Pred :: "('procS, 'procT) Proc \<times> ('procS, 'procT) Proc \<Rightarrow> bool" assumes "\<forall>P. Pred (P, P)"   and "\<forall>P Q R. Pred (P, Q) \<and> Pred (Q, R) \<longrightarrow> Pred (P, R)" shows "(\<forall>(P, Q)\<in>indRelR. Pred (P, Q)) = (\<forall>(P, Q)\<in>indRelRPO. Pred (P, Q))"
invertible_mat_def: fixes A :: "'d mat" shows "invertible_mat A \<equiv> square_mat A \<and> (\<exists>B. inverts_mat A B \<and> inverts_mat B A)"
mat_of_vec_mult_eq_scalar_prod: fixes v :: "'c vec"   and n :: "nat"   and w :: "'c vec" assumes "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "mat 1 (dim_vec v) (\<lambda>(i, j). conjugate v $ j) * mat (dim_vec w) 1 (\<lambda>(i, j). w $ i) = mat 1 1 (\<lambda>k. inner_prod v w)"
amalgamation_in_P1: fixes i :: "nat"   and j :: "nat" shows "amalgamation i j \<in> \<P>"
wp2Conc: fixes x :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "wellformed_policy2Pr (x # xs)" shows "wellformed_policy2Pr xs"
sd_distl_aux: fixes Y :: "('c \<times> 'b) set"   and p :: "'b"   and x :: "'a" assumes "Y \<noteq> {}" shows "p \<squnion> (\<Squnion>y\<in>Y. \<alpha> x (snd y)) = (\<Squnion>y\<in>Y. p \<squnion> \<alpha> x (snd y))"
associated_string_unique: fixes M :: "('a, 'b) message set"   and s :: "('a, 'b option) insert_message list"   and t :: "('a, 'b option) insert_message list" assumes "is_associated_string M s"   and "is_associated_string M t" shows "s = t"
borel_measurable_uminus_eq: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure" shows "((\<lambda>x. - f x) \<in> borel_measurable M) = (f \<in> borel_measurable M)"
JN_Join_diff: fixes i :: "'b"   and I :: "'b set"   and F :: "'b \<Rightarrow> 'c program" assumes "i \<in> I" shows "F i \<squnion> JOIN (I - {i}) F = JOIN I F"
interp_ins_tail_unfold: fixes xs :: "('a \<times> 'a option) list"   and x :: "'a \<times> 'a option" shows "interp_ins (xs @ [x]) = insert_spec (interp_ins xs) x"
subrel_runiq: fixes Q :: "('a \<times> 'b) set"   and P :: "('a \<times> 'b) set" assumes "runiq Q"   and "P \<subseteq> Q" shows "runiq P"
Union_connected_component: fixes S :: "'c set" shows "\<Union> (connected_component_set S ` S) = S"
index_loc_def: fixes m_ops :: "('a, 'b, 'c, 'd) map_ops_scheme"   and s_ops :: "('e, 'b, 'f) set_ops_scheme" shows "index_loc m_ops s_ops \<equiv> StdMap m_ops \<and> StdSet s_ops"
wf_subcls1: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog" assumes "wf_prog wf_md P" shows "wf ((subcls1 P)\<inverse>)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and 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 get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and to_tree_order_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog" assumes "l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs"   and "l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs"   and "l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms to_tree_order_si get_child_nodes get_shadow_root" shows "l_to_tree_order_si\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order_si get_child_nodes get_child_nodes_locs get_shadow_root get_shadow_root_locs type_wf known_ptr"
rec_5: fixes u' :: "'t"   and g' :: "'t \<Rightarrow>\<^sub>0 'b" assumes "u' \<in> Keys (set fs)"   and "u' \<notin> set vs"   and "g' \<in> set gs"   and "lt g' adds\<^sub>t u'" shows "monom_mult (1::'b) (pp_of_term u' - lp g') g' \<in> set fs"
vpairs_eqD: fixes r :: "V"   and x :: "V" assumes "vpairs r = r"   and "x \<in>\<^sub>\<circ> r" shows "\<exists>a b. x = \<langle>a, b\<rangle>"
set_permute_list: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list" assumes "f permutes {..<length xs}" shows "set (permute_list f xs) = set xs"
hyps: shows "\<K> ^ n = (\<Sum>w | length w = n \<and> set w \<subseteq> L. 1 / b ^ cw_len_concat w)"
type_eq_Rty: fixes v :: "val" shows "(type v = Rty) = (\<exists>r. v = Rv r)"
utest_intro: fixes P :: "'a hrel" assumes "\<I>\<I> \<sqsubseteq> P" shows "utp_test P"
elims: fixes x :: "Preliminaries.var \<Rightarrow> 'e trm"   and xa :: "('e, 'f) lit"   and y :: "('e, 'f) lit" assumes "substL x xa = y" obtains \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm" and at :: "('e, 'f) atm" where "x = \<pi>__" and "xa = Pos at__" and "y = Pos (substA \<pi>__ at__)"   | \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm" and at :: "('e, 'f) atm" where "x = \<pi>" and "xa = Neg at" and "y = Neg (substA \<pi> at)"
Px: shows "(\<lambda>x. ennreal (Px x)) \<in> borel_measurable MX"
distinct_14: fixes x31 :: "('a, 'b, 'c) Stmt"   and x32 :: "'a ModeUpd" shows "x31@[x32] \<noteq> Skip"
cod_simp: fixes f :: "'a" assumes "hom.arr f" shows "hom.cod f = cod f"
rep_eq: fixes x :: "'b ttree"   and xa :: "'b ttree" shows "(x \<sqsubseteq> xa) = (Rep_ttree x \<subseteq> Rep_ttree xa)"
E: shows "E = filter (\<lambda>(s, d). runFw s d c rs = Decision FinalAllow) (all_pairs (map getOneIp (build_ip_partition c rs)))"
ltchr: shows "(\<Sum>s\<in>set (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)) \<inter> {s. z I s = - 1}. z I s * rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))) = - rat_of_int (int (card {x. poly p x = 0 \<and> poly (prod_list (retrieve_polys qs I)) x < 0}))"
assms_1: shows "valid_mmsaux args cur (nt, gc, maskL, maskR, data_prev, data_in, tuple_in, tuple_since) auxlist"
ra: shows "range (binaryset A B) \<in> Pow (smallest_ccdi_sets \<Omega> M)"
invar_def: fixes empty :: "'a"   and update :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a"   and delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and lookup :: "'a \<Rightarrow> 'b \<Rightarrow> 'c option"   and inorder :: "'a \<Rightarrow> ('b \<times> 'c) list"   and inv :: "'a \<Rightarrow> bool"   and t :: "'a" assumes "Map_by_Ordered empty update delete lookup inorder inv" shows "Map_by_Ordered.invar inorder inv t \<equiv> inv t \<and> sorted1 (inorder t)"
unit_in_vhom: shows "\<guillemotleft>\<phi> : src g' \<Rightarrow> f \<star> g'\<guillemotright>"
on_ocircline_of_circline: fixes H :: "ocircline"   and z :: "complex_homo" shows "on_circline (of_ocircline H) z = on_ocircline H z"
UNIF1_mono: 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"   and M :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "UNIF1 \<sigma> (E + M) (E' + subst_mset \<sigma> M)"
coprime_normalize_left_iff: 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "algebraic_semidom.coprime times one (normalize a) b = algebraic_semidom.coprime times one a b"
DIAndSG1_def: shows "DIAndSG1 = (Pc pid1 \<rightarrow> ([[$\<alpha> vid1]]Pc pid3) \<rightarrow> [[$\<alpha> vid1]]Pc pid1)"
H_4: assumes "\<And>i. i \<in> F \<Longrightarrow> A i \<in> events" shows "(\<Sum>i\<in>F. prob (A i)) \<le> real (card F) - 1 + prob (\<Inter> (A ` F))"
args_subst: fixes \<rho> :: "'b \<Rightarrow> ('a, 'b) tm"   and s :: "('a, 'b) tm" shows "args (subst \<rho> s) = (case head s of Var x \<Rightarrow> args (\<rho> x) | Sym f \<Rightarrow> []) @ map (subst \<rho>) (args s)"
b_least_def1: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat" shows "b_least f = (\<lambda>x. b_least2 f x x)"
distinct_conv_nth: fixes xs :: "'d list" shows "distinct xs = (\<forall>i<length xs. \<forall>j<length xs. i \<noteq> j \<longrightarrow> xs ! i \<noteq> xs ! j)"
simps_2: fixes x :: "brick list"   and xs :: "wall" shows "is_tangle_diagram (x * xs) = (if is_tangle_diagram xs then codomain_block x = domain_wall xs else False)"
refine: fixes Rv' :: "('a \<times> 'b) set"   and eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and bhc :: "nat \<Rightarrow> 'a \<Rightarrow> nat"   and def_size :: "nat"   and Vi :: "'c list"   and V :: "'d set"   and Rv :: "('c \<times> 'd) set"   and fi :: "'c \<Rightarrow> 'a"   and f :: "'d \<Rightarrow> 'b" assumes "is_bounded_hashcode Rv' eq bhc"   and "is_valid_def_hm_size TYPE('a) def_size"   and "(Vi, V) \<in> \<langle>Rv\<rangle>list_set_rel"   and "(fi, f) \<in> Rv \<rightarrow> Rv'" shows "(RETURN (the_inv_into_map_code eq bhc def_size Vi fi), the_inv_into_map_impl V f) \<in> \<langle>\<langle>Rv', Rv\<rangle>Impl_Array_Hash_Map.ahm_rel bhc\<rangle>nres_rel"
relevant_varsp_relevant_vars_eq: fixes S :: "'node SDG_node set"   and n :: "'node SDG_node" shows "relevant_varsp S n = (\<lambda>x. x \<in> rv S n)"
emeasure_eq_measure: fixes p :: "'a spmf"   and A :: "'a set" shows "emeasure (measure_spmf p) A = ennreal (Sigma_Algebra.measure (measure_spmf p) A)"
is_Atom_def: fixes prot_atom :: "'e prot_atom" shows "prot_atom.is_Atom prot_atom = (\<exists>x1. prot_atom = prot_atom.Atom x1)"
continuous_within_compose3: fixes f :: "'d \<Rightarrow> 'b"   and x :: "'d"   and g :: "'b \<Rightarrow> 'c"   and s :: "'d set" assumes "isCont g (f x)"   and "continuous (at x within s) f" shows "continuous (at x within s) (\<lambda>x. g (f x))"
cod_char: fixes f :: "'b" shows "J.cod f = (if f \<in> Collect J0.arr \<union> Collect J1.arr then f else J.null)"
le_supI1: fixes a :: "'a"   and x :: "'a"   and b :: "'a" assumes "a \<le> x" shows "a \<sqinter> b \<le> x"
inf_r_def: fixes a :: "'b"   and b :: "'b" shows "a \<sqinter> b = (a l\<rightarrow> b) * a"
fixp_unfold: fixes Sup :: "'g set \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'g" assumes "class.ccpo Sup less_eq less"   and "monotone less_eq less_eq f" shows "ccpo.fixp Sup less_eq f = f (ccpo.fixp Sup less_eq f)"
t_def: shows "(sum (\<lambda>x. norm (f x)) \<longlongrightarrow> t) (finite_subsets_at_top A)"
both_mono_ordered_semiring_1_axioms: shows "both_mono_ordered_semiring_1 default (\<succ>) arc_pos"
comp_arr_dom: fixes f :: "('o, 'a) arr"   and a :: "('o, 'a) arr" assumes "arr f"   and "local.dom f = a" shows "COMP f a = f"
order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "orda x y"   and "orda y z" shows "orda x z"
FAssRed1: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and F :: "char list"   and D :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>F{D} := e\<^sub>2,s,b\<rangle> \<rightarrow> \<langle>e'\<bullet>F{D} := e\<^sub>2,s',b'\<rangle>"