hcju/isabelleps huggingface dataset:
irreducible_altdef: fixes x :: "'b" shows "irreducible x = (x \<noteq> (0::'b) \<and> \<not> is_unit x \<and> (\<forall>b. b dvd x \<longrightarrow> x dvd b \<or> is_unit b))"
rep_eq: fixes x :: "'c set"   and xa :: "('c, 'd) fmap" shows "fmlookup (fmdrop_set x xa) = map_drop_set x (fmlookup xa)"
cases: fixes a :: "('b, 'out, 'in) gpv" assumes "local.finite_gpv a" obtains gpv :: "('b, 'out, 'in) gpv" where "a = gpv" and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> local.finite_gpv (c input)"
a_subid_aux1': fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.antidomain_pre_dioid antidomain_op plus times one zero less_eq less" shows "less_eq (times (antidomain_op x) y) y"
untilEA: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'cmp"   and \<gamma>' :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma> :: "(nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "eval c t t' n (\<gamma>' \<UU>\<^sub>b \<gamma>)" shows "\<exists>n'\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<longrightarrow> eval c t t' n'' \<gamma>) \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>. n'' < \<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub> \<longrightarrow> eval c t t' n'' \<gamma>') \<or> \<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> eval c t t' n' \<gamma> \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub>. n'' < n' \<longrightarrow> eval c t t' n'' \<gamma>')"
hom_det: fixes A :: "int mat" shows "det (of_int_hom.mat_hom A) = of_int (det A)"
rebase_poly_self: fixes p :: "'a mod_ring poly" shows "#p = p"
attach_id_converter: fixes res :: "('y, 'z) resource" shows "1\<^sub>C \<rhd> res = res"
non_volatile_owned_or_read_only_read_only_reads_eq: fixes pending_write :: "bool"   and \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "non_volatile_owned_or_read_only pending_write \<S> \<O> sb"   and "\<forall>a\<in>read_only_reads \<O> sb. a \<in> read_only \<S> \<longrightarrow> a \<in> read_only \<S>'" shows "non_volatile_owned_or_read_only pending_write \<S>' \<O> sb"
nat_left_shift_continuous: shows "continuous_on UNIV nat_left_shift"
Ide_implies_Can: fixes t :: "'a local.term" assumes "Ide t" shows "Can t"
homeomorphic_monotone_image_interval: fixes f :: "real \<Rightarrow> 'b" assumes "continuous_on {0..1} f"   and "\<And>y. connected ({0..1} \<inter> f -` {y})"   and "f 1 \<noteq> f 0" shows "f ` {0..1} homeomorphic {0..1}"
submonoid_axioms: shows "submonoid Normal G (\<cdot>) \<one>"
sba1_commutative: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
length_nodes: fixes ls :: "'b tree list"   and xs :: "'b list"   and rs :: "'b tree list" shows "length (nodes ls xs rs) = length xs"
l11_50_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and B' :: "'a"   and A' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B C"   and "Tarski_neutral_dimensionless.CongA Bet Cong B A C B' A' C'"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'"   and "Cong A B A' B'" shows "Cong A C A' C' \<and> Cong B C B' C' \<and> Tarski_neutral_dimensionless.CongA Bet Cong A C B A' C' B'"
reciprocal_real: fixes z :: "complex_homo" assumes "is_real (to_complex z)"   and "z \<noteq> 0\<^sub>h"   and "z \<noteq> \<infinity>\<^sub>h" shows "Re (to_complex (reciprocal z)) = 1 / Re (to_complex z)"
multpw_implies_mult2_ns: fixes X :: "'a multiset"   and Y :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(X, Y) \<in> multpw ns" shows "(X, Y) \<in> mult2_ns ns s"
eventually_weak_subseq: fixes u :: "nat \<Rightarrow> nat"   and P :: "nat \<Rightarrow> bool" assumes "(\<lambda>x. ereal (real (u x))) \<longlonglongrightarrow> \<infinity>"   and "eventually P sequentially" shows "\<forall>\<^sub>F n in sequentially. P (u n)"
next_subset_dfs: fixes S :: "'b"   and xs :: "'a list"   and x :: "'a" assumes "invariant S"   and "list_all is_node xs"   and "x \<in> set xs" shows "memb x (dfs S xs)"
comp_in_homI: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and g :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>local.comp g f : a \<rightarrow> c\<guillemotright>"
dynamic_field_access_ok: fixes G :: "prog"   and stat :: "bool"   and a :: "val"   and s :: "state"   and statC :: "qtname"   and L :: "lenv"   and accC :: "qtname"   and e :: "expr"   and fn :: "vname"   and f :: "qtname \<times> Decl.field"   and dynC :: "qtname" assumes "wf_prog G"   and "\<not> stat \<longrightarrow> a \<noteq> Null"   and "G,snd s\<turnstile>a\<Colon>\<preceq>Class statC"   and "s\<Colon>\<preceq>(G, L)"   and "normal s"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>e\<Colon>-Class statC"   and "accfield G accC statC fn = Some f"   and "if stat then dynC = declclass f else dynC = obj_class (lookup_obj (snd s) a)"   and "if stat then is_static f else \<not> is_static f" shows "table_of (DeclConcepts.fields G dynC) (fn, declclass f) = Some (fld f) \<and> G \<turnstile>Field fn f in dynC dyn_accessible_from accC"
ide_cod: fixes f :: "'a" assumes "A.arr f" shows "A.ide (A.cod f)"
Min_ge_iff: fixes A :: "'b word set"   and x :: "'b word" assumes "finite A"   and "A \<noteq> {}" shows "(x \<le>s signed.Min A) = (\<forall>a\<in>A. x \<le>s a)"
prv_eqv_prv_rev: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "prv \<phi>"   and "prv (eqv \<chi> \<phi>)" shows "prv \<chi>"
detAB_Znm: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat" assumes "A \<in> carrier_mat n m"   and "B \<in> carrier_mat m n" shows "Determinant.det (A * B) = (\<Sum>(f, \<pi>)\<in>Z n m. signof \<pi> * (\<Prod>i = 0..<n. A $$ (i, f i) * B $$ (f i, \<pi> i)))"
Bang_2: fixes P :: "pi"   and u :: "name"   and P'' :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "P \<parallel> !P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'" shows "!P \<Longrightarrow>\<^sub>lu in P''\<rightarrow>a<x> \<prec> P'"
SCallInitRed: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and sh :: "sheap"   and vs :: "val list"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "\<nexists>sfs. sh D = \<lfloor>(sfs, Done)\<rfloor>"   and "M \<noteq> clinit" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),(h, l, sh),False\<rangle> \<rightarrow> \<langle>INIT D ([D],False) \<leftarrow> C\<bullet>\<^sub>sM(map Val vs),(h, l, sh),False\<rangle>"
intro: fixes U\<^sub>1 :: "'a set"   and Inf\<^sub>1 :: "'a set \<Rightarrow> 'a"   and U\<^sub>2 :: "'b set"   and Inf\<^sub>2 :: "'b set \<Rightarrow> 'b" assumes "Inf_ow U\<^sub>1 Inf\<^sub>1"   and "Inf_ow U\<^sub>2 Inf\<^sub>2" shows "Inf_pair_ow U\<^sub>1 Inf\<^sub>1 U\<^sub>2 Inf\<^sub>2"
dynlookup_cases: fixes G :: "prog"   and statT :: "ref_ty"   and dynC :: "qtname"   and sig :: "sig"   and x :: "(qtname \<times> methd) option" assumes "dynlookup G statT dynC sig = x" obtains "statT = NullT" and "None = x"   | I :: "qtname" where "statT = IfaceT I" and "dynimethd G I dynC sig = x"   | statC :: "qtname" where "statT = ClassT statC" and "dynmethd G statC dynC sig = x"   | ty :: "ty" where "statT = ArrayT ty" and "dynmethd G Object dynC sig = x"
convergesE: fixes T :: "'b set set"   and F :: "'b set set"   and x :: "'b" assumes "F \<longlongrightarrow>\<^bsub>T\<^esub> x" obtains "nhds\<^bsub>T\<^esub> x \<subseteq> F"
Cantor_4: shows "sorted_wrt (\<lambda>x y. y < x) \<alpha>s"
infin_imp_iNextStrong_eq_iNext: fixes I :: "nat set"   and t0 :: "nat"   and P :: "nat \<Rightarrow> bool" assumes "infinite I"   and "t0 \<in> I" shows "iNextStrong t0 I P = iNext t0 I P"
preprocess_def: fixes rs :: "'i common_primitive rule list" shows "preprocess rs \<equiv> lower_closure (optimize_matches abstract_for_simple_firewall (lower_closure (packet_assume_new rs)))"
prems_2: shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> t"
exhaust: fixes y :: "PDDL_STRIPS_Semantics.term" obtains x1 :: "variable" where "y = term.VAR x1"   | x2 :: "object" where "y = term.CONST x2"
Cons_1: assumes "distincts xss" shows "lists_succ xss permutes \<Union> (set ` set xss)"
dimacs_model_to_abs_atom: fixes x :: "nat"   and dimacs_M :: "int list"   and M :: "nat \<Rightarrow> bool" assumes "0 < x"   and "int x \<in> set dimacs_M"   and "distinct (map dimacs_lit_to_var dimacs_M)" shows "dimacs_model_to_abs dimacs_M M x"
inject_1: fixes x1 :: "('a, 'b) insert_message"   and y1 :: "('a, 'b) insert_message" shows "(Insert x1 = Insert y1) = (x1 = y1)"
Res: fixes S :: "pi"   and T :: "pi"   and x :: "name" assumes "(S, T) \<in> Rel" shows "(<\<nu>x>S, <\<nu>x>T) \<in> Rel"
inj: fixes y :: "real" assumes "a < y"   and "y < b" shows "f (g y) = y"
TS_b2'': fixes x :: "nat"   and y :: "nat"   and x0 :: "nat"   and y0 :: "nat"   and s :: "(nat list \<times> nat list) pmf"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "TS_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom x, Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom y, Atom y])" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h0)) s qs = real (length qs - 3)"
func: shows "f \<circ> cv \<in> [elts \<beta>]\<^bsup>n\<^esup> \<rightarrow> {..<length \<alpha>}"
weakening_version1: fixes \<Gamma>1 :: "(name \<times> ty) list"   and t :: "lam"   and T :: "ty"   and \<Gamma>2 :: "(name \<times> ty) list" assumes "\<Gamma>1 \<turnstile> t : T"   and "valid \<Gamma>2"   and "\<Gamma>1 \<subseteq> \<Gamma>2" shows "\<Gamma>2 \<turnstile> t : T"
keys_def: fixes t :: "('a, 'b) rbt" shows "keys t = map fst (entries t)"
strict_iff_order: fixes b :: "'a list"   and a :: "'a list" shows "strict_prefix b a = (prefix b a \<and> \<not> a = b)"
equiv_out_equiv: shows "equiv UNIV {(x, y). equiv_out x y}"
subtensor_subtensor_combine: fixes As :: "'b tensor list"   and ds :: "nat list"   and i :: "nat" assumes "\<And>A. A \<in> set As \<Longrightarrow> dims A = ds"   and "i < length As" shows "subtensor (subtensor_combine ds As) i = As ! i"
cong_exp_iff_simps_9: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and m :: "num"   and q :: "num"   and n :: "num" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step" shows "(modulo (numeral.numeral one plus (num.Bit0 m)) (numeral.numeral one plus (num.Bit0 q)) = modulo (numeral.numeral one plus (num.Bit0 n)) (numeral.numeral one plus (num.Bit0 q))) = (modulo (numeral.numeral one plus m) (numeral.numeral one plus q) = modulo (numeral.numeral one plus n) (numeral.numeral one plus q))"
fun_app_def: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "f $ x \<equiv> f x"
summarize_pdevs_list_def: fixes p :: "nat"   and I :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and d :: "nat"   and xs :: "'b pdevs list" shows "summarize_pdevs_list p I d xs = map2 (summarize_pdevs p (\<lambda>i _. I i (pdevs_applys xs i))) [d..<d + length xs] xs"
total_preorder_on_map: fixes f :: "'c \<Rightarrow> 'b" shows "total_preorder_on (f -` f -` carrier) (map_relation f (map_relation f le))"
mset_pos_neg_partition: fixes M :: "'a zmultiset" shows "M = zmset_of (mset_pos M) - zmset_of (mset_neg M)"
ipurge_fail_aux_t_form_aux: fixes Y :: "('b, 'c) ipurge_rec" shows "ipurge_fail_aux_t_form (ipurge_fail_aux_t_aux Y)"
finite_Linear_order_induct: fixes r :: "('d \<times> 'd) set"   and x :: "'d"   and P :: "'d \<Rightarrow> bool" assumes "Linear_order r"   and "x \<in> Field r"   and "finite r"   and "\<And>x. \<lbrakk>x \<in> Field r; \<And>y. y \<in> aboveS r x \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P x"
le_OK_conv: fixes e :: "'b err"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "(e \<sqsubseteq>\<^bsub>Err.le r\<^esub> OK x) = (\<exists>y. e = OK y \<and> y \<sqsubseteq>\<^bsub>r\<^esub> x)"
List_not_Null: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref"   and as :: "ref list" assumes "p \<noteq> Null" shows "List p h as = (\<exists>ps. as = p # ps \<and> List (h p) h ps)"
assoc_is_natural_2: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and \<nu> :: "'c"   and \<tau> :: "'c" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.arr V \<mu>"   and "partial_magma.arr V \<nu>"   and "partial_magma.arr V \<tau>"   and "src \<mu> = trg \<nu>"   and "src \<nu> = trg \<tau>" shows "\<a> \<mu> \<nu> \<tau> = V (\<a> (partial_magma.cod V \<mu>) (partial_magma.cod V \<nu>) (partial_magma.cod V \<tau>)) (H (H \<mu> \<nu>) \<tau>)"
intro: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "\<And>e a. edom (Aexp e\<cdot>a) \<subseteq> fv e" shows "EdomArityAnalysis Aexp"
list_all_take: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and P :: "'a \<Rightarrow> bool"   and n :: "nat"   and s :: "'a list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "list_all P (take n s) = (\<forall>i. i < n \<and> i < length s \<longrightarrow> P (s ! i))"
lit_ord_subst: fixes L :: "'a literal"   and M :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "(L, M) \<in> lit_ord" shows "(subst_lit L \<sigma>, subst_lit M \<sigma>) \<in> lit_ord"
bisim_val_id2: fixes u' :: "'c \<Rightarrow> real"   and l :: "'s"   and u :: "'c \<Rightarrow> real"   and l' :: "'s" assumes "u' \<in> V"   and "(l, u) ~ (l', u')" shows "[u]\<^sub>\<R> = [u']\<^sub>\<R>"
ipassmt_disjoint_inj_k: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface"   and ips :: "('a word \<times> nat) list"   and k :: "iface"   and ips' :: "('a word \<times> nat) list"   and p :: "'a word" assumes "ipassmt_sanity_disjoint ipassmt"   and "ipassmt ifce = Some ips"   and "ipassmt k = Some ips'"   and "p \<in> ipcidr_union_set (set ips)"   and "p \<in> ipcidr_union_set (set ips')" shows "k = ifce"
coprime_preserves: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and ty_q :: "'c itself" assumes "Knuth_ex_4_6_2_22_prime a b u v w" shows "comm_monoid_mult_class.coprime (#(#v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q ty_q)) (#Knuth_ex_4_6_2_22_main.v' ty_q b u v w))) (#(#w + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q ty_q)) (#Knuth_ex_4_6_2_22_main.w' ty_q a b u v w)))"
all_finite_subset_image: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and P :: "'d set \<Rightarrow> bool" shows "(\<forall>B. finite B \<and> B \<subseteq> f ` A \<longrightarrow> P B) = (\<forall>B. finite B \<and> B \<subseteq> A \<longrightarrow> P (f ` B))"
catch_return: fixes x :: "int"   and m :: "'m" shows "catch (return x) m = return x"
2: shows "AE x in lborel. ereal 0 < ereal x \<longrightarrow> ereal x < ereal 1 \<longrightarrow> 0 \<le> f x"
set_mset_inter: fixes A :: "'b multiset"   and B :: "'b multiset" shows "set_mset (A \<inter># B) = set_mset A \<inter> set_mset B"
vwalkE: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "vwalk p G" obtains "set p \<subseteq> verts G" and "set (vwalk_arcs p) \<subseteq> arcs_ends G \<and> p \<noteq> []"
mn: shows "\<pi>' m' = \<pi> m \<and> \<pi>' n' = \<pi> n"
ord_strict_p_recE2: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "p \<noteq> 0"   and "p \<prec>\<^sub>p q"   and "lt p = lt q" shows "lc p = lc q"
intersection_numbers_def: shows "target.intersection_numbers \<equiv> {b1 |\<inter>| b2 |b1 b2. b1 \<in># \<B>' \<and> b2 \<in># remove1_mset b1 \<B>'}"
inject: fixes x1 :: "'f"   and x2 :: "'g"   and x3 :: "('d, 'e, 'f, 'g, 'h) qTerm"   and y1 :: "'f"   and y2 :: "'g"   and y3 :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qAbs x1 x2 x3 = qAbs y1 y2 y3) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3)"
sort_simps_1: fixes cmp :: "'b comparator" shows "Sorting_Algorithms.sort cmp [] = []"
conjugate_square_greater_0_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "((0::'a) < v \<bullet>c v) = (v \<noteq> 0\<^sub>v n)"
success_bind_executeI: fixes f :: "'a Heap"   and h :: "heap"   and x :: "'a"   and h' :: "heap"   and g :: "'a \<Rightarrow> 'b Heap" assumes "execute f h = Some (x, h')"   and "success (g x) h'" shows "success (f \<bind> g) h"
intro: fixes AA :: "('b, 'c, 'd) category_scheme"   and BB :: "('e, 'f, 'g) category_scheme"   and F :: "('b, 'c, 'e, 'f) functor" assumes "two_cats AA BB"   and "functor_axioms AA BB F" shows "Functor F : AA \<longrightarrow> BB"
ordLess_imp_ordLeq: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set" assumes "r <o r'" shows "r \<le>o r'"
openin_Times: fixes S :: "'c set"   and S' :: "'c set"   and T :: "'d set"   and T' :: "'d set" assumes "openin (top_of_set S) S'"   and "openin (top_of_set T) T'" shows "openin (top_of_set (S \<times> T)) (S' \<times> T')"
unfoldr_rbt_keys_generator: fixes t :: "'b set_rbt" shows "list.unfoldr rbt_keys_generator (RBT_Set2.init t) = RBT_Set2.keys t"
divide_nonpos_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less y zero" shows "less_eq zero (divide x y)"
tr_def: shows "{t. path M (io || t) q1 \<and> length io = length t} = {tr}"
A1: shows "pr_conv_2_to_1 f \<in> PrimRec1'"
preserves_hom: fixes f :: "'c"   and a :: "'c"   and b :: "'c" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>G f : G a \<Rightarrow>\<^sub>D G b\<guillemotright>"
subdegree_code: fixes x :: "'b ratfps" shows "subdegree (fps_of_ratfps x) = ratfps_subdegree x"
prems_1: shows "j < e_length (block i (Suc t))"
sorted: fixes v :: "nat"   and p :: "real mpoly" shows "sorted_wrt (\<lambda>(uu_, i) (uu_, i'). i < i') (to_list v p)"
lift_dnfeq_qe_def: shows "R\<^sub>e.lift_dnfeq_qe = R.lift_dnf_qe \<circ> R\<^sub>e.lift_eq_qe"
match_rews_3: fixes k :: "'r match" shows "match_Null\<cdot>Null\<cdot>k = k"
span_superset: fixes S :: "('a, 'b) vec set" shows "S \<subseteq> vec.span S"
and_mask_bintr: fixes w :: "'b word"   and n :: "nat" shows "w AND mask n = word_of_int (take_bit n (uint w))"
sign_hyp: shows "sign = map (\<lambda>q. if 0 < poly q x then 1 else - 1) qs"
similar_sq_mtx_simm: fixes A :: "'a sq_mtx"   and B :: "'a sq_mtx" assumes "A \<sim> B" shows "B \<sim> A"
inf_sup_distrib1: fixes x :: "'c"   and y :: "'c"   and z :: "'c" shows "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z"
intro: fixes alt :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and fail :: "'a" assumes "\<And>m. alt fail m = m"   and "\<And>m. alt m fail = m" shows "monad_fail_alt_axioms fail alt"
sum_unfold_sum_mset: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "sum f A = \<Sum>\<^sub># (image_mset f (mset_set A))"
intros_15: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>c s \<down> Normal s" shows "\<Gamma>\<turnstile>DynCom c \<down> Normal s"
quot_of_fract_mult: fixes x :: "'b fract"   and y :: "'b fract" shows "quot_of_fract (x * y) = (let (a, b) = quot_of_fract x; (c, d) = quot_of_fract y; (e, f) = normalize_quot (a, d); (g, h) = normalize_quot (c, b) in (e * g, f * h))"
sum_clauses_1: fixes f :: "'b \<Rightarrow> 'a" shows "sum f {} = (0::'a)"
Infm_shift: fixes P :: "'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "(\<exists>\<^sub>\<infinity>i. P (shift f n i)) = (\<exists>\<^sub>\<infinity>i. P (f i))"
semilattice_sup_ow_axioms: shows "semilattice_sup_ow U (\<squnion>\<^sub>o\<^sub>w) (\<le>\<^sub>o\<^sub>w) (<\<^sub>o\<^sub>w)"
fds_integral_diff: fixes a :: "'b"   and b :: "'b"   and f :: "'b fds"   and g :: "'b fds" shows "fds_integral (a - b) (f - g) = fds_integral a f - fds_integral b g"
has: shows "has_ell2_norm (Rep_ell2 \<psi>)"
exhaust_sel: fixes llist :: "'b llist" obtains "llist = LNil"   | "llist = LCons (lhd llist) (ltl llist)"
fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_is_fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s: fixes F :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list" shows "fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F = set (fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F)"
mono_def: fixes f :: "'a word \<Rightarrow> 'b" shows "signed.mono f = (\<forall>x y. x \<le>s y \<longrightarrow> f x \<le> f y)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>B) (\<cdot>\<^sub>B) F'G'.FG.map B.map F'G'.\<epsilon>"
r_amalgamation_def: shows "r_amalgamation \<equiv> Cn 3 r_pdec2 [r_parallel]"
of_real_inner_1: fixes x :: "real" shows "of_real x \<bullet> (1::'a) = x"
asset_ii_owner_ii: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, \<lbrace>Num 2, PubKey A\<rbrace>) \<in> s"   and "(Owner n, \<lbrace>Num 1, PubKey A\<rbrace>) \<in> s" shows "False"
l2: fixes p :: "'node" assumes "p \<in> L" shows "\<exists>Q\<subseteq>W. quorum_of p Q"