hcju/isabelleps huggingface dataset:
induced_map_unique: fixes r\<^sub>0 :: "'a"   and \<omega> :: "'a"   and w :: "'a"   and w' :: "'a" assumes "\<tau>.is_induced_by_cell w r\<^sub>0 \<omega>"   and "\<tau>.is_induced_by_cell w' r\<^sub>0 \<omega>" shows "w \<cong> w'"
arbitrage_processI: fixes Mkt :: "('a, 'c) discrete_market"   and p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real" assumes "\<exists>m. self_financing Mkt p \<and> trading_strategy p \<and> (\<forall>w\<in>space M. cls_val_process Mkt p 0 w = 0) \<and> (AE w in M. 0 \<le> cls_val_process Mkt p m w) \<and> 0 < prob {w \<in> space M. 0 < cls_val_process Mkt p m w}" shows "arbitrage_process Mkt p"
mod_0_imp_mod_mult_left_0: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n mod m = 0" shows "k * n mod m = 0"
2: shows "alw (\<lambda>xs. \<phi> xs \<longrightarrow> \<psi> xs \<or> nxt \<phi> xs) xs"
pCons_2: shows "poly (map_poly h p) x = (\<Sum>i\<le>degree p. x ^ i * h (coeff p i))"
assms: shows "\<forall>PROB as s. finite PROB \<and> P PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB s)"
indets_map_indets_subset: fixes f :: "'d \<Rightarrow> 'b"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "indets (map_indets f p) \<subseteq> f ` indets p"
errMOD_igSwapClsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igWlsAllDisj MOD"   and "igWlsDisj MOD"   and "igWlsAbsIsInBar MOD"   and "igConsIPresIGWls MOD"   and "igSwapAllIPresIGWlsAll MOD"   and "igSwapCls MOD" shows "igSwapClsSTR (errMOD MOD)"
mult_oner: fixes x :: "'a" shows "(1::'a) \<cdot> x = x"
singleton_insert_inj_eq: fixes b :: "'a"   and a :: "'a"   and A :: "'a set" shows "({b} = insert a A) = (a = b \<and> A \<subseteq> {b})"
chain_singleton: fixes Sup :: "'c set \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "class.ccpo Sup less_eq less" shows "Complete_Partial_Order.chain less_eq {x}"
f: fixes i :: "nat" shows "f i \<in> borel_measurable M"
simps_1: fixes edge :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and more :: "'l" shows "edge \<lparr>edge = edge, \<dots> = more\<rparr> = edge"
CallParamsThrow: 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es :: "expr list"   and vs :: "val list"   and ex :: "expr"   and es' :: "expr list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<Rightarrow>] \<langle>map Val vs @ throw ex # es',s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw ex,s\<^sub>2\<rangle>"
1: shows "rawpsubstT (substT r (Var u) x) (zip (map Var us) (map snd txs)) = rawpsubstT r (zip (map Var us) (map snd txs) @ [(Var u, x)])"
integral_mult_right: fixes c :: "'b"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'b" assumes "c \<noteq> (0::'b) \<Longrightarrow> integrable M f" shows "LINT x|M. c * f x = c * integral\<^sup>L M f"
item_end_def: fixes item :: "item" shows "item_end item = (case item of Item x1 x2 x3 x4 \<Rightarrow> x4)"
uniformly_convergent_improper_integral: fixes a :: "real"   and G :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real"   and A :: "'a set"   and f :: "'a \<Rightarrow> real \<Rightarrow> 'b" assumes "\<And>x. a \<le> x \<Longrightarrow> (G has_real_derivative g x) (at x within {a..})"   and "\<And>a' b x. \<lbrakk>x \<in> A; a \<le> a'; a' \<le> b\<rbrakk> \<Longrightarrow> f x integrable_on {a'..b}"   and "convergent (\<lambda>x. G (real x))"   and "\<And>y x. \<lbrakk>y \<in> A; a \<le> x\<rbrakk> \<Longrightarrow> norm (f y x) \<le> g x" shows "uniformly_convergent_on A (\<lambda>x xa. integral {a..real x} (f xa))"
eqvt_apply: fixes p :: "perm"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "p \<bullet> f x \<equiv> (p \<bullet> f) (p \<bullet> x)"
regular_Un: fixes S :: "'b list set"   and T :: "'b list set" assumes "regular S"   and "regular T" shows "regular (S \<union> T)"
ivl_diff: fixes i :: "'a"   and n :: "'a"   and m :: "'a" assumes "i \<le> n" shows "{i..<m} - {i..<n} = {n..<m}"
collapse_1: fixes stateful_strand_step :: "('e, 'f) stateful_strand_step" assumes "stateful_strand_step.is_Send stateful_strand_step" shows "send\<langle>the_msg stateful_strand_step\<rangle> = stateful_strand_step"
assms_1: shows "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a d x"
not_less_le: fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Torder D"   and "a \<in> carrier D"   and "b \<in> carrier D" shows "(\<not> a \<prec>\<^bsub>D\<^esub> b) = b \<preceq>\<^bsub>D\<^esub> a"
weakCongruenceWeakBisimulation: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<cong> Q" shows "P \<approx> Q"
remove_child_locs_def: fixes ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr" shows "remove_child_locs ptr owner_document = set_child_nodes_locs ptr \<union> set_disconnected_nodes_locs owner_document"
d: shows "cut_bound n n F \<le> max n n * 2 ^ 2 ^ depth F"
sum_list_us_eq: fixes i :: "nat"   and y :: "nat list" assumes "i < sum_list y" shows "sum_list (\<^bold>u y i) = i + 1"
below_lubmapI: fixes m :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and v :: "'a"   and e :: "'b \<Rightarrow> 'a \<Rightarrow> 'c" assumes "m k = Some v" shows "e k v \<sqsubseteq> lub (mapCollect e m)"
count_Sigma_mset: fixes A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset"   and a :: "'b"   and b :: "'c" shows "count (Sigma_mset A B) (a, b) = count A a * count (B a) b"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'b list" assumes "i < length xs" shows "take (Suc i) xs = take i xs @ [xs ! i]"
intro: fixes NS :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set" assumes "NS O S \<subseteq> S"   and "S O NS \<subseteq> S" shows "compat_pair S NS"
prognosis_IfThenElse: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" shows "prognosis ae (a # as) 0 (\<Gamma>, scrut, Alts e1 e2 # S) \<sqsubseteq> prognosis ae as a (\<Gamma>, scrut ? e1 : e2, S)"
chain_lprefix_lSup: fixes A :: "'f llist set"   and xs :: "'f llist" assumes "Complete_Partial_Order.chain lprefix A"   and "xs \<in> A" shows "lprefix xs (lSup A)"
Collect_minI_ex: fixes I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<exists>k\<in>I. P k" shows "\<exists>x\<in>I. P x \<and> (\<forall>y\<in>I. y < x \<longrightarrow> \<not> P y)"
wellformed_induce_subgraph: fixes vs :: "'a set" shows "wf_digraph ((G \<restriction> {v. u \<rightarrow>\<^sup>* v}) \<restriction> vs)"
OK_imp_ok: fixes I :: "'b set"   and F :: "'b \<Rightarrow> 'c program"   and i :: "'b"   and j :: "'b" assumes "OK I F"   and "i \<in> I"   and "j \<in> I"   and "i \<noteq> j" shows "F i ok F j"
fromsip: fixes dsk1 :: "k"   and dsn2 :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and sip :: "nat"   and dsn1 :: "nat"   and flag1 :: "f"   and hops1 :: "nat"   and nhip1 :: "nat"   and dsk2 :: "k"   and flag2 :: "f"   and hops2 :: "nat"   and nhip2 :: "nat" assumes "dsk1 = unk \<or> Suc 0 \<le> dsn2" shows "\<pi>\<^sub>3 (the (update rt sip (dsn1, dsk1, flag1, hops1, nhip1) sip)) = unk \<or> Suc 0 \<le> sqn (update rt sip (dsn2, dsk2, flag2, hops2, nhip2)) sip"
cases: fixes a1 :: "'d"   and a2 :: "'d" assumes "three TYPE('d)"   and "three.rel_12_23 a1 a2" obtains "a1 = three.three\<^sub>1" and "a2 = three.three\<^sub>2"   | "a1 = three.three\<^sub>2" and "a2 = three.three\<^sub>3"
Ana_fun_subterm: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and f :: "'a"   and T :: "('a, 'b) Term.term list"   and K :: "('a, 'b) Term.term list"   and R :: "('a, 'b) Term.term list" assumes "intruder_model arity Ana"   and "Ana (Fun f T) = (K, R)" shows "set R \<subseteq> set T"
igWlsAbsDisj_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) Iteration.model_scheme" shows "igWlsAbsDisj MOD \<equiv> \<forall>xs s xs' s' A. isInBar (xs, s) \<and> isInBar (xs', s') \<and> igWlsAbs MOD (xs, s) A \<and> igWlsAbs MOD (xs', s') A \<longrightarrow> xs = xs' \<and> s = s'"
seq_char: fixes g :: "'a"   and f :: "'a" shows "S.seq g f = (S.arr f \<and> S.arr g \<and> C.seq g f)"
SN_subset: fixes R :: "('a \<times> 'a) set"   and R' :: "('a \<times> 'a) set" assumes "SN R"   and "R' \<subseteq> R" shows "SN R'"
prems_1: shows "\<lparr>\<nu>*xvec\<rparr>\<lparr>\<nu>x\<rparr>B = \<lparr>\<nu>*yvec\<rparr>\<lparr>\<nu>y\<rparr>B'"
map: fixes f :: "'d \<Rightarrow> 'c"   and x1 :: "'d match_expr"   and x2 :: "action" shows "map_rule f (Rule x1 x2) = Rule (map_match_expr f x1) x2"
case_eq_if: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and prod :: "'b \<times> 'c" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
strict_order_iff: fixes a :: "'c"   and b :: "'c" shows "(a < b) = (a = min a b \<and> a \<noteq> b)"
add_left_comm: fixes y :: "'c"   and x :: "'c"   and z :: "'c" shows "y \<squnion> (x \<squnion> z) = x \<squnion> (y \<squnion> z)"
mat_mult_append_cols: fixes A :: "'b mat"   and nr :: "nat"   and nc1 :: "nat"   and B :: "'b mat"   and nc2 :: "nat"   and v1 :: "'b vec"   and v2 :: "'b vec" assumes "A \<in> carrier_mat nr nc1"   and "B \<in> carrier_mat nr nc2"   and "v1 \<in> carrier_vec nc1"   and "v2 \<in> carrier_vec nc2" shows "(A @\<^sub>c B) *\<^sub>v (v1 @\<^sub>v v2) = A *\<^sub>v v1 + B *\<^sub>v v2"
Cauchy_integral_circlepath: fixes z :: "complex"   and r :: "real"   and f :: "complex \<Rightarrow> complex"   and w :: "complex" assumes "continuous_on (cball z r) f"   and "f holomorphic_on ball z r"   and "cmod (w - z) < r" shows "((\<lambda>u. f u / (u - w)) has_contour_integral 2 * complex_of_real pi * \<i> * f w) (circlepath z r)"
eq2: shows "bind_spmf (TRY map_spmf OK q ELSE return_spmf Fault) (\<lambda>s. obsf_oracle callee2 s x) = TRY bind_spmf q (\<lambda>s. map_spmf (map_prod OK OK) (callee2 s x)) ELSE return_spmf (Fault, Fault)"
test_preserves_equation: fixes p :: "'a"   and x :: "'a" assumes "p \<le> p \<sqinter> p"   and "p \<le> top" shows "(p \<sqinter> x \<le> x \<sqinter> p) = (p \<sqinter> x = p \<sqinter> x \<sqinter> p)"
bangCases: fixes P :: "pi"   and Rs :: "residual" assumes "!P \<longmapsto> Rs" obtains P :: "pi" where "!Pa__ = !P" and "P \<parallel> !P \<longmapsto> Rs"
matchC_ZOC_def2: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set"   and d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchC_ZOC theta d c = (\<forall>s t d' t'. s \<approx> t \<and> (d, t) \<rightarrow>c (d', t') \<longrightarrow> s \<approx> t' \<and> (d', c) \<in> theta \<or> (\<exists>c' s'. (c, s) \<rightarrow>c (c', s') \<and> s' \<approx> t' \<and> (d', c') \<in> theta))"
map_butlast: fixes f :: "'e \<Rightarrow> 'd"   and xs :: "'e list" shows "map f (butlast xs) = butlast (map f xs)"
interp_ins_ref_nonex: fixes ops :: "('a \<times> 'a option) list"   and xs :: "('a \<times> 'a option) list"   and oid :: "'a"   and ref :: "'a"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops ops"   and "ops = xs @ [(oid, Some ref)] @ ys"   and "ref \<notin> set (interp_ins xs)" shows "oid \<notin> set (interp_ins ops)"
inverse_less_1_iff: 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" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less (inverse x) one = (less_eq x zero \<or> less one x)"
indep_sets_def: fixes F :: "'d \<Rightarrow> 'a set set"   and I :: "'d set" shows "indep_sets F I = ((\<forall>i\<in>I. F i \<subseteq> events) \<and> (\<forall>J\<subseteq>I. J \<noteq> {} \<longrightarrow> finite J \<longrightarrow> (\<forall>A\<in>Pi J F. prob (\<Inter> (A ` J)) = (\<Prod>j\<in>J. prob (A j)))))"
insert_bal: fixes t :: "'a btree"   and k :: "nat"   and x :: "'a" assumes "bal t" shows "bal (local.insert k x t)"
induct: fixes x1 :: "'b \<Rightarrow> 'b ttree"   and x2 :: "'b set"   and x3 :: "'b list"   and x4 :: "'b list"   and P :: "('b \<Rightarrow> 'b ttree) \<Rightarrow> 'b set \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "substitute'' x1 x2 x3 x4"   and "\<And>f T. P f T [] []"   and "\<And>zs f x xs' xs T ys. \<lbrakk>zs \<in> paths (f x); xs' \<in> xs \<otimes> zs; substitute'' (f_nxt f T x) T xs' ys; P (f_nxt f T x) T xs' ys\<rbrakk> \<Longrightarrow> P f T (x # xs) (x # ys)" shows "P x1 x2 x3 x4"
interval_split_2: fixes k :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "real" assumes "k \<in> Basis" shows "cbox a b \<inter> {x. c \<le> x \<bullet> k} = cbox (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) b"
compile: shows "rec_ci (Cn n f gs) = (ap, arity, fp)"
segmap_func: fixes D :: "('c, 'd) Order_scheme" shows "segmap D \<in> carrier D \<rightarrow> carrier (SS D)"
valid_involved_to_valid: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and F :: "'f set"   and P :: "'d set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>Z. \<Gamma>\<Turnstile>\<^bsub>/F\<^esub> {s. s = Z \<and> s \<in> P} c {t. Z \<in> P \<longrightarrow> t \<in> Q},{t. Z \<in> P \<longrightarrow> t \<in> A}" shows "\<Gamma>\<Turnstile>\<^bsub>/F\<^esub> P c Q,A"
ALG'_refl: fixes qs :: "'c list"   and i :: "nat"   and x :: "'c"   and A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list" assumes "qs ! i = x" shows "ALG' A qs init i x = 0"
vec_nth: fixes x :: "('b, 'a) vec" shows "($) x \<in> UNIV"
max_min_same_2: fixes x :: "'a"   and y :: "'a" shows "max (min x y) x = x"
similar'_adm: fixes n :: "nat" shows "adm (\<lambda>x. fst x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> snd x)"
mono_inf_fun: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'c" assumes "class.order less_eq less" shows "mono ((\<sqinter>) x)"
lrev_body_final_lrev_body3_eq: shows "lrev_body_final = lrev_body3"
in_language_to_language: fixes L :: "'a list set" shows "Collect (in_language (to_language L)) = L"
C1: fixes y :: "name"   and R :: "ccs"   and T :: "ccs" assumes "y \<sharp> R" shows "(\<lparr>\<nu>y\<rparr>(R \<parallel> T), R \<parallel> \<lparr>\<nu>y\<rparr>T) \<in> Rel"
getTo_def: shows "getTo \<equiv> snd \<circ> snd"
ivl_disj_un_one_1: fixes l :: "'c"   and u :: "'c" assumes "l < u" shows "{..l} \<union> {l<..<u} = {..<u}"
fmap_return: fixes f :: "'f \<rightarrow> 'd"   and x :: "'f" shows "fmap\<cdot>f\<cdot>(return\<cdot>x) = return\<cdot>(f\<cdot>x)"
\<tau>move2FAssRed: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and e :: "'addr expr1"   and F :: "String.literal"   and D :: "String.literal"   and e' :: "'addr expr1" shows "\<tau>move2 P h stk (e\<bullet>F{D} := e') (Suc (length (compE2 e) + length (compE2 e'))) None"
sum_list_distinct_conv_sum_set: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "class.monoid_add plus zero"   and "distinct xs" shows "sum_list (map f xs) = sum f (set xs)"
AsBs: shows "(\<Union>i\<in>Field r. As (succ r i)) \<subseteq> B"
emeasure_UN_countable: fixes I :: "'c set"   and X :: "'c \<Rightarrow> 'd set"   and M :: "'d measure" assumes "\<And>i. i \<in> I \<Longrightarrow> X i \<in> sets M"   and "countable I"   and "disjoint_family_on X I" shows "emeasure M (\<Union> (X ` I)) = \<integral>\<^sup>+ i. emeasure M (X i) \<partial>count_space I"
span_scale: fixes x :: "'a"   and S :: "'a set"   and c :: "complex" assumes "x \<in> cspan S" shows "c *\<^sub>C x \<in> cspan S"
drop_eq_nths: fixes n :: "nat"   and xs :: "'a list" shows "drop n xs = nths xs {i. n \<le> i}"
ageb: fixes u :: "'a" assumes "u \<in> rel_interior S" shows "b \<le> a \<bullet> u"
tensor_mat_unitary: fixes m1 :: "'a mat"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat ps_P.d1 ps_P.d1"   and "m2 \<in> carrier_mat ps_P.d2 ps_P.d2"   and "unitary m1"   and "unitary m2" shows "unitary (ps_P.tensor_mat m1 m2)"
"0_6": shows "\<forall>i\<in>set (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A))). \<forall>j\<in>set (remdups (map (the_LabelN \<circ> fst) (filter (Not \<circ> is_LabelS) A))). i \<noteq> j \<longrightarrow> comp_GSMP_disjoint public arity Ana \<Gamma> (remdups (trms_list\<^sub>s\<^sub>s\<^sub>t (proj_unl i A) @ map (pair' Pair) (setops_list\<^sub>s\<^sub>s\<^sub>t (proj_unl i A)))) (remdups (trms_list\<^sub>s\<^sub>s\<^sub>t (proj_unl j A) @ map (pair' Pair) (setops_list\<^sub>s\<^sub>s\<^sub>t (proj_unl j A)))) (M i) (M j) C"
defs_1: shows "p\<^sub>1 = find_closest p\<^sub>0 (dist (case c\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case c\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y))) (p\<^sub>2 # ps)"
U_irr_monic: shows "U \<subseteq> {q. irreducible q \<and> monic q}"
\<beta>_pos: fixes i :: "nat" assumes "i < length fs" shows "(0::'a) < \<beta> fs i"
is_inf_lower: fixes x :: "'b"   and y :: "'b"   and inf :: "'b" assumes "is_inf x y inf" obtains "inf \<sqsubseteq> x" and "inf \<sqsubseteq> y"
liftPoly_add: fixes w :: "nat"   and z :: "nat"   and a :: "'b mpoly"   and b :: "'b mpoly" shows "liftPoly w z (a + b) = liftPoly w z a + liftPoly w z b"
wp_g_odei: fixes P :: "'a \<Rightarrow> bool"   and I :: "'a \<Rightarrow> bool"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> bool"   and U :: "'a \<Rightarrow> real set"   and S :: "'a set"   and t\<^sub>0 :: "real"   and Q :: "'a \<Rightarrow> bool" assumes "\<lceil>P\<rceil> \<le> \<lceil>I\<rceil>"   and "\<lceil>I\<rceil> \<le> wp x\<acute>= f & G on U S @ t\<^sub>0 \<lceil>I\<rceil>"   and "\<lceil>\<lambda>s. I s \<and> G s\<rceil> \<le> \<lceil>Q\<rceil>" shows "\<lceil>P\<rceil> \<le> wp x\<acute>=f & G on U S @ t\<^sub>0 DINV I  \<lceil>Q\<rceil>"
vec_nth_inject: fixes x :: "('c, 'b) vec"   and y :: "('c, 'b) vec" shows "(($) x = ($) y) = (x = y)"
z_sel: fixes x :: "'a"   and y :: "'b" shows "pmf z (x, y) = enn2real (h x y)"
smallest_dist_unique: fixes M :: "'b set"   and h :: "'b"   and r :: "'b"   and s :: "'b" assumes "convex M"   and "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) r"   and "is_arg_min (\<lambda>x. dist x h) (\<lambda>x. x \<in> M) s" shows "r = s"
is_full: fixes a :: "'a"   and a' :: "'a"   and g :: "('a S.E.term, 'a) S.arr" assumes "ide a"   and "ide a'"   and "\<guillemotleft>g : S.UP a' \<Rightarrow>\<^sub>S S.UP a\<guillemotright>" shows "\<exists>f. \<guillemotleft>f : a' \<Rightarrow> a\<guillemotright> \<and> S.UP f = g"
dyn_equiv_stat: shows "dynamically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F = statically_complete_calculus Bot Inf (\<Turnstile>) Red_Red_I Red_F"
elims: fixes x :: "'b memref \<Rightarrow> bool"   and xa :: "'b memref list"   and y :: "nat set" assumes "outstanding_refs x xa = y" obtains P :: "'b memref \<Rightarrow> bool" where "x = P__" and "xa = []" and "y = {}"   | P :: "'b memref \<Rightarrow> bool" and r :: "'b memref" and rs :: "'b memref list" where "x = P" and "xa = r # rs" and "y = (if P r then address_of r \<union> outstanding_refs P rs else outstanding_refs P rs)"
comp_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<sqinter> x \<le> z \<sqinter> y"
distinct_1: fixes x11 :: "nat"   and x12 :: "'d"   and x13 :: "nat set"   and x21 :: "'d net_state"   and x22 :: "'d net_state" shows "NodeS x11 x12 x13 \<noteq> SubnetS x21 x22"
add_mset_replicate_mset_safe: fixes M :: "'c multiset"   and a :: "'c" assumes "NO_MATCH {#} M" shows "add_mset a M = {#a#} + M"
inverse_nonnegative_iff_nonnegative: fixes a :: "'a" shows "((0::'a) \<le> inverse a) = ((0::'a) \<le> a)"
nat_pow_0: fixes x :: "'d" shows "x ^ 0 = (1::'d)"
gcd_nat_abs_left_eq: fixes k :: "int"   and n :: "nat" shows "gcd (nat \<bar>k\<bar>) n = nat (gcd k (int n))"
eps_1: shows "\<bar>eps\<bar> \<le> snd (trunc_bound_pdevs p (add_pdevs (scaleR_pdevs a (snd X)) (scaleR_pdevs b (snd Y))))"
top_Err: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "Semilat.top (Err.le r) Err"
cexpr_typing_intE: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr"   and t :: "pdf_type"   and t' :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c \<integral>\<^sub>c e \<partial>t : t'" obtains "t' = REAL" and "case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL"
insert_Nil: fixes x :: "'d" shows "List.insert x [] = [x]"
wls_inversion: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set" assumes "wls s X" shows "(\<forall>xs x. X = Var xs x \<longrightarrow> s = asSort xs) \<and> (\<forall>delta inp binp. goodInp inp \<and> goodBinp binp \<and> X = Op delta inp binp \<longrightarrow> stOf delta = s \<and> wlsInp delta inp \<and> wlsBinp delta binp)"
cases: fixes x :: "'b \<times> nat" obtains xa :: "'b" and n :: "nat" where "x = (xa, n)"
TE_arOf_def: shows "TE_arOf \<equiv> ProblemIkTpart.TE_arOf arOf"
B: shows "step_2 (step_1 A) \<in> carrier_mat n n"
comp_permute: fixes f :: "'b"   and g :: "'b"   and k :: "'b"   and l :: "'b"   and h :: "'b" assumes "f \<cdot>\<^sub>D g = k \<cdot>\<^sub>D l"   and "D.seq f g"   and "D.seq g h" shows "f \<cdot>\<^sub>D g \<cdot>\<^sub>D h = k \<cdot>\<^sub>D l \<cdot>\<^sub>D h"
lossless_init_OR: fixes x0 :: "'g"   and x1 :: "'a"   and w :: "'h + 'b" shows "lossless_spmf (init_OR (x0, x1) w)"
unilateral_substitutesD3: fixes f :: "'a set \<Rightarrow> 'a set"   and b :: "'a"   and B :: "'a set"   and a :: "'a" assumes "unilateral_substitutes_on UNIV f"   and "b \<notin> f (insert b B)"   and "b \<in> f (insert a (insert b B))" shows "Xd b \<in> Xd ` B"
alphabet_swapLeaves: fixes t :: "'a tree"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "alphabet (swapLeaves t w\<^sub>a a w\<^sub>b b) = (if a \<in> alphabet t then if b \<in> alphabet t then alphabet t else alphabet t - {a} \<union> {b} else if b \<in> alphabet t then alphabet t - {b} \<union> {a} else alphabet t)"
locality: fixes U :: "'a set"   and I :: "real set"   and V :: "real \<Rightarrow> 'a set"   and s :: "'b" assumes "open_cover_of_open_subset S is_open U I V"   and "\<And>i. i \<in> I \<Longrightarrow> V i \<subseteq> U"   and "s \<in> \<FF> U"   and "\<And>i. i \<in> I \<Longrightarrow> \<rho> U (V i) s = \<zero>\<^bsub>V i\<^esub>" shows "s = \<zero>\<^bsub>U\<^esub>"
vars_minus: fixes p1 :: "linear_poly"   and p2 :: "linear_poly" shows "vars (p1 - p2) \<subseteq> vars p1 \<union> vars p2"