hcju/isabelleps huggingface dataset:
dom_glb: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.domain_near_semiring plus times domain_op less_eq less"   and "less_eq (domain_op x) (domain_op y)"   and "less_eq (domain_op x) (domain_op z)" shows "less_eq (domain_op x) (times (domain_op y) (domain_op z))"
inf_compl_bot: fixes x :: "'d upred" shows "(x \<and> \<not> x) = false"
simps_4: fixes i :: "int"   and c :: "int"   and e :: "MIR.num" shows "d_\<delta> (NDvd i (CN 0 c e)) = (dvd) i"
path_translation_eq: fixes a :: "'a"   and g :: "real \<Rightarrow> 'a" shows "path ((+) a \<circ> g) = path g"
to_mtx_inverse: fixes y :: "((real, 'a) vec, 'a) vec" assumes "y \<in> UNIV" shows "to_vec (to_mtx y) = y"
collapse_1: fixes strand_step :: "('c, 'd) strand_step" assumes "is_Send strand_step" shows "send\<langle>un_Send strand_step\<rangle>\<^sub>s\<^sub>t = strand_step"
finite_inf_greatest: fixes A :: "'a set" assumes "finite A"   and "A \<subseteq> carrier L"   and "A \<noteq> {}" shows "is_glb L (\<Sqinter>A) A"
Fls_fresh: fixes a :: "atom" shows "a \<sharp> Fls"
redT_acquire: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and t :: "'thread_id"   and x :: "'x"   and ln :: "'l \<Rightarrow>f nat"   and n :: "'l"   and s' :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln"   and "0 < ln $ n"   and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)" shows "s -t\<triangleright>(K$ [], [], [], [], [], convert_RA ln)\<rightarrow> s'"
disjoint: shows "insert n ` {K. K \<subseteq> {0..<n} \<and> card K = k} \<inter> {K. K \<subseteq> {0..<n} \<and> card K = Suc k} = {}"
Replacement: fixes y :: "ZF"   and A :: "ZF"   and f :: "ZF \<Rightarrow> ZF option" shows "Elem y (Replacement A f) = (\<exists>x. Elem x A \<and> f x = Some y)"
vprojection_vdomain: fixes I :: "V"   and A :: "V \<Rightarrow> V"   and i :: "V" shows "\<D>\<^sub>\<circ> (vprojection I A i) = vproduct I A"
pair: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "A >>> A'"   and "B >>> B'" shows "SimplyTyped.Pair A B >>> SimplyTyped.Pair A' B'"
select_convs_6: fixes getSATFlag :: "ExtendedBool"   and getF :: "Literal list list"   and getM :: "(Literal \<times> bool) list"   and getConflictFlag :: "bool"   and getConflictClause :: "nat"   and getQ :: "Literal list"   and getReason :: "Literal \<Rightarrow> nat option"   and getWatch1 :: "nat \<Rightarrow> Literal option"   and getWatch2 :: "nat \<Rightarrow> Literal option"   and getWatchList :: "Literal \<Rightarrow> nat list"   and getC :: "Literal list"   and getCl :: "Literal"   and getCll :: "Literal"   and getCn :: "nat"   and more :: "'a" shows "getQ \<lparr>getSATFlag = getSATFlag, getF = getF, getM = getM, getConflictFlag = getConflictFlag, getConflictClause = getConflictClause, getQ = getQ, getReason = getReason, getWatch1 = getWatch1, getWatch2 = getWatch2, getWatchList = getWatchList, getC = getC, getCl = getCl, getCll = getCll, getCn = getCn, \<dots> = more\<rparr> = getQ"
assms_3: fixes k :: "hf" assumes "Ord k"   and "P k" shows "P (succ k)"
assms_2: fixes x :: "'a" shows "open (\<gamma> x)"
nat_to_pr_def: shows "nat_to_pr = (\<lambda>x. sch_to_pr (nat_to_sch x))"
Sup_V_def: fixes X :: "V set" shows "\<Squnion> X \<equiv> if small X \<Rightarrow> set (\<Union> (elts ` X)) | otherwise \<Rightarrow> []\<^sub>\<circ>"
suffix_0: fixes x :: "nat \<Rightarrow> 'c" shows "suffix 0 x = x"
R_def: shows "R \<equiv> real (2 * N + 1) * pi"
member_set_of_seq: fixes x :: "'a"   and xq :: "'a seq" shows "(x \<in> set_of_seq xq) = member xq x"
derivation_ge_\<delta>: shows "derivation_ge D (length \<delta>)"
transD: fixes r :: "('d \<times> 'd) set"   and x :: "'d"   and y :: "'d"   and z :: "'d" assumes "trans r"   and "(x, y) \<in> r"   and "(y, z) \<in> r" shows "(x, z) \<in> r"
intro: fixes K :: "('g, 'h) ring_scheme"   and V :: "('g, 'i, 'j) module_scheme" assumes "module K V"   and "field K" shows "vectorspace K V"
carrier_vs_is_self: fixes K :: "('b, 'c) ring_scheme"   and V :: "('b, 'd, 'e) module_scheme"   and W :: "'d set" assumes "vectorspace K V" shows "carrier (V\<lparr>carrier := W\<rparr>) = W"
nonzero_mat_cnj: fixes A :: "complex \<times> complex \<times> complex \<times> complex" shows "(mat_cnj A = mat_zero) = (A = mat_zero)"
rep_eq: fixes x :: "float interval" shows "bounds_of_interval (floor_float_interval x) = (case bounds_of_interval x of (l, u) \<Rightarrow> (floor_fl l, floor_fl u))"
Basis_zero: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and inner :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and Basis :: "'e set" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "zero \<notin> Basis"
get_pred_def: fixes literal :: "'a literal" shows "get_pred literal = (case literal of Pos x11 x12 \<Rightarrow> x11 | Neg x21 x22 \<Rightarrow> x21)"
dependent_explicit_finite: fixes S :: "('a, 'b) vec set" assumes "finite S" shows "dependent S = (\<exists>u. (\<exists>v\<in>S. u v \<noteq> 0) \<and> (\<Sum>v\<in>S. u v *\<^sub>R v) = 0)"
transfer'_return: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(R ===>\<^sub>T R) Wrap return"
assms_1: shows "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' Z) c (Q' Z),(A' Z)"
exp_conj_mono_left: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "P \<tturnstile> Q" shows "P && R \<tturnstile> Q && R"
code: fixes s1 :: "'d stream"   and s2 :: "'e stream" shows "szip s1 s2 = (shd s1, shd s2) ## szip (stl s1) (stl s2)"
assms: shows "WT_core \<I>_adv \<I>_usr I core"
double_add_le_zero_iff_single_add_le_zero: fixes 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 a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (plus a a) zero = less_eq a zero"
vcg_wlp_wp_conseq: fixes \<pi> :: "char list \<Rightarrow> com option"   and mods :: "char list set"   and P :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and Q' :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "HT_mods \<pi> mods P c Q"   and "P s"   and "\<And>s'. \<lbrakk>modifies mods s' s; Q s s'\<rbrakk> \<Longrightarrow> Q' s'" shows "wlp \<pi> c Q' s"
surj_TM: fixes v :: "'a"   and tm :: "'a tm"   and t :: "nat" assumes "v = Time_Monad.val tm"   and "t = time tm" shows "tm = ??.Time_Monad.tm.TM v t"
phils_state_update_def: fixes N :: "nat"   and ps :: "nat list"   and e :: "dining_event" assumes "DiningPhilosophers N" shows "DiningPhilosophers.phils_state_update N ps e = (let i = phil e in if i = 0 then ps[i := DiningPhilosophers.lphil0_state_update N (ps ! i) e] else ps[i := Up\<^sub>r\<^sub>p i (ps ! i) e])"
eint_minus_comm: fixes d :: "eint"   and b :: "eint"   and c :: "eint" shows "d + b - c = d - c + b"
dom_locals_eval_mono_elim: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)" obtains "dom (locals (snd s0)) \<subseteq> dom (locals (snd s1))" and "\<And>vv s val. \<lbrakk>v = In2 vv; normal s1\<rbrakk> \<Longrightarrow> dom (locals (snd s)) \<subseteq> dom (locals (snd (snd vv val s)))"
arr_char: fixes f :: "'a \<times> 'a" shows "CC.arr f = (arr (fst f) \<and> arr (snd f))"
tl_x_not_x: fixes x :: "'b list" shows "x \<noteq> [] \<longrightarrow> tl x \<noteq> x"
finite_divisors_int: fixes i :: "int" assumes "i \<noteq> 0" shows "finite {d. d dvd i}"
map_def: fixes gf :: "'d \<times> 'd" shows "HomD.map gf = (if DopxD.arr gf then S.mkArr (HomD.set (DopxD.dom gf)) (HomD.set (DopxD.cod gf)) (inD \<circ> (\<lambda>h. snd gf \<cdot>\<^sub>D h \<cdot>\<^sub>D fst gf) \<circ> HomD.\<psi> (DopxD.dom gf)) else S.null)"
Spec: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and r :: "('e \<times> 'e) set"   and Q :: "'e set"   and A :: "'e set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. (\<forall>t. (s, t) \<in> r \<longrightarrow> t \<in> Q) \<and> (\<exists>t. (s, t) \<in> r)} Spec r Q,A"
simps: fixes x :: "rat"   and p :: "int poly"   and l :: "rat"   and r :: "rat" shows "compare_rat_1 x (p, l, r) = (if x < l then Lt else if r < x then Gt else if sgn (ipoly p x) = sgn (ipoly p r) then Gt else Lt)"
add_mono_thms_linordered_semiring_4: fixes i :: "'k"   and j :: "'k"   and k :: "'k"   and l :: "'k" assumes "i = j \<and> k = l" shows "i + k = j + l"
m_o_def: fixes m :: "'b \<Rightarrow> 'a"   and n :: "'a"   and opt :: "'b option" shows "m_o m n opt = (case opt of None \<Rightarrow> n + (1::'a) | Some x \<Rightarrow> m x)"
r_def: shows "r \<equiv> \<lambda>x i. if i = 0 then - x i else x i"
\<alpha>_def: shows "f \<bullet> \<alpha> = \<zero> \<and> \<alpha> \<in> carrier Zp"
simps: fixes a1 :: "'b"   and a2 :: "'b"   and a3 :: "'b \<Rightarrow> real"   and a4 :: "intv'" shows "intv'_elem a1 a2 a3 a4 = ((\<exists>u x y c. a1 = x \<and> a2 = y \<and> a3 = u \<and> a4 = Smaller' c \<and> u x - u y < real_of_int c) \<or> (\<exists>u x y d. a1 = x \<and> a2 = y \<and> a3 = u \<and> a4 = Const' d \<and> u x - u y = real_of_int d) \<or> (\<exists>d u x y. a1 = x \<and> a2 = y \<and> a3 = u \<and> a4 = Intv' d \<and> real_of_int d < u x - u y \<and> u x - u y < real_of_int (d + 1)) \<or> (\<exists>c u x y. a1 = x \<and> a2 = y \<and> a3 = u \<and> a4 = Greater' c \<and> real_of_int c < u x - u y))"
if_eqvt: fixes p :: "perm"   and b :: "bool"   and x :: "'a"   and y :: "'a" shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
execute_lookup: fixes r :: "'b ref"   and h :: "heap" shows "execute (!r) h = Some (Ref.get h r, h)"
v11_vcomp_vconverse: shows "(\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ> \<circ>\<^sub>\<circ> \<FF>\<lparr>ObjMap\<rparr> = vid_on (\<AA>\<lparr>Obj\<rparr>)"
index_mat_swaprows_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and k :: "nat"   and l :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "swaprows k l A $$ (i, j) = (if k = i then A $$ (l, j) else if l = i then A $$ (k, j) else A $$ (i, j))"
card_lists_distinct_length_eq': fixes k :: "nat"   and A :: "'a set" assumes "k < card A" shows "card {xs. length xs = k \<and> distinct xs \<and> set xs \<subseteq> A} = \<Prod>{card A - k + 1..card A}"
PAC_checker_specification_step_spec_def: shows "PAC_checker_specification_step_spec = (\<lambda>(\<V>\<^sub>0, A\<^sub>0) spec (\<V>, A) (b, B). (is_success b \<longrightarrow> \<Union> (vars ` set_mset A\<^sub>0) \<subseteq> \<V>\<^sub>0 \<and> \<Union> (vars ` set_mset A) \<subseteq> \<V> \<and> PAC_Format\<^sup>*\<^sup>* (\<V>\<^sub>0, A\<^sub>0) (\<V>, A) \<and> PAC_Format\<^sup>*\<^sup>* (\<V>, A) B) \<and> (is_found b \<longrightarrow> \<Union> (vars ` set_mset A\<^sub>0) \<subseteq> \<V>\<^sub>0 \<and> \<Union> (vars ` set_mset A) \<subseteq> \<V> \<and> PAC_Format\<^sup>*\<^sup>* (\<V>\<^sub>0, A\<^sub>0) (\<V>, A) \<and> PAC_Format\<^sup>*\<^sup>* (\<V>, A) B \<and> spec \<in> pac_ideal (set_mset A\<^sub>0)))"
atom_injection: fixes y :: "atom" shows "\<exists>x. y = atom x"
assms_2: fixes k :: "'n set" assumes "k \<in> \<D>" shows "f integrable_on k"
orderI: fixes a :: "'c DBMEntry"   and b :: "'c DBMEntry" assumes "a = linordered_monoid.max a b" shows "b \<preceq> a"
flows_reverse: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "flow0 (flow0 x0 t) (0 - t) = x0"
Cons: fixes i :: "nat" shows "i < length zs * length y \<longrightarrow> vec_vec_Tensor zs y ! i = zs ! (i div length y) * y ! (i mod length y)"
secure_refinement_simple: fixes \<R>\<^sub>A :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) set"   and \<R> :: "((((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat)) \<times> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat)) set"   and abs_steps :: "((Eg1.var, aexp, bexp) Stmt \<times> (Mode \<Rightarrow> Eg1.var set)) \<times> (Eg1.var \<Rightarrow> nat) \<Rightarrow> ((var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt \<times> (Mode \<Rightarrow> var\<^sub>C set)) \<times> (var\<^sub>C \<Rightarrow> nat) \<Rightarrow> nat" assumes "secure_refinement_simple \<R>\<^sub>A \<R> abs_steps" shows "secure_refinement_simpler \<R>\<^sub>A \<R> \<I>simple abs_steps"
PLS_state_1_def: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "PLS_state_1 s as \<equiv> length ` {as'. exec_plan s as' = exec_plan s as}"
wfactors_mult: fixes G :: "('c, 'd) monoid_scheme"   and as :: "'c list"   and a :: "'c"   and bs :: "'c list"   and b :: "'c" assumes "comm_monoid_cancel G"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "wfactors G (as @ bs) (a \<otimes>\<^bsub>G\<^esub> b)"
UN_Un_distrib: fixes A :: "'b \<Rightarrow> 'a set"   and B :: "'b \<Rightarrow> 'a set"   and I :: "'b set" shows "(\<Union>i\<in>I. A i \<union> B i) = \<Union> (A ` I) \<union> \<Union> (B ` I)"
p_def: shows "p = map (\<lambda>n. (face_cycle_succ ^^ n) a0) [0..<Suc (funpow_dist face_cycle_succ a0 an)]"
intvs_upper_decr: fixes l :: "int"   and h :: "int" assumes "l < h" shows "{l..<h} = Set.insert (h - 1) {l..<h - 1}"
subgroup_indicator_def: fixes G :: "('c, 'd) monoid_scheme"   and H :: "'c set"   and a :: "'c" shows "subgroup_indicator G H a = (LEAST k. 0 < k \<and> a [^]\<^bsub>G\<^esub> k \<in> H)"
sb: shows "g1 ` {0..1} \<inter> g2 ` {0..1} \<subseteq> {g2 0}"
wait_LEAST: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b stream" shows "wait (holds P) xs = (LEAST n. P (xs !! n))"
trg_cod: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "src (B.cod \<mu>) = src \<mu>"
degree_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and p :: "'b list" assumes "class.semiring_0 plus zero times" shows "semiring_0.degree zero p = length (semiring_0.pnormalize zero p) - 1"
Units_inverse: fixes R :: "('a, 'b) ring_scheme"   and u :: "'a" assumes "ring R"   and "u \<in> Units R" shows "inv\<^bsub>R\<^esub> u \<in> Units R"
set_spmf_G_L: fixes x0 :: "'g"   and x1 :: "'a"   and w0 :: "'h"   and w1 :: "'b" assumes "((x0, x1), w0, w1) \<in> set_spmf G" shows "((x0, x1), w0, w1) \<in> Rel_AND"
finite_reduced_GB_finite: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite F" shows "finite (punit.reduced_GB F)"
extremum: fixes a :: "'b word" shows "a \<le> - 1"
measurable_abs_powr: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and p :: "real" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. \<bar>f x\<bar> powr p) \<in> borel_measurable M"
SBHRMWWrite: fixes cond :: "(nat \<Rightarrow> nat option) \<Rightarrow> bool"   and \<theta> :: "nat \<Rightarrow> nat option"   and t :: "nat"   and m :: "nat \<Rightarrow> nat"   and a :: "nat"   and D :: "nat set"   and f :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and ret :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and "is" :: "instr list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and \<S> :: "nat \<Rightarrow> bool option" assumes "cond (\<theta>(t \<mapsto> m a))" shows "(RMW a t (D, f) cond ret A L R W # is, \<theta>, [], m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is, \<theta>(t \<mapsto> ret (m a) (f (\<theta>(t \<mapsto> m a)))), [], m(a := f (\<theta>(t \<mapsto> m a))), False, \<O> \<union> A - R, Map.empty, \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L)"
add_decreasing: fixes a :: "'b multiset"   and c :: "'b multiset"   and b :: "'b multiset" assumes "a \<subseteq># {#}"   and "c \<subseteq># b" shows "a + c \<subseteq># b"
assms_3: fixes t :: "real" assumes "t \<in> {0..1}" shows "cmod (vector_derivative \<gamma> (at t)) \<le> B"
antisym_subset: fixes r :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "r \<subseteq> s"   and "antisym s" shows "antisym r"
d1c_nzero: shows "\<forall>x. d1 \<le> x \<and> x < c \<longrightarrow> poly p x \<noteq> 0 \<and> poly (pderiv p) x \<noteq> 0"
cl_ivl_Suc: fixes g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and n :: "nat" shows "sum g {m..Suc n} = (if Suc n < m then 0::'a else sum g {m..n} + g (Suc n))"
reachable_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>H\<^esub> v" shows "v \<in> verts H"
deepR_list: fixes n :: "nat"   and m :: "('c, 'd) FingerTreeStruc"   and pr :: "('c, 'd) Digit"   and sf :: "('c, 'd) Node list" assumes "is_leveln_ftree (Suc n) m \<and> is_measured_ftree m"   and "is_leveln_digit n pr \<and> is_measured_digit pr"   and "\<forall>x\<in>set sf. (is_measured_node x \<and> is_leveln_node n x) \<and> length sf \<le> 4" shows "toList (deepR pr m sf) = digitToList pr @ toList m @ nlistToList sf"
fst_swap: fixes x :: "'e \<times> 'd" shows "fst (prod.swap x) = snd x"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b llist \<Rightarrow> 'a"   and llist :: "'b llist" shows "P (case llist of LNil \<Rightarrow> f1 | LCons x xa \<Rightarrow> f2 x xa) = ((llist = LNil \<longrightarrow> P f1) \<and> (\<forall>x21 x22. llist = LCons x21 x22 \<longrightarrow> P (f2 x21 x22)))"
dom_eq_ran: fixes y :: "'index \<times> 'trap" shows "y \<in> set_spmf I \<longrightarrow> domain (fst y) = range (fst y)"
rel_dds_pos_distr: fixes A :: "'b \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and D :: "'g \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_dds A B OO rel_dds C D \<le> rel_dds (A OO C) (B OO D)"
gpv''_def: shows "gpv'' \<equiv> Generative_Probabilistic_Value.Done x"
Collect_case_prod: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "{(a, b). P a \<and> Q b} = Collect P \<times> Collect Q"
mcont_cont: fixes luba :: "'h set \<Rightarrow> 'h"   and orda :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and lubb :: "'i set \<Rightarrow> 'i"   and ordb :: "'i \<Rightarrow> 'i \<Rightarrow> bool"   and f :: "'h \<Rightarrow> 'i" assumes "mcont luba orda lubb ordb f" shows "cont luba orda lubb ordb f"
Un_ac_1: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
f_def: fixes a :: "nat"   and b :: "nat"   and z :: "nat" shows "f a b z = sgn1 b * sgn1 (b * (z + 1) - a) + sgn2 b * sgn2 z"
sorted_wrt_append: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'d list" shows "sorted_wrt P (xs @ ys) = (sorted_wrt P xs \<and> sorted_wrt P ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. P x y))"
lookup_action_def: fixes op :: "(nat, nat) sas_plus_operator" shows "lookup_action op \<equiv> find (\<lambda>(uu_, pres, effs, uu_). precondition_of op = pres \<and> map (\<lambda>(v, a). ([], v, None, a)) (effect_of op) = effs) ast\<delta>"
AG_fp: fixes p :: "'b set" shows "\<^bold>A\<^bold>G p = p \<inter> \<^bold>A\<^bold>X \<^bold>A\<^bold>G p"
UBex: fixes f :: "atom fm"   and x :: "'a"   and xs :: "'a list" assumes "nqfree f"   and "DLO.I f (x # xs)"   and "\<not> DLO.I (inf\<^sub>+ f) xs"   and "x \<notin> EQ f xs" shows "\<exists>u\<in>UB f xs. x < u"
assms: fixes r :: "real" assumes "0 < r"   and "ereal r < R" shows "\<exists>z. norm z = r \<and> summable (\<lambda>n. f n * z ^ n)"
f: shows "(\<lambda>t. [fst t] \<otimes>\<^bsub>K [X]\<^esub> X [^]\<^bsub>K [X]\<^esub> snd t) \<in> set (dense_repr p) \<rightarrow> carrier (K [X])"
Eps_vP_elim: fixes k :: "'k2"   and v :: "'v2" shows "(Eps_Opt (vP k) = Some v) = vP k v"
connected_chain: fixes \<F> :: "'a set set" assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S \<and> connected S"   and "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S" shows "connected (\<Inter> \<F>)"
cases: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" assumes "write_actionsp E a" obtains aa :: "nat" where "a = aa" and "aa \<in> actions E" and "is_write_action (action_obs E aa)"
transpose_mat_plus: fixes nr :: "nat"   and nc :: "nat"   and m1 :: "'c list list"   and m2 :: "'c list list"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr nc m1"   and "mat nr nc m2" shows "Matrix_Legacy.transpose nr (mat_plusI pl m1 m2) = mat_plusI pl (Matrix_Legacy.transpose nr m1) (Matrix_Legacy.transpose nr m2)"
linord_of_list_sorted_on_subseqs: fixes ys :: "'a list"   and xs :: "'a list" assumes "ys \<in> set (subseqs xs)"   and "distinct xs" shows "sorted_on (linord_of_list (rev xs)) ys"
rep_eq: fixes x :: "'a dense_filter_type"   and xa :: "'a dense_filter_type" shows "Rep_dense_filter_type (x \<squnion> xa) = Rep_dense_filter_type x \<squnion> Rep_dense_filter_type xa"
d_def: shows "d \<equiv> \<lambda>k. let (dk, ak, bk, M) = DF (Suc k) in dk k"
Diff_subset: fixes A :: "'e set"   and B :: "'e set" shows "A - B \<subseteq> A"
Im_Ln_tendsto_at_top: fixes a :: "real" shows "((\<lambda>x. Im (Ln (Complex a x))) \<longlongrightarrow> pi / 2) at_top"
deg_add: fixes R :: "('a, 'b) ring_scheme"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "deg R (p \<oplus>\<^bsub>UP R\<^esub> q) \<le> max (deg R p) (deg R q)"
diffg: shows "g differentiable_on sphere (0::'a) 1 \<inter> S"
domain_unique: fixes a :: "'a"   and f :: "'a"   and a' :: "'a" assumes "a \<in> A.domains f"   and "a' \<in> A.domains f" shows "a = a'"
NonUniformExecutionsConstructable: fixes exec :: "('p, 'v, 's) configuration list"   and msg :: "('p, 'v) message"   and p :: "'p"   and trace :: "('p, 'v) message list" assumes "enabled (last exec) msg"   and "isReceiverOf p msg"   and "execution trans sends start exec trace"   and "nonUniform (last exec)"   and "\<And>cfg. reachable (last exec) cfg \<longrightarrow> agreement cfg" shows "\<exists>exec' trace'. execution trans sends start exec' trace' \<and> nonUniform (last exec') \<and> prefixList exec exec' \<and> prefixList trace trace' \<and> (\<forall>cfg. reachable (last exec') cfg \<longrightarrow> agreement cfg) \<and> stepReachable (last exec) msg (last exec') \<and> msg \<in> set (drop (length trace) trace')"