hcju/isabelleps huggingface dataset:
rows_conjugate: fixes m :: "'b mat" shows "rows (conjugate m) = map conjugate (rows m)"
5: shows "invar (case s of (x, y) \<Rightarrow> (real x, y))"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A" shows "sum g (insert x A) = (if x \<in> A then sum g A else g x + sum g A)"
Node_def: shows "Node \<equiv> \<Lambda> app1 app2. btree_abs\<cdot>(sinr\<cdot>(:up\<cdot>app1, up\<cdot>app2:))"
convex_hull_finite: fixes S :: "'c set" assumes "finite S" shows "convex hull S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y}"
dbtm_subst_ignore: fixes u :: "dbtm"   and name :: "name"   and j :: "nat"   and t :: "dbtm" shows "subst_dbtm u name (abst_dbtm name j t) = abst_dbtm name j t"
u: shows "rep_nat_term u = (pp_of_fun (\<lambda>x. if x = 0 then a else if x = 1 then b else 0), 0)"
\<theta>\<delta>_support_2: shows "\<delta> supports \<I>"
dup_code_2: fixes n :: "num" shows "Code_Numeral.dup (Code_Numeral.Pos n) = Code_Numeral.Pos (num.Bit0 n)"
add_le_same_cancel1: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and b :: "'b"   and a :: "'b" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq (plus b a) b = less_eq a zero"
suff_open: fixes r :: "'a llist" assumes "r \<in> A\<^sup>\<clubsuit>" shows "suff A r open"
continuous_map_if_iff: fixes X :: "'d topology"   and Y :: "'e topology"   and P :: "bool"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" shows "continuous_map X Y (\<lambda>x. if P then f x else g x) = continuous_map X Y (if P then f else g)"
conjug_eq_prim_root: fixes x :: "'b list"   and z :: "'b list"   and y :: "'b list" assumes "x \<cdot> z = z \<cdot> y"   and "x \<noteq> \<epsilon>" shows "\<exists>r s i n. (r \<cdot> s) \<^sup>@ Suc i = x \<and> (s \<cdot> r) \<^sup>@ Suc i = y \<and> (r \<cdot> s) \<^sup>@ n \<cdot> r = z \<and> s \<noteq> \<epsilon> \<and> primitive (r \<cdot> s)"
dom_simp: fixes f :: "'c local.term set \<times> 'c local.term set" assumes "CC.arr f" shows "CC.dom f = (local.dom (fst f), local.dom (snd f))"
ord_of_nat_succ_vempty: shows "ZFC_in_HOL.succ []\<^sub>\<circ> = 1\<^sub>\<nat>"
assms_1: shows "vs @ xs \<in> L\<^sub>i\<^sub>n M1 {vs' @ xs'}"
final_face_code_1: fixes vs :: "nat list" shows "final_face (Face vs Final) = True"
simps_2: fixes n :: "nat"   and s :: "'b stream" shows "stake (Suc n) s = shd s # stake n (stl s)"
N: fixes p :: "nat" assumes "N < p" shows "\<exists>m>p. u p < u m"
int_zd: shows "int (nat z - poly_deg f) = z - int (poly_deg f)"
sets_All: fixes A :: "'c \<Rightarrow> 'd set"   and M :: "'c \<Rightarrow> 'd measure"   and i :: "'c" assumes "\<forall>i. A i \<in> sets (M i)" shows "A i \<in> sets (M i)"
iso_is_retraction: fixes f :: "'b" assumes "D.iso f" shows "D.retraction f"
conj_comp_iff: fixes P :: "'d \<Rightarrow> bool"   and Q :: "'d \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'd" shows "(\<lambda>x. P x \<and> Q x) \<circ> g = (\<lambda>x. (P \<circ> g) x \<and> (Q \<circ> g) x)"
put_det: fixes x :: "'c \<Longrightarrow> 'd"   and \<sigma> :: "'d"   and v :: "'c"   and \<rho> :: "'d" assumes "pbij_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
slim_def: shows "slim \<equiv> \<lparr>pverts = slim_verts, parcs = slim_arcs\<rparr>"
seq_nil_right: fixes a :: "'b::refinement_lattice" shows "(a::'b::refinement_lattice) ; nil = a"
transitivity2_L: fixes B :: "'a set"   and A :: "'a set"   and \<psi> :: "'a" assumes "\<forall>\<phi>\<in>B. A \<turnstile> \<phi>"   and "B \<turnstile> \<psi>" shows "A \<turnstile> \<psi>"
icd\<rho>: shows "l icd\<^bsup>\<rho>\<^esup>\<rightarrow> k"
uint_add_lem: fixes x :: "'b word"   and y :: "'b word" shows "(uint x + uint y < 2 ^ LENGTH('b)) = (uint (x + y) = uint x + uint y)"
starfun_star_n: fixes f :: "'b \<Rightarrow> 'a"   and X :: "nat \<Rightarrow> 'b" shows "(*f* f) (star_n X) = star_n (\<lambda>n. f (X n))"
walk_tl: fixes xs :: "'a list" assumes "H.walk xs" shows "H.walk (tl xs)"
has_integral_linear: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b"   and S :: "'a set"   and h :: "'b \<Rightarrow> 'c" assumes "(f has_integral y) S"   and "bounded_linear h" shows "(h \<circ> f has_integral h y) S"
that: fixes v :: "'action stream" assumes "R.run v q"   and "(!!) (smap int (q ## trace u q)) \<approx> (!!) (smap int (q ## trace v q))" shows "thesis"
POR_is_not_definable: shows "\<nexists>f. definable f \<and> appFLv f [POR_arg1_rel, POR_arg2_rel] = POR_result_rel"
c_def: shows "c \<in> carrier Zp \<and> f \<bullet> (\<alpha> \<oplus> h) = f \<bullet> \<alpha> \<oplus> deriv f \<alpha> \<otimes> h \<oplus> c \<otimes> h [^] 2"
simps_2: fixes f :: "'b \<Rightarrow> 'a + unit"   and x :: "'b"   and xs :: "'b list" shows "existsM f (x # xs) = (try f x catch (\<lambda>e. try existsM f xs catch (\<lambda>x. Inl (e # x))))"
cod_char: fixes f :: "'a" shows "V.cod f = (if f \<in> Collect arr then f else V.null)"
rels_def: fixes pointsTo :: "'d \<Rightarrow> 'e" shows "rels pointsTo \<equiv> {((pointsTo, pointsTo l'), pointsTo, Link l') |l'. True}"
image_cong: fixes M :: "'x set"   and N :: "'x set"   and f :: "'x \<Rightarrow> 'y"   and g :: "'x \<Rightarrow> 'y" assumes "M = N"   and "\<And>x. x \<in> N \<Longrightarrow> f x = g x" shows "f ` M = g ` N"
assign_subst: fixes x :: "'b \<Longrightarrow> 'c"   and y :: "'d \<Longrightarrow> 'c"   and u :: "('b, 'c) uexpr"   and v :: "('d, 'c) uexpr" assumes "mwb_lens x"   and "mwb_lens y" shows "(y := v)\<lbrakk>\<lceil>u\<rceil>\<^sub></$x\<rbrakk> = x, y := u, v\<lbrakk>u/x\<rbrakk>"
ltlr_rel_sv: fixes R :: "('a \<times> 'b) set" assumes "single_valued R" shows "single_valued (\<langle>R\<rangle>ltlr_rel)"
graph_le: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and v :: "nat \<times> 'a"   and k :: "nat" assumes "finite (nodes A)"   and "w \<notin> language A"   and "v \<in> graph A w k" shows "k \<le> 2 * card (nodes A)"
S_reps_\<S>: fixes s :: "'s \<times> ('c \<Rightarrow> real) set" assumes "reps s \<in> S" shows "s \<in> \<S>"
reduction_equivalent: fixes q :: "'state"   and u :: "'action stream" assumes "q \<in> nodes"   and "run u q" obtains v :: "'action stream" where "R.run v q" and "(!!) (smap int (q ## trace u q)) \<approx> (!!) (smap int (q ## trace v q))"
AimpB: fixes \<tau> :: "i \<Rightarrow> \<zero>" shows "\<lfloor>(\<lambda>w. (\<forall>x. (\<^bold>\<box>x (\<tau> w) \<^bold>\<rightarrow> \<^bold>\<box>x \<downharpoonleft> \<tau>) w) \<longrightarrow> (\<forall>x. (\<^bold>\<box>x \<downharpoonleft> \<tau> \<^bold>\<rightarrow> \<^bold>\<box>x (\<tau> w)) w))\<rfloor>"
is_interval_oi: fixes r :: "real" shows "is_interval {r<..}"
z_zero_conic_sgn_1: fixes p :: "proj2" assumes "proj2_incident p z_zero" shows "conic_sgn p = 1"
UNION_commute: fixes P :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and B :: "'e set"   and A :: "'d set" shows "(\<Union>x\<in>A. \<Union> (P x ` B)) = (\<Union>y\<in>B. \<Union>x\<in>A. P x y)"
set_attribute_get_disconnected_nodes: fixes ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" shows "\<forall>w\<in>set_attribute_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
C_eq_sortedQ: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and l :: "(int \<times> int \<times> protocol) set set set list" assumes "distinct p"   and "all_in_list p l"   and "singleCombinators p"   and "wellformed_policy1_strong p"   and "wellformed_policy3Pr p"   and "allNetsDistinct p" shows "Cp (list2FWpolicy (qsort p l)) = Cp (list2FWpolicy p)"
initial_safe_free_flowing_implies_safe_delayed: fixes c :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)" assumes "initial c"   and "safe_free_flowing c" shows "safe_delayed c"
pre_4: fixes S :: "'x set"   and \<sigma> :: "'\<sigma>" assumes "S \<subseteq> S0"   and "S \<noteq> S0"   and "\<not> c \<sigma>"   and "I S \<sigma>"   and "\<forall>x\<in>S0 - S. \<forall>y\<in>S. R y x" shows "P \<sigma>"
vint_uminus_closed: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "-\<^sub>\<int> x \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"
less_eq_extended_def: shows "(\<le>) \<equiv> \<lambda>x y. rec_extended (case_extended False (\<lambda>y_0. True) True) (\<lambda>x_0. case_extended False ((<) x_0) True) (case_extended False (\<lambda>y_0. False) False) x y \<or> x = y"
CLBINT_I0c_power_mirror_iexp: fixes x :: "real"   and n :: "nat" shows "CLBINT s=0..ereal x. complex_of_real ((x - s) ^ n) * iexp s = complex_of_real (x ^ Suc n / real (Suc n)) + \<i> / of_nat (Suc n) * (CLBINT s=0..ereal x. complex_of_real ((x - s) ^ Suc n) * iexp s)"
Limit_def: fixes i :: "V" shows "Limit i \<equiv> Ord i \<and> 0 \<in>\<^sub>\<circ> i \<and> (\<forall>y. y \<in>\<^sub>\<circ> i \<longrightarrow> ZFC_in_HOL.succ y \<in>\<^sub>\<circ> i)"
psubstAbs_simp: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "goodEnv rho"   and "freshEnv xs x rho" shows "Abs xs x X $[rho] = Abs xs x X #[rho]"
Longest_common_prefix_eq: fixes L :: "'c list set"   and ps :: "'c list" assumes "L \<noteq> {}"   and "\<forall>xs\<in>L. prefix ps xs"   and "\<forall>qs. (\<forall>xs\<in>L. prefix qs xs) \<longrightarrow> length qs \<le> length ps" shows "Longest_common_prefix L = ps"
that: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and a :: "'c"   and q :: "'a \<Rightarrow>\<^sub>0 'b"   and b :: "'c" assumes "((p, i, a), q, j, b) \<in>\<^sub>p snd ` set ps" shows "thesis"
00: shows "bprv (imp (eql \<langle>neg \<phi>\<rangle> (Var yy)) (imp \<chi> (subst \<chi> \<langle>neg \<phi>\<rangle> yy)))"
of_nat_unat_le_mask_ucast: fixes t :: "'c word"   and w :: "'b word" assumes "word_of_nat (unat t) = w"   and "t \<le> mask LENGTH('b)" shows "t = UCAST('b \<rightarrow> 'c) w"
Span_contains_spanset: fixes ms :: "'v list" assumes "set ms \<subseteq> V" shows "set ms \<subseteq> RSpan ms"
contract_maximal: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "carrier (Vr K v) \<subseteq> carrier (Vr K v')" shows "maximal_ideal (Vr K v) (carrier (Vr K v) \<inter> vp K v')"
msubsteq_def: fixes c :: "poly"   and t :: "tm"   and d :: "poly"   and s :: "tm"   and a :: "poly"   and r :: "tm" shows "msubsteq c t d s a r = evaldjf (\<lambda>(x, y). Parametric_Ferrante_Rackoff.conj x y) [let cd = c *\<^sub>p d in (NEq (CP cd), Eq (tm.Add (tm.Mul (~\<^sub>p a) (tm.Add (tm.Mul d t) (tm.Mul c s))) (tm.Mul ((2)\<^sub>p *\<^sub>p cd) r))), (Parametric_Ferrante_Rackoff.conj (Eq (CP c)) (NEq (CP d)), Eq (tm.Add (tm.Mul (~\<^sub>p a) s) (tm.Mul ((2)\<^sub>p *\<^sub>p d) r))), (Parametric_Ferrante_Rackoff.conj (NEq (CP c)) (Eq (CP d)), Eq (tm.Add (tm.Mul (~\<^sub>p a) t) (tm.Mul ((2)\<^sub>p *\<^sub>p c) r))), (Parametric_Ferrante_Rackoff.conj (Eq (CP c)) (Eq (CP d)), Eq r)]"
nextVertices_def: fixes f :: "face"   and n :: "nat"   and v :: "nat" shows "f\<^bsup>n\<^esup> \<bullet> v \<equiv> (f \<bullet> ^^ n) v"
simps_1: fixes t :: "'a tree23" shows "join_all (T t) = t"
assms: fixes x :: "'a"   and y :: "'a" shows "f (x + y) = f x + f y"
essentially_surjective: fixes b :: "'d" assumes "D.ide b" shows "\<exists>a. C.ide a \<and> D.isomorphic (F a) b"
relabel_action_bound': fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and fxl :: "(nat \<times> nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> nat)"   and p :: "pr_operation' list"   and fxl' :: "(nat \<times> nat \<Rightarrow> 'a) \<times> (nat \<Rightarrow> nat)" assumes "Network c s t"   and "(fxl, p, fxl') \<in> trcl (Network.pr_algo_lts' c s t)" shows "length (filter is_RELABEL' p) \<le> 2 * (card (Graph.V c))\<^sup>2"
abs_summable_on_Times_swap: fixes f :: "'d \<times> 'e \<Rightarrow> 'f"   and A :: "'d set"   and B :: "'e set" shows "(f abs_summable_on A \<times> B) = ((\<lambda>(x, y). f (y, x)) abs_summable_on B \<times> A)"
diff: fixes x :: "'m"   and y :: "'m" shows "f (x - y) = f x - f y"
filter_paths_conv_free_without: fixes y :: "'b"   and t :: "'b ttree" shows "filter (\<lambda>x'. x' \<noteq> y) ` paths t = paths (without y t)"
f: fixes x :: "nat \<Rightarrow> real" assumes "x \<in> topspace (nsphere n)" shows "(f \<circ> (\<lambda>x i. - x i)) x = ((\<lambda>x i. - x i) \<circ> f) x"
A: fixes s :: "'b list"   and t :: "'a event set" assumes "[] \<notin> D P"   and "[] \<notin> D S"   and "s = []"   and "([], t) \<in> F P"   and "([], t) \<in> F S" shows "([], t) \<in> F Q"
iT_Plus_iSince_conv: fixes I :: "nat set"   and k :: "nat"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<S> t2 I \<oplus> k. Q t2) = (P (t1 + k). t1 \<S> t2 I. Q (t2 + k))"
Let_refine: fixes m :: "'c"   and m' :: "'d"   and R' :: "('c \<times> 'd) set"   and f :: "'c \<Rightarrow> ('e + 'f) nres"   and E :: "('e \<times> 'g) set"   and R :: "('f \<times> 'h) set"   and f' :: "'d \<Rightarrow> ('g + 'h) nres" assumes "(m, m') \<in> R'"   and "\<And>x x'. (x, x') \<in> R' \<Longrightarrow> f x \<le> \<Down>\<^sub>E E R (f' x')" shows "Let m f \<le> \<Down>\<^sub>E E R (Let m' f')"
Rel_def: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "Rel r \<equiv> r"
eNorm_eq_lnorm: fixes f :: "'a \<Rightarrow> real"   and p :: "real"   and M :: "'a measure" assumes "f \<in> lspace M (ennreal p)"   and "0 < p" shows "eNorm (\<LL> (ennreal p) M) f = ennreal (lnorm M p f)"
times_commute_eqpoll: fixes A :: "'a set"   and B :: "'b set" shows "A \<times> B \<approx> B \<times> A"
peval_semantics: fixes val :: "'b \<Rightarrow> char list \<Rightarrow> bool"   and w :: "'b"   and r :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and p :: "'a pfm" shows "peval (val w) (psemantics (Kripke val r) w) p = (Kripke val r, w \<Turnstile>\<^sub>! p)"
perm_1: fixes pi :: "(name \<times> name) list"   and x1 :: "trm" shows "pi \<bullet> Base x1 = Base (pi \<bullet> x1)"
map_eq_appendE: fixes f :: "'b \<Rightarrow> 'a"   and ls :: "'b list"   and fl :: "'a list"   and fl' :: "'a list" assumes "map f ls = fl @ fl'" obtains l :: "'b list" and l' :: "'b list" where "ls = l @ l'" and "map f l = fl" and "map f l' = fl'"
y_ex: shows "\<exists>y\<in>fv\<^sub>s\<^sub>e\<^sub>t (set M). \<Gamma> (Var x) = \<Gamma> (Var y)"
alg_F2set: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "('f, 'd, 'e) F1 \<Rightarrow> 'd"   and s2 :: "('f, 'd, 'e) F2 \<Rightarrow> 'e"   and x :: "('f, 'd, 'e) F2" assumes "alg B1 B2 s1 s2"   and "F2set2 x \<subseteq> B1"   and "F2set3 x \<subseteq> B2" shows "s2 x \<in> B2"
ln_2_40_decimals: shows "\<bar>ln 2 - 6931471805599453094172321214581765680755 / 10 ^ 40\<bar> \<le> inverse (10 ^ 40)"
extgsimp: fixes \<sigma> :: "'c"   and l1 :: "'d"   and l2 :: "'e" shows "extg ((\<sigma>, l1), l2) = (\<sigma>, l1, l2)"
eq1: shows "a + punit.lt (rep_list g1) = b + punit.lt (rep_list h)"
f: shows "f = (\<lambda>n. SOME v'. n \<in> V \<longrightarrow> n \<noteq> v \<longrightarrow> adjacent n v' \<and> adjacent v v')"
lsorted_ltlI: fixes xs :: "'a llist" assumes "lsorted xs" shows "lsorted (ltl xs)"
rep_eq: fixes x :: "('b \<times> 'c) list" shows "lookup (the_value x) = (\<lambda>k. case map_of x k of None \<Rightarrow> 0::'c | Some v \<Rightarrow> v)"
lower: shows "alpha * aform_val e X + zeta - delta \<le> inverse (aform_val e X)"
exec_step_ind_Invokestatic_Init: fixes D :: "char list"   and b :: "staticb"   and Ts :: "ty list"   and T :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and ins :: "bytecode"   and xt :: "ex_table"   and P :: "jvm_prog"   and C :: "char list"   and M :: "char list"   and m :: "jvm_method"   and sh :: "sheap"   and ics :: "init_call_status"   and n :: "nat"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and frs :: "frame list" assumes "(D, b, Ts, T, mxs, mxl\<^sub>0, ins, xt) = method P C M"   and "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D"   and "\<forall>sfs. sh D \<noteq> \<lfloor>(sfs, Done)\<rfloor>"   and "\<forall>Cs. ics \<noteq> Called Cs" shows "exec_step_ind (StepI (Invokestatic C M n)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, Calling D []) # frs, sh)"
surjective: fixes r :: "'b transition_scheme" shows "r = \<lparr>Label = Label r, Arity = Arity r, Guards = Guards r, Outputs = Outputs r, Updates = Updates r, \<dots> = more r\<rparr>"
terminates_call_elim: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile>call init p return c \<down> Normal s" obtains bdy :: "('e, 'd, 'f) com" where "\<Gamma> p = Some bdy" and "\<Gamma>\<turnstile>bdy \<down> Normal (init s)" and "\<forall>t. \<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> \<Rightarrow> Normal t \<longrightarrow> \<Gamma>\<turnstile>c s t \<down> Normal (return s t)"   | "\<Gamma> p = None"
disjoint_add_mset_2: fixes A :: "'a multiset"   and b :: "'a"   and B :: "'a multiset" shows "(\<Empt> = A \<inter># add_mset b B) = (b \<notin># A \<and> \<Empt> = A \<inter># B)"
strong_exhaust: fixes y :: "Syntax.term"   and c :: "'a" obtains "y = Unit"   | x2 :: "Syntax.var" where "y = Var x2"   | x31 :: "Syntax.var" and x32 :: "Syntax.term" where "{atom x31} \<sharp>* c" and "y = Lam x31 x32"   | x41 :: "Syntax.var" and x42 :: "Syntax.term" where "{atom x41} \<sharp>* c" and "y = Rec x41 x42"   | x5 :: "Syntax.term" where "y = Inj1 x5"   | x6 :: "Syntax.term" where "y = Inj2 x6"   | x71 :: "Syntax.term" and x72 :: "Syntax.term" where "y = Syntax.Pair x71 x72"   | x81 :: "Syntax.term" and x82 :: "Syntax.var" and x83 :: "Syntax.term" where "{atom x82} \<sharp>* c" and "y = Syntax.Let x81 x82 x83"   | x91 :: "Syntax.term" and x92 :: "Syntax.term" where "y = App x91 x92"   | x101 :: "Syntax.term" and x102 :: "Syntax.term" and x103 :: "Syntax.term" where "y = Case x101 x102 x103"   | x11 :: "Syntax.term" where "y = Prj1 x11"   | x12 :: "Syntax.term" where "y = Prj2 x12"   | x13 :: "Syntax.term" where "y = Roll x13"   | x14 :: "Syntax.term" where "y = Unroll x14"   | x15 :: "Syntax.term" where "y = Auth x15"   | x16 :: "Syntax.term" where "y = Unauth x16"   | x17 :: "hash" where "y = Hash x17"   | x181 :: "hash" and x182 :: "Syntax.term" where "y = Hashed x181 x182"
Lcm_dvdD: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "Lcm A dvd x"   and "y \<in> A" shows "y dvd x"
plus_octo_def: shows "(+) \<equiv> \<lambda>uu uua. corec_octo (\<lambda>(x, y). Ree x + Ree y) (\<lambda>(x, y). Im1 x + Im1 y) (\<lambda>(x, y). Im2 x + Im2 y) (\<lambda>(x, y). Im3 x + Im3 y) (\<lambda>(x, y). Im4 x + Im4 y) (\<lambda>(x, y). Im5 x + Im5 y) (\<lambda>(x, y). Im6 x + Im6 y) (\<lambda>(x, y). Im7 x + Im7 y) (uu, uua)"
measure_Un2: fixes A :: "'b set"   and M :: "'b measure"   and B :: "'b set" assumes "A \<in> fmeasurable M"   and "B \<in> fmeasurable M" shows "Sigma_Algebra.measure M (A \<union> B) = Sigma_Algebra.measure M A + Sigma_Algebra.measure M (B - A)"
simps_1: fixes x21 :: "('a, 'b) rbt"   and x22 :: "color"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "('a, 'b) rbt"   and y21 :: "('a, 'b) rbt"   and y22 :: "color"   and y23 :: "'a"   and y24 :: "'b"   and y25 :: "('a, 'b) rbt" shows "(Node x21 x22 x23 x24 x25 = Node y21 y22 y23 y24 y25) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23 \<and> x24 = y24 \<and> x25 = y25)"
wlp_def: fixes pr :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "wlp pr \<equiv> pr False"
\<R>_elim': fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains "(case \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>"   | "(case \<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>"   | "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>"
listrel_Cons1: fixes y :: "'b"   and ys :: "'b list"   and xs :: "'c list"   and r :: "('b \<times> 'c) set" assumes "(y # ys, xs) \<in> listrel r" obtains ya :: "'c" and ysa :: "'c list" where "xs = ya # ysa" and "(y, ya) \<in> r" and "(ys, ysa) \<in> listrel r"
notin_empty_fset: fixes x :: "'b" shows "x |\<notin>| {||}"
drop_butlast: fixes n :: "nat"   and xs :: "'a list" shows "drop n (butlast xs) = butlast (drop n xs)"
anno_def: fixes C :: "'b acom"   and p :: "nat" shows "anno C p = annos C ! p"
cp_pt_inst: assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "cp TYPE('a) TYPE('b) TYPE('b)"
mset_list_of_A: fixes t :: "'b tree" shows "mset (list_of_A t) = mset_tree t"
oLimit_make_mono_eq: fixes f :: "nat \<Rightarrow> ordinal" assumes "\<forall>n. f n < oLimit f" shows "oLimit (\<lambda>n. f (make_mono f n)) = oLimit f"
acquire: fixes s :: "('b, 'c) locks \<times> (('c \<Rightarrow> ('a \<times> 'b \<Rightarrow>f nat) option) \<times> 'd) \<times> ('c \<Rightarrow> 'e wait_set_status option) \<times> 'c set"   and t :: "'c"   and x :: "'a"   and ln :: "'b \<Rightarrow>f nat"   and final :: "'a \<Rightarrow> bool"   and r :: "('b, 'c, 'a, 'd, 'e, 'f) semantics" assumes "thr s t = \<lfloor>(x, ln)\<rfloor>"   and "ln \<noteq> no_wait_locks"   and "\<not> waiting (wset s t)"   and "may_acquire_all (locks s) t ln" shows "multithreaded_base.active_threadsp final r s t"
adjoint_dim: fixes A :: "'c mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "adjoint A \<in> carrier_mat n n"
uv_2: shows "y = v *\<^sub>R b + (1 - v) *\<^sub>R a"
eval_until_auxlist': fixes I :: "\<I>"   and nt :: "nat"   and auxlist :: "(nat \<times> event_data option list set \<times> event_data option list set) list"   and res :: "event_data option list set list"   and auxlist' :: "(nat \<times> event_data option list set \<times> event_data option list set) list" assumes "eval_until I nt auxlist = (res, auxlist')" shows "auxlist' = drop (length res) auxlist"
strict_iff_order: fixes b :: "'a"   and a :: "'a" shows "lt b a = (le b a \<and> a \<noteq> b)"
lss: shows "\<forall>y\<in>set lss. \<forall>z\<in>set y. z \<le> nmax"
finsum_empty: fixes f :: "'e \<Rightarrow> 'c" shows "finsum V f {} = \<zero>\<^bsub>V\<^esub>"
mono_dp_1: fixes w :: "'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_direct_product w x"   and "is_direct_product y z" shows "\<Phi> w x y z ; y = w"
max_def: fixes ko :: "'b key_order"   and a :: "'b"   and b :: "'b" shows "ko.max ko a b = (if ko.le ko a b then b else a)"
PImp: fixes r :: "'b pfm"   and p :: "'b pfm"   and q :: "'b pfm" shows "\<turnstile>\<^sub>! ([r]\<^sub>! (p \<^bold>\<longrightarrow>\<^sub>! q) \<^bold>\<longleftrightarrow>\<^sub>! [r]\<^sub>! p \<^bold>\<longrightarrow>\<^sub>! [r]\<^sub>! q)"