hcju/isabelleps huggingface dataset:
prime_sum_upto_eqI: fixes a :: "real"   and b :: "real"   and f :: "nat \<Rightarrow> 'a" assumes "a \<le> b"   and "\<And>k. k \<in> {nat \<lfloor>a\<rfloor><..nat \<lfloor>b\<rfloor>} \<Longrightarrow> \<not> prime k" shows "prime_sum_upto f a = prime_sum_upto f b"
dim_gso: fixes i :: "nat" assumes "i < length fs" shows "dim_vec (gso i) = n"
start_def: shows "start \<equiv> OK first # replicate (length is - 1) (OK None)"
simps_3: fixes l1 :: "'b tree"   and a1 :: "'b"   and r1 :: "'b tree"   and l2 :: "'b tree"   and a2 :: "'b"   and r2 :: "'b tree" shows "merge (\<langle>l1, a1, r1\<rangle> =: t1a__) (\<langle>l2, a2, r2\<rangle> =: t2a__) = (if a1 \<le> a2 then \<langle>merge t2a__ r1, a1, l1\<rangle> else \<langle>merge t1a__ r2, a2, l2\<rangle>)"
induct: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and x1 :: "'c rule list"   and x2 :: "state"   and x3 :: "state"   and P :: "'c rule list \<Rightarrow> state \<Rightarrow> state \<Rightarrow> bool" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>x1, x2\<rangle> \<Rightarrow> x3"   and "\<And>t. P [] t t"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Accept] Undecided (Decision FinalAllow)"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Drop] Undecided (Decision FinalDeny)"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Reject] Undecided (Decision FinalDeny)"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Log] Undecided Undecided"   and "\<And>m. matches \<gamma> m p \<Longrightarrow> P [Rule m Empty] Undecided Undecided"   and "\<And>m a. \<not> matches \<gamma> m p \<Longrightarrow> P [Rule m a] Undecided Undecided"   and "\<And>rs X. P rs (Decision X) (Decision X)"   and "\<And>rs\<^sub>1 t rs\<^sub>2 t'. \<lbrakk>\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t; P rs\<^sub>1 Undecided t; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t'; P rs\<^sub>2 t t'\<rbrakk> \<Longrightarrow> P (rs\<^sub>1 @ rs\<^sub>2) Undecided t'"   and "\<And>m chain rs\<^sub>1 m' rs\<^sub>2. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2); matches \<gamma> m' p; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided; P rs\<^sub>1 Undecided Undecided\<rbrakk> \<Longrightarrow> P [Rule m (Call chain)] Undecided Undecided"   and "\<And>m chain rs t. \<lbrakk>matches \<gamma> m p; \<Gamma> chain = Some rs; \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t; P rs Undecided t\<rbrakk> \<Longrightarrow> P [Rule m (Call chain)] Undecided t" shows "P x1 x2 x3"
elims: fixes x :: "('b \<times> 'a \<times> 'b) list option"   and y :: "'a infty" assumes "path_weight' x = y" obtains "x = None" and "y = top"   | p :: "('b \<times> 'a \<times> 'b) list" where "x = Some p" and "y = Num (path_weight p)"
b_rel_subtyping_merge_eqA: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'b"   and P' :: "'a \<Rightarrow> bool" shows "hn_ctxt (b_assn A P) x y \<or>\<^sub>A hn_ctxt (b_assn A P') x y \<Longrightarrow>\<^sub>t hn_ctxt (b_assn A (\<lambda>x. P x \<or> P' x)) x y"
a_subset: fixes H :: "'c set"   and G :: "('c, 'd) ring_scheme" assumes "additive_subgroup H G" shows "H \<subseteq> carrier G"
zero_ikA: shows "is_zero_row_upt_k (mod_type_class.from_nat i) k A"
linorder_lattice_sup_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<le> sup x y"
assms_1: shows "DisjointPaths G v0 v1 paths"
insert_in_sets: fixes x :: "'d"   and M :: "'d measure"   and A :: "'d set" assumes "{x} \<in> sets M"   and "A \<in> sets M" shows "insert x A \<in> sets M"
prems_2: shows "(t # ts) ! i = (p, is, xs, sb, \<D>, \<O>, \<R>)"
lcs_alt_2: fixes a :: "'a"   and x :: "'a"   and b :: "'a"   and y :: "'a" assumes "a + x = b + y" shows "lcs x y = b + y - gcs a b"
Abs1_eq_1: fixes a :: "'b"   and x :: "'a"   and y :: "'a" shows "([{atom a}]set. x = [{atom a}]set. y) = (x = y)"
Thm_def: fixes \<phi> :: "WS1S_Nameful.ws1s" shows "Thm \<phi> = eqv \<phi> T"
subst_identity: fixes P :: "pi"   and a :: "name" shows "P[a::=a] = P"
prob_space_with_indicators_def: fixes M :: "'b measure"   and I :: "'c set"   and A :: "'c \<Rightarrow> 'b set" shows "prob_space_with_indicators M I A \<equiv> prob_space M \<and> prob_space_with_indicators_axioms M I A"
sum_i': fixes i :: "nat"   and x :: "'v"   and y :: "'v" shows "f_i' i (x, y) = (\<Sum>j<i. (k_i' j (x, y) - k_i' j (y, x)) * indicat_real \<^bold>E (x, y))"
suffix_shift: fixes xs :: "'b list"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat" shows "xs @- \<sigma> |\<^sub>s i = drop i xs @- (\<sigma> |\<^sub>s i - length xs)"
IH: fixes f :: "nat \<Rightarrow> 'a" assumes "f \<in> carrier P" shows "monom P ((f [^]\<^bsub>P\<^esub> na__) (deg R (f [^]\<^bsub>P\<^esub> na__))) (deg R (f [^]\<^bsub>P\<^esub> na__)) = monom P (f (deg R f)) (deg R f) [^]\<^bsub>P\<^esub> na__"
compl_zero: shows "NOT (0::'a) = - (1::'a)"
that: fixes xs1 :: "(bool \<times> 'a) list"   and x1 :: "bool \<times> 'a"   and x2 :: "bool \<times> 'a"   and xs2 :: "(bool \<times> 'a) list" assumes "x = xs1 @ x1 # x2 # xs2"   and "y = xs1 @ xs2"   and "canceling x1 x2" shows "thesis"
lbv_axioms: shows "lbv A r f \<top> \<bottom>"
paths_Either: fixes ts :: "'b ttree set" shows "paths (Either ts) = insert [] (\<Union> (paths ` ts))"
remdups_id_iff_distinct: fixes xs :: "'d list" shows "(remdups xs = xs) = distinct xs"
sup_compl_top: fixes x :: "'b" shows "x \<squnion> - x = \<top>"
of_nat_mono: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "of_nat i \<le> of_nat j"
nat_term_comp_def: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" shows "nat_term_comp cmp = ((\<forall>u v. snd (rep_nat_term u) = snd (rep_nat_term v) \<longrightarrow> fst (rep_nat_term u) = 0 \<longrightarrow> cmp u v \<noteq> Gt) \<and> (\<forall>u v. fst (rep_nat_term u) = fst (rep_nat_term v) \<longrightarrow> snd (rep_nat_term u) < snd (rep_nat_term v) \<longrightarrow> cmp u v = Lt) \<and> (\<forall>t u v. cmp u v = Lt \<longrightarrow> cmp (splus t u) (splus t v) = Lt) \<and> (\<forall>u v a b. fst (rep_nat_term u) = fst (rep_nat_term a) \<longrightarrow> fst (rep_nat_term v) = fst (rep_nat_term b) \<longrightarrow> snd (rep_nat_term u) = snd (rep_nat_term v) \<longrightarrow> snd (rep_nat_term a) = snd (rep_nat_term b) \<longrightarrow> cmp a b = Lt \<longrightarrow> cmp u v = Lt))"
conventional'': fixes g :: "'g"   and v :: "'val"   and ns :: "'node list"   and m :: "'node"   and v' :: "'val" assumes "g \<turnstile> defNode g v-ns\<rightarrow>m"   and "defNode g v \<notin> set (tl ns)"   and "v \<in> allUses g m"   and "var g v' = var g v"   and "v \<in> allVars g"   and "v' \<in> allVars g" shows "defNode g v' \<notin> set (tl ns)"
elims_2: fixes x :: "label"   and xa :: "label" assumes "x \<sqsubset>l xa" obtains l :: "label" where "x = Old" and "xa = la__" and "la__ \<noteq> Old"   | l :: "label" where "x = Processed" and "xa = l" and "l = New"
delete_point_group_size: fixes x :: "'a" assumes "{x} \<in> \<G> \<Longrightarrow> 2 < card \<G>" shows "1 < card {g - {x} |g. g \<in> \<G> \<and> g \<noteq> {x}}"
gcd_diff_dvd_right1: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b" shows "gcd a (b - c) = gcd a c"
l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs \<equiv> l_set_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes set_child_nodes_locs set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M \<and> l_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
P1I: fixes r :: "recf"   and f :: "nat \<Rightarrow> nat option" assumes "recfn 1 r"   and "(\<lambda>x. eval r [x]) = f" shows "f \<in> \<P>"
simps_3: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and vb :: "'v"   and vc :: "'v"   and v :: "'v"   and vd :: "'v" shows "split f (Edge vb vc, Edge v vd) = 0"
abs_e1: shows "\<bar>e1\<bar> \<le> snd (trunc_bound_eucl p (fst (fst X) + fst (fst Y)))"
list_def: fixes A :: "'a set" shows "list A \<equiv> SOME al. distinct al \<and> set al = A"
inject_2: fixes x2 :: "'b"   and y2 :: "'b" shows "(Lt x2 = Lt y2) = (x2 = y2)"
gcd_int_int_eq: fixes m :: "nat"   and n :: "nat" shows "gcd (int m) (int n) = int (gcd m n)"
dvd_diff_commute: 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 uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_ring divide plus minus zero times one modulo euclidean_size uminus" shows "dvd.dvd times a (minus c b) = dvd.dvd times a (minus b c)"
len_greater_imp_nonempty: fixes x :: "nat"   and l :: "'b list" assumes "x < length l" shows "l \<noteq> []"
assms: shows "\<Psi> \<rhd> \<tau>.P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"
fds_abs_converges_Re_le: fixes f :: "'a fds"   and z :: "'a"   and z' :: "'a" assumes "fds_abs_converges f z"   and "z \<bullet> (1::'a) \<le> z' \<bullet> (1::'a)" shows "fds_abs_converges f z'"
trace_0: fixes program_step :: "(nat \<Rightarrow> nat option) \<Rightarrow> 'b \<Rightarrow> 'b \<times> instr list \<Rightarrow> bool"   and c :: "nat \<Rightarrow> ('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> unit \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list \<times> (nat \<Rightarrow> nat) \<times> (nat \<Rightarrow> bool option)"   and n :: "nat" shows "program_trace program_step c n 0"
vars_if_subtermeq_set: fixes x :: "'b"   and M :: "('a, 'b) Term.term set" assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "x \<in> fv\<^sub>s\<^sub>e\<^sub>t M"
reachable_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>* v" shows "u \<in> verts G"
simps_1: fixes uu :: "('d \<times> 'd) set"   and uv :: "'c \<Rightarrow> 'd"   and uw :: "'d set" shows "ipurge_tr_rev_aux uu uv uw [] = []"
is_max: fixes \<aa> :: "'a set" assumes "ideal \<aa> A addA multA zeroA oneA"   and "\<aa> \<noteq> A"   and "I \<subseteq> \<aa>" shows "I = \<aa>"
inv_into_f_f: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and x :: "'e" assumes "inj_on f A"   and "x \<in> A" shows "inv_into A f (f x) = x"
Texp_Var: fixes v :: "Vars.var"   and a :: "Arity" shows "single v \<sqsubseteq> Texp (Var v)\<cdot>a"
pex_def: fixes r :: "('b \<times> 'b) set" shows "pex r = {(P1, P2). (Decreasing_Diagrams.measure r P1, Decreasing_Diagrams.measure r P2) \<in> mul r}"
matching_dsts_filterW: fixes filterW :: "'i simple_rule list"   and d :: "'i word"   and s :: "'i word" assumes "filterW = filter (\<lambda>r. simple_conn_matches (match_sel r) c) rs" shows "(d \<in> wordinterval_to_set (matching_dsts s filterW Empty_WordInterval)) = (runFw s d c filterW = Decision FinalAllow)"
assms_5: fixes i :: "'b" assumes "i \<in> J" shows "A i \<in> sets (M' i)"
ord_inj_func: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_inj D E f" shows "f \<in> carrier D \<rightarrow> carrier E"
results'_gpv_Pure: fixes generat :: "('a, 'b, ('a, 'b, 'c) rpv) generat"   and gpv :: "('a, 'b, 'c) gpv"   and x :: "'a" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "x \<in> generat_pures generat" shows "x \<in> results'_gpv gpv"
constructive_security_obsf_lifting: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and adv :: "real"   and \<I>_real' :: "('ae, 'ag) \<I>"   and \<I>_common' :: "('af, 'ah) \<I>"   and \<A> :: "('ae + 'af, 'ag + 'ah) distinguisher_obsf"   and w_adv_real :: "('ae, 'ag, 'y, 'aa) converter"   and w_usr :: "('af, 'ah, 'z, 'ab) converter"   and \<I>_ideal' :: "('ai, 'aj) \<I>"   and w_adv_ideal :: "('ai, 'aj, 'ac, 'ad) converter"   and w_adv_ideal_inv :: "('ac, 'ad, 'ai, 'aj) converter" assumes "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"   and "exception_\<I> (\<I>_real' \<oplus>\<^sub>\<I> \<I>_common') \<turnstile>g \<A> \<surd> \<Longrightarrow> constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) adv"   and "\<I>_common', \<I>_common \<turnstile>\<^sub>C w_usr \<surd>"   and "pfinite_converter \<I>_common' \<I>_common w_usr"   and "\<I>_real', \<I>_real \<turnstile>\<^sub>C w_adv_real \<surd>"   and "\<I>_ideal', \<I>_ideal \<turnstile>\<^sub>C w_adv_ideal \<surd>"   and "\<I>_ideal, \<I>_ideal' \<turnstile>\<^sub>C w_adv_ideal_inv \<surd>"   and "\<I>_ideal,\<I>_ideal \<turnstile>\<^sub>C w_adv_ideal_inv \<odot> w_adv_ideal \<sim> 1\<^sub>C"   and "pfinite_converter \<I>_real' \<I>_real w_adv_real"   and "pfinite_converter \<I>_ideal \<I>_ideal' w_adv_ideal_inv" shows "constructive_security_obsf (w_adv_real |\<^sub>= w_usr \<rhd> real_resource) (w_adv_ideal |\<^sub>= w_usr \<rhd> ideal_resource) (w_adv_real \<odot> sim \<odot> w_adv_ideal_inv) \<I>_real' \<I>_ideal' \<I>_common' \<A> adv"
integrable_cong_AE: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M"   and "AE x in M. f x = g x" shows "integrable M f = integrable M g"
ODNmap_ord_isom: fixes D :: "'c Order"   and x :: "'c Order set" assumes "Worder D"   and "x = ordinal_number D" shows "ord_isom (Iod ODnods (segment ODnods x)) D (restrict (ODNmap D) (carrier (Iod ODnods (segment ODnods x))))"
can_syncI: fixes t :: "'t"   and x :: "'x2"   and m :: "'m2"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x' :: "'x2"   and m' :: "'m2"   and LT :: "('l + 't + 't) set" assumes "r2.r_syntax t x m ta x' m'"   and "LT = collect_waits ta" shows "t \<turnstile> \<langle>x, m\<rangle> LT \<wrong>2"
called_by_chain_def: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option" shows "called_by_chain \<Gamma> = _Collect (callee, caller) (case \<Gamma> caller of None \<Rightarrow> False | Some rs \<Rightarrow> \<exists>m. Rule m (Call callee) \<in> set rs)"
simps: fixes i :: "nat"   and xs :: "'a list" shows "fisher_yates_aux i xs = (if length xs \<le> i + 1 then return_pmf xs else pmf_of_set {i..<length xs} \<bind> (\<lambda>j. fisher_yates_aux (i + 1) (swap xs i j)))"
make_mono_less: fixes f :: "nat \<Rightarrow> ordinal"   and n :: "nat" assumes "\<forall>n. f n < oLimit f" shows "make_mono f n < make_mono f (Suc n)"
replace_min_sift_down: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree"   and x :: "'b" assumes "braun \<langle>l, a, r\<rangle>" shows "replace_min x \<langle>l, a, r\<rangle> = sift_down l x r"
map_acom_Seq: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c acom"   and c1' :: "'b acom"   and c2' :: "'b acom" shows "(ACom.map_acom f c = c1';; c2') = (\<exists>c1 c2. c = c1;; c2 \<and> ACom.map_acom f c1 = c1' \<and> ACom.map_acom f c2 = c2')"
dy_fake_chan_extr_insert_parts: fixes M :: "chan"   and bad :: "agent set"   and IK :: "msg set"   and CH :: "chan set" assumes "M \<in> dy_fake_chan bad IK CH" shows "parts (extr bad IK (insert M CH)) \<subseteq> parts (extr bad IK CH) \<union> dy_fake_msg bad IK CH"
A: shows "Esigma (h * M) = Esigma (h * N)"
not_less: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "class.linorder less_eq less" shows "(\<not> less x y) = less_eq y x"
free_tv_gen: fixes S :: "nat \<Rightarrow> typ"   and A :: "type_scheme list"   and t :: "typ" shows "free_tv (gen ($ S A) t) = free_tv t \<inter> free_tv ($ S A)"
Output: fixes a :: "name"   and b :: "name"   and P :: "pi" shows "a{b}.P \<Longrightarrow>\<^sub>l a[b] \<prec> P"
hd_distinct_neq_last: fixes xs :: "'a list" assumes "distinct xs"   and "1 < length xs" shows "hd xs \<noteq> last xs"
en_eq_7: fixes v :: "i"   and x :: "\<kappa>"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<not>\<lbrace>x,F\<rbrace> \<^bold>\<equiv> \<^bold>\<box>\<^bold>\<not>\<lbrace>x,F\<rbrace> in v]"
bddh_binop: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and l :: "'b bdd"   and r :: "'c bdd" shows "bddh n (bdd_binop f l r) = (bddh n l \<and> bddh n r)"
f_base1: fixes x :: "real" assumes "x\<^sub>0 \<le> x"   and "x \<le> x\<^sub>1" shows "f x \<le> fb1"
f5: fixes ss :: "symbol list" shows "take j ss = take i (take j ss)"
vs: fixes v :: "'a vec" assumes "v \<in> set vvs" shows "v \<in> carrier_vec n"
assms_2: fixes t :: "'a" assumes "unapp t = None" shows "P t"
simulation2: fixes t :: "'t"   and s1 :: "'x1 \<times> 'm1"   and s2 :: "'x2 \<times> 'm2"   and tl2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s2' :: "'x2 \<times> 'm2" assumes "t \<turnstile> s1 \<approx> s2"   and "t \<turnstile> s2 -2-tl2\<rightarrow> s2'"   and "\<not> \<tau>move2 s2 tl2 s2'" shows "\<exists>s1' s1'' tl1. r1.silent_moves t s1 s1' \<and> t \<turnstile> s1' -1-tl1\<rightarrow> s1'' \<and> \<not> \<tau>move1 s1' tl1 s1'' \<and> t \<turnstile> s1'' \<approx> s2' \<and> tl1 \<sim>m tl2"
InterD: fixes A :: "'b"   and C :: "'b set set"   and X :: "'b set" assumes "A \<in> \<Inter> C"   and "X \<in> C" shows "A \<in> X"
cong_def: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr" shows "cong.cong rel eval R \<equiv> equivp R \<and> rel_fun (rel R) R eval eval"
H_def: shows "H = \<lparr>verts = {}, arcs = {}, tail = tail G, head = head G\<rparr>"
bell00_index_3: shows "|\<beta>\<^sub>0\<^sub>0\<rangle> $$ (2, 0) = 0"
Compl_partition: fixes A :: "'c set" shows "A \<union> - A = UNIV"
cases: fixes a1 :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> (Vars.var \<Rightarrow> one\<^sub>\<bottom>) \<times> Arity \<times> Arity list \<times> Vars.var list"   and a2 :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list" assumes "consistent a1 a2" obtains ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" and a :: "Arity" and as :: "Arity list" and r :: "Vars.var list" and \<Gamma> :: "(Vars.var \<times> exp) list" and e :: "exp" and S :: "stack_elem list" and ce :: "Vars.var \<Rightarrow> one\<^sub>\<bottom>" where "a1 = (ae, ce, a, as, r)" and "a2 = (\<Gamma>, e, S)" and "a_consistent (ae, a, as) (restr_conf (- set r) (\<Gamma>, e, S))" and "edom ae = edom ce" and "prognosis ae as a (\<Gamma>, e, S) \<sqsubseteq> ce" and "\<And>x. \<lbrakk>x \<in> thunks \<Gamma>; many \<sqsubseteq> ce x\<rbrakk> \<Longrightarrow> ae x = up\<cdot>0" and "set r \<subseteq> domA \<Gamma> \<union> upds S - edom ce"
borel_eq_sigmaI3: fixes X :: "'c set set"   and F :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and A :: "('d \<times> 'e) set" assumes "borel = sigma UNIV X"   and "\<And>x. x \<in> X \<Longrightarrow> x \<in> sets (sigma UNIV ((\<lambda>(i, j). F i j) ` A))"   and "\<And>i j. (i, j) \<in> A \<Longrightarrow> F i j \<in> sets borel" shows "borel = sigma UNIV ((\<lambda>(i, j). F i j) ` A)"
wlsAbs_disjoint: fixes xs :: "'varSort"   and s :: "'sort"   and A :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set"   and xs' :: "'varSort"   and s' :: "'sort" assumes "wlsAbs (xs, s) A"   and "wlsAbs (xs', s') A" shows "xs = xs' \<and> s = s'"
fold_insert: fixes A :: "'d set"   and x :: "'d"   and z :: "'e" assumes "finite A"   and "x \<notin> A" shows "Finite_Set.fold g z (insert x A) = g x (Finite_Set.fold g z A)"
acyclic_suc_rel: shows "acyclicP (\<lambda>s s'. \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s)"
bij_if_then_else: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set"   and g' :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "inj_on f X"   and "inj_on g' (- X)"   and "- f ` X = g' ` (- X)"   and "h = (\<lambda>z. if z \<in> X then f z else g' z)" shows "inj h \<and> surj h"
apply_guards_append: fixes a :: "vname gexp list"   and a' :: "vname gexp list"   and s :: "vname \<Rightarrow> value option" shows "apply_guards (a @ a') s = (apply_guards a s \<and> apply_guards a' s)"
get_disconnected_document_reads: fixes node_ptr :: "(_) node_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" shows "reads get_disconnected_document_locs (get_disconnected_document node_ptr) h h'"
self_plus_gauss_cnj: fixes z :: "gauss_int" shows "z + gauss_cnj z = of_int (2 * ReZ z)"
measurable_restrict_space2_iff: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and N :: "'d measure"   and \<Omega> :: "'d set" shows "(f \<in> M \<rightarrow>\<^sub>M restrict_space N \<Omega>) = (f \<in> M \<rightarrow>\<^sub>M N \<and> f \<in> space M \<rightarrow> \<Omega>)"
rep_eq: fixes x :: "nat"   and xa :: "integer" shows "int_of_integer (push_bit x xa) = push_bit x (int_of_integer xa)"
Rep_boffa_mon: fixes x :: "('b, 'c) boffa_mon" shows "Rep_boffa_mon x \<in> {f. True}"