hcju/isabelleps huggingface dataset:
wp_pcall_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and p :: "char list"   and c :: "com"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi> p = Some c" shows "wp \<pi> (PCall p) Q s = wp \<pi> c Q s"
real_Gromov_completion_rel_MInf: fixes u :: "nat \<Rightarrow> real"   and v :: "nat \<Rightarrow> real" assumes "(\<lambda>x. ereal (u x)) \<longlonglongrightarrow> - \<infinity>"   and "(\<lambda>x. ereal (v x)) \<longlonglongrightarrow> - \<infinity>" shows "Gromov_completion_rel u v"
in_set_butlast_appendI: fixes x :: "'g"   and xs :: "'g list"   and ys :: "'g list" assumes "x \<in> set (butlast xs) \<or> x \<in> set (butlast ys)" shows "x \<in> set (butlast (xs @ ys))"
nop_v11_vdomain: fixes A :: "V"   and n :: "V"   and f :: "V" assumes "nop_v11 A n f" shows "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"
lossless_R1: fixes R1_12_0T :: "nat \<Rightarrow> bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'v_OT121 spmf"   and n :: "nat"   and m1 :: "bool \<times> bool \<times> bool \<times> bool"   and m2 :: "bool \<times> bool" shows "lossless_spmf (R1_14 R1_12_0T n m1 m2)"
that: fixes m :: "nat" assumes "rotate m (rotate n u) = u" shows "thesis"
thetaParZOL1_def: shows "thetaParZOL1 \<equiv> {(c1 | c2, d) |c1 c2 d. c1 \<approx>01 d \<and> discr c2}"
simps_1: fixes I :: "('b, 'c, 'a) interp"   and c :: "'a" shows "ODE_vars I (OVar c) = ODEBV I c"
Pause_inject: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and out' :: "'e"   and c' :: "('d, 'e, 'f) rpv" shows "(Generative_Probabilistic_Value.Pause out c = Generative_Probabilistic_Value.Pause out' c') = (out = out' \<and> c = c')"
edom_AnalBinds: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" shows "edom (FBinds \<Gamma>\<cdot>ae) \<subseteq> domA \<Gamma> \<inter> edom ae"
assms_2: shows "g \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"
fps_inverse_gp': shows "inverse (Abs_fps (\<lambda>n. 1::'b)) = 1 - fps_X"
elims: fixes x :: "('b \<times> nat) option"   and xa :: "'b"   and y :: "('b \<times> nat) option" assumes "release_all x xa = y" obtains t :: "'b" where "x = None" and "xa = tb__" and "y = None"   | t' :: "'b" and n :: "nat" and t :: "'b" where "x = \<lfloor>(t', n)\<rfloor>" and "xa = t" and "y = (if t = t' then None else \<lfloor>(t', n)\<rfloor>)"
case_prodD: fixes f :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and a :: "'g"   and b :: "'h" assumes "case (a, b) of (c, d) \<Rightarrow> f c d" shows "f a b"
lucas_lehmer_mult'_def: shows "lucas_lehmer_mult' = (\<lambda>(a, b) (c, d). (a * c + 3 * b * d, a * d + b * c))"
mset_fmsetEx: fixes Cs :: "'a set multiset"   and P :: "'a \<Rightarrow> bool" assumes "\<And>X. X \<in># Cs \<Longrightarrow> \<exists>x. P x \<and> X = assocs G x" shows "\<exists>cs. (\<forall>c\<in>set cs. P c) \<and> fmset G cs = Cs"
pop_pow: fixes m :: "nat"   and k :: "nat"   and u :: "'b list" assumes "m \<le> k" shows "u \<^sup>@ m \<cdot> u \<^sup>@ (k - m) = u \<^sup>@ k"
filterlim_uminus_at_bot: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter" shows "filterlim f at_bot F = (LIM x F. - f x :> at_top)"
foldl_Gauss_Jordan_column_k_iarrays_eq: fixes B :: "'b iarray iarray"   and A :: "'b iarray iarray"   and k :: "nat" shows "snd (foldl Gauss_Jordan_column_k_iarrays_PA (B, 0, A) [0..<k]) = foldl Gauss_Jordan_column_k_iarrays (0, A) [0..<k]"
permute_fset_1: fixes p :: "perm" shows "p \<bullet> {||} = {||}"
step: fixes f :: "'c"   and y :: "'a"   and x :: "'b" assumes "\<And>x y. y \<in> U f x \<Longrightarrow> P x y"   and "y \<in> U (F f) x" shows "P x y"
luckyFind_eval: fixes x :: "nat"   and L :: "atom list"   and F :: "atom fm list"   and F' :: "atom fm"   and xs :: "real list"   and \<Gamma> :: "real list" assumes "luckyFind x L F = Some F'"   and "length xs = x" shows "(\<exists>x. eval (list_conj (map fm.Atom L @ F)) (xs @ x # \<Gamma>)) = (\<exists>x. eval F' (xs @ x # \<Gamma>))"
28485: fixes x :: "'a"   and y :: "'a" shows "- x \<sqinter> - y = - y \<sqinter> - x"
Min_eq_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and m :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(linorder.Min less_eq A = m) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq m a))"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'g list" assumes "i < length xs" shows "take (Suc i) xs = take i xs @ [xs ! i]"
plus_adds_left: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.comm_monoid_add plus zero"   and "plus.adds plus (plus a b) c" shows "plus.adds plus a c"
fun_dom_scene_union: fixes A :: "'a set"   and B :: "'a set" shows "fds (A \<union> B) = fds A \<squnion>\<^sub>S fds B"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<sim>\<^sup>s\<^sub>l P \<oplus> <\<nu>x>Q"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'u" assumes "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter" shows "ptr |\<in>| element_ptr_kinds h"
singleton_sub_trivial_uniq: fixes x :: "'b"   and X :: "'b set" assumes "{x} \<subseteq> X"   and "trivial X" shows "x = the_elem X"
pcp_disj: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula set" assumes "F \<^bold>\<or> G \<in> \<Gamma>"   and "(\<forall>xa. xa = F \<or> xa \<in> \<Gamma> \<longrightarrow> is_cnf xa) \<longrightarrow> cnf F \<union> \<Union> (cnf ` \<Gamma>) \<turnstile> \<box>"   and "(\<forall>xa. xa = G \<or> xa \<in> \<Gamma> \<longrightarrow> is_cnf xa) \<longrightarrow> cnf G \<union> \<Union> (cnf ` \<Gamma>) \<turnstile> \<box>"   and "\<forall>x\<in>\<Gamma>. is_cnf x" shows "\<Union> (cnf ` \<Gamma>) \<turnstile> \<box>"
union_less_mono: fixes A :: "'a multiset"   and C :: "'a multiset"   and B :: "'a multiset"   and D :: "'a multiset" assumes "A < C"   and "B < D" shows "A + B < C + D"
interpret_floatarith_fresh_eqI: fixes ea :: "floatarith"   and ys :: "real list"   and zs :: "real list" assumes "\<And>i. fresh_floatarith ea i \<or> i < length ys \<and> i < length zs \<and> ys ! i = zs ! i" shows "interpret_floatarith ea ys = interpret_floatarith ea zs"
c12l: shows "set (labels_conv (fst (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))), snd (conv_mirror (fst \<delta>1''', map (Pair True) (snd \<delta>1'''))) @ snd \<delta>3)) \<subseteq> r \<down>s {fst \<alpha>_step, fst \<beta>_step}"
eval_Lunit: fixes t :: "'c local.term" assumes "Arr t" shows "\<lbrace>\<^bold>\<l>\<^bold>[t\<^bold>]\<rbrace> = \<l>[\<lbrace>Cod t\<rbrace>] \<cdot> (trg \<lbrace>t\<rbrace> \<star> \<lbrace>t\<rbrace>)"
assms: fixes s :: "'a" assumes "invar s" shows "set_iterator (iti s) (\<alpha> s)"
smullyan_double_fixed_point_theorem: fixes g :: "recf"   and h :: "recf" assumes "recfn 2 g"   and "Partial_Recursive.total g"   and "recfn 2 h"   and "Partial_Recursive.total h" shows "\<exists>m n. (\<forall>x. eval r_phi [m, x] = eval r_phi [the (eval g [m, n]), x]) \<and> (\<forall>x. eval r_phi [n, x] = eval r_phi [the (eval h [m, n]), x])"
esssup_mono: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "\<And>x. f x \<le> g x" shows "esssup M f \<le> esssup M g"
ta_seq_consist_coinduct_append: fixes X :: "('a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option) \<Rightarrow> ('a, 'b) obs_event action llist \<Rightarrow> bool"   and vs :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option"   and obs :: "('a, 'b) obs_event action llist"   and P :: "'c prog" assumes "X vs obs"   and "\<And>vs obs. X vs obs \<Longrightarrow> obs = LNil \<or> (\<exists>obs' obs''. obs = lappend obs' obs'' \<and> obs' \<noteq> LNil \<and> ta_seq_consist P vs obs' \<and> (lfinite obs' \<longrightarrow> X (mrw_values P vs (list_of obs')) obs'' \<or> ta_seq_consist P (mrw_values P vs (list_of obs')) obs''))" shows "ta_seq_consist P vs obs"
ctxt_def: fixes p :: "prog" shows "ctxt p \<equiv> set (vars p)"
asymptotics2': fixes x :: "real"   and i :: "nat" assumes "x\<^sub>0 \<le> x"   and "i < k" shows "(1 - hb * inverse (bs ! i) * ln x powr - (1 + e)) powr p * (1 - ln (bs ! i * x + hb * x / ln x powr (1 + e)) powr (- e / 2)) \<le> 1 - ln x powr (- e / 2)"
assms_2: shows "negligible {x \<in> T - S. f x \<noteq> (0::'a)}"
assms_1: shows "X closure_of U \<subseteq> X interior_of T"
bin_op_randomfree_restructure: fixes e :: "expr"   and t :: "pdf_type"   and e' :: "expr"   and t' :: "pdf_type"   and oper :: "pdf_operator"   and tr :: "pdf_type"   and \<rho> :: "nat \<Rightarrow> val" assumes "\<Gamma> \<turnstile> e : t"   and "\<Gamma> \<turnstile> e' : t'"   and "op_type oper (PRODUCT t t') = Some tr"   and "randomfree e'"   and "free_vars e \<subseteq> V \<union> V'"   and "free_vars e' \<subseteq> V'"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> (oper $$ <e ,  e'>)) = distr (dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure tr) (\<lambda>w. op_sem oper <|w, expr_sem_rf \<rho> e'|>)"
le: fixes e :: "'a \<times> 'a" shows "g e \<le> f e"
vimage_Collect: fixes P :: "'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P (f x) = Q x" shows "f -` Collect P = Collect Q"
eq: shows "(r \<star> \<r>[f]) \<cdot> \<a>[r, f, src f] \<cdot> (\<rho> \<star> src f) = ((r \<star> \<theta> \<cdot> (\<theta>' \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[f, w', w]) \<cdot> \<a>[r, f, w' \<star> w] \<cdot> (\<rho> \<star> w' \<star> w)) \<cdot> (\<a>[local.dom \<rho>, w', w] \<cdot> (\<nu>' \<star> w) \<cdot> \<nu>) \<cdot> \<r>[g]"
sim_cfunc_adv: fixes p :: "'s_core spmf"   and q :: "'s_core' spmf"   and ia :: "'iadv_core"   and s1 :: "'s_core"   and s2 :: "'s_core'"   and oa :: "'oadv_core"   and s1' :: "'s_core"   and s2' :: "'s_core'" assumes "S p q"   and "ia \<in> IA"   and "s1 \<in> set_spmf p"   and "s2 \<in> set_spmf q"   and "(oa, s1') \<in> set_spmf (cfunc_adv core1 s1 ia)"   and "(oa, s2') \<in> set_spmf (cfunc_adv core2 s2 ia)" shows "S (cond_spmf_fst (p \<bind> (\<lambda>s1. cfunc_adv core1 s1 ia)) oa) (cond_spmf_fst (q \<bind> (\<lambda>s2. cfunc_adv core2 s2 ia)) oa)"
bounded: fixes f :: "'c \<Rightarrow> 'd" assumes "bounded_linear f" shows "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
vlrestriction_atI: fixes a :: "V"   and A :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"   and "a \<in>\<^sub>\<circ> A"   and "r\<lparr>a\<rparr> = b" shows "(r \<restriction>\<^sup>l\<^sub>\<circ> A)\<lparr>a\<rparr> = b"
lv_rel_def: shows "lv_rel = br eucl_of_list (\<lambda>xs. length xs = DIM('a))"
assms: shows "f \<noteq> 0 \<or> 0 < n"
elims_3: fixes x :: "'e \<times> ('f, 'g, 'h, 'i) SLin_state_scheme" assumes "\<not> P14 x" obtains s1 :: "'e" and s2 :: "('f, 'g, 'h, 'i) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (initialized s2 \<longrightarrow> initVals s2 \<noteq> {})"
simps_2: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'p"   and r :: "'p memref"   and rs :: "'p memref list" shows "history_consistent \<theta> p (r # rs) = (case r of Write\<^sub>s\<^sub>b vol a (D, f) v x xa xb xc \<Rightarrow> D \<subseteq> dom \<theta> \<and> f \<theta> = v \<and> D \<inter> read_tmps rs = {} \<and> history_consistent \<theta> p rs | Read\<^sub>s\<^sub>b vol a t v \<Rightarrow> case \<theta> t of None \<Rightarrow> False | Some v' \<Rightarrow> v = v' \<and> history_consistent \<theta> p rs | Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 is \<Rightarrow> p\<^sub>1 = p \<and> \<theta> |` (dom \<theta> - read_tmps rs)\<turnstile> p\<^sub>1 \<rightarrow>\<^sub>p (p\<^sub>2, is) \<and> history_consistent \<theta> p\<^sub>2 rs | Ghost\<^sub>s\<^sub>b set1 set2 set3 set4 \<Rightarrow> history_consistent \<theta> p rs)"
Start: fixes evsStart :: "event list"   and LID_M :: "nat"   and C :: "agent"   and k :: "nat"   and M :: "agent"   and i :: "nat"   and P :: "agent"   and j :: "nat"   and Transaction :: "msg"   and OrderDesc :: "nat"   and PurchAmt :: "nat" assumes "evsStart \<in> set_pur"   and "Number LID_M \<notin> used evsStart"   and "C = Cardholder k"   and "M = Merchant i"   and "P = PG j"   and "Transaction = \<lbrace>Agent M, Agent C, Number OrderDesc, Number PurchAmt\<rbrace>"   and "LID_M \<notin> range CardSecret"   and "LID_M \<notin> range PANSecret" shows "Notes C \<lbrace>Number LID_M, Transaction\<rbrace> # Notes M \<lbrace>Number LID_M, Agent P, Transaction\<rbrace> # evsStart \<in> set_pur"
Basis_prodD: fixes i :: "'a"   and j :: "'b" assumes "(i, j) \<in> Basis" shows "i \<in> Basis \<and> j = (0::'b) \<or> i = (0::'a) \<and> j \<in> Basis"
edge_preserving: fixes R :: "('e \<times> 'f) set"   and E :: "('g \<times> 'e \<times> 'e) set"   and G :: "('g \<times> 'f \<times> 'f) set" shows "edge_preserving R E G = (on_triple R `` E \<subseteq> G)"
length_vec_A: shows "hd (dims A) * prod_list (tl (dims A)) = length (vec A)"
disjointed_additive: fixes f :: "'a set \<Rightarrow> ennreal"   and A :: "nat \<Rightarrow> 'a set"   and n :: "nat" assumes "positive M f"   and "Measure_Space.additive M f"   and "range A \<subseteq> M"   and "incseq A" shows "(\<Sum>i\<le>n. f (disjointed A i)) = f (A n)"
that: fixes as :: "ct list"   and zs :: "ct list" assumes "ys = as @ zs"   and "ct_list_eq zs xs" shows "thesis"
sel_5: fixes x21 :: "'f"   and x22 :: "('f, 'g) tllist" shows "ttl (TCons x21 x22) = x22"
assms: shows "\<forall>\<phi>\<in>set fs. qfree \<phi>"
efficient_pell_power_def: fixes D :: "nat"   and z :: "nat \<times> nat"   and n :: "nat" shows "efficient_pell_power D z n = efficient_power (pell_mul_nat D) (1, 0) z n"
freshBoundDerivative_1: fixes P :: "pi"   and a :: "subject"   and x :: "name"   and P' :: "pi"   and y :: "name" assumes "P \<longmapsto> a\<guillemotleft>x\<guillemotright> \<prec> P'"   and "y \<sharp> P" shows "y \<sharp> a"
env_delete_def: fixes x :: "'d"   and m :: "'d \<Rightarrow> 'e" shows "env_delete x m = m(x := \<bottom>)"
fixD_def: shows "fixD \<equiv> \<Lambda> (ValF\<cdot>f). fix\<cdot>f"
subdivision_step_def: fixes G :: "('a, 'b) pre_digraph"   and rev_G :: "'b \<Rightarrow> 'b"   and H :: "('a, 'b) pre_digraph"   and rev_H :: "'b \<Rightarrow> 'b" shows "subdivision_step G rev_G H rev_H \<equiv> \<lambda>(u, v, w) (uv, uw, vw). bidirected_digraph G rev_G \<and> bidirected_digraph H rev_H \<and> perm_restrict rev_H (arcs G) = perm_restrict rev_G (arcs H) \<and> compatible G H \<and> verts H = verts G \<union> {w} \<and> w \<notin> verts G \<and> arcs H = {uw, rev_H uw, vw, rev_H vw} \<union> arcs G - {uv, rev_G uv} \<and> uv \<in> arcs G \<and> distinct [uw, rev_H uw, vw, rev_H vw] \<and> arc_to_ends G uv = (u, v) \<and> arc_to_ends H uw = (u, w) \<and> arc_to_ends H vw = (v, w)"
WretrT_def: fixes theta :: "(('test, 'atom) com \<times> ('test, 'atom) com) set" shows "WretrT theta \<equiv> {(c, d). matchC_MC theta c d \<and> matchT_MT c d}"
whisker_left: fixes f :: "'d"   and \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.ide f"   and "D.seq \<nu> \<mu>" shows "f \<star>\<^sub>D \<nu> \<cdot>\<^sub>D \<mu> = (f \<star>\<^sub>D \<nu>) \<cdot>\<^sub>D (f \<star>\<^sub>D \<mu>)"
simps_2: fixes \<Psi> :: "'c \<Rightarrow> 'c \<Rightarrow> 'b \<Rightarrow> 'b"   and x0 :: "'b"   and t :: "nat \<Rightarrow> 'c"   and j :: "nat" shows "grid_function \<Psi> x0 t (Suc j) = \<Psi> (t (Suc j)) (t j) (grid_function \<Psi> x0 t j)"
assms: shows "r O s \<subseteq> s O (s \<union> r)\<^sup>*"
d1: fixes i :: "'a" shows "(Uinv o\<^sub>C\<^sub>L U) *\<^sub>S V i = V i"
intro: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "associative_weak_composition V H \<a>"   and "prebicategory_axioms V H" shows "prebicategory V H \<a>"
dX_singular: fixes X :: "'a set"   and x :: "'a"   and d :: "'b" assumes "inj_on Xd X"   and "x \<in> X"   and "d = Xd x" shows "ManiPref'.dX X d = {x}"
fold_set_fold_remdups: fixes y :: "'b fset set"   and xs :: "'b fset set list" shows "Finite_Set.fold (\<otimes>\<^sub>m) y (set xs) = fold (\<otimes>\<^sub>m) (remdups xs) y"
GF_advice_sync_less: fixes n :: "nat"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and X :: "'b ltln set"   and \<psi> :: "'b ltln" assumes "\<And>i. i < n \<Longrightarrow> \<exists>j. suffix (i + j) w \<Turnstile>\<^sub>n af \<phi> (w [i \<rightarrow> j + i])[X]\<^sub>\<nu>"   and "\<exists>j. suffix (n + j) w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> j + n])[X]\<^sub>\<nu>" shows "\<exists>k\<ge>n. (\<forall>j<n. suffix k w \<Turnstile>\<^sub>n af \<phi> (w [j \<rightarrow> k])[X]\<^sub>\<nu>) \<and> suffix k w \<Turnstile>\<^sub>n af \<psi> (w [n \<rightarrow> k])[X]\<^sub>\<nu>"
snoc_2: shows "xs @ [x] = cs\<^bsup>\<pi>\<^esup> n"
conv_dist_comp: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>T = y\<^sup>T * x\<^sup>T"
tag_name_eq_h2: fixes ptr' :: "(_) element_ptr"   and disc_nodes :: "char list" shows "h2 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes = h3 \<turnstile> get_tag_name ptr' \<rightarrow>\<^sub>r disc_nodes"
fv_sub: shows "fv t \<subseteq> \<Union> (set (map P S)) \<union> V"
exp_pi_i': shows "exp (\<i> * complex_of_real pi) = - 1"
liftT_liftT_1: fixes i :: "nat"   and j :: "nat"   and m :: "nat"   and n :: "nat"   and T :: "POPLmarkRecord.type" assumes "i \<le> j"   and "j \<le> i + m" shows "\<up>\<^sub>\<tau> n j (\<up>\<^sub>\<tau> m i T) = \<up>\<^sub>\<tau> (m + n) i T"
tiTable_ack_st: fixes req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and st_in :: "nat \<Rightarrow> GatewayStatus"   and b :: "nat \<Rightarrow> ECall_Info list"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and st_out :: "nat \<Rightarrow> GatewayStatus"   and t :: "nat" assumes "tiTable_SampleT req a1 stop lose st_in b ack i1 vc st_out"   and "ts lose"   and "msg (Suc 0) a1"   and "msg (Suc 0) stop" shows "ack t = [st_out t]"
prod_encode_eq: fixes x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "(prod_encode x = prod_encode y) = (x = y)"
single_not_bot: fixes x :: "'a" shows "Predicate.single x \<noteq> bot"
alice_step_1_state: fixes \<phi> :: "complex Matrix.mat" assumes "state 1 \<phi>" shows "state 3 (\<phi> \<Otimes> |\<beta>\<^sub>0\<^sub>0\<rangle>)"
disjoint: shows "insert new_el X \<inter> \<Union> (P - {X}) = {}"
right_neutral: fixes a :: "'a" shows "a \<parallel> 1\<^sub>\<pi> = a"
bintr_uint: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "take_bit n (uint w) = uint w"
vector_mul_eq_0: fixes a :: "'d"   and x :: "('d, 'e) vec" shows "(a *s x = 0) = (a = (0::'d) \<or> x = 0)"
sgn_2: fixes x :: "real_alg_2" assumes "invariant_2 x" shows "real_of_rat (sgn_2 x) = sgn (real_of_2 x)"
two_comparisons_into_case_order_4: fixes comp :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and x :: "'b"   and y :: "'b"   and Q :: "'c"   and P :: "'c"   and R :: "'c" assumes "comparator comp" shows "(if le_of_comp comp x y then if lt_of_comp comp x y then Q else P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
sortingSameSet: fixes X :: "'b set" assumes "finite X" shows "set (sorted_list_of_set X) = X"
iprev_nth_atLeastAtMost: fixes a :: "nat"   and n :: "nat"   and m :: "nat" assumes "a \<le> n - m"   and "m \<le> n" shows "{m..n} \<leftarrow> a = n - a"
all_zero: shows "\<forall>m\<ge>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'a)"
simps_2: fixes var :: "nat"   and a :: "atom"   and as :: "atom list"   and xs :: "real list" shows "convert_atom_list var (a # as) xs = (case convert_atom var a xs of None \<Rightarrow> None | Some a \<Rightarrow> case convert_atom_list var as xs of None \<Rightarrow> None | Some as \<Rightarrow> Some (a # as))"
simps: fixes a :: "('s \<times> ('c \<Rightarrow> real) set) pmf"   and st :: "'s \<times> ('c \<Rightarrow> real) set" shows "(a \<in> \<K> st) = ((\<exists>l R R'. a = return_pmf (l, R') \<and> st \<in> \<S> \<and> st = (l, R) \<and> R' \<in> Succ \<R> R \<and> R' \<subseteq> {v. v \<turnstile> PTA.inv_of A l}) \<or> (\<exists>l R g \<mu>. a = map_pmf (\<lambda>(X, l). (l, [X::=0]R)) \<mu> \<and> st \<in> \<S> \<and> st = (l, R) \<and> (l, g, \<mu>) \<in> PTA.trans_of A \<and> R \<subseteq> {v. v \<turnstile> g}) \<or> a = return_pmf st)"
vunion_iff: fixes x :: "V"   and A :: "V"   and B :: "V" shows "(x \<in>\<^sub>\<circ> A \<union>\<^sub>\<circ> B) = (x \<in>\<^sub>\<circ> A \<or> x \<in>\<^sub>\<circ> B)"
set_empty2: fixes xs :: "'d list" shows "({} = set xs) = (xs = [])"
rc_invk_arg: fixes CT :: "nat \<Rightarrow> classDef option"   and ei :: "exp"   and ei' :: "exp"   and e0 :: "exp"   and m :: "nat"   and el :: "exp list"   and er :: "exp list" assumes "CT \<turnstile> ei \<rightarrow> ei'" shows "CT \<turnstile> MethodInvk e0 m (el @ ei # er) \<rightarrow> MethodInvk e0 m (el @ ei' # er)"
lmult_0: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and m :: "'d" assumes "module R M"   and "m \<in> carrier M" shows "\<zero>\<^bsub>R\<^esub> \<odot>\<^bsub>M\<^esub> m = \<zero>\<^bsub>M\<^esub>"
injects: fixes a :: "'b"   and list :: "['b]"   and a' :: "'b"   and list' :: "['b]" shows "(a : list = a' : list') = (a = a' \<and> list = list')"
epiE: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "category C"   and "category.epi C f"   and "partial_magma.arr C (C g f)"   and "partial_magma.arr C (C g' f)"   and "C g f = C g' f" shows "g = g'"
simps_1: fixes r :: "'b rexp" shows "lderivs [] r = r"
assms_1: shows "m \<le>\<^sub>n SPEC \<Phi>"
G_def: shows "(G::'a::refinement_lattice \<Rightarrow> 'a::refinement_lattice) = (\<lambda>x::'a::refinement_lattice. nil \<sqinter> (c::'a::refinement_lattice) ; x)"
M_fin_monotonic: fixes \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<pi>' :: "'a ltl \<Rightarrow> nat option" assumes "dom \<pi> = dom \<pi>'"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) \<le> the (\<pi>' \<chi>)" shows "M_fin \<pi> \<subseteq> M_fin \<pi>'"
hyps_1: shows "(\<Union>a\<in>toplevel_summands r1. toplevel_summands \<guillemotleft>deriv as \<guillemotleft>a\<guillemotright>\<guillemotright>) = toplevel_summands \<guillemotleft>deriv as \<guillemotleft>r1\<guillemotright>\<guillemotright>"
rel_refl: fixes Ra :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "('d, 'c) fmap" assumes "\<And>x. Ra x x" shows "fmrel Ra x x"
minus_zero: shows "\<ominus>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
model_def: shows "model = {z. fst z \<noteq> {}}"
bound_typ_inst_mk_scheme: fixes S :: "nat \<Rightarrow> typ"   and t :: "typ" shows "bound_typ_inst S (mk_scheme t) = t"
a_2: shows "\<forall>n. \<bar>X n\<bar> < a"
base: fixes n :: "nat" assumes "n < length fs" shows "f n = fs ! n"
asList: fixes A :: "'a set" assumes "finite A" shows "set (asList A) = A"
cases: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" assumes "gt_unary a1 a2" obtains t :: "('s, 'v) tm" and s :: "('s, 'v) tm" where "a1 = t" and "a2 = s" and "wt t = wt s" and "\<not> head t \<le>\<ge>\<^sub>h\<^sub>d head s" and "num_args t = 1" and "\<exists>f\<in>local.ground_heads (head t). arity_sym f = 1 \<and> wt_sym f = 0" and "arg t \<ge>\<^sub>t s"
zero: shows "(\<lambda>x. 0) \<in> PrimRec1'"
Hausdorff_space_def: fixes X :: "'a topology" shows "Hausdorff_space X \<equiv> \<forall>x y. x \<in> topspace X \<and> y \<in> topspace X \<and> x \<noteq> y \<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> x \<in> U \<and> y \<in> V \<and> disjnt U V)"