hcju/isabelleps huggingface dataset:
pn_map_impl_def: shows "pn_map_impl \<equiv> \<lambda>x. LIST_FOREACH' (RETURN (id x)) Impl_Array_Map.iam_invar (\<lambda>xa xb. LIST_FOREACH' (RETURN (it_to_list lss.iteratei (old_impl xa))) (\<lambda>xc. True) (\<lambda>xc (xd, xe). case xc of prop\<^sub>r(xf) \<Rightarrow> RETURN (glist_insert (=) xf xd, xe) | nprop\<^sub>r(xf) \<Rightarrow> RETURN (xd, glist_insert (=) xf xe) | _ \<Rightarrow> RETURN (xd, xe)) ([], []) \<bind> (\<lambda>xc. RETURN (Impl_Array_Map.iam_update (name_impl xa) xc xb))) (Impl_Array_Map.iam_empty ())"
differentiable_at_real_polynomial_function: fixes f :: "'c \<Rightarrow> real"   and a :: "'c"   and S :: "'c set" assumes "real_polynomial_function f" shows "f differentiable at a within S"
qiterl_unfoldl: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y \<cdot> qiterl x y = qiterl x y"
PiM_eqI: fixes I :: "'i set"   and P :: "('i \<Rightarrow> 'a) measure" assumes "finite I"   and "sets P = sets (Pi\<^sub>M I M)"   and "\<And>A. (\<And>i. i \<in> I \<Longrightarrow> A i \<in> sets (M i)) \<Longrightarrow> emeasure P (Pi\<^sub>E I A) = (\<Prod>i\<in>I. emeasure (M i) (A i))" shows "P = Pi\<^sub>M I M"
preserves_retraction: fixes e :: "'a" assumes "A.retraction e" shows "B.retraction (F e)"
set_iterator_genord_foldli_correct: fixes xs :: "'c list"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "distinct xs"   and "sorted_wrt R xs" shows "set_iterator_genord (foldli xs) (set xs) R"
lemma1b': fixes tp :: "bool list"   and tq :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and p :: "bool list" assumes "(tp, tq) \<in> (eps (RegExp2NAe.conc L R))\<^sup>*"   and "tp = False # p" shows "\<exists>q. (p, q) \<in> (eps R)\<^sup>* \<and> tq = False # q"
gt_ex: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
lemma_2_6_20_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a r\<rightarrow> b \<sqinter> c = (a r\<rightarrow> b) \<sqinter> (a r\<rightarrow> c)"
A_2: fixes Am :: "'a multiset"   and An :: "'a multiset"   and Bm :: "'a multiset"   and Bn :: "'a multiset" assumes "A = Am + An"   and "B = Bm + Bn"   and "M = Am + Bm"   and "N = An + Bn" shows "P"
hosts: shows "hosts (generate_valid_stateful_policy_IFSACS G M edgesList) = nodes G"
emeasure_eq_AE: fixes A :: "'c set"   and B :: "'c set"   and M :: "'c measure" assumes "AE x in M. (x \<in> A) = (x \<in> B)"   and "A \<in> sets M"   and "B \<in> sets M" shows "emeasure M A = emeasure M B"
brw_invar_add_def: fixes \<delta> :: "('d, 'e) ta_rule set" shows "brw_invar_add \<delta> \<equiv> {(Q, W, rcm). witness_prop \<delta> Q}"
disj3: shows "\<forall>i\<in>I. \<forall>j\<in>I. Xin i \<inter> Xout j = {}"
valof_uminus: fixes a :: "('a, 'b) IEEE.float" assumes "is_finite a" shows "valof (- a) = - valof a"
pnPlus_singleton_PLUS: fixes xs :: "'a rexp list"   and x :: "'a rexp" assumes "xs \<noteq> []"   and "sorted xs"   and "distinct xs"   and "\<forall>x\<in>{x} \<union> set xs. \<not> is_Zero x \<and> \<not> is_Plus x" shows "pnPlus x (PLUS xs) = (if x \<in> set xs then PLUS xs else PLUS (insort x xs))"
outputs_conv_fixp: shows "outputs \<equiv> set.fixp_fun (\<lambda>outputs xs. case xs of Resumption.resumption.Done x \<Rightarrow> {} | Resumption.resumption.Pause out c \<Rightarrow> insert out (\<Union>input. outputs (c input)))"
l2: shows "l\<^sub>2 = override_on l\<^sub>1 l\<^sub>0 (set Vs)"
simps_21: fixes g1 :: "'a gexp"   and g2 :: "'a gexp"   and g1' :: "'a gexp"   and g2' :: "'a gexp" shows "local.less_gexp_aux (Nor g1 g2) (Nor g1' g2') = (local.less_gexp_aux g1 g1' \<or> g1 = g1' \<and> local.less_gexp_aux g2 g2')"
ordinal_less_timesR: fixes x :: "ordinal"   and y :: "ordinal" assumes "0 < x"   and "oSuc 0 < y" shows "x < x * y"
lincomb_Cons: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and rs :: "'a list"   and m :: "'b"   and ms :: "'b list" assumes "RModule R smult M" shows "scalar_mult.lincomb smult (r # rs) (m # ms) = smult r m + scalar_mult.lincomb smult rs ms"
elims_2: fixes x :: "('b \<times> 'c) tree" assumes "Tree2.bst x" obtains "x = \<langle>\<rangle>"   | l :: "('b \<times> 'c) tree" and a :: "'b" and uu_ :: "'c" and r :: "('b \<times> 'c) tree" where "x = \<langle>l, (a, uu_), r\<rangle>" and "(\<forall>x\<in>Tree2.set_tree l. x < a) \<and> Ball (Tree2.set_tree r) ((<) a) \<and> Tree2.bst l \<and> Tree2.bst r"
square_free_rsquarefree: fixes f :: "'a poly" assumes "square_free f" shows "rsquarefree f"
intro: fixes f :: "'b \<Rightarrow> 'b" assumes "bij f" shows "bijection f"
gen_wf_uses_def: shows "gen_wf_uses \<equiv> map_fun Rep_gen_cfg_wf id gen_uses"
simps_3: fixes x1 :: "char list \<Rightarrow> 'b option" shows "Malformed \<noteq> Env x1"
prm_set_singleton: fixes \<pi> :: "'b prm"   and x :: "'b" shows "\<pi> {$} {x} = {\<pi> $ x}"
sum_part_overlap: fixes x :: "i"   and y :: "i" shows "\<forall>w. P w (x \<oplus> y) \<longrightarrow> O w x \<or> O w y"
uminus2: fixes ccw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'b set"   and a :: "'b"   and b :: "'b" assumes "ccw_vector_space ccw S" shows "ccw (0::'b) a (- b) = ccw (0::'b) b a"
qfN_power_div_prime: fixes P :: "int"   and A :: "int"   and n :: "nat"   and p :: "int"   and N :: "int"   and q :: "int"   and a :: "int"   and b :: "int" assumes "prime P \<and> odd P \<and> P dvd A \<and> P ^ n = p\<^sup>2 + N * q\<^sup>2 \<and> A ^ n = a\<^sup>2 + N * b\<^sup>2 \<and> coprime a b \<and> coprime p (N * q) \<and> 0 < n" shows "\<exists>u v. a\<^sup>2 + N * b\<^sup>2 = (u\<^sup>2 + N * v\<^sup>2) * (p\<^sup>2 + N * q\<^sup>2) \<and> coprime u v \<and> (\<exists>e. a = p * u + e * N * q * v \<and> b = p * v - e * q * u \<and> \<bar>e\<bar> = 1)"
1092: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) \<sqinter> y = bot"
homeomorphic_space_prod_topology_sing1: fixes b :: "'d"   and Y :: "'d topology"   and X :: "'e topology" assumes "b \<in> topspace Y" shows "X homeomorphic_space prod_topology X (subtopology Y {b})"
track_Some: fixes gpv :: "('x, 'a, 'b) gpv"   and ss :: "'s"   and s :: "'s" shows "exec_gpv (\<lambda>(ss, s) x. map_spmf (\<lambda>(y, s'). (y, if I s' \<and> ss = None then Some s' else ss, s')) (callee s x)) gpv (Some ss, s) = map_spmf (\<lambda>(a, s). (a, Some ss, s)) (exec_gpv callee gpv s)"
PPs_closed_minus: fixes s :: "'a \<Rightarrow>\<^sub>0 nat"   and X :: "'a set"   and t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "s \<in> .[X]" shows "s - t \<in> .[X]"
0: shows "f \<ominus>\<^bsub>P\<^esub> monom P (f (deg R f)) (deg R f) \<in> carrier P"
rcset_inverse: fixes x :: "'c cset" shows "acset (rcset x) = x"
interior_negations: fixes S :: "'a set" shows "interior (uminus ` S) = uminus ` interior S"
ide_leg1: shows "ide f\<^sub>0g\<^sub>1.p\<^sub>1"
extremum_uniqueI: fixes a :: "nat" assumes "0 dvd a" shows "a = 0"
feq: fixes x :: "complex" assumes "x \<in> S" shows "f x = exp (g x)"
R_lin_independentD_all_scalars: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and rs :: "'a list"   and ms :: "'b list" assumes "RModule R smult M"   and "set rs \<subseteq> R"   and "set ms \<subseteq> M"   and "length rs \<le> length ms"   and "R_scalar_mult.R_lin_independent R smult ms"   and "scalar_mult.lincomb smult rs ms = (0::'b)" shows "set rs \<subseteq> 0"
\<R>\<^sub>2_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>2\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds\<^sub>2, mem\<^sub>2\<rangle>" obtains \<Gamma>\<^sub>1 :: "'Var \<Rightarrow> Sec option" and \<Gamma>\<^sub>2 :: "'Var \<Rightarrow> Sec option" where "mds\<^sub>2 = mds" and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<approx> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" and "\<forall>x\<in>dom \<Gamma>. \<Gamma> x = Some High" and "\<turnstile> \<Gamma>\<^sub>1 {c\<^sub>1} \<Gamma>" and "\<turnstile> \<Gamma>\<^sub>2 {c\<^sub>2} \<Gamma>" and "mds_consistent mds \<Gamma>\<^sub>1" and "mds_consistent mds \<Gamma>\<^sub>2"
sup_pp_semi_commute: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - - y \<le> - - (x \<squnion> y)"
downsets_def: shows "ord.downsets (\<lambda>x y. y \<le> x) = Fix (ord.downset_set (\<lambda>x y. y \<le> x))"
poly_mult_degree_one_monic_imp_same_roots: fixes a :: "'a"   and p :: "'a list" assumes "a \<in> carrier R"   and "p \<in> carrier (poly_ring R)"   and "p \<noteq> []" shows "roots ([\<one>, \<ominus> a] \<otimes>\<^bsub>poly_ring R\<^esub> p) = add_mset a (roots p)"
fps_shift_one: fixes n :: "nat" shows "fps_shift n 1 = (if n = 0 then 1 else 0)"
add_le_cancel_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less" shows "less_eq (plus a c) (plus b c) = less_eq a b"
copy: 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 f :: "'g \<Rightarrow> 'd"   and B1' :: "'g set"   and g :: "'h \<Rightarrow> 'e"   and B2' :: "'h set" assumes "alg B1 B2 s1 s2"   and "bij_betw f B1' B1"   and "bij_betw g B2' B2" shows "\<exists>f' g'. alg B1' B2' f' g' \<and> mor B1' B2' f' g' B1 B2 s1 s2 f g"
Guard: 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 g :: "'e set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and f :: "'f" assumes "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (g \<inter> P) c Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (g \<inter> P) Guard f g c Q,A"
iteration_ref_star: fixes x :: "'a" shows "x\<^sup>\<star> \<le> x\<^sup>\<infinity>"
Quotient_mini_alg_unique: shows "Quotient (eq_onp (\<lambda>r. ma_coeff r = 0 \<and> ma_base r = 0 \<or> ma_coeff r \<noteq> 0 \<and> prime_product (ma_base r))) Abs_mini_alg_unique Rep_mini_alg_unique cr_mini_alg_unique"
pid_system_automaton_def: shows "pid_system_automaton \<equiv> \<lparr>g_V = UNIV, g_E = pid_step, g_V0 = {pid_init}, sa_L = pid_label\<rparr>"
finite_Image_subset: fixes A :: "('c \<times> 'b) set"   and B :: "'c set"   and C :: "('c \<times> 'b) set" assumes "finite (A `` B)"   and "C \<subseteq> A" shows "finite (C `` B)"
elims_3: fixes x :: "real_max"   and xa :: "real_max" assumes "\<not> x < xa" obtains x :: "real" and y :: "real" where "xa__ = Rmax x" and "xa = Rmax y" and "\<not> x < y"
simps_2: fixes t :: "dbtm"   and u :: "dbtm" shows "quot_dbfm (DBEq t u) = Q_Eq (quot_dbtm t) (quot_dbtm u)"
cases: fixes x :: "('a \<times> constraint) list \<times> 'a set \<times> 'a simplex_state" obtains cs :: "('a \<times> constraint) list" and J :: "'a set" and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" where "x = (cs, J, Simplex_State s)"
extremum_unique: fixes a :: "nat" shows "(a \<le> 0) = (a = 0)"
left_neutral: fixes a :: "'b upred" shows "(true \<and> a) = a"
simps_2: fixes a :: "'e"   and n :: "nat" shows "replicate_raw a (Suc n) = Yield a n"
periodic_1: fixes x :: "real" shows "f (x + p) = f x"
simps_1: fixes n :: "nat"   and P :: "'b prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Load n, P, pc, mxs, T\<^sub>r, ST, LT) = (n < length LT \<and> LT ! n \<noteq> Err \<and> length ST < mxs)"
cycle_free_loop_dest: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and neutral :: "'b"   and i :: "nat"   and n :: "nat"   and xs :: "nat list"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "class.linordered_ab_monoid_add plus less_eq less neutral"   and "i \<le> n"   and "set xs \<subseteq> {0..n}"   and "linordered_ab_monoid_add.cycle_free plus less_eq neutral m n" shows "less_eq neutral (linordered_ab_monoid_add.len plus m i i xs)"
suffixes_eq_snoc: fixes ys :: "'b list"   and xs :: "'b list list"   and x :: "'b list" shows "(suffixes ys = xs @ [x]) = ((ys = [] \<and> xs = [] \<or> (\<exists>z zs. ys = z # zs \<and> xs = suffixes zs)) \<and> x = ys)"
intros: fixes X\<^sub>0 :: "'b set"   and X :: "'b set"   and I :: "'b \<Rightarrow> intv"   and r :: "('b \<times> 'b) set"   and k :: "'b \<Rightarrow> nat"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> intv'" assumes "X\<^sub>0 = {x \<in> X. \<exists>d. I x = Intv d}"   and "refl_on X\<^sub>0 r"   and "trans r"   and "total_on X\<^sub>0 r"   and "\<forall>x\<in>X. valid_intv (k x) (I x)"   and "\<forall>x\<in>X. \<forall>y\<in>X. isGreater (I x) \<or> isGreater (I y) \<longrightarrow> valid_intv' (int (k y)) (int (k x)) (J x y)" shows "valid_region X k I J r"
get_dom_component_ok: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_dom_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "ptr |\<in>| object_ptr_kinds h" shows "h \<turnstile> ok get_dom_component ptr"
vfsequence_vcons_vdomain: fixes xs :: "V"   and x :: "V" assumes "vfsequence xs" shows "\<D>\<^sub>\<circ> (xs #\<^sub>\<circ> x) = ZFC_in_HOL.succ (vcard xs)"
path_permute: fixes ab :: "'a set"   and a :: "'a"   and b :: "'a" shows "path ab a b = path ab b a"
augment_Faults: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and F :: "'f set"   and P :: "'e set"   and c :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and F' :: "'f set" assumes "\<Gamma>,{}\<turnstile>\<^bsub>/F \<^esub>P c Q,A"   and "F \<subseteq> F'" shows "\<Gamma>,{}\<turnstile>\<^bsub>/F' \<^esub>P c Q,A"
f\<^sub>2ru: assumes "r \<noteq> u" shows "between (vertices f\<^sub>2) r u = between (vertices f) r u"
connected_convex_1: fixes S :: "real set" shows "connected S = convex S"
llength_less_llength_lselect_less: fixes i :: "nat"   and s :: "nat set"   and w :: "'b llist" shows "(enat i < esize s \<and> enat (nth_least s i) < llength w) = (enat i < llength (lselect s w))"
eq: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Eq) = (x = y)"
tendsto_Im6: fixes g :: "'a \<Rightarrow> octo"   and a :: "octo"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. Im6 (g x)) \<longlongrightarrow> Im6 a) F"
oalist_inv_raw_update_by_pair: fixes x :: "'o"   and xs :: "('a \<times> 'c) list"   and kv :: "'a \<times> 'c" assumes "oalist_inv_raw x xs" shows "oalist_inv_raw x (update_by_pair x kv xs)"
power: fixes n :: "nat"   and m :: "nat" shows "liouville_lambda (n ^ m) = liouville_lambda n ^ m"
r_one: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C" shows "x \<otimes>\<^bsub>C\<^esub> \<one>\<^bsub>C\<^esub> = x"
finite_measure_eq_sum_singleton: fixes M :: "'b measure"   and S :: "'b set" assumes "finite_measure M"   and "finite S"   and "\<And>x. x \<in> S \<Longrightarrow> {x} \<in> sets M" shows "Sigma_Algebra.measure M S = (\<Sum>x\<in>S. Sigma_Algebra.measure M {x})"
inject_1: fixes x11 :: "String.literal"   and x12 :: "typ"   and y11 :: "String.literal"   and y12 :: "typ" shows "(Ct x11 x12 = Ct y11 y12) = (x11 = y11 \<and> x12 = y12)"
reg\<^sub>C_is_not_the_var\<^sub>C_of_anything: fixes x :: "Eg1.var" assumes "reg\<^sub>C = Eg2_var\<^sub>C_of_Eg1 x" shows "False"
weight_random_oracle: fixes A :: "'a set"   and \<sigma> :: "'c \<Rightarrow> 'a option"   and x :: "'c" assumes "finite A"   and "A - ran \<sigma> \<noteq> {}" shows "weight_spmf (random_permutation.random_permutation A \<sigma> x) = 1"
h'_def: fixes n :: "nat" shows "h' n = (\<lambda>(x, y). if x \<le> n \<and> y \<le> n then h n x y else 0)"
simps: fixes a :: "'d"   and ps :: "'d \<times> 'b" shows "(a \<in> scope ps) = (\<exists>v. a = v \<and> v |\<in>| vertices \<and> (\<forall>x xa. path v xa x \<longrightarrow> terminal_vertex xa \<longrightarrow> ps \<in> snd ` set x))"
4: fixes ab :: "32 simple_match"   and x1 :: "char list"   and x2 :: "simple_action"   and bb :: "32 simple_match"   and x4 :: "char list"   and x5 :: "simple_action" assumes "distinct (map fst amr)"   and "(ac, ab, x1, x2) \<in> set amr"   and "(ac, bb, x4, x5) \<in> set amr"   and "ab \<noteq> bb" shows "False"
skl1_inv4D: fixes x :: "skl1_state"   and A :: "agent"   and B :: "agent"   and Ni :: "msg"   and Rb :: "fid_t"   and gnx :: "msg" assumes "0 < signalsResp x (Commit A B \<langle>Ni, NonceF (Rb $ nr), Exp gnx (NonceF (Rb $ ny))\<rangle>)"   and "x \<in> skl1_inv4" shows "guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress x Rb = Some {xny, xni, xnr, xgnx, xgny, xsk, xEnd} \<and> sklvl1.guessed_frame Rb xgnx = Some gnx \<and> sklvl1.guessed_frame Rb xni = Some Ni"
Diff_not_in: fixes a :: "'b"   and A :: "'b set" shows "a \<notin> A - {a}"
mset_subset_eq_exists_conv: fixes A :: "'a multiset"   and B :: "'a multiset" shows "(A \<subseteq># B) = (\<exists>C. B = A + C)"
fml_sem_not_not: fixes I :: "interp"   and \<phi> :: "fml" shows "fml_sem I (! (! \<phi>)) = fml_sem I \<phi>"
restrict_sets_measurable: fixes A :: "('d \<Rightarrow>\<^sub>F 'e) set"   and I :: "'d set set"   and M :: "'d \<Rightarrow> 'e measure"   and J :: "'d set set" assumes "A \<in> sets (Pi\<^sub>F I M)"   and "J \<subseteq> I" shows "A \<inter> {m. domain m \<in> J} \<in> sets (Pi\<^sub>F J M)"
fds_nth_fds_pos: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "0 < n" shows "fds_nth (fds f) n = f n"
ltake_eSuc_LCons: fixes n :: "enat"   and x :: "'b"   and xs :: "'b llist" shows "ltake (eSuc n) (LCons x xs) = LCons x (ltake n xs)"
ind_hyp: fixes m :: "'a \<Rightarrow> 'b option" shows "(map_of l = m) = (set l = map_to_set m)"
in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "R.arr f"   and "R.dom f = a"   and "R.cod f = b" shows "R.in_hom f a b"
compat_NS_S: fixes S :: "('a \<times> 'a) set"   and NS :: "('a \<times> 'a) set" assumes "compat_pair S NS" shows "NS O S \<subseteq> S"
induced_hom_Abs_freeletter_equality: fixes s :: "'a" assumes "s \<in> S" shows "F (BinOpSetGroup.Abs_G_perm (FreeGroup S // Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S)) (+) (Restr (lcoset_rel \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` P'\<rangle>) (FreeGroup S) `` {Abs_freeletter s})) = f s"
Init_def: shows "FIG.Init = (\<lambda>(A, A'). idS A' = {})"
B'_def: shows "B' = (\<lambda>\<epsilon>. (*\<^sub>R) (1 + \<epsilon>) ` B)"
get_elements_by_class_name_is_strongly_dom_component_safe_step: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and idd :: "char list"   and results :: "(_) element_ptr list"   and result :: "(_) element_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_elements_by_class_name ptr' idd \<rightarrow>\<^sub>r results"   and "result \<in> set results" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r result \<in> set c) = (ptr' \<in> set c)"
prod_dividef: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and A :: "'c set" assumes "class.field divide inverse times one plus zero minus uminus" shows "comm_monoid_mult.prod times one (\<lambda>x. divide (f x) (g x)) A = divide (comm_monoid_mult.prod times one f A) (comm_monoid_mult.prod times one g A)"
on2: shows "\<forall>s0\<in>S0. \<exists>b\<ge>0. \<forall>qs\<in>{x. set x \<subseteq> set s0}. \<forall>(x, y)\<in>{(x, y). x \<in> set s0 \<and> y \<in> set s0 \<and> x < y}. T\<^sub>p_on_rand' A (fst A (Lxy s0 {x, y}) \<bind> (\<lambda>is. return_pmf (Lxy s0 {x, y}, is))) (Lxy qs {x, y}) \<le> c * real (T\<^sub>p_opt (Lxy s0 {x, y}) (Lxy qs {x, y})) + b"
one_side_transitivity: fixes P :: "'p"   and Q :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "P Q OS A B"   and "P Q OS B C" shows "P Q OS A C"
fresh_psubst_updEnv: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and xs :: "'varSort"   and x :: "'var" assumes "good X"   and "good Y"   and "goodEnv rho"   and "fresh xs x Y" shows "Y #[rho [x \<leftarrow> X]_xs] = Y #[rho]"
sup_conv_inf: fixes x :: "'b"   and y :: "'b" shows "x \<squnion> y = - (- x \<sqinter> - y)"