hcju/isabelleps huggingface dataset:
mult_right_zero: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b" assumes "class.idempotent_semiring one times sup less_eq less bot" shows "times x bot = bot"
aux_4: fixes x :: "'c DBMEntry"   and y :: "'c DBMEntry"   and z :: "'c DBMEntry" assumes "x \<prec> y"   and "\<not> dbm_add x z \<prec> dbm_add y z" shows "dbm_add x z = dbm_add y z"
albs_ex_AMin: fixes A :: "ant set"   and z :: "int" assumes "A \<subseteq> LBset (ant z)"   and "A \<noteq> {}" shows "\<exists>!m. m \<in> A \<and> (\<forall>x\<in>A. m \<le> x)"
iMOD_div_mod_gr0_not_in_i_set: fixes k :: "nat"   and m :: "nat"   and r :: "nat" assumes "0 < k"   and "k < m"   and "0 < m mod k" shows "[ r, mod m ] \<oslash> k \<notin> i_set"
elims_2: fixes x :: "('fsym, 'psym) atm" assumes "Ik.wtA x" obtains T1 :: "'fsym trm" and T2 :: "'fsym trm" where "x = Eq T1 T2" and "Ik.wt T1 \<and> Ik.wt T2 \<and> Ik.tpOf T1 = Ik.tpOf T2"   | p :: "'psym" and Tl :: "'fsym trm list" where "x = Pr p Tl" and "wtPsym p \<and> list_all Ik.wt Tl \<and> parOf p = map Ik.tpOf Tl"
lmap_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "lmap f (iterates f x) = iterates f (f x)"
hyps_3: assumes "P \<longmapsto> Rs" shows "\<exists>P\<in>S - {Q}. P \<longmapsto> Rs"
update_append_none: fixes env :: "('a, 'b, 'c) env"   and xs :: "'c list"   and y :: "'c"   and ys :: "'c list"   and opt :: "('a, 'b, 'c) env option" assumes "lookup env xs = None" shows "update (xs @ y # ys) opt env = env"
1: fixes \<phi> :: "'b ltln" shows "atoms_ltln (rewrite_modal (rewrite_X \<phi>)) \<subseteq> atoms_ltln \<phi>"
Theorem_1: fixes ds :: "'b set" shows "stable_on ds (cop ds)"
cross7_simps_18: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a - b + c = a + c - b"
proj_in_K': fixes t :: "nat" shows "\<exists>n. \<forall>m\<ge>n. (f m)\<^sub>F t \<in> (\<lambda>k. (k)\<^sub>F t) ` K n"
validCI_cont: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b" assumes "\<exists>i. active cid (t i)"   and "\<not> (\<exists>i'\<ge>n. active cid (t i'))"   and "\<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) (dynamic_component.cnf2bhv tCMP active cid t n)" shows "dynamic_component.eval tCMP active cid t t' n \<gamma>"
that: fixes t :: "'a \<Rightarrow>\<^sub>0 nat" assumes "t \<in> keys p"   and "q = hom_component p (deg_pm t)" shows "thesis"
Ds_mono': fixes es :: "expr list"   and A :: "char list set option"   and A' :: "char list set option" assumes "\<D>s es A"   and "A \<sqsubseteq> A'" shows "\<D>s es A'"
vector_minus_component: fixes x :: "('c, 'd) vec"   and y :: "('c, 'd) vec"   and i :: "'d" shows "(x - y) $ i = x $ i - y $ i"
exp_first_terms: fixes x :: "'a"   and k :: "nat" shows "exp x = (\<Sum>n<k. x ^ n /\<^sub>R fact n) + (\<Sum>n. x ^ (n + k) /\<^sub>R fact (n + k))"
Grp_def: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
K'_le_K: shows "emeasure M K' \<le> emeasure M K"
enn2real_eq_posreal_iff: fixes c :: "real"   and x :: "ennreal" assumes "0 < c" shows "(enn2real x = c) = (x = ennreal c)"
opT: shows "openin (top_of_set (S \<union> T)) T"
join_ti_empty_join: fixes i :: "nat"   and n :: "nat"   and s :: "nat \<Rightarrow> 'a list"   and x :: "nat" assumes "i \<le> n"   and "join_ti s x n = []" shows "s (x + i) = []"
lsc_hull_of_convex_aux: fixes a :: "real"   and b :: "real" shows "Limsup (at 1 within {0..<1}) (\<lambda>m. ereal ((1 - m) * a + m * b)) \<le> ereal b"
ch1_def: shows "ch1 = Cn 4 r_ifeq [Cn 4 rs [Cn 4 r_snoc [recf.Id 4 1, r_constn 3 1]], Cn 4 rs [recf.Id 4 1], r_dummy 3 (r_const z), r_constn 3 1]"
bounded_iff: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a \<le> min b c) = (a \<le> b \<and> a \<le> c)"
eval_F: fixes s :: "complex" assumes "1 < Re s" shows "eval_fds F s = f s"
simps_6: fixes n :: "nat"   and F :: "'b formula"   and G :: "'b formula" shows "cut_bound' n (F \<^bold>\<rightarrow> G) = cut_bound' (cut_bound' n F) G"
locally_antisym_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b multiset" shows "locally_antisym R A \<equiv> \<forall>t u. t \<in># A \<longrightarrow> u \<in># A \<longrightarrow> (t, u) \<in> R \<longrightarrow> (u, t) \<in> R \<longrightarrow> t = u"
infinite_cts_filtration_axioms: shows "infinite_cts_filtration p M F"
msg_sender_def: fixes m :: "msg" shows "msg_sender m \<equiv> case m of Rreq x xa xb xc xd xe xf ipc xg \<Rightarrow> ipc | Rrep x xa xb xc ipc \<Rightarrow> ipc | Rerr x ipc \<Rightarrow> ipc | Pkt x xa ipc \<Rightarrow> ipc"
mat_mult_equiv: fixes m1 :: "int poly fract list list"   and m2 :: "int poly fract list list" shows "rat_poly.matrix_mult m1 m2 = rat_poly.matrix_mult m1 m2"
prems_3: shows "\<not> ps (S t (Suc n)) p \<noteq> None"
cases: fixes a1 :: "s"   and a2 :: "s" assumes "store_extension a1 a2" obtains insts :: "inst list" and insts' :: "inst list" and fs :: "cl list" and fs' :: "cl list" and tclss :: "cl option list list" and tclss' :: "cl option list list" and bss :: "mem list" and bss' :: "mem list" and gs :: "global list" and gs' :: "global list" where "a1 = \<lparr>inst = insts, funcs = fs, tab = tclss, mem = bss, globs = gs\<rparr>" and "a2 = \<lparr>inst = insts', funcs = fs', tab = tclss', mem = bss', globs = gs'\<rparr>" and "insts = insts'" and "fs = fs'" and "tclss = tclss'" and "list_all2 (\<lambda>bs bs'. mem_size bs \<le> mem_size bs') bss bss'" and "gs = gs'"
wf: shows "tyenv_wellformed mds \<Gamma> \<S> P"
iIN_iFROM_subset_conv: fixes n' :: "nat"   and d' :: "nat"   and n :: "nat" shows "([n'\<dots>,d'] \<subseteq> [n\<dots>]) = (n \<le> n')"
precs: shows "precondition (next_state s execs) a \<and> precondition (next_state t' execs) a"
proj_rel_iff: fixes a :: "'b nonzero"   and b :: "'b nonzero" shows "proj_rel a b = (\<exists>c. c \<noteq> 0 \<and> b = c *\<^sub>R a)"
Bcontfun_inverse: fixes y :: "'c \<Rightarrow> 'd" assumes "y \<in> bcontfun" shows "apply_bcontfun (Bcontfun y) = y"
coinduct_strong: fixes R :: "'c stream \<Rightarrow> 'c stream \<Rightarrow> bool"   and stream :: "'c stream"   and stream' :: "'c stream" assumes "R stream stream'"   and "\<And>stream stream'. R stream stream' \<Longrightarrow> shd stream = shd stream' \<and> (R (stl stream) (stl stream') \<or> stl stream = stl stream')" shows "stream = stream'"
maximal_path_cons: fixes Ps :: "'a llist"   and v :: "'a" assumes "\<not> lnull Ps"   and "maximal_path Ps" shows "maximal_path (LCons v Ps)"
group_action_def: fixes G :: "('c, 'd) monoid_scheme"   and E :: "'e set"   and \<phi> :: "'c \<Rightarrow> 'e \<Rightarrow> 'e" shows "group_action G E \<phi> \<equiv> group_hom G (BijGroup E) \<phi>"
lebesgue_affine_measurable: fixes c :: "'b \<Rightarrow> real"   and t :: "'b" assumes "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0" shows "(\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j)) \<in> lebesgue \<rightarrow>\<^sub>M lebesgue"
set_rcs_def: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" shows "set_rcs G H = {C. \<exists>a\<in>carrier G. C = H \<bullet>\<^bsub>G\<^esub> a}"
reds_preserves_lconf: fixes extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and es :: "'addr expr list"   and s :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and ta :: "('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and es' :: "'addr expr list"   and s' :: "'heap \<times> (String.literal \<Rightarrow> 'addr val option)"   and E :: "String.literal \<Rightarrow> ty option"   and Ts :: "ty list" assumes "extTA,P,t \<turnstile> \<langle>es,s\<rangle> [-ta\<rightarrow>] \<langle>es',s'\<rangle>"   and "P,E,hp s \<turnstile> es [:] Ts"   and "P,hp s \<turnstile> lcl s (:\<le>) E" shows "P,hp s' \<turnstile> lcl s' (:\<le>) E"
lookup_update: fixes k :: "'c"   and v :: "'b"   and m :: "('c, 'b) mapping" shows "lookup (update k v m) k = Some v"
res_one_eq: shows "\<one>\<^bsub>R\<^esub> = 1"
IH: fixes f :: "nat \<Rightarrow> int" assumes "maps_to_n (int k) f" shows "\<exists>l. \<forall>m. \<exists>n>m. f n = l"
pw_bind_le_iff: fixes M :: "'d nres"   and f :: "'d \<Rightarrow> 'c nres"   and S :: "'c nres" shows "(M \<bind> f \<le> S) = ((nofail S \<longrightarrow> nofail M) \<and> (\<forall>x. nofail M \<and> inres M x \<longrightarrow> f x \<le> S))"
in_orbit_self: fixes a :: "'a"   and f :: "'a perm" shows "a \<in> orbit f a"
maybe_arith_int_not_None: fixes f :: "int \<Rightarrow> int \<Rightarrow> int"   and a :: "value option"   and b :: "value option" shows "(maybe_arith_int f a b \<noteq> None) = (\<exists>n n'. a = Some (Num n) \<and> b = Some (Num n'))"
product_def: fixes rbt1 :: "'b set_rbt"   and rbt2 :: "'c set_rbt" shows "product rbt1 rbt2 = RBT_Mapping2.product (\<lambda>_ _ _ _. ()) rbt1 rbt2"
fresh_distinct_list: fixes S :: "'a set"   and l :: "'a list" assumes "atom ` S \<sharp>* l" shows "S \<inter> set l = {}"
RN_2: shows "density M (RN_deriv M N) = N"
satC_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "M.satC \<xi> \<equiv> list_ex (M.satL \<xi>)"
disjoint_sublist: fixes t :: "'b set"   and ts :: "'b set list" assumes "disjoint_list (t # ts)" shows "disjoint_list ts"
fv_e: shows "FV e \<subseteq> set (map fst \<rho>)"
shiftpath_def: fixes a :: "real"   and f :: "real \<Rightarrow> 'b" shows "shiftpath a f = (\<lambda>x. if a + x \<le> 1 then f (a + x) else f (a + x - 1))"
hyps: assumes "k \<le> mod_type_class.to_nat a" shows "Hermite_of_upt_row_i A k ass res $ a $ b = A $ a $ b"
add_strict_mono: fixes a :: "'d multiset"   and b :: "'d multiset"   and c :: "'d multiset"   and d :: "'d multiset" assumes "a \<subset># b"   and "c \<subset># d" shows "a + c \<subset># b + d"
seq_lim: fixes x :: "'a" shows "seq x \<longlonglongrightarrow> x"
finite_groupP: shows "finite (carrier (G\<lparr>carrier := P\<rparr>))"
sel_19: fixes x31 :: "nat"   and x32 :: "'b"   and x33 :: "'b"   and x34 :: "'a"   and x35 :: "'a"   and x36 :: "'c" shows "un_Recv5 (Recv x31 x32 x33 x34 x35 x36) = x35"
remove_removes_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r node_ptr # children"   and "h \<turnstile> Shadow_DOM.remove node_ptr \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"
these_insert_Some: fixes x :: "'c"   and A :: "'c option set" shows "Option.these (insert (Some x) A) = insert x (Option.these A)"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   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" assumes "l_set_mode\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs"   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" shows "l_set_mode_get_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_mode set_mode_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs"
ltl_prop_equiv_def: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<phi> \<sim>\<^sub>P \<psi> \<equiv> \<forall>\<A>. \<A> \<Turnstile>\<^sub>P \<phi> = \<A> \<Turnstile>\<^sub>P \<psi>"
CondReds2T: fixes e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>2 :: "bool"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>true,s\<^sub>1,b\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1,b\<^sub>1\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2,b\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>e',s\<^sub>2,b\<^sub>2\<rangle>"
a: fixes x :: "'a extended"   and y :: "nat" assumes "x \<in> set (concat d)"   and "idx w x = Inr y" shows "l < y \<and> y < u"
scale_sum_left: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and x :: "'a" shows "sum f A * x = (\<Sum>a\<in>A. f a * x)"
3133: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - y) \<sqinter> y = y"
sub_fun_top: fixes T1 :: "ty"   and T2 :: "ty" shows "T1 \<rightarrow> T2 <:: \<top>"
inv_f_f: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "inj f" shows "inv f (f x) = x"
zero_closed: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b" assumes "comm_monoid_add_ow S pls z" shows "z \<in> S"
assms_3: fixes a :: "'c"   and b :: "'d" assumes "a \<in> set2_FGl x"   and "b \<in> set2_FGl y"   and "L2 a b" shows "L2' a b"
ide_trg: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "span_vertical_category C"   and "partial_magma.arr (span_vertical_category.vcomp C) \<mu>" shows "partial_magma.ide (span_vertical_category.vcomp C) (span_vertical_category.trg C \<mu>)"
simps_4: fixes n :: "nat"   and k :: "nat"   and T :: "POPLmarkRecord.type"   and U :: "POPLmarkRecord.type" shows "\<up>\<^sub>\<tau> n k (\<forall><:T. U) = (\<forall><:\<up>\<^sub>\<tau> n k T. \<up>\<^sub>\<tau> n (k + 1) U)"
simps_2: fixes t' :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree"   and bq :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" shows "SkewBinomialHeapStruc.ins t' (t # bq) = (if SkewBinomialHeapStruc.rank t' < SkewBinomialHeapStruc.rank t then t' # t # bq else if SkewBinomialHeapStruc.rank t < SkewBinomialHeapStruc.rank t' then t # SkewBinomialHeapStruc.ins t' bq else SkewBinomialHeapStruc.ins (SkewBinomialHeapStruc.link t' t) bq)"
Ker_equality: shows "image.Ker = {a |a. a \<in> G \<and> \<eta> a = \<one>'}"
simps_2: fixes x :: "'b"   and a :: "'b"   and xs :: "'b list" shows "ins_list x (a # xs) = (if x < a then x # a # xs else if x = a then a # xs else a # ins_list x xs)"
ineq_model_single_iff: fixes \<I> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and X :: "'a list"   and F :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list"   and f :: "'b"   and c :: "'b" shows "ineq_model \<I> X F = ineq_model \<I> X [(Fun f (Fun c [] # map fst F), Fun f (Fun c [] # map snd F))]"
distinct_pds_valuation: fixes j :: "nat"   and n :: "nat"   and P :: "nat \<Rightarrow> ('a \<Rightarrow> ant) set" assumes "j \<le> Suc n"   and "distinct_pds K (Suc n) P" shows "valuation K (\<nu>\<^bsub>K P j\<^esub>)"
finite_linorders_on: fixes A :: "'b set" assumes "finite A" shows "finite (linorders_on A)"
DC_refines: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and c :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "a \<sqsubseteq> b"   and "a \<sqsubseteq> c" shows "a \<sqsubseteq> (b \<Sqinter> c)"
objI_src: fixes a :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr a"   and "Span.src a = a" shows "Span.obj a"
subset_fst_snd: fixes A :: "('a \<times> 'b) set" shows "A \<subseteq> fst ` A \<times> snd ` A"
num_params_eval_poly_le: fixes t :: "taylor_model"   and x :: "nat"   and p :: "float poly"   and prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t) \<le> x"   and "num_params p \<le> max 1 x" shows "num_params (tm_poly (eval_poly_at_tm prec ord I a p t)) \<le> x"
converse_iff: fixes a :: "'g"   and b :: "'h"   and r :: "('h \<times> 'g) set" shows "((a, b) \<in> r\<inverse>) = ((b, a) \<in> r)"
f: shows "C1.arr (fst f) \<and> C2.arr (snd f)"
exe_osig_conds_def: fixes a :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" shows "exe_osig_conds a \<equiv> exe_class_conds (fst a) \<and> exe_ars_conds (snd a)"
ex_s2: shows "s \<in> wordinterval_to_set s_range"
\<psi>\<^sub>1_values_odd: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i < dim_row (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"   and "j < dim_col (Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))))"   and "odd i" shows "Matrix.mat (2 ^ (n + 1)) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> if even i then 1 / sqrt 2 ^ (n + 1) else - 1 / sqrt 2 ^ (n + 1))) $$ (i, j) = complex_of_real (- 1 / sqrt 2 ^ (n + 1))"
twiddleVal_compose: fixes \<beta> :: "(Location \<times> Location) set"   and v :: "Val"   and u :: "Val"   and \<gamma> :: "(Location \<times> Location) set"   and w :: "Val" assumes "(\<beta>, v, u) \<in> twiddleVal"   and "(\<gamma>, u, w) \<in> twiddleVal" shows "(Pbij_compose \<beta> \<gamma>, v, w) \<in> twiddleVal"
cnj_Dirichlet_L: fixes s :: "complex" shows "cnj (Dirichlet_L n \<chi> s) = Dirichlet_L n (inv_character \<chi>) (cnj s)"
n_pow_not_prime: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "0 < n" shows "\<not> prime_ideal R (R \<diamondsuit>\<^sub>p a \<^bsup>\<diamondsuit>R Suc n\<^esup>)"
simps_2: fixes c :: "poly"   and n :: "nat"   and p :: "poly" shows "isnpolyh (CN c n p) = (\<lambda>k. k \<le> n \<and> isnpolyh c (Suc n) \<and> isnpolyh p n \<and> p \<noteq> 0\<^sub>p)"
fn_norm_ge_zero: fixes f :: "'a \<Rightarrow> real" assumes "Function_Norm.continuous E f norm" shows "0 \<le> \<parallel>f\<parallel>\<hyphen>E"
dom_inv: fixes f :: "'c local.term set" assumes "\<F>C.iso f" shows "\<F>C.dom (\<F>C.inv f) = \<F>C.cod f"
path_split_second: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and a :: "'a"   and as' :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n (as @ a # as') n'" shows "CFG.path sourcenode targetnode valid_edge (sourcenode a) (a # as') n'"
active_threads_iff: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" shows "active_threads s = {t. \<exists>x ln. thr s t = \<lfloor>(x, ln)\<rfloor> \<and> (if ln = no_wait_locks then \<exists>ta x' m'. t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<and> actions_ok s t ta else \<not> waiting (wset s t) \<and> may_acquire_all (locks s) t ln)}"
assms_2: shows "\<Union> (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2) = U - V"
UN_least: fixes A :: "'d set"   and B :: "'d \<Rightarrow> 'e set"   and C :: "'e set" assumes "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> C" shows "\<Union> (B ` A) \<subseteq> C"
dom_map_upds: fixes xs :: "'a list"   and ys :: "'b list"   and tab :: "'a \<Rightarrow> 'b option" assumes "length xs = length ys" shows "dom (tab(xs [\<mapsto>] ys)) = dom tab \<union> set xs"
simps_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and cs :: "'a list" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main" shows "CFG.valid_path_rev_aux kind get_return_edges cs [] = True"
absorb_iff1: fixes b :: "'c multiset"   and a :: "'c multiset" shows "(b \<subseteq># a) = (a \<union># b = a)"
surj_simplex_map: shows "f \<turnstile> X = X"
LPDs_ok: fixes m :: "nat"   and mr :: "mregex"   and ms :: "mregex" assumes "ok m mr"   and "ms \<in> LPDs mr" shows "ok m ms"
simps: fixes a1 :: "'a list"   and a2 :: "'b sequent" shows "((a1, a2) \<in> upRules) = ((\<exists>c R Fs ps. a1 = ps \<and> a2 = c \<and> c = ( \<LM> Compound R Fs  \<RM> \<Rightarrow>* Em) \<and> ps \<noteq> []) \<or> (\<exists>c F Fs ps. a1 = ps \<and> a2 = c \<and> c = ( \<Empt> \<Rightarrow>* Compound F Fs) \<and> ps \<noteq> []))"
interval_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.interval \<P> betw a b \<equiv> insert b (insert a (MinkowskiBetweenness.segment \<P> betw a b))"
compl_code_2: fixes xs :: "('b, unit) RBT.rbt" shows "- Coset xs = Set xs"
zero_fun: fixes x :: "'c" shows "0 x = (0::'b)"
arr_dom: fixes f :: "'c local.term set" assumes "\<F>C.arr f" shows "\<F>C.arr (\<F>C.dom f)"
subset_mult2_alt: fixes X :: "'b multiset"   and Y :: "'b multiset"   and Z :: "'b multiset"   and b :: "bool"   and ns :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and b' :: "bool" assumes "X \<subseteq># Y"   and "(Y, Z) \<in> mult2_alt b ns s"   and "b \<Longrightarrow> b'" shows "(X, Z) \<in> mult2_alt b' ns s"
homeomorphic_maps_def: fixes X :: "'d topology"   and Y :: "'e topology"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'd" shows "homeomorphic_maps X Y f g \<equiv> continuous_map X Y f \<and> continuous_map Y X g \<and> (\<forall>x\<in>topspace X. g (f x) = x) \<and> (\<forall>y\<in>topspace Y. f (g y) = y)"
assms_3: fixes x :: "'x" assumes "x \<in> X''" shows "x \<in> CD_on ds (X \<union> X'')"
sel_4: fixes x21 :: "variable"   and x22 :: "String.literal set" shows "un_Tv2 (Tv x21 x22) = x22"
prems_3: fixes D :: "'a set" assumes "D \<in> \<D>" shows "\<exists>k a b. D = cbox a b \<and> (\<forall>i\<in>Basis. b \<bullet> i - a \<bullet> i = k)"