hcju/isabelleps huggingface dataset:
is_cindependent_set: fixes scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and scaleC :: "complex \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and canonical_basis :: "'d list" assumes "class.basis_enum scaleR scaleC plus zero minus uminus canonical_basis" shows "\<not> cdependent (set canonical_basis)"
ps_2: shows "y\<^sub>3 = y\<^sub>2"
indexed_eval_index_free: fixes j :: "'d"   and Ps :: "('d multiset \<Rightarrow> 'c multiset \<Rightarrow> 'a) list"   and i :: "'d" assumes "list_all (\<lambda>P. L.index_free P j) Ps"   and "i \<noteq> j" shows "L.index_free (L.indexed_eval Ps i) j"
simps_3: fixes c1 :: "('a, 'b) MWLsCom"   and c2 :: "('a, 'b) MWLsCom" shows "pp (c1;c2) = pp c1"
rev_replicate: fixes n :: "nat"   and x :: "'b" shows "rev (replicate n x) = replicate n x"
\<Lambda>_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "\<Lambda> a b c f = mkArr (local.set a) (local.set (exp b c)) (\<lambda>x. UP (happ (hlam (ide_to_hf a) (ide_to_hf b) (ide_to_hf c) (arr_to_hfun f)) (DOWN x)))"
accmethd_SomeI: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig"   and m :: "qtname \<times> methd"   and S :: "qtname" assumes "methd G C sig = Some m"   and "G \<turnstile>Methd sig m of C accessible_from S" shows "accmethd G S C sig = Some m"
\<R>\<^sub>3_elim: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and 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 mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2' :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'''__ = c\<^sub>1''''__ ;; ca__" and "c\<^sub>2 = c\<^sub>2''__ ;; ca__" and "\<langle>c\<^sub>1''''__, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'''''__\<^esub> \<langle>c\<^sub>2''__, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>'''''__ {ca__} \<Gamma>''''__"   | \<Gamma> :: "'Var \<Rightarrow> Sec option" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2' :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'''__ = c\<^sub>1'''''__ ;; cb__" and "c\<^sub>2 = c\<^sub>2'''__ ;; cb__" and "\<langle>c\<^sub>1'''''__, mds, mem\<^sub>1\<rangle> \<R>\<^sup>2\<^bsub>\<Gamma>''''''__\<^esub> \<langle>c\<^sub>2'''__, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>''''''__ {cb__} \<Gamma>''''__"   | \<Gamma> :: "'Var \<Rightarrow> Sec option" and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and c\<^sub>2' :: "('Var, 'AExp, 'BExp) Stmt" and c :: "('Var, 'AExp, 'BExp) Stmt" where "c\<^sub>1'''__ = c\<^sub>1 ;; c" and "c\<^sub>2 = c\<^sub>2' ;; c" and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2', mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma> {c} \<Gamma>''''__"
set_butlast: fixes x :: "'c"   and xs :: "'c list" assumes "x \<in> set xs"   and "x \<noteq> last xs" shows "x \<in> set (butlast xs)"
assms_2: shows "\<forall>(loc, t)\<in>#\<^sub>z\<Delta>. \<exists>t'. t' \<in>\<^sub>A frontier (c_imp c0 loc) \<and> t' \<le> t"
assms_1: shows "summable (\<lambda>n. c n * K ^ n)"
Union_insert: fixes a :: "'e set"   and B :: "'e set set" shows "\<Union> (insert a B) = a \<union> \<Union> B"
simps: fixes a1 :: "'a list"   and a2 :: "sequent" shows "((a1, a2) \<in> Ax) = ((\<exists>i xs. a1 = [] \<and> a2 = ( \<LM> At i xs  \<RM> \<Rightarrow>* \<LM> At i xs  \<RM>)) \<or> a1 = [] \<and> a2 = ( \<LM> ff  \<RM> \<Rightarrow>* \<Empt>))"
inf_sup_ord_3: fixes x :: "'b"   and y :: "'b" shows "x \<le> x \<squnion> y"
fresh_fun_app: fixes a :: "atom"   and f :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "a \<sharp> f"   and "a \<sharp> x" shows "a \<sharp> f x"
samequot_termination: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and embed :: "'c \<Rightarrow> 'c list"   and singleton :: "'c \<Rightarrow> 'd"   and as :: "'c"   and r :: "'d rexp" assumes "embed2 \<Sigma> wf_atom project lookup embed singleton"   and "while_option fst (embed2.rderiv_and_add lookup embed as) (True, [\<guillemotleft>r\<guillemotright>]) = None" shows "False"
runit_eqI: fixes f :: "'a"   and a :: "'a" assumes "\<guillemotleft>f : C.tensor a \<I> \<rightarrow>\<^sub>C a\<guillemotright>"   and "C.tensor f \<I> = C.tensor a \<iota> \<cdot> C.assoc a \<I> \<I>" shows "f = C.runit a"
Step_CloseId: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and j :: "nat"   and k :: "nat" shows "step_result (SG, C) (i, CloseId j k) = (closeI SG i, C)"
lt_Koszul_syz_comp: fixes i :: "nat"   and j :: "nat" assumes "0 \<notin> set fs"   and "i < length fs" shows "lt (fs ! i \<odot> monomial (1::'b) (term_of_pair (0::'a, j))) = term_of_pair (punit.lt (fs ! i), j)"
divide_nonneg_pos: 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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less zero y" shows "less_eq zero (divide x y)"
r_lifz_recfn: fixes f :: "recf"   and n :: "nat"   and g :: "recf" assumes "recfn n f"   and "recfn n g" shows "recfn (Suc n) (r_lifz f g)"
split_cong: fixes q :: "'g \<times> 'h"   and f :: "'g \<Rightarrow> 'h \<Rightarrow> 'i"   and g :: "'g \<Rightarrow> 'h \<Rightarrow> 'i"   and p :: "'g \<times> 'h" assumes "\<And>x y. (x, y) = q \<Longrightarrow> f x y = g x y"   and "p = q" shows "(case p of (x, xa) \<Rightarrow> f x xa) = (case q of (x, xa) \<Rightarrow> g x xa)"
trm_simps_1: fixes a :: "'a" assumes "ide a" shows "arr \<t>[a]"
poly_gcd_eq_0I: fixes p :: "'a poly"   and x :: "'a"   and q :: "'a poly" assumes "poly p x = (0::'a)"   and "poly q x = (0::'a)" shows "poly (gcd p q) x = (0::'a)"
keys_of_monomial: fixes c :: "'a"   and t :: "'b" assumes "c \<noteq> (0::'a)" shows "keys (monomial c t) = {t}"
qiter_fun_def: fixes x :: "'a"   and y :: "'a" shows "qiter_fun x y = (\<squnion>) x \<circ> (\<cdot>) y"
P1: shows "X Midpoint P' P \<and> Col A B X"
same_mset_distinct_iff: fixes M :: "'a list"   and M' :: "'a list" assumes "mset M = mset M'" shows "distinct M = distinct M'"
partition_on_the_part_mem: fixes A :: "'b set"   and P :: "'b set set"   and x :: "'b" assumes "partition_on A P"   and "x \<in> A" shows "(THE X. x \<in> X \<and> X \<in> P) \<in> P"
fin: fixes M :: "nat"   and N :: "nat" shows "finite {xs. length xs = M \<and> set xs \<subseteq> {0..<N}}"
path_strategies_monotone_Suc: fixes G :: "('c, 'd) ParityGame_scheme"   and S :: "'c set"   and p :: "Player"   and good :: "'c \<Rightarrow> ('c \<Rightarrow> 'c) set"   and r :: "(('c \<Rightarrow> 'c) \<times> ('c \<Rightarrow> 'c)) set"   and P :: "'c llist"   and n :: "nat" assumes "WellOrderedStrategies G S p good r"   and "lset P \<subseteq> S"   and "Digraph.valid_path G P"   and "ParityGame.path_conforms_with_strategy G p P (WellOrderedStrategies.well_ordered_strategy G S good r)"   and "enat (Suc n) < llength P" shows "(WellOrderedStrategies.path_strategies good r P $ Suc n, WellOrderedStrategies.path_strategies good r P $ n) \<in> r"
cases: fixes A :: "'c set"   and n :: "nat"   and a :: "'c list" assumes "nlistsp A n a" obtains xs :: "'c list" where "a = xs" and "set xs \<subseteq> A" and "length xs = n"
bdd_belowI: fixes A :: "'b multiset set"   and m :: "'b multiset" assumes "\<And>x. x \<in> A \<Longrightarrow> m \<subseteq># x" shows "subset_mset.bdd_below A"
maintainedD: fixes A :: "('e, 'f) labeled_graph"   and B :: "('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "maintained (A, B) G"   and "graph_homomorphism A G f" shows "extensible (A, B) G f"
hyps_3: fixes q :: "'Q" assumes "qnh = (q, n)"   and "n \<noteq> n'" shows "\<exists>qa. (qa, n) \<in> set p \<and> n \<in> acc qa"
adv_P1_def: fixes x :: "'grp \<times> 'grp"   and y :: "bool"   and z :: "'d"   and S :: "(('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('e \<Rightarrow> 'f spmf) \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp) \<times> 'e) spmf) \<times> (('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('e \<Rightarrow> 'f spmf) \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'd \<Rightarrow> unit \<Rightarrow> 'e \<Rightarrow> 'f spmf)"   and \<A> :: "('grp \<times> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'd \<Rightarrow> (('grp \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'e \<Rightarrow> ((('grp \<times> 'grp) \<times> 'grp \<times> 'grp) \<times> 'e) spmf) \<times> ('e \<Rightarrow> 'f spmf)"   and D :: "'f \<times> 'grp \<Rightarrow> bool spmf" shows "mal_def.adv_P1 x y z S \<A> D = \<bar>spmf (P1_real_model x y z \<A> \<bind> D) True - spmf (mal_def.ideal_view_1 x y z S \<A> \<bind> D) True\<bar>"
hoare_triple_preI: fixes P :: "assn"   and c :: "'b Heap"   and Q :: "'b \<Rightarrow> assn" assumes "\<And>h. h \<Turnstile> P \<Longrightarrow> <P> c <Q>" shows "<P> c <Q>"
less_eq_matrix_def: fixes A :: "('c, 'd, 'e) matrix"   and B :: "('c, 'd, 'e) matrix" shows "A \<le> B \<equiv> A + B = B"
lifelock_free_v2_def: fixes P :: "'b process" shows "lifelock_free_v2 P \<equiv> CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>F\<^sub>D P"
assms_3: fixes y :: "'a" assumes "Q y" shows "P y"
f4_red_def: fixes gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and sps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" shows "f4_red gs bs ps sps data = (map (\<lambda>h. (h, default)) (f4_red_aux (gs @ bs) sps), snd data)"
R_star: fixes i :: "'a" shows "Ref i i\<^sup>\<star> \<le> Ref i i"
less_num1_def: fixes a :: "num1"   and b :: "num1" shows "(a < b) = (Rep_num1 a < Rep_num1 b)"
fold_mset_add_mset: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set"   and M :: "'e multiset"   and s :: "'e"   and x :: "'e" assumes "comp_fun_commute_on f A"   and "set_mset M \<subseteq> A"   and "s \<in> A"   and "x \<in> A" shows "fold_mset f s (add_mset x M) = f x (fold_mset f s M)"
ar_coset_mem2: fixes I :: "'a set"   and a :: "'a"   and x :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "\<exists>h\<in>I. x = a \<plusminus> h"
SUP_UNION: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "\<Squnion> \<P> f (\<mu> (\<P> g A)) = (\<Squnion>y\<in>A. \<Squnion> \<P> f (g y))"
rotate_3: assumes "round_robin_reschedule t0 (queue @ [t']) n0 s = \<lfloor>(t, \<lfloor>(ta, x', m')\<rfloor>, \<sigma>')\<rfloor>"   and "round_robin_invar (queue @ [t'], n0) (dom (thr s))" shows "round_robin_invar \<sigma>' (dom (thr s) \<union> {t. \<exists>x m. NewThread t x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>})"
self_append_conv: fixes xs :: "'k list"   and ys :: "'k list" shows "(xs = xs @ ys) = (ys = [])"
integrate_insert_commute': fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) message"   and T :: "('a, 'b) message set"   and s :: "String.literal + ('a, 'b option) insert_message list" assumes "consistent M"   and "is_delete m \<or> m \<notin> T"   and "m \<in> M"   and "T \<subseteq> M"   and "deps m \<subseteq> I ` insert_messages T"   and "is_certified_associated_string T s" shows "is_certified_associated_string (T \<union> {m}) (s \<bind> (\<lambda>t. integrate t m))"
list_dfs_def: fixes succ :: "'c \<times> 'd \<times> 'c \<Rightarrow> ('c \<times> 'd \<times> 'c) list"   and S :: "('c \<times> 'd \<times> 'c) list"   and start :: "('c \<times> 'd \<times> 'c) list" shows "list_dfs succ S start \<equiv> gen_dfs succ List.insert (\<lambda>x xs. x \<in> set xs) S start"
is_Ands_def: fixes formula :: "formula" shows "is_Ands formula = (\<exists>x9. formula = Ands x9)"
Pi_inc: fixes i :: "nat" shows "Pi i < Pi (Suc i)"
Sup_set_def: fixes A :: "'b set set" shows "\<Union> A = {x. \<Squnion> ((\<in>) x ` A)}"
wordinterval_optimize_empty_set_eq: fixes r :: "'b wordinterval" shows "wordinterval_to_set (wordinterval_optimize_empty r) = wordinterval_to_set r"
mute_det: fixes A :: "'a mat"   and n :: "nat"   and q :: "'a"   and k :: "nat"   and l :: "nat"   and r :: "'a"   and r' :: "'a"   and A' :: "'a mat" assumes "A \<in> carrier_mat n n"   and "local.mute q k l (r, A) = (r', A')"   and "k < n"   and "l < n"   and "k \<noteq> l" shows "r * det A' = r' * det A"
single_addition_pm: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<^sub>f+ m\<Odot>a\<^bsub>f,i,z\<^esub> = (n + m)\<Odot>a\<^bsub>f,i,z\<^esub>"
index_add_mat_3: fixes A :: "'b mat"   and B :: "'b mat" shows "dim_col (A + B) = dim_col B"
add_rel_class_of: fixes U :: "'a set"   and x :: "'b"   and V :: "'a set"   and y :: "'b"   and W :: "'a set" assumes "(U, x) \<in> Sigma {U. is_open\<^sub>X U \<and> x \<in> U} \<O>\<^sub>X"   and "(V, y) \<in> Sigma {U. is_open\<^sub>X U \<and> x \<in> U} \<O>\<^sub>X"   and "W \<in> {U. is_open\<^sub>X U \<and> x \<in> U}"   and "W \<subseteq> U"   and "W \<subseteq> V" shows "stx.add_rel (stx.class_of U x) (stx.class_of V y) = stx.class_of W (add_str\<^sub>X W (\<rho>\<^sub>X U W x) (\<rho>\<^sub>X V W y))"
succ_of_E_def: fixes E :: "('c \<times> 'd) set" shows "succ_of_E E \<equiv> \<lambda>u. {v. (u, v) \<in> E}"
Ineq'_def: shows "Ineq' \<equiv> \<lambda>\<delta>. list_ex (\<lambda>f. fst f \<cdot> \<delta> \<circ>\<^sub>s \<I> \<noteq> snd f \<cdot> \<I>)"
simps_7: fixes \<phi> :: "'b ltln"   and M :: "'b ltln set" shows "X\<^sub>n \<phi>[M]\<^sub>\<Sigma>\<^sub>2 = X\<^sub>n \<phi>[M]\<^sub>\<Sigma>\<^sub>2"
\<tau>moves_False: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool" shows "\<tau>trsys.silent_move r (\<lambda>s ta s'. False) = (\<lambda>s s'. False)"
period_eq: fixes C :: "'s set"   and x :: "'s" assumes "C \<in> UNIV // communicating"   and "x \<in> C" shows "period C = Gcd (period_set x)"
adjugate_iarrays_def: fixes A :: "'c iarray iarray" shows "adjugate_iarrays A = transpose_iarray (cofactorM_iarrays A)"
2: shows "(\<lambda>z. if 0 \<le> Im z then f z else cnj (f (cnj z))) holomorphic_on S \<inter> {z. Im z < 0}"
rel_envir_conv: fixes p :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e" shows "rel_envir p f g = (\<forall>x. p (f x) (g x))"
f0: shows "w = Tensor.mat_of_cols_list 8 [[\<alpha> / complex_of_real (sqrt 2), 0, 0, \<alpha> / complex_of_real (sqrt 2), 0, \<beta> / complex_of_real (sqrt 2), \<beta> / complex_of_real (sqrt 2), 0]]"
horner_sum_simps_1: fixes f :: "'c \<Rightarrow> 'a"   and a :: "'a" shows "horner_sum f a [] = (0::'a)"
Atom_not_Scons: fixes a :: "'a + nat"   and M :: "('a, 'b) node set"   and N :: "('a, 'b) node set" shows "Atom a \<noteq> Scons M N"
when_add_distrib: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and b :: "'a"   and P :: "bool" assumes "class.monoid_add plus zero" shows "zero.when zero (plus a b) P = plus (zero.when zero a P) (zero.when zero b P)"
mmerge_def: fixes return :: "'f multiset \<Rightarrow> 'e"   and bind :: "'e \<Rightarrow> ('f multiset \<Rightarrow> 'e) \<Rightarrow> 'e"   and A :: "'f multiset"   and f :: "'f \<Rightarrow> 'e" shows "mmerge return bind A f = fold_mset (\<lambda>m1 m2. bind m1 (\<lambda>A. bind m2 (\<lambda>B. return (A + B)))) (return {#}) (image_mset f A)"
ide_leg0: shows "ide f\<^sub>0gh\<^sub>1.p\<^sub>0"
right_lang_def: fixes q :: "hf" shows "right_lang q \<equiv> {u. nextl q u \<in> final M}"
Image_eq_UN: fixes r :: "('b \<times> 'a) set"   and B :: "'b set" shows "r `` B = (\<Union>y\<in>B. r `` {y})"
hyps: fixes y :: "'a list" assumes "length y < length xs"   and "\<And>xs'. \<lbrakk>rtrancl_path r x xs' y; distinct (x # xs'); set xs' \<subseteq> set y\<rbrakk> \<Longrightarrow> thesis"   and "rtrancl_path r x y y" shows "thesis"
comp_a: shows "component_of_term (lt a) = component_of_term u"
intro: fixes num' :: "int \<Rightarrow> 'a"   and plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and \<gamma> :: "'a \<Rightarrow> int set"   and test_num' :: "int \<Rightarrow> 'a \<Rightarrow> bool"   and inv_plus' :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a"   and inv_less' :: "bool \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a" assumes "Val_inv num' plus' \<gamma> test_num' inv_plus' inv_less'" shows "Abs_Int_inv num' plus' test_num' inv_plus' inv_less' \<gamma>"
keys_combine: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and m1 :: "('a, 'b) mapping"   and m2 :: "('a, 'b) mapping" shows "Mapping.keys (Mapping.combine f m1 m2) = Mapping.keys m1 \<union> Mapping.keys m2"
checked_simplex'_def: shows "checked_simplex' = Incremental_NS_Constraint_Ops_To_Ns_For_Incremental_Simplex.checked_cs checked_nsc to_ns"
f2_One: fixes x :: "three"   and y :: "three" assumes "x \<noteq> Two \<and> y \<noteq> Two" shows "f2 x y \<noteq> Two"
bij_phi: shows "bij_betw \<phi> {0..1} {0..1}"
a: fixes x :: "'a" assumes "\<And>y. y < x \<Longrightarrow> P y" shows "P x"
subsumer_not_marked: fixes prb :: "('d, 'e, 'f) pre_RedBlack"   and sub :: "('d \<times> nat) \<times> 'd \<times> nat" assumes "RedBlack prb"   and "sub \<in> subs prb" shows "\<not> marked prb (subsumer sub)"
progress'_1: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T"   and "lambda_closed t"   and "\<forall>s. \<not> t \<longlonglongrightarrow> s" shows "is_nf t"
AL_delete1: fixes L :: "('b \<times> 'a) list"   and a :: "'b" shows "delete L a\<down>a = None"
pderiv_const: fixes b :: "nat \<Rightarrow> int" assumes "b \<in> carrier Zp" shows "pderiv (monom (UP Zp) b 0) = \<zero>\<^bsub>UP Zp\<^esub>"
FactGroup_the_elem_mem: fixes X :: "'a set" assumes "X \<in> carrier (G Mod kernel G H h)" shows "the_elem (h ` X) \<in> carrier H"
stack_set_def: shows "set (stack s) = dom (discovered s) - dom (finished s)"
closed_sum_cspan: fixes X :: "'d set"   and Y :: "'d set" shows "cspan X +\<^sub>M cspan Y = closure (cspan (X \<union> Y))"
induct: fixes P :: "'a prog"   and x1 :: "ty"   and x2 :: "ty"   and Pa :: "ty \<Rightarrow> ty \<Rightarrow> bool" assumes "subtype P x1 x2"   and "\<And>T. Pa T T"   and "\<And>C D. P \<turnstile> C \<preceq>\<^sup>* D \<Longrightarrow> Pa (Class C) (Class D)"   and "\<And>C. Pa NT (Class C)" shows "Pa x1 x2"
infinite_chain_intro': fixes x :: "'a"   and r :: "('a \<times> 'a) set" assumes "\<exists>b. (x, b) \<in> r"   and "\<forall>b. (x, b) \<in> r\<^sup>+ \<longrightarrow> (\<exists>c. (b, c) \<in> r)" shows "\<exists>f. \<forall>i. (f i, f (Suc i)) \<in> r"
carrier_matD_2: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "dim_col A = nc"
comp_def: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "f \<circ> g = (\<lambda>x. f (g x))"
ZObisT_matchT_T_rev: fixes s :: "('test, 'atom) com"   and t :: "('test, 'atom) com" assumes "s \<approx>01T t" shows "matchT_T t s"
hyps_2: fixes ys :: "['a]" assumes "xs \<sqsubseteq> ys" shows "finite_list ys"
\<theta>_nonneg: fixes x :: "real" shows "0 \<le> \<theta> x"
i_expand_1: fixes f :: "nat \<Rightarrow> 'd message_af" shows "f \<odot> Suc 0 = f"
wf_initial_2: shows "\<forall>x<numVars. astI ! x < numVals x"
zero_stalk_def: fixes V :: "'a set" shows "stx.zero_stalk V \<equiv> stx.class_of V (zero_str\<^sub>X V)"
\<Delta>''_bounded_countable_network: assumes "\<And>x. \<lbrakk>x \<in> \<^bold>V; x \<noteq> source \<Delta>\<rbrakk> \<Longrightarrow> d_OUT (capacity \<Delta>) x < \<top>" shows "bounded_countable_network \<Delta>''"
uncurry_preserves_transformations: fixes F :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and G :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and \<tau> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "natural_transformation (\<cdot>\<^sub>J) (\<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>]) F G \<tau>" shows "natural_transformation (\<cdot>\<^sub>J\<^sub>x\<^sub>A) (\<cdot>\<^sub>B) (Curry.uncurry F) (Curry.uncurry G) (Curry.uncurry \<tau>)"
vimage2pI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'c"   and x :: "'e"   and g :: "'f \<Rightarrow> 'd"   and y :: "'f" assumes "R (f x) (g y)" shows "BNF_Def.vimage2p f g R x y"
progress_L0_left: fixes s :: "s"   and vs :: "v list"   and es :: "e list"   and i :: "nat"   and s' :: "s"   and vs' :: "v list"   and es' :: "e list"   and cs :: "e list" assumes "\<lparr>s;vs;es\<rparr> \<leadsto>_ i \<lparr>s';vs';es'\<rparr>"   and "const_list cs" shows "\<lparr>s;vs;cs @ es\<rparr> \<leadsto>_ i \<lparr>s';vs';cs @ es'\<rparr>"
assms_1: shows "eventually_nonneg F (\<lambda>x. - g x)"
Domain: fixes x :: "ZF"   and f :: "ZF" shows "x |\<in>| HOLZF.Domain f = (\<exists>y. Opair x y |\<in>| f)"
ORRIGHT2_eqvt_coname: fixes pi :: "(coname \<times> coname) list"   and A :: "ty"   and B :: "ty"   and X :: "ctrm set" shows "pi \<bullet> ORRIGHT2 (A OR B) X = ORRIGHT2 (A OR B) (pi \<bullet> X)"
assms_2: shows "bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}"
Inf_pres_comp: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b" assumes "Inf_pres f"   and "Inf_pres g" shows "Inf_pres (f \<circ> g)"