hcju/isabelleps huggingface dataset:
CallRedsThrowParams: fixes e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and vs\<^sub>1 :: "val list"   and ex :: "nat \<times> char list list"   and es\<^sub>2 :: "expr list"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and Copt :: "char list option"   and M :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v,s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs\<^sub>1 @ Throw ex # es\<^sub>2,s\<^sub>2\<rangle>" shows "P,E \<turnstile> \<langle>Call e Copt M es,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Throw ex,s\<^sub>2\<rangle>"
valid_countings_Suc_Suc: fixes b :: "nat"   and a :: "nat" assumes "b < a" shows "valid_countings (Suc a) (Suc b) = valid_countings a (Suc b) + valid_countings (Suc a) b"
nth_shiftl': fixes w :: "'b word"   and m :: "nat"   and n :: "nat" shows "(w << m) !! n = (n < size w \<and> m \<le> n \<and> w !! (n - m))"
induct: fixes x1 :: "'node"   and x2 :: "'edge list"   and x3 :: "'node"   and P :: "'node \<Rightarrow> 'edge list \<Rightarrow> 'node \<Rightarrow> bool" assumes "x1 -x2\<rightarrow>* x3"   and "\<And>n. valid_node n \<Longrightarrow> P n [] n"   and "\<And>n'' as n' a n. \<lbrakk>n'' -as\<rightarrow>* n'; P n'' as n'; valid_edge a; sourcenode a = n; targetnode a = n''\<rbrakk> \<Longrightarrow> P n (a # as) n'" shows "P x1 x2 x3"
InvariantConflictClauseCharacterizationAfterApplyLearn: fixes state :: "State" assumes "getConflictFlag state"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)" shows "let state' = applyLearn state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
le_imp_inverse_le_neg: 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 a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq a b"   and "less b zero" shows "less_eq (inverse b) (inverse a)"
inner_prod_eq: fixes i :: "'a" shows "(\<lambda>v. v \<bullet> axis i 1) = (\<lambda>v. v $ i)"
indeps_8: fixes x :: "int \<Longrightarrow> 'a"   and y :: "int \<Longrightarrow> 'a"   and a :: "char list \<Longrightarrow> 'a"   and z :: "char list \<Longrightarrow> 'a"   and v :: "char list \<Longrightarrow> 'a" assumes "myss3 x y a z v" shows "z \<bowtie> v"
sum_upto_inverse_bound_2: shows "\<forall>\<^sub>F x in at_top. sum_upto (\<lambda>i. 1 / real i) x \<le> ln x + 13 / 22"
arrow_of_spans_axioms: shows "arrow_of_spans (\<cdot>) (\<nu> \<star> \<mu>)"
add_BH: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and r' :: "C" assumes "map_of \<Gamma> x = Some e"   and "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>\<Gamma>\<rbrace>\<^esub>)\<cdot>r' \<noteq> \<bottom>" shows "(\<N>\<lbrakk> e \<rbrakk>\<^bsub>\<N>\<lbrace>delete x \<Gamma>\<rbrace>\<^esub>)\<cdot>r' \<noteq> \<bottom>"
mem_in_image3: fixes b :: "'e"   and f :: "'f \<Rightarrow> 'e"   and A :: "'f set" assumes "b \<in> f ` A" shows "\<exists>a\<in>A. b = f a"
C_def: shows "C = Beta (- a - 1 / 2) (1 / 2)"
evalD_evalDdb: fixes e :: "expr"   and \<rho> :: "nat \<rightarrow> ValD" assumes "free e = []" shows "\<lbrakk>e\<rbrakk>\<rho> = evalDdb (transdb e [])\<cdot>\<rho>"
monotonic_fun_mult1: fixes r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'c"   and E1 :: "'c multiset"   and E2 :: "'c multiset" assumes "\<And>t s. (t, s) \<in> r \<Longrightarrow> (f t, f s) \<in> r"   and "(E1, E2) \<in> mult1 r" shows "(image_mset f E1, image_mset f E2) \<in> mult1 r"
test_eq1: fixes p :: "'a"   and y :: "'a"   and x :: "'a" assumes "test p" shows "(y \<le> x) = (p \<cdot> y \<le> x \<and> !p \<cdot> y \<le> x)"
Abs_resumption_inject: fixes x :: "('f + 'g + 'h, 'i) Old_Datatype.node set"   and y :: "('f + 'g + 'h, 'i) Old_Datatype.node set" assumes "x \<in> resumption"   and "y \<in> resumption" shows "(Abs_resumption x = Abs_resumption y) = (x = y)"
tortoise_hare_def: fixes f :: "'b \<Rightarrow> 'b"   and x0 :: "'b" shows "fx0.tortoise_hare f x0 \<equiv> fx0.find_nu f x0 ;; fx0.find_mu f x0 ;; fx0.find_lambda f"
D: shows "ipurge_tr_rev_aux I D U (xs @ zs) = ipurge_tr_rev_aux I D U (xs @ ys @ ipurge_tr_aux I D (D ` set ys) zs)"
size_23: fixes x6 :: "formula" shows "size (Neg x6) = size x6 + Suc 0"
inf_sup_ord_1: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.lattice inf less_eq less sup" shows "less_eq (inf x y) x"
ovr_assoc: fixes F :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "overrider F" shows "F x (F y z) = F (F x y) z"
span_finite: fixes S :: "'c set" assumes "finite S" shows "cspan S = range (\<lambda>u. \<Sum>v\<in>S. u v *\<^sub>C v)"
conc_def: fixes w :: "'c list"   and x :: "nat \<Rightarrow> 'c" shows "w \<frown> x \<equiv> \<lambda>n. if n < length w then w ! n else x (n - length w)"
Borsukian_def: fixes S :: "'b set" shows "Borsukian S \<equiv> \<forall>f. continuous_on S f \<and> f ` S \<subseteq> - {0} \<longrightarrow> (\<exists>a. homotopic_with_canon (\<lambda>h. True) S (- {0}) f (\<lambda>x. a))"
arr_char: fixes f :: "'a \<times> 'a" shows "VV.arr f = (arr (fst f) \<and> arr (snd f) \<and> src (fst f) = trg (snd f))"
divide_eq_0_iff: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = zero) = (a = zero \<or> b = zero)"
If2_def: fixes Q :: "tr"   and x :: "'b"   and y :: "'b" shows "If2 Q x y = If Q then x else y"
filter_union_mset: fixes P :: "'c \<Rightarrow> bool"   and M :: "'c multiset"   and N :: "'c multiset" shows "filter_mset P (M + N) = filter_mset P M + filter_mset P N"
compose_eq: fixes B :: "wall"   and D :: "wall"   and A :: "wall"   and C :: "wall" assumes "B ~ D \<and> A ~ C \<and> is_tangle_diagram A \<and> is_tangle_diagram B \<and> is_tangle_diagram C \<and> is_tangle_diagram D \<and> domain_wall B = codomain_wall A \<and> domain_wall D = codomain_wall C" shows "A \<circ> B ~ C \<circ> D"
ideD_3: fixes a :: "'a" assumes "A.ide a" shows "A.cod a = a"
sample_primes_excl_def: fixes P :: "nat set" shows "sample_primes_excl P = spmf_of_set (prime_set - P)"
Acc: fixes v :: "Term.var"   and Env :: "env"   and B :: "lname set"   and A :: "assigned" assumes "\<forall>vn. v \<noteq> LVar vn"   and "Env\<turnstile> B \<guillemotright>\<langle>v\<rangle>\<guillemotright> A" shows "Env\<turnstile> B \<guillemotright>\<langle>Acc v\<rangle>\<guillemotright> A"
one_fds_def: shows "1 = fds (\<lambda>n. if n = 1 then 1::'a else (0::'a))"
s'_def: fixes t :: "real" shows "ego2.s' t = (if t \<le> ego2.t_stop then ego2.p' t else 0)"
Chinese_remTr2: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set"   and B :: "nat \<Rightarrow> 'a set Ring"   and S :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set" shows "(\<forall>k\<le>Suc n. ideal R (J k)) \<and> (\<forall>k\<le>Suc n. B k = R /\<^sub>r J k) \<and> (\<forall>k\<le>Suc n. S k = pj R (J k)) \<and> (\<forall>i\<le>Suc n. \<forall>j\<le>Suc n. i \<noteq> j \<longrightarrow> coprime_ideals R (J i) (J j)) \<longrightarrow> surjec\<^bsub>R,(r\<Pi>\<^bsub>{j. j \<le> Suc n}\<^esub> B)\<^esub> A_to_prodag R {j. j \<le> Suc n} S B"
right_neq: fixes a :: "real"   and b :: "real"   and m :: "real"   and x :: "real" assumes "rightcut a b m < x"   and "x \<le> b"   and "a \<le> m"   and "m \<le> b"   and "{a..b} \<subseteq> {0..1}" shows "f x \<noteq> f m"
mbisim_deadlocked_preserve2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t" assumes "s1 \<approx>m s2"   and "t \<in> r2.deadlocked s2" shows "t \<in> r1.deadlocked s1"
left_unit: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c" assumes "Group_Theory.monoid M composition unit"   and "a \<in> M" shows "composition unit a = a"
flatten_simps_4: fixes c\<^sub>1 :: "('d, 'e, 'f) com"   and c\<^sub>2 :: "('d, 'e, 'f) com" shows "flatten (Seq c\<^sub>1 c\<^sub>2) = flatten c\<^sub>1 @ flatten c\<^sub>2"
strategy_proofI: fixes ds :: "'d set"   and \<phi> :: "('d \<Rightarrow> ('x \<times> 'x) set) \<Rightarrow> ('h \<Rightarrow> 'x set \<Rightarrow> 'x set) \<Rightarrow> 'd set \<Rightarrow> 'x set" assumes "\<And>Pd Pd' Ch d y. \<lbrakk>ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh Pd Ch; ContractsWithUnilateralSubstitutesAndIRCAndLAD Xd Xh (Pd(d := Pd')) Ch; d \<in> ds; y \<in> \<phi> (Pd(d := Pd')) Ch ds; y \<in> Field (Pd d); \<forall>x\<in>ManiPref.dX (\<phi> Pd Ch ds) d. x \<noteq> y \<and> (x, y) \<in> Pd d\<rbrakk> \<Longrightarrow> False" shows "ManiPref.strategy_proof ds \<phi>"
aencrypt_def: fixes \<alpha> :: "'a"   and msg :: "'a" shows "aencrypt \<alpha> msg = sample_uniform (order \<G>) \<bind> (\<lambda>y. return_spmf (\<^bold>g [^] y, \<alpha> [^] y \<otimes> msg))"
assms_1: shows "path \<sigma> k = path \<sigma>' k'"
\<epsilon>_map_simp: fixes x :: "'c" assumes "C.ide x" shows "transformation_by_components.map (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) \<eta>\<epsilon>.FG.map \<phi>\<psi>.\<epsilon>o x = \<psi> x (G x)"
circline_set_circline_inversion: fixes H :: "circline" shows "circline_set (circline_inversion H) = inversion ` circline_set H"
map_upd_map_add: fixes X :: "'d \<Rightarrow> 'e option"   and V :: "'d"   and v :: "'e" shows "X(V \<mapsto> v) = X ++ [V \<mapsto> v]"
subst_wellformed: fixes n :: "nat"   and t :: "Term_Class.term"   and env :: "(name, Term_Class.term) fmap" assumes "wellformed' n t"   and "fmpred (\<lambda>_. Term.wellformed) env" shows "wellformed' n (subst t env)"
sum_multicount_gen: fixes s :: "'a set"   and t :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "'b \<Rightarrow> nat" assumes "finite s"   and "finite t"   and "\<forall>j\<in>t. card {i \<in> s. R i j} = k j" shows "(\<Sum>i\<in>s. card {j \<in> t. R i j}) = sum k t"
case_eq_if: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and prod :: "'d \<times> 'e" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
intro: fixes p :: "nat" assumes "prime p" shows "residues_prime p"
R: fixes u :: "'a set"   and v :: "'a set" assumes "u \<in> nhds x"   and "v \<in> nhds y"   and "y \<notin> u"   and "x \<notin> v" shows "R"
diagonal_to_Smith_PQ_JNF_canceled_first: fixes n :: "nat"   and m :: "nat"   and A :: "'c mat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and P :: "'c mat"   and S :: "'c mat"   and Q :: "'c mat" assumes "\<exists>Rep Abs. type_definition Rep Abs {0..<int n}"   and "{0..<int m} \<noteq> {}"   and "1 < m"   and "1 < n"   and "isDiagonal_mat A"   and "is_bezout_ext bezout"   and "A \<in> carrier_mat m n"   and "(P, S, Q) = diagonal_to_Smith_PQ_JNF A bezout" shows "S = P * A * Q \<and> invertible_mat P \<and> invertible_mat Q \<and> Smith_normal_form_mat S \<and> P \<in> carrier_mat m m \<and> S \<in> carrier_mat m n \<and> Q \<in> carrier_mat n n"
is_word_cons: fixes x :: "symbol"   and xs :: "symbol list" shows "is_word (x # xs) = (is_terminal x \<and> is_word xs)"
Diff_Int_distrib: fixes C :: "'c set"   and A :: "'c set"   and B :: "'c set" shows "C \<inter> (A - B) = C \<inter> A - C \<inter> B"
hm_delete_impl: shows "imp_map_delete is_hashmap hm_delete"
Arg_exp_diff_2pi: fixes z :: "complex" shows "\<exists>k. Im z - real_of_int k * (2 * pi) = Arg (exp z)"
1155: fixes x :: "'a"   and y :: "'a" shows "- - x \<sqinter> - (- x \<sqinter> y) = - - x"
wf: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.wellorder less_eq less" shows "wf {(x, y). less x y}"
twoposs: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "set xs = {x, y}"   and "x \<noteq> y"   and "distinct xs" shows "xs \<in> {[x, y], [y, x]}"
dvd_mult_left: fixes times :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and one :: "'m"   and a :: "'m"   and b :: "'m"   and c :: "'m" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times (times a b) c" shows "dvd.dvd times a c"
bdd: fixes x :: "'e" assumes "x \<in> space M" shows "bdd_above ((\<lambda>i. F i x) ` I)"
m: assumes "n \<noteq> 0" shows "eval m [f \<triangleright> n] \<down>= amalg01 f"
fls_base_factor_to_fps_shift: fixes m :: "int"   and f :: "'a fls" shows "fls_base_factor_to_fps (fls_shift m f) = fls_base_factor_to_fps f"
cone_decomp_def: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" shows "cone_decomp T ps = direct_decomp T (map cone ps)"
target_in_states: fixes io :: "('d \<times> 'e) list"   and tr :: "'f list"   and q :: "'f" assumes "length io = length tr"   and "0 < length io" shows "last (states (io || tr) q) = target (io || tr) q"
conv_radius_ratio_limit_ereal_nonzero: fixes c :: "ereal"   and f :: "nat \<Rightarrow> 'b" assumes "c \<noteq> 0"   and "(\<lambda>n. ereal (norm (f n) / norm (f (Suc n)))) \<longlonglongrightarrow> c" shows "conv_radius f = c"
SCallParams: fixes es :: "char list exp list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and es' :: "char list exp list"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list" assumes "P \<turnstile> \<langle>es,s,b\<rangle> [\<rightarrow>] \<langle>es',s',b'\<rangle>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(es),s,b\<rangle> \<rightarrow> \<langle>C\<bullet>\<^sub>sM(es'),s',b'\<rangle>"
null_sets_restr_to_subalg: fixes M :: "'b measure"   and F :: "'b measure" assumes "subalgebra M F" shows "null_sets (restr_to_subalg M F) = null_sets M \<inter> sets F"
cases: fixes a1 :: "'c"   and a2 :: "'c"   and U :: "'c set"   and r :: "('c \<times> 'c) set" assumes "(a1, a2) \<in> on U r\<^sup>+" obtains a :: "'c" and b :: "'c" where "a1 = a__" and "a2 = b__" and "a__ \<in> U" and "b__ \<in> U" and "(a__, b__) \<in> r"   | a :: "'c" and b :: "'c" and c :: "'c" where "a1 = a" and "a2 = c" and "a \<in> U" and "b \<in> U" and "c \<in> U" and "(a, b) \<in> on U r\<^sup>+" and "(b, c) \<in> r"
borel_measurable_count_space: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" shows "f \<in> borel_measurable (count_space S)"
elementary_subset_cbox: fixes p :: "'a set set"   and S :: "'a set" assumes "p division_of S" shows "\<exists>a b. S \<subseteq> cbox a b"
iterate_applied: fixes n :: "nat"   and m :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "iterate n m f (f x) = iterate (Suc n) (Suc m) f x"
sum_SDG_ddep_edge: fixes n :: "'node SDG_node"   and V :: "'var"   and n' :: "'node SDG_node" assumes "n influences V in n'" shows "n s-V\<rightarrow>\<^sub>d\<^sub>d n'"
fcompose_add_less: fixes p2 :: "'b poly"   and p1 :: "'b poly"   and q1 :: "'b poly"   and q2 :: "'b poly" assumes "degree p2 < degree p1" shows "fcompose (p1 + p2) q1 q2 = fcompose p1 q1 q2 + q2 ^ (degree p1 - degree p2) * fcompose p2 q1 q2"
finite_dimensional_vector_space_pair_1_with_def: fixes plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and basis\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" shows "finite_dimensional_vector_space_pair_1_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> finite_dimensional_vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 basis\<^sub>1 \<and> vector_space_with plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
dim_gen_eigenspace_order_char_poly: fixes A :: "'b mat"   and n_as :: "(nat \<times> 'b) list"   and lam :: "'b"   and k :: "nat" assumes "jordan_nf A n_as" shows "dim_gen_eigenspace A lam k \<le> order lam (char_poly A)"
set_insort_key: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and xs :: "'c list" shows "set (insort_key f x xs) = insert x (set xs)"
witness_list: fixes p :: "('b, 'c) fm"   and q :: "('b, 'c) fm"   and ps :: "('b, 'c) fm list"   and used :: "'c set" assumes "proper_dia p = Some q" shows "witness_list (p # ps) used = (let i = SOME i. i \<notin> used in (\<^bold>@ i q) # (\<^bold>\<diamond> Nom i) # witness_list ps ({i} \<union> used))"
gtrI: fixes s :: "'a"   and c :: "'a multiset"   and w :: "'b list"   and s' :: "'a"   and c' :: "'a multiset"   and gtrs :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((s, c), w, s', c') \<in> trcl gtrs" shows "(add_mset s c, w, add_mset s' c') \<in> trcl (gtr gtrs)"
plus_aezfun_apply: fixes a :: "('a, 'b) aezfun"   and b :: "('a, 'b) aezfun"   and x :: "'b" shows "aezfun (a + b) x = aezfun a x + aezfun b x"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "nat"   and x4 :: "('a, 'b) BinomialHeapStruc.BinomialTree list"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "nat"   and y4 :: "('a, 'b) BinomialHeapStruc.BinomialTree list" shows "(BinomialHeapStruc.Node x1 x2 x3 x4 = BinomialHeapStruc.Node y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
approx_inverse: fixes x :: "'a star"   and y :: "'a star" assumes "x \<approx> y"   and "y \<in> HFinite - Infinitesimal" shows "inverse x \<approx> inverse y"
get_ancestors_ptr: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and ancestors :: "(_) object_ptr list" assumes "h \<turnstile> Shadow_DOM.get_ancestors ptr \<rightarrow>\<^sub>r ancestors" shows "ptr \<in> set ancestors"
mset_subset_eq_single: fixes a :: "'a"   and B :: "'a multiset" assumes "a \<in># B" shows "{#a#} \<subseteq># B"
valid: shows "\<Gamma>\<Turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
leg0_in_hom_2: shows "\<guillemotleft>m : m \<Rightarrow> m\<guillemotright>"
is_subset_2: fixes c :: "'a" assumes "c \<in> \<delta>_states ta'.\<delta>" shows "c \<in> ta'.Q"
properfactor_fcount: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G"   and "properfactor G a b" shows "factorcount G a < factorcount G b"
simps_5: fixes a\<^sub>1 :: "aexp"   and a\<^sub>2 :: "aexp"   and s :: "char list \<Rightarrow> int option" shows "paval (Div a\<^sub>1 a\<^sub>2) s = paval a\<^sub>1 s div paval a\<^sub>2 s"
left_residual_bl: fixes x :: "'a"   and a :: "'a"   and b :: "'a" shows "(x * a \<le> b) = (x \<le> a l\<rightarrow> b)"
ldrop_simps_1: fixes n :: "enat" shows "ldrop n LNil = LNil"
simps_1: fixes I :: "\<I>"   and mr :: "mregex"   and nt :: "nat" shows "eval_matchF I mr nt [] = ([], [])"
invertible_right_inverse: fixes u :: "'a" assumes "f.source.additive.invertible u"   and "u \<in> A" shows "f.source.subtraction u u = zeroA"
stable_constrains_Int: fixes F :: "'c program"   and C :: "'c set"   and A :: "'c set"   and A' :: "'c set" assumes "F \<in> stable C"   and "F \<in> C \<inter> A co A'" shows "F \<in> C \<inter> A co C \<inter> A'"
streams_def: shows "streams \<equiv> \<lambda>A. {x. streamsp (\<lambda>x. x \<in> A) x}"
sup_state_refl: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and s :: "ty list \<times> ty err list" shows "P \<turnstile> s \<le>\<^sub>i s"
lincomb_def: fixes a :: "'a vec \<Rightarrow> 'a"   and A :: "'a vec set" shows "NC.lincomb a A = (\<Oplus>\<^bsub>NC.V\<^esub>v\<in>A. a v \<cdot>\<^sub>v v)"
poly_condI: fixes p :: "int poly" assumes "0 < lead_coeff p"   and "irreducible p" shows "poly_cond p"
nres_more_simps_4: fixes X :: "'b set"   and Y :: "'b set" shows "(RES X = RES Y) = (X = Y)"
simps_2: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and ys :: "'a list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (x # ys) = (Ball (set ys) (less_eq x) \<and> linorder.sorted less_eq ys)"
newman: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "wfP R\<inverse>\<inverse>"   and "\<And>a b c. \<lbrakk>R a b; R a c\<rbrakk> \<Longrightarrow> \<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"   and "R\<^sup>*\<^sup>* a b"   and "R\<^sup>*\<^sup>* a c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
elims_1: fixes x :: "label"   and xa :: "label"   and y :: "bool" assumes "(x \<sqsubset>l xa) = y" obtains l :: "label" where "x = Old" and "xa = l__" and "y = (l__ \<noteq> Old)"   | l :: "label" where "x = Processed" and "xa = la__" and "y = (la__ = New)"   | l :: "label" where "x = New" and "xa = l" and "\<not> y"
gext_cong1: fixes l :: "lname \<Rightarrow> val option"   and s1 :: "st"   and s2 :: "st" shows "set_locals l s1\<le>|s2 = s1\<le>|s2"
seq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c\<^sub>1' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" assumes "\<langle>c\<^sub>1, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1', mds', mem'\<rangle>" shows "\<langle>c\<^sub>1 ;; c\<^sub>2, mds, mem\<rangle> \<leadsto> \<langle>c\<^sub>1' ;; c\<^sub>2, mds', mem'\<rangle>"