hcju/isabelleps huggingface dataset:
summable_of_real: fixes X :: "nat \<Rightarrow> real" assumes "summable X" shows "summable (\<lambda>n. of_real (X n))"
H_2: fixes q :: "'a" assumes "q \<in> H" shows "infdist q G \<le> K"
in_homI: fixes f :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr" assumes "S.arr f"   and "S.dom f = a"   and "S.cod f = b" shows "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"
IH: fixes P :: "indi \<Rightarrow> alt \<Rightarrow> real" assumes "D = {i. b \<cdot>< P i}"   and "Extreme P b"   and "b \<cdot>< F P" shows "\<exists>i. pivotal i b"
member_le_sum: fixes i :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "i \<in> A"   and "\<And>x. x \<in> A - {i} \<Longrightarrow> (0::'b) \<le> f x"   and "finite A" shows "f i \<le> sum f A"
calcChainAtom: fixes xvec :: "name list"   and yvec :: "name list"   and x :: "name" assumes "length xvec = length yvec"   and "x \<sharp> xvec"   and "x \<sharp> yvec" shows "[xvec yvec] \<bullet>\<^sub>v x = x"
cases: fixes a :: "('Q set, 'L) ta_rule" assumes "a \<in> \<delta>ss" obtains f :: "'L" and ss :: "'Q set list" and s :: "'Q set" where "a = s \<rightarrow> f ss" and "A f = Some (length ss)" and "listsp (\<lambda>s. s \<subseteq> Q) ss" and "s = \<delta>ss_lhs f ss"
norm_nn_in_cf: fixes n :: "'a itrm"   and n' :: "'a itrm" assumes "n \<in> CF"   and "n' \<in> CF" shows "norm_nn n n' \<in> CF"
wand_assnI: fixes h :: "heap"   and as :: "nat set"   and Q :: "assn"   and R :: "assn" assumes "in_range (h, as)"   and "\<And>h' as'. \<lbrakk>as \<inter> as' = {}; relH as h h'; in_range (h', as); (h', as') \<Turnstile> Q\<rbrakk> \<Longrightarrow> (h', as \<union> as') \<Turnstile> R" shows "(h, as) \<Turnstile> Q -* R"
cmp_rfn0: fixes G :: "('e, 'f) Group_scheme"   and r :: "nat"   and s :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and g :: "nat \<Rightarrow> 'e set"   and i :: "nat"   and j :: "nat" assumes "Group G"   and "0 < r"   and "0 < s"   and "compseries G r f"   and "compseries G s g"   and "i \<le> r - 1"   and "j \<le> s - 1" shows "G \<guillemotright> f (Suc i) \<diamondop>\<^bsub>G\<^esub> f i \<inter> g j "
SD_iff_expected_utilities_le: fixes carrier :: "'a set"   and le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a pmf"   and q :: "'a pmf" assumes "finite_total_preorder_on carrier le"   and "p \<in> lotteries_on carrier"   and "q \<in> lotteries_on carrier" shows "SD le p q = (\<forall>u. vnm_utility carrier le u \<longrightarrow> measure_pmf.expectation p u \<le> measure_pmf.expectation q u)"
assms_4: shows "subst_domain \<I> \<inter> X = {}"
inducts: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> parts H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>K X. \<lbrakk>Crypt K X \<in> parts H; P (Crypt K X)\<rbrakk> \<Longrightarrow> P X" shows "P x"
esqrt_mult: fixes a :: "ereal"   and b :: "ereal" assumes "0 \<le> a" shows "esqrt (a * b) = esqrt a * esqrt b"
assms_9: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "x \<notin> domA \<Gamma>"   and "isVal e"   and "I (\<Gamma>, e, Upd x # S)" shows "P (\<Gamma>, e, Upd x # S) ((x, e) # \<Gamma>, e, S)"
error_bind: fixes e :: "'a"   and f :: "'c \<Rightarrow> 'a + 'b" shows "Inl e \<bind> f = Inl e"
supp_rel_tAct: fixes t :: "('e, 'f, 'g) Tree"   and \<alpha> :: "'g" assumes "finite (supp_rel (=\<^sub>\<alpha>) t)" shows "supp_rel (=\<^sub>\<alpha>) (tAct \<alpha> t) = supp \<alpha> \<union> supp_rel (=\<^sub>\<alpha>) t - bn \<alpha>"
assms_4: fixes u :: "'a"   and a :: "'a"   and v :: "'a"   and b :: "'a" assumes "fst (rep_nat_term u) = fst (rep_nat_term a)"   and "fst (rep_nat_term v) = fst (rep_nat_term b)"   and "snd (rep_nat_term u) = snd (rep_nat_term v)"   and "snd (rep_nat_term a) = snd (rep_nat_term b)"   and "cmp a b = Lt" shows "cmp u v = Lt"
callee_invariant_extend_state_oracle_const': fixes "oracle" :: "'b \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'b) spmf"   and I :: "'a \<Rightarrow> bool" shows "callee_invariant \<dagger>oracle (\<lambda>s. I (fst s))"
cov_invertible_nonneg_eq: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and T :: "(real, 'a) vec set"   and h :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and h' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> real"   and b :: "real" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S)"   and "\<And>y. y \<in> T \<Longrightarrow> (h has_derivative h' y) (at y within T)"   and "\<And>y. y \<in> T \<Longrightarrow> 0 \<le> f y"   and "\<And>x. x \<in> S \<Longrightarrow> g x \<in> T \<and> h (g x) = x"   and "\<And>y. y \<in> T \<Longrightarrow> h y \<in> S \<and> g (h y) = y"   and "\<And>y. y \<in> T \<Longrightarrow> h' y \<circ> g' (h y) = id" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> * f (g x)) has_integral b) S = (f has_integral b) T"
eq_sym: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "\<phi> \<sim>\<^sub>Q \<psi>" shows "\<psi> \<sim>\<^sub>Q \<phi>"
not_essentialD: fixes G :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and x :: "'a"   and p :: "'a list"   and y :: "'a" assumes "\<not> essential G B S x"   and "path G x p y"   and "y \<in> B" shows "x \<noteq> y \<and> (\<exists>z\<in>set p. z \<noteq> x \<and> z \<in> S)"
list_all_iff: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e list" shows "list_all P x = Ball (set x) P"
elims: fixes x :: "'b trm"   and y :: "Preliminaries.var set" assumes "nv2T x = y" obtains x :: "Preliminaries.var" where "xa__ = Var x" and "y = {x}"   | f :: "'b" and Tl :: "'b trm list" where "xa__ = Fn f Tl" and "y = {}"
Dom_dom: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Dom (local.dom f) = Dom f"
simps_1: fixes i :: "nat" shows "Koszul_syz_sigs_aux [] i = []"
assms_2: fixes r :: "'a" assumes "r2 = Some r" shows "r \<noteq> i1"
simps_5: fixes v :: "'b rexp"   and va :: "'b rexp" shows "toplevel_summands (Times v va) = {Times v va}"
cor_squared: fixes x :: "real" shows "(cor x)\<^sup>2 = cor (x\<^sup>2)"
shift_zero: fixes r :: "real_int" shows " shift r 0 = r"
has_parametrized_subprob_densityI: fixes M :: "'c measure"   and N :: "'c \<Rightarrow> 'd measure"   and R :: "'d measure"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> N x = density R (f x)"   and "\<And>x. x \<in> space M \<Longrightarrow> subprob_space (N x)"   and "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M R)" shows "has_parametrized_subprob_density M N R f"
in_combination_in_component_2: fixes r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option" assumes "r \<in> RID\<^sub>S (\<sigma>;;\<tau>)"   and "r \<notin> RID\<^sub>S \<tau>" shows "r \<in> RID\<^sub>S \<sigma>"
irrefl: shows "\<forall>x\<in>A. \<not> gt x x"
simps_4: fixes r :: "'a ifex list" shows "lowest_tops (Falseif # r) = lowest_tops r"
NUM_subseteq_TOTAL: shows "NUM \<subseteq> TOTAL"
logical_instr_sub1_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and instr_name :: "sparc_operation"   and result :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = snd (fst (logical_instr_sub1 instr_name result s1)) \<and> t2 = snd (fst (logical_instr_sub1 instr_name result s2))" shows "low_equal t1 t2"
m_rcancel: fixes a :: "'a list"   and b :: "'a list"   and c :: "'a list" assumes "a \<noteq> \<zero>\<^bsub>poly_ring R\<^esub>"   and "a \<in> carrier (poly_ring R)"   and "b \<in> carrier (poly_ring R)"   and "c \<in> carrier (poly_ring R)" shows "(b \<otimes>\<^bsub>poly_ring R\<^esub> a = c \<otimes>\<^bsub>poly_ring R\<^esub> a) = (b = c)"
OrE: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<or> G"   and "F \<triangleright> \<Gamma> \<turnstile> H"   and "G \<triangleright> \<Gamma> \<turnstile> H" shows "\<Gamma> \<turnstile> H"
measure_spmf_def: fixes p :: "'c spmf" shows "measure_spmf p = distr (restrict_space (measure_pmf p) (range Some)) (count_space UNIV) the"
asIMOD_igFreshIGVar: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igFreshIGVar (asIMOD SEM)"
the_elem_id: fixes S :: "'a set" assumes "finite S" shows "the_elem' S = the_elem S"
eq: shows "exi k' ((k' \<leftrightarrow> k) \<bullet> A) = exi k A"
characteriseLast: fixes C :: "'b sequent"   and m :: "nat"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "(C, m + 1) \<in> derivable R" shows "\<exists>Ps. Ps \<noteq> [] \<and> (Ps, C) \<in> R \<and> (\<forall>p\<in>set Ps. \<exists>n\<le>m. (p, n) \<in> derivable R)"
cases: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'c pmf"   and B :: "('d, 'd cset Monomorphic_Monad.id) nondetT" assumes "cr_prob_ndi A p B" obtains "rel_set A (set_pmf p) (rcset (extract (run_nondet B)))"
prems_4: shows "n \<notin> fst (pn_atoms G)"
landau_product_preprocess_33: shows "neg_numeral_class.dbl (1::'a) = (2::'a)"
polymul: fixes bs :: "'b list"   and p :: "'b poly"   and q :: "'b poly" shows "Ipoly bs (p *\<^sub>p q) = Ipoly bs p * Ipoly bs q"
essentialE_RF: fixes \<Gamma> :: "('a, 'b) graph_scheme"   and B :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "essential \<Gamma> B S x" obtains p :: "'a list" and y :: "'a" where "path \<Gamma> x p y" and "y \<in> B" and "distinct (x # p)" and "\<And>z. z \<in> set p \<Longrightarrow> z \<notin> roofed_gen \<Gamma> B S"
pelims: fixes x :: "('c, 'd) lit"   and y :: "Preliminaries.var set" assumes "varsL x = y"   and "varsL_dom x" obtains at :: "('c, 'd) atm" where "x = Pos at__" and "y = varsA at__" and "varsL_dom (Pos at__)"   | at :: "('c, 'd) atm" where "x = Neg at" and "y = varsA at" and "varsL_dom (Neg at)"
hyps: shows "(\<lambda>m. x (m + n)) \<longlonglongrightarrow> y = x \<longlonglongrightarrow> y"
mk_safe_def: fixes X :: "'c set" shows "mk_safe X = ASSERT (wd TYPE('c)) \<bind> (\<lambda>_. safe_set (X ::: appr_rel) \<bind> (\<lambda>s. if s then RETURN (X ::: appr_rel) else SUCCEED))"
rep_eq: fixes x :: "'e \<Rightarrow> 'd"   and xa :: "('e, 'c) lgenerator'" shows "lgenerator (lmap_trans' x xa) = lmap_trans x (lgenerator xa)"
foundation25: fixes \<tau> :: "'b state \<times> 'b state"   and P :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Q :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" assumes "\<tau> \<Turnstile> P" shows "\<tau> \<Turnstile> P or Q"
iMODb_inext_nth: fixes a :: "nat"   and c :: "nat"   and r :: "nat"   and m :: "nat" assumes "a \<le> c" shows "[ r, mod m, c ] \<rightarrow> a = r + m * a"
prems_1: shows "obtains 1 c1 drinks2 0 <> (a # t)"
qItem_versus_item_simps_1: fixes f :: "('c, 'd, 'varSort, 'var, 'e) qTerm \<Rightarrow> 'f"   and Xa :: "('c, 'd, 'varSort, 'var, 'e) qTerm" assumes "f respectsP alphaGood"   and "qGood Xa" shows "Equiv_Relation2.univ f (asTerm Xa) = f Xa"
cint_fw: shows "(\<lambda>w. f w / (w - z)) contour_integrable_on \<gamma>"
r_minus: fixes x :: "'c \<Rightarrow> 'a"   and y :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F"   and "y \<in> carrier F" shows "x \<otimes>\<^bsub>F\<^esub> \<ominus>\<^bsub>F\<^esub> y = \<ominus>\<^bsub>F\<^esub> (x \<otimes>\<^bsub>F\<^esub> y)"
sin_times_sin: fixes w :: "'a"   and z :: "'a" shows "sin w * sin z = (cos (w - z) - cos (w + z)) / (2::'a)"
lm09: fixes X :: "'b set" assumes "trivial X" shows "Pow X \<subseteq> {{}, X}"
nn_transfer_operator_zero: shows "AE x in M. nn_transfer_operator (\<lambda>x. 0) x = 0"
simps_117: fixes x111 :: "('d, 'e, 'f) com"   and x112 :: "('d, 'e, 'f) com"   and x91 :: "'f"   and x92 :: "'d set"   and x93 :: "('d, 'e, 'f) com" shows "Catch x111 x112 \<noteq> Guard x91 x92 x93"
less_pair_def: fixes a :: "'a \<times> 'b"   and b :: "'a \<times> 'b" shows "(a < b) = (fst a < fst b \<or> fst a = fst b \<and> snd a < snd b)"
quicksort_poly_def: fixes x :: "nat"   and y :: "nat"   and z :: "(char list list \<times> int) list" shows "quicksort_poly x y z = quicksort_ref (\<le>) fst (x, y, z)"
simps_1: fixes p1 :: "('a, 'b, 'c, 'd) seqp"   and p2 :: "('a, 'b, 'c, 'd) seqp" shows "stermsl (p1 \<oplus> p2) = stermsl p1 \<union> stermsl p2"
usup_and: fixes P :: "'a upred"   and Q :: "'a upred" shows "P \<squnion> Q = (P \<and> Q)"
mlex_eq: fixes f :: "'b \<Rightarrow> nat"   and R :: "('b \<times> 'b) set" shows "f <*mlex*> R = {(x, y). f x < f y \<or> f x \<le> f y \<and> (x, y) \<in> R}"
renamings_apart_inj: fixes Cs :: "('a, nat) Term.term literal multiset list" shows "\<forall>\<sigma>\<in>set (renamings_apart Cs). inj \<sigma>"
desargues_config_2D_rkA'B'PQ_rkA'C'PQ_rkB'C'PQ_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}" shows "rk {B', C', P, Q} = 4"
ivl_subset_existence_ivl: fixes t :: "real"   and t0 :: "real"   and x0 :: "'a" assumes "t - t0 \<in> Y.existence_ivl0 x0" shows "{t0..t} \<subseteq> (+) t0 ` Y.existence_ivl0 x0"
igSwapInpIPresIGWlsInpSTR_def: fixes MOD :: "('i, 'j, 'k, 'sort, 'l, 'm, 'n, 'o, 'p) model_scheme" shows "igSwapInpIPresIGWlsInpSTR MOD \<equiv> \<forall>zs z1 z2 delta inp. igWlsInp MOD delta (igSwapInp MOD zs z1 z2 inp) = igWlsInp MOD delta inp"
mono: fixes x :: "'a"   and y :: "'a" assumes "leq x y"   and "\<not> leq x bound" shows "ord (f x) (f y)"
distinct_2: fixes x2 :: "'b" shows "Some x2 \<noteq> None"
cnf_\<alpha>_def: fixes cnfi :: "(lit \<times> lit) list" shows "cnf_\<alpha> cnfi = uncurry Upair ` set cnfi"
cases: fixes a :: "'a llist" assumes "valid_path a" obtains "a = LNil"   | v :: "'a" where "a = LCons va__ LNil" and "va__ \<in> V"   | v :: "'a" and w :: "'a" and Ps :: "'a llist" where "a = LCons v Ps" and "v \<in> V" and "w \<in> V" and "v \<rightarrow> w" and "valid_path Ps" and "\<not> lnull Ps" and "lhd Ps = w"
IH_1: assumes "Fun f T \<sqsubseteq> t"   and "\<not> P f T" shows "\<exists>l\<le>n. \<langle>M; l\<rangle> \<turnstile>\<^sub>n Fun f T \<and> (\<forall>t\<in>set T. \<exists>j<l. \<langle>M; j\<rangle> \<turnstile>\<^sub>n t)"
of_int_code_3: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (Int.Pos k) = numeral.numeral one plus k"
dagger_prod_unfold: fixes x :: "'a"   and y :: "'a" shows "(1::'a) + x \<cdot> (y \<cdot> x)\<^sup>\<dagger> \<cdot> y = (x \<cdot> y)\<^sup>\<dagger>"
Abs_num1_induct: fixes P :: "1 \<Rightarrow> bool"   and x :: "1" assumes "\<And>y. y \<in> UNIV \<Longrightarrow> P (Abs_num1 y)" shows "P x"
assign_load_rel: fixes c\<^sub>A :: "(Eg1.var, aexp, bexp) Stmt"   and x :: "Eg1.var"   and y :: "Eg1.var"   and c\<^sub>A_tail :: "(Eg1.var, aexp, bexp) Stmt"   and c\<^sub>C :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and c\<^sub>C_tail :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds\<^sub>A :: "Mode \<Rightarrow> Eg1.var set"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and mem\<^sub>A :: "Eg1.var \<Rightarrow> nat"   and mem\<^sub>C :: "var\<^sub>C \<Rightarrow> nat" assumes "c\<^sub>A = (x \<leftarrow> aexp.Load y) ;; c\<^sub>A_tail"   and "c\<^sub>C = (Eg2_var\<^sub>C_of_Eg1 x \<leftarrow> aexp\<^sub>C.Load (Eg2_var\<^sub>C_of_Eg1 y)) ;; c\<^sub>C_tail"   and "mds\<^sub>A = mds\<^sub>A_of mds\<^sub>C"   and "mem\<^sub>A = mem\<^sub>A_of mem\<^sub>C"   and "\<forall>v\<^sub>C. v\<^sub>C \<notin> range Eg2_var\<^sub>C_of_Eg1 \<longrightarrow> v\<^sub>C \<in> mds\<^sub>C AsmNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> mds\<^sub>A GuarNoWrite"   and "y \<notin> mds\<^sub>A GuarNoReadOrWrite"   and "x \<notin> \<C>"   and "y \<notin> \<C>"   and "\<forall>mds\<^sub>A' mem\<^sub>A' mem\<^sub>A'' mds\<^sub>C'' mem\<^sub>C' mem\<^sub>C''. mds\<^sub>A' = mds\<^sub>A_of mds\<^sub>C'' \<and> mem\<^sub>A' = mem\<^sub>A_of mem\<^sub>C' \<and> mem\<^sub>A'' = mem\<^sub>A_of mem\<^sub>C'' \<and> eval_abv\<^sub>A \<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A''\<rangle>\<^sub>A \<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A \<and> eval_abv\<^sub>C \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C''\<rangle>\<^sub>C \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<langle>Stop ;; c\<^sub>A_tail, mds\<^sub>A', mem\<^sub>A'\<rangle>\<^sub>A, \<langle>Stop ;; c\<^sub>C_tail, mds\<^sub>C'', mem\<^sub>C'\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2" shows "(\<langle>c\<^sub>A, mds\<^sub>A, mem\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>C, mds\<^sub>C, mem\<^sub>C\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2"
sorted_sparse_matrix_def: fixes A :: "(nat \<times> (nat \<times> 'a) list) list" shows "sorted_sparse_matrix A = (sorted_spvec A \<and> sorted_spmat A)"
w_reach_3: shows "v \<rightarrow>\<^sup>*\<^bsub>H\<^esub> w"
final_Sk: shows "is_final_dstate ((deterministic_RP_step ^^ k) (deterministic_RP_step St))"
exhaust: fixes y :: "three" obtains "y = Zero"   | "y = One"   | "y = Two"
simps_2: fixes a :: "'b \<Rightarrow> 'b"   and n :: "nat" shows "fpower a (Suc n) = a \<circ> fpower a n"
\<tau>red1r_max_vars: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list" assumes "\<tau>red1gr uf P t h (e, xs) (e', xs')" shows "max_vars e' \<le> max_vars e"
tmps_distinct_axioms: shows "tmps_distinct ts\<^sub>s\<^sub>b"
possible_steps_0_invalid: fixes l :: "String.literal"   and i :: "value list" assumes "\<not> (l = STR ''select'' \<and> length i = 1)" shows "possible_steps drinks 0 <> l i = {||}"
snapshot_stable_ver_2: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and p :: "'a"   and j :: "nat" assumes "trace init t final"   and "ps (S t i) p \<noteq> None"   and "i \<le> j" shows "ps (S t j) p \<noteq> None"
compact_eq_seq_compact_metric: fixes S :: "'b set" shows "compact S = seq_compact S"
ev_imp_shift: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream" assumes "ev \<phi> xs" shows "\<exists>xl xs2. xs = xl @- xs2 \<and> \<phi> xs2"
M': fixes i :: "'i" assumes "i \<in> I" shows "sets (M' i) = sigma_sets (space (M' i)) (E i)"
that: fixes q1 :: "gauss_int"   and q2 :: "gauss_int" assumes "prime q1"   and "prime q2"   and "prime_factorization (of_nat p) = {#q1, q2#}"   and "gauss_int_norm q1 = p"   and "gauss_int_norm q2 = p"   and "q2 = \<i>\<^sub>\<int> * gauss_cnj q1"   and "0 < ReZ q1"   and "0 < ImZ q1"   and "0 < ReZ q1"   and "0 < ImZ q2" shows "thesis"
sym_in_hom: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<guillemotleft>\
[a, b] : a \<otimes> b \<rightarrow> b \<otimes> a\<guillemotright>"
edge_pres_faceAt: fixes g :: "graph"   and u :: "nat"   and f :: "face"   and v :: "nat" assumes "minGraphProps g"   and "u \<in> \<V> g"   and "f \<in> set (facesAt g u)"   and "(u, v) \<in> \<E> f" shows "f \<in> set (facesAt g v)"
FSwSbImorph_def: fixes h :: "'g \<Rightarrow> 'h"   and hA :: "'i \<Rightarrow> 'j"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'g, 'i, 'l) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'k, 'h, 'j, 'm) model_scheme" shows "FSwSbImorph h hA MOD MOD' \<equiv> FSwImorph h hA MOD MOD' \<and> ipresIGSubstAll h hA MOD MOD'"
h1b: shows "((\<Sum>i = 0..<deg + 1. f i / D ^ i) = 0) = ((\<Sum>i = 0..<deg + 1. f i * D ^ (deg - i)) = 0)"
mono: fixes less :: "'g \<Rightarrow> 'g \<Rightarrow> bool" shows "mono (\<lambda>p x1 x2. (\<exists>y ys. x1 = [] \<and> x2 = y # ys) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> less x y) \<or> (\<exists>x y xs ys. x1 = x # xs \<and> x2 = y # ys \<and> \<not> less x y \<and> \<not> less y x \<and> p xs ys))"
dvd_prod_eqI: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'c set"   and a :: "'c"   and b :: "'b"   and f :: "'c \<Rightarrow> 'b" assumes "class.comm_semiring_1 times one plus zero"   and "finite A"   and "a \<in> A"   and "b = f a" shows "dvd.dvd times b (comm_monoid_mult.prod times one f A)"
usubstr_length: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and R :: "fml list \<times> fml" assumes "usubstr \<sigma> R \<noteq> None" shows "length (fst (the (usubstr \<sigma> R))) = length (fst R)"
simps_1: fixes u :: "'b trm"   and v :: "'b" shows "u \<prec> Var v = False"
S: shows "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in net. f x \<in> S)"
merge_correct: assumes "distinct l1 \<and> sorted l1"   and "distinct l2 \<and> sorted l2" shows "distinct (merge l1 l2) \<and> sorted (merge l1 l2) \<and> set (merge l1 l2) = set l1 \<union> set l2"
le: fixes i :: "'n" shows "HMA_Connect.spectral_radius A * (\<chi>i. cmod (local.max_v $h i)) $h i \<le> (local.B *v (\<chi>i. cmod (local.max_v $h i))) $h i"
execl_drop: fixes P :: "instr list"   and cfs :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list"   and n :: "nat" assumes "P \<Turnstile> cfs" shows "P \<Turnstile> drop n cfs"
ring_mat_simps_3: fixes ty :: "'a itself"   and n :: "nat"   and b :: "'b" shows "\<one>\<^bsub>ring_mat ty n b\<^esub> = 1\<^sub>m n"
asymp_equiv_eventually_neg_iff: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "f \<sim>[F] g" shows "\<forall>\<^sub>F x in F. (f x < 0) = (g x < 0)"
index_col: fixes i :: "nat"   and A :: "'b mat"   and j :: "nat" assumes "i < dim_row A"   and "j < dim_col A" shows "col A j $v i = A $$ (i, j)"
simps: fixes i :: "'d ifex"   and t :: "'d ifex"   and e :: "'d ifex" shows "ifex_ite_opt i t e = (case param_opt i t e of None \<Rightarrow> case lowest_tops [i, t, e] of None \<Rightarrow> case i of Trueif \<Rightarrow> t | Falseif \<Rightarrow> e | Some x \<Rightarrow> IFC x (ifex_ite_opt (restrict_top i x True) (restrict_top t x True) (restrict_top e x True)) (ifex_ite_opt (restrict_top i x False) (restrict_top t x False) (restrict_top e x False)) | Some b \<Rightarrow> b)"
LT5: fixes F :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and G :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<turnstile> \<box>(F \<longrightarrow> \<diamond>G) \<longrightarrow> \<diamond>F \<longrightarrow> \<diamond>G"
degree_zero: fixes v :: "nat" shows "degree 0 v = 0"
empty: fixes c :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" shows "(c, [], c) \<in> trcl t"
quotient_of_int: fixes i :: "int" shows "quotient_of (rat_of_int i) = (i, 1)"
vreal_of_real_impl'_def: shows "vreal_of_real_impl' = (SOME f. bij_betw f (range vreal_of_real_impl) (elts vreal_impl))"