hcju/isabelleps huggingface dataset:
pw_ords_iff_1: fixes S :: "'e nres"   and S' :: "'e nres" shows "(S \<le> S') = (nofail S' \<longrightarrow> nofail S \<and> (\<forall>x. inres S x \<longrightarrow> inres S' x))"
eval_dioph_add_map: fixes ks :: "int list"   and f :: "nat \<Rightarrow> nat"   and xs :: "nat list"   and g :: "nat \<Rightarrow> nat" shows "eval_dioph ks (map f xs) + eval_dioph ks (map g xs) = eval_dioph ks (map (\<lambda>x. f x + g x) xs)"
simps_1: fixes p :: "('a, 'b) form" shows "put_imps p [] = p"
conjugate_vCons: fixes a :: "'a"   and v :: "'a vec" shows "conjugate (vCons a v) = vCons (conjugate a) (conjugate v)"
iiws_imp_iipwn: fixes c :: "('loc, 't) Propagate.configuration" assumes "inv_imps_work_sum c" shows "inv_imp_plus_work_nonneg c"
id_func_preserves_dom: shows "preserves_dom (id_func AA)"
left_tree_separates': fixes s :: "'a"   and t :: "'a"   and u :: "'a"   and u' :: "'a" assumes "s \<rightarrow> t"   and "u \<in> left_tree s t"   and "u' \<in> left_tree t s" shows "t \<in> set (u \<leadsto> u')"
Domain_insert: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" shows "Domain (insert (a, b) r) = insert a (Domain r)"
eval_var_no_jump: fixes Env :: "env"   and s0 :: "state"   and var :: "Term.var"   and w :: "val"   and upd :: "val \<Rightarrow> state \<Rightarrow> state"   and s1 :: "state"   and j :: "jump"   and T :: "ty"   and s :: "state"   and j' :: "jump"   and val :: "val" assumes "prg Env\<turnstile>s0 \<midarrow>var=\<succ>(w, upd)\<rightarrow> s1"   and "fst s0 \<noteq> Some (Jump j)"   and "Env,empty_dt\<Turnstile>var\<Colon>=T"   and "wf_prog (prg Env)" shows "fst s1 \<noteq> Some (Jump j) \<and> (normal s1 \<longrightarrow> fst s \<noteq> Some (Jump j') \<longrightarrow> fst (upd val s) \<noteq> Some (Jump j'))"
npk: shows "SN (n[x::=p] \<star> k)"
finite_extension_mem: fixes K :: "'a set"   and xs :: "'a list" assumes "subring K R"   and "set xs \<subseteq> carrier R" shows "set xs \<subseteq> finite_extension K xs"
f_Exec_append: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and xs :: "'b list"   and ys :: "'b list"   and c :: "'a" shows "f_Exec_Comp trans_fun (xs @ ys) c = f_Exec_Comp trans_fun ys (f_Exec_Comp trans_fun xs c)"
inverse_mult_eq_1: fixes f :: "'a fps" assumes "fps_nth f 0 \<noteq> (0::'a)" shows "inverse f * f = 1"
le_eq_neg: fixes a :: "'a"   and b :: "'a" shows "(a \<le> - b) = (a + b \<le> (0::'a))"
is_adjoint_sym: fixes F :: "'b \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'b" assumes "is_cadjoint F G" shows "is_cadjoint G F"
AtSeqExpr1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a1 :: "('e, 'd, 'f) ann"   and c1 :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com" assumes "atomicsR \<Gamma> \<Theta> a1 c1 a" shows "atomicsR \<Gamma> \<Theta> (AnnComp a1 a2) (Seq c1 c2) a"
simps_2: fixes t :: "'a tree"   and ts :: "'a tree list"   and b :: "'a" shows "freq\<^sub>F (t # ts) b = freq t b + freq\<^sub>F ts b"
SINV_pre_def: fixes par :: "Var list"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and A :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "SINV_pre par T A = (\<lambda>s0 s. \<exists>ops1 ops2 S R h k w. (ops1, par, R, ops2) \<in> Frame \<and> T (R, k) (h, w) \<and> A s0 (ops1, S, k) \<and> s = (w # ops2, S, h))"
env_len_def: fixes env :: "'c list set"   and l :: "nat" shows "env_len env l = (\<forall>xs\<in>env. length xs = l)"
dense_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and z :: "'e" assumes "class.dense_linorder less_eq less"   and "\<And>x. less x y \<Longrightarrow> less_eq x z" shows "less_eq y z"
assms_2: shows "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H"
simps_1: fixes i :: "nat"   and \<Gamma> :: "nat list" shows "transdb (Var i) \<Gamma> = DBVar (index \<Gamma> i 0)"
entry: shows "entry = (\<tau> \<sigma> i, rels, rel)"
rep_eq: fixes x :: "'h \<Rightarrow> 'f"   and xa :: "('g, 'f) generator"   and xb :: "('h, 'e) generator" shows "generator (flatten x xa xb) = flatten_raw x (generator xa) (generator xb)"
attach_wiring_right_def: shows "attach_wiring_right = (\<lambda>(f, g). map_fun id (map_fun id (map_gpv' id f g)))"
V_simps_2: fixes \<phi> :: "('a, 'b, 'c) trans \<Rightarrow> bool"   and trn :: "('a, 'b, 'c) trans"   and f :: "('a, 'b, 'c) trans \<Rightarrow> 'd"   and tr :: "('a, 'b, 'c) trans list" assumes "\<not> \<phi> trn" shows "BD_Security.V \<phi> f (trn # tr) = BD_Security.V \<phi> f tr"
lookup_single_not_eq: fixes k :: "'b"   and k' :: "'b"   and v :: "'c" assumes "k \<noteq> k'" shows "lookup (single k v) k' = (0::'c)"
Cons_7: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) Q"
impl_elimC: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "(P \<longrightarrow> Q) x" obtains "\<not> P x"   | "Q x"
\<sigma>_full: fixes f :: "'f"   and k :: "nat" shows "set (\<sigma> (f, k)) = {0..<k}"
inc: shows "\<forall>n. I n \<subseteq> I (n + 1)"
supp_sum_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and f :: "'d \<Rightarrow> 'c"   and S :: "'d set" assumes "class.comm_monoid_add plus zero" shows "comm_monoid_add.supp_sum plus zero f S = comm_monoid_add.sum plus zero f (monoid_add.support_on zero S f)"
sort_key_less_eq_def: fixes x :: "'b sort_key"   and y :: "'b sort_key" shows "(x \<le> y) = (sort_key_embedding x \<le> sort_key_embedding y)"
unique_1: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly"   and V2 :: "'b mod_ring poly"   and W2 :: "'b mod_ring poly"   and ty_q :: "'c itself" assumes "Knuth_ex_4_6_2_22_prime a b u v w"   and "v = #V2"   and "w = #W2"   and "u = V2 * W2"   and "degree V2 = degree v"   and "degree W2 = degree w"   and "lead_coeff V2 = @lead_coeff v" shows "V2 = #v + Polynomial.smult (of_nat (Knuth_ex_4_6_2_22_base.q ty_q)) (#Knuth_ex_4_6_2_22_main.v' ty_q b u v w)"
unit_simps_5: shows "trg \<eta>\<^sub>1 = src e\<^sub>1"
hpsafe: shows "hpsafe (EvolveODE ODE \<phi>)"
ChainsI: fixes Y :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>a b. \<lbrakk>a \<in> Y; b \<in> Y\<rbrakk> \<Longrightarrow> (a, b) \<in> r \<or> (b, a) \<in> r" shows "Y \<in> Chains r"
kronecker_factorization_rat_main_complete: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and bnd :: "nat"   and p :: "rat poly" assumes "divisors_fun df"   and "divisors_pos_fun dp"   and "kronecker_factorization_rat_main df dp bnd p = None"   and "2 \<le> degree p" shows "\<nexists>q. 1 \<le> degree q \<and> degree q \<le> bnd \<and> q dvd p"
f_def: fixes x :: "'v" shows "f x = (if x = xx then integral\<^sup>N (count_space (\<^bold>E `` X)) (weight \<Gamma>) - integral\<^sup>N (count_space (X - {xx})) (weight \<Gamma>) else if x \<in> X then weight \<Gamma> x else 0)"
simps_20: fixes t :: "'b local.term"   and u :: "'b local.term"   and va :: "'b" shows "(t \<^bold>\<otimes> u) \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<langle>va\<^bold>\<rangle> = t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<langle>va\<^bold>\<rangle>"
simps_3: fixes v :: "vname aexp"   and va :: "vname aexp" shows "enumerate_gexp_inputs (Gt v va) = enumerate_aexp_inputs v \<union> enumerate_aexp_inputs va"
def_hashmap_size_unit_def: shows "def_hashmap_size = (\<lambda>_. 2)"
aSh_simps: fixes sh :: "int" shows "aSh (shortV sh) = sh"
OEA: fixes \<AA> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula" assumes "\<AA> # \<Gamma> \<Rightarrow> \<Theta>"   and "\<BB> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<AA> \<^bold>\<or> \<BB> # \<Gamma> \<Rightarrow> \<Theta>"
var: fixes t :: "'trm" assumes "set (xs @ us @ vs) \<subseteq> var"   and "set ts \<subseteq> trm"   and "set [\<phi>] \<subseteq> fmla"   and "t \<in> set ts" shows "set (getFrN (xs @ us @ vs) ts [\<phi>] (length xs)) \<inter> set (xs @ us @ vs) = {}"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> kerbV" shows "K \<notin> keysFor (parts (knows Spy evs))"
ext_fieldname_simp: fixes f :: "fspec" shows "(fldname f, fldclass f) = f"
simps: fixes S :: "'a set"   and t :: "real"   and a1 :: "'a"   and a2 :: "(real \<times> 'a) stream" shows "trace_in S t a1 a2 = ((\<exists>t' s s' \<omega>. a1 = s \<and> a2 = (t', s') ## \<omega> \<and> t < t' \<and> s \<in> S) \<or> (\<exists>t' s' \<omega> s. a1 = s \<and> a2 = (t', s') ## \<omega> \<and> t' \<le> t \<and> trace_in S t s' \<omega>))"
sas_plus_problem_has_serial_solution_iff_i: fixes \<Psi> :: "('a, 'b) sas_plus_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_sas_plus \<Psi>"   and "\<A> \<Turnstile> \<Phi>\<^sub>\<forall> \<phi> \<Psi>  t" shows "SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> (map (strips_op_to_sasp \<Psi>) (concat (\<Phi>\<inverse> \<phi> \<Psi>  \<A> t)))"
square_integrable_imp_measurable: fixes f :: "real \<Rightarrow> real"   and S :: "real set" assumes "f square_integrable S" shows "f \<in> borel_measurable (lebesgue_on S)"
opt_bst_simps_2: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "local.opt_bst i j = argmin (local.wpl i j) (map (\<lambda>k. \<langle>local.opt_bst i (k - 1), k, local.opt_bst (k + 1) j\<rangle>) [i..j])"
ground_subst_no_var: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and x :: "'a" assumes "ground (subst_range s)" shows "x \<notin> range_vars s"
deg_le_sect_alt: fixes X :: "'a set"   and d :: "nat" shows "deg_le_sect X d = .[X] \<inter> {t. deg_pm t \<le> d}"
exhaust: fixes y :: "'e\<cdot>('f, 'g) writerT" obtains "y = \<bottom>"   | app :: "'e\<cdot>'g writer\<cdot>'f" where "y = WriterT\<cdot>app" and "app \<noteq> \<bottom>"
set_to_map_dom: fixes S :: "('g \<times> 'h) set" shows "dom (set_to_map S) = fst ` S"
IH2: fixes C :: "int poly"   and Gs :: "int poly list" assumes "C =m \<Prod>\<^sub># (factors_of_factor_tree r)"   and "\<And>F. F \<in># factors_of_factor_tree r \<Longrightarrow> Mp F = F \<and> monic F"   and "hensel_main C r = Gs"   and "monic C"   and "pn.Mp C = C"   and "square_free_m C"   and "\<And>t f. \<lbrakk>t \<in> sub_trees r; factor_node_info t = f\<rbrakk> \<Longrightarrow> f = Mp (\<Prod>\<^sub># (factors_of_factor_tree t))" shows "pn.eq_m C (prod_list Gs) \<and> factors_of_factor_tree r = mset (map Mp Gs) \<and> (\<forall>G. G \<in> set Gs \<longrightarrow> monic G \<and> pn.Mp G = G)"
img_is_add_subgroup: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g"   and H :: "'e set" assumes "ring_hom_ring R S h"   and "subgroup H (add_monoid R)" shows "subgroup (h ` H) (add_monoid S)"
fminus_funion: fixes A :: "'a fset"   and B :: "'a fset"   and C :: "'a fset" shows "A |-| (B |\<union>| C) = A |-| B |\<inter>| (A |-| C)"
ipresIGSwapAll_termMOD: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and h :: "('b, 'c, 'd, 'a, 'f) qTerm set \<Rightarrow> 'g"   and hA :: "('b, 'c, 'd, 'a, 'f) qAbs set \<Rightarrow> 'h"   and MOD :: "('i, 'j, 'd, 'k, 'l, 'a, 'g, 'h, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta" shows "ipresIGSwapAll h hA (FixSyn.termMOD Delta) MOD = FixSyn.ipresSwapAll Delta h hA MOD"
not_dbm_le_lt_impl: fixes a :: "'c"   and b :: "'c" assumes "\<not> Le a \<prec> Lt b" shows "b \<le> a"
span_explicit': fixes b :: "('a \<Rightarrow>\<^sub>0 'b) set" shows "phull b = {\<Sum>v | f v \<noteq> (0::'b). f v \<cdot> v |f. finite {v. f v \<noteq> (0::'b)} \<and> (\<forall>v. f v \<noteq> (0::'b) \<longrightarrow> v \<in> b)}"
Int_atLeastAtMost: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "class.linorder less_eq less" shows "ord.atLeastAtMost less_eq a b \<inter> ord.atLeastAtMost less_eq c d = ord.atLeastAtMost less_eq (ord.max less_eq a c) (ord.min less_eq b d)"
drop_all_iff: fixes n :: "nat"   and xs :: "'a list" shows "(drop n xs = []) = (length xs \<le> n)"
fmrestrict_set_insert_notin: fixes xa :: "'c"   and N :: "('c, 'd) fmap"   and l1 :: "'c set" assumes "xa \<notin> fset (fmdom N)" shows "fmrestrict_set (insert xa l1) N = fmrestrict_set l1 N"
secure_implies_c_secure: fixes step :: "'h \<Rightarrow> 'e \<Rightarrow> 'h"   and out :: "'h \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'h"   and I :: "('g \<times> 'g) set"   and D :: "'e \<Rightarrow> 'g" assumes "secure (c_process step out s\<^sub>0) I (c_dom D)" shows "c_secure step out s\<^sub>0 I D"
Cos: shows "((\<lambda>x. cos (eval C x)) expands_to Cos) basis"
path_append_eq_up_to: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<pi>' :: "nat \<Rightarrow> 'a" assumes "IFC prob" shows "IFC_def.eq_up_to (IFC_def.path_append \<pi> k \<pi>') k \<pi>"
moebius_rotation_zero: shows "moebius_rotation 0 = id_moebius"
bheight_paint_Red: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "color t = Black" shows "bheight (paint Red t) = bheight t - 1"
wp2Conc: fixes x :: "((int \<times> int) set set, int) Combinators"   and xs :: "((int \<times> int) set set, int) Combinators list" assumes "wellformed_policy2 (x # xs)" shows "wellformed_policy2 xs"
simps_1: fixes f :: "'c \<Rightarrow> 'b" shows "ListSum [] f = (0::'b)"
p_def: fixes s :: "'a \<times> 'b"   and t :: "'a \<times> 'b"   and n :: "nat" shows "p s t n = T.prob s {\<omega> \<in> space (T s). (s ## \<omega>) !! n = t}"
map_attributes_def: fixes f :: "'a \<Rightarrow> 'a"   and "file" :: "('a \<times> 'b, 'a, 'c) env" shows "map_attributes f file = (case file of Val (att, text) \<Rightarrow> Val (f att, text) | Env att dir \<Rightarrow> Env (f att) dir)"
set_list_R1_list_to_list_R1: fixes xs :: "'c list" shows "list_R1.set_list_R1 (list_R1.list_to_list_R1 xs) = set xs"
d_OUT_ge_point: fixes f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k"   and y :: "'k" shows "f (x, y) \<le> d_OUT f x"
n'_def: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" shows "n' N M y = distr (n N M y) (stream_space (count_space UNIV)) ((##) y)"
r_neg1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus> x \<oplus> (x \<oplus> y) = y"
closed_others_RelS_Eg1Eg2: shows "closed_others RelS_Eg1Eg2"
backward_terminating_iff4: fixes x :: "'a" assumes "path x" shows "backward_terminating x = (x \<le> - (1 ; x) ; x\<^sup>\<star>)"
mark_guards_simps_5: fixes f :: "'f"   and b :: "'d set"   and c\<^sub>1 :: "('d, 'e, 'g) com"   and c\<^sub>2 :: "('d, 'e, 'g) com" shows "mark_guards f (Cond b c\<^sub>1 c\<^sub>2) = Cond b (mark_guards f c\<^sub>1) (mark_guards f c\<^sub>2)"
ideI: fixes a :: "'a" assumes "Arr a"   and "C.ide a" shows "ide a"
wls_cases: fixes x1 :: "'sort"   and x2 :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and P1 :: "'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> bool"   and P3 :: "'opSym \<Rightarrow> ('index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option) \<Rightarrow> bool"   and P4 :: "'opSym \<Rightarrow> ('bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option) \<Rightarrow> bool"   and P2 :: "'varSort \<times> 'sort \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> bool" assumes "wls x1 x2"   and "\<And>xs x. P1 (asSort xs) (Var xs x)"   and "\<And>delta inp binp. \<lbrakk>wlsInp delta inp; P3 delta inp; wlsBinp delta binp; P4 delta binp\<rbrakk> \<Longrightarrow> P1 (stOf delta) (Op delta inp binp)"   and "\<And>delta inp. \<lbrakk>wlsOpS delta; \<And>i. arOf delta i = None \<and> inp i = None \<or> (\<exists>s X. arOf delta i = Some s \<and> inp i = Some X \<and> wls s X \<and> P1 s X)\<rbrakk> \<Longrightarrow> P3 delta inp"   and "\<And>delta binp. \<lbrakk>wlsOpS delta; \<And>i. barOf delta i = None \<and> binp i = None \<or> (\<exists>us s A. barOf delta i = Some (us, s) \<and> binp i = Some A \<and> wlsAbs (us, s) A \<and> P2 (us, s) A)\<rbrakk> \<Longrightarrow> P4 delta binp"   and "\<And>xs s X x. \<lbrakk>isInBar (xs, s); wls s X; P1 s X\<rbrakk> \<Longrightarrow> P2 (xs, s) (Abs xs x X)" shows "P1 x1 x2"
Rep_eint_inverse: fixes x :: "eint" shows "Abs_eint (Rep_eint x) = x"
com_validity_def: fixes p :: "'b set"   and c :: "'b com"   and q :: "'b set" shows "\<parallel>= p  c q \<equiv> SEM c p \<subseteq> q"
LatticeDisjunction: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and H :: "'b \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) = ((F \<leadsto> H) \<and> (G \<leadsto> H))"
cospan: shows "trg s\<^sub>1 = trg r\<^sub>0"
X_inv: shows "X * X = 1\<^sub>m 2"
D1: shows "\<forall>mo. Deliver (i, Expunge e2 mo i) \<notin> set pre"
listrelp_imp_listsp2: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list" assumes "listrelp (\<lambda>x. P) xs ys" shows "listsp P ys"
infsuff_mono2: fixes s :: "'b llist"   and t :: "'b llist"   and A :: "'b set" assumes "s \<le> t" shows "infsuff A t \<subseteq> infsuff A s"
span_eq: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'c set"   and T :: "'c set" assumes "module scale" shows "(module.span scale S = module.span scale T) = (S \<subseteq> module.span scale T \<and> T \<subseteq> module.span scale S)"
appPutstatic: fixes C :: "char list"   and F :: "char list"   and D :: "char list"   and P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Putstatic C F D, P, pc, mxs, T\<^sub>r, s) = (\<exists>vT vT' ST LT. s = (vT # ST, LT) \<and> P \<turnstile> C sees F,Static:vT' in D \<and> subtype P vT vT')"
alw_inv_init_imp_prop_safe: fixes s :: "('b, 't, 'loc) Combined.configuration stream" assumes "FullSpec s" shows "alw (holds inv_init_imp_prop_safe) s"
H_on_ket_zero: shows "H * |Deutsch.zero\<rangle> = Matrix.mat (2 ^ 1) 1 (\<lambda>x. complex_of_real (case x of (i, j) \<Rightarrow> 1 / sqrt 2 ^ 1))"
r_findr_def: shows "r_findr \<equiv> let g = Cn 3 r_ifz [Cn 3 r_nth [recf.Id 3 2, recf.Id 3 0], Cn 3 r_ifeq [recf.Id 3 0, recf.Id 3 1, Cn 3 S [recf.Id 3 0], recf.Id 3 1], recf.Id 3 0] in Cn 1 (Pr 1 Z g) [Cn 1 r_length [recf.Id 1 0], recf.Id 1 0]"
is_red_monic: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red B (monic p) = is_red B p"
run_alt_option: fixes alt :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "('b, 'a) optionT"   and m2 :: "('b, 'a) optionT" shows "run_option (alt_option alt m1 m2) = alt (run_option m1) (run_option m2)"
simps_1: fixes a :: "'c" shows "Binomial_Queue.find a [] = None"
count_single: fixes b :: "'k"   and a :: "'k" shows "count {#b#} a = (if b = a then 1 else 0)"
HPhase1or2ReadThen_def: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" shows "HPhase1or2ReadThen s s' p d q = (Phase1or2ReadThen s s' p d q \<and> HNextPart s s')"
matrix_mult_1_left: fixes A :: "('c, 'd, 'e) matrix" shows "1 *\<^sub>M A = A"
P\<^sub>1_lower_bound_card: fixes P :: "'a set set"   and P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "bp P"   and "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))" shows "card P\<^sub>1 + 1 \<le> card P"
Elem_Elem_Ext_ZF_hull: fixes R :: "(ZF \<times> ZF) set"   and x :: "ZF"   and S :: "ZF"   and y :: "ZF" assumes "set_like R"   and "Elem x (Ext_ZF_hull R S)"   and "(y, x) \<in> R" shows "Elem y (Ext_ZF_hull R S)"
prime_elemI: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "p \<noteq> zero"   and "\<not> dvd.dvd times p one"   and "\<And>a b. dvd.dvd times p (times a b) \<Longrightarrow> dvd.dvd times p a \<or> dvd.dvd times p b" shows "comm_semiring_1.prime_elem times one zero p"
Osum_wf_Id: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Total r"   and "Total r'"   and "Field r \<inter> Field r' = {}"   and "wf (r - Id)"   and "wf (r' - Id)" shows "wf ((r \<union>o r') - Id)"
swap_lens_fst: shows "fst\<^sub>L ;\<^sub>L swap\<^sub>L = snd\<^sub>L"
mod_eqE: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.euclidean_ring_cancel divide plus minus zero times one modulo euclidean_size uminus"   and "modulo a c = modulo b c" obtains d :: "'a" where "b = plus a (times c d)"
size_YS1: shows "size YS1 \<le> m div 2"
jkbpCn_jkbpC_subset: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview" shows "PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n \<subseteq> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"
nu_fusion_1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq h"   and "order.has_greatest_postfixpoint less_eq g"   and "order.has_greatest_fixpoint less_eq h"   and "less_eq (h (u (l (order.the_greatest_fixpoint less_eq h)))) (u (g (l (order.the_greatest_fixpoint less_eq h))))" shows "less_eq (order.the_greatest_fixpoint less_eq h) (u (order.the_greatest_postfixpoint less_eq g))"
vifintersectionE1: fixes a :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V"   and i :: "V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)" obtains "a \<in>\<^sub>\<circ> f i"   | "i \<notin>\<^sub>\<circ> I"
simps_1: fixes i :: "nat"   and j :: "nat"   and l :: "nat"   and T :: "bool list list" shows "mk_eqcl' [] i j l T = []"
permutations_of_list: fixes xs :: "'a list" shows "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))"
borel_eq_sigmaI5: fixes G :: "'d \<Rightarrow> 'c set"   and F :: "'e \<Rightarrow> 'f \<Rightarrow> 'c set" assumes "borel = sigma UNIV (range G)"   and "\<And>i. G i \<in> sets (sigma UNIV (range (\<lambda>(i, j). F i j)))"   and "\<And>i j. F i j \<in> sets borel" shows "borel = sigma UNIV (range (\<lambda>(i, j). F i j))"
findim: shows "\<exists>ms. set ms \<subseteq> V \<and> RSpan ms = V"
errMOD_igAbsCongSSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model" assumes "igSwapIPresIGWls MOD"   and "igWlsDisj MOD"   and "igAbsCongS MOD" shows "igAbsCongSSTR (errMOD MOD)"
insertion_four: fixes xs :: "'a list" shows "insertion (nth_default (0::'a) xs) 4 = (4::'a)"