hcju/isabelleps huggingface dataset:
1: fixes \<tau> :: "'\<AA> state \<times> '\<AA> state"   and X :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'a"   and Y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'b" assumes "\<tau> \<Turnstile> d\<^sub>x X"   and "\<tau> \<Turnstile> d\<^sub>y Y" shows "\<tau> \<Turnstile> \<delta> f X Y"
simps_4: fixes v :: "'a"   and va :: "'a"   and vb :: "'b" shows "first_destNet (AllowPortFromTo v va vb) = destNet (AllowPortFromTo v va vb)"
even_fourtimes_len: shows "is_even (length t_fourtimes)"
lenext_compat_list: fixes gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and xs :: "'b list"   and y :: "'b"   and xs' :: "'b list"   and x :: "'b" assumes "gts (xs @ y # xs') (xs @ x # xs')" shows "lenext gts (xs @ y # xs') (xs @ x # xs')"
map_of_Cons_code_2: fixes l :: "'b"   and v :: "'a"   and ps :: "('b \<times> 'a) list"   and k :: "'b" shows "map_of ((l, v) # ps) k = (if l = k then Some v else map_of ps k)"
True: fixes F :: "('e \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
list_digraph_ext_def: fixes ext :: "'c"   and G_list :: "'a list \<times> ('a \<times> 'a) list" shows "list_digraph_ext ext G_list \<equiv> \<lparr>pverts = set (fst G_list), parcs = set (snd G_list), \<dots> = ext\<rparr>"
itrm_bracket_ap: fixes i :: "nat"   and x :: "nat itrm"   and x' :: "nat itrm" assumes "itrm_bracket i x = Some x'" shows "x' \<diamondop> Opaque i \<simeq>\<^sup>+ x"
axioms_4: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "functor A B G"
power_strict_increasing_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and b :: "'c"   and x :: "nat"   and y :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one b" shows "less (power.power one times b x) (power.power one times b y) = (x < y)"
finsert_absorb: fixes a :: "'c"   and A :: "'c fset" assumes "a |\<in>| A" shows "finsert a A = A"
eq_iff: fixes a :: "'c list"   and b :: "'c list" shows "(a = b) = (suffix b a \<and> suffix a b)"
extensional_funcset_def: fixes S :: "'a set"   and T :: "'b set" shows "S \<rightarrow>\<^sub>E T = (S \<rightarrow> T) \<inter> extensional S"
inf_set_def: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B = Collect ((\<lambda>x. x \<in> A) \<sqinter> (\<lambda>x. x \<in> B))"
finrank_Span: fixes vs :: "'v list" assumes "set vs \<subseteq> V" shows "R_finrank (RSpan vs)"
scalar_invertible: fixes k :: "real"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k \<noteq> 0"   and "invertible A" shows "invertible (k *\<^sub>R A)"
UNlift01_inj_on: fixes cl :: "('test, 'atom, 'choice) cmd list"   and dl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and F :: "nat \<Rightarrow> nat set \<Rightarrow> nat set" assumes "length cl = length dl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "\<And>n. n < length dl \<Longrightarrow> part {..<brn (dl ! n)} (F n ` P n) \<and> {} \<notin> F n ` P n"   and "\<And>n. n < length cl \<Longrightarrow> inj_on (F n) (P n)" shows "inj_on (UNlift01 cl dl P F) (UNpart01 cl dl P)"
bind_rpv_def: fixes rpv :: "('d, 'c, 'a) rpv"   and f :: "'d \<Rightarrow> ('b, 'c, 'a) gpv" shows "rpv \<bind> f = (\<lambda>input. rpv input \<bind> f)"
fim: shows "f ` span ((+) (- a) ` S) = span ((+) (- b) ` T)"
nb\<^sub>2: shows "are_all_operators_non_interfering ops"
\<sigma>_def: shows "\<sigma> = (\<lambda>x. case map_of (Tableau S) x of None \<Rightarrow> lp_monom 1 x | Some p \<Rightarrow> p)"
vars_pivot_eq: fixes x\<^sub>j :: "nat"   and eq :: "nat \<times> linear_poly" assumes "x\<^sub>j \<in> rvars_eq eq"   and "lhs eq \<notin> rvars_eq eq" shows "let eq' = pivot_eq eq x\<^sub>j in lhs eq' = x\<^sub>j \<and> rvars_eq eq' = {lhs eq} \<union> (rvars_eq eq - {x\<^sub>j})"
cblinfun_compose_image: fixes A :: "'e \<Rightarrow>\<^sub>C\<^sub>L 'c"   and B :: "'d \<Rightarrow>\<^sub>C\<^sub>L 'e"   and S :: "'d ccsubspace" shows "(A o\<^sub>C\<^sub>L B) *\<^sub>S S = A *\<^sub>S B *\<^sub>S S"
\<I>_2: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
SC_WbisT_Seq: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c1 :: "('b, 'c) com"   and c2 :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.SC_WbisT tval aval indis (c1 ;; c2) = (PL_Indis.SC_WbisT tval aval indis c1 \<and> PL_Indis.SC_WbisT tval aval indis c2)"
rank_of_mono: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<subseteq> Y"   and "Y \<subseteq> carrier" shows "rank_of X \<le> rank_of Y"
numeral_2_eq_2: shows "nx = Suc (Suc 0)"
append_assoc: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
Scons_inject1: fixes M :: "('c, 'd) Old_Datatype.node set"   and N :: "('c, 'd) Old_Datatype.node set"   and M' :: "('c, 'd) Old_Datatype.node set"   and N' :: "('c, 'd) Old_Datatype.node set" assumes "Scons M N = Scons M' N'" shows "M = M'"
laplace_integrand_integrable: fixes M :: "real"   and c :: "real"   and f :: "real \<Rightarrow> complex"   and B :: "real \<Rightarrow> real"   and s :: "complex" assumes "exponential_order M c f"   and "\<And>k. AE x\<in>{0..k} in lebesgue. cmod (f x) \<le> B k"   and "\<And>k. f integrable_on {0..k}"   and "c < Re s" shows "laplace_integrand f s integrable_on {0..}"
poly_y_x_def: fixes p :: "'a poly poly" shows "poly_y_x p \<equiv> \<Sum>i\<le>degree p. \<Sum>j\<le>degree (coeff p i). monom (monom (coeff (coeff p i) j) i) j"
fresh_star_bij_2: fixes pi :: "(name \<times> name) list"   and b :: "name list"   and x :: "'c" shows "(pi \<bullet> b) \<sharp>* (pi \<bullet> x) = b \<sharp>* x"
set2_var_def: shows "set2_var \<equiv> map_fun rep_var id set2_var0"
takeWhile_append1: fixes x :: "'g"   and xs :: "'g list"   and P :: "'g \<Rightarrow> bool"   and ys :: "'g list" assumes "x \<in> set xs"   and "\<not> P x" shows "takeWhile P (xs @ ys) = takeWhile P xs"
Mismatch_1: fixes P :: "pi"   and b :: "name"   and x :: "name"   and P' :: "pi"   and a :: "name"   and c :: "name" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^b<\<nu>x> \<prec> P'"   and "a \<noteq> c" shows "[a\<noteq>c]P \<Longrightarrow>\<^sub>l\<^sup>^b<\<nu>x> \<prec> P'"
CallUndefined: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and s :: "'d" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow> Stuck"
norm_normalize: fixes x :: "'b" assumes "x \<noteq> (0::'b)" shows "norm (Projections.normalize x) = 1"
filtercomap_principal: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'d set" shows "filtercomap f (principal A) = principal (f -` A)"
h_def: shows "h \<equiv> zipWithStep\<cdot>f\<cdot>ha\<cdot>hb"
ass: shows "prime P \<and> odd P \<and> P dvd A \<and> P ^ n = p\<^sup>2 + N * q\<^sup>2 \<and> A ^ n = a\<^sup>2 + N * b\<^sup>2 \<and> coprime a b \<and> coprime p (N * q) \<and> 0 < n"
mono: fixes f :: "'b \<Rightarrow> 'b" shows "mono (\<lambda>p x. (\<exists>xa. x = f xa \<and> p xa) \<or> (\<exists>M. x = lub' M \<and> Complete_Partial_Order.chain ord' M \<and> (\<forall>x\<in>M. p x)))"
2: shows "(\<lambda>f. if prerequisites f m then Some (match_no_prereq f p) else None) ` m = (\<lambda>f. Some (match_no_prereq f p)) ` m"
infinite_iff_natLeq_ordLeq: fixes A :: "'f set" shows "finite A \<noteq> (natLeq \<le>o |A|)"
right_neutral: fixes a :: "'d \<Rightarrow> ('e, 'd) Term.term" shows "a \<circ>\<^sub>s Var = a"
distr_pair_swap: shows "M1 \<Otimes>\<^sub>M M2 = distr (M2 \<Otimes>\<^sub>M M1) (M1 \<Otimes>\<^sub>M M2) (\<lambda>(x, y). (y, x))"
lub_set: fixes S :: "'a set set" shows "lub S = \<Union> S"
derived_is_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier G" shows "subgroup (derived G H) G"
prems_2: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
nat_pow_Suc2: fixes x :: "int vec"   and n :: "nat" assumes "x \<in> carrier_vec n" shows "[Suc n] \<cdot>\<^bsub>V\<^esub> x = x + [n] \<cdot>\<^bsub>V\<^esub> x"
set_zip: fixes xs :: "'a list"   and ys :: "'b list" shows "set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
line_intersect_S_at_most_twice: fixes l :: "proj2_line" shows "\<exists>p q. \<forall>r\<in>S. proj2_incident r l \<longrightarrow> r = p \<or> r = q"
is_interval_oo: fixes r :: "real"   and s :: "real" shows "is_interval {r<..<s}"
distinct_265: shows "BinXor \<noteq> ShiftRightSigned"
elem_exists_non_empty_set: fixes A :: "'a set" assumes "0 < card A" obtains x :: "'a" where "x \<in> A"
IDE_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "free_monoidal_category.IDE C f \<equiv> free_monoidal_category.ARR C f \<and> (\<forall>t. t \<in> f \<longrightarrow> monoidal_language.Ide C (monoidal_language.Diagonalize C t)) \<and> monoidal_language.Dom C (free_monoidal_category.rep C f) = monoidal_language.Cod C (free_monoidal_category.rep C f)"
length_concat: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xss :: "'c list list" assumes "class.monoid_add plus zero" shows "\<^bold>|concat xss\<^bold>| = sum_list (map length xss)"
oalist_inv_raw_ConsD3: fixes x :: "'o"   and k :: "'a"   and v :: "'c"   and xs :: "('a \<times> 'c) list"   and k' :: "'a" assumes "oalist_inv_raw x ((k, v) # xs)"   and "k' \<in> fst ` set xs" shows "lt x k k'"
Red: shows "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
pr_pri_agrk_parts: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and x :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Pri_AgrK x \<notin> U" shows "Pri_AgrK x \<notin> parts (A \<union> spies evs)"
lnth_iterates: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c"   and n :: "nat" shows "lnth (iterates f x) n = (f ^^ n) x"
a': shows "( \<Gamma> \<Rightarrow>* \<Delta>, n) \<in> derivable R*"
stack_set_def: fixes G :: "('c, 'd) graph_rec_scheme"   and param :: "('c, ('c, 'e) state_scheme, 'e) gen_parameterization"   and s :: "('c, 'e) state_scheme" assumes "DFS_invar G param s" shows "set (stack s) = dom (discovered s) - dom (finished s)"
assms_1: shows "u \<le>p v \<cdot> w"
aux_comp_with_sqrt2: fixes n :: "nat" shows "sqrt 2 ^ n * sqrt 2 ^ n = 2 ^ n"
has_integral: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and a :: "'c"   and b :: "'c" shows "(f has_integral y) (cbox a b) = (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - y) < e))"
reflect_poly_power: fixes p :: "'a poly"   and n :: "nat" shows "reflect_poly (p ^ n) = reflect_poly p ^ n"
inf_llist_lnth: fixes xs :: "'b llist" assumes "linfinite xs" shows "inf_llist ((?!) xs) = xs"
vdunion_def: fixes I :: "V"   and A :: "V \<Rightarrow> V" shows "VSigma I A = \<Union>\<^sub>\<circ> ((\<lambda>a\<in>\<^sub>\<circ>I. \<Union>\<^sub>\<circ> ((\<lambda>aa\<in>\<^sub>\<circ>A a. set {\<langle>a, aa\<rangle>}) `\<^sub>\<circ> A a)) `\<^sub>\<circ> I)"
mult_adjugate_det: fixes A :: "(('b, 'c) vec, 'c) vec" shows "A ** adjugate A = mat (det A)"
unit_group_unique: fixes G :: "('c, 'd) monoid_scheme" assumes "Group.group G"   and "card (carrier G) = 1" shows "\<exists>h. h \<in> Group.iso G unit_group"
strict_iff_order: fixes b :: "'a Word.word"   and a :: "'a Word.word" shows "(b <s a) = (b \<le>s a \<and> a \<noteq> b)"
simps_3: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and proper_interval :: "'a option \<Rightarrow> 'a option \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list" shows "proper_intrvl.proper_interval_set_aux less proper_interval (x # xs) (y # ys) = (if less x y then False else if less y x then proper_interval (Some y) (Some x) \<or> ys \<noteq> [] \<or> \<not> proper_intrvl.exhaustive_above proper_interval x xs else proper_intrvl.proper_interval_set_aux less proper_interval xs ys)"
ground_subst_fm: fixes A :: "fm"   and i :: "name"   and S :: "atom set"   and t :: "tm" assumes "ground_fm_aux A (insert (atom i) S)"   and "ground t" shows "ground_fm_aux (A(i::=t)) S"
min_wpl_simps_2: fixes i :: "int"   and j :: "int"   and w :: "int \<Rightarrow> int \<Rightarrow> nat" assumes "i \<le> j" shows "Optimal_BST.min_wpl w i j = (MIN k\<in>{i..j}. Optimal_BST.min_wpl w i (k - 1) + Optimal_BST.min_wpl w (k + 1) j + w i j)"
ws_subcls1_induct: fixes G :: "prog"   and C :: "qtname"   and P :: "qtname \<Rightarrow> bool" assumes "is_class G C"   and "ws_prog G"   and "\<And>C c. \<lbrakk>class G C = Some c; C \<noteq> Object \<longrightarrow> G\<turnstile>C\<prec>\<^sub>C1super c \<and> P (super c) \<and> is_class G (super c)\<rbrakk> \<Longrightarrow> P C" shows "P C"
Sep: fixes b :: "ZF"   and A :: "ZF"   and p :: "ZF \<Rightarrow> bool" shows "Elem b (Sep A p) = (Elem b A \<and> p b)"
inject_2: fixes x3 :: "char list list"   and y3 :: "char list list" shows "(Called x3 = Called y3) = (x3 = y3)"
eigenvalue_char_matrix: fixes A :: "'b mat"   and n :: "nat"   and e :: "'b" assumes "A \<in> carrier_mat n n" shows "eigenvalue A e = (\<exists>v. v \<in> carrier_vec n \<and> v \<noteq> 0\<^sub>v n \<and> char_matrix A e *\<^sub>v v = 0\<^sub>v n)"
8: shows "\<forall>n\<ge>s. (\<Prod>j. d (n + j)) < A / (MAX j\<in>{s..n}. real_of_int (a j) powr (1 / real_of_int (2 ^ j)))"
th_def: shows "th = (bis \<union> bis\<inverse>)\<^sup>+"
NPreflow_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a" shows "NPreflow c s t f \<equiv> Network c s t \<and> Preflow c s t f"
vcard_transfer: shows "(cr_vfsequence ===> cr_omega) vcard length"
F_b2_c: shows "LeftDerivation b2 F (drop (Suc j) c)"
l2_inv4I: fixes x :: "l2_state" assumes "\<And>Ra A B gny Nr. \<lbrakk>hmac \<langle>Number 0, Exp Gen (NonceF (Ra $ nx)), gny, Messages.Agent B, Messages.Agent A\<rangle> (hmac (NonceF (Ra $ ni)) Nr) \<in> parts (extr (bad x) (ik x) (chan x)); guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; A \<notin> bad x \<and> B \<notin> bad x\<rbrakk> \<Longrightarrow> \<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> in_progressS (progress x Rb) {xny, xni, xnr, xgnx, xgny, xsk} \<and> sklvl1.guessed_frame Rb xgny = Some gny \<and> sklvl1.guessed_frame Rb xnr = Some Nr \<and> sklvl1.guessed_frame Rb xni = Some (NonceF (Ra $ ni)) \<and> sklvl1.guessed_frame Rb xgnx = Some (Exp Gen (NonceF (Ra $ nx)))" shows "x \<in> l2_inv4"
inclusion_is_functor: shows "functor C (\<cdot>) \<F>C.inclusion_of_generators"
prv_eqv_trans: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "\<psi> \<in> fmla"   and "prv (eqv \<phi> \<chi>)"   and "prv (eqv \<chi> \<psi>)" shows "prv (eqv \<phi> \<psi>)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) FG.Dop.comp F' F' F'"
P: shows "P \<subseteq> {q. irreducible\<^sub>d q \<and> monic q}"
pq: fixes x :: "'a \<times> 'b" assumes "x \<in> set_spmf pq" shows "case x of (x, xa) \<Rightarrow> A x xa"
that: fixes u :: "Lambda_Z.term" assumes "t = Abs x u"   and "s \<rightarrow>\<^sub>\<beta> u" shows "thesis"
simps_1: fixes f :: "'c \<Rightarrow> 'd" shows "groupF f [] = []"
eval_Runit'_Ide: fixes t :: "'a E.term" assumes "E.Ide t" shows "EVAL \<^bold>\<r>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = B.runit' (EVAL t)"
left_diff_distrib: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "(a - b) * c = a * c - b * c"
plain_bind: fixes x :: "'c"   and M :: "'c nres"   and f :: "'c \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd nres" assumes "RETURN x \<le> M"   and "\<And>x. RETURN (f x) \<le> F x" shows "RETURN (Let x f) \<le> M \<bind> F"
Bot_map_not_empty: fixes B :: "'f" assumes "B \<in> Bot_F" shows "\<G>_F B \<noteq> {}"
mult_vv_mv: fixes x :: "complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and y :: "complex \<times> complex" shows "x *\<^sub>v\<^sub>v (A *\<^sub>m\<^sub>v y) = x *\<^sub>v\<^sub>m A *\<^sub>v\<^sub>v y"
rotate_rev: fixes n :: "nat"   and xs :: "'c list" shows "rotate n (rev xs) = rev (rotate (|xs| - n mod |xs|) xs)"
less_imp_not_less: fixes y :: "'d"   and x :: "'d" assumes "y < x" shows "(\<not> x < y) = True"
un_reduced_homology_group: fixes p :: "int"   and X :: "'a topology" assumes "p \<noteq> 0" shows "reduced_homology_group p X = homology_group p X"
subst_comp_upd1: fixes \<theta> :: "'a \<Rightarrow> ('b, 'd) Term.term"   and v :: "'a"   and t :: "('b, 'd) Term.term"   and \<sigma> :: "'d \<Rightarrow> ('b, 'c) Term.term" shows "\<theta>(v := t) \<circ>\<^sub>s \<sigma> = (\<theta> \<circ>\<^sub>s \<sigma>)(v := t \<cdot> \<sigma>)"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "redT a1 a2 a3" obtains t :: "'t" and x :: "'x" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a1 = sa__" and "a2 = (ta__, ta)" and "a3 = s'a__" and "ta__ \<turnstile> \<langle>x__, shr sa__\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "thr sa__ ta__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "actions_ok sa__ ta__ ta" and "redT_upd sa__ ta__ ta x' m' s'a__"   | s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x" and n :: "'l" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ln :: "'l \<Rightarrow>f nat" where "a1 = s" and "a2 = (t, K$ [], [], [], [], [], convert_RA ln)" and "a3 = s'" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
quasi_isometry_onD_3: fixes lambda :: "real"   and C :: "real"   and X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "lambda C -quasi_isometry_on X f" shows "1 \<le> lambda"
assms_8: shows "has_least_fixpoint (h \<circ> g)"
beta_eta_lift: fixes s :: "dB"   and t :: "dB"   and k :: "nat" assumes "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t" shows "lift s k \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* lift t k"
pair_declclass_simp: fixes c :: "'b"   and x :: "'c" shows "declclass (c, x) = declclass c"
Object_is_class: fixes G :: "prog" assumes "wf_prog G" shows "is_class G Object"
aform_err_sum_list': fixes y :: "real"   and e :: "nat \<Rightarrow> real"   and c :: "real \<times> real pdevs"   and ds :: "real list"   and p :: "nat" assumes "y \<in> aform_err e (c, sum_list ds)" shows "y \<in> aform_err e (c, sum_list' p ds)"
len_mono: fixes x :: "nat"   and y :: "nat" assumes "x \<le> y" shows "len x \<le> len y"
mono: fixes c :: "'b"   and a :: "'b"   and d :: "'b"   and b :: "'b" assumes "c \<le> a"   and "d \<le> b" shows "c \<squnion> d \<le> a \<squnion> b"
simps_1: fixes w :: "nat"   and a :: "'a" shows "weight (Leaf w a) = w"
simps_1: fixes M :: "'b nfa"   and Q :: "hf set" assumes "nfa M" shows "nfa.nextl M Q [] = nfa.epsclo M Q"
prems: shows "x\<^sub>\<real> = y\<^sub>\<real>"
call\<^sub>C_def: fixes M :: "'f \<Rightarrow> 'd control_state_scheme \<Rightarrow> ('e \<times> 'd control_state_scheme) option"   and A\<^sub>1 :: "'d control_state_scheme \<Rightarrow> 'f" shows "call\<^sub>C M A\<^sub>1 = (\<lambda>\<sigma>. if exec_stop \<sigma> then Some (undefined, \<sigma>) else M (A\<^sub>1 \<sigma>) \<sigma>)"
eSuc_def: fixes i :: "enat" shows "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
no_Private_stat_override: fixes G :: "prog"   and new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl" assumes "G \<turnstile> new overrides\<^sub>S old" shows "accmodi old \<noteq> Private"
strand_subst_vars_union_bound: fixes S :: "('b, 'a) strand_step list"   and \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term" shows "vars\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>) \<subseteq> vars\<^sub>s\<^sub>t S \<union> range_vars \<delta>"
scaleR: fixes r :: "real"   and b :: "'a" shows "F (r *\<^sub>R b) = r *\<^sub>R F b"
wf_dbtm_trans_tm: fixes t :: "tm" shows "wf_dbtm (trans_tm [] t)"
seq_compact_imp_totally_bounded: fixes S :: "'b set" assumes "seq_compact S" shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> S \<and> S \<subseteq> (\<Union>x\<in>k. ball x e)"
prems: shows "as1 @ as2 = filter f1 (map f2 (a # p))"
standard_simplex_def: fixes p :: "nat" shows "standard_simplex p \<equiv> {x. (\<forall>i. 0 \<le> x i \<and> x i \<le> 1) \<and> (\<forall>i>p. x i = 0) \<and> sum x {..p} = 1}"
ahm_update_correct: fixes k :: "'a"   and v :: "'b"   and hm :: "('a, 'b) hashmap" shows "ahm_\<alpha> (ahm_update k v hm) = ahm_\<alpha> hm(k \<mapsto> v)"
sumlist_nth: fixes xs :: "'a vec list"   and i :: "nat" assumes "\<forall>x\<in>set xs. dim_vec x = n"   and "i < n" shows "M.sumlist xs $ i = (\<Sum>j = 0..<length xs. xs ! j $ i)"
list_update_code_2: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" shows "(x # xs)[0 := y] = y # xs"
ord_of_nat_oexp: fixes m :: "nat"   and n :: "nat" shows "ord_of_nat (m ^ n) = ord_of_nat m \<up> ord_of_nat n"
lt: shows "lt (monom_mult c (0::'a) f) = lt f"
double_SNa_aux: fixes a :: "trm"   and b :: "trm"   and P :: "trm \<Rightarrow> trm \<Rightarrow> bool" assumes "SNa a"   and "SNa b"   and "\<And>x z. \<lbrakk>\<And>y. x \<longrightarrow>\<^sub>a y \<Longrightarrow> SNa y; \<And>y. x \<longrightarrow>\<^sub>a y \<Longrightarrow> P y z; \<And>u. z \<longrightarrow>\<^sub>a u \<Longrightarrow> SNa u; \<And>u. z \<longrightarrow>\<^sub>a u \<Longrightarrow> P x u\<rbrakk> \<Longrightarrow> P x z" shows "P a b"
T_mtf_eq: fixes as :: "(nat \<times> nat list) list"   and rs :: "'b list"   and s0 :: "'b list" assumes "MTF_Off as rs s0" shows "MTF_Off.T_mtf rs s0 (length rs) = int (T_on MTF s0 rs)"
fw'_def: fixes m :: "nat \<times> nat \<Rightarrow> 'b"   and n :: "nat"   and k :: "nat" shows "fw' m n k = nfoldli [0..<k + 1] (\<lambda>_. True) (\<lambda>k M. for_rec2 (\<lambda>M. fw_upd' M k) M n n n) m"
sem_5c: shows "\<forall>X Y Z. \<I>X \<sqinter> Y \<sqinter> Z \<and> ob X Y \<and> ob X Z \<longrightarrow> ob X (Y \<sqinter> Z)"
len_mult2: fixes x :: "nat" shows "len (2 * x) = (if x = 0 then 0 else Suc (len x))"
refines_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "refines x y"   and "refines y z" shows "refines x z"
fact_diff_Suc: fixes n :: "nat"   and m :: "nat" assumes "n < Suc m" shows "fact (Suc m - n) = (Suc m - n) * fact (m - n)"