hcju/isabelleps huggingface dataset:
simps_2: fixes name :: "mname"   and parTs :: "ty list"   and more :: "'a" shows "parTs \<lparr>name = name, parTs = parTs, \<dots> = more\<rparr> = parTs"
less_eq_def: fixes x :: "'a"   and y :: "'a" shows "(x \<le> y) = (x \<squnion> y = y)"
left_step: shows "revision_step r s\<^sub>1 s\<^sub>2"
MGT_Body: fixes G :: "state triple set"   and Procs :: "pname set" assumes "G \<union> (\<lambda>pn. {=}.BODY pn.{->}) ` Procs||-(\<lambda>pn. {=}.the (body pn).{->}) ` Procs"   and "finite Procs" shows "G||-(\<lambda>pn. {=}.BODY pn.{->}) ` Procs"
lookup_sum: fixes pp :: "'s \<Rightarrow> 'r \<Rightarrow>\<^sub>0 'q"   and X :: "'s set"   and i :: "'r" shows "lookup (sum pp X) i = (\<Sum>x\<in>X. lookup (pp x) i)"
assms_1: shows "picard_lindeloef (\<lambda>t. f) UNIV UNIV 0"
next_subseqs2_set_def: fixes tail :: "'a iarray"   and v :: "'b"   and js :: "nat list" shows "next_subseqs2_set tail v js \<equiv> {(j, f (tail !! j) v) |j. j \<in> set js}"
E: shows "Crypt (sesK (d * e)) (Passwd n) \<in> synth (analz (A \<union> spies evsFC5))"
add_job: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and x :: "nat" assumes "lb T A j"   and "x \<in> {1..m}" shows "lb (T(x := T x + t (Suc j))) (A(x := A x \<union> {Suc j})) (Suc j)"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" shows "((a1, a2) \<in> encoding.indRel Enc) = ((\<exists>S. a1 = SourceTerm S \<and> a2 = TargetTerm (Enc S)) \<or> (\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S))"
mono_sup: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b"   and B :: "'b" assumes "mono f" shows "f A \<sqinter> f B \<le> f (A \<sqinter> B)"
assms_4: shows "q \<cdot> l \<le> l \<cdot> q"
compiler_composition_pow: fixes step :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and final :: "'d \<Rightarrow> bool"   and load :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and order :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and match :: "'f \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and compile :: "'e \<Rightarrow> 'e option"   and n :: "nat" assumes "compiler step step final final load load order match compile" shows "compiler step step final final load load (lexp order\<^sup>+\<^sup>+) (rel_comp_pow match) (option_comp_pow compile n)"
asm: shows "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') = 0"
term_variants_Fun: fixes T :: "('e, 'f) Term.term list"   and S :: "('e, 'f) Term.term list"   and P :: "'e \<Rightarrow> 'e list"   and f :: "'e" assumes "length T = length S"   and "\<And>i. i < length T \<Longrightarrow> term_variants_pred P (T ! i) (S ! i)" shows "term_variants_pred P (Fun f T) (Fun f S)"
wf_eq_wfzf: fixes R :: "(ZF \<times> ZF) set" shows "(wf R \<and> set_like R) = wfzf R"
in_\<R>\<^sub>A': shows "(\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>1\<^sub>C [\<parallel>\<^sub>1 A\<^sub>A_of A\<^sub>C]\<rangle>\<^sub>A, \<langle>c\<^sub>2\<^sub>A, mds\<^sub>A_of mds\<^sub>C, mem\<^sub>A_of mem\<^sub>2\<^sub>C [\<parallel>\<^sub>2 A\<^sub>A_of A\<^sub>C]\<rangle>\<^sub>A) \<in> \<R>\<^sub>A"
a: shows "a \<in> R\<^sup>* `` set_of init"
awalk_hd_in_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" shows "u \<in> pverts (subdivide G e w)"
quotient_of_div: fixes r :: "rat"   and n :: "int"   and d :: "int" assumes "quotient_of r = (n, d)" shows "r = rat_of_int n / rat_of_int d"
phi_def: fixes i :: "nat"   and x :: "nat" shows "\<phi> i x \<equiv> eval r_phi [i, x]"
ChiPlus_ChiPlus_NoCycles: fixes S :: "'a"   and T :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(S, T) \<in> ChiPlus A" shows "(T, S) \<notin> ChiPlus A"
scene_union_commute: fixes X :: "'a scene"   and Y :: "'a scene" shows "X \<squnion>\<^sub>S Y = Y \<squnion>\<^sub>S X"
f_Exec_Stream_append_last: fixes xs :: "'d list"   and trans_fun :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and ys :: "'d list"   and c :: "'e" assumes "xs \<noteq> []" shows "f_Exec_Comp_Stream trans_fun (xs @ ys) c = f_Exec_Comp_Stream trans_fun xs c @ f_Exec_Comp_Stream trans_fun ys (last (f_Exec_Comp_Stream trans_fun xs c))"
single_valuedp_iff_Uniq: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "single_valuedp r = (\<forall>x. \<exists>\<^sub>\<le>\<^sub>1 y. r x y)"
rep: shows "int (size (filter_mset ((\<in>) x) \<B>)) = \<Lambda> * (\<v> - 1) div (\<k> - 1)"
coboundedI: fixes A :: "'d set"   and a :: "'d" assumes "finite A"   and "a \<in> A" shows "Min A \<le> a"
assms_7: shows "InvariantClCurrentLevel (getCl state) (getM state)"
euclidean_all_zero_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and x :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "(\<forall>u\<in>Basis. inner x u = 0) = (x = zero)"
zip_eq_ConsE: fixes xs :: "'a list"   and ys :: "'b list"   and xy :: "'a \<times> 'b"   and xys :: "('a \<times> 'b) list" assumes "zip xs ys = xy # xys" obtains x :: "'a" and xs' :: "'a list" and y :: "'b" and ys' :: "'b list" where "xs = x # xs'" and "ys = y # ys'" and "xy = (x, y)" and "xys = zip xs' ys'"
two_comparisons_into_case_order_10: fixes x :: "'a"   and y :: "'a"   and P :: "'c"   and Q :: "'c"   and R :: "'c" shows "(if x = y then P else if lt x y then Q else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
snd_map_prod: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'b"   and x :: "'e \<times> 'f" shows "snd (map_prod f g x) = g (snd x)"
center: assumes "c1' = c1"   and "qs' = qs"   and "c2' = c2" shows "thesis"
ba3_top_unique: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> - x = y \<squnion> - y"
nfv_def: fixes \<phi> :: "'b formula" shows "nfv \<phi> = Max (insert 0 (Suc ` fv \<phi>))"
nprv_eql_substE: fixes F :: "'fmla set"   and t1 :: "'trm"   and t2 :: "'trm"   and \<phi> :: "'fmla"   and y :: "'var"   and \<psi> :: "'fmla" assumes "nprv F (eql t1 t2)"   and "nprv F (subst \<phi> t2 y)"   and "nprv (insert (subst \<phi> t1 y) F) \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "\<phi> \<in> fmla"   and "y \<in> var" shows "nprv F \<psi>"
midpoint_idem: fixes x :: "'a" shows "midpoint x x = x"
Try_\<tau>red1t_xt: 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"   and C :: "String.literal"   and V :: "nat"   and e2 :: "'addr expr1" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (try e catch(C V) e2, xs) (try e' catch(C V) e2, xs')"
valid_path_edges: fixes P :: "'a llist"   and n :: "nat" assumes "valid_path P"   and "enat (Suc n) < llength P" shows "P $ n \<rightarrow> P $ Suc n"
hyps: fixes f :: "nat \<Rightarrow> 'a" shows "lnth (inf_llist f) n = f n"
distinct_15: fixes x1 :: "int"   and x91 :: "int"   and x92 :: "MIR.num"   and x93 :: "MIR.num" shows "C x1 \<noteq> CF x91 x92 x93"
hyps_3: assumes "\<And>i. i \<in> I \<Longrightarrow> f i \<in> hom_deg_set z A" shows "sum f I \<in> hom_deg_set z A"
st': shows "(Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m') / 2 = Inum (x # bs) (Add (Mul m' t') (Mul n' s')) / real_of_int (2 * n' * m')"
invmode_def: fixes m :: "'b member_scheme"   and e :: "expr" shows "invmode m e = (if is_static m then Static else if e = Super then SuperM else IntVir)"
divides_cong_r: fixes x :: "'a"   and y :: "'a"   and y' :: "'a" assumes "x divides y"   and "y \<sim> y'"   and "x \<in> carrier G" shows "x divides y'"
order_trans_rules_25: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "b = c" shows "a \<le> c"
adjacent_Cons: fixes w :: "nat \<times> nat"   and t :: "'b tree"   and ts :: "'b tree list" shows "adjacent w (t # ts) = (t \<noteq> Leaf \<and> r t = snd w \<and> (case ts of [] \<Rightarrow> l t = fst w | u # us \<Rightarrow> adjacent (fst w, r u) ts \<and> l t = Suc (r u)))"
cb1: shows "c - b + (c + b) = 2 * c"
dests_bigger_than_sqn: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). l \<in> {PAodv-:15..PAodv-:19} \<union> {PPkt-:7..PPkt-:11} \<union> {PRreq-:9..PRreq-:13} \<union> {PRreq-:21..PRreq-:25} \<union> {PRrep-:9..PRrep-:13} \<union> {PRerr-:1..PRerr-:5} \<longrightarrow> (\<forall>ip\<in>dom (dests \<xi>). ip \<in> kD (rt \<xi>) \<and> sqn (rt \<xi>) ip \<le> the (dests \<xi> ip)))"
var_map_of_subst_def: fixes \<sigma> :: "nat \<Rightarrow> ('a, nat) Term.term"   and v :: "nat" shows "var_map_of_subst \<sigma> v = the_Var (\<sigma> v)"
prob_space_C: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} M)" shows "prob_space (C n m x)"
dom: fixes s :: "(char list \<Rightarrow> int option) \<times> nat" shows "I s \<longrightarrow> vars b \<subseteq> dom (fst s)"
eq\<rho>_join: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae' :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<rho>1 :: "Vars.var \<Rightarrow> Value"   and \<rho>2 :: "Vars.var \<Rightarrow> Value" shows "eq\<rho> (ae \<squnion> ae') \<rho>1 \<rho>2 = (eq\<rho> ae \<rho>1 \<rho>2 \<and> eq\<rho> ae' \<rho>1 \<rho>2)"
s11_inj: fixes p :: "nat" shows "inj (\<lambda>x. smn 1 p [x])"
ex_ex1I: fixes P :: "'d \<Rightarrow> bool" assumes "\<exists>x. P x"   and "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> x = y" shows "\<exists>!x. P x"
simps_1: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "restricted_subformulas (\<phi> and\<^sub>n \<psi>) = restricted_subformulas \<phi> \<union> restricted_subformulas \<psi>"
rel_nondetT_simps: fixes M :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and m :: "'d"   and m' :: "'e" shows "rel_nondetT M (NondetT m) (NondetT m') = M m m'"
singlPart_def: fixes I :: "'b set" shows "singlPart I \<equiv> {{i} |i. i \<in> I}"
tagged_division_split_left_inj: fixes d :: "('a \<times> 'a set) set"   and i :: "'a set"   and x1 :: "'a"   and K1 :: "'a set"   and x2 :: "'a"   and K2 :: "'a set"   and k :: "'a"   and c :: "real" assumes "d tagged_division_of i"   and "(x1, K1) \<in> d"   and "(x2, K2) \<in> d"   and "K1 \<noteq> K2"   and "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}" shows "interior (K1 \<inter> {x. x \<bullet> k \<le> c}) = {}"
lift_sup_distrib: fixes x :: "V"   and a :: "V"   and b :: "V" shows "lift x (a \<squnion> b) = lift x a \<squnion> lift x b"
bounds_lg_4: fixes c :: "'a"   and b :: "'a option"   and x :: "'a" assumes "c <\<^sub>l\<^sub>b b"   and "x \<ge>\<^sub>l\<^sub>b b" shows "c < x"
simps_3: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a"   and x :: "'b"   and y :: "'c" shows "lift2_option f (Some x) (Some y) = Some (f x y)"
encode_goal_state_valid: fixes sv :: "sat_plan_variable"   and Prob :: "'a strips_problem"   and t :: "nat" assumes "sv \<in> atoms ((\<Phi>\<^sub>G Prob) t)"   and "t < Suc h" shows "valid_state_var sv"
quotient_of_unique: fixes r :: "rat" shows "\<exists>!p. r = Rat.Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)"
BInit_def: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "BInit ic q oc \<equiv> $q = #[] \<and> $ic = $oc"
Inr_Inl_False: fixes x :: "'b"   and y :: "'a" shows "(Inr x = Inl y) = False"
antisym: fixes b :: "'b list"   and a :: "'b list" assumes "b \<le>p a"   and "a \<le>p b" shows "a = b"
butlast: shows "butlast (D # E # Es) = D # rev Cs'"
strict_implies_not_eq: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<subset># b" shows "a \<noteq> b"
map_spmf_conv_bind_spmf: fixes f :: "'m \<Rightarrow> 'l"   and p :: "'m spmf" shows "map_spmf f p = p \<bind> (\<lambda>x. return_spmf (f x))"
permutes_bij: fixes p :: "'a \<Rightarrow> 'a"   and s :: "'a set" assumes "p permutes s" shows "bij p"
lhopital_right_at_top_at_bot: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and g :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" assumes "filterlim f at_top (at_right a)"   and "filterlim g at_bot (at_right a)"   and "\<forall>\<^sub>F x in at_right a. (f has_real_derivative f' x) (at x)"   and "\<forall>\<^sub>F x in at_right a. (g has_real_derivative g' x) (at x)"   and "LIM x at_right a. f' x / g' x :> at_bot" shows "LIM x at_right a. f x / g x :> at_bot"
comp_fun_commute_Image_fold: fixes S :: "'b set" shows "comp_fun_commute (\<lambda>(x, y) A. if x \<in> S then insert y A else A)"
red_single_0_lt: fixes f :: "'a \<Rightarrow>\<^sub>0 'c"   and h :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "punit.red_single f 0 h t" shows "punit.lt f = local.punit.splus t (punit.lt h)"
lnth_lappend2: fixes xs :: "'a llist"   and k :: "nat"   and n :: "nat"   and ys :: "'a llist" assumes "llength xs = enat k"   and "k \<le> n" shows "lnth (xs @\<^sub>l ys) n = lnth ys (n - k)"
in_closed_segment_iff_rot: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<noteq> b" shows "(x \<in> {a--b}) = ((x - a) \<bullet> rot (b - a) = 0 \<and> x \<bullet> (b - a) \<in> {a \<bullet> (b - a)..b \<bullet> (b - a)})"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "\<langle>'b\<rangle>\<^sub>\<bottom>" shows "P (case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = (\<not> (option = \<bottom> \<and> \<not> P f1 \<or> (\<exists>x2. option = \<lfloor>x2\<rfloor> \<and> \<not> P (f2 x2))))"
map_le_map_add: fixes f :: "'d \<Rightarrow> 'e option"   and g :: "'d \<Rightarrow> 'e option" shows "f \<subseteq>\<^sub>m g ++ f"
transE: fixes r :: "('c \<times> 'c) set"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "trans r"   and "x \<^bsub>r\<^esub>\<preceq> y"   and "y \<^bsub>r\<^esub>\<preceq> z" obtains "x \<^bsub>r\<^esub>\<preceq> z"
mono_ndet_D: fixes P :: "'a process"   and P' :: "'a process"   and S :: "'a process"   and S' :: "'a process" assumes "P \<sqsubseteq>\<^sub>D P'"   and "S \<sqsubseteq>\<^sub>D S'" shows "(P |-| S) \<sqsubseteq>\<^sub>D (P' |-| S')"
canonical_embedding_ring_hom: fixes K :: "'a set" assumes "subring K R" shows "ring_hom_ring (R\<lparr>carrier := K\<rparr>) (K [X]) poly_of_const"
neg_0_le_iff_le: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq zero (uminus a) = less_eq a zero"
split_case: fixes t :: "('c \<times> nat) tree" shows "split t = (case t of \<langle>\<rangle> \<Rightarrow> \<langle>\<rangle> | \<langle>t1, (x, lvx), \<langle>\<rangle>\<rangle> \<Rightarrow> \<langle>t1, (x, lvx), \<langle>\<rangle>\<rangle> | \<langle>t1, (x, lvx), \<langle>t2, (y, lvy), \<langle>\<rangle>\<rangle>\<rangle> \<Rightarrow> \<langle>t1, (x, lvx), \<langle>t2, (y, lvy), \<langle>\<rangle>\<rangle>\<rangle> | \<langle>t1, (x, lvx), \<langle>t2, (y, lvy), \<langle>t3, (z, lvz), t4\<rangle>\<rangle>\<rangle> \<Rightarrow> if lvx = lvy \<and> lvy = lvz then \<langle>\<langle>t1, (x, lvx), t2\<rangle>, (y, lvx + 1), \<langle>t3, (z, lvx), t4\<rangle>\<rangle> else t)"
scene_union_pres_compat: fixes A :: "'a scene"   and B :: "'a scene"   and C :: "'a scene" assumes "A ##\<^sub>S B"   and "A ##\<^sub>S C" shows "A ##\<^sub>S B \<squnion>\<^sub>S C"
wf_until_aux_def: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi> :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi> :: "'a MFOTL.formula"   and aux :: "(nat \<times> 'a option list set \<times> 'a option list set) list"   and ne :: "nat" shows "wf_until_aux \<sigma> n R pos \<phi> I \<psi> aux ne = list_all2 (\<lambda>x i. case x of (t, r1, r2) \<Rightarrow> t = \<tau> \<sigma> i \<and> qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. if pos then \<forall>k\<in>{i..<ne + length aux}. MFOTL.sat \<sigma> (map the v) k \<phi> else \<exists>k\<in>{i..<ne + length aux}. MFOTL.sat \<sigma> (map the v) k \<phi>) r1 \<and> qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. \<exists>j\<ge>i. j < ne + length aux \<and> mem (\<tau> \<sigma> j - \<tau> \<sigma> i) I \<and> MFOTL.sat \<sigma> (map the v) j \<psi> \<and> (\<forall>k\<in>{i..<j}. if pos then MFOTL.sat \<sigma> (map the v) k \<phi> else \<not> MFOTL.sat \<sigma> (map the v) k \<phi>)) r2) aux [ne..<ne + length aux]"
D_hiding: fixes P :: "'b process"   and A :: "'b set" shows "D (P \ A) = div_hide P A"
SC3: fixes p :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "singleCombinators p" shows "singleCombinators (FWNormalisationCore.sort p l)"
cycle: fixes G :: "('a, 'b) graph_scheme"   and v :: "'a"   and p :: "'a list" assumes "path G v p v"   and "p \<noteq> []"   and "distinct p" shows "cycle G p"
s1_s2_P2: fixes x :: "nat"   and xa :: "nat"   and xb :: "nat"   and xc :: "nat"   and y :: "nat" assumes "x < q"   and "xa < q"   and "xb < q"   and "xc < q"   and "y < q" shows "((y, xa, (xb * xa + q - xc) mod q, (x + xb) mod q), (x * ((y + q - xa) mod q) + q - xc) mod q, ((x + xb) mod q * xa + q - (xb * xa + q - xc) mod q) mod q) = ((y, xa, (xb * xa + q - xc) mod q, (x + xb) mod q), (x * ((y + q - xa) mod q) + q - xc) mod q, (x * y + q - (x * ((y + q - xa) mod q) + q - xc) mod q) mod q)"
dord_fun_antisym: fixes ord :: "('a \<Rightarrow> 'c) \<Rightarrow> ('a \<Rightarrow> 'c) \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'c"   and t :: "'a \<Rightarrow> 'c" assumes "\<lbrakk>ord s t; ord t s\<rbrakk> \<Longrightarrow> s = t"   and "dord_fun ord s t"   and "dord_fun ord t s" shows "s = t"
snap_p_1: shows "\<not> ps (S t i) p \<noteq> None"
alw_mono: fixes \<phi> :: "'a stream \<Rightarrow> bool"   and xs :: "'a stream"   and \<psi> :: "'a stream \<Rightarrow> bool" assumes "alw \<phi> xs"   and "\<And>xs. \<phi> xs \<Longrightarrow> \<psi> xs" shows "alw \<psi> xs"
simps_1: fixes a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term" shows "tfr\<^sub>s\<^sub>t\<^sub>p \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t = ((\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t')"
bound_indices: shows "f 0 = a\<^sub>1 \<and> f (card Y - 1) = a\<^sub>n"
HeronConf_interp_stepwise_tagrel_cases: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and R :: "'c tag_const \<times> 'c tag_const \<Rightarrow> bool"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g"
Quotient3_rel: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and r :: "'b"   and s :: "'b" assumes "Quotient3 R Abs Rep" shows "(R r r \<and> R s s \<and> Abs r = Abs s) = R r s"
atms_of_plus: fixes C :: "'a literal multiset"   and D :: "'a literal multiset" shows "atms_of (C + D) = atms_of C \<union> atms_of D"
subset_iff: fixes A :: "'f set"   and B :: "'f set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
bpred_def: fixes p :: "'b discr \<Rightarrow> 'b discr \<Rightarrow> bool" shows "bpred p \<equiv> \<Lambda> x y. unbox\<cdot>x >>= (\<Lambda> x'. unbox\<cdot>y >>= (\<Lambda> y'. if p x' y' then TT else FF))"
SubsetGuards_sound: fixes c :: "('d, 'e, 'f) com"   and c' :: "('d, 'e, 'f) com"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and P :: "'d set"   and Q :: "'d set"   and A :: "'d set" assumes "c \<subseteq>\<^sub>g c'"   and "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c' Q,A" shows "\<Gamma>,\<Theta>\<Turnstile>\<^sub>t\<^bsub>/{}\<^esub> P c Q,A"
assms_3: shows "msgs (S t (i + 1)) cid \<noteq> []"
sum: shows "(\<Sum>i\<le>n. p i * i) = n"
simps_4: fixes F :: "'b set \<Rightarrow> 'a"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and M :: "'b acom set" shows "lift F (IF b THEN c1 ELSE c2) M = IF b THEN lift F c1 (sub\<^sub>1 ` M) ELSE lift F c2 (sub\<^sub>2 ` M) {F (post ` M)}"
inject_8: fixes x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly"   and y81 :: "poly"   and y82 :: "nat"   and y83 :: "poly" shows "(CN x81 x82 x83 = CN y81 y82 y83) = (x81 = y81 \<and> x82 = y82 \<and> x83 = y83)"
eqv_class_eq: fixes a :: "'b"   and b :: "'b" shows "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
snull_FF_conv: fixes xxs :: "[:'r:]" shows "(snull\<cdot>xxs = FF) = (\<exists>x xs. xxs \<noteq> \<bottom> \<and> xxs = x :# xs)"
One_fract_def: shows "1 = Fract (1::'b) (1::'b)"
b_def: shows "b \<in> carrier R \<and> f = monom P b n"
substitutes_onI: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "\<And>B a b. \<lbrakk>B \<subseteq> A; a \<in> A; b \<in> A; b \<notin> f (insert b B)\<rbrakk> \<Longrightarrow> b \<notin> f (insert a (insert b B))" shows "substitutes_on A f"
update_convs_3: fixes tj_stack' :: "'c list \<Rightarrow> 'c list"   and counter :: "nat"   and discovered :: "'c \<Rightarrow> nat option"   and finished :: "'c \<Rightarrow> nat option"   and pending :: "('c \<times> 'c) set"   and stack :: "'c list"   and tree_edges :: "('c \<times> 'c) set"   and back_edges :: "('c \<times> 'c) set"   and cross_edges :: "('c \<times> 'c) set"   and sccs :: "'c set set"   and lowlink :: "'c \<Rightarrow> nat option"   and tj_stack :: "'c list"   and more :: "'d" shows "tj_stack_update tj_stack' \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack, \<dots> = more\<rparr> = \<lparr>counter = counter, discovered = discovered, finished = finished, pending = pending, stack = stack, tree_edges = tree_edges, back_edges = back_edges, cross_edges = cross_edges, sccs = sccs, lowlink = lowlink, tj_stack = tj_stack' tj_stack, \<dots> = more\<rparr>"
squarefree_power_iff: fixes n :: "'b"   and m :: "nat" shows "squarefree (n ^ m) = (m = 0 \<or> is_unit n \<or> squarefree n \<and> m = 1)"