hcju/isabelleps huggingface dataset:
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and x :: "name"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>l a[b] \<prec>\<^sub>l P'"   and "Q \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l Q'" shows "P \<parallel> Q \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P' \<parallel> Q'[x::=b]"
det_mul: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and B :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "Determinants.det (A ** B) = Determinants.det A * Determinants.det B"
Sup_least: fixes Inf :: "'b::type set \<Rightarrow> 'b::type"   and Sup :: "'b::type set \<Rightarrow> 'b::type"   and inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and bot :: "'b::type"   and top :: "'b::type"   and A :: "'b::type set"   and z :: "'b::type" assumes "class.complete_lattice (Inf::'b::type set \<Rightarrow> 'b::type) (Sup::'b::type set \<Rightarrow> 'b::type) (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (bot::'b::type) (top::'b::type)"   and "\<And>x::'b::type. x \<in> (A::'b::type set) \<Longrightarrow> (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) x (z::'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) ((Sup::'b::type set \<Rightarrow> 'b::type) (A::'b::type set)) (z::'b::type)"
elims_2: fixes x :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" assumes "is_final_dstate x" obtains N :: "('a literal list \<times> nat) list" and P :: "('a literal list \<times> nat) list" and Q :: "('a literal list \<times> nat) list" and n :: "nat" where "x = (N, P, Q, n)" and "N = [] \<and> P = []"
zero_vec_def: shows "0 \<equiv> \<chi>i. (0::'c)"
wt_hd_le: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<epsilon> :: "nat"   and \<delta> :: "nat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and wt_sym :: "'b \<Rightarrow> nat"   and s :: "('b, 'a) tm" assumes "kbo_std ground_heads_var gt_sym \<epsilon> \<delta> extf arity_sym arity_var wt_sym" shows "kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym (Hd (head s)) \<le> kbo_std.wt ground_heads_var \<delta> arity_sym wt_sym s"
induct: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a list"   and x2 :: "'b list"   and P :: "'a list \<Rightarrow> 'b list \<Rightarrow> bool" assumes "listrelp r x1 x2"   and "P [] []"   and "\<And>x y xs ys. \<lbrakk>r x y; listrelp r xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (y # ys)" shows "P x1 x2"
check2: shows "\<not> (w1 = POS_INF \<and> w2 = POS_INF)"
I_axiom_lemma: fixes I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" assumes "is_interp I"   and "\<nu> \<in> fml_sem I ([[$\<alpha> vid1**]]Pc pid1 \<rightarrow> [[$\<alpha> vid1]]Pc pid1)"   and "\<nu> \<in> fml_sem I (Pc pid1)" shows "\<nu> \<in> fml_sem I ([[$\<alpha> vid1**]]Pc pid1)"
subset_reorder: fixes a :: "'c"   and b :: "'c"   and c :: "'c set" assumes "{a, b} \<subseteq> c" shows "{b, a} \<subseteq> c"
assms: shows "\<turnstile> p : T \<Rightarrow> \<Delta>"
Rep_nat_int_inverse: fixes x :: "nat_int" shows "Abs_nat_int (Rep_nat_int x) = x"
logical_new_psr_val_def: fixes result :: "word32"   and s :: "'b sparc_state" shows "logical_new_psr_val result s \<equiv> let psr_val = cpu_reg_val PSR s; n_val = ucast (result >> 31); z_val = if result = 0 then 1 else 0; v_val = 0; c_val = 0 in update_PSR_icc n_val z_val v_val c_val psr_val"
le_RES_nofailI: fixes a :: "'b nres"   and x :: "'b set" assumes "a \<le> RES x" shows "nofail a"
sat\<^sub>b_def: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula" shows "sat\<^sub>b \<AA> \<phi> \<equiv> sat_vars_gen (\<lambda>k P. (\<le>) (len P)) UNIV \<AA> \<phi>"
span_explicit': fixes b :: "'b set" shows "span b = {\<Sum>v | f v \<noteq> (0::'a). f v *s v |f. finite {v. f v \<noteq> (0::'a)} \<and> (\<forall>v. f v \<noteq> (0::'a) \<longrightarrow> v \<in> b)}"
lt_comm: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "A B Lt C D" shows "B A Lt D C"
mult_sign_intros_5: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) < a"   and "(0::'c) < b" shows "(0::'c) < a * b"
isTrue_implies_bprv_PPf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (PPf n \<langle>\<phi>\<rangle>)" shows "bprv (PPf n \<langle>\<phi>\<rangle>)"
simps_2: fixes p :: "nat"   and a :: "nat"   and r :: "nat"   and y :: "nat"   and ys :: "nat list" shows "build_fpc p a r (y # ys) = Triple p a r # build_fpc p a (r div y) ys"
adhoc_le: fixes d :: "real"   and c :: "real" assumes "\<not> d * 4 \<le> 3"   and "0 \<le> c"   and "c \<le> 1" shows "c * 4 \<le> 1 + c * (d * 4)"
simps_1: fixes P :: "'f \<Rightarrow> bool" shows "takeWhile P [] = []"
psubstT_def: fixes t :: "'trm"   and txs :: "('trm \<times> 'var) list" shows "psubstT t txs = (let xs = map snd txs; ts = map fst txs; us = getFrN xs (t # ts) [] (length xs) in rawpsubstT (rawpsubstT t (zip (map Var us) xs)) (zip ts us))"
empty_neutral_2: fixes x :: "'d multiset" shows "x + {#} = x"
cone_decomp_singleton: fixes t :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" shows "cone_decomp (cone (t, U)) [(t, U)]"
assms: shows "(subst_domain \<delta> \<union> range_vars \<delta>) \<inter> bvars\<^sub>s\<^sub>s\<^sub>t S = {}"
eucl_rel_poly_mult: fixes x :: "'b poly"   and y :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and z :: "'b poly"   and q' :: "'b poly"   and r' :: "'b poly" assumes "eucl_rel_poly x y (q, r)"   and "eucl_rel_poly q z (q', r')" shows "eucl_rel_poly x (y * z) (q', y * r' + r)"
imp_igSubstInpIPresIGWlsInpSTR: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme" assumes "igSubstIPresIGWlsSTR MOD" shows "igSubstInpIPresIGWlsInpSTR MOD"
extended_Ax: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<inter># \<Delta> \<noteq> {#}" shows "\<Gamma> \<Rightarrow> \<Delta>"
star_def: fixes A :: "'b list set" shows "star A = \<Union> (range ((^^) A))"
rel_those_compile_env_entries: fixes \<A> :: "'fun \<Rightarrow> nat option"   and \<O> :: "'fun \<Rightarrow> nat \<Rightarrow> Unboxed.type option"   and xs :: "('fun \<times> ('b, 'c, 'fun, 'd, 'e) Inca.instr fundef) list"   and ys :: "('fun \<times> ('b, 'c, 'fun, 'd, 'e, 'g, 'h, 'i) Ubx.instr fundef) list" assumes "Result.those (map (compile_env_entry \<A> \<O>) xs) = Ok ys" shows "rel_fundefs (Finca_get (Sinca.Fenv.from_list xs)) (Fubx_get (Subx.Fenv.from_list ys))"
simps_122: fixes f1 :: "char list \<Rightarrow> 'b"   and f2 :: "48 word \<Rightarrow> 'b"   and f3 :: "48 word \<Rightarrow> 'b"   and f4 :: "16 word \<Rightarrow> 'b"   and f5 :: "16 word \<Rightarrow> 'b"   and f6 :: "16 word \<Rightarrow> 'b"   and f7 :: "32 prefix_match \<Rightarrow> 'b"   and f8 :: "32 prefix_match \<Rightarrow> 'b"   and f9 :: "8 word \<Rightarrow> 'b"   and f10 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and f11 :: "16 word \<Rightarrow> 16 word \<Rightarrow> 'b"   and x1 :: "char list" shows "(case IngressPort x1 of IngressPort x \<Rightarrow> f1 x | EtherSrc x \<Rightarrow> f2 x | EtherDst x \<Rightarrow> f3 x | EtherType x \<Rightarrow> f4 x | VlanId x \<Rightarrow> f5 x | VlanPriority x \<Rightarrow> f6 x | IPv4Src x \<Rightarrow> f7 x | IPv4Dst x \<Rightarrow> f8 x | IPv4Proto x \<Rightarrow> f9 x | L4Src x xa \<Rightarrow> f10 x xa | L4Dst x xa \<Rightarrow> f11 x xa) = f1 x1"
fold_A_chamber_system_image_fixespointwise': fixes X :: "'b set set"   and \<A> :: "'b set set set"   and A :: "'b set set"   and B :: "'b set set"   and B' :: "'b set set"   and C :: "'b set"   and D :: "'b set"   and E :: "'b set"   and z :: "'b set"   and F :: "'b set" assumes "ChamberComplexApartmentSystemTriangle X \<A> A B B' C D E z"   and "F \<in> {F \<in> ChamberComplex.chamber_system A. ChamberComplex.face_distance X z F = ChamberComplex.chamber_distance X C F}" shows "fixespointwise (ChamberComplexWithApartmentSystem.canonical_retraction X \<A> A D \<circ> ChamberComplexWithApartmentSystem.canonical_retraction X \<A> B C) F"
unat_eq_of_nat: fixes n :: "nat"   and x :: "'b word" assumes "n < 2 ^ LENGTH('b)" shows "(unat x = n) = (x = word_of_nat n)"
simps_4: fixes xc' :: "int \<Rightarrow> int"   and xc :: "int"   and yc :: "int"   and more :: "'a" shows "xc_update xc' \<lparr>xc = xc, yc = yc, \<dots> = more\<rparr> = \<lparr>xc = xc' xc, yc = yc, \<dots> = more\<rparr>"
uniformity_fun_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
succs_def: fixes P :: "instr list"   and n :: "int" shows "succs P n = {s. \<exists>i\<ge>0. i < size P \<and> s \<in> isuccs (P !! i) (n + i)}"
UP_car_memE_1: fixes g :: "nat \<Rightarrow> nat \<Rightarrow> int"   and x :: "nat" assumes "g \<in> carrier (UP Zp)" shows "g x \<in> carrier Zp"
upd_callD: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and c :: "char list"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and m :: "'c match_expr"   and t :: "state" assumes "\<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> [Rule m (Call c)] \<Rightarrow>\<^sub>z t"   and "matches \<gamma> m p" shows "\<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t \<or> (\<exists>rs\<^sub>1 rs\<^sub>2 m'. rs = rs\<^sub>1 @ Rule m' Return # rs\<^sub>2 \<and> matches \<gamma> m' p \<and> \<Gamma>(c \<mapsto> rs),\<gamma>,p\<turnstile> rs\<^sub>1 \<Rightarrow>\<^sub>z Undecided \<and> t = Undecided)"
assms: fixes U :: "'v set" assumes "RModule FG (\<cdot>) U \<and> U \<subseteq> V" shows "U = 0 \<or> U = V"
distinct_disc_16: fixes event :: "('a, 'b, 'c) event" assumes "isSnapshot event" shows "\<not> isRecv event"
ct_suffix_nil: fixes ts :: "ct list" shows "ct_suffix [] ts"
OrdIndH: fixes j :: "name"   and i :: "name"   and A :: "fm"   and H :: "fm set" assumes "atom j \<sharp> (i, A)"   and "H \<turnstile> SyntaxN.All i (OrdP (Var i) IMP All2 j (Var i) (A(i::=Var j)) IMP A)" shows "insert (OrdP (Var i)) H \<turnstile> A"
store_sub1_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and rd :: "word5"   and s_val :: "word1"   and s :: "'b sparc_state" assumes "s' = snd (fst (store_sub1 instr rd s_val s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
monotone_extreme_extreme_boundI: fixes f :: "'d \<Rightarrow> 'c"   and I :: "'d set"   and A :: "'c set"   and ir :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and e :: "'d"   and x :: "'c" assumes "f ` I \<subseteq> A"   and "monotone_on I ir r f"   and "extreme I ir e"   and "x = f e" shows "extreme_bound A r (f ` I) x"
less_le_not_le: fixes x :: "complex"   and y :: "complex" shows "(x <\<^sub>\<complex> y) = (x \<le>\<^sub>\<complex> y \<and> \<not> y \<le>\<^sub>\<complex> x)"
shift_snth: fixes xs :: "'d list"   and s :: "'d stream"   and n :: "nat" shows "(xs @- s) !! n = (if n < length xs then xs ! n else s !! (n - length xs))"
veqpoll_vsingleton: fixes x :: "V"   and y :: "V" shows "set {x} \<approx>\<^sub>\<circ> set {y}"
even_succ_div_2: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and a :: "'a" assumes "class.semiring_bits plus minus zero times one divide modulo bit"   and "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a" shows "divide (plus one a) (numeral.numeral one plus (num.Bit0 num.One)) = divide a (numeral.numeral one plus (num.Bit0 num.One))"
distinct_set_conv_list: fixes xs :: "'b list"   and g :: "'b \<Rightarrow> 'a" assumes "distinct xs" shows "sum g (set xs) = sum_list (map g xs)"
pdom_res_override: fixes A :: "'a set"   and f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "A \<lhd>\<^sub>p (f + g) = A \<lhd>\<^sub>p f + A \<lhd>\<^sub>p g"
assms_1: shows "x \<in>\<^sub>\<circ> A"
simps_1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option"   and xs :: "'varSort"   and x :: "'var" shows "qVar xs x #[[rho]] = (case rho xs x of None \<Rightarrow> qVar xs x | Some X \<Rightarrow> X)"
coboundedI1: fixes a :: "nat"   and c :: "nat"   and b :: "nat" assumes "a dvd c" shows "gcd a b dvd c"
try_gpv_bind_gpv: fixes gpv :: "('k, 'i, 'j) gpv"   and f :: "'k \<Rightarrow> ('h, 'i, 'j) gpv"   and gpv' :: "('h, 'i, 'j) gpv" shows "TRY gpv \<bind> f ELSE gpv' = TRY map_gpv Some id gpv ELSE Done None \<bind> (\<lambda>x. case x of None \<Rightarrow> gpv' | Some x' \<Rightarrow> TRY f x' ELSE gpv')"
subst_bvs_def: fixes args :: "Core.term list"   and t :: "Core.term" shows "subst_bvs args t \<equiv> subst_bvs1 t 0 args"
iadd_ileD1: fixes m :: "enat"   and k :: "enat"   and n :: "enat" assumes "m + k \<le> n" shows "m \<le> n"
semidecomposition: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and a :: "'b" assumes "p permutes S"   and "finite S" shows "(\<lambda>y. if y \<in> S - set (support p a) then p y else y) permutes S - set (support p a)"
ord_le_eq_trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "b \<le> a"   and "b = c" shows "c \<le> a"
lookup_lower_when: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and u :: "'t"   and v :: "'t" shows "lookup (lower p u) v = (lookup p v when v \<prec>\<^sub>t u)"
is_shadow_root_ptr_def: fixes ptr :: "(_) shadow_root_ptr" shows "is_shadow_root_ptr\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case ptr of shadow_root_ptr.Ref x \<Rightarrow> True | shadow_root_ptr.Ext shadow_root_ptr \<Rightarrow> False)"
rep_eq: 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 x :: "'c word" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus x = ring_1.of_int one plus zero minus (signed_take_bit (LENGTH('c) - Suc 0) (Word.rep x))"
assms: shows "P \<longmapsto>\<^sub>l \<tau> \<prec>\<^sub>l P'"
abs_mult: fixes abs :: "'d \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs (times a b) = times (abs a) (abs b)"
not_in_keys_iff_lookup_eq_zero: fixes k :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'c" shows "(k \<notin> keys f) = (lookup f k = (0::'c))"
subo: shows "Subobjs P (last Cs) (last Cs # Ds')"
coinduct: fixes X :: "(real \<Rightarrow> real) \<Rightarrow> real msstream \<Rightarrow> bool"   and x1 :: "real \<Rightarrow> real"   and x2 :: "real msstream" assumes "X x1 x2"   and "\<And>x1 x2. X x1 x2 \<Longrightarrow> \<exists>f cs f' c. x1 = f' \<and> x2 = MSSCons c cs \<and> (X f cs \<or> local.asymp_powser f cs) \<and> f' \<in> O[F](\<lambda>_. 1) \<and> (\<forall>\<^sub>F x in F. f' x = c + f x * h x)" shows "local.asymp_powser x1 x2"
med23s_def: fixes t :: "l3_state" shows "med23s t \<equiv> \<lparr>ik = ik t \<inter> payload, secret = secret t, progress = progress t, signals = l1_state.signals t, chan = local.abs (ik t), bad = l3_state.bad t\<rparr>"
IdenticCycleLength_def: fixes n :: "nat"   and nC :: "nat \<Rightarrow> Config" shows "IdenticCycleLength n nC \<equiv> \<forall>i j. i < n \<and> j < n \<longrightarrow> cycleLength (nC i) = cycleLength (nC j)"
shift2_inv_init: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs" shows "shift2_inv k qs"
reachable_snapshot_inv_def: fixes m :: "'a" shows "mut_m.reachable_snapshot_inv m = (\<lambda>s. \<forall>x. mut_m.reachable m x s \<longrightarrow> in_snapshot x s)"
eventually_division_filter: fixes P :: "('b \<times> 'b set) set \<Rightarrow> bool"   and s :: "'b set" shows "eventually P (division_filter s) = (\<exists>g. gauge g \<and> (\<forall>p. p tagged_division_of s \<and> g fine p \<longrightarrow> P p))"
lm012: fixes N :: "'c set" shows "allAllocations N {} \<subseteq> {{}}"
integral_le: fixes f :: "real \<Rightarrow> real"   and a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> real" assumes "integrable f a b"   and "integrable g a b"   and "\<And>x. x \<in> {a..b} \<Longrightarrow> f x \<le> g x" shows "integral f a b \<le> integral g a b"
in_set_tl: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
winker: shows "\<beta> \<squnion> \<alpha> = \<beta>"
inject: fixes x2 :: "ty"   and x1 :: "ty"   and y2 :: "ty"   and y1 :: "ty" shows "(x2\<rightarrow>x1 = y2\<rightarrow>y1) = (x2 = y2 \<and> x1 = y1)"
l2_inv4I: fixes x :: "l2_state" assumes "\<And>Ra A B gny. \<lbrakk>guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>; in_progress (progress x Ra) xsk; guessed_frame Ra xgny = Some gny\<rbrakk> \<Longrightarrow> guessed_frame Ra xsk = Some (Exp gny (NonceF (Ra $ nx)))" shows "x \<in> l2_inv4"
if_distrib: fixes f :: "'f \<Rightarrow> 'e"   and c :: "bool"   and x :: "'f"   and y :: "'f" shows "f (if c then x else y) = (if c then f x else f y)"
InvariantWatchListsContainOnlyClausesFromF_def: fixes Wl :: "Literal \<Rightarrow> nat list"   and F :: "Literal list list" shows "InvariantWatchListsContainOnlyClausesFromF Wl F = (\<forall>l c. c \<in> set (Wl l) \<longrightarrow> 0 \<le> c \<and> c < length F)"
r_amalgamation: fixes i :: "nat"   and j :: "nat"   and x :: "nat" shows "eval r_amalgamation [i, j, x] = amalgamation i j x"
lin_indpt_full_rank: fixes A :: "'b mat"   and n :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat n nc"   and "distinct (cols A)"   and "\<not> module.lin_dep class_ring (module_vec TYPE('b) n) (set (cols A))" shows "vec_space.rank n A = nc"
wt_tNN: fixes T :: "'fsym trm" assumes "Ik.wt T" shows "TE.wt (tNN T)"
homotopic_with_def: fixes P :: "('a \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'a topology"   and Y :: "'c topology"   and f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c" shows "homotopic_with P X Y f g \<equiv> \<exists>h. continuous_map (prod_topology (top_of_set {0..1}) X) Y h \<and> (\<forall>x. h (0, x) = f x) \<and> (\<forall>x. h (1, x) = g x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. h (t, x)))"
pr_conv_1_to_2_lm: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec1" shows "pr_conv_1_to_2 f \<in> PrimRec2"
2: shows "hbt \<langle>l, (a, n), r\<rangle>"
map_of_rev_distinct: fixes m :: "('d \<times> 'e) list" assumes "distinct (map fst m)" shows "map_of (rev m) = map_of m"
Invariants_sets: shows "sets I.Invariants = {A \<in> sets M. Tinv -` A \<inter> space M = A}"
poly_of_vars_remdups_mset: fixes xs :: "char list multiset" shows "poly_of_vars (remdups_mset xs) - poly_of_vars xs \<in> More_Modules.ideal polynomial_bool"
nres_monad3: fixes M :: "'e nres"   and f :: "'e \<Rightarrow> 'd nres"   and g :: "'d \<Rightarrow> 'c nres" shows "M \<bind> f \<bind> g = M \<bind> (\<lambda>x. f x \<bind> g)"
num'_plus_den'_minus_chop_den': shows "num' + den' - tree_chop den' = 2 * num_mod_den'"
acc_C: shows "G \<turnstile> m in C dyn_accessible_from accC"
fresh_4: fixes a :: "atom"   and fm :: "fm" shows "a \<sharp> Neg fm = a \<sharp> fm"
supt_subst: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('d, 'f) Term.term" assumes "s \<rhd> t" shows "s \<cdot> \<sigma> \<rhd> t \<cdot> \<sigma>"
simps_1: fixes x2 :: "'b"   and y2 :: "'b" shows "(Some x2 = Some y2) = (x2 = y2)"
gfp_F_lfp_F: fixes ds :: "'b set" shows "gfp_F ds = (F1 (lfp (F2 ds \<circ> F1)), lfp (F2 ds \<circ> F1))"
Exit_label_incr: fixes n :: "w_node"   and i :: "nat" assumes "(_Exit_) = n \<oplus> i" shows "n = (_Exit_)"
atMost_def: fixes u :: "'c list" shows "prefix_order.atMost u \<equiv> {x. prefix x u}"
ideal_def: fixes I :: "'a set"   and R :: "('a, 'b) ring_scheme" shows "ideal I R \<equiv> additive_subgroup I R \<and> ring R \<and> ideal_axioms I R"
wf_eq_defl: fixes x :: "'a" shows "(\<forall>y. d y \<le> d (x \<cdot> y) \<longrightarrow> d y = (0::'a)) = (\<forall>y. y \<le> x \<cdot> y \<longrightarrow> y = (0::'a))"
scaleR_right: fixes a :: "'c \<Rightarrow>\<^sub>L 'b"   and r :: "real"   and b :: "'c" shows "a *\<^sub>v r *\<^sub>R b = r *\<^sub>R (a *\<^sub>v b)"
reals_Archimedean: fixes x :: "'a" assumes "(0::'a) < x" shows "\<exists>n. inverse (of_nat (Suc n)) < x"
size_filter_neg: fixes P :: "'a \<Rightarrow> bool"   and A :: "'a multiset" shows "size (filter_mset P A) = size A - size {#a \<in># A. \<not> P a#}"
simps_4: fixes \<sigma> :: "nat \<Rightarrow> val"   and oper :: "pdf_operator"   and e :: "cexpr" shows "cexpr_sem \<sigma> (oper $$\<^sub>c e) = op_sem oper (cexpr_sem \<sigma> e)"
emax_pos_2: fixes x :: "('a, 'b) IEEE.float itself" shows "emax x \<noteq> 0"
conc_Diff_if_Nil2: fixes B :: "'b list set"   and A :: "'b list set" assumes "[] \<in> B" shows "A @@ B = A @@ (B - {[]}) \<union> A"
cases: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat" assumes "read_actionsp E a" obtains aa :: "nat" and ad :: "'b" and al :: "addr_loc" and v :: "'b val" where "a = aa" and "aa \<in> actions E" and "action_obs E aa = NormalAction (ReadMem ad al v)"
rc_4: shows "\<exists>!x. root_cond (p, l, r) x"
special_cf_len: shows "fst (C\<^sub>0 S) = 0"
mod_ring_inverse: shows "(mod_ring_rel ===> mod_ring_rel) (inverse_p p) inverse"
v_2: shows "us ! i + c \<cdot>\<^sub>v us ! j \<in> carrier_vec n"
rep_eq: fixes x :: "'b regular"   and xa :: "'b regular" shows "(x \<le> xa) = (Rep_regular x \<le> Rep_regular xa)"
simps_4: fixes b :: "bexp"   and c1 :: "'a acom"   and c2 :: "'a acom"   and S :: "'a"   and b' :: "bexp"   and c1' :: "'a acom"   and c2' :: "'a acom"   and S' :: "'a" shows "(IF b THEN c1 ELSE c2 {S} \<le> IF b' THEN c1' ELSE c2' {S'}) = (b = b' \<and> c1 \<le> c1' \<and> c2 \<le> c2' \<and> S \<le> S')"
blinop_apply_times_blinop: fixes f :: "'b blinop"   and g :: "'b blinop"   and x :: "'b" shows "blinop_apply (f * g) x = blinop_apply f (blinop_apply g x)"
StartBallot_HInv3_t: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "StartBallot s s' t"   and "HInv3_inner s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "HInv3_inner s' p q d"