hcju/isabelleps huggingface dataset:
induct: fixes P :: "('c \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> 'c list \<Rightarrow> 'c list \<Rightarrow> bool"   and a0 :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c list"   and a2 :: "'c list" assumes "\<And>eq. P eq [] []"   and "\<And>eq a l a' l'. (eq a a' \<Longrightarrow> P eq l l') \<Longrightarrow> P eq (a # l) (a' # l')"   and "\<And>uu_ v va. P uu_ (v # va) []"   and "\<And>uu_ v va. P uu_ [] (v # va)" shows "P a0 a1 a2"
singleton: fixes x :: "'b" shows "join.Sup_fin {x} = x"
BlockRedNone: fixes e :: "char list exp"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and V :: "char list"   and e' :: "char list exp"   and h' :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l' :: "char list \<Rightarrow> val option"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and T :: "ty" assumes "P \<turnstile> \<langle>e,(h, l(V := None))\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "l' V = None"   and "\<not> assigned V e" shows "P \<turnstile> \<langle>{V:T; e},(h, l)\<rangle> \<rightarrow> \<langle>{V:T; e'},(h', l'(V := l V))\<rangle>"
setdist_pos_le: fixes s :: "'c set"   and t :: "'c set" shows "0 \<le> setdist s t"
m_o2: fixes m :: "'a \<Rightarrow> nat"   and h :: "nat"   and X :: "char list set"   and o1 :: "'a st option"   and o2 :: "'a st option" assumes "Measure m h"   and "finite X"   and "top_on\<^sub>o o1 (- X)"   and "top_on\<^sub>o o2 (- X)"   and "o1 < o2" shows "Measure1.m_o m h o2 X < Measure1.m_o m h o1 X"
transpose_infmatrix_twice: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b" shows "transpose_infmatrix (transpose_infmatrix A) = A"
assms_3: shows "borel_adapt_stoch_proc F (cls_val_process Mkt p)"
nlex_pdevs_zero_pdevs: shows "nlex_pdevs zero_pdevs = zero_pdevs"
jajajaj: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], rev a @ h) b = ([y, x], rev b @ rev a @ h)"
atomics_def: fixes i :: "nat"   and j :: "nat" shows "local.atomics i j \<equiv> map local.atomic (rev [i..<Suc j])"
bit_take_bit_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "nat"   and a :: "'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "bit (take_bit m a) n = (n < m \<and> bit a n)"
rcs_fixed2: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> H " shows "\<forall>h\<in>H. H \<bullet>\<^bsub>G\<^esub> h = H"
singleton_orbit_eq_fixed_point: fixes m :: "'c" assumes "m \<in> M" shows "(card (orbit m) = 1) = (m \<in> fixed_points)"
assms_2: shows "RModuleHom R (\<star>) N smult'' S"
All2I: fixes A :: "fm"   and i :: "name"   and t :: "tm" assumes "se_fm A"   and "atom i \<sharp> t" shows "se_fm (All2 i t A)"
deg_fun_def: fixes s :: "'b \<Rightarrow> 'a" shows "deg_fun s \<equiv> sum s (supp_fun s)"
lift_lemma_2: fixes e :: "nat \<Rightarrow> 'a"   and z :: "'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and ts :: "'b FOL_Fitting.term list" shows "evalts (e\<langle>0:z\<rangle>) f (liftts ts) = evalts e f ts"
less_not_refl: fixes n :: "nat" shows "\<not> n \<sqsubset> n"
dom_map_add: fixes m :: "'a \<Rightarrow> 'b option"   and n :: "'a \<Rightarrow> 'b option" shows "dom (Map.map_add m n) = dom n \<union> dom m"
f_join_insert: fixes n :: "nat"   and xs :: "'b list"   and I :: "nat set" assumes "n < length xs" shows "xs \<Join>\<^sub> insert n I = xs \<Join>\<^sub> (I \<down>< n) @ xs ! n # xs \<Join>\<^sub> (I \<down>> n)"
knows_Notes: fixes A :: "agent"   and A' :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows A (Notes A' X # evs) = (if A = A' then insert X (knows A evs) else knows A evs)"
derivative_norm_bounded_subset: fixes T :: "real set"   and X :: "'b set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and f' :: "real \<times> 'b \<Rightarrow> real \<times> 'b \<Rightarrow> 'b"   and B :: "real"   and B' :: "real"   and X' :: "'b set" assumes "derivative_norm_bounded T X f f' B B'"   and "X' \<noteq> {}"   and "X' \<subseteq> X"   and "convex X'" shows "derivative_norm_bounded T X' f f' B B'"
r_zero: fixes x :: "'c \<Rightarrow> 'a" assumes "x \<in> carrier F" shows "x \<oplus>\<^bsub>F\<^esub> \<zero>\<^bsub>F\<^esub> = x"
g_orbital_eq: fixes f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> bool"   and U :: "'b \<Rightarrow> real set"   and S :: "'b set"   and t\<^sub>0 :: "real"   and s :: "'b" shows "g_orbital f G U S t\<^sub>0 s = {X t |t X. t \<in> U s \<and> \<P> X (down (U s) t) \<subseteq> {s. G s} \<and> X \<in> Sols f U S t\<^sub>0 s}"
assms_1: shows "monom_mult c1 t p = monom_mult c2 t p"
minus_carrier_mat: fixes B :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and A :: "'b mat" assumes "B \<in> carrier_mat nr nc" shows "A - B \<in> carrier_mat nr nc"
cp0': fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and a3 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>3"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f self a1 a2 a3 \<tau> = f (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. a2 \<tau>) (\<lambda>_. a3 \<tau>) \<tau>"
rbt_insert_is_rbt: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t :: "('c, 'd) rbt"   and k :: "'c"   and v :: "'d" assumes "class.linorder less_eq less"   and "ord.is_rbt less t" shows "ord.is_rbt less (ord.rbt_insert less k v t)"
ref_older: fixes opset :: "('c \<times> 'd) set"   and deps :: "'d \<Rightarrow> 'c set"   and oid :: "'c"   and oper :: "'d"   and ref :: "'c" assumes "opset opset deps"   and "(oid, oper) \<in> opset"   and "ref \<in> deps oper" shows "ref < oid"
elims_2: fixes x :: "'b formula" assumes "is_literal_formula x" obtains uu_ :: "'b" where "x = Atom uu_"   | uv_ :: "'b" where "x = \<^bold>\<not> (Atom uv_)"
occs_subst_elim: fixes v :: "'b"   and t :: "('a, 'b) Term.term" assumes "\<not> Var v \<sqsubset> t" shows "subst_elim (Var(v := t)) v \<or> Var(v := t) = Var"
Le_le_LtI: fixes a :: "'c"   and b :: "'c" assumes "a < b" shows "Le a \<le> Lt b"
sup_Un_eq: fixes R :: "'b set"   and S :: "'b set" shows "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
ev_ev: fixes \<phi> :: "'c stream \<Rightarrow> bool" shows "ev (ev \<phi>) = ev \<phi>"
accept\<^sub>R_def: fixes R :: "('b \<Rightarrow> 'c \<Rightarrow> 'b) \<times> 'b \<times> (('b \<times> 'c \<times> 'b) set \<times> ('b \<times> 'c \<times> 'b) set) set"   and w :: "nat \<Rightarrow> 'c" shows "accept\<^sub>R R w \<equiv> \<exists>P\<in>snd (snd R). accepting_pair\<^sub>R (fst R) (fst (snd R)) P w"
i_f_join_length: fixes I :: "nat set"   and f :: "nat \<Rightarrow> 'b" assumes "finite I" shows "length (f \<Join>\<^sub> I) = card I"
product_in_factors: fixes x :: "i"   and y :: "i" assumes "CM"   and "O x y" shows "P (x \<otimes> y) x \<and> P (x \<otimes> y) y"
f: shows "f = (\<lambda>t. \<Sum>i\<in>K. z1 i * \<chi> (C i) t)"
disc_1: fixes x1 :: "nat" shows "shadow_root_ptr.is_Ref (shadow_root_ptr.Ref x1)"
except_inj_on_keys_homogeneous: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and x :: "'b" assumes "homogeneous p" shows "inj_on (\<lambda>t. except t {x}) (keys p)"
less_def: fixes A :: "'a matrix"   and B :: "'a matrix" shows "(A < B) = (A \<le> B \<and> \<not> B \<le> A)"
final: fixes \<sigma> :: "'\<sigma>" assumes "I \<sigma> l0 []" shows "P \<sigma>"
fold_finfun_update_finite_univ: fixes b' :: "'b"   and b :: "'b" assumes "finite UNIV" shows "Finite_Set.fold (\<lambda>a f. finfun_update f a b') (finfun_const b) UNIV = finfun_const b'"
simps_4: fixes x2 :: "'b blindable\<^sub>h"   and x1 :: "'a" shows "Blinded x2 \<noteq> Unblinded x1"
lt_1_iff_eq_0_hmset: fixes M :: "hmultiset" shows "(M < 1) = (M = 0)"
isomorphic_unit_left: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f" shows "Span.isomorphic (Span.trg f \<circ> f) f"
simps_2: fixes b :: "nat"   and x :: "C"   and xs :: "C list" shows "evalC b (C (x # xs)) = b ^ evalC b x + evalC b (C xs)"
in_\<R>': shows "(\<langle>c\<^sub>A', mds\<^sub>A_of mds\<^sub>C', mem\<^sub>A_of mem\<^sub>C'\<rangle>\<^sub>A, \<langle>c\<^sub>C', mds\<^sub>C', mem\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>"
zero_min: fixes cmp_term :: "'b nat_term_order"   and t :: "'b" shows "le_of_nat_term_order cmp_term (fst (0::'b, the_min)) (fst (t, the_min))"
soundness_def: fixes ES :: "('b list set \<times> 'b trm set) set" shows "soundness ES \<equiv> \<forall>(X, rhs)\<in>ES. X = lang_rhs rhs"
pmf_None_eq_weight_spmf: fixes p :: "'c spmf" shows "pmf p None = 1 - weight_spmf p"
encode_problem_serializable_complete: fixes \<Pi> :: "'a strips_problem"   and \<pi> :: "'a strips_operator list list" assumes "is_valid_problem_strips \<Pi>"   and "STRIPS_Semantics.is_parallel_solution_for_problem \<Pi> \<pi>"   and "\<forall>k<length \<pi>. are_all_operators_non_interfering (\<pi> ! k)" shows "valuation_for_plan \<Pi> \<pi> \<Turnstile> \<Phi>\<^sub>\<forall> \<Pi> length \<pi>"
intro: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and ass_bnd :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and chk :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and update :: "nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state" assumes "AssertAllState'' init ass_bnd chk"   and "Update update"   and "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"   and "AssertAllState'''_axioms ass_bnd chk update eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update" shows "AssertAllState''' init ass_bnd chk update eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"
v_4: fixes i :: "nat"   and x :: "'a" shows "v i x < top"
exhaust: fixes y :: "'a fundef" obtains x1 :: "'a list" and x2 :: "nat" where "y = Fundef x1 x2"
inverts_components: fixes a :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.ide a" shows "B.inverse_arrows \<a>\<^sub>B[fst a, fst (snd a), snd (snd a)] (B.\<alpha>' a)"
inf_Assertion_def: fixes p :: "'c Assertion"   and q :: "'c Assertion" shows "p \<sqinter> q = abs_wpt ({\<cdot> p } \<sqinter> {\<cdot> q })"
cod_cod: fixes f :: "'d" assumes "D.arr f" shows "D.cod (D.cod f) = D.cod f"
unique_PPV_def: fixes V :: "('a, 'b) MWLsCom list" shows "unique_PPV V = distinct (PPV V)"
Arg2pi_le_div_sum: fixes w :: "complex"   and z :: "complex" assumes "w \<noteq> 0"   and "z \<noteq> 0"   and "Arg2pi w \<le> Arg2pi z" shows "Arg2pi z = Arg2pi w + Arg2pi (z / w)"
r: shows "r @ xrs \<in> set_of_appr (b @ xs)"
thetaSSeqI_Sretr: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'c \<Rightarrow> 'b \<Rightarrow> bool" assumes "PL_Indis cval indis" shows "PL_Indis.thetaSSeqI aval tval cval indis \<subseteq> PL_Indis.Sretr aval tval cval indis (PL_Indis.thetaSSeqI aval tval cval indis \<union> PL_Indis.Sbis aval tval cval indis)"
assms: shows "(\<lambda>x. \<pi> x * ln (\<pi> x)) \<sim>[at_top] (\<lambda>x. x)"
index_finsum_vec: fixes F :: "'a set"   and i :: "nat"   and n :: "nat"   and vs :: "'a \<Rightarrow> 'b vec" assumes "finite F"   and "i < n"   and "vs \<in> F \<rightarrow> carrier_vec n" shows "finsum_vec TYPE('b) n vs F $ i = (\<Sum>f\<in>F. vs f $ i)"
map_inj_pair_non_membership: fixes k :: "'c"   and ps :: "('c \<times> 'd) list"   and f :: "'c \<Rightarrow> 'e" assumes "k \<notin> set (map fst ps)"   and "inj f" shows "f k \<notin> set (map fst (map (\<lambda>(k, v). (f k, v)) ps))"
a1: shows "low_equal s1 s2 \<and> t1 = store_barrier_pending_mod False s1 \<and> t2 = store_barrier_pending_mod False s2"
equivalence_in_bicategory_axioms: shows "internal_equivalence e' e (local.inv \<psi>') (local.inv \<phi>)"
unstream_lift_update_trans_None: fixes g :: "('e, 'f) generator"   and b :: "'e"   and s :: "'f" shows "unstream (list_update_trans g b) (0, s) = unstream g s"
zhmset_of_enat_of_nat: fixes n :: "nat" shows "zhmset_of_enat (of_nat n) = of_nat n"
Inv_mtf: fixes xs :: "'b list"   and ys :: "'b list"   and z :: "'b" assumes "set xs = set ys"   and "z \<in> set ys" shows "Inv xs (mtf z ys) = Inv xs ys \<union> {(x, z) |x. x < z in xs \<and> x < z in ys} - {(z, x) |x. z < x in xs \<and> x < z in ys}"
of_int_of_nat_eq: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and n :: "nat" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (int n) = semiring_1.of_nat one plus zero n"
Is_valid: fixes i :: "nat" assumes "i \<le> len" shows "Is i \<in> valid_I"
XLN: shows "(redT_updTs ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>(t \<mapsto> (x', ln'))) T = \<lfloor>(X, LN)\<rfloor>"
NullPointerC_def: shows "NullPointerC \<equiv> (NullPointer, Object, [], [])"
set_delete_subset: fixes k :: "'a"   and al :: "('a \<times> 'b) list" shows "set (delete k al) \<subseteq> set al"
ringed_space_axioms: shows "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
homeomorphic_imp_injective_map: fixes X :: "'a topology"   and Y :: "'b topology"   and f :: "'a \<Rightarrow> 'b" assumes "homeomorphic_map X Y f" shows "inj_on f (topspace X)"
f_def: shows "f \<equiv> \<lambda>x. (1 - norm (x - a) / r) *\<^sub>R (u - a) + x"
div_int_then_rqp_1: fixes p :: "int poly"   and q :: "int poly"   and r :: "int poly" assumes "div_int_poly p q = Some r" shows "r * q = p"
algebraic_int_plus: fixes x :: "'a"   and y :: "'a" assumes "algebraic_int x"   and "algebraic_int y" shows "algebraic_int (x + y)"
simps_13: fixes f11 :: "'b"   and f12 :: "X \<Rightarrow> 'b \<Rightarrow> 'b"   and f13 :: "Y \<Rightarrow> 'a \<Rightarrow> 'b"   and f21 :: "X \<Rightarrow> 'b \<Rightarrow> 'a"   and f22 :: "Y \<Rightarrow> 'a \<Rightarrow> 'a"   and f23 :: "'a"   and x22 :: "Y" shows "rec_Y f11 f12 f13 f21 f22 f23 (Y.E x22) = f22 x22 (rec_Y f11 f12 f13 f21 f22 f23 x22)"
escape_rate_pos: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and x :: "'d" assumes "transition_rates R" shows "0 < transition_rates.escape_rate R x"
next_between2: fixes f :: "face"   and v :: "nat"   and u :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "u \<in> \<V> f"   and "u \<noteq> v" shows "v \<in> set (between (vertices f) u (f \<bullet> v))"
cr_pp_def: shows "cr_pp \<equiv> \<lambda>x y. x = pp.mapping_of y"
ccApprox_ccTTree: fixes S :: "Vars.var set"   and G :: "CoCalls" shows "ccApprox (ccTTree S G) = G G|` S"
simps_2: fixes v :: "'b"   and va :: "'b list" shows "diff_sorted (v # va) [] = v # va"
left_iff_left_inv: fixes \<mu> :: "'a"   and \<tau> :: "'a" assumes "local.iso \<mu>" shows "left \<tau> \<mu> = left \<tau> (local.inv \<mu>)"
minGraphProps11': fixes g :: "graph" assumes "minGraphProps g" shows "distinct (faces g)"
intro: fixes f :: "'b \<Rightarrow> 'c"   and period :: "'b" assumes "\<And>x. f (x + period) = f x" shows "periodic_fun_simple f period"
vimage_subset_eq: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'c set"   and A :: "'b set" assumes "bij f" shows "(f -` B \<subseteq> A) = (B \<subseteq> f ` A)"
vec_nth_inverse: fixes x :: "('d, 'e) vec" shows "vec_lambda (($) x) = x"
ToTC_def: fixes f :: "'a \<times> 'a \<times> 'a" shows "C.T.ToTC f \<equiv> if C.CCC.arr f then C.tensor (C.tensor (fst f) (fst (snd f))) (snd (snd f)) else C.null"
obtain_fresh_variable: fixes \<Gamma> :: "Core.term set"   and \<tau> :: "typ"   and t :: "Core.term" assumes "finite \<Gamma>" obtains x :: "variable" where "(x, \<tau>) \<notin> fv t \<union> FV \<Gamma>"
homeomorphism_on_sequentially: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x u. \<lbrakk>x \<in> S; \<forall>n. u n \<in> S\<rbrakk> \<Longrightarrow> u \<longlonglongrightarrow> x = (\<lambda>n. f (u n)) \<longlonglongrightarrow> f x" shows "homeomorphism_on S f"
d: shows "degree (poly_of_list (map g [0..<k])) \<le> k - 1"
max_def: fixes a :: "'b"   and b :: "'b" shows "max a b = (if a \<le> b then b else a)"
dot_norm: fixes x :: "'a"   and y :: "'a" shows "inner x y = ((norm (x + y))\<^sup>2 - (norm x)\<^sup>2 - (norm y)\<^sup>2) / 2"
maximalideal_def: fixes I :: "'e set"   and R :: "('e, 'f) ring_scheme" shows "maximalideal I R \<equiv> ideal I R \<and> maximalideal_axioms I R"
distinct_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "cycle (map f xs) = (cycle xs \<and> inj_on f (set xs))"
IH: assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> coprime (g x) (g y)" shows "f (prod g A) = (\<Prod>x\<in>A. f (g x))"
DynProc: fixes P :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'d \<Rightarrow> 'e \<Rightarrow> 'd set"   and Q' :: "'d \<Rightarrow> 'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'d \<Rightarrow> 'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and p :: "'d \<Rightarrow> 'f" assumes "P \<subseteq> {s. \<exists>Z. init s \<in> P' s Z \<and> (\<forall>t. t \<in> Q' s Z \<longrightarrow> return s t \<in> R s t) \<and> (\<forall>t. t \<in> A' s Z \<longrightarrow> return s t \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (R s t) c s t Q,A"   and "\<forall>s\<in>P. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> (P' s Z) Call (p s) (Q' s Z),(A' s Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
bezout_matrix_def: fixes 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, 'f) vec, 'e) vec"   and a :: "'e"   and b :: "'e"   and j :: "'f"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" assumes "class.bezout_ring times one plus zero minus uminus" shows "bezout_ring.bezout_matrix one zero A a b j bezout = (\<chi>x y. let (p, q, u, v, d) = bezout (A $ a $ j) (A $ b $ j) in if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then one else zero)"
tendsto_ennrealD: fixes f :: "'d \<Rightarrow> real"   and x :: "real"   and F :: "'d filter" assumes "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 \<le> x" shows "(f \<longlongrightarrow> x) F"
simps_2: fixes xs :: "'a msstream" shows "un_MSLCons1 (msllist_of_msstream xs) = (case xs of MSSCons x xs \<Rightarrow> x)"
dom_simp: fixes f :: "'c local.term set" assumes "arr f" shows "local.dom f = mkarr (DOM f)"
simps_2: fixes w :: "nat"   and t\<^sub>1 :: "'a tree"   and t\<^sub>2 :: "'a tree" shows "alphabet (Node w t\<^sub>1 t\<^sub>2) = alphabet t\<^sub>1 \<union> alphabet t\<^sub>2"
mono: fixes c :: "'b"   and a :: "'b"   and d :: "'b"   and b :: "'b" assumes "c \<le> a"   and "d \<le> b" shows "sup_class.sup c d \<le> sup_class.sup a b"
rel_chain_take: fixes l :: "'b list"   and R :: "('b \<times> 'b) set"   and n :: "nat" assumes "rel_chain l R" shows "rel_chain (take n l) R"
f_2: shows "\<forall>x\<in>A. \<forall>y\<in>A. (f x < f y) = ((x, y) \<in> r)"
integral_pmf_restrict: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b pmf" assumes "f \<in> borel_measurable (count_space UNIV)" shows "measure_pmf.expectation M f = integral\<^sup>L (restrict_space (measure_pmf M) (set_pmf M)) f"
tv: shows "\<langle>tv w\<rangle> = \<langle>w\<rangle>(x := p \<lbrace> \<langle>w\<rangle> \<rbrace>)"
Cons_3: shows "lin_ts_mmuaux aux = (ts # tss) @ tss'"
in_homE: fixes f :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and a :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr"   and b :: "('c \<Rightarrow> 's, 'c \<Rightarrow> 's) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>C\<^sub>o\<^sub>p\<^sub>,\<^sub>S\<^sub>] b\<guillemotright>" obtains "Cop_S.arr f" and "Cop_S.dom f = a" and "Cop_S.cod f = b"
discrete_def: fixes X :: "'a set" shows "discrete X = (I = id)"
subgame_strategy_stays_in_subgame: fixes V' :: "'a set"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and v :: "'a" assumes "ParityGame.strategy (subgame V') p \<sigma>"   and "v \<in> (if p = Even then V0\<^bsub>subgame V'\<^esub> else V\<^bsub>subgame V'\<^esub> - V0\<^bsub>subgame V'\<^esub>)"   and "\<not> \<not> (\<exists>w\<in>V\<^bsub>subgame V'\<^esub>. v \<rightarrow>\<^bsub>subgame V'\<^esub> w)" shows "\<sigma> v \<in> V'"
simps_1: fixes ao :: "'a option"   and n :: "nat" shows "proper_interval_set_Compl_aux ao n [] [] = (n + 1 < CARD('a))"
simps: fixes i :: "Integer"   and xs :: "[:'r:]" shows "sdrop\<cdot>i\<cdot>xs = If le\<cdot>i\<cdot>0 then xs else (case xs of [::] \<Rightarrow> [::] | y :# ys \<Rightarrow> sdrop\<cdot>(i - 1)\<cdot>ys)"
accept_gba_alt: fixes m :: "'b" shows "Automata.gba.accept (to_gba_ext m) = accept"
diff_right_commute: fixes a :: "'a"   and c :: "'a"   and b :: "'a" shows "a - c - b = a - b - c"
fic_Ax_elim: fixes x :: "name"   and a :: "coname"   and b :: "coname" assumes "fic (Ax x a) b" shows "a = b"
simps_2: fixes i :: "nat" shows "Entry \<oplus> i = Entry"
consistent_def3: shows "consistent = (\<forall>\<phi>\<in>fmla. \<not> (prv \<phi> \<and> prv (neg \<phi>)))"
tail_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs (subdivide G e w)" shows "fst e \<in> pverts (subdivide G e w)"
finite_proj_wf: fixes E :: "'b set"   and x :: "'b"   and r :: "('b \<times> 'b) set" assumes "finite E"   and "x \<in> E"   and "acyclic r" shows "\<exists>y. y \<in> E \<and> (\<forall>z. (z, y) \<in> r \<longrightarrow> z \<notin> E)"
sharing_consis_tl: fixes \<S> :: "nat \<Rightarrow> bool option"   and t :: "'b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "sharing_consis \<S> (t # ts)" shows "sharing_consis \<S> ts"