hcju/isabelleps huggingface dataset:
alpha_tAct: fixes f1 :: "'d"   and \<alpha>1 :: "'c"   and t1 :: "('a, 'b, 'c, 'd) Tree"   and f2 :: "'d"   and \<alpha>2 :: "'c"   and t2 :: "('a, 'b, 'c, 'd) Tree" shows "(tAct f1 \<alpha>1 t1 =\<^sub>\<alpha> tAct f2 \<alpha>2 t2) = (f1 = f2 \<and> (\<exists>p. (bn \<alpha>1, t1) \<approx>set (=\<^sub>\<alpha>) supp_rel (=\<^sub>\<alpha>) p (bn \<alpha>2, t2) \<and> (bn \<alpha>1, \<alpha>1) \<approx>set (=) supp p (bn \<alpha>2, \<alpha>2)))"
good_item_simps_51: fixes inp :: "'a \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and binpa :: "'b \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qAbs set option"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var"   and delta :: "'c" assumes "goodInp inp"   and "goodBinp binpa" shows "Op delta inp binpa #[y1 // y]_ys = Op delta (inp %[y1 // y]_ys) (binpa %%[y1 // y]_ys)"
ttree_restr_nxt_subset: fixes x :: "'a"   and S :: "'a set"   and t :: "'a ttree" assumes "x \<in> S" shows "paths (ttree_restr S (nxt t x)) \<subseteq> paths (nxt (ttree_restr S t) x)"
prems_1: shows "length m1 = Suc nn \<and> (\<forall>a\<in>set m1. vec nr a)"
ag_r_inv1: fixes x :: "'a" assumes "x \<in> carrier M" shows "x \<plusminus> -\<^sub>a x = \<zero>"
R21_1: shows "pmf (sds R21) d = 1 - pmf (sds R21) a"
Ana_invar_subst': fixes \<S> :: "('fun, 'var) Term.term set" shows "Ana_invar_subst \<S>"
cmp_simps_4: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.dom (\<Phi> (f, g)) = F f \<star>\<^sub>D F g"
h2: shows "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h2"
simps_1: fixes x1 :: "'b"   and y1 :: "'b" shows "(\<^bold>\<langle>x1\<^bold>\<rangle>\<^sub>0 = \<^bold>\<langle>y1\<^bold>\<rangle>\<^sub>0) = (x1 = y1)"
evaldjf_def: fixes f :: "'a \<Rightarrow> fm"   and ps :: "'a list" shows "evaldjf f ps = foldr (djf f) ps F"
charact_cond_exp': fixes Y :: "bool stream \<Rightarrow> 'b"   and X :: "bool stream \<Rightarrow> real"   and N :: "'b measure" assumes "disc_fct Y"   and "integrable M X"   and "random_variable N Y"   and "\<forall>r\<in>range Y \<inter> space N. \<exists>A\<in>sets N. range Y \<inter> A = {r}" shows "AE w in M. real_cond_exp M (fct_gen_subalgebra M N Y) X w = expl_cond_expect M Y X w"
A: shows "weak_reduction_simulation (TRel\<^sup>+) Target"
has_derivative_eq_rhs: fixes f :: "'d \<Rightarrow> 'e"   and f' :: "'d \<Rightarrow> 'e"   and F :: "'d filter"   and g' :: "'d \<Rightarrow> 'e" assumes "(f has_derivative f') F"   and "f' = g'" shows "(f has_derivative g') F"
case_2: fixes f1 :: "'a"   and f2 :: "qtname \<Rightarrow> 'a"   and f3 :: "qtname \<Rightarrow> 'a"   and f4 :: "ty \<Rightarrow> 'a"   and x2 :: "qtname" shows "(case IfaceT x2 of NullT \<Rightarrow> f1 | IfaceT x \<Rightarrow> f2 x | ClassT x \<Rightarrow> f3 x | ArrayT x \<Rightarrow> f4 x) = f2 x2"
strong_induct: fixes x1 :: "trm"   and x2 :: "trm"   and P :: "'a \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<Down> x2"   and "\<And>x e y. x \<sharp> y \<Longrightarrow> P y (Lam [x].e) (Lam [x].e)"   and "\<And>x e\<^sub>1 e\<^sub>2 e' e e\<^sub>2' y. \<lbrakk>x \<sharp> y; x \<sharp> (e\<^sub>1, e\<^sub>2, e'); e\<^sub>1 \<Down> Lam [x].e; \<And>z. P z e\<^sub>1 (Lam [x].e); e\<^sub>2 \<Down> e\<^sub>2'; \<And>z. P z e\<^sub>2 e\<^sub>2'; e[x::=e\<^sub>2'] \<Down> e'; \<And>z. P z (e[x::=e\<^sub>2']) e'\<rbrakk> \<Longrightarrow> P y (App e\<^sub>1 e\<^sub>2) e'" shows "P z x1 x2"
SC_ZObis_imp_SC_Wbis: fixes c :: "('test, 'atom) com" assumes "SC_ZObis c" shows "SC_Wbis c"
pow_Suc2_list: fixes u :: "'b list"   and n :: "nat" shows "u \<^sup>@ Suc n = u \<^sup>@ n \<cdot> u"
upd_elim': fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and upd :: "'Var ModeUpd"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mem' :: "'Var \<Rightarrow> 'Val"   and P :: "bool" assumes "(c@[upd], mem) \<leadsto>\<^sub>s (c', mem')" shows "P"
flatten_superset: fixes A :: "'b set"   and p :: "'b \<Rightarrow>\<^sub>0 'b \<Rightarrow>\<^sub>0 'c" assumes "finite A"   and "keys p \<subseteq> A" shows "flatten p = (\<Sum>t\<in>A. MPoly_Type_Class.punit.monom_mult (1::'c) t (lookup p t))"
inv_naturality: fixes f :: "'d" assumes "D.arr f" shows "D.map f \<cdot>\<^sub>D D.inv (\<eta> (D.dom f)) = D.inv (\<eta> (D.cod f)) \<cdot>\<^sub>D (G \<circ> F) f"
nrows_diagonal_to_Smith_aux: fixes A :: "(('a, 'b) vec, 'c) vec"   and xs :: "nat list"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "nrows (diagonal_to_Smith_aux A xs bezout) = nrows A"
size_gen_3: fixes x3 :: "v list" shows "size_res_step (RSReturn x3) = 0"
fw_int_aux_Suc_a: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and a :: "nat"   and b :: "nat"   and c :: "nat"   and i :: "nat"   and j :: "nat" assumes "dbm_int (fw M n a n n) n"   and "Suc a \<le> n"   and "b \<le> n"   and "c \<le> n"   and "i \<le> n"   and "j \<le> n"   and "fw M n (Suc a) b c i j \<noteq> \<infinity>" shows "get_const (fw M n (Suc a) b c i j) \<in> \<int>"
left_right_simps_3: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "pre_term.is_app unapp t" shows "app (pre_term.left unapp t) (pre_term.right unapp t) = t"
Diag_Diagonalize_2: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Dom C (monoidal_language.Diagonalize C t) = monoidal_language.Diagonalize C (monoidal_language.Dom C t)"
assms_2: shows "g \<in> B \<rightarrow> A"
cases: fixes x :: "'b bro \<times> 'b \<times> 'b bro" obtains a1 :: "'b" and a2 :: "'b" and t :: "'b bro" where "x = (L2 a1__, a2__, t)"   | t1 :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and t4 :: "'b bro" where "x = (N3 t1__ a1a__ t2__ a2a__ t3__, a3__, N1 t4__)"   | t1 :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" where "x = (N3 t1a__ a1b__ t2a__ a2b__ t3a__, a3a__, N0)"   | t1 :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and v :: "'b bro" and va :: "'b" and vb :: "'b bro" where "x = (N3 t1b__ a1c__ t2b__ a2c__ t3b__, a3b__, N2 v__ va__ vb__)"   | t1 :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and v :: "'b" where "x = (N3 t1c__ a1d__ t2c__ a2d__ t3c__, a3c__, L2 vc__)"   | t1 :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and v :: "'b bro" and va :: "'b" and vb :: "'b bro" and vc :: "'b" and vd :: "'b bro" where "x = (N3 t1d__ a1e__ t2d__ a2e__ t3d__, a3d__, N3 vd__ vaa__ vba__ vca__ vda__)"   | a1 :: "'b" and a2 :: "'b" where "x = (N0, a1f__, L2 a2f__)"   | v :: "'b bro" and a1 :: "'b" and a2 :: "'b" where "x = (N1 ve__, a1g__, L2 a2g__)"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" and a1 :: "'b" and a2 :: "'b" where "x = (N2 vf__ vab__ vbb__, a1h__, L2 a2h__)"   | t1 :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and t4 :: "'b bro" where "x = (N1 t1, a1i__, N3 t2e__ a2i__ t3e__ a3e__ t4a__)"   | a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and t4 :: "'b bro" where "x = (N0, a1j__, N3 t2f__ a2j__ t3f__ a3f__ t4b__)"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" and a1 :: "'b" and t2 :: "'b bro" and a2 :: "'b" and t3 :: "'b bro" and a3 :: "'b" and t4 :: "'b bro" where "x = (N2 vg__ vac__ vbc__, a1, N3 t2 a2 t3 a3 t4)"   | a :: "'b" where "x = (N0, aa__, N0)"   | a :: "'b" and v :: "'b bro" where "x = (N0, ab__, N1 vh__)"   | a :: "'b" and v :: "'b bro" and va :: "'b" and vb :: "'b bro" where "x = (N0, ac__, N2 vi__ vad__ vbd__)"   | v :: "'b bro" and a :: "'b" where "x = (N1 vj__, ad__, N0)"   | v :: "'b bro" and a :: "'b" and va :: "'b bro" where "x = (N1 vk__, ae__, N1 vae__)"   | v :: "'b bro" and a :: "'b" and va :: "'b bro" and vb :: "'b" and vc :: "'b bro" where "x = (N1 vl__, af__, N2 vaf__ vbe__ vcb__)"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" and a :: "'b" where "x = (N2 vm__ vag__ vbf__, ag__, N0)"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" and a :: "'b" and vc :: "'b bro" where "x = (N2 vn__ vah__ vbg__, ah__, N1 vcc__)"   | v :: "'b bro" and va :: "'b" and vb :: "'b bro" and a :: "'b" and vc :: "'b bro" and vd :: "'b" and ve :: "'b bro" where "x = (N2 v va vb, a, N2 vc vd ve)"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc (x ::= a) Q = True"
simps: fixes F :: "'fenv_std"   and H :: "'c"   and main :: "'d" shows "compile (Prog F H main) = Some (Prog (compile_env F) H main)"
inv_inv_lens: fixes X :: "'d \<Longrightarrow> 'e" assumes "bij_lens X" shows "inv\<^sub>L (inv\<^sub>L X) = X"
vsv_vimageI2': fixes b :: "V"   and a :: "V" assumes "b = s\<lparr>a\<rparr>"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> s" shows "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> s"
nec_exist_unique: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>x. \<phi> x \<^bold>\<rightarrow> \<^bold>\<box>\<phi> x) \<^bold>\<rightarrow> ((\<^bold>\<exists>!x. \<phi> x) \<^bold>\<rightarrow> (\<^bold>\<exists>!x. \<^bold>\<box>\<phi> x)) in v]"
pivot_code_def: shows "pivot_code = Pivot'.pivot eq_idx_for_lvar pivot_eq subst_var"
iT_Mult_1: fixes I :: "nat set" shows "I \<otimes> Suc 0 = I"
J_3: fixes n :: "nat" shows "B n \<in> sets (Pi\<^sub>M (J n) (\<lambda>i. borel))"
lookup_ev_Some: fixes ev :: "'b"   and j :: "nat"   and A :: "'b mat"   and i :: "nat" assumes "??.Jordan_Normal_Form_Existence.lookup_ev ev j A = Some i" shows "i < j \<and> A $$ (i, i) = ev \<and> (\<forall>k>i. k < j \<longrightarrow> A $$ (k, k) \<noteq> ev)"
weakStatImpStatEq: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and \<Psi>' :: "'b"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<lessapprox><Rel> Q"   and "\<Psi> \<simeq> \<Psi>'"   and "\<And>\<Psi>' R S \<Psi>''. \<lbrakk>(\<Psi>', R, S) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', R, S) \<in> Rel'" shows "\<Psi>' \<rhd> P \<lessapprox><Rel'> Q"
stutter_equiv_sym: fixes \<sigma> :: "nat \<Rightarrow> 'b"   and \<tau> :: "nat \<Rightarrow> 'b" assumes "\<sigma> \<approx> \<tau>" shows "\<tau> \<approx> \<sigma>"
assms_1: shows "x \<in> fv (t \<cdot> \<delta>)"
find_elem_def: fixes x :: "int"   and xs :: "int list" shows "find_elem x xs \<equiv> WHILE\<^sub>T\<^bsup>\<lambda>i. i \<le> length xs \<and> x \<notin> set (take i xs)\<^esup> (\<lambda>i. i < length xs \<and> xs ! i \<noteq> x) (\<lambda>i. RETURN (i + 1)) 0"
kzkk0: shows "ket_zero_k (Suc k) $ 0 = 1"
assms_2: shows "(e = Unop_i t uu \<or> e = Testop t uv) \<and> is_int_t t \<or> e = Unop_f t uw \<and> is_float_t t \<or> e = Cvtop t1 Convert t sx \<and> convert_cond t1 t sx \<or> e = Cvtop t1 Reinterpret t sx \<and> t1 \<noteq> t \<and> t_length t1 = t_length t \<and> sx = None"
a1: shows "\<forall>p\<in>set Ps. \<exists>\<Phi>' \<Psi>'. p = ( \<Phi>' \<oplus> Modal M Ms \<Rightarrow>* \<Psi>')"
right_inverse: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c"   and inverse :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "Groups.group f z inverse" shows "f a (inverse a) = z"
redT_updT'_Some: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'h"   and xw :: "'f \<times> 'g \<Rightarrow>f nat"   and ta :: "('h, 'i, 'j) new_thread_action" assumes "ts t = \<lfloor>xw\<rfloor>"   and "thread_ok ts ta" shows "redT_updT' ts ta t = \<lfloor>xw\<rfloor>"
f_foldl_abs_alt_def: fixes \<phi> :: "'a ltl"   and w :: "'b list" shows "f_foldl_abs (Abs \<phi>) w = foldl f_abs (Abs \<phi>) w"
in_hom_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a"   and a :: "'a"   and b :: "'a" assumes "partial_magma C" shows "partial_magma.in_hom C f a b \<equiv> a \<in> partial_magma.domains C f \<and> b \<in> partial_magma.codomains C f"
wfp_on_def: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" shows "wfp_on P A = (\<nexists>f. \<forall>i. f i \<in> A \<and> P (f (Suc i)) (f i))"
unbounded_Greater: fixes X :: "'a set"   and k :: "'a \<Rightarrow> nat"   and I :: "'a \<Rightarrow> Regions.intv"   and r :: "('a \<times> 'a) set"   and c :: "'a" assumes "Regions.valid_region X k I r"   and "c \<in> X"   and "unbounded c (Regions.region X I r)" shows "I c = Regions.intv.Greater (k c)"
Minimal_Solutions_subset_solve: shows "local.Minimal_Solutions \<subseteq> set (solve a b)"
residue_primroot_iff_Carmichael: fixes n :: "nat" shows "(\<exists>g. residue_primroot n g) = (Carmichael n = totient n \<and> 0 < n)"
BVG_assign_other: fixes x :: "variable"   and y :: "variable"   and \<theta> :: "trm" assumes "x \<noteq> y" shows "y \<notin> BVG (x := \<theta>)"
prj_chine_assoc_1: shows "SPN_fgh.Prj\<^sub>1 \<odot> SPN_fgh.chine_assoc = SPN_fgh.Prj\<^sub>1\<^sub>1"
simps_4: fixes e :: "name list"   and A :: "fm" shows "trans_fm e (Neg A) = DBNeg (trans_fm e A)"
secure: fixes \<eta> :: "nat" shows "\<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>res S_CHAN.res \<surd>"
image_empty: fixes f :: "'e \<Rightarrow> 'd" shows "f ` {} = {}"
equal_ccsubspace_def: fixes A :: "'b ccsubspace"   and B :: "'b ccsubspace" shows "equal_class.equal A B = (A = B)"
power_less_imp_less_exp: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "less (power.power one times a m) (power.power one times a n)" shows "m < n"
linear_inj_on_span_iff_independent_image: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec"   and B :: "('a, 'b) vec set" assumes "linear (*s) (*s) f"   and "vec.independent (f ` B)" shows "inj_on f (vec.span B) = inj_on f B"
embedS_defs_1: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "embedS r r' f \<equiv> embed r r' f \<and> \<not> bij_betw f (Field r) (Field r')"
iso_eval_Can: fixes t :: "'c local.term" assumes "Can t" shows "D.iso \<lbrace>t\<rbrace>"
ord_spmf_map_spmf1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'a"   and p :: "'a spmf" shows "ord_spmf R (map_spmf f p) = ord_spmf (\<lambda>x. R (f x)) p"
gp_less: shows "\<forall>t\<in>{0..1}. cmod (g t - p t) < d"
strict_boundedE: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a < min b c" obtains "a < b" and "a < c"
assms_6: shows "h \<turnstile> P x \<rightarrow>\<^sub>r True"
rsp: shows "eq_onp bounded_clinear (\<lambda>x. x) (\<lambda>x. x)"
det30_zero_scaleR2: fixes e :: "real"   and P :: "real \<times> real"   and xr :: "real \<times> real" assumes "0 < e"   and "coll 0 P xr" shows "coll 0 P (e *\<^sub>R xr)"
fail_option_def: fixes return :: "'a option \<Rightarrow> 'b" shows "fail_option return = OptionT (return None)"
lexext_Cons_eq: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list"   and x :: "'b" assumes "lexext gt ys xs" shows "lexext gt (x # ys) (x # xs)"
rewrite: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b spmf" shows "sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. sample_uniform (order \<G>) \<bind> (\<lambda>z. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) (\<^bold>g [^] z \<otimes> \<^bold>g)))) = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. sample_uniform (order \<G>) \<bind> (\<lambda>z. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) (\<^bold>g [^] z))))"
subset_of_opens: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and I :: "'d set set"   and U :: "'d set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "U \<in> I" shows "is_open U"
subst_apply_pairs_def: fixes F :: "(('a, 'c) Term.term \<times> ('a, 'c) Term.term) list"   and \<theta> :: "'c \<Rightarrow> ('a, 'b) Term.term" shows "F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta> \<equiv> map (\<lambda>f. f \<cdot>\<^sub>p \<theta>) F"
rGamma_neg_of_nat: fixes m :: "nat" shows "rGamma (- of_nat m) = (0::'a)"
assms: shows "f \<approx>>\<^bsub>SET\<^esub> g"
vec_of_basis_enum_def: fixes v :: "'b" shows "vec_of_basis_enum v = vec_of_list (map (crepresentation (set canonical_basis) v) canonical_basis)"
Nats_1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" assumes "class.semiring_1 one times plus zero" shows "one \<in> semiring_1.Nats one plus zero"
RR: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "RR t1 t2 \<in> fmla"
fg: shows "homeomorphic_maps (Euclidean_space n) (top_of_set (span B)) f g"
permutations_of_list: fixes xs :: "'b list" shows "permutations_of_set (set xs) = set (permutations_of_set_list (remdups xs))"
simps_3: fixes S :: "(char list \<Rightarrow> int) set"   and c1 :: "(char list \<Rightarrow> int) set acom"   and c2 :: "(char list \<Rightarrow> int) set acom" shows "step S (c1;; c2) = step S c1;; step (post c1) c2"
chain_def: fixes below :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Y :: "nat \<Rightarrow> 'c" assumes "class.po below" shows "po.chain below Y = (\<forall>i. below (Y i) (Y (Suc i)))"
assms: fixes C :: "'a set" assumes "chain_on P C A"   and "C \<noteq> {}" shows "\<exists>x\<in>A. glb P C x"
closed_in_weaken: fixes T :: "ty"   and \<Delta> :: "binding list"   and \<Gamma> :: "binding list"   and B :: "binding" assumes "T closed_in (\<Delta> @ \<Gamma>)" shows "T closed_in (\<Delta> @ B # \<Gamma>)"
AssignE: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int" assumes "(x ::= a, s) \<Rightarrow> t" obtains "t = s(x := aval a s)"
build_relp_def: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and I :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'d" shows "brp \<alpha> I x y \<equiv> y = \<alpha> x \<and> I x"
simps_12: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> R\<^sub>c \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>c \<psi> \<or> (\<exists>j<i. suffix j \<xi> \<Turnstile>\<^sub>c \<phi>))"
correctCompositionDiffLevelsS3: shows "correctCompositionDiffLevels sS3"
consec_in_exclusive1: fixes i :: "nat_int"   and j :: "nat_int"   and n :: "nat" shows "consec i j \<and> n \<^bold>\<in> i \<longrightarrow> n \<^bold>\<notin> j"
3: shows "w * F * e\<^sup>T * e \<le> F"
orthogonal_matrix_def: fixes Q :: "(('d, 'e) vec, 'e) vec" shows "orthogonal_matrix Q = (Finite_Cartesian_Product.transpose Q ** Q = mat (1::'d) \<and> Q ** Finite_Cartesian_Product.transpose Q = mat (1::'d))"
disjoint_insert_2: fixes A :: "'a set"   and b :: "'a"   and B :: "'a set" shows "({} = A \<inter> insert b B) = (b \<notin> A \<and> {} = A \<inter> B)"
mann6: fixes x :: "'a" shows "- (- 3 \<times> x \<squnion> x) = - (- (- (- 3 \<times> x \<squnion> x) \<squnion> - 3 \<times> x) \<squnion> - (- (- 3 \<times> x \<squnion> x) \<squnion> 5 \<times> x))"
vpairsD_1: fixes x :: "V"   and r :: "V" assumes "x \<in>\<^sub>\<circ> vpairs r" shows "x \<in>\<^sub>\<circ> r"
nth_replicate: fixes i :: "nat"   and n :: "nat"   and x :: "'f" assumes "i < n" shows "replicate n x ! i = x"
mat0I_def: fixes ze :: "'b"   and nr :: "nat"   and nc :: "nat" shows "mat0I ze nr nc = replicate nc (vec0I ze nr)"
pderiv_eval_deriv: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "deriv f a = to_fun (pderiv f) a"
assms_1: shows "factorize_polys ps = (ftrs, data)"
PlusE: fixes u :: "'d + 'e"   and A :: "'d set"   and B :: "'e set" assumes "u \<in> A <+> B" obtains x :: "'d" where "x \<in> A" and "u = Inl x"   | y :: "'e" where "y \<in> B" and "u = Inr y"
State_IdleD: fixes localState :: "'d \<Rightarrow> 'e"   and output_fun :: "'d \<Rightarrow> 'f message_af"   and trans_fun :: "'g message_af \<Rightarrow> 'd \<Rightarrow> 'd"   and state :: "'e"   and c :: "'d" assumes "State_Idle localState output_fun trans_fun state"   and "localState c = state" shows "localState (trans_fun NoMsg c) = state \<and> output_fun (trans_fun NoMsg c) = NoMsg"
geodesic_segment_param_7: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and s :: "real"   and t :: "real" assumes "geodesic_segment_between G x y"   and "s \<in> {0..dist x y}"   and "t \<in> {0..dist x y}" shows "dist (geodesic_segment_param G x s) (geodesic_segment_param G x t) = \<bar>s - t\<bar>"
wf_tuple_restrict_simple: fixes n :: "nat"   and B :: "nat set"   and v :: "'a option list"   and A :: "nat set" assumes "wf_tuple n B v"   and "A \<subseteq> B" shows "wf_tuple n A (restrict A v)"
ordinal_of_hmset_suc: fixes k :: "hmultiset" shows "ordinal_of_hmset (k + 1) = ordinal_of_hmset k + 1"
project_def: fixes slices :: "'c \<Rightarrow> 'b set set"   and S :: "'b set"   and n :: "'c" shows "project slices S n \<equiv> {Sl \<inter> S |Sl. Sl \<in> slices n}"
expand: fixes labeled_graph :: "('e, 'f) labeled_graph"   and labeled_graph' :: "('e, 'f) labeled_graph" assumes "edges labeled_graph = edges labeled_graph' \<and> vertices labeled_graph = vertices labeled_graph'" shows "labeled_graph = labeled_graph'"
birkhoff_sum_nn_integral: fixes f :: "'a \<Rightarrow> ennreal"   and n :: "nat" assumes "f \<in> borel_measurable M"   and "\<And>x. 0 \<le> f x" shows "integral\<^sup>N M (birkhoff_sum f n) = of_nat n * integral\<^sup>N M f"
erf_real_le_cancel: fixes a :: "real"   and b :: "real" shows "(erf a \<le> erf b) = (a \<le> b)"
poly_roots_finite: 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 p :: "'b list" assumes "class.idom_char_0 one times plus zero minus uminus" shows "(semiring_0.poly plus zero times p \<noteq> semiring_0.poly plus zero times []) = finite {x. semiring_0.poly plus zero times p x = zero}"
is_interval_translationI: fixes X :: "'b set"   and x :: "'b" assumes "is_interval X" shows "is_interval ((+) x ` X)"
subtype_Collection_x: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type" assumes "Collection \<tau> \<sqsubset> \<sigma>" obtains \<sigma>' :: "'a OCL_Types.type" where "\<sigma> = Collection \<sigma>'" and "\<tau> \<sqsubset> \<sigma>'"   | "\<tau> = OclSuper" and "\<sigma> = OclSuper"
preserves_cod: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.cod (fst f \<star>\<^sub>C snd f) = fst (C.VV.cod f) \<star>\<^sub>C snd (C.VV.cod f)"
prems: fixes s :: "'a" assumes "s \<in> insert s__ ss" shows "\<exists>rs. s = s0 \<star> rs \<and> set rs \<subseteq> rset"