hcju/isabelleps huggingface dataset:
tensor_relations3_def: fixes R :: "('g, 'h) Ring_scheme"   and M :: "('e, 'g, 'i) Module_scheme"   and N :: "('f, 'g, 'i) Module_scheme"   and P :: "('e \<times> 'f, 'g, 'i) Module_scheme" shows "TR3 R M N P = {x. \<exists>m\<in>carrier M. \<exists>n\<in>carrier N. \<exists>a\<in>carrier R. x = (a \<cdot>\<^sub>s\<^bsub>M\<^esub> m, n) \<plusminus>\<^bsub>P\<^esub> -\<^sub>a\<^bsub>P\<^esub> a \<cdot>\<^sub>s\<^bsub>P\<^esub> (m, n)}"
resr_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y \<le> z) = (y \<le> inv_resr x z)"
supset_ld_is_ld: fixes R :: "('g, 'h) ring_scheme"   and M :: "('g, 'i, 'j) module_scheme"   and A :: "'i set"   and B :: "'i set" assumes "module R M"   and "LinearCombinations.module.lin_dep R M A"   and "A \<subseteq> B" shows "LinearCombinations.module.lin_dep R M B"
div_mod_imp_dvdm: fixes b :: "int poly"   and a :: "int poly"   and m :: "int" assumes "\<exists>q r. b = q * a + smult m r" shows "poly_mod.dvdm m a b"
elims: fixes x :: "acom"   and y :: "char list set" assumes "varacom x = y" obtains C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = C\<^sub>1__;; C\<^sub>2__" and "y = varacom C\<^sub>1__ \<union> varacom C\<^sub>2__"   | b :: "bexp" and C\<^sub>1 :: "acom" and C\<^sub>2 :: "acom" where "x = IF b__ THEN C\<^sub>1 ELSE C\<^sub>2" and "y = varacom C\<^sub>1 \<union> varacom C\<^sub>2"   | P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" and Qannot :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" and uu_ :: "(char list \<Rightarrow> int) \<Rightarrow> nat" and C :: "acom" where "x = {P/Qannot/uu_} CONSEQ Ca__" and "y = support P \<union> varacom Ca__ \<union> support Qannot"   | I :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" and S :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int" and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat" and b :: "bexp" and C :: "acom" where "x = {(I, S, E)} WHILE b DO C" and "y = support I \<union> varacom C"   | "x = SKIP" and "y = {}"   | v :: "char list" and va :: "aexp" where "x = v ::= va" and "y = {}"
finsum_empty: fixes f :: "'c \<Rightarrow> nat \<Rightarrow> 'a" shows "finsum P f {} = \<zero>\<^bsub>P\<^esub>"
min_elems_charn: fixes t :: "'c list"   and A :: "'c list set" assumes "t \<in> A" shows "\<exists>t' r. t = t' @ r \<and> t' \<in> min_elems A"
U0_V0_not_learn_bc: shows "\<not> learn_bc \<phi> (U\<^sub>0 \<union> V\<^sub>0) s'"
sat_the_restrict: fixes \<phi> :: "Formula.formula"   and A :: "nat set"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data option list"   and i :: "nat" assumes "fv \<phi> \<subseteq> A" shows "Formula.sat \<sigma> V (map the (restrict A v)) i \<phi> = Formula.sat \<sigma> V (map the v) i \<phi>"
bind_assoc': fixes m :: "'a \<Rightarrow> ('d \<times> 'a) option"   and k :: "'a \<Rightarrow> ('c \<times> 'a) option"   and h :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "(m ;- k) ;- h = m ;- k ;- h"
sound_def: fixes P :: "'a \<Rightarrow> real" shows "sound P \<equiv> Expectations.bounded P \<and> nneg P"
in_subterms_Union: fixes t :: "('c, 'd) Term.term"   and M :: "('c, 'd) Term.term set" assumes "t \<in> M" shows "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
en_fin_word: fixes p :: "'process"   and s :: "'state"   and w :: "'action list" assumes "\<And>r a b. \<lbrakk>r \<in> nodes; a \<in> psen p s - {a. en a s}; b \<in> {a. en a r} - pac p; en a (ex b r)\<rbrakk> \<Longrightarrow> en a r"   and "s \<in> nodes"   and "path w s"   and "pac p \<inter> set w = {}" shows "pac p \<inter> {a. en a (target w s)} \<subseteq> pac p \<inter> {a. en a s}"
tmsubst: fixes bs :: "'b list"   and a :: "tm"   and n :: "nat"   and vs :: "'b list"   and t :: "tm" assumes "tmboundslt (length bs) a"   and "n \<le> length bs" shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n := Itm vs bs t]) a"
lex_ext_iff: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and m :: "nat"   and xs :: "'b list"   and ys :: "'b list" shows "lex_ext f m xs ys = ((length xs = length ys \<or> length ys \<le> m) \<and> ((\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys < length xs), (length xs = length ys \<or> length ys \<le> m) \<and> ((\<exists>i<length xs. i < length ys \<and> (\<forall>j<i. snd (f (xs ! j) (ys ! j))) \<and> fst (f (xs ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (xs ! i) (ys ! i))) \<and> length ys \<le> length xs))"
norm_scaleR: 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 a :: "real"   and x :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (scaleR a x) = \<bar>a\<bar> * norm x"
inv_add: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "\<ominus> (a \<oplus> b) = \<ominus> a \<ominus> b"
image_ring_def: fixes f :: "'e \<Rightarrow> 'd"   and R :: "('e, 'f) ring_scheme" shows "image_ring f R \<equiv> monoid.extend (image_group f R) \<lparr>zero = f \<zero>\<^bsub>R\<^esub>, add = \<lambda>a b. f (inv_into (carrier R) f a \<oplus>\<^bsub>R\<^esub> inv_into (carrier R) f b)\<rparr>"
inverse_unit: fixes M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "monoid M composition unit" shows "monoid.inverse M composition unit unit = unit"
indep_inI: fixes X :: "'a set"   and \<E> :: "'a set" assumes "X \<subseteq> \<E>"   and "indep_in (insert x X) X" shows "X_insert.indep_in \<E> X"
greatest: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat" shows "(if triangle (Suc x\<^sub>0) \<le> x\<^sub>1 then Suc x\<^sub>0 else GREATEST y. y \<le> x\<^sub>0 \<and> triangle y \<le> x\<^sub>1) = (GREATEST y. y \<le> Suc x\<^sub>0 \<and> triangle y \<le> x\<^sub>1)"
arr_Set_length: shows "vcard S = 3\<^sub>\<nat>"
mod_base: fixes n :: "nat"   and d :: "nat \<Rightarrow> nat" assumes "\<And>i. i < n \<Longrightarrow> d i < base"   and "0 < n" shows "from_digits n d mod base = d 0"
no_grey_refsD_1: fixes s :: "'d process_name \<Rightarrow> ('e, 'd, 'f, 'g) Model.local_state"   and r :: "'g"   and p :: "'d process_name" assumes "no_grey_refs s" shows "r \<notin> W (s p)"
assms_8: shows "arb_pf = (\<lambda>x n w. 0)(x := \<lambda>n w. - 1, pos_stock := \<lambda>n w. diff_inv)"
tCn: shows "t \<in> SPR.jkbpCn n"
bit_Suc: 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"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "bit a (Suc n) = bit (divide a (numeral.numeral one plus (num.Bit0 num.One))) n"
igSwapIPresIGWls_def: fixes MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" shows "igSwapIPresIGWls MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> igWls MOD s (igSwap MOD zs z1 z2 X)"
NSLIMSEQ_unique: fixes X :: "nat \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>N\<^sub>S a"   and "X \<longlonglongrightarrow>\<^sub>N\<^sub>S b" shows "a = b"
dims_matricize_2: fixes rmodes :: "nat set"   and T :: "'a tensor" shows "dim_col (matricize rmodes T) = prod_list (nths (dims T) (- rmodes))"
use_iso2: fixes A :: "('f, 'g) gorder_scheme"   and B :: "('h, 'i) gorder_scheme"   and f :: "'f \<Rightarrow> 'h"   and x :: "'f"   and y :: "'f" assumes "isotone A B f"   and "x \<in> carrier A"   and "y \<in> carrier A"   and "x \<sqsubseteq>\<^bsub>A\<^esub> y" shows "f x \<sqsubseteq>\<^bsub>B\<^esub> f y"
\<F>_3: shows "S \<subseteq> \<Union> \<F>"
dhequality_alternative_def: fixes d :: "'d"   and PP :: "nat set"   and \<iota> :: "nat"   and m :: "'id \<Rightarrow> 'val"   and m' :: "'id \<Rightarrow> 'val" shows "dhequality_alternative d PP \<iota> m m' \<equiv> m \<sim>\<^bsub>d,htchLocSet PP\<^esub> m' \<or> \<not> htchLoc \<iota> \<subseteq> htchLocSet PP"
cyc_free_obtains_valuation: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and v :: "'d \<Rightarrow> nat" assumes "\<forall>i xs. i \<le> n \<and> set xs \<subseteq> {0..n} \<longrightarrow> \<one> \<le> len M i i xs"   and "\<forall>c. v c \<le> n \<longrightarrow> 0 < v c" shows "\<exists>u. u \<turnstile>\<^bsub>v,n\<^esub> M"
maxU12: shows "wmax (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2) \<equiv>\<^sub>U max (rl1 * rl2) (ru1 * rl2)"
C0NAreUniform: fixes cX :: "('p, 'v, 's) configuration" assumes "cX = c0 \<or> cX = cN" shows "vUniform (cX = cN) cX"
FAccThrow: fixes e :: "char list exp"   and F :: "char list"   and D :: "char list"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" shows "P \<turnstile> \<langle>throw e\<bullet>F{D},s,b\<rangle> \<rightarrow> \<langle>throw e,s,b\<rangle>"
step_commute: fixes ws :: "('a list \<times> 's \<times> 's) list"   and ps :: "('s \<times> 's) list"   and N :: "('s \<times> 's) set" assumes "ws \<noteq> []" shows "(case step_invariant (ws, ps, N) of (ws', ps', N') \<Rightarrow> (map snd ws', N')) = step (map snd ws, N)"
matrix_mul_rid: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" shows "A ** Finite_Cartesian_Product.mat (1::'b) = A"
fps_unit_factor_mult_fps_X_power_2: fixes f :: "'a fps"   and n :: "nat" shows "unit_factor (f * fps_X ^ n) = unit_factor f"
atm_imp_pos_or_neg_lit: fixes A :: "'a"   and C :: "'a literal multiset" assumes "A \<in> atms_of C" shows "Pos A \<in># C \<or> Neg A \<in># C"
hyps_1: fixes Z :: "'b" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>{s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F)} c\<^sub>1 {t. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c\<^sub>1,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
B: shows "(is_NegChecks u \<and> bvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p u = [] \<and> the_eqs u = [] \<and> length (the_ins u) = 1 \<and> is_Var (fst (hd (the_ins u))) \<and> is_Fun_Set (snd (hd (the_ins u)))) \<and> the_Set (the_Fun (snd (hd (the_ins u)))) = s \<and> the_Var (fst (hd (the_ins u))) = x"
at_bot_def: shows "at_bot = (INF k. principal {..k})"
emeasure_PiM_exponential_Ioi_sequence: fixes R :: "nat \<Rightarrow> real"   and x :: "real" assumes "summable R"   and "\<And>i. 0 < R i"   and "0 \<le> x" shows "emeasure (Pi\<^sub>M UNIV (\<lambda>i. exponential (R i))) (UNIV \<rightarrow> {x<..}) = ennreal (exp (- x * suminf R))"
empty_comp1: fixes \<sigma> :: "char list \<Rightarrow> fterm" shows "\<sigma> \<cdot> \<epsilon> = \<sigma>"
assms_1: shows "app\<^sub>i (i, P, pc, mxs, T\<^sub>r, ST, LT)"
map_option_cong: fixes x :: "'a option"   and y :: "'a option"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "x = y"   and "\<And>a. y = Some a \<Longrightarrow> f a = g a" shows "map_option f x = map_option g y"
weakly_precedes_e: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>"
case_eq_if: fixes f1 :: "'b"   and f2 :: "'d \<Rightarrow> 'b"   and option :: "'d option" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (if option = None then f1 else f2 (the option))"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "le x y"   and "le y z" shows "le x z"
between_imp_collinear: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "between (a, b) x" shows "collinear {a, x, b}"
AssertAllState''_precond: fixes init :: "(nat \<times> linear_poly) list \<Rightarrow> ('a, 'b) state"   and assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and t :: "(nat \<times> linear_poly) list"   and ats :: "('a \<times> 'b atom) list" assumes "AssertAllState'' init assert_bound check"   and "\<triangle> t" shows "let s' = AssertAllState''.assert_bound_loop assert_bound ats (init t) in \<triangle> (\<T> s') \<and> \<nabla> s' \<and> (\<not> \<U> s' \<longrightarrow> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s' \<and> \<diamond> s')"
simps_3: fixes x :: "num"   and y :: "num" shows "Num.pow x (num.Bit1 y) = Num.sqr (Num.pow x y) * x"
fA_1: shows "OrderingSetIso (\<^bold>\<le>) (\<^bold><) dual_order.greater_eq dual_order.greater P f"
simps_2: fixes p :: "pat"   and q :: "pat"   and t :: "btrm" shows "(\<lambda>[\<langle>\<langle>p, q\<rangle>\<rangle>]. t) = (\<lambda>[p]. \<lambda>[q]. t)"
bin_rep_index_0_geq: fixes n :: "nat"   and m :: "nat" assumes "2 ^ n \<le> m"   and "m < 2 ^ (n + 1)" shows "bin_rep (n + 1) m ! 0 = 1"
learn_cons_wrt_goedel: fixes \<chi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and s :: "nat \<Rightarrow> nat option" assumes "goedel_numbering \<chi>"   and "learn_cons \<psi> U s" shows "\<exists>t. learn_cons \<chi> U t"
atLeastLessThan_iff: fixes i :: "'c word"   and l :: "'c word"   and u :: "'c word" shows "(i \<in> signed.atLeastLessThan l u) = (l \<le>s i \<and> i <s u)"
obj_char: fixes a :: "'a" shows "S.obj a = (S.arr a \<and> obj a)"
pos_divide_less_eq: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sgn :: "'e \<Rightarrow> 'e"   and abs :: "'e \<Rightarrow> 'e"   and c :: "'e"   and b :: "'e"   and a :: "'e" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less zero c" shows "less (divide b c) a = less b (times a c)"
assms_2: shows "1 / p + 1 / q = 1"
exhaust: fixes y :: "'b dres" obtains "y = dSUCCEEDi"   | "y = dFAILi"   | x3 :: "'b" where "y = dRETURN x3"
simps_5: fixes t :: "'c local.term" shows "Inv \<^bold>\<l>\<^bold>[t\<^bold>] = \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[Inv t\<^bold>]"
A8: shows "\<forall>TQ TR. TQ \<in>T Q \<and> TR \<in>T R \<longrightarrow> (TQ, TR) \<in> TRel"
intros_3: fixes m1 :: "'a"   and m2 :: "'a"   and m3 :: "'a" assumes "hb m1 m2"   and "hb m2 m3" shows "hb m1 m3"
sum_in_smallo_2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> o[F](h)"   and "g \<in> o[F](h)" shows "(\<lambda>x. f x - g x) \<in> o[F](h)"
not_matches_add_match_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state" assumes "\<not> matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>add_match m rs, Undecided\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>[], Undecided\<rangle> \<Rightarrow> t"
countable_converse: fixes A :: "('b \<times> 'a) set" shows "countable (A\<inverse>) = countable A"
pseudofunctor_axioms: shows "pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C F \<Phi>\<^sub>F"
eq2: shows "2 \<cdot>\<^sub>m proj_psi *\<^sub>v psi'_l l = 2 \<cdot>\<^sub>v (proj_psi *\<^sub>v psi'_l l)"
simps_3: fixes uu :: "preal" shows "preal_plus uu preal.PInfty = preal.PInfty"
has_codomain_char: fixes f :: "'a arrow_of_spans_data \<times> 'a arrow_of_spans_data" shows "(VxV.codomains f \<noteq> {}) = (arr (fst f) \<and> arr (snd f))"
false_PL: fixes p :: "'a" shows "{\<bottom>} \<turnstile> p"
rec_lo_Maxr_lor: fixes x :: "nat"   and y :: "nat" assumes "Suc 0 < x"   and "Suc 0 < y" shows "rec_exec rec_lo [x, y] = Maxr loR [x, y] x"
dense_complement_affine: fixes T :: "'a set"   and S :: "'a set" assumes "aff_dim T < aff_dim S"   and "affine S" shows "closure (S - T) = S"
simps_25: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a"   and f5 :: "pdf_type \<Rightarrow> pdf_type \<Rightarrow> 'a" shows "(case REAL of UNIT \<Rightarrow> f1 | BOOL \<Rightarrow> f2 | INTEG \<Rightarrow> f3 | REAL \<Rightarrow> f4 | PRODUCT x xa \<Rightarrow> f5 x xa) = f4"
assms_2: shows "set signs = consistent_sign_vectors_R qs UNIV"
abs_eq: fixes x :: "'d \<Rightarrow> 'e" assumes "eq_onp bounded_linear x x" shows "norm (Blinfun x) = onorm x"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> int \<Rightarrow> int list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat"   and a2 :: "int"   and a3 :: "int list" assumes "\<And>n m c bs. (\<And>x xa y. \<lbrakk>\<not> (m = 0 \<or> n = 0); x = bin_split n c; (xa, y) = x\<rbrakk> \<Longrightarrow> P n (m - n) xa (y # bs)) \<Longrightarrow> P n m c bs" shows "P a0 a1 a2 a3"
Darapti: fixes A :: "'a set"   and C :: "'a set"   and B :: "'a set" assumes "A Q C"   and "B Q C"   and "C \<noteq> {}" shows "A I B"
locally_determined: fixes A :: "'a set" assumes "A \<subseteq> space M"   and "\<And>B. \<lbrakk>B \<in> sets M; emeasure M B < \<infinity>\<rbrakk> \<Longrightarrow> A \<inter> B \<in> sets M" shows "A \<in> sets M"
Oadmit_Var_simps: fixes \<sigma> :: "('a, 'b, 'c) subst"   and c :: "'c"   and U :: "('c + 'c) set" shows "Oadmit \<sigma> (OVar c) U = True"
rs_def: shows "rs \<equiv> {t. (f t = 0 \<or> t \<in> s) \<and> x < t \<and> t < b}"
test_def: fixes pTs :: "ty list" shows "test pTs = e:==NewC Ext;; Try Expr ({Main,ClassT Base,IntVir}!! e\<cdot>foo( {pTs}[Lit Null])) Catch(SXcpt NullPointer z) (lab1\<bullet> While(Acc (Acc (arr_viewed_from Main Ext).[Lit (Intg 2)])) Skip)"
snd_sym_preproc_aux_superset: fixes fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and vs :: "'t list" shows "set fs \<subseteq> set (snd (sym_preproc_aux gs ks (vs, fs)))"
iT_Div_icard: fixes d :: "nat"   and I :: "nat set"   and n :: "nat" assumes "0 < d"   and "finite I \<Longrightarrow> Max I div d \<le> n" shows "icard (I \<oslash> d) = (if finite I then enat (\<Sum>k\<le>n. if I \<inter> [k * d\<dots>,d - Suc 0] = {} then 0 else Suc 0) else \<infinity>)"
norm_less_p1: fixes x :: "'c" shows "norm x < norm (of_real (norm x) + (1::'c))"
lem1: fixes Y :: "msg"   and G :: "msg set"   and H :: "msg set" assumes "Y \<in> synth (analz (synth G \<union> H) \<inter> (analz (G \<union> H) \<union> synth G))" shows "Y \<in> synth (analz (G \<union> H))"
2: shows "forest_components h * (j \<sqinter> - choose_component (forest_components h) j) = j \<sqinter> forest_components h * - choose_component (forest_components h) j"
lfinite_lmapD: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d llist" assumes "lfinite (lmap f xs)" shows "lfinite xs"
cart_flip: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and B :: "'c set"   and A :: "'d set" shows "(\<lambda>p. f (snd p) (fst p)) ` (B \<times> A) = (\<lambda>p. f (fst p) (snd p)) ` (A \<times> B)"
simps_3: fixes c :: "int"   and s :: "MIR.num"   and t :: "MIR.num" shows "dvdnumcoeff (CF c s t) = (\<lambda>g. g dvd c \<and> dvdnumcoeff t g)"
locally_open_subset: fixes P :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and t :: "'b set" assumes "locally P S"   and "openin (top_of_set S) t" shows "locally P t"
concat_equal_length: fixes xss :: "'b list list"   and l :: "nat" assumes "\<And>xs. xs \<in> set xss \<Longrightarrow> length xs = l" shows "length (concat xss) = length xss * l"
sup_aci_3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
tl_in_cptn: fixes a :: "'b com option \<times> 'b"   and xs :: "('b com option \<times> 'b) list" assumes "a # xs \<in> cptn"   and "xs \<noteq> []" shows "xs \<in> cptn"
double_diff: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "order.greater_eq B A"   and "order.greater_eq C B" shows "B - (C - A) = A"
elementary_bounded: fixes p :: "'c set set"   and S :: "'c set" assumes "p division_of S" shows "bounded S"
diagram_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and d :: "('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> ('b \<times> 'a) list" shows "diagram ars d = (let (\<tau>, \<sigma>, \<sigma>', \<tau>') = d in {\<sigma>, \<tau>, \<sigma>', \<tau>'} \<subseteq> seq ars \<and> fst \<sigma> = fst \<tau> \<and> lst \<sigma> = fst \<tau>' \<and> lst \<tau> = fst \<sigma>' \<and> lst \<sigma>' = lst \<tau>')"
assms_1: shows "bld \<in># \<B>\<^sup>R"
1: fixes x :: "nat \<Rightarrow> 'a" shows "submatrix (A x) I J $$ (i, j) = A x $$ (pick I i, pick J j)"
f: shows "(\<lambda>(x, y). f x y) \<in> borel \<Otimes>\<^sub>M borel \<rightarrow>\<^sub>M M"
bound_intersect_2d_ud_def: fixes prec :: "nat"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs"   and xl :: "real" shows "bound_intersect_2d_ud prec X xl = (case segments_of_aform X of [] \<Rightarrow> if fst (fst X) = xl then let m = snd (fst X) in Some (m, m) else None | [x] \<Rightarrow> case bound_intersect_2d prec (filter (\<lambda>((x1, y1), x2, y2). x1 < x2) [x]) xl of None \<Rightarrow> None | Some (m, M') \<Rightarrow> case bound_intersect_2d prec (filter (\<lambda>((x1, y1), x2, y2). x2 < x1) [x]) xl of None \<Rightarrow> None | Some (m', M) \<Rightarrow> Some (min m m', max M M') | [x, y] \<Rightarrow> intersect_segment_xline prec x xl | x # y # ad # list \<Rightarrow> case bound_intersect_2d prec (filter (\<lambda>((x1, y1), x2, y2). x1 < x2) (x # y # ad # list)) xl of None \<Rightarrow> None | Some (m, M') \<Rightarrow> case bound_intersect_2d prec (filter (\<lambda>((x1, y1), x2, y2). x2 < x1) (x # y # ad # list)) xl of None \<Rightarrow> None | Some (m', M) \<Rightarrow> Some (min m m', max M M'))"
submanifold_axioms: shows "submanifold charts_eucl k (codomain c)"
\<zeta>: shows "\<forall>\<^sub>F x in at_right 0. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
SPEC_rule_nested2: fixes m :: "('b \<times> 'c) nres"   and P :: "'b \<times> 'c \<Rightarrow> bool"   and g :: "'b \<times> 'c \<Rightarrow> ('b \<times> 'c) nres" assumes "m \<le> SPEC P"   and "\<And>r1 r2. P (r1, r2) \<Longrightarrow> g (r1, r2) \<le> SPEC P" shows "m \<le> SPEC (\<lambda>r'. g r' \<le> SPEC P)"
strongest_direction_def: fixes T :: "'c" shows "strongest_direction T = (let b = fold (\<lambda>b' b. if \<bar>T \<bullet> b\<bar> \<le> \<bar>T \<bullet> b'\<bar> then b' else b) Basis_list (0::'c) in (sgn (T \<bullet> b) *\<^sub>R b, \<bar>T \<bullet> b\<bar>))"