hcju/isabelleps huggingface dataset:
H': fixes p :: "(char list list \<times> int) list"   and pa :: "int mpoly" assumes "(p, pa) \<in> sorted_poly_rel O mset_poly_rel" shows "weak_equality_l p spec \<le> SPEC (\<lambda>eqa. eqa \<longrightarrow> pa = spec')"
real_cond_exp_add: fixes f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "integrable M f"   and "integrable M g" shows "AE x in M. real_cond_exp M Invariants (\<lambda>x. f x + g x) x = real_cond_exp M Invariants f x + real_cond_exp M Invariants g x"
rev_rev_ident: fixes xs :: "'k list" shows "rev (rev xs) = xs"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "Sublist.prefix a b" shows "strict_prefix a b"
\<I>'1: shows "\<forall>X F \<delta>. \<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t \<in> set S \<longrightarrow> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F - set X \<subseteq> subst_domain \<I>'"
ldrop_eq_LNil: fixes n :: "enat"   and xs :: "'a llist" shows "(ldrop n xs = []\<^sub>l) = (llength xs \<le> n)"
sem_def: fixes n :: "'n"   and \<sigma> :: "'var \<Rightarrow> 'val" shows "sem n \<sigma> = snd (local.step (n, \<sigma>))"
part_UNpart: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set" assumes "properL cl"   and "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n)" shows "part {..<brnL cl (length cl)} (UNpart cl P)"
interior_Int: fixes S :: "'b set"   and T :: "'b set" shows "interior (S \<inter> T) = interior S \<inter> interior T"
zor_poly_inverse: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z"   and "\<exists>\<^sub>F w in at z. f w \<noteq> 0" shows "\<forall>\<^sub>F w in at z. zor_poly (\<lambda>w. inverse (f w)) z w = inverse (zor_poly f z w)"
yh: shows "YMap C X \<eta> |\<in>| (Hom\<^bsub>C\<^esub> X Y)"
prime_elem_not_unit: fixes p :: "'a" assumes "prime_elem p" shows "\<not> is_unit p"
closure_inside_subset: fixes S :: "'a set" assumes "closed S" shows "closure (inside S) \<subseteq> S \<union> inside S"
h_def: shows "h \<equiv> \<lambda>(x, y). \<langle>f x, g y\<rangle>"
mult_right_mono: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.ordered_semiring plus zero less_eq less times"   and "less_eq a b"   and "less_eq zero c" shows "less_eq (times a c) (times b c)"
nN_6: shows "no \<in> set_of (Node lt po rt)"
linorder_sorted_wrt_exists: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and B :: "'b set" assumes "linorder_on A R"   and "finite B"   and "B \<subseteq> A" shows "\<exists>xs. set xs = B \<and> distinct xs \<and> Linorder_Relations.sorted_wrt R xs"
hyps_1: shows "vars a \<union> {x} \<subseteq> dom ps"
structCongBisim: fixes P :: "pi"   and Q :: "pi" assumes "P \<equiv>\<^sub>s Q" shows "P \<sim>\<^sub>l Q"
quasinorm_subsetI': fixes N1 :: "'b quasinorm"   and N2 :: "'b quasinorm"   and C :: "real" assumes "\<And>f. f \<in> space\<^sub>N N1 \<Longrightarrow> f \<in> space\<^sub>N N2"   and "\<And>f. f \<in> space\<^sub>N N1 \<Longrightarrow> Norm N2 f \<le> C * Norm N1 f" shows "N1 \<subseteq>\<^sub>N N2"
b: shows "liminf (v \<circ> r \<circ> s) \<noteq> - \<infinity>"
gauss_int_norm_eq_Suc_0_iff: fixes x :: "gauss_int" shows "(gauss_int_norm x = Suc 0) = is_unit x"
div_mod_lm: fixes y :: "nat"   and x :: "nat"   and u :: "nat" assumes "y < x" shows "(u + 2 ^ x) div 2 ^ y mod 2 = u div 2 ^ y mod 2"
affine_alt: fixes s :: "'a set" shows "affine s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u. (1 - u) *\<^sub>R x + u *\<^sub>R y \<in> s)"
span_clauses_2: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale" shows "(0::'e) \<in> module.span scale S"
subtermeqI'': fixes t :: "('a, 'b) Term.term"   and T :: "('a, 'b) Term.term list"   and f :: "'a" assumes "t \<in> set T" shows "t \<sqsubseteq> Fun f T"
bf_and_def: fixes a :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool" shows "bf_and a b \<equiv> bf_ite a b bf_False"
ag_pOp_commute: fixes A :: "('d, 'e) aGroup_scheme"   and x :: "'d"   and y :: "'d" assumes "aGroup A"   and "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> y = y \<plusminus>\<^bsub>A\<^esub> x"
take_bit_horner_sum_bit_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and bs :: "bool list" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "take_bit n (comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) bs) = comm_semiring_0.horner_sum plus zero times (zero_neq_one.of_bool one zero) (numeral.numeral one plus (num.Bit0 num.One)) (take n bs)"
order_iff_strict: fixes a :: "'k multiset"   and b :: "'k multiset" shows "(a \<subseteq># b) = (a \<subset># b \<or> a = b)"
distinct_27: fixes x6 :: "ty" shows "NT \<noteq> x6\<lfloor>\<rceil>"
h1d: shows "MPoly_Type.degree (Var var) var = 1"
linear_system_homogenous_3_2: fixes f1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a11 :: "'a"   and a12 :: "'a"   and a13 :: "'a"   and f2 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a21 :: "'a"   and a22 :: "'a"   and a23 :: "'a"   and x1 :: "'a"   and y1 :: "'a"   and z1 :: "'a"   and x2 :: "'a"   and y2 :: "'a"   and z2 :: "'a" assumes "f1 = (\<lambda>x y z. a11 * x + a12 * y + a13 * z)"   and "f2 = (\<lambda>x y z. a21 * x + a22 * y + a23 * z)"   and "f1 x1 y1 z1 = (0::'a)"   and "f2 x1 y1 z1 = (0::'a)"   and "f1 x2 y2 z2 = (0::'a)"   and "f2 x2 y2 z2 = (0::'a)"   and "x2 \<noteq> (0::'a) \<or> y2 \<noteq> (0::'a) \<or> z2 \<noteq> (0::'a)"   and "det2 a11 a12 a21 a22 \<noteq> (0::'a) \<or> det2 a11 a13 a21 a23 \<noteq> (0::'a) \<or> det2 a12 a13 a22 a23 \<noteq> (0::'a)" shows "\<exists>k. x1 = k * x2 \<and> y1 = k * y2 \<and> z1 = k * z2"
\<alpha>_ide_simp: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and h :: "'b" assumes "prebicategory_with_homs_and_units V H \<a> \<i> src trg"   and "partial_magma.ide V f"   and "partial_magma.ide V g"   and "partial_magma.ide V h"   and "src f = trg g"   and "src g = trg h" shows "prebicategory_with_homs_and_units.\<alpha> V H \<a> src trg f g h = \<a> f g h"
smc_is_arrD_4: fixes a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<BB>\<^esub> b" shows "\<BB>\<lparr>Dom\<rparr>\<lparr>f\<rparr> = a"
ncop__ncol: fixes A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "\<not> Coplanar A B C D" shows "\<not> Col A B C"
frpar_def: fixes p :: "fm" shows "frpar p = simpfm (qelim p ferrack)"
cla_mono': fixes Z' :: "('c \<Rightarrow> real) set"   and Z :: "('c \<Rightarrow> real) set" assumes "Z' \<subseteq> V"   and "Z \<subseteq> Z'" shows "Closure\<^sub>\<alpha> Z \<subseteq> Closure\<^sub>\<alpha> Z'"
incl_incl_of_1: fixes a :: "'s"   and b :: "'s" assumes "incl_in a b" shows "incl (incl_of a b)"
b: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"   and "s = Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow>\<^sub>s t"
isomorphic_hcomp_REP: fixes F :: "('a, 'a set) arr"   and G :: "('a, 'a set) arr" assumes "seq F G" shows "REP F \<star> REP G \<cong>\<^sub>B REP (F \<odot> G)"
hash_tree_simps: fixes h :: "'b \<Rightarrow> 'a"   and x :: "('b \<times> ('b, 'a) rose_tree\<^sub>m list, 'a \<times> 'a rose_tree\<^sub>h list) blindable\<^sub>m" shows "hash_tree h (Tree\<^sub>m x) = Tree\<^sub>h (hash_rt_F\<^sub>m h (hash_tree h) x)"
assms: shows "\<turnstile>\<^sub>! (p \<^bold>\<longleftrightarrow>\<^sub>! p')"
gamma2_vec_deriv_within: fixes x :: "real" assumes "x \<in> {0..1} - s" shows "vector_derivative (\<gamma>2 \<circ> \<phi>) (at x) = vector_derivative (\<gamma>2 \<circ> \<phi>) (at x within {0..1})"
phl_cons1: fixes x :: "'a"   and w :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> w"   and "\<lbrace>w\<rbrace>y\<lbrace>z\<rbrace>" shows "\<lbrace>x\<rbrace>y\<lbrace>z\<rbrace>"
leg1_simps_6: shows "cod f\<^sub>0gh\<^sub>1.p\<^sub>1 = f\<^sub>0gh\<^sub>1.p\<^sub>1"
DOM_comp: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "ARR f"   and "ARR g"   and "DOM f = COD g" shows "DOM (f \<cdot> g) = DOM g"
cases: fixes x :: "('a list \<times> 's \<times> 's) list \<times> ('s \<times> 's) list \<times> ('s \<times> 's) set" obtains ws :: "('a list \<times> 's \<times> 's) list" and uu_ :: "('s \<times> 's) list" and uv_ :: "('s \<times> 's) set" where "x = (ws, uu_, uv_)"
hyps_1: fixes p :: "ref" assumes "Dag p low high lt"   and "ordered lt var"   and "no \<in> set_of lt"   and "low no \<noteq> Null"   and "high no \<noteq> Null" shows "var (low no) < var no \<and> var (high no) < var no"
dvd_power: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "nat"   and x :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "0 < n \<or> x = one" shows "dvd.dvd times x (power.power one times x n)"
distinct_1: fixes x2 :: "name" shows "zer \<noteq> Var x2"
vaflow2: fixes F :: "real \<Rightarrow> variable \<Rightarrow> real"   and \<theta> :: "trm"   and \<zeta> :: "real" assumes "solves_ODE (USubst.adjoint \<sigma> I \<omega>) F x \<theta>" shows "Uvariation (F \<zeta>) (F 0) {RVar x, DVar x}"
mor_beh: fixes s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2" shows "mor UNIV UNIV s1 s2 carT1 carT2 strT1 strT2 (beh1 s1 s2) (beh2 s1 s2)"
fset_inverse: fixes x :: "'b fset" shows "Abs_fset (fset x) = x"
unit_disc_fix_conjugate_moebius: fixes M :: "moebius" assumes "unit_disc_fix M" shows "unit_disc_fix (conjugate_moebius M)"
v_x_rt: shows "(v, x) \<in> r\<^sup>*"
eq_swap_range: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 =\<^sub>S w\<^sub>2" shows "set w\<^sub>1 = set w\<^sub>2"
A_5: shows "Z \<subseteq> {u \<in> V. u x - u y < real_of_int m}"
coprime_common_divisor: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "coprime a b"   and "c dvd a"   and "c dvd b" shows "is_unit c"
snd1: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P >>> P'"   and "\<And>A B. P \<noteq> SimplyTyped.Pair A B" shows "Snd P >>> Snd P'"
analz_Crypt_if: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "analz (insert (Crypt K X) H) = (if Key (invKey K) \<in> analz H then insert (Crypt K X) (analz (insert X H)) else insert (Crypt K X) (analz H))"
between_commute: fixes a :: "'b"   and b :: "'b" shows "between (a, b) = between (b, a)"
queue_invar_simps_2: fixes t :: "('c, 'd) SkewBinomialTree"   and q :: "('c, 'd) SkewBinomialTree list" shows "queue_invar (t # q) = (tree_invar t \<and> queue_invar q)"
absorb_iff1: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (min a b = a)"
A: shows "euclidean_size (gcd a b) = euclidean_size a"
indep_sets_mono_sets: fixes F :: "'e \<Rightarrow> 'a set set"   and I :: "'e set"   and G :: "'e \<Rightarrow> 'a set set" assumes "indep_sets F I"   and "\<And>i. i \<in> I \<Longrightarrow> G i \<subseteq> F i" shows "indep_sets G I"
Ptl_LCons: shows "ltl P = LCons w0 (ltl (ltl P))"
C0_notin: shows "(C0, active) \<notin> lnth Ns n"
terminal_tllist_of_llist: fixes y :: "'d"   and xs :: "'e llist" shows "terminal (tllist_of_llist y xs) = (if lfinite xs then y else undefined)"
subtopology_eq_discrete_topology_sing: fixes X :: "'c topology"   and a :: "'c" shows "(X = discrete_topology {a}) = (topspace X = {a})"
span_Int_subset: fixes A :: "'b set"   and B :: "'b set" shows "span (A \<inter> B) \<subseteq> span A \<inter> span B"
ts: fixes i :: "nat" assumes "i < length bs" shows "akra_bazzi_term x\<^sub>0 x\<^sub>1 (bs ! i) (ts ! i)"
postdominate_implies_inner_path: fixes n' :: "'node"   and n :: "'node" assumes "n' postdominates n" obtains as :: "'edge list" where "n -as\<rightarrow>\<^sub>\<iota>* n'" and "n' \<notin> set (sourcenodes as)"
simps_2: fixes x :: "char list"   and a :: "aexp"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "pre (x ::= a) Q = (\<lambda>s. Q (s[a/x]))"
id_galcon_def: fixes A :: "'a gorder" shows "I\<^sub>g A = \<lparr>orderA = A, orderB = A, lower = id, upper = id\<rparr>"
finite_succeed_t': fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "q\<^sub>0 \<notin> F" shows "finite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i) = finite (mojmir_def.succeed_t \<Sigma> \<delta> q\<^sub>0 w F i)"
entry_entry_same_proc: fixes p :: "'p"   and p' :: "'p" assumes "entry fg p = entry fg p'" shows "p = p'"
assms: shows "\<exists>e. f e \<noteq> bot \<and> (\<forall>d. e \<noteq> d \<longrightarrow> f d = bot)"
dg_prod_ArrD_2: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>" shows "\<D>\<^sub>\<circ> f = I"
vsv_value: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Arr\<rparr>" obtains b :: "V" where "\<FF>\<lparr>ArrMap\<rparr>\<lparr>a\<rparr> = b" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ArrMap\<rparr>)"
eval_dioph_mod: fixes ks :: "int list"   and xs :: "int list"   and n :: "int" shows "eval_dioph ks xs mod n = eval_dioph ks (map (\<lambda>x. x mod n) xs) mod n"
i_1: shows "\<forall>m<card A. \<forall>x. m = card x \<longrightarrow> finite x \<longrightarrow> x \<subseteq> carrier G \<longrightarrow> (\<exists>gs. set gs \<subseteq> generate G x \<and> cycle gs \<and> is_idirprod (generate G x) (\<lambda>g. generate G {g}) (set gs) \<and> successively (dvd) (map ord gs) \<and> card (set gs) \<le> card x)"
gallery_Cons_reduce: fixes x :: "'a set"   and xs :: "'a set list" assumes "folding_g.gallery (x # xs)" shows "folding_g.gallery xs"
succeed: fixes x :: "'a"   and h' :: "heap" assumes "execute f h = Some (x, h')" shows "P"
vcomp_ide_dom: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e" assumes "natural_transformation A B F G \<tau>" shows "vertical_composite.map A B F \<tau> = \<tau>"
termi_strip: shows "strip F \<Gamma>\<turnstile>c \<down> s"
lipschitz_on_def: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "C-lipschitz_on U f = (0 \<le> C \<and> (\<forall>x\<in>U. \<forall>y\<in>U. dist (f x) (f y) \<le> C * dist x y))"
fmeasurable_Int_fmeasurable: fixes S :: "'b set"   and M :: "'b measure"   and T :: "'b set" assumes "S \<in> fmeasurable M"   and "T \<in> sets M" shows "S \<inter> T \<in> fmeasurable M"
cases: fixes a :: "'d \<Rightarrow> 'e tree option"   and \<delta> :: "('d, 'e) ta_rule set" assumes "a \<in> brw_iq \<delta>" obtains Q :: "'d \<Rightarrow> 'e tree option" where "a = Q" and "\<forall>q t. Q q = Some t \<longrightarrow> (\<exists>r\<in>\<delta>. rhsq r = [] \<and> q = lhs r \<and> t = NODE (rhsl r) [])" and "\<forall>r\<in>\<delta>. rhsq r = [] \<longrightarrow> Q (lhs r) \<noteq> None"
\<X>_def: fixes k :: "nat" shows "\<X> k = {uu_. \<exists>t l \<omega>. uu_ = (t, \<omega>) \<and> t \<in> \<TT> \<and> l \<in> Lex t Doc k \<and> \<omega> = take l (drop k Doc)}"
oconf_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and obj :: "char list \<times> (char list \<times> char list \<Rightarrow> val option)" shows "P,h \<turnstile> obj \<surd> \<equiv> let (C, fs) = obj in \<forall>F D T. P \<turnstile> C has F:T in D \<longrightarrow> (\<exists>v. fs (F, D) = \<lfloor>v\<rfloor> \<and> P,h \<turnstile> v :\<le> T)"
bounded_linear_axioms_def: fixes f :: "'b \<Rightarrow> 'c" shows "bounded_linear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
reads_insert_writes_set_right: fixes getter :: "'l \<Rightarrow> 'l \<Rightarrow> bool"   and S :: "('l \<Rightarrow> 'l \<Rightarrow> bool) set"   and f :: "('l, 'm, 'n) prog"   and h :: "'l"   and h' :: "'l" assumes "reflp getter"   and "transp getter"   and "reads S f h h'" shows "reads (insert getter S) f h h'"
simps_4: fixes r :: "real" shows "ereal r * - \<infinity> = (if r = 0 then 0 else if 0 < r then - \<infinity> else \<infinity>)"
L6: fixes k :: "nat"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "k < length xs" shows "map f xs ! k = f (xs ! k)"
nat_0_less_mult_iff: fixes m :: "nat"   and n :: "nat" shows "(0 < m *n n) = (0 < m \<and> 0 < n)"
assms_2: fixes s :: "(char list \<Rightarrow> int option) \<times> nat" assumes "Q s" shows "Q' s"
gState_inv_def: fixes prog :: "program"   and g :: "'b gState_scheme" shows "gState_inv prog g = (length (procs g) \<le> max_procs \<and> (\<forall>p\<in>set (procs g). pState_inv prog p \<and> cl_inv (g, p)) \<and> length (gState.channels g) \<le> max_channels \<and> set (gState.channels g) \<subseteq> Collect channel_inv \<and> lm.ball (gState.vars g) (\<lambda>(k, v). variable_inv v))"
vcpower_two_ftimes: fixes A :: "V" shows "A ^\<^sub>\<times> 2\<^sub>\<nat> = A \<times>\<^sub>\<bullet> A"
rcos_const_imp_mem: fixes i :: "'a" assumes "i \<in> carrier R"   and "(\<Inter>i\<le>n. I (Suc i)) +> i = (\<Inter>i\<le>n. I (Suc i))" shows "i \<in> (\<Inter>i\<le>n. I (Suc i))"
mult_pos_neg: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less b zero" shows "less (times a b) zero"
steps: shows "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (Skip, t)"
finite_measure_lebesgue_on: fixes S :: "'a set" assumes "S \<in> lmeasurable" shows "finite_measure (lebesgue_on S)"
sem_behaves_deterministic: fixes s :: "'state"   and b1 :: "'state behaviour"   and b2 :: "'state behaviour" assumes "\<And>x y z. \<lbrakk>step x y; step x z\<rbrakk> \<Longrightarrow> y = z"   and "s \<down> b1"   and "s \<down> b2" shows "b1 = b2"
continuous_on_Im: fixes s :: "'a set"   and g :: "'a \<Rightarrow> complex" assumes "continuous_on s g" shows "continuous_on s (\<lambda>x. Im (g x))"
bin_mask_conv_pow2: fixes n :: "nat" shows "mask n = 2 ^ n - 1"
ord_option_map1: fixes ord :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'a"   and x :: "'c option"   and y :: "'b option" shows "ord_option ord (map_option f x) y = ord_option (\<lambda>x. ord (f x)) x y"
continuous_within_compose2: fixes x :: "'d"   and s :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'e \<Rightarrow> 'f" assumes "continuous (at x within s) f"   and "continuous (at (f x) within f ` s) g" shows "continuous (at x within s) (\<lambda>x. g (f x))"
unfold_eq_fix: fixes h :: "'b \<rightarrow> ('c, 'b) Step" shows "unfold\<cdot>h = fix\<cdot>(unfoldF\<cdot>h)"
s_Prj2_inv: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and v :: "Syntax.term" assumes "\<lless>\<pi>, Prj2 e\<ggreater> m\<rightarrow> \<lless>\<pi>', v\<ggreater>" obtains e' :: "Syntax.term" where "v = Prj2 e'" and "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>"   | v\<^sub>1 :: "Syntax.term" where "e = Syntax.Pair v\<^sub>1 v" and "\<pi>' = \<pi>" and "value v\<^sub>1" and "value v"
pmf_pmf_of_list: fixes xs :: "('b \<times> real) list"   and x :: "'b" assumes "pmf_of_list_wf xs" shows "pmf (pmf_of_list xs) x = sum_list (map snd (filter (\<lambda>z. fst z = x) xs))"
addition_inc_add0: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "A \<subseteq> addition_set f A"
eq0: shows "(pp_of_term u - lp a) \<oplus> lt a = u"
those_Cons_OkD: fixes x :: "('l, 'm) result"   and xs :: "('l, 'm) result list"   and ys :: "'m list" assumes "Result.those (x # xs) = Ok ys" shows "\<exists>y ys'. ys = y # ys' \<and> x = Ok y \<and> Result.those xs = Ok ys'"
WCR_on_def: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" shows "WCR_on r A = (\<forall>a\<in>A. \<forall>b c. (a, b) \<in> r \<and> (a, c) \<in> r \<longrightarrow> (b, c) \<in> r\<^sup>\<down>)"
is_tm_dghm_op: shows "op_dghm \<GG> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
Icc_subset_Iic_iff: fixes l :: "'a"   and h :: "'a"   and h' :: "'a" shows "({l..h} \<subseteq> {..h'}) = (\<not> l \<le> h \<or> h \<le> h')"
ereal_truncation_real_bottom: fixes x :: "ereal" assumes "x \<noteq> \<infinity>" shows "(\<lambda>x. ereal (real_of_ereal (max x (ereal (- real x))))) \<longlonglongrightarrow> x"
Provability: fixes \<alpha> :: "fm" assumes "Sigma_fm \<alpha>"   and "ground_fm \<alpha>" shows "{\<alpha>} \<turnstile> PfP \<guillemotleft>\<alpha>\<guillemotright>"
simps_2: fixes g :: "'f \<Rightarrow> ('e, 'f) step"   and h :: "'g \<Rightarrow> ('e, 'g) step"   and sg :: "'f"   and sh :: "'g" shows "splice_raw g h (splice_state.Left sg sh) = (case g sg of Done \<Rightarrow> Skip (Right_only sh) | Skip sg' \<Rightarrow> Skip (splice_state.Left sg' sh) | Yield a sg' \<Rightarrow> Yield a (splice_state.Right sg' sh))"
sorted_append: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (xs @ ys) = (linorder.sorted less_eq xs \<and> linorder.sorted less_eq ys \<and> (\<forall>x\<in>set xs. \<forall>y\<in>set ys. less_eq x y))"
viewRn_empty: fixes t :: "('a, 'b) FingerTreeStruc" shows "(t \<noteq> Empty) = (FingerTreeStruc.viewRn t \<noteq> None)"
shared_weight_net_Conv: fixes m :: "real mat convnet"   and A :: "real mat" assumes "shared_weight_net m" shows "shared_weight_net (Conv A m)"
FinFunc_def: fixes r :: "('d \<times> 'd) set"   and s :: "('c \<times> 'c) set" shows "FinFunc r s = Func (Field s) (Field r) \<inter> fin_support (zero r) (Field s)"
remove_vertex_E'': fixes v :: "'a"   and w :: "'a"   and x :: "'a" assumes "v \<rightarrow>\<^bsub>remove_vertex x\<^esub> w"   and "v \<noteq> x"   and "w \<noteq> x" shows "v \<rightarrow>\<^bsub>H.remove_vertex x\<^esub> w"
simps_8: fixes uu :: "'b ltln set" shows "true\<^sub>n[uu]\<^sub>\<Sigma>\<^sub>1 = true\<^sub>n"
simps_20: fixes v :: "Core.term"   and va :: "Core.term"   and vb :: "nat"   and uw :: "variable \<times> String.literal set \<Rightarrow> typ option" shows "raw_match_term (v $ va) (Bv vb) uw = None"