hcju/isabelleps huggingface dataset:
eq_iff: fixes x :: "nat list"   and y :: "nat list" shows "(x = y) = (x \<le>\<^sub>v y \<and> y \<le>\<^sub>v x)"
wt_term_def: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term" shows "wt_term \<Sigma> t \<equiv> wf_term \<Sigma> t \<and> (\<exists>T. \<turnstile>\<^sub>\<tau> t : T)"
fin: shows "finite {i \<in> I. u i \<noteq> 0}"
DISAMBIG_def: fixes n :: "nat" shows "DISAMBIG n \<equiv> True"
equivpE: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "equivp R" obtains "reflp R" and "symp R" and "transp R"
id_subst_comp_subst: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and id_subst :: "'c"   and comp_subst :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and renamings_apart :: "'b literal multiset list \<Rightarrow> 'c list"   and atm_of_atms :: "'b list \<Rightarrow> 'b"   and \<sigma> :: "'c" assumes "substitution subst_atm id_subst comp_subst renamings_apart atm_of_atms" shows "comp_subst id_subst \<sigma> = \<sigma>"
dom_inv: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C"   and "category.iso C f" shows "partial_magma.dom C (category.inv C f) = partial_magma.cod C f"
semigroup_def: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" shows "semigroup f \<equiv> \<forall>a b c. f (f a b) c = f a (f b c)"
self_in_orbit_step: fixes s :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "s \<in> orbit f s" shows "orbit f (f s) = orbit f s"
simps_2: fixes j :: "nat"   and xs :: "nat list"   and A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "diagonal_to_Smith_i (j # xs) A i bezout = (if A $ from_nat i $ from_nat i dvd A $ from_nat j $ from_nat j then diagonal_to_Smith_i xs A i bezout else let (p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat j $ from_nat j); A' = diagonal_step A i j d v in diagonal_to_Smith_i xs A' i bezout)"
sep_conjI: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c"   and Q :: "'c \<Rightarrow> bool"   and y :: "'c"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "P x"   and "Q y"   and "sep_disj x y"   and "h = plus x y" shows "sep_algebra.sep_conj plus sep_disj P Q h"
simps: fixes n :: "nat"   and l :: "nat" shows "invar (n, l) = (l0 \<le> l \<and> (int l0 \<le> \<lfloor>real l / c\<rfloor> \<longrightarrow> f1 * real l \<le> real n) \<and> real n \<le> f2 * real l)"
singleton_eq_conv: fixes a :: "'a"   and b :: "'a" shows "({a} = {b}) = (a = b)"
insert_remove: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "finite A" shows "sum g (insert x A) = g x + sum g (A - {x})"
2: shows "[(\<^bold>\<exists>\<alpha>. \<^bold>\<not>\<^bold>\<box>\<^bold>\<not>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<not>\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<^bold>\<not>\<phi> \<alpha>) in v]"
eint_ord_code_3: fixes q :: "eint" shows "(q \<le> \<infinity>) = True"
integral_mono: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> real"   and g :: "'d \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> f x \<le> g x" shows "integral\<^sup>L M f \<le> integral\<^sup>L M g"
maxt12: shows "wmax (wtimes l\<^sub>1 l\<^sub>2) (wtimes u\<^sub>1 l\<^sub>2) \<equiv>\<^sub>E max (rl1 * rl2) (ru1 * rl2)"
utp_partial_order: fixes T :: "'a hrel \<Rightarrow> 'a hrel" shows "partial_order (utp_order T)"
trace_callee_eq_def: fixes callee1 :: "'h \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'h) spmf"   and callee2 :: "'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'k) spmf"   and A :: "'i set"   and p :: "'h spmf"   and q :: "'k spmf" shows "trace_callee_eq callee1 callee2 A p q = (\<forall>xs. set xs \<subseteq> A \<times> UNIV \<longrightarrow> (\<forall>x\<in>A. trace_callee callee1 p xs x = trace_callee callee2 q xs x))"
has_vderiv_on_add: fixes f :: "real \<Rightarrow> 'b"   and f' :: "real \<Rightarrow> 'b"   and T :: "real set"   and g :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b" assumes " D f = f' on T"   and " D g = g' on T" shows " D (\<lambda>x. f x + g x) = (\<lambda>x. f' x + g' x) on T"
TOTALS: shows "\<forall>a\<in>Field r. \<forall>b\<in>Field r. (a, b) \<in> r \<or> (b, a) \<in> r"
is_total_def: fixes x :: "'a" shows "is_total x \<equiv> 1' \<le> x ; x\<^sup>\<smile>"
deriv_cong_ev: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "\<forall>\<^sub>F x in nhds x. f x = g x"   and "x = y" shows "deriv f x = deriv g y"
validE_NF_alt_def: fixes P :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> (('e + 'f) \<times> 'd) \<times> bool"   and Q :: "'f \<Rightarrow> 'd \<Rightarrow> bool"   and E :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> B \<lbrace>Q\<rbrace>, \<lbrace>E\<rbrace>! = \<lbrace>P\<rbrace> B \<lbrace>\<lambda>v s. case v of Inl e \<Rightarrow> E e s | Inr r \<Rightarrow> Q r s\<rbrace>!"
qiter_fun_exp: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "qiter_fun x y z = x \<squnion> y \<cdot> z"
eq_t12: shows "lhs eq \<notin> lhs ` (set t1 \<union> set t2)"
atom_eqvt: fixes permute :: "perm \<Rightarrow> 'b \<Rightarrow> 'b"   and atom :: "'b \<Rightarrow> atom"   and p :: "perm"   and a :: "'b" assumes "class.at_base permute atom" shows "p \<bullet> atom a = atom (permute p a)"
the_\<nu>_props_1: shows "\<guillemotleft>the_\<nu> : r\<^sub>1 \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright>"
rtrancl_induct2: fixes ax :: "'b"   and ay :: "'c"   and bx :: "'b"   and "by" :: "'c"   and r :: "(('b \<times> 'c) \<times> 'b \<times> 'c) set"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "((ax, ay), bx, by) \<in> r\<^sup>*"   and "P ax ay"   and "\<And>a b aa ba. \<lbrakk>((ax, ay), a, b) \<in> r\<^sup>*; ((a, b), aa, ba) \<in> r; P a b\<rbrakk> \<Longrightarrow> P aa ba" shows "P bx by"
integral_null: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "content (cbox a b) = 0" shows "integral (cbox a b) f = (0::'d)"
zero_flow: shows "flow \<Delta> (\<lambda>_. 0)"
fps_radical_power_nth: fixes r :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and k :: "nat"   and a :: "'a fps" assumes "r k (fps_nth a 0) ^ k = fps_nth a 0" shows "fps_nth (fps_radical r k a ^ k) 0 = (if k = 0 then 1::'a else fps_nth a 0)"
ket_vec_def: fixes v :: "complex Matrix.vec" shows "|v\<rangle> \<equiv> Matrix.mat (dim_vec v) 1 (\<lambda>(i, j). v $ i)"
complex_of_real_of_rat: fixes q :: "rat" shows "complex_of_real (real_of_rat q) = of_rat q"
continuous_on_real_polynomial_function: fixes p :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "real_polynomial_function p" shows "continuous_on S p"
typeof_OutOfMemory: fixes h :: "'heap" assumes "preallocated h" shows "typeof_addr h (addr_of_sys_xcpt OutOfMemory) = \<lfloor>Class_type OutOfMemory\<rfloor>"
le_utrans_trans: fixes x :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and y :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and z :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "le_utrans x y"   and "le_utrans y z" shows "le_utrans x z"
add_eq_if: fixes m :: "nat"   and n :: "nat" shows "m + n = (if m = 0 then n else Suc (m - 1 + n))"
norm_FW_empty: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "valid_dbm M"   and "[M]\<^bsub>v,n\<^esub> = {}" shows "[DBM_Normalization.norm (FW M n) (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> = {}"
Gamma_times_zeta_integral: fixes z :: "complex" assumes "1 < Re z" shows "Gamma z * zeta z = CLINT x:{0<..}|lebesgue. complex_of_real x powr (z - 1) / complex_of_real (exp x - 1)"
of_int_0: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus 0 = zero"
lc_g1: shows "punit.lc (rep_list g1) \<noteq> (0::'b)"
Graph3: fixes T :: "nat"   and E :: "(nat \<times> nat) list"   and R :: "nat" assumes "T \<in> Reach E"   and "R < length E" shows "Reach (E[R := (fst (E ! R), T)]) \<subseteq> Reach E"
steps_preserve_SN_on: fixes a :: "'b"   and b :: "'b"   and A :: "('b \<times> 'b) set" assumes "(a, b) \<in> A\<^sup>*"   and "SN_on A {a}" shows "SN_on A {b}"
c_tr_futures: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and xs :: "'e list"   and ys :: "'e list" shows "(c_tr step out (foldl step s\<^sub>0 xs) ys, {(x, p). p \<noteq> out (foldl step (foldl step s\<^sub>0 xs) ys) x}) \<in> futures (c_process step out s\<^sub>0) (c_tr step out s\<^sub>0 xs)"
assms: fixes A :: "'a mat"   and B :: "'a mat" assumes "(A, B) \<in> set Ms" shows "similar_mat A B"
le_zero: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C C" shows "A = B"
sums_Suc_iff: fixes s :: "'a" shows "(\<lambda>n. f (Suc n)) sums s = f sums (s + f 0)"
app\<^sub>i_Invokestatic: fixes C :: "char list"   and M :: "char list"   and n :: "nat"   and P :: "'b prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and ST :: "ty list"   and LT :: "ty err list" shows "app\<^sub>i (Invokestatic C M n, P, pc, mxs, T\<^sub>r, ST, LT) = (length ST - n < mxs \<and> n \<le> length ST \<and> M \<noteq> clinit \<and> (\<exists>D Ts T m. P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = m in D \<and> P \<turnstile> rev (take n ST) [\<le>] Ts))"
iso_tuple_surjective_proof_assist_idE: fixes x :: "'b"   and y :: "'b" assumes "Record.iso_tuple_surjective_proof_assist x y id" shows "x = y"
subset_preserves_o: fixes v :: "'c \<Rightarrow> 'd"   and w :: "'d \<Rightarrow> 'e" shows "preserves v \<subseteq> preserves (w \<circ> v)"
demonic_demonic: fixes r :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and r' :: "'f \<Rightarrow> 'd \<Rightarrow> bool" shows "[: r :] \<circ> [: r' :] = [: r OO r' :]"
alpha_l_def: fixes l :: "nat" shows "alpha_l l = complex_of_real (cos ((real l + 1 / 2) * \<theta>))"
single_less_eq_eval: fixes x :: "'a"   and P :: "'a Predicate.pred" shows "(Predicate.single x \<le> P) = pred.eval P x"
occuring_gens_in_element: fixes x :: "(bool \<times> 'e) list"   and gens :: "'e set" assumes "x \<in> carrier \<F>\<^bsub>gens\<^esub>" shows "x \<in> lists (UNIV \<times> gens)"
finite_g_IN: fixes x :: "'v" shows "d_IN (\<lambda>(x, y). g (\<langle>y\<rangle>, \<langle>x\<rangle>)) x \<noteq> \<top>"
induct: fixes P :: "('e, 'f) T\<^sub>m \<Rightarrow> ('e, 'f) T\<^sub>m \<Rightarrow> bool"   and a0 :: "('e, 'f) T\<^sub>m"   and a1 :: "('e, 'f) T\<^sub>m" assumes "\<And>x y. (\<And>a b. a \<in> set3_F\<^sub>m x \<Longrightarrow> P a b) \<Longrightarrow> P (T\<^sub>m x) (T\<^sub>m y)" shows "P a0 a1"
distinct_disc_18: fixes event :: "('a, 'b, 'c) event" assumes "isRecvMarker event" shows "\<not> isRecv event"
intro: fixes R :: "('e, 'f) ring_scheme"   and S :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "cring R"   and "cring S"   and "ring_hom_cring_axioms R S h" shows "ring_hom_cring R S h"
wf: shows "wf_matchF_invar \<sigma> V n R I r st (Suc i)"
decomp_Polys: shows "direct_decomp P[X] [ideal {f} \<inter> P[X], P, N]"
isosceles_def: fixes l :: "triangle" shows "isosceles l = (Rep_triangle l ! 0 \<sim> Rep_triangle l ! 1 \<or> Rep_triangle l ! 1 \<sim> Rep_triangle l ! Suc 1)"
asInp_equal_imp_alphaInp: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option"   and qinp' :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" assumes "qGoodInp qinp"   and "asInp qinp = asInp qinp'" shows "(qinp, qinp') \<in> alphaInp"
msed_map_invR: fixes f :: "'b \<Rightarrow> 'a"   and M :: "'b multiset"   and b :: "'a"   and N :: "'a multiset" assumes "f `# M = add_mset b N" shows "\<exists>M1 a. M = add_mset a M1 \<and> f a = b \<and> f `# M1 = N"
set_remove1_eq: fixes xs :: "'c list"   and x :: "'c" assumes "cycle xs" shows "set (remove1 x xs) = set xs - {x}"
irreducible_cong: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "normalize a = normalize b" shows "comm_semiring_1.irreducible times one zero a = comm_semiring_1.irreducible times one zero b"
term_powerprod_axioms: shows "term_powerprod pair_of_term term_of_pair"
schroeder_3_p: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x * y \<le> - z) = (x\<^sup>T * z \<le> - y)"
vertex_map: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "ChamberComplexEndomorphism X f"   and "x \<in> \<Union> X" shows "f x \<in> \<Union> X"
update_cases: fixes r :: "nat \<times> k \<times> f \<times> nat \<times> nat"   and ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option) \<Rightarrow> bool" assumes "(\<pi>\<^sub>2 r = 0) = (\<pi>\<^sub>3 r = unk)"   and "ip \<notin> kD rt \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>ip \<in> kD rt; sqn rt ip < \<pi>\<^sub>2 r\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>ip \<in> kD rt; sqn rt ip = \<pi>\<^sub>2 r; \<pi>\<^sub>5 r < the (dhops rt ip)\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>ip \<in> kD rt; sqn rt ip = \<pi>\<^sub>2 r; the (flag rt ip) = Aodv_Basic.inv\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> r))"   and "\<lbrakk>ip \<in> kD rt; \<pi>\<^sub>3 r = unk\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> (\<pi>\<^sub>2 (the (rt ip)), \<pi>\<^sub>3 r, \<pi>\<^sub>4 r, \<pi>\<^sub>5 r, \<pi>\<^sub>6 r)))"   and "\<lbrakk>ip \<in> kD rt; \<pi>\<^sub>2 r \<le> sqn rt ip; \<pi>\<^sub>3 r = kno; sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val\<rbrakk> \<Longrightarrow> P (rt(ip \<mapsto> the (rt ip)))" shows "P (update rt ip r)"
iso_set_refl: fixes G :: "('c, 'd) monoid_scheme" shows "(\<lambda>x. x) \<in> Group.iso G G"
simps_4: fixes disc :: "'b \<Rightarrow> bool"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "has_disc disc (MatchAnd m1 m2) = (has_disc disc m1 \<or> has_disc disc m2)"
axioms_4: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'b \<Rightarrow> 'a"   and G :: "'a \<Rightarrow> 'b"   and \<eta> :: "'b \<Rightarrow> 'b"   and \<epsilon> :: "'a \<Rightarrow> 'a" assumes "equivalence_of_categories C D F G \<eta> \<epsilon>" shows "functor C D G"
inject_2: fixes x2 :: "funty"   and y2 :: "funty" shows "(TFun x2 = TFun y2) = (x2 = y2)"
sel_2: fixes x1 :: "'a"   and x2 :: "'a msstream" shows "un_MSSCons2 (MSSCons x1 x2) = x2"
less_eq_prod_def: shows "(\<sqsubseteq>) = (\<lambda>(x1, x2) (y1, y2). x1 \<sqsubset>\<^sub>a y1 \<or> x1 \<sqsubseteq>\<^sub>a y1 \<and> x2 \<sqsubseteq>\<^sub>b y2)"
convex_linear_vimage: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'c set" assumes "linear f"   and "convex S" shows "convex (f -` S)"
scopeExtSum: fixes x :: "name"   and P :: "pi"   and Q :: "pi" assumes "x \<sharp> P" shows "<\<nu>x>(P \<oplus> Q) \<sim>\<^sub>e P \<oplus> <\<nu>x>Q"
homogeneous_ideal: fixes F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and n :: "nat" assumes "\<And>f. f \<in> F \<Longrightarrow> homogeneous f"   and "p \<in> ideal F" shows "hom_component p n \<in> ideal F"
add_less_imp_less_right: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less (plus a c) (plus b c)" shows "less a b"
real_poly_pCons: fixes c :: "'b"   and p :: "'b poly" assumes "c \<in> \<real>"   and "set (coeffs p) \<subseteq> \<real>" shows "set (coeffs (pCons c p)) \<subseteq> \<real>"
distinct_3: fixes x11 :: "String.literal"   and x12 :: "typ"   and x3 :: "nat" shows "Ct x11 x12 \<noteq> Bv x3"
BisT_Sym: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com" assumes "c \<approx>T d" shows "d \<approx>T c"
leq_funI: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. f x \<sqsubseteq> g x" shows "f \<sqsubseteq> g"
vars_B: shows "\<Union> (vars ` set_mset (ran_m A)) \<subseteq> \<V>"
snd_corec_prod: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'c"   and s :: "'e" shows "snd (corec_prod f g s) = g s"
elims: fixes x :: "trieP"   and y :: "trie" assumes "abs_trieP x = y" obtains "x = LfP" and "y = Lf"   | ps :: "bool list" and b :: "bool" and l :: "trieP" and r :: "trieP" where "x = NdP ps b (l, r)" and "y = prefix_trie ps (Nd b (abs_trieP l, abs_trieP r))"
svar_deriv: fixes x :: "'d"   and \<nu> :: "(real, 'd) vec" shows "((\<lambda>v. v $ x) has_derivative (\<lambda>v'. v' \<bullet> (\<chi>i. if i = x then 1 else 0))) (at \<nu>)"
hunion_hinsert_right: fixes B :: "hf"   and x :: "hf"   and A :: "hf" shows "B \<squnion> (A \<triangleleft> x) = B \<squnion> A \<triangleleft> x"
bconfs_nonVals: fixes P :: "'c prog"   and sh :: "sheap"   and es :: "'d exp list" assumes "P,sh \<turnstile>\<^sub>b (es,True) \<surd>" shows "map_vals_of es = None"
natural_cases: fixes m :: "natural" obtains n :: "nat" where "m = of_nat n"
xfer_reachable_aux2: fixes cs :: "'c" assumes "C.reachable cs" obtains as :: "'a" where "R as cs" and "A.reachable as"
Rep_step: fixes \<Phi> :: "'b ltl_prop_equiv_quotient"   and \<nu> :: "'b set" shows "Rep (\<up>step \<Phi> \<nu>) \<equiv>\<^sub>P step (Rep \<Phi>) \<nu>"
associativity: fixes M1 :: "'a list list"   and M2 :: "'a list list"   and M3 :: "'a list list" assumes "mat (row_length M1) (length M1) M1 \<and> mat (row_length M2) (length M2) M2 \<and> mat (row_length M3) (length M3) M3" shows "M1 \<otimes> (M2 \<otimes> M3) = M1 \<otimes> M2 \<otimes> M3"
pcr_cr_eq: shows "pcr_transaction\<^sub>m = cr_transaction\<^sub>m"
SubspaceD1: fixes U :: "'b set" assumes "Subspace U" shows "VectorSpace (\<cdot>) U"
ord_resolve_atms_of_concl_subset: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and CAs :: "'b literal multiset list"   and DA :: "'b literal multiset"   and AAs :: "'b multiset list"   and As :: "'b list"   and E :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.ord_resolve S CAs DA AAs As E" shows "atms_of E \<subseteq> \<Union> (atms_of ` set CAs) \<union> atms_of DA"
term_poly_list_rel_empty_iff: fixes q' :: "char list multiset" shows "(([], q') \<in> term_poly_list_rel) = (q' = {#})"
nospoof: shows "no_spoofing TYPE('a) (ipassmt::iface \<Rightarrow> ('i word \<times> nat) list option) ((preprocess::'i common_primitive rule list \<Rightarrow> 'i common_primitive rule list) (rs::'i common_primitive rule list))"
Universe_def: fixes U :: "ZF" shows "Universe U \<equiv> (Elem Empty U \<and> (\<forall>u. Elem u U \<longrightarrow> HOLZF.subset u U) \<and> (\<forall>u. Elem u U \<longrightarrow> Elem (Singleton u) U)) \<and> (\<forall>u. Elem u U \<longrightarrow> Elem (Power u) U) \<and> (\<forall>I u. Elem I U \<longrightarrow> Elem u (Fun I U) \<longrightarrow> Elem (HOLZF.Sum (HOLZF.Range u)) U) \<and> Elem HOLZF.Nat U"
diagram_in_functor_category_axioms: shows "diagram_in_functor_category (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) J (local.map \<circ> D)"
sdrop_cycle_eq: fixes u :: "'b list" assumes "u \<noteq> []" shows "sdrop (length u) (cycle u) = cycle u"
forest_maximally_connected_incl_max2: fixes H :: "('v, 'w) graph" assumes "forest H"   and "subgraph H G"   and "maximally_connected H G" shows "\<forall>(a, w, b)\<in>E - edges H. \<not> forest (add_edge a w b H)"
completeTA_def: shows "dt.completeTA \<equiv> \<lparr>ta_initial = Some ` dt.Qi, ta_rules = dt.\<delta>complete\<rparr>"
asAbs_def: fixes qA :: "('e, 'f, 'varSort, 'var, 'g) qAbs" shows "asAbs qA \<equiv> proj alphaAbsGood qA"
ri: shows "(if p = 0 then [] else if x = 0 then [0] else if x < 0 \<and> even p then [] else case root_int_main p x of (y, True) \<Rightarrow> if even p then [y, - y] else [sgn x * y] | (y, False) \<Rightarrow> []) = y # ys"
web_flowD_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal" assumes "web_flow \<Gamma> f" shows "current \<Gamma> f"
inorder_baliL: fixes l :: "('c \<times> color) tree"   and a :: "'c"   and r :: "('c \<times> color) tree" shows "Tree2.inorder (baliL l a r) = Tree2.inorder l @ a # Tree2.inorder r"
run\<^sub>t_finite: fixes \<Sigma> :: "'c set"   and \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>0 :: "'b"   and w :: "nat \<Rightarrow> 'c" assumes "finite (reach \<Sigma> \<delta> q\<^sub>0)"   and "finite \<Sigma>"   and "range w \<subseteq> \<Sigma>" shows "finite (range (run\<^sub>t \<delta> q\<^sub>0 w))"
mset_replicate_replicate_mset: fixes n :: "nat"   and L :: "'a" shows "mset (replicate n L) = replicate_mset n L"
l2: fixes x :: "'a" shows "f (projection (orthogonal_complement (f -` {0})) x) = \<langle>(cnj (f t) / \<langle>t, t\<rangle>) *\<^sub>C t, x\<rangle>"
fn1: fixes A :: "'b ptrm"   and B :: "'b ptrm"   and x :: "'b"   and T :: "PreSimplyTyped.type" assumes "A \<approx> B" shows "PFn x T A \<approx> PFn x T B"
eq_refl: fixes x :: "('a, 'b) Term.term"   and y :: "('a, 'b) Term.term" assumes "x = y" shows "x \<sqsubseteq> y"
prime_power: fixes p :: "nat"   and k :: "nat"   and n :: "nat" assumes "prime p"   and "0 < k" shows "ind (is_nth_power n) (p ^ k) = (if n dvd k then 1::'b else (0::'b))"
GB_alt_3_finite: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite G" shows "punit.is_Groebner_basis G = (\<forall>f\<in>local.punit.pmdl G. f \<noteq> 0 \<longrightarrow> (\<exists>g\<in>G. g \<noteq> 0 \<and> local.punit.adds_term (lpp g) (lpp f)))"
Pi_def: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" shows "Pi A B = {f. \<forall>x. x \<in> A \<longrightarrow> f x \<in> B x}"
ereal_times_3: shows "1 \<noteq> - \<infinity>"
lpfpc_termination: fixes f :: "'b option acom \<Rightarrow> 'b option acom"   and c :: "com" assumes "acc {(x, y). x \<sqsubseteq> y}"   and "\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y"   and "\<And>c. strip (f c) = strip c" shows "\<exists>c'. lpfp\<^sub>c f c = Some c'"
dirichlet_inverse_Suc_0: fixes f :: "nat \<Rightarrow> 'b"   and i :: "'b" shows "dirichlet_inverse f i (Suc 0) = i"
G1: shows "G1 = \<lparr>nodes = G1_nodes, edges = {(n, e, n'). (n, e, n') \<in> edges (del_unEdge v w v' G) \<and> n \<in> G1_nodes \<and> n' \<in> G1_nodes}\<rparr>"
dom_dom_restrict: fixes S :: "'a set"   and p :: "'a \<Rightarrow> 'b decision option" shows "dom (S \<triangleleft> p) = S \<inter> dom p"
precondition_after_cswitch: shows "\<forall>s d n a. rprecondition s d a \<longrightarrow> rprecondition (rcswitch n s) d a"
assms_2: fixes t :: "real" assumes "t \<in> K" shows "x t \<in> X"
notfin_inf: fixes x :: "'b llist" shows "(x \<notin> UNIV\<^sup>\<star>) = (x \<in> UNIV\<^sup>\<omega>)"