hcju/isabelleps huggingface dataset:
trim1_def: fixes g :: "'v \<times> 'v \<Rightarrow> ennreal" shows "trim1 g = (if local.trimming g then g else let z = SOME z. z \<in> RF\<^sup>\<circ> (TER g) \<and> z \<notin> A \<Gamma> \<and> d_OUT g z \<noteq> d_IN g z; factor = d_OUT g z / d_IN g z in (\<lambda>(y, x). (if x = z then factor else 1) * g (y, x)))"
comp_min_basis_is_minimal_basis: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "is_minimal_basis (set (comp_min_basis xs))"
lexord_eq_length: fixes xs :: "'b list"   and ys :: "'b list"   and R :: "('b \<times> 'b) set" assumes "(xs, ys) \<in> lexord R"   and "length xs = length ys" shows "\<exists>i. (xs ! i, ys ! i) \<in> R \<and> i < length xs \<and> (\<forall>j<i. xs ! j = ys ! j)"
dfs_reachable_def: fixes D :: "'a \<Rightarrow> bool"   and w :: "'a list" shows "local.dfs_reachable D w \<equiv> let (V, w, brk) = while (\<lambda>(V, w, brk). \<not> brk \<and> w \<noteq> []) (\<lambda>(V, w, uu_). case w of v # w \<Rightarrow> if D v then (V, v # w, True) else if v \<in> V then (V, w, False) else let V = insert v V; w = succ v @ w in (V, w, False)) ({}, w, False) in brk"
corthogonal_is_orthogonal: fixes xs :: "'a vec list" shows "corthogonal xs = orthogonal xs"
homeomorphic_map_imp_homeomorphic_space: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" assumes "homeomorphic_map X Y f" shows "X homeomorphic_space Y"
quasinorm_intersection_minimum: fixes N :: "'c quasinorm"   and N1 :: "'c quasinorm"   and N2 :: "'c quasinorm" assumes "N \<subseteq>\<^sub>N N1"   and "N \<subseteq>\<^sub>N N2" shows "N \<subseteq>\<^sub>N N1 \<inter>\<^sub>N N2"
prems_2: shows "\<forall>t\<in>set (a # bq). rank t < rank t'"
ahm_iteratei_def: fixes hm :: "('e, 'f) hashmap" shows "ahm_iteratei hm = ArrayHashMap_Impl.ahm_iteratei (impl_of hm)"
infinite_hyp_wrong_not_Lim: fixes f :: "nat \<Rightarrow> nat option"   and U :: "(nat \<Rightarrow> nat option) set"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and s :: "nat \<Rightarrow> nat option" assumes "f \<in> U"   and "\<forall>n. \<exists>m>n. \<psi> (the (s (f \<triangleright> m))) \<noteq> f" shows "\<not> learn_lim \<psi> U s"
forest_subE: fixes E' :: "'edge set" assumes "forest E'" shows "E' \<subseteq> E"
qTermLess_modulo_def: fixes rel :: "(('a, 'b, 'c, 'd, 'e) qTerm \<times> ('a, 'b, 'c, 'd, 'e) qTerm) set" shows "qTermLess_modulo rel \<equiv> {uu_. \<exists>X delta inp binp i. uu_ = (termIn X, termIn (qOp delta inp binp)) \<and> inp i = Some X} \<union> {uu_. \<exists>A delta inp binp j. uu_ = (absIn A, termIn (qOp delta inp binp)) \<and> binp j = Some A} \<union> {(termIn Y, absIn (qAbs xs x X)) |X Y xs x. (X, Y) \<in> rel}"
splitting_lemma_left_gen: fixes C :: "'b monoid"   and B :: "'b monoid"   and A :: "'b monoid"   and g :: "'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b"   and f' :: "'b \<Rightarrow> 'b" assumes "exact_seq ([C, B, A], [g, f])"   and "f' \<in> hom B A"   and "f' \<circ> f \<in> Group.iso A A"   and "inj_on f (carrier A)"   and "g ` carrier B = carrier C" obtains H :: "'b set" and K :: "'b set" where "H \<lhd> B" and "K \<lhd> B" and "H \<inter> K \<subseteq> {\<one>\<^bsub>B\<^esub>}" and "H <#>\<^bsub>B\<^esub> K = carrier B" and "f \<in> Group.iso A (subgroup_generated B H)" and "g \<in> Group.iso (subgroup_generated B K) C"
ideduct_ik_eq: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and M :: "('a, 'b) Term.term set"   and public :: "'a \<Rightarrow> bool"   and M' :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "\<forall>t\<in>M. intruder_model.intruder_deduct arity public Ana M' t" shows "intruder_model.intruder_deduct arity public Ana M' t = intruder_model.intruder_deduct arity public Ana (M' \<union> M) t"
numeral_le_real_of_nat_iff: fixes n :: "num"   and m :: "nat" shows "(numeral n \<le> real m) = (numeral n \<le> m)"
affine_basis_exists: fixes V :: "'b set" shows "\<exists>B\<subseteq>V. \<not> affine_dependent B \<and> affine hull V = affine hull B"
span_is_subset2: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "local.module.span S \<subseteq> carrier V"
equivalence_functor_F: shows "equivalence_functor hom.comp hom'.comp (\<lambda>\<mu>. e\<^sub>1 \<star> \<mu> \<star> d\<^sub>0)"
write_cpu_PSR_CWP_privilege: fixes s' :: "'b sparc_state"   and cwp_val :: "word5"   and s :: "'b sparc_state" assumes "s' = snd (fst (write_cpu (update_CWP cwp_val (cpu_reg_val PSR s)) PSR s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
non_exec_call_1': fixes \<sigma> :: "'a control_state_scheme"   and M :: "'c \<Rightarrow> 'a control_state_scheme \<Rightarrow> ('b \<times> 'a control_state_scheme) option"   and A\<^sub>1 :: "'a control_state_scheme \<Rightarrow> 'c"   and M' :: "'a control_state_scheme \<Rightarrow> (bool \<times> 'a control_state_scheme) option" assumes "\<not> exec_stop \<sigma>" shows "(\<sigma> \<Turnstile> call_1\<^sub>C M A\<^sub>1 ;- M') = (\<sigma> \<Turnstile> M (A\<^sub>1 \<sigma>) ;- M')"
quasiconvex_empty: fixes C :: "real" assumes "0 \<le> C" shows "quasiconvex C {}"
simps_1: fixes \<mu> :: "'a" shows "E.Ide \<^bold>\<langle>\<mu>\<^bold>\<rangle>\<^sub>0 = B.obj \<mu>"
sum_Un_eq_sum_sum: fixes P :: "'a set set" assumes "bp P" shows "W U = sum W P"
intros: fixes u :: "'e \<Rightarrow> 'f"   and A :: "('g \<times> ('e, 'f) cconstraint \<times> 'h \<times> 'e list \<times> 'g) set \<times> ('g \<Rightarrow> ('e, 'f) cconstraint)"   and l :: "'g"   and d :: "'f" assumes "u \<turnstile> inv_of A l"   and "u \<oplus> d \<turnstile> inv_of A l"   and "(0::'f) \<le> d" shows "A \<turnstile> \<langle>l, u\<rangle> \<rightarrow>\<^bsup>d\<^esup> \<langle>l, u \<oplus> d\<rangle>"
simple_function_def: fixes M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" shows "simple_function M g = (finite (g ` space M) \<and> (\<forall>x\<in>g ` space M. g -` {x} \<inter> space M \<in> sets M))"
g2: fixes z :: "real" assumes "0 < z"   and "z < 1"   and "z \<notin> s2" shows "vector_derivative (\<lambda>x. if x * 2 \<le> 1 then g1 (2 * x) else g2 (2 * x - 1)) (at (z / 2 + 1 / 2)) = 2 *\<^sub>R vector_derivative g2 (at z)"
cs: shows "(c, s) \<rightarrow>*c (d', t')"
TransLet_subst: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and y :: "Vars.var"   and b :: "'b"   and e :: "exp" assumes "atom ` domA \<Gamma> \<sharp>* (x, y)" shows "(TransLet b \<Gamma> e)[x::=y] = TransLet b \<Gamma>[x::h=y] e[x::=y]"
inside_bounded_complement_connected_empty: fixes S :: "'a set" assumes "connected (- S)"   and "bounded S" shows "inside S = {}"
isomorphicE: fixes a :: "'b"   and a' :: "'b" assumes "D.isomorphic a a'" obtains f :: "'b" where "\<guillemotleft>f : a \<Rightarrow>\<^sub>D a'\<guillemotright> \<and> D.iso f"
wellformed_1: fixes e :: "'a \<times> 'a" assumes "e \<in> parcs G" shows "fst e \<in> pverts G"
esize_prefixes: fixes w :: "'b llist" shows "esize {v. v \<le> w} = eSuc (llength w)"
infsetsum'_converges_finite: fixes F :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "finite F" shows "infsetsum'_converges f F"
symevalb_def: fixes P :: "(char list \<Rightarrow> int option) \<times> 'a \<Rightarrow> bool"   and e :: "bexp"   and v :: "bool" shows "symevalb P e v \<equiv> \<forall>s n. P (s, n) \<longrightarrow> pbval' e s = Some v"
faccs_correct_aux_1: fixes q :: "'d"   and \<delta> :: "('d, 'e) ta_rule set"   and n :: "'e tree" shows "(q \<in> faccs \<delta> n) = accs \<delta> n q"
intro: fixes A :: "V"   and f :: "V" assumes "pnop A (2\<^sub>\<nat>) f" shows "pbinop A f"
compat_add_triple: fixes a :: "'a"   and b :: "'a"   and p :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "nonsingular a b"   and "on_curve a b p"   and "p \<noteq> opp p"   and "local.add a p p \<noteq> opp p" shows "local.add a (local.add a p p) (opp p) = p"
a_1: fixes uu :: "('a \<times> 'c) list"   and k :: "'a"   and v :: "'c" assumes "distinct (map fst (rev uu))"   and "(k, v) \<in> set uu" shows "Some v = fold (\<lambda>(x, v) m. m(f x \<mapsto> v)) uu Map.empty (f k)"
induct: fixes P :: "'b set \<Rightarrow> 'b set list \<Rightarrow> bool \<Rightarrow> bool"   and a0 :: "'b set"   and a1 :: "'b set list"   and a2 :: "bool" assumes "\<And>X b. P X [] b"   and "\<And>X x xs b. (\<not> (x \<subseteq> X \<or> b \<and> X \<subseteq> x) \<Longrightarrow> P X xs b) \<Longrightarrow> P X (x # xs) b" shows "P a0 a1 a2"
simps_1: shows "invar \<langle>\<rangle> = True"
h1: shows "\<turnstile> F \<longrightarrow> \<box>F"
is_pole_def: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a" shows "is_pole f a = filterlim f at_infinity (at a)"
dim_diag_block_mat_2: fixes As :: "'b mat list" shows "dim_col (diag_block_mat As) = sum_list (map dim_col As)"
antiderivative_integral_continuous: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'c" assumes "continuous_on {a..b} f" obtains g :: "real \<Rightarrow> 'c" where "\<forall>u\<in>{a..b}. \<forall>v\<in>{a..b}. u \<le> v \<longrightarrow> (f has_integral g v - g u) {u..v}"
map_of_zip_upto_length_eq_nth: fixes i :: "nat"   and B :: "'d list"   and d :: "nat" assumes "i < length B"   and "d = length B" shows "map_of (zip [0..<d] B) i = Some (B ! i)"
connectedE: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^sup>* w" obtains xs :: "'a list" where "walk xs" and "xs \<noteq> []" and "hd xs = v" and "last xs = w"
Limit_add_Limit: fixes \<mu> :: "V"   and \<beta> :: "V" assumes "Limit \<mu>"   and "Ord \<beta>" shows "Limit (\<beta> + \<mu>)"
A: shows "u \<in> unaffected_domains I D {D y} zs"
finite_glitOfC: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and protCl :: "'c \<Rightarrow> bool"   and c :: "('a, 'b) lit list" assumes "ProblemIkTpartG wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw protCl" shows "finite (ProblemIkTpartG.glitOfC prot protFw c)"
hyps: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and n :: "nat" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g"   and "one_final_but g (if u = v then {} else Edges (u # between (vertices f) u v @ [v]))" shows "one_final (subdivFace' g f v n ovs)"
func_space_is_module: fixes S :: "'f set" shows "module K (func_space S)"
iMinI_ex2: fixes I :: "'b set" assumes "I \<noteq> {}" shows "iMin I \<in> I"
has_integral_eq_rhs: fixes f :: "'b \<Rightarrow> 'c"   and j :: "'c"   and S :: "'b set"   and i :: "'c" assumes "(f has_integral j) S"   and "i = j" shows "(f has_integral i) S"
braun_list_not_Nil: fixes xs :: "'b list"   and t :: "'b tree" assumes "xs \<noteq> []" shows "braun_list t xs = (\<exists>l x r. t = \<langle>l, x, r\<rangle> \<and> x = hd xs \<and> braun_list l (take_nths 1 1 xs) \<and> braun_list r (take_nths 2 1 xs))"
simps_17: fixes f :: "float"   and vs :: "real list" shows "interpret_floatarith (Num f) vs = real_of_float f"
token_squats_def: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat" shows "semi_mojmir_def.token_squats \<Sigma> \<delta> q\<^sub>0 w x = (\<forall>n. \<not> semi_mojmir_def.sink \<Sigma> \<delta> q\<^sub>0 (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n))"
induct: fixes P1 :: "('b \<Rightarrow> ('c, 'd) trm) \<Rightarrow> ('c + 'b, 'e, 'd) hp \<Rightarrow> bool"   and P2 :: "('b \<Rightarrow> ('c, 'd) trm) \<Rightarrow> ('c + 'b, 'e, 'd) formula \<Rightarrow> bool"   and x1 :: "'b \<Rightarrow> ('c, 'd) trm"   and x2 :: "('c + 'b, 'e, 'd) hp"   and x3 :: "'b \<Rightarrow> ('c, 'd) trm"   and x4 :: "('c + 'b, 'e, 'd) formula" assumes "\<And>\<sigma> a. P1 \<sigma> ($\<alpha> a)"   and "\<And>\<sigma> a b. \<lbrakk>NPadmit \<sigma> a; P1 \<sigma> a; NPadmit \<sigma> b; P1 \<sigma> b; PUadmitFO \<sigma> b (BVP (PsubstFO a \<sigma>)); hpsafe (PsubstFO a \<sigma>)\<rbrakk> \<Longrightarrow> P1 \<sigma> (a ;; b)"   and "\<And>\<sigma> a. \<lbrakk>NPadmit \<sigma> a; P1 \<sigma> a; PUadmitFO \<sigma> a (BVP (PsubstFO a \<sigma>)); hpsafe (PsubstFO a \<sigma>)\<rbrakk> \<Longrightarrow> P1 \<sigma> a**"   and "\<And>\<sigma> ODE \<phi>. \<lbrakk>OadmitFO \<sigma> ODE (BVO ODE); NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; FUadmitFO \<sigma> \<phi> (BVO ODE); fsafe (FsubstFO \<phi> \<sigma>); osafe (OsubstFO ODE \<sigma>)\<rbrakk> \<Longrightarrow> P1 \<sigma> (EvolveODE ODE \<phi>)"   and "\<And>\<sigma> a b. \<lbrakk>NPadmit \<sigma> a; P1 \<sigma> a; NPadmit \<sigma> b; P1 \<sigma> b\<rbrakk> \<Longrightarrow> P1 \<sigma> (a \<union>\<union> b)"   and "\<And>\<sigma> \<theta> x. TadmitFO \<sigma> \<theta> \<Longrightarrow> P1 \<sigma> (x := \<theta>)"   and "\<And>\<sigma> \<theta> x. TadmitFO \<sigma> \<theta> \<Longrightarrow> P1 \<sigma> (DiffAssign x \<theta>)"   and "\<And>\<sigma> \<phi>. \<lbrakk>NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>\<rbrakk> \<Longrightarrow> P1 \<sigma> (? \<phi>)"   and "\<And>\<sigma> \<theta>1 \<theta>2. \<lbrakk>TadmitFO \<sigma> \<theta>1; TadmitFO \<sigma> \<theta>2\<rbrakk> \<Longrightarrow> P2 \<sigma> (Geq \<theta>1 \<theta>2)"   and "\<And>\<sigma> args f. (\<And>i. TadmitFO \<sigma> (args i)) \<Longrightarrow> P2 \<sigma> ($\<phi> f args)"   and "\<And>\<sigma> \<phi>. \<lbrakk>NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>\<rbrakk> \<Longrightarrow> P2 \<sigma> (! \<phi>)"   and "\<And>\<sigma> \<phi> \<psi>. \<lbrakk>NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; NFadmit \<sigma> \<psi>; P2 \<sigma> \<psi>\<rbrakk> \<Longrightarrow> P2 \<sigma> (\<phi> && \<psi>)"   and "\<And>\<sigma> \<phi> x. \<lbrakk>NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; FUadmitFO \<sigma> \<phi> {Inl x}\<rbrakk> \<Longrightarrow> P2 \<sigma> (Exists x \<phi>)"   and "\<And>\<sigma> \<phi> a. \<lbrakk>NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; NPadmit \<sigma> a; P1 \<sigma> a; FUadmitFO \<sigma> \<phi> (BVP (PsubstFO a \<sigma>)); hpsafe (PsubstFO a \<sigma>)\<rbrakk> \<Longrightarrow> P2 \<sigma> (\<langle> a \<rangle> \<phi>)"   and "\<And>\<sigma> \<phi> C. \<lbrakk>NFadmit \<sigma> \<phi>; P2 \<sigma> \<phi>; FUadmitFO \<sigma> \<phi> UNIV\<rbrakk> \<Longrightarrow> P2 \<sigma> (InContext C \<phi>)" shows "(NPadmit x1 x2 \<longrightarrow> P1 x1 x2) \<and> (NFadmit x3 x4 \<longrightarrow> P2 x3 x4)"
measure_countable_negligible_Union: fixes S :: "nat \<Rightarrow> 'c set"   and B :: "real" assumes "\<And>n. S n \<in> lmeasurable"   and "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"   and "\<And>n. (\<Sum>k\<le>n. Sigma_Algebra.measure lebesgue (S k)) \<le> B" shows "(\<lambda>n. Sigma_Algebra.measure lebesgue (S n)) sums Sigma_Algebra.measure lebesgue (\<Union> (range S))"
minAB: fixes x :: "'a" assumes "x \<in> A"   and "x \<notin> B" shows "ab \<le> x \<and> (x \<le> ab \<longrightarrow> ab = x)"
seqsim_trans: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "s \<approx> t"   and "t \<approx> z" shows "s \<approx> z"
mset_subst_cls_list_subst_cls_mset: fixes Cs :: "'a literal multiset list"   and \<sigma> :: "'s" shows "mset (Cs \<cdot>cl \<sigma>) = mset Cs \<cdot>cm \<sigma>"
int_pow_closed: fixes G :: "('a, 'b) monoid_scheme"   and x :: "'a"   and i :: "int" assumes "Group.group G"   and "x \<in> carrier G" shows "x [^]\<^bsub>G\<^esub> i \<in> carrier G"
ex_post_efficient_sds_axioms: shows "ex_post_efficient_sds agents alts sds"
dirichlet_prod_fds_nth_fds_left: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" shows "dirichlet_prod (fds_nth (fds f)) g = dirichlet_prod f g"
tllist_all2_coinduct: fixes X :: "('a, 'b) tllist \<Rightarrow> ('c, 'd) tllist \<Rightarrow> bool"   and xs :: "('a, 'b) tllist"   and ys :: "('c, 'd) tllist"   and R :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'a \<Rightarrow> 'c \<Rightarrow> bool" assumes "X xs ys"   and "\<And>xs ys. X xs ys \<Longrightarrow> is_TNil xs = is_TNil ys \<and> (is_TNil xs \<longrightarrow> is_TNil ys \<longrightarrow> R (terminal xs) (terminal ys)) \<and> (\<not> is_TNil xs \<longrightarrow> \<not> is_TNil ys \<longrightarrow> P (thd xs) (thd ys) \<and> (X (ttl xs) (ttl ys) \<or> tllist_all2 P R (ttl xs) (ttl ys)))" shows "tllist_all2 P R xs ys"
measurable_RealVal: shows "RealVal \<in> borel \<rightarrow>\<^sub>M stock_measure REAL"
exhaust_sel: fixes transaction\<^sub>h :: "transaction\<^sub>h" obtains "transaction\<^sub>h = Transaction\<^sub>h (the_Transaction\<^sub>h transaction\<^sub>h)"
i: shows "t \<inter> s \<union> (s - t \<inter> s) = s"
splitted_def: fixes R :: "('c, 'd) ring_scheme"   and p :: "'c list" assumes "ring R" shows "ring.splitted R p = (size (ring.roots R p) = degree p)"
insert: fixes A :: "'c set"   and x :: "'c" assumes "finite A"   and "A \<noteq> {}" shows "Max (insert x A) = max x (Max A)"
U2: shows "(\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> 0 < n}. set (case (p, n, s) of (p, n, s) \<Rightarrow> if n = 0 then [(p, 0, Floor s)] else map (case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) (if 0 < n then [0..n] else [n..0]))) = (\<Union>(p, n, s)\<in>{(p, n, s). (p, n, s) \<in> set (rsplit0 a) \<and> 0 < n}. set (map (case (p, n, s) of (p, n, s) \<Rightarrow> \<lambda>j. ((case (p, n, s) of (p, n, s) \<Rightarrow> fp p n s) j, 0, Add (Floor s) (C j))) [0..n]))"
bound_compare_defs_16: fixes c :: "'a"   and b :: "'a option" shows "c >\<^sub>l\<^sub>b b = \<rhd>\<^sub>l\<^sub>b (<) c b"
rel_pmf_mono: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'b pmf"   and g :: "'d pmf"   and B :: "'b \<Rightarrow> 'd \<Rightarrow> bool" assumes "rel_pmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_pmf B f g"
Map_comp: fixes f :: "('obj, 'obj) concrete_category.arr"   and g :: "('obj, 'obj) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Map (g \<cdot> f) = Dom f"
sparse\<^sub>0_def: fixes xs :: "('a \<times> 'b) list" shows "sparse\<^sub>0 xs = Pm_fmap (fmap_of_list xs)"
least_map_leD: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and y :: "'a" assumes "x \<in> least_map f S"   and "y \<in> S" shows "f x \<le> f y"
assms: shows "run_one_step d i (s, vs, ves, e) = (s', vs', RSBreak n res)"
inv_ipp_def: fixes z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" shows "inv_ipp z i f A = (\<forall>a\<in>addition_set f (aug_pm_set z i A). zeroA z i f A (\<^sub>i- a \<^sub>f+ a))"
b: shows " N \<longrightarrow>\<^sub>1 N'"
has_terminal_arrows_from_functor_1: fixes x :: "'d" assumes "D.ide x" shows "terminal_arrow_from_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) F (G x) x (Adj.\<epsilon> x)"
2: shows "((\<lambda>x. ln (real (nat \<lfloor>x\<rfloor>)) - ln x) \<longlongrightarrow> 0) at_top"
in_prime_factors_iff: 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 p :: "'b"   and x :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "(p \<in># factorial_semiring.prime_factorization zero times one normalize x) = (x \<noteq> zero \<and> dvd.dvd times p x \<and> normalization_semidom.prime zero times one normalize p)"
wf_eq_check_append'': fixes V :: "'d set"   and S :: "('e, 'd) strand_step list"   and t :: "('e, 'd) Term.term"   and t' :: "('e, 'd) Term.term" assumes "wf\<^sub>s\<^sub>t V S" shows "wf\<^sub>s\<^sub>t V (S @ [\<langle>check: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t])"
filtermap_filtermap: fixes f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'e"   and F :: "'f filter" shows "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
set_pmf_eq': fixes p :: "'b pmf" shows "set_pmf p = {x. 0 < pmf p x}"
minf_3: fixes t :: "'a" shows "\<exists>z. \<forall>x<z. (x = t) = False"
f_less_t: fixes y :: "nat" assumes "y < t" shows "eval (Cn 3 r_not [r_result]) [y, i, x] \<down>\<noteq> 0"
represents_imp_degree: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "degree p \<noteq> 0"
that_1: fixes a :: "'form" assumes "n = Assumption a"   and "p = Reg a" shows "thesis"
RModule_Im: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and T :: "'b \<Rightarrow> 'c" assumes "RModuleHom R smult M smult' T" shows "RModule R smult' (T ` M)"
is_acc_iface_def: fixes G :: "prog"   and P :: "pname"   and I :: "qtname" shows "is_acc_iface G P I = (is_iface G I \<and> G \<turnstile> Iface I accessible_in P)"
comp_def: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "f \<cdot> g = (\<lambda>x. f (g x))"
diamond_fin_word_inf_word: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and v :: "'a list"   and w :: "'a stream"   and p :: "'b" assumes "transition_system_traces ex en ind"   and "\<forall>a\<in>set v. \<forall>b\<in>sset w. ind a b"   and "transition_system.path ex en v p"   and "transition_system.run ex en w p" shows "transition_system.run ex en w (fold ex v p)"
subst_bv2_lift: fixes t :: "Core.term"   and k :: "nat"   and s :: "Core.term" shows "subst_bv2 (lift t k) k s = t"
sym: fixes A :: "wall"   and B :: "wall" assumes "A ~ B" shows "B ~ A"
sums_mult2: fixes f :: "nat \<Rightarrow> 'c"   and a :: "'c"   and c :: "'c" assumes "f sums a" shows "(\<lambda>n. f n * c) sums (a * c)"
add_less_imp_less_right: fixes a :: "'b zmultiset"   and c :: "'b zmultiset"   and b :: "'b zmultiset" assumes "a + c \<subset>#\<^sub>z b + c" shows "a \<subset>#\<^sub>z b"
comparator_of_def: fixes x :: "'c"   and y :: "'c" shows "comparator_of x y = (if x < y then Lt else if x = y then Eq else Gt)"
trivial_group_subgroup_generated_eq: fixes s :: "'a set" shows "trivial_group (subgroup_generated G s) = (carrier G \<inter> s \<subseteq> {\<one>})"
not_FinalAllow: fixes foo :: "state" shows "(foo \<noteq> Decision FinalAllow) = (foo = Decision FinalDeny \<or> foo = Undecided)"
combine_common_factor: fixes a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<parallel> e + (b \<parallel> e + c) = (a + b) \<parallel> e + c"
assms_3: shows "g \<in> borel_measurable (F t)"
spick_def: fixes p :: "real"   and M :: "bool stream measure"   and w :: "bool stream"   and n :: "nat"   and v :: "bool" assumes "infinite_coin_toss_space p M" shows "spick w n v = stake n w @- v ## sconst True"
pt_swap_bij': fixes a :: "'b"   and b :: "'b"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "[(a, b)] \<bullet> [(b, a)] \<bullet> x = x"
measurable_proj_state_measure: fixes i :: "nat"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "i \<in> V" shows "(\<lambda>x. x i) \<in> state_measure V \<Gamma> \<rightarrow>\<^sub>M stock_measure (\<Gamma> i)"
eq2: shows "monom 1 b = (\<Prod>i = 0..<b. monom 1 1) mod f"
c_futures_tr: fixes yps :: "('e \<times> 'f) list"   and Y :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and xps :: "('e \<times> 'f) list" assumes "(yps, Y) \<in> futures (c_process step out s\<^sub>0) xps" shows "yps = c_tr step out (foldl step s\<^sub>0 (map fst xps)) (map fst yps)"
nz: shows "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0"
isomorphicE: fixes a :: "'a"   and a' :: "'a" assumes "L.isomorphic a a'" obtains f :: "'a" where "L.in_hom f a a' \<and> L.iso f"