hcju/isabelleps huggingface dataset:
prv_eql_refl: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and x :: "'a" assumes "Deduct var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv"   and "x \<in> var" shows "prv (eql (Var x) (Var x))"
convex_Epigraph: fixes S :: "'b set"   and f :: "'b \<Rightarrow> ereal" assumes "convex S" shows "convex (Epigraph S f) = Lower_Semicontinuous.convex_on S f"
coefficients_normh: fixes p :: "'b poly"   and n0 :: "nat" assumes "isnpolyh p n0" shows "\<forall>q\<in>set (coefficients p). isnpolyh q n0"
succ_def: fixes G :: "('d, 'c) graph"   and v :: "'d" shows "succ G v \<equiv> {(w, v'). (v, w, v') \<in> edges G}"
iT_Plus_neg_singleton: fixes a :: "nat"   and k :: "nat" shows "{a} \<oplus>- k = (if k \<le> a then {a - k} else {})"
ab_semigroup_mult_sep_conj: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "class.ab_semigroup_mult (sep_algebra.sep_conj plus sep_disj)"
remove_vertex_Digraph: fixes v :: "'a" shows "Digraph (H.remove_vertex v)"
gr_implies_not_zero: 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 m :: "'d"   and n :: "'d" assumes "class.canonically_ordered_monoid_add plus zero less_eq less"   and "less m n" shows "n \<noteq> zero"
mon_c_unconc_1: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and a :: "'b list multiset"   and b :: "'b list multiset" shows "mon_c fg (a + b) = mon_c fg a \<union> mon_c fg b"
rjoin_deterministic: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option"   and r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<E> :: "('a, 'b, 'c) cntxt"   and r' :: "'a"   and \<sigma>' :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau>' :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and v :: "('a, 'b, 'c) val"   and s' :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst"   and "s r = Some (\<sigma>, \<tau>, \<E> [Rjoin (VE (Rid r'))])"   and "s r' = Some (\<sigma>', \<tau>', VE v)" shows "substitution.revision_step subst r s s' = (s' = (s(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r' := None))"
rep_eq: fixes x :: "('a, 'b) lifted_pair"   and xa :: "('a, 'b) lifted_pair" shows "Rep_lifted_pair (x \<squnion> xa) = (\<lambda>f. triple.pairs_sup (Rep_lifted_pair x f) (Rep_lifted_pair xa f))"
linear_def: fixes f :: "'c \<Rightarrow> 'd" shows "linear f \<equiv> Vector_Spaces.linear (*\<^sub>R) (*\<^sub>R) f"
comp_ide_self: fixes a :: "'a" assumes "C.ide a" shows "a \<cdot> a = a"
prep_arrTr1: fixes A :: "'b set"   and h :: "nat \<Rightarrow> 'a"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and n1 :: "nat"   and s :: "nat \<Rightarrow> 'b"   and m :: "'a" assumes "ideal R A"   and "h \<in> {j. j \<le> Suc n} \<rightarrow> carrier M"   and "f \<in> {j. j \<le> n1} \<rightarrow> h ` {j. j \<le> Suc n}"   and "s \<in> {j. j \<le> n1} \<rightarrow> A"   and "m = l_comb R M n1 s f" shows "\<exists>l\<in>{j. j \<le> Suc n}. \<exists>s\<in>{j. j \<le> l} \<rightarrow> A. \<exists>g\<in>{j. j \<le> l} \<rightarrow> h ` {j. j \<le> Suc n}. m = l_comb R M l s g \<and> bij_to g {j. j \<le> l} (f ` {j. j \<le> n1})"
bij_extend: fixes h :: "'c \<Rightarrow> 'd" assumes "bij h" shows "bij (extend (\<lambda>(x, u). h x))"
prod_set_simps_1: fixes x :: "'c"   and y :: "'d" shows "fsts (x, y) = {x}"
meas: shows "(\<rightarrow>\<^sub>M) (dens_ctxt_measure ({}, V', \<Gamma>, \<lambda>_. 1) \<rho>) = (\<rightarrow>\<^sub>M) (state_measure V' \<Gamma>)"
mset_le_add_mset_decr_left2: fixes c :: "'a"   and a :: "'a multiset"   and b :: "'a multiset" assumes "add_mset c a \<subseteq># b" shows "{#c#} \<subseteq># b"
simps_2: fixes \<phi> :: "fml option" shows "Diamondo undefg \<phi> = undeff"
path_decomp': fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "path (xs @ x # ys)" shows "path (xs @ [x])"
fds_totient_times_zeta': shows "fds totient * fds_zeta = fds id"
of_int: fixes n :: "int" shows "fds_perzeta (real_of_int n) = fds_perzeta 0"
comm_semiring_hom_axioms: shows "comm_semiring_hom hom"
unit_all_impI: fixes P :: "unit \<Rightarrow> bool"   and Q :: "unit \<Rightarrow> bool" assumes "P () \<Longrightarrow> Q ()" shows "\<forall>x. P x \<longrightarrow> Q x"
mgu_eliminates: fixes M :: "('b, 'a) Term.term"   and N :: "('b, 'a) Term.term"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "mgu M N = Some \<sigma>" shows "(\<exists>v\<in>fv M \<union> fv N. subst_elim \<sigma> v) \<or> \<sigma> = Var"
bprules_2: shows "{} \<notin> P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> V}"
assms: shows "P \<equiv> \<lambda>l u x. f x \<in> {l1 x..u1 x} \<longrightarrow> g x \<in> {l2 x..u2 x} \<longrightarrow> f x powr g x \<in> {l..u}"
emeasure_P: fixes I :: "'b set"   and P :: "'b set \<Rightarrow> ('b \<Rightarrow> 'c) measure"   and M :: "'b \<Rightarrow> 'c measure"   and J :: "'b set"   and K :: "'b set"   and X :: "('b \<Rightarrow> 'c) set" assumes "projective_family I P M"   and "J \<subseteq> K"   and "finite K"   and "K \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure (P K) (prod_emb K M J X) = emeasure (P J) X"
54: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> y \<squnion> (x \<sqinter> z \<squnion> (x \<sqinter> (y \<squnion> z) \<squnion> u)) = x \<sqinter> (y \<squnion> z) \<squnion> u"
R_lin_independent_ConsD1: fixes m :: "'m"   and ms :: "'m list" assumes "R_lin_independent (m # ms)" shows "R_lin_independent ms"
takeWhile_append1: fixes x :: "'d"   and xs :: "'d list"   and P :: "'d \<Rightarrow> bool"   and ys :: "'d list" assumes "x \<in> set xs"   and "\<not> P x" shows "takeWhile P (xs @ ys) = takeWhile P xs"
CD_on_card: fixes ds :: "'b set"   and X :: "'a set" shows "card (CD_on ds X) = (\<Sum>d\<in>ds. card (Cd d X))"
assms_4: shows "prefs_from_table_wf agents alts ys"
invariant_run: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c stream" assumes "Graph_Invariant E P"   and "Graph_Defs.run E (x ## xs)"   and "P x" shows "pred_stream P (x ## xs)"
Mp_coeff: fixes f :: "int poly"   and i :: "nat" shows "coeff (pl.Mp f) i = pl.M (coeff f i)"
posliveness_def: fixes A :: "'b set"   and P :: "'b llist set" shows "posliveness A P \<equiv> \<forall>t\<in>A\<^sup>\<clubsuit>. \<exists>s\<in>A\<^sup>\<infinity>. t @@ s \<in> P"
indspaceD_supp: fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V" shows "supp f \<subseteq> FH"
ring_distribs_2: fixes a :: "'d"   and b :: "'d"   and c :: "'d" shows "(a + b) * c = a * c + b * c"
ls_nds_image_autoref_aux: fixes fi :: "'g \<Rightarrow> 'h"   and f :: "('i, 'j) node_scheme \<Rightarrow> ('k, 'l) node_scheme"   and Ra :: "('g \<times> ('i, 'j) node_scheme) set"   and Rb :: "('h \<times> ('k, 'l) node_scheme) set"   and l :: "'g list"   and s :: "('i, 'j) node_scheme set" assumes "(fi, f) \<in> Ra \<rightarrow> Rb"   and "(l, s) \<in> \<langle>Ra\<rangle>ls_nds_rel"   and "\<forall>x. name (f x) = name x" shows "(map fi l, f ` s) \<in> \<langle>Rb\<rangle>ls_nds_rel"
length_zip: fixes xs :: "'c list"   and ys :: "'d list" shows "length (zip xs ys) = ord_class.min (length xs) (length ys)"
mult_zero_right: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<otimes> mbot = mbot"
cl_in_lattice: fixes L :: "'b set set"   and r :: "'b set" assumes "lattice L" shows "cl L r \<in> L"
ANDConc: fixes a :: "('a set set, 'b) Combinators"   and p :: "('a set set, 'b) Combinators list" assumes "allNetsDistinct (a # p)" shows "allNetsDistinct p"
check_no_overlap2_def: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and ft :: "('a, 'c) flow_entry_match list" shows "check_no_overlap2 \<gamma> ft = (\<forall>a\<in>set ft. \<forall>b\<in>set ft. a \<noteq> b \<and> ofe_prio a = ofe_prio b \<longrightarrow> \<not> (\<exists>p\<in>UNIV. \<gamma> (ofe_fields a) p \<and> \<gamma> (ofe_fields b) p))"
intro: fixes C :: "('e, 'd, 'f) LSCategory_scheme" assumes "\<And>x y. \<lbrakk>x \<in> mor\<^bsub>C\<^esub>; y \<in> mor\<^bsub>C\<^esub>; m2z\<^bsub>C\<^esub>x = m2z\<^bsub>C\<^esub>y\<rbrakk> \<Longrightarrow> x = y"   and "\<And>X Y. \<lbrakk>X \<in> obj\<^bsub>C\<^esub>; Y \<in> obj\<^bsub>C\<^esub>\<rbrakk> \<Longrightarrow> HOMCollection C X Y \<in> range explode"   and "mor2ZF C \<in> extensional (mor\<^bsub>C\<^esub>)" shows "LSCategory_axioms C"
suffix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "xs \<le>s ys = (\<exists>zs. ys = zs \<cdot> xs)"
not_eq_order_implies_strict: fixes a :: "'a multiset"   and b :: "'a multiset" assumes "a \<noteq> b"   and "a \<subseteq># b" shows "a \<subset># b"
simps: fixes a :: "Syntax.term" shows "value a = (a = Unit \<or> (\<exists>uu_. a = Var uu_) \<or> (\<exists>uv_ uw_. a = Lam uv_ uw_) \<or> (\<exists>ux_ uy_. a = Rec ux_ uy_) \<or> (\<exists>v. a = Inj1 v \<and> value v) \<or> (\<exists>v. a = Inj2 v \<and> value v) \<or> (\<exists>v\<^sub>1 v\<^sub>2. a = Syntax.Pair v\<^sub>1 v\<^sub>2 \<and> value v\<^sub>1 \<and> value v\<^sub>2) \<or> (\<exists>v. a = Roll v \<and> value v) \<or> (\<exists>uz_. a = Hash uz_) \<or> (\<exists>v va_. a = Hashed va_ v \<and> value v))"
kingman_theorem_nonergodic_1: fixes u :: "nat \<Rightarrow> (int \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow> real" assumes "I.subcocycle u"   and "- \<infinity> < I.subcocycle_avg_ereal u" shows "AE x in MI. (\<lambda>n. u n x / real n) \<longlonglongrightarrow> I.subcocycle_lim u x"
simE2: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and \<alpha> :: "act"   and Q' :: "ccs" assumes "(P, Q) \<in> Rel"   and "Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> Q'"   and "\<And>R S. (R, S) \<in> Rel \<Longrightarrow> R \<leadsto>\<^sup>^<Rel> S" obtains P' :: "ccs" where "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'" and "(P', Q') \<in> Rel"
min_step_iprev: fixes x :: "nat"   and y :: "nat"   and I :: "nat set" assumes "x < y"   and "x \<in> I"   and "y \<in> I"   and "\<And>k. \<lbrakk>x < k; k < y\<rbrakk> \<Longrightarrow> k \<notin> I" shows "iprev y I = x"
cols_dim: fixes A :: "'b mat" shows "set (cols A) \<subseteq> carrier_vec (dim_row A)"
subset: fixes Q :: "'a set" assumes "order.greater_eq P Q" shows "OrderingSetMap (\<^bold>\<le>) (\<^bold><) (\<^bold>\<le>*) (\<^bold><*) Q f"
ZFAxioms_def: fixes Ax :: "(ZF, ZF) Axioms" shows "ZFAxioms Ax \<equiv> Axioms Ax \<and> ZFAxioms_axioms Ax"
monic_v: fixes a :: "'a mod_ring poly"   and b :: "'a mod_ring poly"   and u :: "'b mod_ring poly"   and v :: "'c mod_ring poly"   and w :: "'c mod_ring poly" assumes "Knuth_ex_4_6_2_22_main a b u v w" shows "monic v"
reachable_nodes_step'_1: fixes u :: "'b"   and g :: "'b ugraph"   and r :: "'b"   and v :: "'b" assumes "u \<in> reachable_nodes g r"   and "(u, v) \<in> edges g" shows "v \<in> reachable_nodes g r"
pnet_sos_never_newpkt: fixes st :: "'c net_state"   and a :: "'d node_action"   and st' :: "'c net_state"   and S :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and T :: "('c net_state \<times> 'd node_action \<times> 'c net_state) set"   and i :: "nat"   and d :: "nat"   and di :: "nat" assumes "(st, a, st') \<in> pnet_sos S T"   and "\<And>i d di a s s'. (s, a, s') \<in> S \<Longrightarrow> a \<noteq> i:newpkt(d, di)"   and "\<And>i d di a t t'. (t, a, t') \<in> T \<Longrightarrow> a \<noteq> i:newpkt(d, di)" shows "a \<noteq> i:newpkt(d, di)"
affine_hull_convex_hull: fixes S :: "'b set" shows "affine hull (convex hull S) = affine hull S"
assms_3: shows "ide w' \<and> \<guillemotleft>\<theta>' : f \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu>' : g' \<Rightarrow> g \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> (r \<star> \<theta>') \<cdot> \<a>[r, f, w'] \<cdot> (\<rho> \<star> w') \<cdot> \<nu>' = \<rho>'"
big_clocked_timeout_0_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3 :: "'b state \<times> (v, v) result" assumes "evaluate ck env s e r3" shows "\<forall>s'. r3 = (s', Rerr (Rabort Rtimeout_error)) \<and> ck = True \<longrightarrow> clock s' = 0"
residue_primroot_iff_in_cyclic_moduli: fixes m :: "nat" shows "(\<exists>g. residue_primroot m g) = (m \<in> cyclic_moduli)"
no_override_in_Object: fixes G :: "prog"   and statC :: "qtname"   and dynC :: "qtname"   and sig :: "sig"   and dynM :: "qtname \<times> methd"   and statM :: "qtname \<times> methd" assumes "dynmethd G statC dynC sig = Some dynM"   and "is_class G dynC"   and "ws_prog G"   and "methd G statC sig = Some statM"   and "dynM \<noteq> statM" shows "dynC \<noteq> Object"
Osum_Well_order: fixes r :: "('b \<times> 'b) set"   and r' :: "('b \<times> 'b) set" assumes "Field r \<inter> Field r' = {}"   and "Well_order r"   and "Well_order r'" shows "Well_order (r \<union>o r')"
bij_mono_map_set_to_nat_def: fixes A :: "'b set" shows "bij_mono_map_set_to_nat A = (\<lambda>x. if x \<in> A then the_inv_into {..<card A} ((!) (sorted_list_of_set A)) x else card A)"
nadds_red_rtrancl_lt: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "\<And>q. q \<in> B \<Longrightarrow> \<not> lt q adds\<^sub>t lt p"   and "(red B)\<^sup>*\<^sup>* p r" shows "lt r = lt p"
test_kat_1: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "test p"   and "test q" shows "(p \<cdot> x \<cdot> q = x \<cdot> q) = (!p \<cdot> x \<cdot> q = (0::'a))"
arr_char: fixes f :: "'a \<times> 'a \<times> 'a" shows "Cop.VVV.arr f = (arr (fst f) \<and> Cop.VV.arr (snd f) \<and> Cop.src (fst f) = Cop.trg (fst (snd f)))"
nprv_clear2_1: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and dsj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and num :: "'b set"   and prv :: "'c \<Rightarrow> bool"   and \<phi>2 :: "'c"   and \<psi> :: "'c"   and \<phi>1 :: "'c" assumes "Deduct_with_False_Disj var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv"   and "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>2} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<psi> \<in> fmla" shows "Deduct_with_False_Disj.nprv cnj imp fls prv {\<phi>1, \<phi>2} \<psi>"
set_to_nat_def: shows "set_to_nat = sum ((^) 2)"
of_weak_ranking_permute: fixes f :: "'b \<Rightarrow> 'b"   and xs :: "'b set list" assumes "f permutes \<Union> (set xs)" shows "of_weak_ranking (map ((`) f) xs) = map_relation (inv f) (of_weak_ranking xs)"
Forall: shows "wf_dfa (dfa_of_pf n (Neg (Exist (Neg p)))) n"
sectionE: fixes f :: "'a arrow_of_spans_data" assumes "V.section f" obtains g :: "'a arrow_of_spans_data" where "V.ide (g \<bullet> f)"
continuous_closed_graph_eq: fixes T :: "'a set"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set" assumes "compact T"   and "f ` S \<subseteq> T" shows "continuous_on S f = closedin (top_of_set (S \<times> T)) ((\<lambda>x. (x, f x)) ` S)"
complete_eq_closed: fixes S :: "'a set" shows "complete S = closed S"
corthogonal_inv_result: fixes A :: "'b mat" assumes "corthogonal_mat A" shows "inverts_mat (corthogonal_inv A) A"
Assign: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and x :: "char list"   and a :: "aexp" shows "\<turnstile> {\<lambda>s. P (s[a/x])} x ::= a {P}"
step_Pr_correct: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "valid ((Pr n f g, xs, []) # rest)"   and "\<And>xs rest. valid ((f, xs, []) # rest) \<Longrightarrow> correct ((f, xs, []) # rest, None)"   and "\<And>xs rest. valid ((g, xs, []) # rest) \<Longrightarrow> correct ((g, xs, []) # rest, None)" shows "correct ((Pr n f g, xs, []) # rest, None)"
integrableD_2: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M \<noteq> \<infinity>"
Red_I_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and Q :: "'b set"   and entails_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a inference set"   and Red_F_q :: "'b \<Rightarrow> 'a set \<Rightarrow> 'a set"   and N :: "'a set" assumes "intersection_calculus Bot Inf Q entails_q Red_I_q Red_F_q" shows "intersection_calculus.Red_I Q Red_I_q N = (\<Inter>q\<in>Q. Red_I_q q N)"
distinct_95: fixes x6 :: "arith_type"   and x9 :: "ctrl_type" shows "arith_type x6 \<noteq> ctrl_type x9"
l2product_sym: fixes S :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" shows "l2product S f g = l2product S g f"
ipathD: fixes E :: "('b \<times> 'b) set"   and r :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "ipath E r" shows "(r i, r (Suc i)) \<in> E"
ode_slps_of_def: fixes ode_ops :: "ode_ops" shows "ode_slps_of ode_ops = (ode_slp ode_ops, euler_incr_slp ode_ops, euler_slp ode_ops, rk2_slp ode_ops, D ode_ops)"
list_conj_nb': fixes ps :: "fm list" assumes "\<forall>p\<in>set ps. bound0 p" shows "bound0 (list_conj ps)"
padd_closed: fixes a :: "'b"   and b :: "'b"   and p\<^sub>1 :: "'b \<times> 'b \<times> 'b"   and p\<^sub>2 :: "'b \<times> 'b \<times> 'b" assumes "on_curvep a b p\<^sub>1"   and "on_curvep a b p\<^sub>2" shows "on_curvep a b (padd a p\<^sub>1 p\<^sub>2)"
codomains_comp: fixes g :: "'a"   and f :: "'a" assumes "A.seq g f" shows "A.codomains (g \<cdot>\<^sub>A f) = A.codomains g"
boundedE: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b \<sqinter> c" obtains "a \<le> b" and "a \<le> c"
nlcons_inv_1: fixes n :: "nat"   and t :: "('c, 'd) FingerTreeStruc"   and nd :: "('c, 'd) Node" assumes "is_leveln_ftree n t"   and "is_leveln_node n nd" shows "is_leveln_ftree n (nlcons nd t)"
square_free_factorization_int'_2: fixes f :: "int poly"   and d :: "int"   and fs :: "(int poly \<times> nat) list"   and fi :: "int poly"   and i :: "nat" assumes "square_free_factorization_int' f = (d, fs)"   and "(fi, i) \<in> set fs" shows "content fi = 1 \<and> 0 < lead_coeff fi"
assms_3: shows "\<C> \<turnstile> [e] : t_in _> t_out"
one_not_zero: fixes R :: "('c, 'd) ring_scheme" assumes "domain R" shows "\<one>\<^bsub>R\<^esub> \<noteq> \<zero>\<^bsub>R\<^esub>"
IH: assumes "(V, asx, asx) \<in> dependent_live_vars n'"   and "asx \<noteq> [] \<longrightarrow> n' = last (targetnodes asx)" shows "V \<in> Use n' \<and> (\<forall>as' a as''. asx = as' @ a # as'' \<longrightarrow> \<not> sourcenode a -{V}a # as''\<rightarrow>\<^sub>d\<^sub>d n')"
cases: fixes a1 :: "nat \<Rightarrow> Syntax.type"   and a2 :: "nat \<Rightarrow> Syntax.type"   and a3 :: "cmd" assumes "a1 , a2 \<turnstile>\<^sub>C a3" obtains \<Gamma> :: "nat \<Rightarrow> Syntax.type" and \<Delta> :: "nat \<Rightarrow> Syntax.type" and t :: "trm" and T :: "Syntax.type" and x :: "nat" where "a1 = \<Gamma>" and "a2 = \<Delta>" and "a3 = <x>t" and "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T" and "\<Delta> x = T"
goodI: fixes i :: "nat"   and j :: "nat"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'a" assumes "i < j"   and "P (f i) (f j)" shows "good P f"
fps_integral_def: fixes a :: "'a fps"   and a0 :: "'a" shows "fps_integral a a0 = Abs_fps (\<lambda>n. if n = 0 then a0 else inverse (of_nat n) * fps_nth a (n - 1))"
hyps: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set" assumes "no_write_to_read_only_memory \<S> sb"   and "non_volatile_owned_or_read_only True \<S> \<O> sb"   and "sharing_consistent \<S> \<O> sb"   and "a \<in> read_only_reads \<O> sb"   and "a \<notin> \<O> \<union> all_acquired sb" shows "a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b sb"
isTrue_neg_neg: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (neg (neg \<phi>))" shows "isTrue \<phi>"
jipsen4l: fixes x :: "'a" shows "(x \<leftarrow> x) \<cdot> x = x"
P1: shows "Bet A B A0 \<and> Cong A B B A0"
not_numeral_less_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (numeral.numeral one plus n) zero"
assm1: shows "f (a\<lparr>new := new1 xa \<union> (new a - {xa}), old := insert xa (old a), next := next1 xa \<union> next a\<rparr>, b) \<le> SPEC (\<lambda>c. case c of (uu_, nds) \<Rightarrow> old_next_pair ` snd (a, b) \<subseteq> old_next_pair ` nds \<and> expand_inv_result \<phi> nds)"
inject_4: fixes x2 :: "('d, 'e, 'f) psi"   and x1 :: "('d, 'e, 'f) psi"   and y2 :: "('d, 'e, 'f) psi"   and y1 :: "('d, 'e, 'f) psi" shows "(x2 \<parallel> x1 = y2 \<parallel> y1) = (x2 = y2 \<and> x1 = y1)"
char_approx3: fixes \<sigma>2 :: "real"   and t :: "real" assumes "integrable \<mu> (\<lambda>x. x)"   and "\<mu>.expectation (\<lambda>x. x) = 0"   and "integrable \<mu> power2"   and "\<mu>.expectation (\<lambda>x. (x - \<mu>.expectation (\<lambda>x. x))\<^sup>2) = \<sigma>2" shows "cmod (char \<mu> t - complex_of_real (1 - t\<^sup>2 * \<sigma>2 / 2)) \<le> t\<^sup>2 / 6 * \<mu>.expectation (\<lambda>x. min (6 * x\<^sup>2) (\<bar>t\<bar> * \<bar>x\<bar> ^ 3))"
deg_monom: fixes a :: "nat \<Rightarrow> int"   and n :: "nat" assumes "a \<noteq> \<zero>"   and "a \<in> carrier Zp" shows "deg Zp (monom (UP Zp) a n) = n"
sum_character: shows "sum (principal_char G) (carrier G) = (if principal_char G = principal_char G then of_nat (order G) else 0)"