hcju/isabelleps huggingface dataset:
assms: shows "r\<^sup>\<noteq>\<^sup>\<noteq> a b"
fps_of_nat: fixes c :: "nat" shows "fps_const (of_nat c) = of_nat c"
preserves_iso: fixes f :: "'d \<times> 'd \<times> 'd" assumes "CCC.iso f" shows "local.iso (\<alpha> f)"
distinct_35: fixes x31 :: "poly"   and x32 :: "poly"   and x81 :: "poly"   and x82 :: "nat"   and x83 :: "poly" shows "Add x31 x32 \<noteq> CN x81 x82 x83"
rotate10: fixes A10 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and A8 :: "fmla"   and A9 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A10 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 H))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 H))))))))) \<turnstile> B"
circ_circ_mult: fixes x :: "'a" shows "(1::'a)\<^sup>\<star> * x\<^sup>\<star> = x\<^sup>\<star>\<^sup>\<star>"
simps_1: fixes w\<^sub>c :: "nat"   and c :: "'a"   and w\<^sub>a :: "nat"   and a :: "'a"   and w\<^sub>b :: "nat"   and b :: "'a" shows "splitLeaf (Leaf w\<^sub>c c) w\<^sub>a a w\<^sub>b b = (if c = a then Node w\<^sub>c (Leaf w\<^sub>a a) (Leaf w\<^sub>b b) else Leaf w\<^sub>c c)"
simps_8: fixes f1 :: "brick list \<Rightarrow> 'a"   and f2 :: "brick list \<Rightarrow> wall \<Rightarrow> 'a \<Rightarrow> 'a"   and x21 :: "brick list"   and x22 :: "wall" shows "rec_wall f1 f2 (x21 * x22) = f2 x21 x22 (rec_wall f1 f2 x22)"
wprepare_goto_start_pos_Bk: fixes m :: "nat"   and lm :: "nat list"   and b :: "cell list"   and list :: "cell list" assumes "wprepare_erase m lm (b, Bk # list)" shows "wprepare_goto_start_pos m lm (Bk # b, list)"
vbrelation_vcomp_vid_on_left: fixes A :: "V" assumes "\<R>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>) \<subseteq>\<^sub>\<circ> A" shows "vid_on A \<circ>\<^sub>\<circ> F\<lparr>ArrVal\<rparr> = F\<lparr>ArrVal\<rparr>"
simps: fixes uu :: "String.literal \<Rightarrow> typ option"   and uv :: "String.literal \<Rightarrow> nat option"   and oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" shows "osig (uu, uv, oss) = oss"
sat_vars_RESTRICT_VARS: fixes ks :: "'k list"   and \<phi> :: "('a, 'k) aformula"   and r :: "'k \<Rightarrow> 'v \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'i" assumes "\<forall>k\<in>set ks. finite (FV \<phi> k)" shows "sat_vars_gen r (set ks) \<AA> \<phi> = satisfies_gen r \<AA> (RESTRICT_VARS ks (sorted_list_of_set \<circ> FV \<phi>) \<phi>)"
homotopic_with_sym: fixes P :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "homotopic_with P X Y f g = homotopic_with P X Y g f"
insert_not_elem: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<notin> A"   and "A \<noteq> {}" shows "\<Sqinter> insert x A = x \<sqinter> \<Sqinter> A"
ResOutput': fixes x :: "name"   and b :: "name"   and P :: "pi" shows "<\<nu>x>x{b}.P \<equiv>\<^sub>e \<zero>"
sets_lebesgue_inner_closed: fixes S :: "'b set"   and e :: "real" assumes "S \<in> sets lebesgue"   and "0 < e" obtains T :: "'b set" where "closed T" and "T \<subseteq> S" and "S - T \<in> lmeasurable" and "emeasure lebesgue (S - T) < ennreal e"
lookup_pair_filter: fixes xs :: "('a \<times> 'b) list"   and P :: "'a \<times> 'b \<Rightarrow> bool"   and k :: "'a" assumes "tc.oalist_inv_raw xs" shows "lookup_pair_tc (filter P xs) k = (let v = lookup_pair_tc xs k in if P (k, v) then v else (0::'b))"
a: shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
lexord_cons_cons: fixes a :: "'a"   and x :: "'a list"   and b :: "'a"   and y :: "'a list"   and r :: "('a \<times> 'a) set" shows "((a # x, b # y) \<in> lexord r) = ((a, b) \<in> r \<or> a = b \<and> (x, y) \<in> lexord r)"
Brouwer_degree2_def: fixes p :: "nat"   and f :: "(nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real" shows "Brouwer_degree2 p f \<equiv> SOME d. \<forall>x\<in>carrier (reduced_homology_group (int p) (nsphere p)). hom_induced (int p) (nsphere p) {} (nsphere p) {} f x = x [^]\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub> d"
ilist_unslice_def: fixes f :: "nat \<Rightarrow> 'b list" shows "ilist_unslice f \<equiv> \<lambda>n. f (n div length (f 0)) ! (n mod length (f 0))"
vimageI2: fixes f :: "'f \<Rightarrow> 'e"   and a :: "'f"   and A :: "'e set" assumes "f a \<in> A" shows "a \<in> f -` A"
subgraph_induce_subgraphI: fixes V :: "'a set" assumes "V \<subseteq> verts G" shows "subgraph (G \<restriction> V) G"
integrable_on_cnj_iff: fixes f :: "'a \<Rightarrow> complex"   and A :: "'a set" shows "((\<lambda>x. cnj (f x)) integrable_on A) = (f integrable_on A)"
in_diffD: fixes a :: "'b"   and M :: "'b multiset"   and N :: "'b multiset" assumes "a \<in># M - N" shows "a \<in># M"
nop_onto_vrange: shows "\<R>\<^sub>\<circ> f = A"
basis_in_indep_in_card: fixes \<E> :: "'a set"   and X :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> \<E>"   and "\<E>.indep_in \<E> X"   and "\<E>.basis_in \<E> B" shows "card X \<le> card B"
freess_single: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and x :: "'b" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.freess frees [x] = frees x"
Inf_as_limit: fixes A :: "'c set" assumes "A \<noteq> {}" shows "\<exists>u. (\<forall>n. u n \<in> A) \<and> u \<longlonglongrightarrow> \<Sqinter> A"
Rep_boffa_mon_inject: fixes x :: "('b, 'c) boffa_mon"   and y :: "('b, 'c) boffa_mon" shows "(Rep_boffa_mon x = Rep_boffa_mon y) = (x = y)"
to_fun_is_Fun: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "to_fun x \<in> carrier (function_ring (carrier R) R)"
assms_6: fixes n :: "nat" assumes "n \<le> 2 ^ k" shows "psi n \<le> c * ln 2 * real n"
comp_char: fixes g :: "'a"   and f :: "'a" shows "hhom\<^sub>C.comp g f = (if hhom\<^sub>C.arr f \<and> hhom\<^sub>C.arr g \<and> C.seq g f then g \<cdot>\<^sub>C f else C.null)"
strict_implies_order: fixes b :: "'b"   and a :: "'b" assumes "order.greater a b" shows "order.greater_eq a b"
pw_conc_nofail: fixes R :: "('b \<times> 'c) set"   and S :: "'c nres" shows "nofail (\<Down> R S) = nofail S"
simps_2: fixes uv :: "typ"   and v :: "String.literal"   and va :: "typ" shows "eta_reducible (Abs uv (Ct v va)) = eta_reducible (Ct v va)"
less_imp_neq: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and x :: "'k"   and y :: "'k" assumes "class.order less_eq less"   and "less x y" shows "x \<noteq> y"
dds_of_parametric: fixes S :: "'b \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'c \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'g \<Rightarrow> bool" shows "((S ===> A ===> rel_prod B S) ===> S ===> rel_dds A B) dds_of dds_of"
SK_lappend_distrib: fixes xs :: "nat discr llist"   and ys :: "nat discr llist"   and s :: "nat discr \<rightarrow> nat discr llist \<rightarrow> nat discr llist"   and f :: "nat discr llist" shows "SK\<cdot>(lappend\<cdot>xs\<cdot>ys)\<cdot>s\<cdot>f = SK\<cdot>xs\<cdot>s\<cdot>(SK\<cdot>ys\<cdot>s\<cdot>f)"
RBT_lookup_empty: fixes t :: "('b, unit) rbt" assumes "ID ccompare \<noteq> None" shows "(ord.rbt_lookup cless t = Map.empty) = (t = rbt.Empty)"
framed_uncross_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "framed_uncross w1 w2" shows "kauff_mat w1 = kauff_mat w2"
invrules_1: shows "bp (P\<^sub>1 \<union> wrap B\<^sub>1 \<union> P\<^sub>2 \<union> wrap B\<^sub>2 \<union> {{v} |v. v \<in> S \<union> L})"
smap_Union_append: fixes m1 :: "('a \<Rightarrow> 'b set) list"   and m2 :: "('a \<Rightarrow> 'b set) list" shows "\<Union>.m1 @ m2 = \<Union>.m1 \<union>. \<Union>.m2"
Inf_insert: fixes a :: "'a"   and A :: "'a set" shows "\<Sqinter> insert a A = a \<sqinter> \<Sqinter> A"
intp: shows "integrable M (\<lambda>x. fp x * g x)"
\<phi>_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'c \<Rightarrow> 'b"   and G :: "'b \<Rightarrow> 'c"   and \<phi> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and \<psi> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and y :: "'c"   and f :: "'b"   and x :: "'b" assumes "meta_adjunction C D F G \<phi> \<psi>"   and "partial_magma.ide D y"   and "partial_magma.in_hom C f (F y) x" shows "partial_magma.in_hom D (\<phi> y f) y (G x)"
simps_6: fixes \<iota> :: "nat"   and V :: "('a, 'b) MWLsCom list" shows "PPc (spawn\<^bsub>\<iota>\<^esub> V) = [\<iota>] @ PPV V"
fmsubset_filter_mono: fixes m :: "('c, 'd) fmap"   and n :: "('c, 'd) fmap"   and P :: "'c \<Rightarrow> bool" assumes "m \<subseteq>\<^sub>f n" shows "fmfilter P m \<subseteq>\<^sub>f fmfilter P n"
star_circ_simulate_left_plus: fixes x :: "'a"   and z :: "'a"   and y :: "'a"   and w :: "'a" assumes "x * z \<le> z * y\<^sup>\<star> \<squnion> w" shows "x\<^sup>\<star> * z \<le> (z \<squnion> x\<^sup>\<star> * w) * y\<^sup>\<star>"
assms_2: shows "infs (degen cs) (w ||| k ## sscan (count cs) w k)"
hcomplex_hcnj_inverse: fixes z :: "complex star" shows "hcnj (inverse z) = inverse (hcnj z)"
c_d: shows "(c, d) \<in> thetaZOCh ch c1 c2 d1 d2"
eq_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" assumes "bi_unique A" shows "(A ===> A ===> (=)) (=) (=)"
rcos_assoc_lcos: fixes H :: "'a set"   and K :: "'a set"   and x :: "'a" assumes "H \<subseteq> carrier G"   and "K \<subseteq> carrier G"   and "x \<in> carrier G" shows "H #> x <#> K = H <#> (x <# K)"
inject_2: fixes x21 :: "'d net_state"   and x22 :: "'d net_state"   and y21 :: "'d net_state"   and y22 :: "'d net_state" shows "(SubnetS x21 x22 = SubnetS y21 y22) = (x21 = y21 \<and> x22 = y22)"
basic_monos_6: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
varMonotoneAux: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and \<Psi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool" assumes "(b, \<Phi>, \<beta>, s, t) \<in> var"   and "\<And>s t \<gamma>. \<Phi> (s, t, \<gamma>) \<Longrightarrow> \<Psi> (s, t, \<gamma>)" shows "(b, \<Psi>, \<beta>, s, t) \<in> var"
"2_7": fixes x :: "real" assumes "a \<le> x"   and "x \<le> c" shows "\<delta> x \<le> \<delta>' x"
assigned_nodes_different_ptr: fixes h :: "(_) heap"   and slot :: "(_) element_ptr"   and nodes :: "(_) node_ptr list"   and slot' :: "(_) element_ptr"   and nodes' :: "(_) node_ptr list" assumes "h \<turnstile> assigned_nodes slot \<rightarrow>\<^sub>r nodes"   and "h \<turnstile> assigned_nodes slot' \<rightarrow>\<^sub>r nodes'"   and "slot \<noteq> slot'" shows "set nodes \<inter> set nodes' = {}"
f: shows "pdevs_val f (pdevs_of_list xs) = (\<Sum>i<length xs. (if i < length xs then f i else g (i - length xs)) *\<^sub>R pdevs_apply (pdevs_of_list (xs @ ys)) i)"
gkernTr6: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g" assumes "Group F"   and "Group G"   and "f \<in> gHom F G" shows "\<one>\<^bsub>F\<^esub> \<in> gker\<^bsub>F,G\<^esub> f"
linorder_from_keys_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "linorder_from_keys A f = {(x, y). (x, y) \<in> A \<times> A \<and> f x \<le> f y}"
omit_block_id: fixes block :: "('c, 'd) fm list \<times> 'd" shows "omit_block {} block = block"
fw_upd_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and k :: "nat"   and i :: "nat"   and j :: "nat" assumes "class.linordered_ab_monoid_add plus zero less_eq less" shows "linordered_ab_monoid_add.fw_upd plus less_eq m k i j \<equiv> linordered_ab_monoid_add.upd m i j (ord.min less_eq (m i j) (plus (m i k) (m k j)))"
indep_in_iff_rank_of: fixes \<E> :: "'a set"   and X :: "'a set" assumes "\<E> \<subseteq> carrier"   and "X \<subseteq> \<E>" shows "indep_in \<E> X = (rank_of X = card X)"
simps_10: fixes vl :: "'u \<Rightarrow> 'v fundef option"   and opinl :: "'f"   and \<Sigma> :: "Unboxed.type option list" shows "sp_instr vl (IOpInl opinl) \<Sigma> = sp_gen_pop_push (replicate (\<AA>\<rr>\<ii>\<tt>\<yy> (\<DD>\<ee>\<II>\<nn>\<ll> opinl)) None, None) \<Sigma>"
lselect_least: fixes w :: "'b llist"   and s :: "nat set" assumes "\<not> lnull (lselect s w)" shows "lselect s w = w ?! least s % lselect (s - {least s}) w"
vrange_atD: fixes b :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vreal_uminus" shows "\<exists>a\<in>elts (\<D>\<^sub>\<circ> vreal_uminus). -\<^sub>\<real> a = b"
inject_1: fixes x2 :: "'d"   and y2 :: "'d" shows "(Skip x2 = Skip y2) = (x2 = y2)"
fin_A: shows "finite {l. length l = m - 1 \<and> sum_list l = N}"
to_abs_alt_def: fixes D :: "((('b, 'c, 'd) prot_fun, 'e) Term.term \<times> (('f, 'g, 'a) prot_fun, 'h) Term.term) list"   and n :: "nat \<times> bool" shows "\<alpha>\<^sub>0 D n = {s. \<exists>S. (Fun (Val n) [], Fun (Set s) S) \<in> set D}"
nlex_half_segments_of_aform: fixes a :: "real \<times> real"   and b :: "real \<times> real"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "(a, b) \<in> set (half_segments_of_aform X)" shows "Counterclockwise_2D_Arbitrary.lex b a"
supp_diff_subset_union_supp: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "supp (f - g) \<subseteq> supp f \<union> supp g"
le_measure: fixes M :: "'b measure"   and N :: "'b measure" assumes "sets M = sets N" shows "(M \<le> N) = (\<forall>A\<in>sets M. emeasure M A \<le> emeasure N A)"
if_weak_cong: fixes b :: "bool"   and c :: "bool"   and x :: "'b"   and y :: "'b" assumes "b = c" shows "(if b then x else y) = (if c then x else y)"
Domain_Un_eq: fixes A :: "('e \<times> 'f) set"   and B :: "('e \<times> 'f) set" shows "Domain (A \<union> B) = Domain A \<union> Domain B"
uadmit_sterm_ntadjoint': fixes \<sigma> :: "'b \<Rightarrow> ('sf, 'sz) trm"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<omega> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and \<theta> :: "('sf + 'b, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp" assumes "\<And>i. dsafe (\<sigma> i)"   and "Vagree \<nu> \<omega> (\<Union>i\<in>{i. Inr i \<in> SIGT \<theta>}. FVT (\<sigma> i))" shows "sterm_sem (adjointFO I \<sigma> \<nu>) \<theta> = sterm_sem (adjointFO I \<sigma> \<omega>) \<theta>"
insert_def: fixes e :: "'a"   and a :: "'b"   and q :: "('a, 'b) BinomialHeap" shows "BinomialHeap_loc.insert e a q \<equiv> Abs_BinomialHeap (BinomialHeapStruc.insert e a (Rep_BinomialHeap q))"
2: shows "list_all tfr\<^sub>s\<^sub>t\<^sub>p A'"
card_proots_unbounded_line: assumes "st \<noteq> tt" shows "card (proots_within p (unbounded_line st tt)) = (let pc = p \<circ>\<^sub>p [:st, tt - st:]; pR = map_poly Re pc; pI = map_poly Im pc; g = gcd pR pI in nat (changes_R_smods g (pderiv g)))"
1: shows "a * b * s1 + b * r1 = (a * s1 + r1) * b"
trancl_multi_insert: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set"   and X :: "'a set"   and m :: "'a" assumes "(a, b) \<in> (r \<union> X \<times> {m})\<^sup>+" obtains "(a, b) \<in> r\<^sup>+"   | x :: "'a" where "x \<in> X" and "(a, x) \<in> r\<^sup>*" and "(m, b) \<in> r\<^sup>*"
map_poly_div: fixes p :: "'a poly"   and q :: "'a poly" shows "map_poly hom (p div q) = map_poly hom p div map_poly hom q"
deleted_not_domA: fixes x :: "'b"   and \<Gamma> :: "('b \<times> 'c) list" shows "x \<notin> domA (delete x \<Gamma>)"
p23: shows "q \<in> UNIV - {p} \<and> bk \<in> blocksOf s q"
parts_idem: fixes H :: "msg set" shows "parts (parts H) = parts H"
IF2rel_def: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "IF2rel R = (BNF_Def.Grp (IF2in {(x, y). R x y}) (IF2map fst))\<inverse>\<inverse> OO BNF_Def.Grp (IF2in {(x, y). R x y}) (IF2map snd)"
distrib: fixes g :: "'c \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "(\<Prod>x\<in>A. g x * h x) = prod g A * prod h A"
card_UN_disjoint: fixes I :: "'c set"   and A :: "'c \<Rightarrow> 'd set" assumes "finite I"   and "\<forall>i\<in>I. finite (A i)"   and "\<forall>i\<in>I. \<forall>j\<in>I. i \<noteq> j \<longrightarrow> A i \<inter> A j = {}" shows "card (\<Union> (A ` I)) = (\<Sum>i\<in>I. card (A i))"
inf_assertion: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top"   and "x \<in> mbt_algebra.assertion one times inf less_eq dual top"   and "y \<in> mbt_algebra.assertion one times inf less_eq dual top" shows "inf x y \<in> mbt_algebra.assertion one times inf less_eq dual top"
hyps: shows "distinct xs = (\<forall>a. count (mset xs) a = (if a \<in> set xs then 1 else 0))"
coerce_cfun: fixes f :: "'c \<rightarrow> 'd" shows "COERCE('c \<rightarrow> 'd, 'a \<rightarrow> 'b)\<cdot>f = COERCE('d, 'b) oo f oo COERCE('a, 'c)"
atLeastAtMost_singleton: fixes a :: "'a" shows "ordered_powerprod_lin.atLeastAtMost a a = {a}"
distinct_5: fixes x1 :: "bool"   and x41 :: "aexp"   and x42 :: "aexp" shows "Bc x1 \<noteq> Less x41 x42"
Keys_bad_insert_Fake: fixes IK :: "msg set"   and Bad :: "agent set"   and X :: "msg" assumes "Keys_bad IK Bad"   and "parts IK \<inter> range LtK \<subseteq> IK"   and "X \<in> synth (analz IK)" shows "Keys_bad (insert X IK) Bad"
proj_psi_l_def: fixes l :: "nat" shows "proj_psi_l l = outer_prod (psi_l l) (psi_l l)"
correct_state_impl_Some_method: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and \<Phi> :: "char list \<Rightarrow> char list \<Rightarrow> (ty list \<times> ty err list) option list"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list" assumes "P,\<Phi> |- (None, h, (stk, loc, C, M, pc) # frs) [ok]" shows "\<exists>m Ts T. P \<turnstile> C sees M: Ts\<rightarrow>T = m in C"
ordinal_rec0_mono: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a"   and s :: "ordinal \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "ordinal"   and y :: "ordinal"   and z :: "'a" assumes "omega_complete le lub"   and "\<forall>p x. le x (s p x)"   and "x \<le> y" shows "le (ordinal_rec0 z s lub x) (ordinal_rec0 z s lub y)"
equal_refl: fixes equal :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "class.equal equal" shows "equal x x = True"
sum_abs_complex: fixes h :: "nat \<Rightarrow> nat \<Rightarrow> complex"   and B :: "nat set"   and A :: "nat set" shows "cmod (\<Sum>x\<in>A. sum (h x) B) \<le> (\<Sum>x\<in>A. \<Sum>y\<in>B. cmod (h x y))"
simps_3: fixes a :: "('a, 'b) rbt"   and w :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and s :: "'a"   and t :: "'b"   and c :: "('a, 'b) rbt"   and y :: "'a"   and z :: "'b"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt" shows "balance (Branch R (Branch R a w x b) s t c) y z (Branch B va vb vc vd) = Branch R (Branch B a w x b) s t (Branch B c y z (Branch B va vb vc vd))"
P\<^sub>1_B\<^sub>1_lower_bound_card: fixes P :: "'a set set"   and P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set" assumes "bp P"   and "inv\<^sub>1 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "bij_exists P\<^sub>1 (\<Union> (P\<^sub>2 \<union> wrap B\<^sub>2))" shows "card (P\<^sub>1 \<union> wrap B\<^sub>1) \<le> card P"
DERIV_inverse: fixes x :: "'c"   and s :: "'c set" assumes "x \<noteq> (0::'c)" shows "(inverse has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
SUP_le_iff: fixes U\<^sub>2 :: "'c set"   and f :: "'c \<Rightarrow> 'al"   and A :: "'c set"   and u :: "'al" assumes "\<forall>x\<in>U\<^sub>2. f x \<in> U"   and "A \<subseteq> U\<^sub>2"   and "u \<in> U" shows "(\<Squnion>\<^sub>o\<^sub>w (f ` A) \<le>\<^sub>o\<^sub>w u) = (\<forall>x\<in>A. f x \<le>\<^sub>o\<^sub>w u)"
evaluate_iff_3: fixes env :: "v sem_env"   and st :: "'a state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and v' :: "v"   and r :: "'a state \<times> (v, v) result" shows "evaluate_match True env st v pes v' r = (r = eval_match env v pes v' st)"
ex_inj: assumes "class.countable TYPE('a)" shows "\<exists>to_nat. inj to_nat"
move_uminus: fixes x :: "real"   and y :: "real" shows "(- x \<le> y) = (- y \<le> x)"
Rats_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<rat>"   and "b \<in> \<rat>" shows "a * b \<in> \<rat>"
simps_8: fixes c :: "int"   and e :: "Cooper.num" shows "iszlfm (Ge (CN 0 c e)) = (0 < c \<and> numbound0 e)"
ext_irrefl_before_trans_axioms: shows "ext_irrefl_before_trans cwiseext"
aMvC: shows "\<bar>M\<bar> \<cdot>\<^sub>v v \<in> cone C"
Abs_cars_inverse: fixes y :: "nat" assumes "y \<in> {n. True}" shows "Rep_cars (Abs_cars y) = y"
assms: shows "a \<notin>\<^sub>\<circ> B"
healthy_fixes_bot: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "healthy t" shows "t (\<lambda>s. 0) = (\<lambda>s. 0)"
length_take: fixes n :: "nat"   and xs :: "'b list" shows "length (xs \<down> n) = min (length xs) n"
uniformity_real_def: shows "uniformity = (INF e\<in>{0<..}. principal {(x, y). dist x y < e})"
asymmetric_bot_closed: shows "bot \<sqinter> id bot = bot"
orderE: fixes a :: "'d DBMEntry"   and b :: "'d DBMEntry" assumes "a \<preceq> b" obtains "a = linordered_monoid.min a b"
sum_is_pr: fixes g :: "nat \<Rightarrow> nat" assumes "g \<in> PrimRec1" shows "(\<lambda>u. sum g (nat_to_set u)) \<in> PrimRec1"
set_rel_def: fixes R :: "('d \<times> 'e) set" shows "\<langle>R\<rangle>set_rel \<equiv> {(A, B). (\<forall>x\<in>A. \<exists>y\<in>B. (x, y) \<in> R) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. (x, y) \<in> R)}"
aform_euclarithform_refine: fixes optns :: "nat" shows "(nres_of o2 aform_form optns, approx_form_spec) \<in> Id \<rightarrow> aforms_rel \<rightarrow> \<langle>bool_rel\<rangle>nres_rel"
assms: shows "C.arr \<mu> \<and> C.arr \<nu> \<and> src\<^sub>C \<mu> = trg\<^sub>C \<nu>"
fbox_star_unfold_var: fixes z :: "'a"   and x :: "'a" shows "d z \<cdot>  |x]  |x\<^sup>\<star>] z =  |x\<^sup>\<star>] z"