hcju/isabelleps huggingface dataset:
beukers_integral1_def: fixes r :: "nat"   and s :: "nat" shows "beukers_integral1 r s = LBINT (x, y):{0<..<1} \<times> {0<..<1}. - ln (x * y) / (1 - x * y) * x ^ r * y ^ s"
simps_3: fixes b :: "'ubx2" shows "norm_instr (IPushUbx2 b) = Inca.instr.IPush (box_ubx2 b)"
subset_cl: fixes r :: "'b set"   and L :: "'b set set" shows "r \<subseteq> cl L r"
simps_14: fixes sh :: "sheap"   and e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp" shows "iconf sh (e\<^sub>1;; e\<^sub>2) = (case val_of e\<^sub>1 of None \<Rightarrow> case lass_val_of e\<^sub>1 of None \<Rightarrow> iconf sh e\<^sub>1 \<and> \<not> sub_RI e\<^sub>2 | \<lfloor>p\<rfloor> \<Rightarrow> iconf sh e\<^sub>2 | \<lfloor>v\<rfloor> \<Rightarrow> iconf sh e\<^sub>2)"
inorder_update: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (update x y t) = upd_list x y (Tree2.inorder t)"
fi_dvd_prod_vc: shows "fi dvd (\<Prod>c\<in>UNIV. v - [:c:])"
maximally_connected_impl_connected: fixes G :: "('c, 'd) graph"   and H :: "('c, 'd) graph" assumes "connected_graph G"   and "maximally_connected H G"   and "subgraph H G" shows "connected_graph H"
pred_set: fixes P :: "'d \<Rightarrow> bool" shows "pred_stream P = (\<lambda>x. Ball (sset x) P)"
remove_unknowns_generic_simps2_6: fixes \<beta> :: "'b \<Rightarrow> 'c \<Rightarrow> ternaryvalue"   and \<alpha> :: "action \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "action"   and m1 :: "'b match_expr"   and m2 :: "'b match_expr" shows "remove_unknowns_generic (\<beta>, \<alpha>) a (MatchAnd m1 m2) = MatchAnd (remove_unknowns_generic (\<beta>, \<alpha>) a m1) (remove_unknowns_generic (\<beta>, \<alpha>) a m2)"
r_result_nonempty_stack: fixes i :: "nat"   and f :: "recf"   and x :: "nat"   and t :: "nat" assumes "i = encode f"   and "recfn (e_length x) f"   and "fst (iterate t step ([(f, Partial_Recursive.list_decode x, [])], None)) \<noteq> []" shows "eval r_result [t, i, x] \<down>= 0"
inv_f_f: fixes x :: "'c poly" shows "inv (map_poly to_fract) (map_poly to_fract x) = x"
intersect'_nodes_finite: fixes A :: "('e, 'd) dbta"   and B :: "('e, 'f) dbta" assumes "finite (DBTA.nodes A)"   and "finite (DBTA.nodes B)" shows "finite (DGBTA.nodes (intersect' A B))"
0: shows "val_Zp c = val_Zp (c \<ominus> a \<oplus> a)"
inverse_mk_dimvec: fixes xs :: "'a list" shows "inverse (mk_dimvec xs) = (if length xs = CARD('b) then mk_dimvec (map uminus xs) else 1)"
image_ident: fixes Y :: "'e set" shows "(\<lambda>x. x) ` Y = Y"
simps_17: fixes v :: "nat"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fa (Var v) M slp = slp_of_fa_cnst (Var v) (Var (v + length slp)) M slp"
union'_nodes_card: fixes A :: "('d, 'c) dca"   and B :: "('d, 'e) dca" assumes "finite (DCA.nodes A)"   and "finite (DCA.nodes B)" shows "card (DGCA.nodes (union' A B)) \<le> card (DCA.nodes A) * card (DCA.nodes B)"
l_get_dom_component_get_element_by_id\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_get_element_by_id\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptr DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_class_name get_elements_by_tag_name first_in_tree_order get_attribute get_attribute_locs"
fnzf: fixes Ax :: "(ZF, ZF) Axioms" assumes "ZFAxioms Ax" shows "Fn\<^bsub>aS\<^bsub>Ax\<^esub>\<^esub> \<in> range explode"
ltl_to_generalized_rabin_correct: fixes \<Sigma> :: "'b set set"   and w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" assumes "ltl_to_rabin_af_unf \<Sigma> w" shows "w \<Turnstile> \<phi> = accept\<^sub>G\<^sub>R (ltl_to_rabin_base_def.ltl_to_generalized_rabin \<up>af\<^sub>\<UU> \<up>af\<^sub>G\<^sub>\<UU> (Abs \<circ> Unf) (Abs \<circ> Unf\<^sub>G) M\<^sub>\<UU>_fin \<Sigma> \<phi>) w"
cfs_monom: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "a \<in> carrier R" shows "monom P a m n = (if m = n then a else \<zero>)"
nextl_state: fixes q :: "hf"   and xs :: "'a list" assumes "q \<in> dfa.states Power_dfa" shows "Power.nextl q xs \<in> dfa.states Power_dfa"
fold_set_fold: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and xs :: "'a list" assumes "comp_fun_idem f" shows "Finite_Set.fold f y (set xs) = fold f xs y"
LCons_all: fixes l :: "'b llist"   and A :: "'b set"   and a :: "'b" assumes "l \<in> A\<^sup>\<infinity>"   and "a \<in> A" shows "a ## l \<in> A\<^sup>\<infinity>"
lunit_hcomp_1: fixes f :: "'b"   and g :: "'b" assumes "ide f"   and "ide g"   and "src f = trg g" shows "\<l>[f \<star> g] \<cdot> \<a>[trg f, f, g] = \<l>[f] \<star> g"
intros_15: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Prj2 e\<ggreater> m\<rightarrow> \<lless>\<pi>', Prj2 e'\<ggreater>"
lift_comp: fixes g :: "'d \<Rightarrow> 'c"   and f :: "'e \<Rightarrow> 'd"   and inp :: "'b \<Rightarrow> 'e option" shows "lift g (lift f inp) = lift (g \<circ> f) inp"
bisim1CastFail: fixes P :: "'a prog"   and h :: "'heap"   and T :: "ty"   and e :: "'addr expr1"   and xs :: "'addr val list"   and v :: "'addr val" shows "P,Cast T e,h \<turnstile> (Throw (addr_of_sys_xcpt ClassCast), xs) \<leftrightarrow> ([v], xs, length (compE2 e), \<lfloor>addr_of_sys_xcpt ClassCast\<rfloor>)"
assms_3: fixes a :: "'a" assumes "a \<sqsubseteq> x"   and "a \<sqsubseteq> y" shows "a \<sqsubseteq> z"
simpfm: fixes bbs :: "bool list"   and bs :: "int list"   and p :: "fm" shows "Ifm bbs bs (simpfm p) = Ifm bbs bs p"
pos_lit_in_atms_of: fixes A :: "'b"   and C :: "'b literal multiset" assumes "Pos A \<in># C" shows "A \<in> atms_of C"
FIXvarFIX': fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and b :: "BExpr"   and \<Phi> :: "(Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "PhiWhileP A b \<Phi> = (\<lambda>(s, t). (b, A, \<Phi>, s, t) \<in> var)"
integrableI_real_bounded: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M < \<infinity>" shows "integrable M f"
flat_lub_def: fixes b :: "'b"   and A :: "'b set" shows "flat_lub b A = (if A \<subseteq> {b} then b else THE x. x \<in> A - {b})"
not_less_bot: fixes a :: "'b list" shows "\<not> a <p \<epsilon>"
preserves_cod: fixes f :: "'a1" assumes "A1.arr f" shows "A2_B.cod (F f) = F (A1.cod f)"
0: fixes j :: "nat" assumes "j < n" shows "A $$ (j, i) = (0::'a)"
dickson_grading_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and lcs :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.dickson_powerprod plus minus zero lcs" shows "plus.dickson_grading plus (\<lambda>_. 0)"
abs_fresh_6: fixes b :: "W.var"   and a :: "W.tvar"   and x :: "'a" shows "b \<sharp> [a].x = b \<sharp> x"
Or\<^sub>n_dnf: fixes \<Phi> :: "'b ltln set" assumes "finite \<Phi>" shows "dnf (Or\<^sub>n \<Phi>) = Finite_Set.fold (\<lambda>\<phi>. (\<union>) (dnf \<phi>)) {} \<Phi>"
dim_row_polys_to_mat: fixes ts :: "'e list"   and ps :: "('e \<Rightarrow>\<^sub>0 'd) list" shows "dim_row (polys_to_mat ts ps) = length ps"
to_Gromov_completion_range_dense: shows "closure (range to_Gromov_completion) = UNIV"
right_neutral: fixes a :: "'c" shows "a OR (0::'c) = a"
inL2: fixes F :: "'b formula"   and G :: "'b formula"   and H :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "F, G, H, \<Gamma> \<Rightarrow> \<Delta>" shows "G, H, F, \<Gamma> \<Rightarrow> \<Delta>"
inf_last_ti_nonempty_k: fixes dt :: "nat \<Rightarrow> 'a list"   and t :: "nat"   and k :: "nat" assumes "inf_last_ti dt t \<noteq> []" shows "inf_last_ti dt (t + k) \<noteq> []"
elims: fixes x :: "nat"   and xa :: "'a list"   and y :: "'a list list" assumes "chop x xa = y" obtains uu_ :: "'a list" where "x = 0" and "xa = uu_" and "y = []"   | v :: "nat" where "x = Suc v__" and "xa = []" and "y = []"   | v :: "nat" and va :: "'a" and vb :: "'a list" where "x = Suc v" and "xa = va # vb" and "y = take (Suc v) (va # vb) # chop (Suc v) (drop (Suc v) (va # vb))"
head_arev: fixes a :: "'b" assumes "a \<in> arcs G" shows "head G (arev a) = tail G a"
assms_1: shows "a \<in> \<Union> (set (x # xs))"
N: fixes n :: "nat" assumes "N \<le> n" shows "f (- int n) = (0::'a)"
le_fun_def: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l" shows "(f \<le> g) = (\<forall>x. f x \<le> g x)"
simple_extension_incl: fixes K :: "'a set"   and x :: "'a" assumes "K \<subseteq> carrier R"   and "x \<in> carrier R" shows "K \<subseteq> simple_extension K x"
distinct_inl: fixes X :: "'a \<times> (real \<times> real) pdevs" shows "distinct (inl (snd X))"
HBL1_PP: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "prv \<phi>" shows "prv (PP \<langle>\<phi>\<rangle>)"
greatest_cong: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and x' :: "'a"   and A :: "'a set" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> x'"   and "x \<in> carrier L"   and "x' \<in> carrier L"   and "is_closed\<^bsub>L\<^esub> A" shows "greatest L x A = greatest L x' A"
predicate1D: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "P \<le> Q"   and "P x" shows "Q x"
lset_llist_of_stream: fixes xs :: "'b stream" shows "lset (llist_of_stream xs) = sset xs"
span_insert_zero: fixes B :: "'b set" shows "local.span (insert (0::'b) B) = local.span B"
integrable_self_fin_uvp: fixes pf :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real"   and M :: "'d measure"   and Mkt :: "('d, 'c) discrete_market"   and n :: "nat" assumes "\<forall>asset\<in>support_set pf. integrable M (\<lambda>w. prices Mkt asset n w * pf asset (Suc n) w)"   and "self_financing Mkt pf" shows "integrable M (cls_val_process Mkt pf n)"
foo: fixes x :: "'a" shows "foo' (A x) B (A x)"
ennreal_top_neq_one: shows "top \<noteq> 1"
enum_all_finite_1_def: fixes P :: "finite_1 \<Rightarrow> bool" shows "enum_all P = P a\<^sub>1"
RR_def: fixes t1 :: "'trm"   and t2 :: "'trm" shows "RR t1 t2 = psubst R [(t1, yy), (t2, xx)]"
Lt_le_LeD: fixes a :: "'c"   and b :: "'c" assumes "Lt a \<le> Le b" shows "a \<le> b"
subdomain_iff: fixes H :: "'a set" assumes "H \<subseteq> carrier R" shows "subdomain H R = domain (R\<lparr>carrier := H\<rparr>)"
height_del: fixes t :: "'b tree234"   and x :: "'b" assumes "bal t" shows "height (del x t) = height t"
cot_in_Reals: fixes z :: "'a" assumes "z \<in> \<real>" shows "cot z \<in> \<real>"
assms: shows "(E, {#}) \<notin> unif\<^sup>*"
walk_decomp_2: fixes xs :: "'a list"   and ys :: "'a list" assumes "walk (xs @ ys)" shows "walk ys"
inv: shows "adjoint (tensor_P (1\<^sub>m N) (mat_incr K)) * tensor_P (1\<^sub>m N) (mat_incr K) = 1\<^sub>m d"
cases: fixes a1 :: "nat \<Rightarrow> LambdaType.type"   and a2 :: "dB"   and a3 :: "LambdaType.type" assumes "a1 \<turnstile> a2 : a3" obtains env :: "nat \<Rightarrow> LambdaType.type" and x :: "nat" and T :: "LambdaType.type" where "a1 = env__" and "a2 = Var x" and "a3 = Ta__" and "env__ x = Ta__"   | env :: "nat \<Rightarrow> LambdaType.type" and T :: "LambdaType.type" and t :: "dB" and U :: "LambdaType.type" where "a1 = enva__" and "a2 = Abs t__" and "a3 = Tb__ \<Rightarrow> U__" and "enva__\<langle>0:Tb__\<rangle> \<turnstile> t__ : U__"   | env :: "nat \<Rightarrow> LambdaType.type" and s :: "dB" and T :: "LambdaType.type" and U :: "LambdaType.type" and t :: "dB" where "a1 = env" and "a2 = s \<degree> t" and "a3 = U" and "env \<turnstile> s : T \<Rightarrow> U" and "env \<turnstile> t : T"
rewrite_B_None: fixes \<alpha> :: "'a \<Rightarrow> bool" shows "None \<rightharpoonup> \<alpha> = True"
assms_1: shows "K01 = I0 \<union> K1"
DProp_def: fixes B :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" shows "DProp B = (\<exists>A R S \<phi>. B = Sec A R S (\<phi> (FIX \<phi>)) \<and> Monotone \<phi>)"
Cantor_\<omega>\<omega>: fixes \<gamma> :: "V" assumes "\<gamma> \<in> elts (\<omega> \<up> \<omega>)" shows "omega_sum (Cantor_\<omega>\<omega> \<gamma>) = \<gamma>"
greaterThan_Int_greaterThan: fixes a :: "'a"   and b :: "'a" shows "{..<a} \<inter> {..<b} = {..<min a b}"
sum_of_2_squares_nat_gauss_int_norm: fixes z :: "gauss_int" shows "sum_of_2_squares_nat (gauss_int_norm z)"
rep_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g" shows "free_monoidal_category.rep C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g) = free_monoidal_category.norm C (free_monoidal_category.rep C f \<^bold>\<otimes> free_monoidal_category.rep C g)"
normalize_replicate_zero: fixes R :: "('c, 'd) ring_scheme"   and n :: "nat"   and p :: "'c list" assumes "ring R" shows "ring.normalize R (replicate n \<zero>\<^bsub>R\<^esub> @ p) = ring.normalize R p"
tick_T_F: fixes t :: "'a event list"   and P :: "'a process"   and X :: "'a event set" assumes "t @ [tick] \<in> T P" shows "(t @ [tick], X) \<in> F P"
fix_eq4: fixes f :: "'b"   and F :: "'b \<rightarrow> 'b" assumes "f = fix\<cdot>F" shows "f = F\<cdot>f"
arev_dom: fixes a :: "'b" shows "(a \<in> arcs G) = (edge_rev GM a \<noteq> a)"
streams_mono: fixes s :: "'a stream"   and A :: "'a set"   and B :: "'a set" assumes "s \<in> streams A"   and "A \<subseteq> B" shows "s \<in> streams B"
bound: fixes n :: "nat" shows "\<bar>Sigma_Algebra.measure M (fst (A n)) - Sigma_Algebra.measure M (snd (A n))\<bar> < 1 / real (Suc n)"
star_inductr: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> y" shows "y \<cdot> x\<^sup>\<star> \<le> y"
dvd_mult: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a c" shows "dvd.dvd times a (times b c)"
CatIdInMor: fixes C :: "('d, 'e, 'f) Category_scheme"   and X :: "'d" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X \<in> mor\<^bsub>C\<^esub>"
demorgans2: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> y) = - x \<sqinter> - y"
case_prodI2': fixes p :: "'a \<times> 'b"   and c :: "'a \<Rightarrow> 'b \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" assumes "\<And>a b. (a, b) = p \<Longrightarrow> c a b x" shows "(case p of (a, b) \<Rightarrow> c a b) x"
N: fixes n :: "nat" shows "(n \<in> N) = (n \<in> M \<and> rejects \<F> (insert n S) M \<and> Sup S < n)"
norm2_def: shows "norm2 \<equiv> vec_of_basis_enum a \<bullet>c vec_of_basis_enum a"
MWLsSteps_det_cases_3: fixes c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom"   and m :: "'id \<Rightarrow> 'val"   and \<alpha> :: "('exp, 'id) MWLsCom list"   and p :: "('exp, 'id) MWLsCom option"   and m' :: "'id \<Rightarrow> 'val" assumes "SRabbr (c1;c2) m \<alpha> p m'" obtains "p = Some c2" and "SRabbr c1 m \<alpha> None m'"   | c1' :: "('exp, 'id) MWLsCom" where "p = Some (c1';c2)" and "SRabbr c1 m \<alpha> (Some c1') m'"
isomorphic_trans: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and W :: "'a set"   and smult'' :: "('f, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set" assumes "isomorphic smult' W"   and "\<exists>T. FGModuleIso G smult' W smult'' T X" shows "isomorphic smult'' X"
H\<^sub>R_preserved_along_iso: fixes a :: "'a"   and a' :: "'a" assumes "weak_unit a"   and "a \<cong> a'" shows "endofunctor (local.Right a) (H\<^sub>R a')"
keys_frag_of: fixes a :: "'a" shows "keys (monomial 1 a) = {a}"
ozero_ordLeq: fixes r :: "('d \<times> 'd) set" assumes "Well_order r" shows "ozero \<le>o r"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list"   and a2 :: "char list" assumes "subclsRp P a1 a2" obtains C :: "char list" and Bs :: "base list" and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list" and D :: "char list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(Bs, rest)\<rfloor>" and "Repeats D \<in> set Bs"
b'_im: fixes i :: "'a" assumes "i \<in> Basis" shows "b' i < n"
first_countable_topology_transfer_h: fixes B :: "'c \<Rightarrow> 'd set"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and \<tau> :: "'d set \<Rightarrow> bool" shows "(\<forall>i. B i \<subseteq> Collect (Domainp A) \<and> x \<in> B i \<and> \<tau> (B i)) = (B ` Collect top \<subseteq> {Aaa. Aaa \<subseteq> Collect (Domainp A)} \<and> (\<forall>i. x \<in> B i \<and> \<tau> (B i)))"
dims1_def: shows "b.dims1 = nths dims (vars1 \<union> vars2)"
N_entails_B: shows "no_labels.\<G>_Fset N \<Turnstile>G no_labels.\<G>_Fset {B}"
in_joinE: fixes v :: "'b option list"   and X :: "'b option list set"   and b :: "bool"   and Y :: "'b option list set"   and n :: "nat"   and A :: "nat set"   and B :: "nat set" assumes "v \<in> join X b Y"   and "table n A X"   and "table n B Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A" obtains "wf_tuple n (A \<union> B) v" and "restrict A v \<in> X" and "if b then restrict B v \<in> Y else restrict B v \<notin> Y"
append_butlast_last_id: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
map_of_map_transform: fixes t :: "'a \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'a\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var" shows "map_of (map_transform t ae \<Gamma>) x = map_option (lift_transform t (ae x)) (map_of \<Gamma> x)"
times_mat_def: fixes A :: "'b mat"   and B :: "'b mat" shows "A * B \<equiv> Matrix.mat (dim_row A) (dim_col B) (\<lambda>(i, j). Matrix.row A i \<bullet> col B j)"
jumpF_cong: fixes F :: "real filter"   and G :: "real filter"   and f :: "real \<Rightarrow> real"   and g :: "real \<Rightarrow> real" assumes "F = G"   and "\<forall>\<^sub>F x in F. f x = g x" shows "jumpF f F = jumpF g G"
simps_1: fixes p :: "fm"   and q :: "fm" shows "\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)"
cball_eq_sing: fixes x :: "'b"   and e :: "real" shows "(cball x e = {x}) = (e = 0)"
is_list_UNIV_def: fixes xs :: "'a list" shows "is_list_UNIV xs = (let c = CARD('a) in if c = 0 then False else length (remdups xs) = c)"
has_subprob_density_imp_has_density: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "has_subprob_density M N f" shows "has_density M N f"
config'_embed: fixes A :: "('a list \<Rightarrow> 'b) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> (nat \<times> nat list) \<times> 'b)"   and s0 :: "'a list \<times> 'b"   and qs :: "'a list" shows "Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed A) (return_pmf s0) qs = return_pmf (Partial_Cost_Model.config' A s0 qs)"
2: shows "fmdom' (snd a) \<inter> (prob_dom PROB - vs) = fmdom' (snd a) - fmdom' (snd a) \<inter> vs"
asum_mem: fixes a :: "'a"   and n :: "nat" assumes "a \<in> carrier A" shows "aSum A n a \<in> carrier A"
funpow_0: fixes f :: "'k \<Rightarrow> 'k"   and x :: "'k" shows "(f ^^ 0) x = x"
dom_subid_aux1'': fixes x :: "'a"   and y :: "'a" shows "ad (ad x) \<cdot> y \<le> y"
b: shows "\<forall>w. 0 \<le> (\<Sum>i. X w * indpre i w)"
subcls1I: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and D :: "char list"   and rest :: "(char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "P \<turnstile> C \<prec>\<^sup>1 D"
T_eq: fixes \<Gamma> :: "binding list"   and t :: "trm"   and T :: "POPLmark.type"   and T' :: "POPLmark.type" assumes "\<Gamma> \<turnstile> t : T"   and "T = T'" shows "\<Gamma> \<turnstile> t : T'"
assms_2: fixes k :: "'a" assumes "k \<in> K - {\<zero>}" shows "inv k \<in> K"
image_subsetI: fixes A :: "'h set"   and f :: "'h \<Rightarrow> 'i"   and B :: "'i set" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "f ` A \<subseteq> B"
delete_point_strong_block_in_orig: fixes bl :: "'a set"   and p :: "'a" assumes "bl \<in># str_del_point_blocks p" shows "bl \<in># \<B>"
graph_homomorphismI: fixes s :: "('b, 'a) labeled_graph"   and h :: "('a \<times> 'c) set"   and t :: "('b, 'c) labeled_graph" assumes "vertices s = Domain h"   and "h `` vertices s \<subseteq> vertices t"   and "univalent h"   and "edge_preserving h (edges s) (edges t)"   and "graph s"   and "graph t" shows "graph_homomorphism s t h"
logically_equivalent_def: fixes P :: "('state, 'effect) L_state"   and Q :: "('state, 'effect) L_state" shows "L_transform.logically_equivalent P Q \<equiv> \<forall>x. L_transform.valid P x = L_transform.valid Q x"
lr_of_tran_fbs_def: fixes rt :: "('a, 'b) routing_rule_scheme list"   and fw :: "'a simple_rule list"   and ifs :: "'c" shows "lr_of_tran_fbs rt fw ifs \<equiv> let gfw = map simple_rule_dtor fw; frt = lr_of_tran_s1 rt; prd = generalized_fw_join frt gfw in prd"
emh: shows "emh \<in> mheads G accC statT sig"
triple_def: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" shows "triple phi \<equiv> bounded_lattice_homomorphism phi"
possible_steps_can_take_transition: fixes s2 :: "nat"   and t1 :: "transition"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list" assumes "(s2, t1) |\<in>| possible_steps e1 s1 r l i" shows "can_take_transition t1 i r"
cases: fixes a1 :: "'a"   and a2 :: "'a multiset"   and a3 :: "'b el_step"   and a4 :: "'a"   and a5 :: "'a multiset"   and S :: "(('a \<times> 'a multiset) \<times> 'b \<times> 'a \<times> 'a multiset) set" assumes "((a1, a2), a3, a4, a5) \<in> gtrp S" obtains s :: "'a" and c :: "'a multiset" and e :: "'b" and s' :: "'a" and c' :: "'a multiset" where "a1 = s__" and "a2 = c__" and "a3 = LOC e__" and "a4 = s'__" and "a5 = c'__" and "((s__, c__), e__, s'__, c'__) \<in> S"   | s :: "'a" and sl :: "'a" and c :: "'a multiset" and e :: "'b" and s' :: "'a" and c' :: "'a multiset" where "a1 = sl" and "a2 = add_mset s c" and "a3 = ENV e" and "a4 = sl" and "a5 = add_mset s' c'" and "((s, add_mset sl c), e, s', add_mset sl c') \<in> S"
that: fixes m :: "'a set"   and p :: "'a set" assumes "m \<subseteq> c"   and "p \<subseteq> c"   and "m \<inter> p = {}"   and "convex hull m \<inter> convex hull p \<noteq> {}" shows "thesis"
simps_105: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x4 :: "val" shows "Call x101 x102 x103 x104 \<noteq> Val x4"
fun_eq_on_subset_and_diff_imp_eq_on: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "order.greater_eq B A"   and "fun_eq_on f g A"   and "fun_eq_on f g (B - A)" shows "fun_eq_on f g B"
nths_vars1': shows "nths st2.dims0 st2.vars1' = st2.dims1"
transform_filter_img_empty_rm: fixes G :: "'a set set"   and x :: "'a" assumes "\<And>g. g \<in> G \<Longrightarrow> g \<noteq> {}" shows "{g - {x} |g. g \<in> G \<and> g \<noteq> {x}} = {g - {x} |g. g \<in> G} - {{}}"
assms_2: shows "X \<in> obj\<^bsub>CatDom F\<^esub>"
Fun_mapsto: fixes f :: "'t" assumes "S'.arr f" shows "S'.Fun f \<in> extensional (S'.set (S'.dom f)) \<inter> (S'.set (S'.dom f) \<rightarrow> S'.set (S'.cod f))"
Port_Graph_Signature_Scoped_axioms: shows "Port_Graph_Signature_Scoped inPorts outPorts hyps"