hcju/isabelleps huggingface dataset:
\<psi>_in_hom: fixes x :: "'a"   and g :: "'b"   and y :: "'b" assumes "Adj.C.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>D G' x\<guillemotright>" shows "\<guillemotleft>\<psi>' x g : F y \<rightarrow>\<^sub>C x\<guillemotright>"
lookup'_def: fixes k :: "'a" shows "lookup' k \<equiv> State (\<lambda>heap. the (execute (lookup k) heap))"
add_closed: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "semigroup_add_ow S pls"   and "a \<in> S"   and "b \<in> S" shows "pls a b \<in> S"
not_final_thread_conv: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "r2.not_final_thread s t = (thr s t \<noteq> None \<and> \<not> r2.final_thread s t)"
abs_add_abs: fixes abs :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (plus (abs a) (abs b)) = plus (abs a) (abs b)"
lcons_inv: fixes t :: "('c, 'd) FingerTreeStruc"   and a :: "'c \<times> 'd" assumes "ft_invar t" shows "ft_invar (a \<lhd> t)"
sequence_to_failure_succ: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and io :: "('d \<times> 'e) list" assumes "sequence_to_failure M1 M2 io" shows "\<forall>q\<in>io_targets M2 (initial M2) (butlast io). succ M2 (last io) q = {}"
bcontfun_normI: fixes f :: "'c \<Rightarrow> 'd"   and b :: "real" assumes "continuous_on UNIV f"   and "\<And>x. norm (f x) \<le> b" shows "f \<in> bcontfun"
list_relE3: fixes x :: "'a"   and xs :: "'a list"   and l' :: "'b list"   and R :: "('a \<times> 'b) set" assumes "(x # xs, l') \<in> \<langle>R\<rangle>list_rel" obtains x' :: "'b" and xs' :: "'b list" where "l' = x' # xs'" and "(x, x') \<in> R" and "(xs, xs') \<in> \<langle>R\<rangle>list_rel"
U_arOf_def: shows "U_arOf \<equiv> length \<circ> GE_arOf"
convex_hulls_are_convex: fixes A :: "'a vec set" assumes "A \<subseteq> carrier_vec n" shows "convex (convex_hull A)"
add_right: fixes f :: "'a \<Rightarrow> real" assumes "\<And>x y. le x y \<Longrightarrow> f x \<le> f y" shows "vnm_utility carrier le (\<lambda>x. u x + f x)"
assms_3: shows "\<not> i cd\<^bsup>\<pi>\<^esup>\<rightarrow> k"
assms: fixes \<pi> :: "('state \<times> 'aprop set) stream" assumes "wfp AP' \<pi>"   and "\<pi>l \<noteq> [] \<Longrightarrow> stateOf \<pi> = stateOf (last \<pi>l)"   and "\<pi>l = [] \<Longrightarrow> stateOf \<pi> = s0" shows "F \<pi> \<pi>l"
assms_2: fixes fp :: "'x set" assumes "P ds fp" shows "P' ds fp"
bconf: shows "P,shp s \<turnstile>\<^sub>b (e,b) \<surd>"
inf_top_right: fixes x :: "'a" shows "x \<cdot> 1 = x"
dg_dghm_id_is_dghm: shows "dghm_id \<DD> : \<DD> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<DD>"
simps: fixes a :: "'b"   and fs :: "'b list"   and rs :: "'b list" shows "enq a (fs, rs) = norm (fs, a # rs)"
SEval_def2: fixes m :: "U set \<times> (nat \<Rightarrow> U list \<Rightarrow> bool)"   and e :: "nat \<Rightarrow> U"   and s :: "form list" shows "SEval m e s = (\<exists>f. f \<in> set s \<and> FEval m e f)"
base_red: fixes ns :: "nat list"   and n :: "nat"   and i :: "nat"   and b :: "nat"   and ms :: "nat list"   and m :: "nat"   and j :: "nat" assumes "\<And>n'. n' \<in> set ns \<Longrightarrow> n < n'"   and "i < b"   and "i \<noteq> 0"   and "\<And>m'. m' \<in> set ms \<Longrightarrow> m < m'"   and "j < b"   and "j \<noteq> 0"   and "i * b ^ n + sum_list (map ((^) b) ns) = j * b ^ m + sum_list (map ((^) b) ms)" shows "i = j \<and> n = m"
PROTECT2_def: fixes x :: "'c"   and y :: "prop" shows "PROTECT2 x PROP y \<equiv> x"
eq: shows "\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)"
mono: fixes c :: "'b multiset"   and a :: "'b multiset"   and d :: "'b multiset"   and b :: "'b multiset" assumes "c \<subseteq># a"   and "d \<subseteq># b" shows "c \<union># d \<subseteq># a \<union># b"
strengthen_pre: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat"   and c :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> enat" assumes "\<forall>s. P s \<le> P' s"   and "\<turnstile>\<^sub>2\<^sub>' {P} c {Q}" shows "\<turnstile>\<^sub>2\<^sub>' {P'} c {Q}"
star_def: fixes X :: "'a set" shows "X\<^sup>\<star> = \<Union> (range ((^) X))"
traverse_tree_ap: fixes path :: "dir list"   and f :: "('b \<Rightarrow> 'a) tree"   and x :: "'b tree" shows "traverse_tree path (f \<diamondop> x) = traverse_tree path f \<diamondop> traverse_tree path x"
valid_Conj: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and xset :: "('d, 'b, 'c) formula set['d]"   and P :: "'a" assumes "nominal_ts satisfies transition"   and "finite (supp xset)" shows "nominal_ts.valid satisfies transition P (Conj xset) = (\<forall>x\<in>set_bset xset. nominal_ts.valid satisfies transition P x)"
Un_0_Suc: fixes f :: "nat \<Rightarrow> 'a set" shows "\<Union> (range f) = f 0 \<union> (\<Union>n. f (Suc n))"
ax_do_not_a_reduce: fixes x :: "name"   and a :: "coname"   and M :: "trm" assumes "Ax x a \<longrightarrow>\<^sub>a M" shows "False"
cot_zero: shows "cot (0::'a) = (0::'a)"
le_less_trans: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<le> y"   and "y < z" shows "x < z"
seq_msg_ok: fixes i :: "nat" shows "ptoy i \<TTurnstile>\<^sub>A onll \<Gamma>\<^sub>T\<^sub>O\<^sub>Y (\<lambda>((\<xi>, uu_), a, uu_). anycast (\<lambda>m. case m of Pkt num' sid' \<Rightarrow> num' = no \<xi> \<and> sid' = i | Newpkt nat1 nat2 \<Rightarrow> True) a)"
intros_2: fixes f :: "int \<Rightarrow> int" assumes "Ex1p f" shows "Ex1p (upOne \<circ> f)"
tuple_decode_length: fixes n :: "nat"   and i :: "nat" shows "length (tuple_decode n i) = Suc n"
unit_circle_fix_moebius_add: fixes M1 :: "moebius"   and M2 :: "moebius" assumes "unit_circle_fix M1"   and "unit_circle_fix M2" shows "unit_circle_fix (M1 + M2)"
conv_radius_leI: fixes f :: "nat \<Rightarrow> 'b"   and z :: "'b" assumes "\<not> summable (\<lambda>n. norm (f n * z ^ n))" shows "conv_radius f \<le> ereal (norm z)"
neville_aitken_def: shows "local.neville_aitken = local.neville_aitken_main 0"
nth_append_length: fixes xs :: "'c list"   and x :: "'c"   and ys :: "'c list" shows "(xs @ x # ys) ! length xs = x"
newton_poly_def: fixes xs :: "'b list"   and fs :: "'b list"   and n :: "nat" shows "newton_poly xs fs n = b xs fs 0 n"
test: fixes P :: "ccs"   and P' :: "ccs" assumes "P \<Longrightarrow>\<^sub>\<tau> P'" shows "P = P' \<or> (\<exists>P''. P \<longmapsto>\<tau> \<prec> P'' \<and> P'' \<Longrightarrow>\<^sub>\<tau> P')"
join_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<squnion> y \<in> carrier L"
absorb_iff1: fixes b :: "'c DBMEntry"   and a :: "'c DBMEntry" shows "(b \<preceq> a) = (linordered_monoid.max a b = a)"
wlsInp_imp_goodInp: fixes delta :: "'opSym"   and inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set option" assumes "wlsInp delta inp" shows "goodInp inp"
round: shows "168 \<le> roundup (length (nat_to_bv (p * q)) - Suc 0) 8 * 8"
simps_5: fixes M :: "('b, 'c) kripke"   and s :: "'c"   and p :: "'b fm"   and q :: "'b fm" shows "(M, s \<Turnstile> p \<^bold>\<longrightarrow> q) = (M, s \<Turnstile> p \<longrightarrow> M, s \<Turnstile> q)"
f: shows "f \<in> restrict_space M A \<rightarrow>\<^sub>M subprob_algebra N"
spmf_geometric_ge_1: fixes p :: "real" assumes "1 \<le> p" shows "geometric_spmf p = return_spmf 0"
isTrue_PPf_Implies_bprv_PPf: fixes "prf" :: "'proof"   and \<phi> :: "'fmla" assumes "prf \<in> proof"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}"   and "isTrue (PPf (encPf prf) \<langle>\<phi>\<rangle>)" shows "bprv (PPf (encPf prf) \<langle>\<phi>\<rangle>)"
try_spmf_def: fixes p :: "'a spmf"   and q :: "'a spmf" shows "TRY p ELSE q = p \<bind> (\<lambda>x. case x of None \<Rightarrow> q | Some y \<Rightarrow> return_spmf y)"
comm_ring_ow_def: fixes U :: "'c set"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "comm_ring_ow U plus zero minus uminus times \<equiv> comm_semiring_ow U plus times \<and> ab_group_add_ow U plus zero minus uminus"
eSubstAbs_nchotomy: fixes eY :: "'a withERR"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'a, 'b) model"   and ys :: "'varSort"   and eA :: "'b withERR"   and y :: "'var" shows "(\<exists>Y. eY = OK Y \<and> igWls MOD (asSort ys) Y) \<and> (\<exists>us s A. eA = OK A \<and> igWlsAbs MOD (us, s) A) \<or> eSubstAbs MOD ys eY y eA = ERR"
sas_plus_equivalent_to_strips_i_a_IV: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and ops' :: "('c \<times> 'd) strips_operator list"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) ops' \<and> STRIPS_Semantics.are_all_operator_effects_consistent ops'" shows "are_all_operators_applicable_in (\<phi>\<^sub>S\<inverse> \<Psi> \<phi>\<^sub>S \<Psi> s) (map (strips_op_to_sasp \<Psi>) ops') \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent (map (strips_op_to_sasp \<Psi>) ops')"
subst_eq_def: fixes s1 :: "('b \<times> 'b trm) list"   and s2 :: "('b \<times> 'b trm) list" shows "s1 \<doteq> s2 = (\<forall>t. t \<lhd> s1 = t \<lhd> s2)"
smult_monom: fixes c :: "'b"   and x :: "nat \<Rightarrow>\<^sub>0 nat"   and c' :: "'b" shows "smult c (monom x c') = monom x (c * c')"
lepoll_Pow_self: fixes A :: "'a set" shows "A \<lesssim> Pow A"
splitAt_simpA: fixes ram :: "'a"   and b :: "'a list" shows "fst (splitAt ram (ram # b)) = []"
comp_PRE_def: fixes R :: "('b \<times> 'a) set"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'a \<Rightarrow> bool" shows "comp_PRE R P Q S \<equiv> \<lambda>x. S x \<longrightarrow> P x \<and> (\<forall>y. (y, x) \<in> R \<longrightarrow> Q x y)"
power_not_zero: fixes a :: "'c"   and n :: "nat" assumes "a \<noteq> (0::'c)" shows "a ^ n \<noteq> (0::'c)"
App_base_def: fixes K :: "('c, 'd) carrier_scheme"   and vv :: "nat \<Rightarrow> 'c \<Rightarrow> ant"   and n :: "nat" shows "App_base K vv n = (\<lambda>j\<in>{h. h \<le> n}. SOME x. x \<in> carrier K \<and> (vv \<circ> \<tau>\<^bsub>0 j\<^esub>) 0 x = 1 \<and> (\<forall>k\<in>nset (Suc 0) n. (vv \<circ> \<tau>\<^bsub>0 j\<^esub>) k x = 0))"
one_integer_def: shows "1 \<equiv> integer_of_int 1"
inverse_distrib_swap: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and inverse :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "group f z inverse" shows "inverse (f a b) = f (inverse b) (inverse a)"
prj0_simps_1: fixes f :: "('a, 'a set) Maps.arr"   and g :: "('a, 'a set) Maps.arr" assumes "Maps.cospan f g" shows "Maps.arr (Maps.PRJ\<^sub>0 f g)"
comp_subst_terms: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<theta> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and t :: "'b trm" assumes "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>" shows "t \<lhd> \<sigma> = t \<lhd> \<theta> \<lhd> \<eta>"
Re_Arccos_bound: fixes z :: "complex" shows "\<bar>Re (Arccos z)\<bar> \<le> pi"
not_in_dom: fixes i :: "'agent"   and x :: "'alt pmf"   and y :: "'alt pmf" assumes "i \<notin> agents" shows "\<not> (SD \<circ> R) i x y"
f: fixes z :: "'a" assumes "z \<in> S" shows "f z \<noteq> 0"
dinit_def: shows "dinit \<equiv> SPEC (\<lambda>(wl, res). wl = V \<and> res v0 = Some [] \<and> (\<forall>v\<in>V - {v0}. res v = None))"
assms: shows "P \<longmapsto>\<^sub>l a<\<nu>x> \<prec>\<^sub>l P'"
inversions_subset: fixes xs :: "'a list" shows "inversions xs \<subseteq> (SIGMA i:{..<length xs}. {i<..<length xs})"
inj_map_component_plus: fixes n :: "nat" shows "inj (map_component (\<lambda>k. k + n))"
col_1: fixes j :: "nat" assumes "j < nc1" shows "col (four_block_mat A2 B2 C2 D2) j = col A2 j @\<^sub>v col C2 j"
NatTransP_def: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "NatTransP \<eta> \<equiv> (Functor (NTDom \<eta>) \<and> Functor (NTCod \<eta>) \<and> NTCatDom \<eta> = CatDom (NTCod \<eta>)) \<and> NTCatCod \<eta> = CatCod (NTDom \<eta>) \<and> (\<forall>X. X \<in> obj\<^bsub>NTCatDom \<eta>\<^esub> \<longrightarrow> \<eta> $$ X maps\<^bsub>NTCatCod \<eta>\<^esub> NTDom \<eta> @@ X to NTCod \<eta> @@ X) \<and> (\<forall>f X Y. f maps\<^bsub>NTCatDom \<eta>\<^esub> X to Y \<longrightarrow> NTDom \<eta> ## f ;;\<^bsub>NTCatCod \<eta>\<^esub> (\<eta> $$ Y) = \<eta> $$ X ;;\<^bsub>NTCatCod \<eta>\<^esub> (NTCod \<eta> ## f))"
c_traces: fixes xps :: "('a \<times> 'b) list"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c" shows "(xps \<in> traces (c_process step out s\<^sub>0)) = (\<exists>X. (xps, X) \<in> c_failures step out s\<^sub>0)"
L2_set_constant: fixes y :: "real"   and A :: "'a set" shows "L2_set (\<lambda>x. y) A = sqrt (real (card A)) * \<bar>y\<bar>"
Int_ac_3: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> B = B \<inter> A"
tendsto_divide: fixes f :: "'e \<Rightarrow> 'f"   and a :: "'f"   and F :: "'e filter"   and g :: "'e \<Rightarrow> 'f"   and b :: "'f" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "b \<noteq> (0::'f)" shows "((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
deg_rep_nat_pp: fixes t :: "('a, 'b) pp" shows "deg_pp (rep_nat_pp t) = rep_nat (deg_pp t)"
in_hom_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and a :: "'d"   and b :: "'d" assumes "partial_magma C" shows "partial_magma.in_hom C f a b \<equiv> a \<in> partial_magma.domains C f \<and> b \<in> partial_magma.codomains C f"
monadic_nfoldli_assert_aux: fixes l :: "'c list"   and S :: "'c set"   and c :: "'d \<Rightarrow> bool nres"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and s :: "'d" assumes "set l \<subseteq> S" shows "monadic_nfoldli l c (\<lambda>x s. ASSERT (x \<in> S) \<bind> (\<lambda>_. f x s)) s = monadic_nfoldli l c f s"
pred_subset_eq: fixes R :: "'a set"   and S :: "'a set" shows "((\<lambda>x. x \<in> R) \<le> (\<lambda>x. x \<in> S)) = (R \<subseteq> S)"
that: fixes T :: "'a set" assumes "open T"   and "closedin (top_of_set T) S" shows "thesis"
silent_moves_intra_path_no_obs: fixes m :: "'node"   and S :: "'node SDG_node set"   and m' :: "'node"   and s :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and msx' :: "'node list" assumes "obs_intra m \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"   and "method_exit m'"   and "get_proc m = get_proc m'"   and "valid_node m"   and "length s = length (m # msx')"   and "Ball (set msx') return_node" obtains as :: "'edge list" where "S,slice_kind S \<turnstile> (m # msx',s) =as\<Rightarrow>\<^sub>\<tau> (m' # msx',s)"
shiftpath_alt_def: fixes a :: "real"   and f :: "real \<Rightarrow> 'a" shows "shiftpath a f = (\<lambda>x. if x \<le> 1 - a then f (a + x) else f (a + x - 1))"
Darii: fixes A :: "'b set"   and B :: "'b set"   and C :: "'b set" assumes "A Q B"   and "B I C" shows "A I C"
normalize_dst_ports_def: fixes m :: "'b common_primitive match_expr" shows "normalize_dst_ports m = normalize_ports_generic normalize_positive_dst_ports rewrite_negated_dst_ports m"
card_partitions_of: fixes F :: "('c \<Rightarrow> 'd) set"   and A :: "'c set"   and B :: "'d set" assumes "F \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B"   and "univ (\<lambda>f. f ` A = B) F" shows "card (partitions_of A B F) = card B"
eucl_of_list_if: fixes xs :: "nat list"   and x :: "nat" assumes "length xs = DIM('a)"   and "distinct xs" shows "eucl_of_list (map (\<lambda>xa. if xa = x then 1 else 0) xs) = (if x \<in> set xs then Basis_list ! index xs x else (0::'a))"
iso_runit: fixes f :: "'b" assumes "D.ide f" shows "D.iso \<r>\<^sub>D[f]"
D_mono: fixes A :: "char list set"   and c :: "com"   and M :: "char list set"   and A' :: "char list set" assumes "D A c M"   and "A \<subseteq> A'" shows "\<exists>M'. D A' c M' \<and> M \<subseteq> M'"
generator: shows "carrier (G\<lparr>carrier := generate G {a}\<rparr>) = generate (G\<lparr>carrier := generate G {a}\<rparr>) {a}"
nodup_all_def: fixes l :: "'a list" shows "nodup_all l \<equiv> \<forall>x\<in>set l. nodup x l"
assms_3: shows "vec_mat_Tensor v (y # ys) = x # xs"
subset_mset_def: fixes A :: "'b multiset"   and B :: "'b multiset" shows "(A \<subset># B) = (A \<subseteq># B \<and> A \<noteq> B)"
lhd_conv_lnth: fixes xs :: "'c llist" assumes "\<not> lnull xs" shows "lhd xs = lnth xs 0"
igWlsBinp_termMOD_wlsBinp: fixes delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set option" shows "igWlsBinp termMOD delta binp = wlsBinp delta binp"
whileLoopE_unroll: fixes C :: "'e \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'c \<Rightarrow> (('d + 'e) \<times> 'c) set \<times> bool"   and r :: "'e" shows "whileLoopE C   B r = condition (C r)   (B r >>=E whileLoopE C   B)   (returnOk r)"
deg_pm_sum: fixes t :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'b"   and A :: "'d set" shows "deg_pm (sum t A) = (\<Sum>a\<in>A. deg_pm (t a))"
residual_galoisI2: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<le> residual f y" shows "f x \<le> y"
le_add_same_cancel2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
rev_subst_var: fixes e :: "exp"   and y :: "nat"   and v :: "exp"   and e' :: "exp"   and v' :: "val"   and \<rho> :: "(nat \<times> val) list" assumes "e = EVar y \<and> v = e'"   and "is_val v"   and "v' \<in> E e' \<rho>" shows "\<exists>\<rho>' v''. v'' \<in> E v [] \<and> v' \<in> E e \<rho>' \<and> \<rho>' \<approx> (y, v'') # \<rho>"
e_1: fixes x :: "float" assumes "x \<in> U" shows "0 < e x"
simps_14: fixes b :: "expr"   and e :: "expr" shows "fv (while (b) e) = fv b \<union> fv e"
return_count_space_eq_density: fixes M :: "'a set"   and x :: "'a" shows "return (count_space M) x = density (count_space M) (indicator {x})"
word_prev_unfold: fixes w :: "'a word" shows "word_prev w = (if w = 0 then 0 else w - 1)"
nterm_to_term_eq: fixes u :: "nterm"   and \<Gamma> :: "name list"   and \<Gamma>' :: "name list" assumes "frees u |\<subseteq>| fset_of_list (common_prefix \<Gamma> \<Gamma>')" shows "nterm_to_term \<Gamma> u = nterm_to_term \<Gamma>' u"
fv2: shows "fv exp2 \<subseteq> set (Vs @ [V])"
supr_Above: fixes B :: "'a set" assumes "B \<subseteq> Field r"   and "local.Above B \<noteq> {}" shows "supr B \<in> local.Above B"
simps: fixes a1 :: "(name \<times> ty) list"   and a2 :: "trm"   and a3 :: "ty" shows "a1 \<turnstile> a2 : a3 = ((\<exists>\<Gamma> x T. a1 = \<Gamma> \<and> a2 = Var x \<and> a3 = T \<and> valid \<Gamma> \<and> (x, T) \<in> set \<Gamma>) \<or> (\<exists>\<Gamma> e\<^sub>1 T\<^sub>1 T\<^sub>2 e\<^sub>2. a1 = \<Gamma> \<and> a2 = App e\<^sub>1 e\<^sub>2 \<and> a3 = T\<^sub>2 \<and> \<Gamma> \<turnstile> e\<^sub>1 : T\<^sub>1\<rightarrow>T\<^sub>2 \<and> \<Gamma> \<turnstile> e\<^sub>2 : T\<^sub>1) \<or> (\<exists>x \<Gamma> T\<^sub>1 e T\<^sub>2. a1 = \<Gamma> \<and> a2 = Lam [x].e \<and> a3 = T\<^sub>1\<rightarrow>T\<^sub>2 \<and> x \<sharp> \<Gamma> \<and> (x, T\<^sub>1) # \<Gamma> \<turnstile> e : T\<^sub>2))"
H: shows "H \<equiv> {f \<turnstile> folding_g.\<C>, g \<turnstile> folding_g.\<C>}"
swap_swap: fixes p :: "'c \<times> 'd" shows "prod.swap (prod.swap p) = p"
map_simp: fixes f :: "'d \<times> 'd \<times> 'd" assumes "D.CCC.arr f" shows "D.\<alpha>' f = D.inv (\<alpha>\<^sub>D (D.CCC.cod f)) \<cdot>\<^sub>D D.T.ToCT f"
obtain_sorted_carrier_def: shows "obtain_sorted_carrier \<equiv> SPEC (\<lambda>L. wsorted L \<and> set L = carrier)"
normalized_Cons: fixes x :: "('c, 'd) bintree option"   and xs :: "('c, 'd) bintree option list" assumes "normalized (x # xs)" shows "normalized xs"
mult_inverse_of_nat_commute: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and xa :: "nat"   and x :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "times (inverse (semiring_1.of_nat one plus zero xa)) x = times x (inverse (semiring_1.of_nat one plus zero xa))"
pre_bfs_correct: fixes src :: "nat"   and dst :: "nat" assumes "src \<in> V"   and "src \<noteq> dst"   and "finite V" shows "pre_bfs src dst \<le> SPEC (bfs_spec src dst)"
finsertCI: fixes a :: "'a"   and B :: "'a fset"   and b :: "'a" assumes "a |\<notin>| B \<Longrightarrow> a = b" shows "a |\<in>| finsert b B"
convex_lincomb_list_def: fixes c :: "nat \<Rightarrow> 'a"   and Vs :: "'a vec list"   and b :: "'a vec" shows "m.convex_lincomb_list c Vs b = (m.nonneg_lincomb_list c Vs b \<and> sum c {0..<length Vs} = (1::'a))"
other_localD: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and i :: "'b"   and \<sigma> :: "'b \<Rightarrow> 'a"   and \<sigma>' :: "'b \<Rightarrow> 'a" assumes "other P {i} \<sigma> \<sigma>'" shows "\<sigma>' i = \<sigma> i"
assms: fixes i :: "nat" assumes "i < 2" shows "\<exists>j. suffix j w \<Turnstile>\<^sub>n af (if i = 0 then \<phi> else \<psi>) (prefix j w)[X]\<^sub>\<nu>"
simps: fixes x :: "nat"   and y :: "nat"   and u :: "nat"   and v :: "nat" shows "((x, y) \<approx> (u, v)) = (x + v = u + y)"
linear_simps_2: fixes f :: "'c \<Rightarrow> 'd"   and a :: "'c"   and b :: "'c" assumes "bounded_linear f" shows "f (a - b) = f a - f b"
density_context_equiv: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and \<delta>' :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "\<And>\<sigma>. \<sigma> \<in> space (state_measure (V \<union> V') \<Gamma>) \<Longrightarrow> \<delta> \<sigma> = \<delta>' \<sigma>"   and "\<delta>' \<in> borel_measurable (state_measure (V \<union> V') \<Gamma>)"   and "density_context V V' \<Gamma> \<delta>" shows "density_context V V' \<Gamma> \<delta>'"
terminal_path_is_path: fixes v :: "'form \<times> nat list"   and v' :: "'form \<times> nat list"   and pth :: "((('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent) list" assumes "terminal_path v v' pth" shows "path v v' pth"
hyps: shows "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
dvd_choose_prime: fixes k :: "nat"   and n :: "nat" assumes "k < n"   and "k \<noteq> 0"   and "n \<noteq> 0"   and "prime n" shows "n dvd n choose k"
prems_1: shows "(WHILE b DO c, t1) \<Rightarrow> t3"
two_reduced_heads_imp_nzero_relfun: fixes s :: "'w"   and t :: "'w"   and w :: "'w"   and as :: "'w list"   and bs :: "'w list" assumes "s \<noteq> t"   and "S_reduced_for w (s # as)"   and "S_reduced_for w (t # bs)" shows "add_order (s + t) \<noteq> 0"
retractionE: fixes g :: "'a arrow_of_spans_data" assumes "V.retraction g" obtains f :: "'a arrow_of_spans_data" where "V.ide (g \<bullet> f)"