hcju/isabelleps huggingface dataset:
inf_mono: fixes a :: "'b \<times> 'c"   and c :: "'b \<times> 'c"   and b :: "'b \<times> 'c"   and d :: "'b \<times> 'c" assumes "less_eq_prod a c"   and "less_eq_prod b d" shows "less_eq_prod (inf_prod a b) (inf_prod c d)"
akra_bazzi_lower_def: fixes x\<^sub>0 :: "nat"   and x\<^sub>1 :: "nat"   and k :: "nat"   and as :: "real list"   and bs :: "real list"   and ts :: "(nat \<Rightarrow> nat) list"   and f :: "nat \<Rightarrow> real"   and integrable :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> bool"   and integral :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and g' :: "real \<Rightarrow> real" shows "akra_bazzi_lower x\<^sub>0 x\<^sub>1 k as bs ts f integrable integral g g' \<equiv> akra_bazzi_function x\<^sub>0 x\<^sub>1 k as bs ts f integrable integral g \<and> akra_bazzi_lower_axioms as bs f integrable g g'"
M_in: shows "M \<in># hmsetmset (local.encode e n)"
sup_Sup_var: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "sup_prod x y = Sup_prod {z. z \<in> {x, y}}"
from_hma\<^sub>v_add: fixes v :: "('d, 'e) Finite_Cartesian_Product.vec"   and w :: "('d, 'e) Finite_Cartesian_Product.vec" shows "Mod_Type_Connect.from_hma\<^sub>v v + Mod_Type_Connect.from_hma\<^sub>v w = Mod_Type_Connect.from_hma\<^sub>v (v + w)"
simps_3: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "incr (x # y # xs) = (x < y \<and> incr (y # xs))"
5: shows "infs (degen cs) (v ||| l' ## sscan (count cs) v l')"
intro: fixes r :: "('b \<times> 'b) set" assumes "Well_order r" shows "wo_rel r"
seq_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "CCC.seq g f = (seq (fst g) (fst f) \<and> CC.seq (snd g) (snd f))"
icd_path_shift: fixes l :: "nat"   and k :: "nat"   and \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "l \<le> k"   and "is_path \<pi>" shows "(i icd\<^bsup>\<pi>\<^esup>\<rightarrow> k) = (i - l icd\<^bsup>\<pi> \<guillemotleft> l\<^esup>\<rightarrow> k - l)"
Vagree_def: fixes \<nu> :: "(real, 'f) vec \<times> (real, 'f) vec"   and \<nu>' :: "(real, 'f) vec \<times> (real, 'f) vec"   and V :: "('f + 'f) set" shows "Vagree \<nu> \<nu>' V \<equiv> (\<forall>i. Inl i \<in> V \<longrightarrow> fst \<nu> $ i = fst \<nu>' $ i) \<and> (\<forall>i. Inr i \<in> V \<longrightarrow> snd \<nu> $ i = snd \<nu>' $ i)"
emeasure_eq_1_AE: fixes S :: "'s stream set"   and s :: "'s" assumes "S \<in> T.events s"   and "AE x in T s. x \<in> S" shows "emeasure (T s) S = 1"
DirProds_def: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set" shows "DirProds G I = \<lparr>carrier = Pi\<^sub>E I (carrier \<circ> G), monoid.mult = \<lambda>x y. \<lambda>i\<in>I. x i \<otimes>\<^bsub>G i\<^esub> y i, one = \<lambda>i\<in>I. \<one>\<^bsub>G i\<^esub>\<rparr>"
cases: fixes x :: "'fsym trm" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl"
callee_invariant_on_def: fixes callee :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and I :: "'d \<Rightarrow> bool"   and \<I> :: "('e, 'f) \<I>" shows "callee_invariant_on callee I \<I> \<equiv> (\<forall>s x y s'. (y, s') \<in> set_spmf (callee s x) \<longrightarrow> I s \<longrightarrow> x \<in> outs_\<I> \<I> \<longrightarrow> I s') \<and> (\<forall>s. I s \<longrightarrow> \<I> \<turnstile>c callee s \<surd>)"
unique_factorization_m_cong: fixes f :: "int poly"   and cfs :: "int \<times> int poly multiset"   and g :: "int poly" assumes "p.unique_factorization_m f cfs"   and "p.eq_m f g" shows "p.unique_factorization_m g cfs"
locally_similar_def: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g" shows "locally_similar \<sigma> \<tau> \<equiv> \<forall>p. (\<lambda>n. \<sigma> n p) \<approx> (\<lambda>n. \<tau> n p)"
lappend_code_2: fixes xa :: "'c"   and x :: "'c llist"   and ys :: "'c llist" shows "lappend (LCons xa x) ys = LCons xa (lappend x ys)"
Ord_pred: fixes k :: "hf" assumes "Ord k"   and "k \<noteq> 0" shows "succ (\<Squnion>k) = k"
fin_g: shows "finite {b. g b \<noteq> (0::'b)}"
elims: fixes x :: "('fun, 'var) extstrand_step list"   and y :: "('fun, 'var) strand_step list" assumes "to_st x = y" obtains "xa__ = []" and "y = []"   | x :: "('fun, 'var) strand_step" and S :: "('fun, 'var) extstrand_step list" where "xa__ = Step x # S__" and "y = x # to_st S__"   | t :: "('fun, 'var) Term.term" and S :: "('fun, 'var) extstrand_step list" where "xa__ = Decomp t # S" and "y = decomp t @ to_st S"
simps_1: fixes x1 :: "bool"   and x2 :: "bool"   and x3 :: "bool"   and x4 :: "bool"   and x5 :: "bool"   and x6 :: "bool"   and x7 :: "bool"   and x8 :: "bool"   and y1 :: "bool"   and y2 :: "bool"   and y3 :: "bool"   and y4 :: "bool"   and y5 :: "bool"   and y6 :: "bool"   and y7 :: "bool"   and y8 :: "bool" shows "(Char x1 x2 x3 x4 x5 x6 x7 x8 = Char y1 y2 y3 y4 y5 y6 y7 y8) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4 \<and> x5 = y5 \<and> x6 = y6 \<and> x7 = y7 \<and> x8 = y8)"
inf_commute: fixes x :: "'d multiset"   and y :: "'d multiset" shows "x \<inter># y = y \<inter># x"
wt_cont_eff_Par_2: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and i :: "nat"   and s :: "'state" assumes "properL cl"   and "n < length cl"   and "i < brn (cl ! n)" shows "local.cont (Par cl) s (brnL cl n + i) = Par (cl[n := local.cont (cl ! n) s i])"
add_mono1: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero"   and "less a b" shows "less (plus a one) (plus b one)"
winning_region_extends_VVp: fixes v :: "'a"   and p :: "Player"   and w :: "'a" assumes "v \<in> VV p"   and "v \<rightarrow> w"   and "w \<in> winning_region p" shows "v \<in> winning_region p"
basic_monos_1: fixes A :: "'a set" shows "A \<subseteq> A"
exp_series_stream_def: shows "exp_series_stream = exp_series_stream_aux 1 1"
f_neg: shows "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>join M \<noteq> \<infinity>"
lb_impl_job_sum: fixes m :: "nat"   and t :: "nat \<Rightarrow> nat"   and T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" assumes "LoadBalancing m"   and "LoadBalancing.lb t m T A j" shows "sum T {1..m} = sum t {1..j}"
counit_in_hom_2: shows "\<guillemotleft>counit a : \<tau>\<^sub>0 a \<star>\<^sub>D map\<^sub>0 a \<Rightarrow>\<^sub>D trg\<^sub>D (\<tau>\<^sub>0 a)\<guillemotright>"
Quotient3_refl1: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Abs :: "'a \<Rightarrow> 'b"   and Rep :: "'b \<Rightarrow> 'a"   and r :: "'a"   and s :: "'a" assumes "Quotient3 R Abs Rep"   and "R r s" shows "R r r"
nth_map_upt: fixes i :: "nat"   and n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'g" assumes "i < n - m" shows "map f [m..<n] ! i = f (m + i)"
insertion_chop: fixes f :: "nat \<Rightarrow> real"   and p :: "real mpoly"   and v :: "nat" shows "insertion f p = (\<Sum>(term, i)\<leftarrow>chop (to_list v p). insertion f term * f v ^ i)"
bind_assoc: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'b \<Rightarrow> 'c measure"   and R :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "M \<bind> f \<bind> g = M \<bind> (\<lambda>x. f x \<bind> g)"
eq1: shows "ind_in_set A ` ({0..<M} \<inter> A) - ind_in_set A ` B = ind_in_set A ` ({0..<M} \<inter> A) - ind_in_set A ` ({0..<M} \<inter> B)"
nonzero_idempotent: fixes p :: "pexpr"   and ps :: "pexpr list"   and xs :: "'a list" assumes "p \<in> set ps" shows "(peval xs p \<noteq> \<zero> \<and> nonzero xs ps) = nonzero xs ps"
steps_c\<^sub>1: shows "\<Gamma> \<turnstile> (c\<^sub>1, Fault f) \<rightarrow>\<^sup>* (Skip, Fault f)"
i_Exec_Stream_Init_eq_i_Exec_Stream_Cons: fixes trans_fun :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'b"   and c :: "'a" shows "i_Exec_Comp_Stream_Init trans_fun input c = [c] \<frown> i_Exec_Comp_Stream trans_fun input c"
that_2: fixes u :: "'a" assumes "uv = Inl u"   and "uv' = Inr u"   and "u \<in> \<^bold>V"   and "u \<notin> A \<Gamma>"   and "u \<notin> B \<Gamma>" shows "thesis"
mic2mic: fixes i :: "nat"   and Y :: "nat \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> 'b" assumes "max_in_chain i Y" shows "max_in_chain i (\<lambda>i. f (Y i))"
nth_default_beyond: fixes xs :: "'b list"   and n :: "nat"   and dflt :: "'b" assumes "length xs \<le> n" shows "nth_default dflt xs n = dflt"
assms_2: fixes i :: "'a" assumes "i \<in> Basis" shows "continuous_on S (\<lambda>x. f' x i)"
assms_1: shows "SubnetS s t \<in> reachable (pnet np (p1 \<parallel> p2)) I"
replicate_mset_eq_empty_iff: fixes n :: "nat"   and a :: "'c" shows "(replicate_mset n a = {#}) = (n = 0)"
shift_aux_def: shows "shift_mmuaux args nt aux = (tp, tss, len, maskL, maskR, a1_map, a2_map, done, done_length)"
awalk_cost_def: fixes f :: "'b \<Rightarrow> real"   and es :: "'b list" shows "awalk_cost f es = sum_list (map f es)"
finalE: fixes a :: "('a, 'b, 'c) exp" assumes "final a" obtains v :: "'c val" where "a = Val v"   | aa :: "'c" where "a = Throw aa"
set_intros_2: fixes y :: "'e"   and x22 :: "'e list"   and x21 :: "'e" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
AL_update1: fixes L :: "('b \<times> 'a) list"   and a :: "'b"   and b :: "'a" shows "L[a\<mapsto>b]\<down>a = Some b"
"2_1": shows "u\<^sub>2 \<le>\<^sub>F\<^sub>I w"
order_class_def: shows "OFCLASS('a, order_class) \<equiv> (OFCLASS('a, preorder_class) &&& class.order_axioms (\<le>))"
rep_eq: fixes x :: "uint" shows "Rep_uint64 (uint64_of_uint x) = ucast (Rep_uint x)"
that: fixes I :: "'a \<Rightarrow>\<^sub>L 'a" assumes "I o\<^sub>L 1\<^sub>L + w = 1\<^sub>L"   and "1\<^sub>L + w o\<^sub>L I = 1\<^sub>L"   and "norm (I - 1\<^sub>L + w) \<le> (norm w)\<^sup>2 / (1 - norm w)" shows "thesis"
mat_plus_left_mono: fixes A :: "'b mat"   and B :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and C :: "'b mat" assumes "A \<ge>\<^sub>m B"   and "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "C \<in> carrier_mat nr nc" shows "A + C \<ge>\<^sub>m B + C"
premb: shows "b \<in> set (wordinterval_CIDR_split_prefixmatch rs)"
filter_prop: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool"   and ys :: "'a list" assumes "x \<in> set (filter P ys)" shows "P x"
U: fixes ss :: "'s list"   and f :: "'op" assumes "\<forall>s\<in>set ss. inv s"   and "length ss = arity f" shows "real (cost f ss) + \<Phi> (exec f ss) - sum_list (map \<Phi> ss) \<le> U f ss"
sign_r_pos_power: fixes a :: "real"   and n :: "nat" shows "sign_r_pos ([:- a, 1:] ^ n) a"
order_trans: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "z \<le> y" shows "z \<le> x"
a_less_i: shows "a < mod_type_class.from_nat i"
FAssRedsVal: fixes e\<^sub>1 :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs' :: "char list list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 E :: "char list \<Rightarrow> ty option"   and e\<^sub>2 :: "expr"   and v :: "val"   and h\<^sub>2 :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l\<^sub>2 :: "char list \<Rightarrow> val option"   and D :: "char list"   and S :: "(char list list \<times> (char list \<Rightarrow> val option)) set"   and F :: "char list"   and T :: "ty"   and Cs :: "char list list"   and v' :: "val"   and Ds :: "char list list"   and fs :: "char list \<Rightarrow> val option" assumes "P,E \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<rightarrow>* \<langle>ref (a, Cs'),s\<^sub>1\<rangle>"   and "P,E \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<rightarrow>* \<langle>Val v,(h\<^sub>2, l\<^sub>2)\<rangle>"   and "h\<^sub>2 a = \<lfloor>(D, S)\<rfloor>"   and "P \<turnstile> last Cs' has least F:T via Cs"   and "P \<turnstile> T casts v to v' "   and "Ds = Cs' @\<^sub>p Cs"   and "(Ds, fs) \<in> S" shows "P,E \<turnstile> \<langle>e\<^sub>1\<bullet>F{Cs} := e\<^sub>2,s\<^sub>0\<rangle> \<rightarrow>* \<langle>Val v',(h\<^sub>2(a \<mapsto> (D, insert (Ds, fs(F \<mapsto> v')) (S - {(Ds, fs)}))), l\<^sub>2)\<rangle>"
cobounded2: fixes a :: "'b multiset"   and b :: "'b multiset" shows "a \<inter># b \<subseteq># b"
frees_all_frees_term: fixes t :: "Term_Class.term" shows "all_frees t = frees t"
prod_ord_pair_eq_lex_ord_pair: fixes P :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "('a \<times> 'd) list"   and ys :: "('a \<times> 'e) list" shows "prod_ord_pair P xs ys = (lex_ord_pair (\<lambda>k x y. if P k x y then Some Eq else None) xs ys = Some Eq)"
focus_prod: fixes X :: "'a set"   and f :: "'c \<Rightarrow> ('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and I :: "'c set" shows "focus X (prod f I) = (\<Prod>i\<in>I. focus X (f i))"
rep_eq: fixes x :: "('g, 'e) generator"   and xa :: "('h, 'f) generator" shows "generator (zip_trans x xa) = zip_raw (generator x) (generator xa)"
assert_bound_tableau_valuated: fixes assert_bound :: "'a \<times> 'b atom \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state"   and a :: "'a \<times> 'b atom" assumes "AssertBound assert_bound"   and "\<not> \<U> s"   and "\<Turnstile> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<nabla> (assert_bound a s)"
ordsetmap_strict: fixes a :: "'a"   and b :: "'a" assumes "a \<in> P"   and "b \<in> P"   and "b \<succ> a" shows "codomain.greater (f b) (f a)"
lm02: fixes y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "y \<in> f ` A" shows "A \<inter> f -` {y} \<noteq> {}"
id: shows "fmlookup (linear_poly_map (p1 + p2)) x = (case fmlookup (linear_poly_map p1) x of None \<Rightarrow> fmlookup (linear_poly_map p2) x | Some y1 \<Rightarrow> case fmlookup (linear_poly_map p2) x of None \<Rightarrow> Some y1 | Some y2 \<Rightarrow> if y1 + y2 = 0 then None else Some (y1 + y2))"
tendsto_norm_zero_cancel: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> (0::'b)) F"
\<pi>\<rho>: shows "\<pi> =\<^bsub>n\<^esub> \<rho>"
divide_divide_eq_right: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and inverse :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a (divide b c) = divide (times a c) b"
rep_eq: fixes x :: "int" shows "Rep_proper_base (into_base x) = (if 2 \<le> x then x else 2)"
intro: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category C"   and "category J"   and "functor_category J C" shows "diagonal_functor J C"
filterlim_of_nat_at_top: fixes f :: "real \<Rightarrow> 'a"   and F :: "'a filter" assumes "filterlim f F at_top" shows "LIM x sequentially. f (real x) :> F"
Zorn_lemmaTr: fixes C :: "'a set"   and M :: "'a set"   and b :: "'a" assumes "Chain D C"   and "M \<in> carrier fTo D"   and "C \<subseteq> M"   and "maximal\<^bsub>fTo D\<^esub> M"   and "b \<in> carrier D"   and "\<forall>s\<in>M. s \<preceq> b" shows "maximal b \<and> b \<in> upper_bounds D C"
opp_half_apartment_subset_complex: shows "order.greater_eq X Y"
SEQ_def: fixes A :: "'b set" shows "SEQ A = {f. \<forall>i. f i \<in> A}"
lenext_compat_snoc: fixes ys :: "'b list"   and xs :: "'b list"   and gts :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b" assumes "\<lbrakk>length ys = length xs; gts ys xs\<rbrakk> \<Longrightarrow> gts (ys @ [x]) (xs @ [x])"   and "lenext gts ys xs" shows "lenext gts (ys @ [x]) (xs @ [x])"
oldest_token_monotonic_Suc: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x n) n = Some i"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w (semi_mojmir_def.token_run \<delta> q\<^sub>0 w x (Suc n)) (Suc n) = Some j" shows "j \<le> i"
less_assn_def: fixes a :: "assn"   and b :: "assn" shows "a < b \<equiv> a \<le> b \<and> a \<noteq> b"
sub: shows "(\<Union>x\<in>{x. Inl (Inr x) \<in> SIGF \<phi>}. FVT (\<sigma> x)) \<subseteq> - U"
pcompose_conv_poly: fixes p :: "'a poly"   and q :: "'a poly" shows "p \<circ>\<^sub>p q = poly (poly_lift p) q"
dvd: shows "q dvd (\<Prod>(a, i)\<in>set bs. a ^ Suc i)"
ag_pOp_add_l: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> carrier M"   and "b \<in> carrier M"   and "c \<in> carrier M"   and "a = b" shows "c \<plusminus> a = c \<plusminus> b"
max_bag_card_def: fixes G :: "('a, 'b) Graph_scheme"   and T :: "('c, 'd) Graph_scheme"   and bag :: "'c \<Rightarrow> 'a set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.max_bag_card T bag \<equiv> Max {card (bag t) |t. t \<in> V\<^bsub>T\<^esub>}"
Ks_2: shows "S.combine Ks (map h Us) = \<zero>\<^bsub>S\<^esub>"
vec_cong: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "n = m"   and "\<And>i. i < m \<Longrightarrow> f i = g i" shows "vec n f = vec m g"
l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set" shows "l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<equiv> l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms type_wf"
cexpr_sem_integral_nonneg: fixes t :: "pdf_type"   and \<sigma> :: "nat \<Rightarrow> val"   and e :: "cexpr"   and V :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type" assumes "\<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (case_nat x \<sigma>) e)) \<partial>stock_measure t < \<infinity>"   and "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"   and "case_nat t \<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> shift_var_set V"   and "\<sigma> \<in> space (state_measure V \<Gamma>)" shows "ennreal (extract_real (cexpr_sem \<sigma> (\<integral>\<^sub>c e \<partial>t))) = \<integral>\<^sup>+ x. ennreal (extract_real (cexpr_sem (case_nat x \<sigma>) e)) \<partial>stock_measure t"
A1: shows "c_fst (c_fst key) mod 7 = 6"
base: fixes \<sigma> :: "'state" shows "R' P \<sigma> \<sigma> collect_id"
continuous_on_eq_continuous_within: fixes s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on s f = (\<forall>x\<in>s. continuous (at x within s) f)"
eq_class_to_rel: fixes r :: "'a"   and s :: "'a"   and r' :: "'a"   and s' :: "'a" assumes "(r, s) \<in> carrier R \<times> S"   and "(r', s') \<in> carrier R \<times> S"   and "(r |\<^bsub>rel\<^esub> s) = (r' |\<^bsub>rel\<^esub> s')" shows "(r, s) .=\<^bsub>rel\<^esub> (r', s')"
insert_in_sets: fixes x :: "'b"   and M :: "'b measure"   and A :: "'b set" assumes "{x} \<in> null_sets M"   and "A \<in> null_sets M" shows "insert x A \<in> null_sets M"
rep_eq: fixes x :: "'b multiset"   and xa :: "'b multiset" shows "count (x + xa) = (\<lambda>a. count x a + count xa a)"
ID_restricted_store_subset_store_2: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and l :: "'a" shows "LID\<^sub>S (\<sigma>(l := None)) \<subseteq> LID\<^sub>S \<sigma>"
char_poly_upper_triangular: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "upper_triangular A" shows "char_poly A = (\<Prod>a\<leftarrow>diag_mat A. [:- a, 1::'b:])"
power_not_zero: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a"   and n :: "nat" assumes "class.semiring_1_no_zero_divisors one times plus zero"   and "a \<noteq> zero" shows "power.power one times a n \<noteq> zero"
POR_sym: fixes x :: "ValD"   and y :: "ValD" shows "(x por y) = (y por x)"
assert_bound_nolhs_atoms_imply_bounds_index: fixes s :: "('i, 'a) state"   and ats :: "('i \<times> 'a atom) set"   and a :: "'i \<times> 'a atom" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<triangle> (\<T> s)"   and "\<nabla> s"   and "\<diamond> s"   and "ats \<Turnstile>\<^sub>i \<B>\<I> s" shows "insert a ats \<Turnstile>\<^sub>i \<B>\<I> (assert_bound a s)"
hypnat_of_nat_zero_not_prime: shows "hypnat_of_nat 0 \<notin> starprime"
assms_5: shows "p\<^sup>+\<^sup>+ (x ! i) (x ! j)"
nat_term_compare_splus: fixes to :: "'b nat_term_order"   and t :: "'b"   and u :: "'b"   and v :: "'b" shows "nat_term_compare to (splus t u) (splus t v) = nat_term_compare to u v"
ex_terminal_arrow: fixes y :: "'c" assumes "C.ide y" shows "\<exists>x e. terminal_arrow_from_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F x y e"
psubstEnv_substEnv_compose: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho' :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and ys :: "'varSort"   and y :: "'var" assumes "goodEnv rho"   and "good Y"   and "goodEnv rho'" shows "rho &[Y / y]_ys &[rho'] = rho &[rho' [y \<leftarrow> Y #[rho']]_ys]"
g_imp_f: shows "\<forall>x\<in>{a..b}. g x = 0 \<longrightarrow> f x \<noteq> 0"
strong_induct: fixes x1 :: "lam"   and x2 :: "lam"   and P :: "'a \<Rightarrow> lam \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a" assumes "x1 \<rightarrow> x2"   and "\<And>a y. P y (Var a) (Var a)"   and "\<And>t1 t2 y. P y (App t1 t2) (App t1 t2)"   and "\<And>t t' x y. \<lbrakk>x \<sharp> y; t \<rightarrow> t'; \<And>z. P z t t'\<rbrakk> \<Longrightarrow> P y (Lam [x].t) t'" shows "P z x1 x2"
unique_decomposition: fixes R :: "('a, 'b) ring_scheme"   and K :: "'a set"   and Us :: "'a list"   and a :: "'a" assumes "ring R"   and "subfield K R"   and "ring.independent R K Us"   and "a \<in> ring.Span R K Us" shows "\<exists>!Ks. set Ks \<subseteq> K \<and> length Ks = length Us \<and> a = ring.combine R Ks Us"
fr_p_1: shows "atom (p \<bullet> a) \<sharp> p \<bullet> h"
v11_vintersection: fixes s :: "V" shows "v11 (r \<inter>\<^sub>\<circ> s)"
coeff_mult: fixes p :: "'a poly"   and q :: "'a poly"   and n :: "nat" shows "poly.coeff (p * q) n = (\<Sum>i\<le>n. poly.coeff p i * poly.coeff q (n - i))"
ordinal_not_0_less: fixes x :: "ordinal" shows "(\<not> 0 < x) = (x = 0)"