hcju/isabelleps huggingface dataset:
elims: fixes x :: "('a, 'b) Combinators list"   and y :: "('a, 'b) Combinators" assumes "list2policyR x = y" obtains xa :: "('a, 'b) Combinators" where "x = [xa__]" and "y = xa__"   | xa :: "('a, 'b) Combinators" and v :: "('a, 'b) Combinators" and va :: "('a, 'b) Combinators list" where "x = xa # v # va" and "y = list2policyR (v # va) \<oplus> xa"   | "x = []" and "y = undefined"
if_SE_split': fixes \<sigma> :: "'d"   and P :: "'d \<Rightarrow> bool"   and B\<^sub>1 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and B\<^sub>2 :: "'d \<Rightarrow> ('e \<times> 'd) option"   and M :: "'d \<Rightarrow> (bool \<times> 'd) option" shows "(\<sigma> \<Turnstile> _if_SE P B\<^sub>1 B\<^sub>2 ;- M) = ((P \<sigma> \<longrightarrow> (\<sigma> \<Turnstile> B\<^sub>1 ;- M)) \<and> (\<not> P \<sigma> \<longrightarrow> (\<sigma> \<Turnstile> B\<^sub>2 ;- M)))"
edom_env_delete_subset: fixes x :: "'c"   and m :: "'c \<Rightarrow> 'd" shows "edom (env_delete x m) \<subseteq> edom m"
fast_dice_roll_alt: shows "local.fdr_alt = local.fast_dice_roll"
"1_2": fixes x2 :: "'t \<Rightarrow>\<^sub>0 'b" assumes "p \<noteq> 0"   and "find_adds fs (lt p) = Some x2"   and "\<not> is_red (set fs) r" shows "\<not> is_red (set fs) (trd_aux fs (tail p - monom_mult (lc p / lc x2) (lp p - lp x2) (tail x2)) r)"
vector_space_prod_def: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f" shows "vector_space_prod s1 s2 \<equiv> vector_space_pair s1 s2"
hom_prod: fixes f :: "'c \<Rightarrow> 'b poly"   and X :: "'c set"   and p :: "'b poly" shows "prod f X \<circ>\<^sub>p p = (\<Prod>x\<in>X. f x \<circ>\<^sub>p p)"
out_diff2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Out Bet A B C" shows "C \<noteq> A"
subset_iff_psubset_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (A \<subset> B \<or> A = B)"
Path_upd_same: fixes p :: "ref"   and f :: "ref \<Rightarrow> ref"   and q :: "ref"   and qs :: "ref list" shows "Path p (f(p := p)) q qs = (p = Null \<and> q = Null \<and> qs = [] \<or> p \<noteq> Null \<and> q = p \<and> (\<forall>x\<in>set qs. x = p))"
mismatchCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "residual" assumes "[a\<noteq>b]P \<longmapsto> Rs" obtains P :: "pi" and aa :: "name" and ba :: "name" where "a = aa \<and> b = ba \<and> Pb__ = P" and "P \<longmapsto> Rs" and "aa \<noteq> ba"
sum_ideals_cont: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and A :: "'c set"   and B :: "'c set" assumes "Ring R"   and "ideal R I"   and "A \<subseteq> I"   and "B \<subseteq> I" shows "A \<minusplus>\<^bsub>R\<^esub> B \<subseteq> I"
avl_fib_bound: fixes t :: "('b \<times> nat) tree" assumes "avl t" shows "fib (height t + 2) \<le> size1 t"
rep_eq: fixes x :: "circline_mat" shows "Rep_circline_mat (opposite_ocircline_clmat x) = opposite_ocircline_cmat (Rep_circline_mat x)"
total: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool"   and A :: "'a set"   and gt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list"   and xs :: "'a list" assumes "ext_total ext"   and "\<forall>y\<in>A. \<forall>x\<in>A. gt y x \<or> gt x y \<or> y = x"   and "ys \<in> lists A"   and "xs \<in> lists A" shows "ext gt ys xs \<or> ext gt xs ys \<or> ys = xs"
simps_3: fixes a :: "name"   and x :: "'d" shows "[(a, a)] \<bullet> x = [] \<bullet> x"
cands: shows "set cands \<subseteq> S (lu, []) vs d"
preserves_reflects_arr: fixes f :: "'d" shows "D.arr (src\<^sub>D f) = D.arr f"
epsclo_eq_Image: fixes Q :: "hf set" shows "epsclo Q = nfa.states M \<inter> (eps M)\<^sup>* `` Q"
err_def': fixes A :: "'c set" shows "err A = insert Err {x. \<exists>y\<in>A. x = OK y}"
Units_inverse: fixes u :: "nat \<Rightarrow> int" assumes "u \<in> Units Zp" shows "inv u \<in> Units Zp"
VSigma_empty1: fixes B :: "V \<Rightarrow> V" shows "VSigma 0 B = 0"
FTCi: fixes i :: "nat" shows "interval_lebesgue_integral lborel (ereal (l i)) (ereal (u i)) f = F (u i) - F (l i)"
is_faithful: fixes f :: "'c"   and f' :: "'c" assumes "C.par f f'"   and "F f = F f'" shows "f = f'"
LeftDerives1_skip_prefix: fixes a :: "symbol list"   and i :: "nat"   and b :: "symbol list"   and r :: "symbol \<times> symbol list"   and c :: "symbol list" assumes "length a \<le> i"   and "LeftDerives1 (a @ b) i r (a @ c)" shows "LeftDerives1 b (i - length a) r c"
power_add_numeral: fixes a :: "'a"   and m :: "num"   and n :: "num" shows "a ^ numeral m * a ^ numeral n = a ^ numeral (m + n)"
common_boundary_sudivision_commutative: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "common_boundary_sudivision_exists one_chain1 one_chain2 = common_boundary_sudivision_exists one_chain2 one_chain1"
that: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" assumes "h = f \<parallel> g" shows "thesis"
cinner_diff_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "\<langle>x, y - z\<rangle> = \<langle>x, y\<rangle> - \<langle>x, z\<rangle>"
i_in: shows "\<iota> \<in> Inf_from (Liminf_llist (lmap snd Ns))"
map_of_delete_insert: fixes \<Gamma> :: "('b \<times> 'a) list"   and x :: "'b"   and e :: "'a" assumes "map_of \<Gamma> x = Some e" shows "map_of ((x, e) # delete x \<Gamma>) = map_of \<Gamma>"
polyn_expr_def: fixes R :: "('e, 'f) Ring_scheme"   and X :: "'e"   and k :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'e)" shows "polyn_expr R X k c \<equiv> \<Sigma>\<^sub>e R (\<lambda>j. snd c j \<cdot>\<^sub>r\<^bsub>R\<^esub> X^\<^bsup>R j\<^esup>) k"
\<tau>exec_moves_conv_\<tau>exec_meth: fixes addr2thread_id :: "'b \<Rightarrow> 'c"   and thread_id2addr :: "'c \<Rightarrow> 'b"   and spurious_wakeups :: "bool"   and empty_heap :: "'a"   and allocate :: "'a \<Rightarrow> htype \<Rightarrow> ('a \<times> 'b) set"   and typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and heap_read :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> bool"   and heap_write :: "'a \<Rightarrow> 'b \<Rightarrow> addr_loc \<Rightarrow> 'b val \<Rightarrow> 'a \<Rightarrow> bool"   and ci :: "('b, 'a) check_instr"   and P :: "'b J1_prog"   and t :: "'c"   and es :: "'b expr1 list" shows "JVM_heap_base.\<tau>exec_moves addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci P t es = JVM_heap_base.\<tau>exec_meth addr2thread_id thread_id2addr spurious_wakeups empty_heap allocate typeof_addr heap_read heap_write ci (compP2 P) (compEs2 es) (compxEs2 es 0 0) t"
negateI: fixes P :: "'c \<Rightarrow> bool"   and s :: "'c" assumes "\<not> P s" shows "\<N> P s"
nth_visible_inc': fixes a :: "'d extended \<Rightarrow> 'c"   and s :: "('d, 'e option) insert_message list"   and n :: "nat"   and i :: "'d extended"   and j :: "'d extended" assumes "sorted_wrt (<) (map a (ext_ids s))"   and "nth_visible s n = Inr i"   and "nth_visible s (Suc n) = Inr j" shows "a i < a j"
WellDefinedVar: fixes S :: "('a, 'b) Signature"   and A :: "'a" assumes "Sig S \<triangleright> \<turnstile> A Type" shows "Sig S \<triangleright> Vx : A \<turnstile> Vx : A"
SeqStTermP_unique: fixes v :: "tm"   and a :: "tm"   and t :: "tm"   and u :: "tm"   and s :: "tm"   and kk :: "tm"   and u' :: "tm"   and s' :: "tm"   and kk' :: "tm" shows "{SeqStTermP v a t u s kk, SeqStTermP v a t u' s' kk'} \<turnstile> u' EQ u"
intros_6: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and G :: "'b formula" assumes "\<Gamma> \<Rightarrow> F, \<Delta>"   and "\<Gamma> \<Rightarrow> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta>"
CR6_Says_imp_Notes: fixes i :: "nat"   and C :: "agent"   and KC2 :: "nat"   and NC3 :: "nat"   and Y :: "nat"   and cardSK :: "nat"   and X :: "nat"   and evs :: "event list" assumes "Says (CA i) C (Crypt KC2 \<lbrace>sign (priSK (CA i)) \<lbrace>Agent C, Nonce NC3, Agent (CA i), Nonce Y\<rbrace>, certC (pan C) cardSK X onlySig (priSK (CA i)), cert (CA i) (pubSK (CA i)) onlySig (priSK RCA)\<rbrace>) \<in> set evs"   and "evs \<in> set_cr" shows "Notes (CA i) (Key cardSK) \<in> set evs"
Prefixes_altdef: fixes A :: "'b list set" shows "Prefixes A = (\<Union>w\<in>A. set (prefixes w))"
sum_iso: fixes B :: "'b set"   and A :: "'b set" assumes "finite B" shows "A \<subseteq> B \<longrightarrow> \<Sum> A \<le> \<Sum> B"
mult_minus_distrib_mat: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and C :: "'b mat" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc"   and "C \<in> carrier_mat n nc" shows "A * (B - C) = A * B - A * C"
rep_eq: shows "Rep_ell2 1 = (\<lambda>_. 1)"
real_in_int_intervals: fixes m :: "int"   and x :: "real"   and n :: "int" assumes "real_of_int m \<le> x \<and> x < real_of_int (n + 1)" shows "\<exists>j\<in>{m..n}. real_of_int j \<le> x \<and> x < real_of_int (j + 1)"
lookup_in_empty: fixes f :: "'b \<Rightarrow> bool" shows "lookup f [] = None"
verit_comp_simplify_3: fixes b' :: "'b"   and a' :: "'b" shows "(\<not> b' \<le> a') = (a' < b')"
d2c_nzero: shows "\<forall>x. c < x \<and> x \<le> d2 \<longrightarrow> poly p x \<noteq> 0 \<and> poly (pderiv p) x \<noteq> 0"
surjective: fixes r :: "'a state_t_scheme" shows "r = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj r, sp_impl_subj_obj = sp_impl_subj_obj r, current = current r, obj = obj r, thread = thread r, \<dots> = state_t.more r\<rparr>"
dBox_false: fixes \<Phi> :: "'b upred" shows "\<^bold>[false\<^bold>]\<Phi> = true"
ctr_2: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "lappend xs ys = LCons (case xs of LNil \<Rightarrow> lhd ys | LCons x xs' \<Rightarrow> x) (case xs of LNil \<Rightarrow> case ys of LCons uu uua \<Rightarrow> uua | LCons x xs' \<Rightarrow> lappend xs' ys)"
comb: fixes \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma>' :: "(name, value) fmap"   and u :: "sterm"   and u' :: "value"   and env :: "(name, value) fmap"   and uu :: "Term_Class.term"   and rhs :: "sterm"   and val :: "value" assumes "\<Gamma> \<turnstile>\<^sub>v t \<down> Vabs cs \<Gamma>'"   and "\<Gamma> \<turnstile>\<^sub>v u \<down> u'"   and "vfind_match cs u' = Some (env, uu, rhs)"   and "\<Gamma>' ++\<^sub>f env \<turnstile>\<^sub>v rhs \<down> val" shows "\<Gamma> \<turnstile>\<^sub>v t $\<^sub>s u \<down> val"
set_bit_int_def: fixes i :: "int"   and n :: "nat"   and b :: "bool" shows "set_bit_class.set_bit i n b = bin_sc n b i"
proj_rep_set: fixes n :: "nat" shows "proj_rep_set n = range (pseudo_proj_True n)"
mconn_trans': fixes E :: "('b \<times> 'b) set" shows "trans (mconn E)"
P_S: fixes ps :: "('a \<times> bool) list" assumes "ps \<in> P" shows "order.greater_eq S (fst ` set ps)"
ennreal_approx_INF: fixes A :: "'b set"   and x :: "ennreal"   and f :: "'b \<Rightarrow> ennreal" assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> f i"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>i\<in>A. f i \<le> x + ennreal e" shows "x = Inf (f ` A)"
in_rel_Fbc: fixes B :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'Fb"   and y :: "'Fc" shows "rel_Fbc B x y = (\<exists>z. z \<in> {x. set_Fbc x \<subseteq> {(x, y). B x y}} \<and> \<pi>_Fbcb z = x \<and> \<pi>_Fbcc z = y)"
rep_eq: fixes x :: "'b mpoly"   and xa :: "'b mpoly" shows "mapping_of (x * xa) = mapping_of x * mapping_of xa"
fMax_ge: fixes x :: "'c"   and A :: "'c fset" assumes "x |\<in>| A" shows "x \<le> fMax A"
Decompose: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term"   and K :: "('fun, 'var) Term.term list"   and M :: "('fun, 'var) Term.term list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> S"   and "ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>"   and "Ana t = (K, M)"   and "\<And>k. k \<in> set K \<Longrightarrow> ik\<^sub>e\<^sub>s\<^sub>t S \<union> M\<^sub>0 \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> k \<cdot> \<I>" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> (S @ [Decomp t])"
is_app_def: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and t :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_app unapp t = (unapp t \<noteq> None)"
power_sum_puzzle_has_solution: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex" shows "\<exists>Root. \<forall>k\<in>{1..n}. (\<Sum>i<n. Root i ^ k) = f k"
inverse_p64_def: fixes p :: "uint64"   and x :: "uint64" shows "inverse_p64 p x = (if x = 0 then 0 else power_p64 p x (p - 2))"
empty_upd_none: fixes x :: "'a" shows "\<epsilon>(x := None) = \<epsilon>"
sets_def: fixes M :: "'c measure" shows "sets M = fst (snd (Rep_measure M))"
exhaust: fixes y :: "('c, 'd) strand_step" obtains x1 :: "('c, 'd) Term.term" where "y = send\<langle>x1\<rangle>\<^sub>s\<^sub>t"   | x2 :: "('c, 'd) Term.term" where "y = receive\<langle>x2\<rangle>\<^sub>s\<^sub>t"   | x31 :: "poscheckvariant" and x32 :: "('c, 'd) Term.term" and x33 :: "('c, 'd) Term.term" where "y = \<langle>x31: x32 \<doteq> x33\<rangle>\<^sub>s\<^sub>t"   | x41 :: "'d list" and x42 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" where "y = \<forall>x41\<langle>\<or>\<noteq>: x42\<rangle>\<^sub>s\<^sub>t"
inpt_str_comp_a: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "nat \<Rightarrow> 'a \<times> 'd"   and x :: "nat \<Rightarrow> 'b"   and r' :: "'d \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> bool" shows "(prec_st (inpt_st r) r (fst \<circ> u) x \<and> (\<forall>y. (\<box> lft_rel_st r) (fst \<circ> u) x y \<longrightarrow> prec_st (inpt_st r') r' (snd \<circ> u) y)) = prec_st (\<lambda>u u' x. inpt_st r (fst u) (fst u') x \<and> (\<forall>y. r (fst u) (fst u') x y \<longrightarrow> inpt_st r' (snd u) (snd u') y)) (\<lambda>(u, v) (u', v'). r u u' OO r' v v') u x"
RelS_Eg1Eg2p_RelS_Eg1Eg2_eq: shows "RelS_Eg1Eg2p = (\<lambda>x xa xb xc xd xe. (\<langle>x, xa, xb\<rangle>\<^sub>A, \<langle>xc, xd, xe\<rangle>\<^sub>C) \<in> RelS_Eg1Eg2)"
simps: fixes f :: "'e \<Rightarrow> 'c"   and as :: "'e list"   and bs :: "'d"   and cs :: "('e \<times> 'e + nat) list"   and n :: "nat" shows "app_args f ((as, bs, cs), n) = ((map f as, bs, map (app_pair f) cs), n)"
satisfies_bounded_nFOR: fixes \<AA> :: "'i"   and \<phi>s :: "('a, 'k) aformula list" shows "(\<AA> \<Turnstile>\<^sub>b nFOR \<phi>s) = (\<exists>\<phi>\<in>set \<phi>s. \<AA> \<Turnstile>\<^sub>b \<phi>)"
inst\<^sub>2: shows "instance_of\<^sub>l\<^sub>s D' D"
Sublists_conv_Suffixes: fixes A :: "'b list set" shows "Sublists A = Suffixes (Prefixes A)"
Cauchy_Schwarz_ineq2: fixes x :: "'c"   and y :: "'c" shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
reflexive: fixes A :: "'a trm" shows "A >>\<^sup>* A"
inverse_le_iff_sgn: fixes x :: "real"   and y :: "real" assumes "sgn x = sgn y" shows "(inverse x \<le> inverse y) = (y \<le> x)"
AE_discrete_difference: fixes X :: "'c set"   and M :: "'c measure" assumes "countable X"   and "\<And>x. x \<in> X \<Longrightarrow> emeasure M {x} = 0"   and "\<And>x. x \<in> X \<Longrightarrow> {x} \<in> sets M" shows "AE x in M. x \<notin> X"
iMOD_iff: fixes x :: "nat"   and r :: "nat"   and m :: "nat" shows "(x \<in> [ r, mod m ]) = (x mod m = r mod m \<and> r \<le> x)"
run_altc_env: fixes altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and C :: "'c cset"   and f :: "'c \<Rightarrow> ('b, 'a) envT"   and r :: "'b" shows "run_env (altc_env altc C f) r = altc C (\<lambda>c. run_env (f c) r)"
continuous_constant_on_closure: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and a :: "'b"   and x :: "'a" assumes "continuous_on (closure S) f"   and "\<And>x. x \<in> S \<Longrightarrow> f x = a"   and "x \<in> closure S" shows "f x = a"
F_def: fixes x :: "real" shows "F x = Inf {lim (\<lambda>k. f (d k) (r n)) |n. x < r n}"
cwiseext_def: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ys :: "'b list"   and xs :: "'b list" shows "cwiseext gt ys xs = (length ys = length xs \<and> (\<forall>i<length ys. gt (ys ! i) (xs ! i) \<or> ys ! i = xs ! i) \<and> (\<exists>i<length ys. gt (ys ! i) (xs ! i)))"
N: fixes sa :: "'s" assumes "sa \<in> S"   and "sa \<notin> N" shows "sa \<in> S1"
twoCisTypeI_4: shows "twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b)))"
assms_6: shows "w \<le> v * v\<^sup>T"
Id_on_def: fixes A :: "'d set" shows "Id_on A = (\<Union>x\<in>A. {(x, x)})"
trm_prm_simp_6: fixes \<pi> :: "'b prm"   and P :: "'b trm" shows "\<pi> \<cdot> Fst P = Fst (\<pi> \<cdot> P)"
mult: shows "(a * d + b * c) * (a * d - b * c) = (a\<^sup>2 + N * b\<^sup>2) * (d\<^sup>2 - b\<^sup>2)"
lists_accD: fixes xs :: "'g list"   and r :: "('g \<times> 'g) set" assumes "xs \<in> lists (Wellfounded.acc r)" shows "xs \<in> Wellfounded.acc (listrel1 r)"
init_s: fixes t :: "real" assumes "t \<le> 0" shows "other.s t = s\<^sub>o"
nonempty_path_components_of: fixes C :: "'a set"   and X :: "'a topology" assumes "C \<in> path_components_of X" shows "C \<noteq> {}"
AnnoFAss: fixes is_lub :: "ty \<Rightarrow> ty \<Rightarrow> ty \<Rightarrow> bool"   and P :: "'b J_prog"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'b expr"   and e1' :: "'b expr"   and e2 :: "'b expr"   and e2' :: "'b expr"   and U :: "ty"   and C :: "String.literal"   and F :: "String.literal"   and T :: "ty"   and fm :: "fmod"   and D :: "String.literal" assumes "is_lub,P,E \<turnstile> e1 \<leadsto> e1'"   and "is_lub,P,E \<turnstile> e2 \<leadsto> e2'"   and "is_lub,P,E \<turnstile> e1' :: U"   and "class_type_of' U = \<lfloor>C\<rfloor>"   and "P \<turnstile> C sees F:T (fm) in D"   and "is_Array U \<longrightarrow> F \<noteq> length" shows "is_lub,P,E \<turnstile> e1\<bullet>F := e2 \<leadsto> e1'\<bullet>F{D} := e2'"
encode_operator_effect_is_cnf: fixes \<Pi> :: "'b strips_problem"   and t :: "nat"   and op :: "'b strips_operator" shows "is_cnf (encode_operator_effect \<Pi> t op)"
sublist_until_end_eq_drop: fixes hi :: "nat"   and xs :: "'c list"   and lo :: "nat" assumes "Suc hi = length xs" shows "WB_Sort.sublist xs lo hi = drop lo xs"
M_def: shows "M = {s \<in> S \<union> N. \<forall>t\<in>S \<union> N. \<Delta> t \<le> \<Delta> s}"
cases: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" assumes "a1 \<rhd> a2 \<sim> a3" obtains P :: "('a, 'b, 'c) psi" and \<Psi> :: "'b" and Q :: "('a, 'b, 'c) psi" where "a1 = \<Psi>" and "a2 = P" and "a3 = Q" and "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>" and "\<Psi> \<rhd> P \<leadsto>[bisim] Q" and "\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<sim> Q" and "\<Psi> \<rhd> Q \<sim> P"
going_to_inf: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'d filter"   and G :: "'d filter"   and A :: "'c set" shows "f going_to inf F G within A = inf (f going_to F within A) (f going_to G within A)"
count_list_append: fixes xs :: "'b list"   and ys :: "'b list"   and a :: "'b" shows "count_list (xs @ ys) a = count_list xs a + count_list ys a"
greatest_closed: fixes L :: "('a, 'b) gorder_scheme"   and l :: "'a"   and A :: "'a set" assumes "greatest L l A" shows "l \<in> carrier L"
Sbis_pres_discr_L: fixes c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "c \<approx>s d"   and "discr d" shows "discr c"
comparator_list_simps_2: fixes comp\<^sub>'\<^sub>a :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and y :: "'b"   and ya :: "'b list" shows "comparator_list comp\<^sub>'\<^sub>a [] (y # ya) = Lt"
env_restr_env_restr: fixes d1 :: "'d set"   and d2 :: "'d set"   and x :: "'d \<Rightarrow> 'e" shows "x f|` d2 f|` d1 = x f|` (d1 \<inter> d2)"
assms_2: shows "\<Psi>' \<hookrightarrow> \<Psi>''"
Ifun_star_n: fixes F :: "nat \<Rightarrow> 'b \<Rightarrow> 'a"   and X :: "nat \<Rightarrow> 'b" shows "star_n F \<star> star_n X = star_n (\<lambda>n. F n (X n))"
carrier_DirProd: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "carrier (G \<times>\<times> H) = carrier G \<times> carrier H"
simps: fixes uu :: "'b list"   and uv :: "('c, 'd, 'a, 'b, 'e) qTerm set list"   and uw :: "('c, 'd, 'a, 'b, 'e) qAbs set list"   and rhoL :: "('a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'a, 'b, 'e) qTerm set option) list" shows "envsOf (Par uu uv uw rhoL) = set rhoL"
ccpo_Sup_below_iff: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Y :: "'b set"   and x :: "'b" assumes "class.ccpo Sup less_eq less"   and "Complete_Partial_Order.chain less_eq Y" shows "less_eq (Sup Y) x = (\<forall>y\<in>Y. less_eq y x)"
wset_Suspend_okD2: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "(('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set) set"   and t :: "'t"   and w :: "'w wait_set_status" assumes "s \<in> wset_Suspend_ok I"   and "wset s t = \<lfloor>w\<rfloor>" shows "\<exists>s0\<in>I. \<exists>s1\<in>I. \<exists>ttas x x0 ta w' ln' ln''. s0 -t\<triangleright>ta\<rightarrow> s1 \<and> s1 -\<triangleright>ttas\<rightarrow>* s \<and> thr s0 t = \<lfloor>(x0, no_wait_locks)\<rfloor> \<and> t \<turnstile> \<langle>x0, shr s0\<rangle> -ta\<rightarrow> \<langle>x, shr s1\<rangle> \<and> Suspend w' \<in> set \<lbrace>ta\<rbrace>\<^bsub>w\<^esub> \<and> actions_ok s0 t ta \<and> thr s1 t = \<lfloor>(x, ln')\<rfloor> \<and> thr s t = \<lfloor>(x, ln'')\<rfloor>"
pp_init_height_bound: shows "Height_Bounded_Labeling c s t pp_init_f pp_init_l"
compl_imp_diff_cosets: fixes H :: "'a set"   and J :: "'a set"   and a :: "'a"   and b :: "'a" assumes "subgroup H G"   and "subgroup J G"   and "finite H"   and "finite J"   and "complementary H J"   and "a \<in> J"   and "b \<in> J"   and "a \<noteq> b" shows "H #> a \<noteq> H #> b"
pmf_of_list_wfI: fixes xs :: "('c \<times> 'b) list" assumes "\<And>x. x \<in> set (map snd xs) \<Longrightarrow> (0::'b) \<le> x"   and "sum_list (map snd xs) = (1::'b)" shows "pmf_of_list_wf xs"
if_subst: fixes indep :: "'c set \<Rightarrow> bool"   and e :: "'c"   and T :: "'c set" shows "(if indep (insert e T) then RETURN (insert e T) else RETURN T) = (if e \<notin> T \<and> indep (insert e T) then RETURN (insert e T) else RETURN T)"
Pareto_strict_iff: fixes x :: "'alt pmf"   and y :: "'alt pmf" shows "x \<prec>[Pareto (SD \<circ> R)] y = ((\<forall>i\<in>agents. (SD \<circ> R) i x y) \<and> (\<exists>i\<in>agents. x \<prec>[(SD \<circ> R) i] y))"
m2zz2m: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d" assumes "LSCategory C"   and "f maps\<^bsub>C\<^esub> X to Y" shows "m2z\<^bsub>C\<^esub>f |\<in>| (Hom\<^bsub>C\<^esub> X Y)"
iT_Minus_iWeakUntil_conv: fixes k :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1. t1 \<W> t2 k \<ominus> I. Q t2) = (P (k - t1). t1 \<B> t2 I \<down>\<le> k. Q (k - t2))"
F_def: shows "F' \<equiv> make_F (insert i V) p'"