hcju/isabelleps huggingface dataset:
InvGlobPointstampsEq_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('d, 'c, 'a) Combined.configuration" assumes "dataflow_topology summary results_in" shows "dataflow_topology.InvGlobPointstampsEq c = (\<forall>p loc t. zcount (c_pts (prop_config c p) loc) t = zcount (c_glob (exchange_config c) p) (loc, t))"
dg_Arr_vsubset_Vset: shows "\<CC>\<lparr>Arr\<rparr> \<subseteq>\<^sub>\<circ> Vset \<alpha>"
PP: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and prv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and P :: "'c"   and t :: "'b" assumes "HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv enc P"   and "t \<in> trm" shows "HBL1.PP var subst P t \<in> fmla"
cards_def: shows "cards = map (\<lambda>k. (k, 2 * as ! k - as ! (k - 1) - as ! Suc k)) [1..<Suc kk]"
vcpower_vrange: fixes f :: "V"   and A :: "V"   and n :: "V" assumes "f \<in>\<^sub>\<circ> A ^\<^sub>\<times> n" shows "\<R>\<^sub>\<circ> f \<subseteq>\<^sub>\<circ> A"
state_unchanged: fixes i :: "nat"   and t :: "nat"   and y :: "nat" assumes "gap i t - 1 \<le> y"   and "y \<le> t" shows "gap i t = gap i y"
of_nat_neq_0: fixes k :: "nat" assumes "0 < k"   and "k < 2 ^ LENGTH('b)" shows "word_of_nat k \<noteq> 0"
sol: shows "(sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (sol 0, b) x \<in> fml_sem I \<phi>}"
B: shows "\<forall>\<^sub>F x in F. h x / f x / h x = inverse (f x)"
plus_current_def: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal" shows "plus_current f g = (\<lambda>e. f e + g e)"
distinct_disc_2: fixes pr_operation :: "pr_operation" assumes "is_RELABEL pr_operation" shows "\<not> is_PUSH pr_operation"
abs_eq: fixes xa :: "int \<times> int"   and x :: "int \<times> int" assumes "ratrel xa xa"   and "ratrel x x" shows "Abs_Rat xa + Abs_Rat x = Abs_Rat (fst xa * snd x + fst x * snd xa, snd xa * snd x)"
unwrapForall'': fixes i :: "nat"   and j :: "nat"   and A :: "atom fm"   and L :: "real list" shows "eval (AllN (i + j) A) L = eval (AllN i (AllN j A)) L"
awalk_verts_in_verts: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "u \<in> pverts sG"   and "set p \<subseteq> parcs sG"   and "v \<in> set (pawalk_verts u p)" shows "v \<in> pverts sG"
neq_init_forall_ge: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat"   and g :: "nat \<Rightarrow> nat option"   and m :: "nat" assumes "f \<triangleright> n \<noteq> g \<triangleright> n"   and "n \<le> m" shows "f \<triangleright> m \<noteq> g \<triangleright> m"
plus_oracle_stop_split: fixes P :: "(('b + 'd) option \<times> 's) spmf \<Rightarrow> bool"   and lr :: "'a + 'c" shows "P (local.plus_oracle_stop lr) = ((\<forall>x. lr = Inl x \<longrightarrow> P (map_spmf (apfst (map_option Inl)) (left s x))) \<and> (\<forall>y. lr = Inr y \<longrightarrow> P (map_spmf (apfst (map_option Inr)) (right s y))))"
eq: shows "x \<cdot> x\<inverse>\<^sup>>w = w\<^sup><\<inverse>x  \<cdot> x"
iT_Plus_neg_inext_nth: fixes I :: "nat set"   and k :: "nat"   and n :: "nat" assumes "I \<oplus>- k \<noteq> {}" shows "(I \<oplus>- k) \<rightarrow> n = (I \<down>\<ge> k \<rightarrow> n) - k"
le_infE: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream"   and v\<^sub>1 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"   and "v\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>1" obtains "v\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2"
semilattice_axioms_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semilattice_axioms f \<equiv> \<forall>a. f a a = a"
Ssegment_not_inc: fixes D :: "'c Order"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b" shows "b \<notin> Ssegment D a"
t1_def: shows "t1 = TaylorModel p1 e1"
A1a: shows "[\<lambda>w. \<forall>x. (P (\<lambda>xa. m\<not> (x xa)) m\<rightarrow> m\<not> (P x)) w]"
sums_0: fixes f :: "nat \<Rightarrow> 'b" assumes "\<And>n. f n = (0::'b)" shows "f sums (0::'b)"
g_size_def: fixes m :: "'s" shows "g_size m \<equiv> iterate m (\<lambda>_. Suc) 0"
assms_2: shows "\<not> i - Suc (length x) < Suc (length y)"
irreducible_m_dvdm_prod_list: fixes a :: "int poly"   and xs :: "int poly list" assumes "p.irreducible_m a"   and "p.dvdm a (prod_list xs)" shows "\<exists>b\<in>set xs. p.dvdm a b"
is_isoI: fixes h :: "'e \<Rightarrow> 'f"   and G :: "('e, 'g) monoid_scheme"   and H :: "('f, 'h) monoid_scheme" assumes "h \<in> Group.iso G H" shows "G \<cong> H"
bd_invI: fixes l :: "('a, 'b) rbt"   and r :: "('a, 'b) rbt"   and c :: "color"   and k :: "'a"   and v :: "'b" assumes "bd_inv l"   and "bd_inv r"   and "black_depth l = black_depth r" shows "bd_inv (Node l c k v r)"
elims_3: fixes x :: "('b, 'c) Combinators list" assumes "\<not> noDenyAll x" obtains xa :: "('b, 'c) Combinators" and xs :: "('b, 'c) Combinators list" where "x = xa # xs" and "\<not> (\<not> member DenyAll xa \<and> noDenyAll xs)"
trans_inv_image: fixes r :: "('a \<times> 'a) set"   and f :: "'b \<Rightarrow> 'a" assumes "trans r" shows "trans (inv_image r f)"
takeWhile_eq_take: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "takeWhile P xs = take (length (takeWhile P xs)) xs"
sints_def: shows "sints 64 = range (signed_take_bit (64 - 1))"
preserves_ide: fixes a :: "'c" assumes "C.ide a" shows "C.ide (C'.T (C.cod \<iota>\<^sub>C, a))"
eq_implies: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "reflexive A less_eq"   and "x = y"   and "x \<in> A" shows "less_eq x y"
obj_trg: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.obj (src \<mu>)"
Nonce_Hash: fixes X :: "nonce_t"   and X' :: "msg" shows "Nonce X \<noteq> Hash X'"
PiE_mem: fixes f :: "'e \<Rightarrow> 'f"   and S :: "'e set"   and T :: "'e \<Rightarrow> 'f set"   and x :: "'e" assumes "f \<in> Pi\<^sub>E S T"   and "x \<in> S" shows "f x \<in> T x"
n_zero_nonempty: fixes n :: "nat" assumes "n \<noteq> 0" shows "{0..<n} \<noteq> {}"
not_gr_zero: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and n :: "'e" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
opp': shows "OpposedThinChamberComplexFoldings X f' g' C'"
1: shows "(x, y) = Rep_lifted_pair (Rep_dense pf) f \<and> (x, y) \<in> triple.pairs (Rep_phi f) \<and> triple.pairs_uminus (Rep_phi f) (x, y) = (bot, Filters.filter.Abs_filter UNIV)"
inv_meldUniq: shows "queue_invar (meldUniq (uniqify q) (uniqify q'))"
distributed_borel_measurable: fixes M :: "'a measure"   and N :: "'b measure"   and X :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> ennreal" assumes "distributed M N X f" shows "f \<in> borel_measurable N"
prems: shows "accs \<delta>2 (NODE l ts) q2"
capacity'_source_in: fixes y :: "'v node" shows "capacity' (y, Inner (source \<Delta>)) = (if y = SOURCE then \<alpha> else 0)"
is_tm_dghm_axioms: shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> \<BB>"
good_context_5: fixes s :: "'b sparc_state"   and s'' :: "'b sparc_state"   and v1 :: "word32"   and v2 :: "sparc_operation \<times> inst_operand list" assumes "good_context s \<and> s'' = delayed_pool_write s \<and> fetch_instruction s'' = Inr v1 \<and> decode_instruction v1 = Inr v2 \<and> annul_val s'' = False \<and> supported_instruction (fst v2) = True \<and> fst v2 = ctrl_type RETT \<and> get_ET (cpu_reg_val PSR s'') \<noteq> 1 \<and> get_S (cpu_reg_val PSR s'') \<noteq> 0 \<and> get_WIM_bit (nat ((uint (get_CWP (cpu_reg_val PSR s'')) + 1) mod NWINDOWS)) (cpu_reg_val WIM s'') \<noteq> 0" shows "False"
OO_Grp_alt: fixes A :: "'i set"   and f :: "'i \<Rightarrow> 'g"   and g :: "'i \<Rightarrow> 'h" shows "(BNF_Def.Grp A f)\<inverse>\<inverse> OO BNF_Def.Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
streams_Stream: fixes x :: "'b"   and s :: "'b stream"   and A :: "'b set" shows "(x ## s \<in> streams A) = (x \<in> A \<and> s \<in> streams A)"
map_decomp: shows "[xa # xs [\<mapsto>] y # ys] = [xa \<mapsto> y] ++ [xs [\<mapsto>] ys]"
check_unsat_id: fixes check :: "('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "Check check"   and "\<U> s" shows "check s = s"
wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s_M: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"
MakeCatMor: fixes C :: "('e, 'd, 'f) Category_scheme" shows "mor\<^bsub>MakeCat C\<^esub> = mor\<^bsub>C\<^esub>"
vec_tensor: fixes v :: "'b list"   and d :: "nat list" assumes "length v = prod_list d" shows "vec (tensor_from_vec d v) = v"
intro: fixes f :: "'a \<Rightarrow> 'a" assumes "bij f" shows "bijection f"
exp_div_exp_eq: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = times (zero_neq_one.of_bool one zero (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m \<noteq> zero \<and> n \<le> m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (m - n))"
rderivs_replicate_diff: fixes i :: "nat"   and as :: "'a"   and r :: "'b rexp"   and j :: "nat" assumes "\<guillemotleft>rderivs (replicate i as) r\<guillemotright> = \<guillemotleft>rderivs (replicate j as) r\<guillemotright>"   and "j < i" shows "\<guillemotleft>rderivs (replicate (i - j) as) (rderivs (replicate j as) r)\<guillemotright> = \<guillemotleft>rderivs (replicate j as) r\<guillemotright>"
finite_subset_Union: fixes f :: "'b \<Rightarrow> 'a set"   and A :: "'b set" assumes "finite (\<Union> (f ` A))" shows "\<exists>B. finite B \<and> B \<subseteq> A \<and> \<Union> (f ` B) = \<Union> (f ` A)"
T'_I_T: fixes s :: "'b"   and as :: "'b list"   and os :: "'a list"   and o :: "'a" shows "emeasure (T' (I s)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>o \<omega>'. \<omega> = (s, o) ## zip as os @- \<omega>'} = emeasure (T (s, o)) {\<omega> \<in> space (stream_space (count_space UNIV)). \<exists>\<omega>'. \<omega> = zip as os @- \<omega>'}"
mono_update: fixes a :: "'b"   and a' :: "'b"   and S :: "'c \<Rightarrow> 'b"   and S' :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "a \<le> a'"   and "S \<le> S'" shows "S(x := a) \<le> S'(x := a')"
e: shows "abc_step_l (sa, lma) (abc_fetch sa (A [+] B)) = abc_step_l (sa, lma) (abc_fetch sa A)"
times_ms_aux_MSLNil_iff: fixes F :: "('a \<times> real) msllist"   and G :: "('a \<times> real) msllist" shows "(times_ms_aux F G = MSLNil) = (F = MSLNil \<or> G = MSLNil)"
jkbpC_traces_of_length: fixes n :: "nat" shows "rel_ext (\<lambda>t. t \<in> Clock.jkbpC \<and> tLength t = n) = Clock.jkbpCn n"
add_eq_0_iff_both_eq_0: fixes x :: "'d multiset"   and y :: "'d multiset" shows "(x + y = {#}) = (x = {#} \<and> y = {#})"
natural_transformation_\<alpha>: shows "natural_transformation VVV.comp (\<bullet>) HoHV HoVH (\<lambda>\<mu>\<nu>\<pi>. \<a>[fst \<mu>\<nu>\<pi>, fst (snd \<mu>\<nu>\<pi>), snd (snd \<mu>\<nu>\<pi>)])"
keep_drop_sels_1: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "fst (R\<^sup>k) = R"
const_def: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set" assumes "wo_rel2 r s" shows "wo_rel2.const r s = (\<lambda>x. if x \<in> Field s then zero r else undefined)"
3: shows "(deriv f \<alpha> \<oplus> c \<otimes> h) \<otimes> h = \<zero>"
m1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "m y \<le> m x"
assms_1: shows "liftAll2 (\<lambda>v v'. phi v v' \<longrightarrow> chi v v') inp inp'"
fold_insert_idem2: fixes A :: "'c set"   and z :: "('c, real) mapping"   and x :: "'c" assumes "finite A" shows "Finite_Set.fold (\<lambda>x. Mapping.update x p) z (insert x A) = Finite_Set.fold (\<lambda>x. Mapping.update x p) (Mapping.update x p z) A"
column_Gram_Schmidt_column_k': fixes i :: "'a"   and k :: "'a"   and A :: "(('b, 'a) vec, 'c) vec" assumes "i \<noteq> k" shows "column i (Gram_Schmidt_column_k A (mod_type_class.to_nat k)) = column i A"
div_positive: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"   and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"   and b :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring_numeral divide plus minus zero times one modulo euclidean_size division_segment less_eq less divmod divmod_step"   and "less zero b"   and "less_eq b a" shows "less zero (divide a b)"
typeofv: shows "typeof\<^bsub>h\<^esub> v = \<lfloor>Tv\<rfloor>"
one_nonzero: shows "\<one> \<in> nonzero Zp"
demonic_def: fixes Q :: "'a \<Rightarrow> 'b"   and p :: "'b" shows "[: Q :] p = {s. Q s \<sqsubseteq> p}"
intro: fixes T :: "'a \<Rightarrow> 'a"   and V :: "'a set" assumes "T ` V \<subseteq> V" shows "VectorSpaceEnd_axioms V T"
seqseq_fold_reduce: fixes P :: "nat \<Rightarrow> (nat \<Rightarrow> nat) \<Rightarrow> bool"   and n :: "nat" assumes "Diagonal_Subsequence.subseqs P" shows "subseqs.seqseq P n = subseqs.fold_reduce P 0 n"
assms: shows "\<exists>u\<in>U. (u, D x) \<in> I"
v1_def: shows "u = \<lparr>ext = Abs_real_int (left (ext v), left (ext v) + x), lan = lan v, own = own v\<rparr>"
b: shows "PreFunctor (HomP\<^bsub>C\<^esub>[X,\<midarrow>])"
inf0: fixes a :: "'b" shows "a \<equiv> a \<sqinter> \<top>"
CompDiag_in_Hom: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Diag t"   and "Diag u"   and "Dom t = Cod u" shows "t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u \<in> Hom (Dom u) (Cod t)"
terminates_on_stop: shows "terminates_on (\<lambda>_. Done) = UNIV"
bij_region_coregion: fixes S :: "<'d, 'e> \<Longleftrightarrow> 'f" assumes "sym_lens S" shows "bij_lens (\<V>\<^bsub>S\<^esub> +\<^sub>L \<C>\<^bsub>S\<^esub>)"
induct: fixes P :: "nat list \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b) \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat \<Rightarrow> bool"   and a0 :: "nat list"   and a1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a2 :: "'b mat \<times> 'b mat \<times> 'b mat" assumes "\<And>bezout P A Q. P [] bezout (P, A, Q)"   and "\<And>i xs bezout P A Q. P xs bezout (Diagonal_to_Smith_row_i_PQ_JNF i bezout (P, A, Q)) \<Longrightarrow> P (i # xs) bezout (P, A, Q)" shows "P a0 a1 a2"
bounded_clinear_intro: fixes f :: "'c \<Rightarrow> 'd"   and K :: "real" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>r x. f (r *\<^sub>C x) = r *\<^sub>C f x"   and "\<And>x. norm (f x) \<le> norm x * K" shows "bounded_clinear f"
zero_vec_def: shows "0 \<equiv> \<chi>i. (0::'a)"
g0: fixes w :: "complex" assumes "w \<in> ball \<xi> r1" shows "g w \<noteq> 0"
interrupt_action_ok_imp_interrupt_action_ok': fixes "is" :: "'b set"   and ia :: "'b interrupt_action" assumes "interrupt_action_ok is ia" shows "interrupt_action_ok' is ia"
set_eq_subset: fixes A :: "'b set"   and B :: "'b set" shows "(A = B) = (A \<subseteq> B \<and> B \<subseteq> A)"
eval_Pr_diverg_add: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list"   and x :: "nat"   and y :: "nat" assumes "recfn (Suc n) (Pr n f g)"   and "n = length xs"   and "eval (Pr n f g) (x # xs) \<up>" shows "eval (Pr n f g) ((x + y) # xs) \<up>"
bounds: fixes i :: "nat" assumes "p i \<noteq> 0" shows "1 \<le> i \<and> i \<le> n"
aZ: shows "a \<in> Z \<rightarrow> {\<zero>\<^bsub>K\<^esub>}"
lim_i: shows "Liminf_upto_llist Ns (enat i) = lnth Ns i"
periodic_set_tan_poly: fixes p :: "real poly"   and c :: "real" assumes "p \<noteq> 0"   and "c \<noteq> 0" shows "periodic_set (roots (\<lambda>x. poly p (tan (x / c)))) (c * pi)"
univ_def: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set" shows "Equiv_Relation2.univ f X \<equiv> f (SOME x. x \<in> X)"
concat_eq_append_conv: fixes xss :: "'c list list"   and ys :: "'c list"   and zs :: "'c list" shows "(concat xss = ys @ zs) = (if xss = [] then ys = [] \<and> zs = [] else \<exists>xss1 xs xs' xss2. xss = xss1 @ (xs @ xs') # xss2 \<and> ys = concat xss1 @ xs \<and> zs = xs' @ concat xss2)"
prj_simps_4: shows "C.dom fgf.Prj\<^sub>1\<^sub>1 = Chn ((f \<star> g) \<star> f)"
HAStates_HARoot_CompFun: fixes S :: "'a"   and HA :: "('a, 'b, 'c) hierauto" assumes "S \<in> HAStates HA" shows "HARoot HA \<notin> the (CompFun HA S)"
dirichlet_prod_neutral_def: fixes n :: "nat" shows "dirichlet_prod_neutral n = (if n = 1 then 1::'a else (0::'a))"
reduce_mset_eq: fixes C :: "'a literal list"   and C' :: "'a literal list"   and Ds :: "'a literal list list"   and E :: "'a literal list" assumes "mset C = mset C'" shows "reduce Ds C E = reduce Ds C' E"
from_list_correct_1: assumes "G_valid_wgraph_repr l" shows "G.invar (G.from_list l)"
ereal_inverse_minus: fixes x :: "ereal" shows "inverse (- x) = (if x = 0 then \<infinity> else - inverse x)"
homotopic_paths_imp_subset: fixes s :: "'a set"   and p :: "real \<Rightarrow> 'a"   and q :: "real \<Rightarrow> 'a" assumes "homotopic_paths s p q" shows "path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
simps_2: fixes y :: "name"   and L :: "stack"   and n :: "trm" assumes "y \<sharp> L" shows "|[y]n\<ggreater>L| = 1 + |L|"
fmlookup_drop: fixes a :: "'n"   and m :: "('n, 'm) fmap"   and x :: "'n" shows "fmlookup (fmdrop a m) x = (if x \<noteq> a then fmlookup m x else None)"
StatImp: fixes \<Psi>' :: "'b"   and R :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi" assumes "(\<Psi>', R, S) \<in> Rel" shows "insertAssertion (extractFrame S) \<Psi>' \<hookrightarrow>\<^sub>F insertAssertion (extractFrame R) \<Psi>'"
1: shows "card (reachable_s PROB s) < card (state_set (state_list s as))"
app_inject: fixes u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a"   and u\<^sub>3 :: "'a"   and u\<^sub>4 :: "'a" assumes "app u\<^sub>1 u\<^sub>2 = app u\<^sub>3 u\<^sub>4" shows "u\<^sub>1 = u\<^sub>3 \<and> u\<^sub>2 = u\<^sub>4"
DOM_comp: 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 "free_monoidal_category.ARR C f"   and "free_monoidal_category.ARR C g"   and "monoidal_language.Dom C (free_monoidal_category.rep C f) = monoidal_language.Cod C (free_monoidal_category.rep C g)" shows "monoidal_language.Dom C (free_monoidal_category.rep C (free_monoidal_category.comp C f g)) = monoidal_language.Dom C (free_monoidal_category.rep C g)"
simps_3: fixes x :: "name"   and t :: "nterm" shows "frees (\<Lambda>\<^sub>n x. t) = frees t |-| {|x|}"
scaleR_le_cancel_left_neg: fixes c :: "real"   and a :: "'a"   and b :: "'a" assumes "c < 0" shows "(c *\<^sub>R a \<le> c *\<^sub>R b) = (b \<le> a)"
c_tr_map: fixes step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s :: "'c"   and xs :: "'a list" shows "map fst (c_tr step out s xs) = xs"
sel_1: fixes x1 :: "nat" shows "shadow_root_ptr.the_ref (shadow_root_ptr.Ref x1) = x1"
elims_2: fixes x :: "'b trm" assumes "is_compound x" obtains xa :: "'b trm" and y :: "'b trm" where "x = xa \<cdot> y"
simulation_Wakeup2: fixes t :: "'t"   and x1 :: "'x1"   and m1 :: "'m1"   and x2 :: "'x2"   and m2 :: "'m2"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x2' :: "'x2"   and m2' :: "'m2" assumes "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "x1 \<approx>w x2"   and "r2.r_syntax t x2 m2 ta2 x2' m2'"   and "Notified \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub> \<or> WokenUp \<in> set \<lbrace>ta2\<rbrace>\<^bsub>w\<^esub>" shows "\<exists>ta1 x1' m1'. r1.r_syntax t x1 m1 ta1 x1' m1' \<and> t \<turnstile> (x1', m1') \<approx> (x2', m2') \<and> ta1 \<sim>m ta2"
subprob_space_axioms_def: fixes M :: "'c measure" shows "subprob_space_axioms M \<equiv> emeasure M (space M) \<le> 1 \<and> space M \<noteq> {}"
dist_norm: fixes x :: "'a"   and y :: "'a" shows "dist x y = norm (x - y)"
box_diamond_galois_1: fixes p :: "'a"   and q :: "'a"   and x :: "'a" assumes "d p = p"   and "d q = q" shows "(\<langle>x| p \<le> q) = (p \<le>  |x] q)"
iT_Mult_assoc: fixes I :: "nat set"   and a :: "nat"   and b :: "nat" shows "I \<otimes> a \<otimes> b = I \<otimes> a * b"
symbolic_checker: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real" assumes "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o" shows "safe_distance.collision a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o {0..} = (quadroot_in 0 (min (movement.t_stop a\<^sub>e v\<^sub>e) (movement.t_stop a\<^sub>o v\<^sub>o)) (1 / 2 * (a\<^sub>o - a\<^sub>e)) (v\<^sub>o - v\<^sub>e) (s\<^sub>o - s\<^sub>e) \<or> quadroot_in (movement.t_stop a\<^sub>e v\<^sub>e) (movement.t_stop a\<^sub>o v\<^sub>o) (1 / 2 * a\<^sub>o) v\<^sub>o (s\<^sub>o - movement.s a\<^sub>e v\<^sub>e s\<^sub>e (movement.t_stop a\<^sub>e v\<^sub>e)) \<or> quadroot_in (movement.t_stop a\<^sub>o v\<^sub>o) (movement.t_stop a\<^sub>e v\<^sub>e) (1 / 2 * a\<^sub>e) v\<^sub>e (s\<^sub>e - movement.s a\<^sub>o v\<^sub>o s\<^sub>o (movement.t_stop a\<^sub>o v\<^sub>o)))"
simple_match_ip_Drop: fixes s :: "'a word" assumes "simple_match_ip (dst m) d" shows "(runFw s d c (SimpleRule m Drop # rs) = Decision FinalAllow) = (\<not> simple_match_ip (src m) s \<and> runFw s d c rs = Decision FinalAllow)"
complex_of_real_nn_iff: fixes y :: "real" shows "(0 \<le> complex_of_real y) = (0 \<le> y)"
path2_hd_in_ns: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "n \<in> set ns"
finish_simps_6: fixes u :: "'v"   and s :: "'v tarjan_state" shows "tree_edges (finish u s) = tree_edges s"
bound: fixes j :: "nat" shows "AE x in M. norm (u j x) \<le> w x"
wuntil_coinduct: fixes \<chi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" assumes "\<chi> \<pi>l"   and "\<And>\<pi>l. \<chi> \<pi>l \<Longrightarrow> \<psi> \<pi>l \<or> \<phi> \<pi>l \<and> \<chi> (map stl \<pi>l)" shows "wuntil \<phi> \<psi> \<pi>l"
quotient_map_product_projection: fixes i :: "'c"   and I :: "'c set"   and X :: "'c \<Rightarrow> 'd topology" assumes "i \<in> I" shows "quotient_map (product_topology X I) (X i) (\<lambda>x. x i) = (topspace (product_topology X I) = {} \<longrightarrow> topspace (X i) = {})"
relAPP_def: fixes f :: "('c \<times> 'd) set \<Rightarrow> 'a"   and x :: "('c \<times> 'd) set" shows "\<langle>x\<rangle>f \<equiv> f x"