hcju/isabelleps huggingface dataset:
numeral_eq_iff: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and m :: "num"   and n :: "num" assumes "class.semiring_char_0 one times plus zero" shows "(numeral.numeral one plus m = numeral.numeral one plus n) = (m = n)"
aux3: fixes M :: "trm"   and M' :: "trm"   and M'' :: "trm" assumes "M = M'"   and "M' \<longrightarrow>\<^sub>a* M''" shows "M \<longrightarrow>\<^sub>a* M''"
assms_3: shows "class1 \<inter> class2 \<noteq> {}"
aGroup_def: fixes A :: "('h, 'i) aGroup_scheme" shows "aGroup A \<equiv> ((\<plusminus>\<^bsub>A\<^esub>) \<in> carrier A \<rightarrow> carrier A \<rightarrow> carrier A \<and> (\<forall>a b c. a \<in> carrier A \<longrightarrow> b \<in> carrier A \<longrightarrow> c \<in> carrier A \<longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c)) \<and> (\<forall>a b. a \<in> carrier A \<longrightarrow> b \<in> carrier A \<longrightarrow> a \<plusminus>\<^bsub>A\<^esub> b = b \<plusminus>\<^bsub>A\<^esub> a)) \<and> (mop A \<in> carrier A \<rightarrow> carrier A \<and> (\<forall>a. a \<in> carrier A \<longrightarrow> -\<^sub>a\<^bsub>A\<^esub> a \<plusminus>\<^bsub>A\<^esub> a = \<zero>\<^bsub>A\<^esub>)) \<and> \<zero>\<^bsub>A\<^esub> \<in> carrier A \<and> (\<forall>a. a \<in> carrier A \<longrightarrow> \<zero>\<^bsub>A\<^esub> \<plusminus>\<^bsub>A\<^esub> a = a)"
equality_axioms_def: shows "equality_axioms = {refl_ax, eq_cong_ax, syc_cong_ax, pls_cong_ax, tms_cong_ax}"
subst_comp_scheme_list: fixes R :: "nat \<Rightarrow> typ"   and S :: "nat \<Rightarrow> typ"   and A :: "type_scheme list" shows "$ R ($ S A) = $ (\<lambda>x. $ R (S x)) A"
csj: shows "cs\<^bsup>\<pi>\<^esup> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k)) = map \<pi> (sorted_list_of_set ({i. (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k)) cd\<^bsup>\<pi>\<^esup>\<rightarrow> i} \<union> {GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> ipd (\<pi> k)}))"
critD: fixes U :: "'a set"   and s :: "'a" assumes "crit U s" shows "s \<notin> U"
vec_p_prod_meet: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> nc) \<cdot> nc \<parallel> ((y \<sqinter> nc) \<cdot> nc) = (x \<sqinter> nc) \<cdot> nc \<sqinter> (y \<sqinter> nc) \<cdot> nc"
smallest_closed_cdi1: shows "M \<subseteq> smallest_ccdi_sets \<Omega> M"
simps: fixes Q_pos :: "(nat set \<times> 'b option list set) set"   and Q_neg :: "(nat set \<times> 'b option list set) set" shows "New_max_getIJ_wrapperGenericJoin Q_pos Q_neg = (if (\<exists>(A, y)\<in>Q_pos. Set.is_empty y) \<or> (\<exists>(A, X)\<in>Q_neg. Set.is_empty A \<and> \<not> Set.is_empty X) then {} else let Q = Set.filter (\<lambda>(A, uu_). \<not> Set.is_empty A) Q_pos in if Set.is_empty Q then (\<Inter>(A, X)\<in>Q_pos. X) - (\<Union>(A, X)\<in>Q_neg. X) else let V = \<Union>(A, X)\<in>Q. A in Let (Set.filter (\<lambda>(A, uu_). A \<subseteq> V \<and> 1 \<le> card A) Q_neg) (New_max_getIJ_genericJoin V Q))"
Grp_def: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'g" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
subpath_def: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> 'a" shows "subpath a b g \<equiv> \<lambda>x. g ((b - a) * x + a)"
Collect_disj_Un: fixes f :: "'b \<Rightarrow> 'a"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" shows "{f i |i. P i \<or> Q i} = {f i |i. P i} \<union> {f i |i. Q i}"
OnlyOccurenceDisables: fixes cfg1 :: "('p, 'v, 's) configuration"   and m :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration"   and m' :: "('p, 'v) message" assumes "cfg1 \<turnstile> m \<mapsto> cfg2"   and "enabled cfg1 m'"   and "\<not> enabled cfg2 m'" shows "m = m'"
less_le_not_le: fixes x :: "'b multiset"   and y :: "'b multiset" shows "(x \<subset># y) = strict (\<subseteq>#) x y"
uniformly_convergent_on_def: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" shows "uniformly_convergent_on X f = (\<exists>l. uniform_limit X f l sequentially)"
A1: fixes x :: "'a list"   and y :: "'a list" assumes "x \<noteq> []"   and "y \<noteq> []" shows "h (f x y) = g (h x) (h y)"
assms_1: shows "QD('u\<^sub>1) = QD('u\<^sub>2)"
axioms_2: fixes f :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f" shows "bounded_clinear_axioms f"
bisimE_2: fixes P :: "ccs"   and Q :: "ccs" assumes "P \<sim> Q" shows "Q \<sim> P"
arr_cod_iff_arr: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "category C" shows "partial_magma.arr C (partial_magma.cod C f) = partial_magma.arr C f"
abs_eq_mult: 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 times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.ordered_ring_abs abs plus zero minus uminus less_eq less times"   and "(less_eq zero a \<or> less_eq a zero) \<and> (less_eq zero b \<or> less_eq b zero)" shows "abs (times a b) = times (abs a) (abs b)"
Complete_elem_in_Gen: fixes I :: "item set"   and k :: "nat"   and u :: "nat"   and x :: "item" assumes "I \<subseteq> Gen (\<P> k u)"   and "k \<le> length Doc"   and "x \<in> Complete k I" shows "x \<in> Gen (\<P> k u)"
0: shows "one_chain_line_integral F basis (one_chain1 - ps1) = one_chain_line_integral F basis subdiv"
cnf_of_encode_initial_state_set_i: fixes \<Pi> :: "'b strips_problem" shows "cnf (\<Phi>\<^sub>I \<Pi>) = \<Union> {cnf (encode_state_variable 0 (index (\<Pi>\<^sub>\<V>) v) ((\<Pi>\<^sub>I) v)) |v. v \<in> set (\<Pi>\<^sub>\<V>) \<and> (\<Pi>\<^sub>I) v \<noteq> None}"
part_implies_overlap: fixes x :: "i"   and y :: "i" assumes "P x y" shows "O x y"
borel_cantelli_limsup1: fixes A :: "nat \<Rightarrow> 'b set"   and M :: "'b measure" assumes "\<And>n. A n \<in> sets M"   and "\<And>n. emeasure M (A n) < \<infinity>"   and "summable (\<lambda>n. Sigma_Algebra.measure M (A n))" shows "limsup A \<in> null_sets M"
simps_2: fixes I :: "nat set"   and n :: "nat" shows "I \<rightarrow> Suc n = inext (I \<rightarrow> n) I"
vector_space_on_axioms: shows "vector_space_on U\<^sub>M\<^sub>_\<^sub>1 (*s\<^sub>1)"
dickson_le_def: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ord_strict :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" assumes "gd_powerprod ord ord_strict" shows "gd_powerprod.dickson_le ord d m s t = (d s \<le> m \<and> d t \<le> m \<and> ord s t)"
range_inj_infinite: fixes f :: "nat \<Rightarrow> 'c" assumes "inj f" shows "infinite (range f)"
ln_2_le': shows "ln 2 \<le> 16 / 23"
Cons_outer_2: shows "\<not> no_matching_Goto \<gamma> p (r # rs)"
atU_union: fixes U :: "'b set"   and c1 :: "'b list multiset"   and c2 :: "'b list multiset" shows "atU U (c1 + c2) = (atU U c1 \<or> atU U c2)"
derives_empty_implies_LeftDerivation: fixes \<delta> :: "symbol list" assumes "derives \<delta> []" shows "\<exists>D. LeftDerivation \<delta> D []"
a: shows "concat (tms_of ap) = tp1 @ ci ly (start_of ly as) ins @ tp2 \<and> tp1 = concat (take as (tms_of ap)) \<and> tp2 = concat (drop (Suc as) (tms_of ap))"
suppD_contra: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "f x = (0::'b)" shows "x \<notin> supp f"
convex_closure: fixes S :: "'c set" assumes "convex S" shows "convex (closure S)"
IH: fixes x2aa :: "fterm" assumes "x2aa \<in> set ts"   and "ground\<^sub>t x2aa" shows "eval\<^sub>t E F x2aa = eval\<^sub>t E' F x2aa"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a arrow_of_spans_data \<Rightarrow> 'a arrow_of_spans_data" assumes "natural_isomorphism (\<bullet>) (\<bullet>) L local.map \<phi>" shows "equivalence_functor (\<bullet>) (\<bullet>) L"
emeasure_neq_0_sets: fixes M :: "'a measure"   and A :: "'a set" assumes "emeasure M A \<noteq> 0" shows "A \<in> sets M"
swap_Send_Snapshot: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isSend ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
cmod_cos_le_exp: fixes u :: "real"   and z :: "complex" assumes "0 \<le> u"   and "u \<le> 1" shows "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
vdiffD2: fixes x :: "V"   and A :: "V"   and B :: "V"   and P :: "bool" assumes "x \<in>\<^sub>\<circ> A -\<^sub>\<circ> B"   and "x \<in>\<^sub>\<circ> B" shows "P"
continuous_map_closedin: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "continuous_map X Y f = ((\<forall>x\<in>topspace X. f x \<in> topspace Y) \<and> (\<forall>C. closedin Y C \<longrightarrow> closedin X {x \<in> topspace X. f x \<in> C}))"
sum_var_monom_var: fixes x :: "'b"   and y :: "'b" shows "sum_var (var_monom x) y = (if x = y then 1 else 0)"
varE: fixes x :: "'b"   and Y :: "'b ptrm" assumes "PVar x \<approx> Y" obtains "Y = PVar x"
thetaIfWT_def: shows "thetaIfWT \<equiv> {(if tst then c1 else c2, if tst then d1 else d2) |tst c1 c2 d1 d2. compatTst tst \<and> c1 \<approx>wT d1 \<and> c2 \<approx>wT d2}"
counit_simps_4: shows "src \<epsilon>\<^sub>1 = trg e\<^sub>1"
alw_eq_sconst: fixes h :: "'c"   and t :: "'c stream" shows "alw (HLD {h}) t = (t = sconst h)"
append_assoc: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(xs @ ys) @ zs = xs @ ys @ zs"
assms: shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<circ> A"
elims: fixes x :: "nat"   and y :: "PrimScheme" assumes "nat_to_sch x = y" obtains "x = 0" and "y = Base_zero"   | "x = Suc 0" and "y = Base_zero"   | va :: "nat" where "x = Suc (Suc va)" and "y = (let u = mod7 (c_fst (Suc (Suc va))); v = c_snd (Suc (Suc va)); v1 = c_fst v; v2 = c_snd v; sch1 = nat_to_sch v1 in Let (nat_to_sch v2) (loc_f u sch1))"
unfold2: fixes E :: "(\<AA> state \<times> \<AA> state \<Rightarrow> Set(\<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)) \<Rightarrow> \<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state"   and self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and POST' :: "(\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>) \<Rightarrow> \<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and BODY :: "(\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>) \<Rightarrow> \<AA> state \<times> \<AA> state \<Rightarrow> Set(\<langle>\<langle>int\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>)" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self"   and "\<tau> \<Turnstile> true"   and "\<tau> \<Turnstile> POST' self"   and "\<And>res. (res \<triangleq> if self.boss \<doteq> null then Set{self.salary} else self.boss.contents()->including\<^sub>S\<^sub>e\<^sub>t(self.salary) endif) = (POST' self and (res \<triangleq> BODY self))" shows "(\<tau> \<Turnstile> E (self.contents())) = (\<tau> \<Turnstile> E (BODY self))"
cond1: shows "A $ mod_type_class.from_nat (k - 1) $ mod_type_class.from_nat (k - 1) dvd A $ mod_type_class.from_nat k $ mod_type_class.from_nat k"
times_closed': shows "\<forall>x\<in>U. \<forall>y\<in>U. x *\<^sub>o\<^sub>w y \<in> U"
induct: fixes x1 :: "('a, 'b, 'c, 'd) Tree"   and x2 :: "('a, 'b, 'c, 'd) Tree"   and P :: "('a, 'b, 'c, 'd) Tree \<Rightarrow> ('a, 'b, 'c, 'd) Tree \<Rightarrow> bool" assumes "(x1, x2) \<in> Tree_wf"   and "\<And>t tset. t \<in> set_bset tset \<Longrightarrow> P t (tConj tset)"   and "\<And>t. P t (tNot t)"   and "\<And>t f \<alpha>. P t (tAct f \<alpha> t)" shows "P x1 x2"
obt_2: shows "S \<subseteq> {x - a |x. x \<in> s}"
assms: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "b (x, y) = b (x, z)"
powser_zero: fixes f :: "nat \<Rightarrow> 'b" shows "(\<Sum>n. f n * (0::'b) ^ n) = f 0"
null_def: fixes xs :: "'a list" shows "List.null xs = (xs = [])"
set_child_nodes_get_child_nodes: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "DocumentClass.known_ptr ptr"   and "DocumentClass.type_wf h"   and "h \<turnstile> set_child_nodes ptr children \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
eqv_class_eq': fixes a :: "'b"   and b :: "'b" assumes "a \<in> domain" shows "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
K_def: shows "K = (\<Sum>n. (inverse (real (Suc n)))\<^sup>2)"
F: shows "\<forall>a\<in>A. |F a| <o r"
rtranclp_rtrancl_eq: fixes r :: "('a \<times> 'a) set" shows "(\<lambda>x xa. (x, xa) \<in> r)\<^sup>*\<^sup>* = (\<lambda>x xa. (x, xa) \<in> r\<^sup>*)"
simps_8: fixes vb :: "H1" shows "CX4h vb {X4.Xd1', X4.Xd2} = {X4.Xd1'}"
rel_fundef_arities: fixes r :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and gd1 :: "'l fundef"   and gd2 :: "'m fundef" assumes "rel_fundef r gd1 gd2" shows "arity gd1 = arity gd2"
flow_leaves_compact_ivl_left: fixes x0 :: "'a"   and K :: "'a set" assumes "0 \<in> UNIV"   and "x0 \<in> X"   and "bdd_below (existence_ivl0 x0)"   and "Inf (existence_ivl0 x0) \<in> UNIV"   and "compact K"   and "K \<subseteq> X" obtains t :: "real" where "t \<le> 0" and "t \<in> existence_ivl0 x0" and "flow0 x0 t \<notin> K"
Ord_Least_le: fixes k :: "V"   and P :: "V \<Rightarrow> bool" assumes "Ord k"   and "P k" shows "(LEAST i. Ord i \<and> P i) \<le> k"
times_set_def: fixes A :: "'a set"   and B :: "'a set" shows "A * B = {u * v |u v. u \<in> A \<and> v \<in> B}"
trancl_trans: fixes x :: "'a"   and y :: "'a"   and r :: "('a \<times> 'a) set"   and z :: "'a" assumes "(x, y) \<in> r\<^sup>+"   and "(y, z) \<in> r\<^sup>+" shows "(x, z) \<in> r\<^sup>+"
poly_N_xi: fixes i :: "nat"   and j :: "nat" shows "poly (local.N i) (local.x j) = local.pprod j 0 i"
d_\<beta>_mono: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and l :: "int"   and l' :: "int" assumes "iszlfm p (a # bs)"   and "d_\<beta> p l"   and "l dvd l'" shows "d_\<beta> p l'"
Rep_mod_ring_inverse: fixes x :: "'b mod_ring" shows "Abs_mod_ring (Rep_mod_ring x) = x"
assms_4: fixes b :: "real" assumes "x0 \<le> b" shows "bounded (f ` {x0..b})"
assign_eval\<^sub>w': fixes eval\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and eval\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and mds :: "Mode \<Rightarrow> 'a set"   and mds' :: "Mode \<Rightarrow> 'a set"   and mem' :: "'a \<Rightarrow> 'b"   and mem :: "'a \<Rightarrow> 'b"   and x :: "'a"   and e :: "'c" assumes "sifum_lang_no_dma eval\<^sub>A eval\<^sub>B aexp_vars bexp_vars"   and "mds = mds'"   and "mem' = mem(x := eval\<^sub>A mem e)" shows "(((x \<leftarrow> e, mds), mem), (Stop, mds'), mem') \<in> sifum_lang_no_dma.eval\<^sub>w eval\<^sub>A eval\<^sub>B"
aforms_of_ivl_leI1: fixes en :: "real"   and xsn :: "real"   and ysn :: "real" assumes "- 1 \<le> en"   and "xsn \<le> ysn" shows "xsn \<le> (xsn + ysn) / 2 + (ysn - xsn) * en / 2"
S_def: shows "S = insert 1 {\<bar>x - y\<bar> |x y. poly (real_of_int_poly p) x = 0 \<and> poly (real_of_int_poly p) y = 0 \<and> x \<noteq> y}"
wens_single_finite_eq_Union: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and n :: "nat" shows "wens_single_finite act B n = \<Union> (wens_single_finite act B ` {..n})"
finvimage_vsingleton_in: fixes b :: "V"   and A :: "V"   and a :: "V" assumes "b \<in>\<^sub>\<circ> A" shows "set {[a, b]\<^sub>\<circ>} -`\<^sub>\<bullet> A = set {a}"
sel_rews_4: shows "stail\<cdot>[::] = \<bottom>"
chaos_seq_chaos: shows "chaos \<sqsubseteq> chaos ; chaos"
typing_cond_prot_stateful_def: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list set" shows "typing_cond_prot_stateful \<P> \<equiv> wf\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>s \<P> \<and> tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t ` \<P>) \<union> pair ` \<Union> (setops\<^sub>s\<^sub>s\<^sub>t ` unlabel ` \<P>)) \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t ` \<P>)) \<and> (\<forall>S\<in>\<P>. list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (unlabel S))"
le_supE: fixes a :: "'a"   and b :: "'a"   and x :: "'a" assumes "a \<squnion> b \<le> x" obtains "a \<le> x" and "b \<le> x"
dg_prod_is_arrI: fixes f :: "V"   and a :: "V"   and b :: "V" assumes "vsv f"   and "\<D>\<^sub>\<circ> f = 2\<^sub>\<nat>"   and "vsv a"   and "\<D>\<^sub>\<circ> a = 2\<^sub>\<nat>"   and "vsv b"   and "\<D>\<^sub>\<circ> b = 2\<^sub>\<nat>"   and "\<And>i. i \<in>\<^sub>\<circ> 2\<^sub>\<nat> \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>\<^esub> b\<lparr>i\<rparr>" shows "f : a \<mapsto>\<^bsub>\<Prod>\<^sub>D\<^sub>Gi\<in>\<^sub>\<circ>2\<^sub>\<nat>. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>\<^esub> b"
simps: fixes prec :: "nat"   and x :: "float" shows "lb_arctan prec x = (let ub_horner = \<lambda>x. float_round_up prec (x * ub_arctan_horner prec (get_odd (prec div 4 + 1)) 1 (float_round_down (Suc prec) (x * x))); lb_horner = \<lambda>x. float_round_down prec (x * lb_arctan_horner prec (get_even (prec div 4 + 1)) 1 (float_round_up (Suc prec) (x * x))) in if x < 0 then - ub_arctan prec (- x) else if x \<le> Float 1 (- 1) then lb_horner x else if x \<le> Float 1 1 then Float 1 1 * lb_horner (float_divl prec x (float_plus_up prec 1 (ub_sqrt prec (float_plus_up prec 1 (float_round_up prec (x * x)))))) else let inv = float_divr prec 1 x in if 1 < inv then 0 else float_plus_down prec (lb_pi prec * Float 1 (- 1)) (- ub_horner inv))"
dsafe_Var_simps: fixes x :: "'b" shows "dsafe (Var x) = True"
reflexive: fixes Rel :: "(ccs \<times> ccs) set"   and P :: "ccs" assumes "Id \<subseteq> Rel" shows "P \<leadsto>[Rel] P"
WTrtVar: fixes E :: "char list \<Rightarrow> ty option"   and V :: "char list"   and T :: "ty"   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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option" assumes "E V = \<lfloor>T\<rfloor>" shows "P,E,h \<turnstile> Var V : T"
independent_rotate1_aux: fixes u :: "'a"   and Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K (u # Us @ Vs)" shows "local.independent K ((Us @ [u]) @ Vs)"
Sup_UNIV: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup UNIV = top"
items_crypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" shows "items (insert (Crypt K X) H) = insert (Crypt K X) (items (insert X H))"
NOT_mask: fixes n :: "nat" shows "NOT (mask n) = - (2 ^ n)"
insert_greaterThan: fixes n :: "nat" shows "insert n {n<..} = {n..}"
to_gba_alt_L: fixes m :: "'b" shows "gba_L (to_gba_ext m) = L"
eq_fin_concat_end: fixes w\<^sub>1 :: "'action list"   and w :: "'action list"   and w\<^sub>2 :: "'action list" shows "(w\<^sub>1 @ w =\<^sub>F w\<^sub>2 @ w) = (w\<^sub>1 =\<^sub>F w\<^sub>2)"
lcm_mult_distrib': fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times (normalize c) (lcm a b) = lcm (times c a) (times c b)"
K_statefun_apply: fixes c :: "'e"   and x :: "'f" shows "K_statefun c x = c"
commute: fixes a :: "'b upred"   and b :: "'b upred" shows "(a \<and> b) = (b \<and> a)"
word_less_nat_alt: fixes a :: "'b word"   and b :: "'b word" shows "(a < b) = (unat a < unat b)"
sigma_sets_mono'': fixes A :: "'b set"   and C :: "'b set"   and D :: "'b set set"   and B :: "'b set set" assumes "A \<in> sigma_sets C D"   and "B \<subseteq> D"   and "D \<subseteq> Pow C" shows "sigma_sets A B \<subseteq> sigma_sets C D"
s_def: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event list"   and i :: "nat" shows "s c t i = (THE c'. trace c (take i t) c')"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> set xs"   and "y \<in> set ys"   and "snd (P x y)" shows "snd (P' x y)"
currentD_weight_IN: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal"   and x :: "'k" assumes "current \<Gamma> f" shows "d_IN f x \<le> weight \<Gamma> x"
bisim_inv_red_other: fixes t' :: "'t"   and x :: "'x1"   and m1 :: "'m1"   and xx :: "'x2"   and m2 :: "'m2"   and t :: "'t"   and x1 :: "'x1"   and x2 :: "'x2"   and x1' :: "'x1"   and ta1 :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and x1'' :: "'x1"   and m1' :: "'m1"   and x2' :: "'x2"   and ta2 :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and x2'' :: "'x2"   and m2' :: "'m2" assumes "t' \<turnstile> (x, m1) \<approx> (xx, m2)"   and "t \<turnstile> (x1, m1) \<approx> (x2, m2)"   and "r1.silent_moves t (x1, m1) (x1', m1)"   and "r1.r_syntax t x1' m1 ta1 x1'' m1'"   and "\<not> \<tau>move1 (x1', m1) ta1 (x1'', m1')"   and "r2.silent_moves t (x2, m2) (x2', m2)"   and "r2.r_syntax t x2' m2 ta2 x2'' m2'"   and "\<not> \<tau>move2 (x2', m2) ta2 (x2'', m2')"   and "t \<turnstile> (x1'', m1') \<approx> (x2'', m2')"   and "ta1 \<sim>m ta2" shows "t' \<turnstile> (x, m1') \<approx> (xx, m2')"
erH_rHom_0: fixes A :: "('g, 'h) Ring_scheme"   and B :: "('g, 'i) Ring_scheme"   and Y :: "'g"   and h :: "'a \<Rightarrow> 'g" assumes "PolynRg A B Y"   and "h \<in> rHom S B" shows "erH R S X A B Y h \<zero> = \<zero>\<^bsub>A\<^esub>"
completeI: fixes A :: "'b set"   and CC :: "'b set set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>X. \<lbrakk>X \<subseteq> A; X \<in> CC\<rbrakk> \<Longrightarrow> Ex (extreme_bound A r X)" shows "CC-complete A r"
elims_1: fixes x :: "('state, 'effect) L_state"   and xa :: "'pred"   and y :: "bool" assumes "x \<turnstile>\<^sub>L xa = y" obtains uu_ :: "'effect" and uv_ :: "'effect fs_set" and P :: "'state" and \<phi> :: "'pred" where "x = AC (uu_, uv_, P)" and "xa = \<phi>'__" and "y = P \<turnstile> \<phi>'__"   | uw_ :: "'effect fs_set \<times> 'state" and \<phi> :: "'pred" where "x = EF uw_" and "xa = \<phi>" and "\<not> y"
permutes_implies_image_mset_eq: fixes p :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b" assumes "p permutes A"   and "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)" shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"