hcju/isabelleps huggingface dataset:
neg_one_eq_numeral_iff: fixes one :: "'b"   and times :: "'b \<Rightarrow> '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 n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus one = uminus (numeral.numeral one plus n)) = (n = num.One)"
bigoI: fixes f :: "'a \<Rightarrow> 'b"   and c :: "real"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)" shows "f \<in> O[F](g)"
disjoint_sum: fixes B :: "'d set set"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "finite B"   and "finite A"   and "Congruence.partition A B" shows "sum (sum f) B = sum f A"
continuous_within_sequentiallyI: fixes a :: "'b"   and s :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "\<And>u. \<lbrakk>u \<longlonglongrightarrow> a; \<forall>n. u n \<in> s\<rbrakk> \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a" shows "continuous (at a within s) f"
set_qsort: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list" shows "set (qsort xs l) = set xs"
that: fixes q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q"   and "\<not> is_sig_red sing_reg top_tail F q" shows "thesis"
l_neg: fixes x :: "int" assumes "x \<in> UNIV" shows "\<ominus>\<^bsub>\<Z>\<^esub> x + x = 0"
inject: 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)"
elims: fixes x :: "nat \<times> nat list"   and xa :: "abc_inst list"   and xb :: "nat"   and y :: "nat \<times> nat list" assumes "abc_steps_l x xa xb = y" obtains s :: "nat" and lm :: "nat list" and p :: "abc_inst list" where "x = (s__, lm__)" and "xa = p__" and "xb = 0" and "y = (s__, lm__)"   | s :: "nat" and lm :: "nat list" and p :: "abc_inst list" and n :: "nat" where "x = (s, lm)" and "xa = p" and "xb = Suc n" and "y = abc_steps_l (abc_step_l (s, lm) (abc_fetch s p)) p n"
ring_hom_restrict: fixes f :: "'a \<Rightarrow> 'd"   and S :: "('d, 'e) ring_scheme"   and g :: "'a \<Rightarrow> 'd" assumes "f \<in> ring_hom R S"   and "\<And>r. r \<in> carrier R \<Longrightarrow> f r = g r" shows "g \<in> ring_hom R S"
final_threadE: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" assumes "r2.final_thread s t" obtains x :: "'x2" where "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" and "final2 x" and "wset s t = None"
of_rat_eq_0_iff: fixes a :: "rat" shows "(of_rat a = (0::'b)) = (a = 0)"
has_unary_products: fixes I :: "'b set" assumes "card I = 1"   and "I \<noteq> UNIV" shows "has_products I"
fin_sat_antimono: fixes F :: "'b formula set"   and G :: "'b formula set" assumes "fin_sat F"   and "G \<subseteq> F" shows "fin_sat G"
const_poly_dvd: fixes a :: "'a"   and b :: "'a" shows "([:a:] dvd [:b:]) = (a dvd b)"
integral_less_AE: fixes M :: "'a pmf"   and X :: "'a \<Rightarrow> real"   and Y :: "'a \<Rightarrow> real"   and A :: "'a set" assumes "integrable (measure_pmf M) X"   and "integrable (measure_pmf M) Y"   and "emeasure (measure_pmf M) A \<noteq> 0"   and "A \<in> measure_pmf.events M"   and "AE x\<in>A in measure_pmf M. X x \<noteq> Y x"   and "AE x in measure_pmf M. X x \<le> Y x" shows "measure_pmf.expectation M X < measure_pmf.expectation M Y"
prec_after_IPC_step: fixes s :: "rstate_t"   and aseq :: "int_point_t list"   and n :: "nat" assumes "rprecondition s (rcurrent s) (aseq ! n)"   and "Suc n < length aseq"   and "is_SK_IPC aseq"   and "\<not> raborting s (rcurrent s) (aseq ! n)"   and "\<not> rwaiting s (rcurrent s) (aseq ! n)" shows "rprecondition (rstep s (aseq ! n)) (rcurrent s) (aseq ! Suc n)"
lhd_llist_of: fixes xs :: "'c list" shows "lhd (llist_of xs) = hd xs"
extended_simps_4: fixes x' :: "'b" shows "\<not> \<lbrakk>x'\<rbrakk> < \<turnstile>"
sorted_keys: fixes t :: "('a, 'b) RBT.rbt" shows "sorted (RBT.keys t)"
EX_RG: shows "\<exists>f. NFlow c s t f \<and> cf = cf_of f"
positions_strict_border: fixes l :: "nat"   and w :: "'a list" assumes "l < length w"   and "\<forall>j<l. w ! j = w ! (length w - l + j)" shows "strict_border (take l w) w"
sign_swap_id: fixes a :: "'b"   and b :: "'b" shows "sign (Fun.swap a b id) = (if a = b then 1 else - 1)"
inject: fixes x2 :: "'d"   and y2 :: "'d" shows "(Some x2 = Some y2) = (x2 = y2)"
fmpt_def: fixes M :: "'c measure"   and T :: "'c \<Rightarrow> 'c" shows "fmpt M T \<equiv> mpt M T \<and> finite_measure M"
elims: fixes x :: "('b \<times> bal) tree"   and y :: "('b \<times> bal) tree \<times> 'b" assumes "split_max x = y" obtains l :: "('b \<times> bal) tree" and a :: "'b" and ba :: "bal" and r :: "('b \<times> bal) tree" where "x = \<langle>l, (a, ba), r\<rangle>" and "y = (if r = \<langle>\<rangle> then (l, a) else let (r', a') = split_max r; t' = if decr r r' then balL l a ba r' else \<langle>l, (a, ba), r'\<rangle> in (t', a'))"   | "x = \<langle>\<rangle>" and "y = undefined"
getFrN_var: fixes xs :: "'var list"   and ts :: "'trm list"   and \<phi>s :: "'fmla list"   and t :: "'trm"   and n :: "nat" assumes "set xs \<subseteq> var"   and "set ts \<subseteq> trm"   and "set \<phi>s \<subseteq> fmla"   and "t \<in> set ts" shows "set (getFrN xs ts \<phi>s n) \<inter> set xs = {}"
islimpt_sequential: fixes x :: "'a"   and S :: "'a set" shows "x islimpt S = (\<exists>f. (\<forall>n. f n \<in> S - {x}) \<and> f \<longlonglongrightarrow> x)"
Fubini: fixes f :: "'a \<times> 'b \<Rightarrow> ennreal" assumes "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M1 \<partial>M2 = \<integral>\<^sup>+ x. \<integral>\<^sup>+ y. f (x, y) \<partial>M2 \<partial>M1"
preservation: fixes M :: "'a trm"   and M' :: "'a trm"   and \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and \<tau> :: "PreSimplyTyped.type" assumes "M \<rightarrow>\<beta>\<^sup>* M'"   and "\<Gamma> \<turnstile> M : \<tau>" shows "\<Gamma> \<turnstile> M' : \<tau>"
prob_space_PiM: fixes I :: "'b set"   and M :: "'b \<Rightarrow> 'c measure" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)" shows "prob_space (Pi\<^sub>M I M)"
append_butlast_last_id: fixes xs :: "'f list" assumes "xs \<noteq> []" shows "butlast xs @ [last xs] = xs"
eq0: shows "lst (fst \<gamma>1', snd \<gamma>1' @ snd \<sigma>1') = lst \<sigma>1'"
reindex_nontrivial: fixes A :: "'c set"   and h :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'a" assumes "finite A"   and "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y; h x = h y\<rbrakk> \<Longrightarrow> g (h x) = \<^bold>1" shows "F g (h ` A) = F (g \<circ> h) A"
connectedI_app_iso: fixes hom :: "('a, 'a \<times> nat \<times> 'a, 'c, 'd) digraph_isomorphism" assumes "Digraph_Component.connected (app_iso iso_f G)"   and "H.digraph_isomorphism hom" shows "Digraph_Component.connected (app_iso hom (app_iso iso_f G))"
lb2_sgn: shows "\<forall>x\<le>lb2. sgn (poly p x) = sgn_neg_inf p"
Mult: shows "\<forall>x\<in>P. \<forall>y\<in>P. x * y \<in> P"
cases: fixes a1 :: "('form, 'rule) graph_node"   and a2 :: "nat"   and a3 :: "'subst"   and a4 :: "'form fset \<times> 'form" assumes "local_fresh_check a1 a2 a3 a4" obtains \<Gamma> :: "'form fset" and i :: "nat" and n :: "('form, 'rule) graph_node" and s :: "'subst" and c :: "'form" where "a1 = n" and "a2 = i" and "a3 = s" and "a4 = (\<Gamma> \<turnstile> c)" and "\<And>f. f |\<in>| \<Gamma> \<Longrightarrow> freshenLC i ` all_local_vars n \<inter> lconsts f = {}" and "freshenLC i ` all_local_vars n \<inter> subst_lconsts s = {}"
solution2_def: fixes A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat"   and x :: "nat \<Rightarrow> 'b" shows "solution2 A m n x = (\<forall>i<m. (\<Sum>j = 0..<m. A i j * x j) = A i n)"
nonnegs_6: shows "0 \<le> det3 t r p"
exists_eq_sort: fixes sort_fun :: "'b \<Rightarrow> atom_sort" shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}"
is_empty_def: fixes t :: "('c, 'd) RBT.rbt" shows "is_empty t = (case impl_of t of rbt.Empty \<Rightarrow> True | Branch color rbt1 a b rbt2 \<Rightarrow> False)"
assms_2: shows "\<forall>x. g x \<le> f x"
ur2: shows "\<exists>!x. root_cond (p2, l2, r2) x"
assms_3: shows "\<rho> \<in> carrier_mat d d"
leg_simps_1: shows "C.arr HH\<mu>\<nu>\<pi>.leg0"
b: shows "lincomb (\<lambda>w. if w \<in> A then a w else \<zero>) S = \<zero>\<^bsub>M\<^esub>"
integrable_const_bound: fixes f :: "real \<Rightarrow> 'b"   and B :: "real" assumes "AE x in M. norm (f x) \<le> B"   and "random_variable borel f" shows "integrable M f"
less_numeral_extra_1: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less zero one"
conjr2_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x \<rhd> z \<le> y \<rhd> z"
rem_0_coeff: fixes p :: "(char list multiset \<times> int) multiset"   and q :: "(char list multiset \<times> int) multiset"   and xs :: "char list multiset" assumes "normalize_poly_p p q" shows "normalize_poly_p (add_mset (xs, 0) p) q"
psubstT_Var_in: fixes y :: "'var"   and txs :: "('trm \<times> 'var) list"   and s :: "'trm" assumes "y \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)"   and "(s, y) \<in> set txs" shows "psubstT (Var y) txs = s"
simps: fixes t :: "'a"   and a1 :: "'b"   and a2 :: "'c"   and ts' :: "'a"   and tp' :: "'d"   and f :: "'d \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'd \<Rightarrow> 'c" shows "triple_eq_pair (t, a1, a2) (ts', tp') f g = (t = ts' \<and> a1 = f tp' \<and> a2 = g ts' tp')"
prems_3: shows "i - 1 \<le> k + 1"
SemAssign: fixes t :: "Var \<Rightarrow> Val"   and s :: "Var \<Rightarrow> Val"   and x :: "Var"   and e :: "Expr" assumes "t = update s x (evalE e s)" shows " s , Assign x e \<rightarrow>\<^sub>1  t "
fact_reduce: fixes n :: "nat" assumes "0 < n" shows "fact n = of_nat n * fact (n - 1)"
Lcm_factorial_eq_0_iff: fixes A :: "'a set" shows "(Lcm_factorial A = (0::'a)) = ((0::'a) \<in> A \<or> \<not> subset_mset.bdd_above (prime_factorization ` A))"
mpoly_eq_iff: fixes p :: "'a mpoly"   and q :: "'a mpoly" shows "(p = q) = (\<forall>m. MPoly_Type.coeff p m = MPoly_Type.coeff q m)"
intros_7: shows "CK_nf_pos (K \<circ> I \<circ> K)"
fls_inverse_subdegree: fixes f :: "'a fls" shows "fls_subdegree (inverse f) = - fls_subdegree f"
contains_propagates_natoms: fixes n :: "nat"   and p :: "nat"   and vs :: "nat list"   and q :: "nat" assumes "infinite (deriv s)"   and "contains f n (0, NAtom p vs)" shows "contains f (n + q) (0, NAtom p vs)"
simps_10: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and f3 :: "'a" shows "(case \<stileturn> of \<turnstile> \<Rightarrow> f1 | \<lbrakk>x\<rbrakk> \<Rightarrow> f2 x | \<stileturn> \<Rightarrow> f3) = f3"
Un_Diff_cancel: fixes A :: "'a set"   and B :: "'a set" shows "A \<union> (B - A) = A \<union> B"
get_odd_ex: fixes n :: "nat" shows "\<exists>k. Suc k = get_odd n \<and> odd (Suc k)"
inverse_def: shows "inverse = (\<lambda>u\<in>M. THE v. v \<in> M \<and> u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
domains_comp: fixes g :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "A_B.seq g f" shows "A_B.domains (g \<cdot>\<^sub>[\<^sub>A\<^sub>,\<^sub>B\<^sub>] f) = A_B.domains f"
less_bot_Bot_Value: fixes x :: "'a" shows "Bot < Value x"
interpret_ode_d_fa: fixes xs :: "floatarith list"   and vs :: "real list"   and ds :: "floatarith list"   and n :: "nat" assumes "einterpret xs vs \<in> Csafe"   and "length ds = DIM('c)"   and "length xs = DIM('c)" shows "einterpret (ode_d_fa n xs ds) vs = blinfun_apply (ode_d n (einterpret xs vs) (einterpret ds vs)) (einterpret ds vs)"
hm_it_adjust_rule: fixes i :: "nat"   and l :: "('a \<times> 'b) list list"   and ht :: "('a, 'b) hashtable" assumes "i < length l" shows "<is_hashtable l ht> hm_it_adjust i ht <\<lambda>j. is_hashtable l ht * \<up> (j \<le> i \<and> concat (take (Suc i) l) = concat (take (Suc j) l) \<and> (j = 0 \<or> l ! j \<noteq> []))>"
translation_graph: fixes X :: "'a allegorical_term" shows "graph (translation X)"
smc_Comp_vdomain: fixes gf :: "V" shows "(gf \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (\<BB>\<lparr>Comp\<rparr>)) = (\<exists>g f b c a. gf = [g, f]\<^sub>\<circ> \<and> g : b \<mapsto>\<^bsub>\<BB>\<^esub> c \<and> f : a \<mapsto>\<^bsub>\<BB>\<^esub> b)"
powhr_powhr: fixes x :: "real star"   and a :: "real star"   and b :: "real star" shows "(x powhr a) powhr b = x powhr (a * b)"
antisymI: fixes r :: "('c \<times> 'c) set" assumes "\<And>x y. \<lbrakk>(x, y) \<in> r; (y, x) \<in> r\<rbrakk> \<Longrightarrow> x = y" shows "antisym r"
Diff_infinite_finite: fixes T :: "'e set"   and S :: "'e set" assumes "finite T"   and "infinite S" shows "infinite (S - T)"
quasi_inverses_isomorphic_right: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and g' :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "bicategory.quasi_inverses V H \<a> \<i> src trg f g" shows "bicategory.quasi_inverses V H \<a> \<i> src trg f g' = category.isomorphic V g g'"
of_nat_word_less_eq_iff: fixes m :: "nat"   and n :: "nat" shows "(Word.of_nat m \<le> Word.of_nat n) = (take_bit LENGTH('k) m \<le> take_bit LENGTH('k) n)"
C_TApp: fixes E :: "trm \<Rightarrow> trm"   and T :: "POPLmarkRecord.type" assumes "E \<in> ctxt" shows "(\<lambda>t. E t \<bullet>\<^sub>\<tau> T) \<in> ctxt"
eq_neutr_iff: fixes a :: "'c \<times> 'c \<Rightarrow> 'd"   and b :: "'c \<times> 'c \<Rightarrow> 'd" shows "(a \<oplus> b = mbot) = (a = mbot \<and> b = mbot)"
vfield_iff: fixes a :: "V"   and r :: "V" shows "(a \<in>\<^sub>\<circ> \<F>\<^sub>\<circ> r) = (\<exists>b. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<or> \<langle>b, a\<rangle> \<in>\<^sub>\<circ> r)"
distinct_2: fixes x2 :: "enat" shows "eSuc x2 \<noteq> 0"
linear_span_subModule: fixes A :: "'b set"   and H :: "'a set" assumes "ideal R A"   and "H \<subseteq> carrier M" shows "submodule R M (linear_span R M A H)"
partial_state2_def: fixes vars1 :: "nat set"   and vars2 :: "nat set" shows "partial_state2 vars1 vars2 \<equiv> vars1 \<inter> vars2 = {}"
abs_eq: fixes xa :: "nat"   and x :: "real_alg_3" shows "root_real_alg xa (Real_Alg_Quotient x) = Real_Alg_Quotient (root_3 xa x)"
add_mset_remove_trivial: fixes x :: "'a"   and M :: "'a multiset" shows "add_mset x M \<ominus> x = M"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains s :: "name" and x :: "tm" and y :: "tm" and z :: "tm" where "atom s \<sharp> (x, y, z)" and "xa__ = (x, y, z)"
fps_divide_cancel: fixes h :: "'b fps"   and f :: "'b fps"   and g :: "'b fps" assumes "h \<noteq> 0" shows "f * h / (g * h) = f / g"
integral_mult: fixes f :: "'a \<Rightarrow> real"   and X :: "'a set"   and K :: "real" assumes "f integrable_on X" shows "K * integral X f = integral X (\<lambda>x. K * f x)"
approx_floatarith_Elem: fixes p :: "nat"   and ra :: "floatarith"   and VS :: "((real \<times> real pdevs) \<times> real) list"   and X :: "(real \<times> real pdevs) \<times> real"   and e :: "nat \<Rightarrow> real"   and vs :: "real list" assumes "approx_floatarith p ra VS = Some X"   and "e \<in> UNIV \<rightarrow> {- 1..1}"   and "vs \<in> aforms_err e VS" shows "interpret_floatarith ra vs \<in> aform_err e X"
vector_space_on_def: fixes U\<^sub>V\<^sub>S :: "'c set"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "vector_space_on U\<^sub>V\<^sub>S scale \<equiv> module_on U\<^sub>V\<^sub>S scale"
weak_supplementation: fixes y :: "i"   and x :: "i" assumes "MM"   and "PP y x" shows "\<exists>z. P z x \<and> \<not> O z y"
assms_1: shows "\<forall>s\<in>\<RR> Tr Up s\<^sub>0. Tr s \<noteq> {}"
A: shows "sinks_aux I D U (ipurge_tr_aux I' D' U' xs) \<subseteq> sinks_aux I D U xs"
intro: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'d set" assumes "map \<alpha> S T"   and "injective_map_axioms \<alpha> S" shows "injective_map \<alpha> S T"
inject: fixes x2 :: "name"   and x1 :: "name"   and y2 :: "name"   and y1 :: "name" shows "(x2[x1] = y2[y1]) = (x2 = y2 \<and> x1 = y1)"
vars\<^sub>s\<^sub>tI: fixes x :: "'a"   and S :: "('b, 'a) strand_step list" assumes "x \<in> fv\<^sub>s\<^sub>t S" shows "x \<in> vars\<^sub>s\<^sub>t S"
right_translate_bij: fixes g :: "'g" assumes "g \<in> G" shows "bij_betw (\<lambda>x. x + g) G G"
one_induced_iff_principal: shows "principal.induced_modulus 1 = (principal_dchar n = principal_dchar n)"
state_to_strips_state_range_is: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and v :: "'c"   and a :: "'d"   and s :: "'c \<Rightarrow> 'd option" assumes "is_valid_problem_sas_plus \<Psi>"   and "(v, a) \<in> dom (\<phi>\<^sub>S \<Psi> s)" shows "(\<phi>\<^sub>S \<Psi> s) (v, a) = Some (the (s v) = a)"
simps_1: fixes uu :: "(real, 'b) vec" shows "KD_Tree.complete (Leaf uu) = True"
join: fixes a :: "'b\<^sub>m"   and b :: "'b\<^sub>m" assumes "rh a = rh b"   and "a \<in> A" shows "\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u)"
sg_i_induced2: fixes H :: "'a set"   and K :: "'a set"   and a :: "'a" assumes "G \<guillemotright> H "   and "G \<guillemotright> K "   and "a \<in> H \<inter> K" shows "\<rho>\<^bsub>\<natural>H \<inter> K\<^esub> a = \<rho>\<^bsub>\<natural>K\<^esub> a"
negation_implies: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> y) = - - x \<sqinter> - y"
normal_form: fixes f :: "recf"   and n :: "nat" assumes "recfn n f" obtains i :: "nat" where "\<forall>x. e_length x = n \<longrightarrow> eval r_normal_form [i, x] = eval f (Partial_Recursive.list_decode x)"
elims: fixes x :: "int option"   and y :: "nat" assumes "int_option_enumeration x = y" obtains n :: "int" where "x = Some n" and "y = (if 0 \<le> n then nat (2 * (n + 1)) else nat (2 * - n + 1))"   | "x = None" and "y = 0"
phis'_code_correct: fixes g :: "'g" assumes "uninst_code.ssa.redundant (u_g g s1_uses) (p_g g s1_phis) g" shows "uninst.phis' ((usesOf \<circ>\<circ>\<circ> u_g) g s1_uses) (\<lambda>ga. Mapping.lookup (p_g g s1_phis ga)) g = Mapping.lookup (uninst_code.phis'_code (p_g g s1_phis) g (uninst_code.chooseNext' (u_g g s1_uses) (p_g g s1_phis) g))"
all_relation_is_weak_reduction_bisimulation: fixes Cal :: "'a processCalculus" shows "weak_reduction_bisimulation (_Collect (a, b) True) Cal"
simps_5: fixes v :: "poscheckvariant"   and va :: "('c, 'd) Term.term"   and vb :: "('c, 'd) Term.term" shows "simple\<^sub>s\<^sub>t\<^sub>p \<langle>v: va \<doteq> vb\<rangle>\<^sub>s\<^sub>t = False"
fnA: fixes n :: "nat" shows "f^n \<bottom> \<in> A"
intro: fixes carrier :: "'c set" assumes "finite carrier" shows "finite_total_preorder_on_axioms carrier"
eval_tm_SUCC: fixes e :: "(name, hf) finfun"   and t :: "tm" shows "\<lbrakk>SUCC t\<rbrakk>e = succ \<lbrakk>t\<rbrakk>e"
Sup_aform: fixes e :: "nat \<Rightarrow> real"   and X :: "'a \<times> 'a pdevs" assumes "e \<in> funcset UNIV {- 1..1}" shows "aform_val e X \<le> Sup_aform X"
next_commutes_5: fixes f :: "bool \<Rightarrow> bool"   and x :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "\<circle>f<x> \<equiv> f<\<circle>x>"
evalEffect_pState_inv: fixes prog :: "program"   and p :: "pState"   and g :: "gState\<^sub>I"   and e :: "edgeEffect" assumes "pState_inv prog p"   and "gState_inv prog g"   and "cl_inv (g, p)"   and "e \<in> effect ` edgeSet (states prog !! pState.idx p)" shows "pState_inv prog (snd (evalEffect e prog g p))"
reads_consistent_snoc_Ghost\<^sub>s\<^sub>b: fixes pending_write :: "bool"   and \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" assumes "reads_consistent pending_write \<O> m sb" shows "reads_consistent pending_write \<O> m (sb @ [Ghost\<^sub>s\<^sub>b A L R W])"
Inv1_def: fixes s :: "state" shows "Inv1 s = (\<forall>p. inpt s p \<in> Inputs \<and> phase s p \<le> 3 \<and> finite (allRdBlks s p))"
M2_def: shows "M2 \<equiv> PM \<and> M2_axioms"
InvariantWatchListsCharacterization_def: fixes Wl :: "Literal \<Rightarrow> nat list"   and w1 :: "nat \<Rightarrow> Literal option"   and w2 :: "nat \<Rightarrow> Literal option" shows "InvariantWatchListsCharacterization Wl w1 w2 = (\<forall>c l. (c \<in> set (Wl l)) = (Some l = w1 c \<or> Some l = w2 c))"
has_integral_spike: fixes S :: "'c set"   and T :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "(f has_integral y) T" shows "(g has_integral y) T"
dfs_invariant: fixes S :: "nat option list list \<times> (nat \<times> nat) list"   and xs :: "(nat \<times> nat) list"   and I :: "nat option list list \<times> (nat \<times> nat) list \<Rightarrow> bool" assumes "prod_invariant A B S"   and "list_all (prod_is_node A B) xs"   and "I S"   and "\<And>S x. \<lbrakk>\<not> prod_memb x S; prod_is_node A B x; prod_invariant A B S; I S\<rbrakk> \<Longrightarrow> I (prod_ins x S)" shows "I (dfs S xs)"
lens_put_irr2: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and u :: "'e" assumes "X \<bowtie> Y" shows "get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>"
that: fixes V :: "'a set set"   and a :: "'a"   and f :: "'a"   and b :: "'a"   and g :: "'a" assumes "is_zariski_open V"   and "V \<subseteq> U"   and "\<pp> \<in> V"   and "a \<in> R"   and "f \<in> R"   and "b \<in> R"   and "g \<in> R"   and "f \<notin> \<pp>"   and "g \<notin> \<pp>"   and "\<And>\<qq>. \<qq> \<in> V \<Longrightarrow> f \<notin> \<qq> \<and> s \<qq> = quotient_ring.frac (R\<setminus>\<qq>) R (+) (\<cdot>) \<zero> a f"   and "\<And>\<qq>. \<qq> \<in> V \<Longrightarrow> g \<notin> \<qq> \<and> t \<qq> = quotient_ring.frac (R\<setminus>\<qq>) R (+) (\<cdot>) \<zero> b g" shows "thesis"
valid_wgraph_repr_def: fixes l :: "(('c \<times> 'c) \<times> 'd) list" shows "valid_wgraph_repr l = (valid_graph_repr (map fst l) \<and> valid_weight_repr l)"
assms_2: shows "h \<in> \<R>\<^sup>2"
red1Call: fixes e :: "'addr expr1"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and h :: "'heap"   and U :: "htype"   and P :: "'addr J1_prog"   and Ts :: "ty list"   and T :: "ty"   and body :: "'addr expr1"   and D :: "String.literal"   and uf :: "bool"   and t :: "'thread_id"   and x :: "'addr val list"   and exs :: "('addr expr1 \<times> 'addr val list) list" assumes "call1 e = \<lfloor>(a, M, vs)\<rfloor>"   and "typeof_addr h a = \<lfloor>U\<rfloor>"   and "P \<turnstile> class_type_of U sees M: Ts\<rightarrow>T = \<lfloor>body\<rfloor> in D"   and "length vs = length Ts" shows "uf,P,t \<turnstile>1 \<langle>(e, x)/exs,h\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>(blocks1 0 (Class D # Ts) body, Addr a # vs @ replicate (max_vars body) undefined_value)/(e, x) # exs,h\<rangle>"
all_edges_def: fixes S :: "nat set" shows "all_edges S \<equiv> mk_uedge ` {uv \<in> S \<times> S. fst uv \<noteq> snd uv}"