hcju/isabelleps huggingface dataset:
A_trusts_NS5: fixes K :: "nat"   and NB :: "nat"   and evs :: "event list"   and A :: "agent"   and NA :: "nat"   and B :: "agent"   and X :: "msg" assumes "Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace> \<in> parts (knows Spy evs)"   and "Crypt (shrK A) \<lbrace>Nonce NA, Agent B, Key K, X\<rbrace> \<in> parts (knows Spy evs)"   and "Key K \<notin> analz (knows Spy evs)"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_shared" shows "Says A B (Crypt K \<lbrace>Nonce NB, Nonce NB\<rbrace>) \<in> set evs"
valid: fixes g :: "('a, 'c, 'b) gen_cfg_wf" assumes "True" shows "valid_graph (gen_wf.\<alpha> g)"
subfield_iff_1: fixes K :: "'a set" assumes "field (R\<lparr>carrier := K\<rparr>)"   and "K \<subseteq> carrier R" shows "subfield K R"
empty_set_less_eq_def: fixes B :: "'a set" shows "({} \<sqsubseteq>'' B) = (B \<in> infinite_complement_partition)"
bot_finfun_apply: shows "($) bot = (\<lambda>_. bot)"
final_iff_SKIP: fixes c :: "com"   and s :: "char list \<Rightarrow> int" shows "final (c, s) = (c = SKIP)"
one_minus_of_nat_in_nonpos_Ints_iff: fixes n :: "nat" shows "((1::'a) - of_nat n \<in> \<int>\<^sub>\<le>\<^sub>0) = (0 < n)"
collect_interrupts_interrupted: fixes "is" :: "'b set"   and ias :: "'b interrupt_action list"   and t' :: "'b" assumes "interrupt_actions_ok is ias"   and "t' \<in> collect_interrupts ias" shows "t' \<in> is"
Domainp_eq_on: fixes X :: "'d set" shows "Domainp (eq_on X) = (\<lambda>x. x \<in> X)"
s_Roll: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e' :: "Syntax.term" assumes "\<lless>\<pi>, e\<ggreater> m\<rightarrow> \<lless>\<pi>', e'\<ggreater>" shows "\<lless>\<pi>, Roll e\<ggreater> m\<rightarrow> \<lless>\<pi>', Roll e'\<ggreater>"
par_beta_cases_4: fixes f :: "sterm"   and l :: "Label"   and t :: "sterm"   and u :: "sterm" assumes "Upd f l t => u" obtains t' :: "sterm" and L :: "char list set" and u' :: "sterm" where "u = Upd t'__ l u'" and "f => t'__" and "lc f" and "finite L__" and "\<forall>s p. s \<notin> L__ \<and> p \<notin> L__ \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> u' = \<sigma>[s,p] t'')" and "body t"   | fa :: "Label \<Rightarrow> sterm option" and T :: "Sigma.type" and f' :: "Label \<Rightarrow> sterm option" and L :: "char list set" and t' :: "sterm" where "f = Obj fa T" and "u = Obj (f'(l \<mapsto> t')) T" and "Obj fa T => Obj f' T" and "finite L" and "\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> => t'' \<and> t' = \<sigma>[s,p] t'')" and "l \<in> dom fa" and "lc (Obj fa T)" and "body t"
ide_src: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "B.ide (src \<mu>)"
vec_to_iarray_column: fixes i :: "'e"   and A :: "(('c, 'e) vec, 'd) vec" shows "vec_to_iarray (column i A) = column_iarray (mod_type_class.to_nat i) (matrix_to_iarray A)"
elims: fixes x :: "'a Literal set set"   and xa :: "'a Literal set"   and y :: "'a Literal set set" assumes "partial_evaluation x xa = y" obtains S :: "'a Literal set set" and C :: "'a Literal set" where "x = S" and "xa = C" and "y = {E. \<exists>D. D \<in> S \<and> E = D - C \<and> (\<nexists>L. L \<in> C \<and> complement L \<in> D)}"
UNIV_I: fixes x :: "'b" shows "s2p UNIV x"
simps_12: fixes a :: "floatarith" shows "fold_const_fa (Sqrt a) = Sqrt (fold_const_fa a)"
simps_3: fixes a :: "name"   and x :: "'a" shows "[(a, a)] \<bullet> x = [] \<bullet> x"
convolution_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "convolution f g = (\<lambda>x. \<Sum>y | x - y \<in> supp f \<and> y \<in> supp g. f (x - y) * g y)"
p2_1: fixes i :: "nat" shows "local.coeff p2 i \<in> carrier R"
sifum_example_axioms_def: shows "sifum_example_axioms \<equiv> \<forall>lc lc' lc''. eval_abv\<^sub>A lc lc' \<longrightarrow> eval_abv\<^sub>A lc lc'' \<longrightarrow> lc' = lc''"
find_min_def: fixes xs :: "('c, 'd) bintree option list" shows "find_min xs = (case Binomial_Queue.min xs of None \<Rightarrow> None | Some a \<Rightarrow> Binomial_Queue.find a xs)"
compat: fixes y :: "'a"   and x :: "'a"   and z :: "'a" assumes "y \<le> x"   and "gt y z" shows "gt x z"
cyclic_on_edge_succ: fixes x :: "'a" assumes "x \<in> verts G"   and "out_arcs G x \<noteq> {}" shows "cyclic_on (edge_succ GM) (out_arcs G x)"
spp_of_def: fixes r :: "'t \<Rightarrow>\<^sub>0 'b" shows "spp_of r = (lt r, rep_list r)"
setbydecision_all_appendAccept: fixes rs :: "'a common_primitive rule list"   and r :: "'a common_primitive match_expr"   and iface :: "iface" assumes "simple_ruleset (rs @ [Rule r Accept])" shows "setbydecision_all TYPE('b) iface rs FinalDeny = setbydecision_all TYPE('b) iface (rs @ [Rule r Accept]) FinalDeny"
bounds_consistent_def: fixes s :: "('a, 'b) state" shows "\<diamond> s \<equiv> \<forall>x. if \<B>\<^sub>l s x = None \<or> \<B>\<^sub>u s x = None then True else the (\<B>\<^sub>l s x) \<le> the (\<B>\<^sub>u s x)"
map_bset_bdoubleton: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e"   and y :: "'e" shows "map_bset f (bdoubleton x y) = bdoubleton (f x) (f y)"
assms_4: shows "g \<in> punit.reduced_GB (monomial (1::'a) ` S)"
dependent_insertD: fixes a :: "'c"   and S :: "'c set" assumes "a \<notin> vs2.span S"   and "vs2.dependent (insert a S)" shows "vs2.dependent S"
path_cons: fixes \<pi> :: "nat \<Rightarrow> 'n"   and \<pi>' :: "nat \<Rightarrow> 'n"   and m :: "nat" assumes "is_path \<pi>"   and "is_path \<pi>'"   and "\<pi> m = \<pi>' 0" shows "is_path (\<pi> @\<^bsup>m\<^esup> \<pi>')"
mismatchId: fixes a :: "name"   and b :: "name"   and P :: "pi" assumes "a \<noteq> b" shows "[a\<noteq>b]P \<sim> P"
subcls1_wfD: fixes C :: "char list"   and D :: "char list"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and wf_md :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> 'b \<Rightarrow> bool" assumes "P \<turnstile> C \<prec>\<^sup>1 D"   and "wf_prog wf_md P" shows "D \<noteq> C \<and> (D, C) \<notin> (subcls1 P)\<^sup>+"
finite_set: fixes xs :: "'c list" shows "finite (ran\<^sub>l xs)"
distinct_1: fixes x1 :: "nat"   and x2 :: "nat" shows "Pos x1 \<noteq> Neg x2"
prime_number_theorem'': shows "(\<lambda>x. real (card {p. prime p \<and> real p \<le> x})) \<sim>[at_top] (\<lambda>x. x / ln x)"
irreducible_altdef: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.idom times one plus zero minus uminus" shows "comm_semiring_1.irreducible times one zero x = (x \<noteq> zero \<and> \<not> dvd.dvd times x one \<and> (\<forall>b. dvd.dvd times b x \<longrightarrow> dvd.dvd times x b \<or> dvd.dvd times b one))"
face_cycle_set_self: fixes a :: "'b" shows "a \<in> face_cycle_set a"
G'_def: shows "G' = \<lparr>pverts = pverts G - {w}, parcs = {(u, v), (v, u)} \<union> (parcs G - {(u, w), (w, u), (v, w), (w, v)})\<rparr>"
bind_rotate: fixes C :: "'a stream \<Rightarrow> 'b \<Rightarrow> 'g measure"   and xa :: "'a"   and ya :: "'b"   and N :: "'g measure" assumes "T1K2.random_variable xa ya (subprob_algebra N) (\<lambda>(x, y). C x y)" shows "K1.T xa \<bind> (\<lambda>x. measure_pmf (K2 ya) \<bind> C x) = measure_pmf (K2 ya) \<bind> (\<lambda>y. K1.T xa \<bind> (\<lambda>x. C x y))"
subprob_space_axioms: fixes M :: "'a pmf" shows "subprob_space (measure_pmf M)"
smooth_compat_trans: fixes c1 :: "('a, 'b) chart"   and c2 :: "('a, 'b) chart" assumes "\<And>c. c \<in> charts \<Longrightarrow> k-smooth_compat c1 c"   and "\<And>c. c \<in> charts \<Longrightarrow> k-smooth_compat c2 c"   and "domain c1 \<inter> domain c2 \<subseteq> carrier" shows "k-smooth_compat c1 c2"
mult_mm_inv_r: fixes B :: "complex \<times> complex \<times> complex \<times> complex"   and A :: "complex \<times> complex \<times> complex \<times> complex"   and C :: "complex \<times> complex \<times> complex \<times> complex" assumes "mat_det B \<noteq> 0"   and "A *\<^sub>m\<^sub>m B = C" shows "A = C *\<^sub>m\<^sub>m mat_inv B"
sel_8: fixes x8 :: "'b ltl" shows "theG (G x8) = x8"
assms_2: shows "x |\<in>| HOLZF.Domain f"
Collect_case_prodD: fixes x :: "'e \<times> 'f"   and A :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "x \<in> {(x, y). A x y}" shows "A (fst x) (snd x)"
Dij: fixes x :: "nat"   and y :: "nat" assumes "x < n"   and "y < n"   and "x \<noteq> y" shows "D $$ (x, y) = (0::'a)"
of_nat_CHAR: shows "of_nat CHAR = (0::'a)"
check_poly_weak_mono_discrete: fixes p :: "('b monom \<times> 'a) list"   and v :: "'b" assumes "discrete"   and "check_poly_weak_mono_discrete p v" shows "poly_weak_mono p v"
series_and_derivative_comparison: fixes S :: "complex set"   and h :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> complex \<Rightarrow> complex"   and f' :: "nat \<Rightarrow> complex \<Rightarrow> complex" assumes "open S"   and "summable h"   and "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"   and "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. cmod (f n x) \<le> h n" obtains g :: "complex \<Rightarrow> complex" and g' :: "complex \<Rightarrow> complex" where "\<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (\<lambda>n. f' n x) sums g' x \<and> (g has_field_derivative g' x) (at x)"
coinductive_eq_iff_lnth_eq: fixes a :: "'b llist"   and b :: "'b llist" shows "(a = b) = (llength a = llength b \<and> (\<forall>n. enat n < llength a \<longrightarrow> a $ n = b $ n))"
can_occur_def: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration" assumes "distributed_system channel" shows "distributed_system.can_occur channel trans send recv ev c \<equiv> case ev of Trans p s s' \<Rightarrow> states c p = s \<and> trans p s s' | Send i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (p, q) \<and> send i p q s s' msg | Recv i p q s s' msg \<Rightarrow> states c p = s \<and> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Msg msg \<and> recv i p q s s' msg | Snapshot p \<Rightarrow> \<not> ps c p \<noteq> None | RecvMarker i p q \<Rightarrow> channel i = Some (q, p) \<and> 0 < length (msgs c i) \<and> hd (msgs c i) = Marker"
fact_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero 2 = numeral.numeral one plus (num.Bit0 num.One)"
bind_spmf_assoc: fixes x :: "'c spmf"   and f :: "'c \<Rightarrow> 'b spmf"   and g :: "'b \<Rightarrow> 'a spmf" shows "x \<bind> f \<bind> g = x \<bind> (\<lambda>y. f y \<bind> g)"
fun_type_inv_1: fixes t :: "('fun, 'var) Term.term"   and f :: "'fun"   and T :: "('fun, 'atom) Term.term list" assumes "\<Gamma> t = Fun f T" shows "0 < arity f"
element_in_carrier: fixes a :: "'a mod_ring poly" shows "(a \<in> carrier R) = (degree a < degree f)"
hyp2: shows "matrix_to_iarray (snd (Gauss_Jordan_upt_k_PA A k)) = snd (Gauss_Jordan_upt_k_iarrays_PA (matrix_to_iarray A) k)"
wf_map_prod_image: fixes r :: "('a \<times> 'a) set"   and f :: "'a \<Rightarrow> 'b" assumes "wf r"   and "inj f" shows "wf (map_prod f f ` r)"
simps_4: fixes x21 :: "'d \<times> 'e"   and x22 :: "('d, 'e, 'f) ctree fset"   and x1 :: "'f" shows "cNode x21 x22 \<noteq> Link x1"
eq: shows "(f =m Polynomial.smult c (\<Prod>\<^sub># fs)) = (F = Polynomial.smult C (\<Prod>\<^sub># Fs))"
ne: shows "(\<lambda>i. if i = n then 1 else 0) \<in> topspace (subtopology (nsphere (Suc n)) {x. x (Suc n) \<le> 0}) \<inter> {x. x (Suc n) = 0}"
lenv: shows "length v mod 2 = 1 \<and> 3 \<le> length v \<and> last v = x"
mod_add_self1: 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 b :: "'a"   and a :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (plus b a) b = modulo a b"
wf_m\<mu>t: fixes m :: "'m" shows "wfP (m\<mu>t m)"
not_trimmed_ms_aux_MSLNil: shows "\<not> trimmed_ms_aux MSLNil"
smallest_prime_between_def: fixes m :: "nat"   and n :: "nat" shows "smallest_prime_between m n = (if \<exists>p. prime p \<and> m \<le> p \<and> p \<le> n then Some (smallest_prime_beyond m) else None)"
sel_2: fixes r :: "real"   and x :: "quat" shows "Im1 (r *\<^sub>R x) = r * Im1 x"
igSubstClsSTR_imp_igSubstCls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igSubstClsSTR Delta MOD" shows "FixSyn.igSubstCls Delta MOD"
lincomb_list_Cons: fixes c :: "nat \<Rightarrow> 'a"   and v :: "'a vec"   and vs :: "'a vec list" shows "lincomb_list c (v # vs) = c 0 \<cdot>\<^sub>v v + lincomb_list (c \<circ> Suc) vs"
T_def: shows "T \<equiv> \<lambda>x. \<Sum>i<N. (bernoulli' (Suc i) / fact (Suc i)) *\<^sub>R fs i x"
add_mset_not_empty: fixes a :: "'b"   and A :: "'b multiset" shows "add_mset a A \<noteq> {#}"
prems_5: shows "Marker \<in> set (msgs (S t j) cid)"
inverse_aform'E: fixes e :: "nat \<Rightarrow> real"   and p :: "nat"   and X :: "real \<times> real pdevs"   and x :: "real" assumes "e \<in> UNIV \<rightarrow> {- 1..1}"   and "0 < Inf_aform' p X"   and "x = aform_val e X" shows "inverse x \<in> aform_err e (inverse_aform' p X)"
bdd_above_Icc: fixes a :: "'b multiset"   and b :: "'b multiset" shows "subset_mset.bdd_above (subset_mset.atLeastAtMost a b)"
set_minus_singleton_eq: fixes x :: "'b"   and X :: "'b set" assumes "x \<notin> X" shows "X - {x} = X"
rat_of_int_div_dvd: fixes d :: "int"   and n :: "int" assumes "d dvd n" shows "rat_of_int (n div d) = rat_of_int n / rat_of_int d"
cts_bot: shows "bd_cts (\<lambda>P s. 0)"
prems: fixes x :: "'a" assumes "x \<in> set (x__ # xs)" shows "(0::'a) \<le> x"
implies_antitone_inf: fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "x \<leadsto> z \<le> x \<sqinter> y \<leadsto> z"
iso_is_arr: fixes f :: "'a" assumes "L.iso f" shows "L.arr f"
neq_iff: fixes x :: "'b"   and y :: "'b" shows "(x \<noteq> y) = (y < x \<or> x < y)"
IH: fixes x :: "('a, 'c) Term.term literal"   and x2 :: "'b \<Rightarrow> ('a, 'c) Term.term option" assumes "x \<in> set Ks"   and "match_term_list [(atm_of L, atm_of x)] \<sigma> = Some x2"   and "subsumes_list Ls (remove1 x Ks) x2"   and "\<And>L K \<sigma> \<tau>. \<lbrakk>L \<in> set Ls; match_term_list [(atm_of L, atm_of K)] \<sigma> = Some \<tau>; is_pos L = is_pos K\<rbrakk> \<Longrightarrow> P K" shows "subsumes_list Ls (filter P (remove1 x Ks)) x2"
bounded_clinear_zero: shows "bounded_clinear (\<lambda>x. 0::'c)"
le_less: fixes x :: "'d word"   and y :: "'d word" shows "(x \<le>s y) = (x <s y \<or> x = y)"
proj_llength: fixes tCMP :: "'b \<Rightarrow> cnf \<Rightarrow> 'a"   and active :: "'b \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'b"   and t :: "cnf llist" shows "llength (dynamic_component.proj tCMP active c t) \<le> llength t"
simps: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and a1 :: "Core.term set"   and a2 :: "Core.term" shows "(\<Theta>,a1 \<turnstile> a2) = ((\<exists>A \<rho> \<Gamma>. a1 = \<Gamma> \<and> a2 = tsubst A \<rho> \<and> wf_theory \<Theta> \<and> A \<in> axioms \<Theta> \<and> wf_inst \<Theta> \<rho>) \<or> (\<exists>A \<Gamma>. a1 = \<Gamma> \<and> a2 = A \<and> wf_term (sig \<Theta>) A \<and> \<turnstile>\<^sub>\<tau> A : propT \<and> A \<in> \<Gamma>) \<or> (\<exists>\<Gamma> B x \<tau>. a1 = \<Gamma> \<and> a2 = mk_all x \<tau> B \<and> wf_theory \<Theta> \<and> \<Theta>,\<Gamma> \<turnstile> B \<and> (x, \<tau>) \<notin> FV \<Gamma> \<and> wf_type (sig \<Theta>) \<tau>) \<or> (\<exists>\<Gamma> \<tau> B a. a1 = \<Gamma> \<and> a2 = subst_bv a B \<and> \<Theta>,\<Gamma> \<turnstile> Ct STR ''Pure.all'' ((\<tau> \<rightarrow> propT) \<rightarrow> propT) $ Abs \<tau> B \<and> \<turnstile>\<^sub>\<tau> a : \<tau> \<and> wf_term (sig \<Theta>) a) \<or> (\<exists>\<Gamma> B A. a1 = \<Gamma> - {A} \<and> a2 = A \<longmapsto> B \<and> wf_theory \<Theta> \<and> \<Theta>,\<Gamma> \<turnstile> B \<and> wf_term (sig \<Theta>) A \<and> \<turnstile>\<^sub>\<tau> A : propT) \<or> (\<exists>\<Gamma>\<^sub>1 A B \<Gamma>\<^sub>2. a1 = \<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<and> a2 = B \<and> \<Theta>,\<Gamma>\<^sub>1 \<turnstile> A \<longmapsto> B \<and> \<Theta>,\<Gamma>\<^sub>2 \<turnstile> A) \<or> (\<exists>c T \<Gamma>. a1 = \<Gamma> \<and> a2 = mk_of_class T c \<and> wf_theory \<Theta> \<and> const_type (sig \<Theta>) (const_of_class c) = Some (itselfT Core.aT \<rightarrow> propT) \<and> wf_type (sig \<Theta>) T \<and> has_sort (osig (sig \<Theta>)) T (insert c full_sort)) \<or> (\<exists>T t u \<Gamma>. a1 = \<Gamma> \<and> a2 = mk_eq (Abs T t $ u) (subst_bv u t) \<and> wf_theory \<Theta> \<and> wt_term (sig \<Theta>) (Abs T t) \<and> wf_term (sig \<Theta>) u \<and> \<turnstile>\<^sub>\<tau> u : T) \<or> (\<exists>t \<tau> \<tau>' \<Gamma>. a1 = \<Gamma> \<and> a2 = mk_eq (Abs \<tau> (t $ Bv 0)) t \<and> wf_theory \<Theta> \<and> wf_term (sig \<Theta>) t \<and> \<turnstile>\<^sub>\<tau> t : \<tau> \<rightarrow> \<tau>'))"
strong_cases: fixes a1 :: "(name \<times> ty) list"   and a2 :: "lam"   and a3 :: "ty"   and x :: "name"   and xa :: "name" assumes "a1 \<turnstile> a2 : a3" obtains \<Gamma> :: "(name \<times> ty) list" and x :: "name" and T :: "ty" where "a1 = \<Gamma>__" and "a2 = VAR x" and "a3 = T__" and "valid \<Gamma>__" and "(x, T__) \<in> set \<Gamma>__"   | \<Gamma> :: "(name \<times> ty) list" and t\<^sub>1 :: "lam" and T\<^sub>1 :: "ty" and T\<^sub>2 :: "ty" and t\<^sub>2 :: "lam" where "a1 = \<Gamma>'__" and "a2 = APP t\<^sub>1__ t\<^sub>2__" and "a3 = T\<^sub>2__" and "\<Gamma>'__ \<turnstile> t\<^sub>1__ : T\<^sub>1__ \<rightarrow> T\<^sub>2__" and "\<Gamma>'__ \<turnstile> t\<^sub>2__ : T\<^sub>1__"   | \<Gamma> :: "(name \<times> ty) list" and T\<^sub>1 :: "ty" and t :: "lam" and T\<^sub>2 :: "ty" where "\<lbrakk>x__ \<sharp> a1; x__ \<sharp> a2; x__ \<sharp> a3\<rbrakk> \<Longrightarrow> a1 = \<Gamma>''__ \<and> a2 = LAM [x__].t__ \<and> a3 = T\<^sub>1 \<rightarrow> T\<^sub>2 \<and> x__ \<sharp> \<Gamma>''__ \<and> (x__, T\<^sub>1) # \<Gamma>''__ \<turnstile> t__ : T\<^sub>2"   | \<Gamma> :: "(name \<times> ty) list" and n :: "nat" where "a1 = \<Gamma>'''__" and "a2 = NUM n" and "a3 = tINT"   | \<Gamma> :: "(name \<times> ty) list" and t\<^sub>1 :: "lam" and t\<^sub>2 :: "lam" where "a1 = \<Gamma>''''__" and "a2 = t\<^sub>1'__ -- t\<^sub>2'__" and "a3 = tINT" and "\<Gamma>''''__ \<turnstile> t\<^sub>1'__ : tINT" and "\<Gamma>''''__ \<turnstile> t\<^sub>2'__ : tINT"   | \<Gamma> :: "(name \<times> ty) list" and t\<^sub>1 :: "lam" and t\<^sub>2 :: "lam" where "a1 = \<Gamma>'''''__" and "a2 = t\<^sub>1 ++ t\<^sub>2" and "a3 = tINT" and "\<Gamma>'''''__ \<turnstile> t\<^sub>1 : tINT" and "\<Gamma>'''''__ \<turnstile> t\<^sub>2 : tINT"   | \<Gamma> :: "(name \<times> ty) list" where "a1 = \<Gamma>''''''__" and "a2 = TRUE" and "a3 = tBOOL"   | \<Gamma> :: "(name \<times> ty) list" where "a1 = \<Gamma>'''''''__" and "a2 = FALSE" and "a3 = tBOOL"   | \<Gamma> :: "(name \<times> ty) list" and t1 :: "lam" and t2 :: "lam" and T :: "ty" and t3 :: "lam" where "a1 = \<Gamma>''''''''__" and "a2 = IF t1__ t2__ t3" and "a3 = Ta__" and "\<Gamma>''''''''__ \<turnstile> t1__ : tBOOL" and "\<Gamma>''''''''__ \<turnstile> t2__ : Ta__" and "\<Gamma>''''''''__ \<turnstile> t3 : Ta__"   | \<Gamma> :: "(name \<times> ty) list" and t :: "lam" where "a1 = \<Gamma>'''''''''__" and "a2 = ZET ta__" and "a3 = tBOOL" and "\<Gamma>'''''''''__ \<turnstile> ta__ : tINT"   | \<Gamma> :: "(name \<times> ty) list" and t1 :: "lam" and t2 :: "lam" where "a1 = \<Gamma>''''''''''__" and "a2 = EQI t1 t2" and "a3 = tBOOL" and "\<Gamma>''''''''''__ \<turnstile> t1 : tINT" and "\<Gamma>''''''''''__ \<turnstile> t2 : tINT"   | \<Gamma> :: "(name \<times> ty) list" and T :: "ty" and t :: "lam" where "\<lbrakk>xa \<sharp> a1; xa \<sharp> a2; xa \<sharp> a3\<rbrakk> \<Longrightarrow> a1 = \<Gamma> \<and> a2 = FIX [xa].t \<and> a3 = T \<and> xa \<sharp> \<Gamma> \<and> (xa, T) # \<Gamma> \<turnstile> t : T"
good_context_4: 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'') = 0" shows "False"
IH: fixes \<sigma> :: "nat \<Rightarrow> val"   and \<sigma>' :: "nat \<Rightarrow> val" assumes "\<And>x. x \<in> free_vars e \<Longrightarrow> \<sigma> x = \<sigma>' x" shows "cexpr_sem \<sigma> e = cexpr_sem \<sigma>' e"
M: fixes i :: "'a" assumes "i \<in> I" shows "prob_space (M i)"
is_unit_div_mult_cancel_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero"   and "dvd.dvd times b one" shows "divide a (times b a) = divide one b"
Extend_in_C: fixes C :: "(nat, 'a) form set set"   and S :: "(nat, 'a) form set"   and f :: "nat \<Rightarrow> (nat, 'a) form" assumes "alt_consistency C"   and "finite_char C"   and "S \<in> C"   and "infinite (- \<Union> (params ` S))" shows "Extend S C f \<in> C"
simps_1: fixes x1 :: "prim_ty"   and y1 :: "prim_ty" shows "(PrimT x1 = PrimT y1) = (x1 = y1)"
apps_append: fixes s :: "('c, 'd) tm"   and ss :: "('c, 'd) tm list"   and ts :: "('c, 'd) tm list" shows "apps s (ss @ ts) = apps (apps s ss) ts"
IH: shows "\<forall>ys. length ys < length xs \<longrightarrow> set ys \<subseteq> A \<longrightarrow> distinct ys \<longrightarrow> map_pmf snd (rquicksort R ys) = rqs_cost (length ys)"
unity_root_conv_exp: fixes k :: "nat"   and n :: "int" shows "unity_root k n = exp (complex_of_real (2 * pi * real_of_int n / real k) * \<i>)"
assms: shows "exec_plan s (as1 @ as2) = exec_plan s as1"
inject_2: fixes atom_ordering :: "('a \<times> 'a) set"   and x2 :: "'b"   and y2 :: "'b" assumes "propositional_atoms atom_ordering" shows "(Neg x2 = Neg y2) = (x2 = y2)"
is_rat_complex_def: fixes x :: "complex" shows "is_rat x = (is_rat (Re x) \<and> Im x = 0)"
in_codomains_char: fixes b :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "(b \<in> A_B.codomains f) = ((f \<noteq> A_B.Null \<and> A_B.Dom f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> A_B.Cod f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B)) \<and> A_B.Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Dom f) (A_B.Cod f))) \<and> b = A_B.MkIde (A_B.Cod f))"
inj_Inr: fixes A :: "'a set" shows "inj_on Inr A"