hcju/isabelleps huggingface dataset:
distinct_3: fixes name :: "name"   and name' :: "name"   and lam' :: "lam" shows "Var name \<noteq> Lam [name'].lam'"
distinct_agents: shows "distinct [A1, A2, A3, A4]"
outputCases': fixes a :: "name"   and b :: "name"   and P :: "pi"   and Rs :: "Early_Semantics.residual" assumes "a{b}.P \<longmapsto>\<^sub>e Rs" obtains aa :: "name" and ba :: "name" and P :: "pi" where "a = aa \<and> b = ba \<and> Pb__ = P" and "Rs = aa[ba] \<prec>\<^sub>e P"
countably_additive_iff_continuous_from_below: fixes M :: "'b measure"   and f :: "'b set \<Rightarrow> ennreal" assumes "positive (sets M) f"   and "Measure_Space.additive (sets M) f" shows "countably_additive (sets M) f = (\<forall>A. range A \<subseteq> sets M \<longrightarrow> incseq A \<longrightarrow> \<Union> (range A) \<in> sets M \<longrightarrow> (\<lambda>i. f (A i)) \<longlonglongrightarrow> f (\<Union> (range A)))"
cwiseext_irrefl: fixes xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> gt x x" shows "\<not> cwiseext gt xs xs"
indep_vars_def: fixes M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'a \<Rightarrow> 'e"   and I :: "'d set" shows "indep_vars M' X I = ((\<forall>i\<in>I. random_variable (M' i) (X i)) \<and> indep_sets (\<lambda>i. sigma_sets (space M) {X i -` A \<inter> space M |A. A \<in> sets (M' i)}) I)"
exhaust: fixes y :: "('a, 'b) Proc" obtains x1 :: "'a" where "y = SourceTerm x1"   | x2 :: "'b" where "y = TargetTerm x2"
map_poly_homo_polymul: fixes f :: "'a \<Rightarrow> 'b"   and p1 :: "'a poly"   and p2 :: "'a poly" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>x. (f x = (0::'b)) = (x = (0::'a))"   and "\<And>x y. f (x * y) = f x * f y" shows "map_poly f (p1 *\<^sub>p p2) = map_poly f p1 *\<^sub>p map_poly f p2"
exec_step_ind_Getstatic_NonStatic: fixes P :: "jvm_prog"   and C :: "char list"   and F :: "char list"   and t :: "ty"   and D :: "char list"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "P \<turnstile> C has F,NonStatic:t in D" shows "exec_step_ind (StepI (Getstatic C F D)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (\<lfloor>addr_of_sys_xcpt IncompatibleClassChangeError\<rfloor>, h, (stk, loc, C\<^sub>0, M\<^sub>0, pc, ics) # frs, sh)"
simps_1: fixes x1 :: "'d"   and y1 :: "'d" shows "(ln x1 = ln y1) = (x1 = y1)"
12: shows "r\<^sup>T \<le> v\<^sup>T"
prems_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> insert x__ A"   and "y \<in> insert x__ A"   and "x \<noteq> y" shows "coprime (f x) (f y)"
is_igb_graph: shows "igb_graph (ngba_igbg A)"
eval_fds_deriv: fixes f :: "'b fds"   and s :: "'b" assumes "conv_abscissa f < ereal (s \<bullet> (1::'b))" shows "eval_fds (fds_deriv f) s = deriv (eval_fds f) s"
gf_F: fixes x :: "'s"   and y :: "'s" shows "(gf_F x y \<longlongrightarrow> F x y) (at_left 1)"
simple_bochner_integrable_eq_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "simple_bochner_integrable M f" shows "simple_bochner_integral M f = integral\<^sup>L M f"
measurable_on_const: fixes S :: "'a set"   and k :: "'b" assumes "S \<in> sets lebesgue" shows "(\<lambda>x. k) measurable_on S"
isAbrE: fixes S :: "('d, 'e) xstate" assumes "isAbr S" obtains s :: "'d" where "S = Abrupt s"
nop_onto_vdomain: shows "\<D>\<^sub>\<circ> vrat_plus = \<rat>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and option :: "'d option" shows "h (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | Some x \<Rightarrow> h (f2 x))"
elims_3: fixes x :: "('a, 'b) Term.term"   and xa :: "('a, 'b) Term.term" assumes "\<not> subterm_of x xa" obtains t :: "('a, 'b) Term.term" and y :: "'b" where "x = ta__" and "xa = Var y" and "ta__ \<noteq> Var y"   | t :: "('a, 'b) Term.term" and f :: "'a" and T :: "('a, 'b) Term.term list" where "x = t" and "xa = Fun f T" and "\<not> (t = Fun f T \<or> list_ex (subterm_of t) T)"
comlist_snoc: fixes \<Pi> :: "('b, 'a) chain"   and x :: "'a"   and p :: "'b" shows "comlist (cSnoc \<Pi> x p) = comlist \<Pi> @ [x]"
power_strict_mono: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and b :: "'a"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less a b"   and "less_eq zero a" shows "0 < n \<longrightarrow> less (power.power one times a n) (power.power one times b n)"
proves_eq_combination_rule: 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 f :: "Core.term"   and \<tau> :: "typ"   and \<tau>' :: "typ"   and g :: "Core.term"   and x :: "Core.term"   and y :: "Core.term"   and \<Gamma> :: "Core.term set" assumes "wf_theory \<Theta>"   and "term_ok \<Theta> f"   and "typ_of f = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> g"   and "typ_of g = Some (\<tau> \<rightarrow> \<tau>')"   and "term_ok \<Theta> x"   and "typ_of x = Some \<tau>"   and "term_ok \<Theta> y"   and "typ_of y = Some \<tau>"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq f g"   and "\<Theta>,\<Gamma> \<turnstile> mk_eq x y"   and "finite \<Gamma>"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq (f $ x) (g $ y)"
has_left_adjoint_functor: shows "left_adjoint_functor (\<cdot>\<^sub>D) (\<cdot>\<^sub>C) F"
isSimplePath_fwd: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" shows "cf.isSimplePath s p t = (cf.isPath s p t \<and> distinct (cf.pathVertices_fwd s p))"
zero_le_scaleR_iff: fixes a :: "real"   and b :: "'c" shows "((0::'c) \<le> a *\<^sub>R b) = (0 < a \<and> (0::'c) \<le> b \<or> a < 0 \<and> b \<le> (0::'c) \<or> a = 0)"
induced_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "induced A r \<equiv> {(a, b). a \<in> A \<and> b \<in> A \<and> (a, b) \<in> r}"
ANNOT_def: fixes x :: "'b"   and a :: "annot" shows "Autoref_Tagging.ANNOT x a \<equiv> x"
M: fixes n :: "nat" shows "norm (a n) * r0 ^ n \<le> M"
sup_commute: fixes x :: "'a"   and y :: "'a" shows "x \<squnion>1 y = y \<squnion>1 x"
both_contains_arg2: fixes t' :: "'a ttree"   and t :: "'a ttree" shows "paths t' \<subseteq> paths (t \<otimes>\<otimes> t')"
rep_eq: fixes x :: "nat"   and xa :: "nat"   and xb :: "(nat \<Rightarrow>\<^sub>0 nat, 'b) fmap" shows "fmlookup (lowerPoly\<^sub>f x xa xb) = (\<lambda>mon. fmlookup xb (lowerPowers x xa mon))"
assms_1: shows "set (ps ()) = (\<lambda>p. (p, multiplicity p n - 1)) ` prime_factors n"
nextMax_set: fixes xs :: "nat list" assumes "sorted xs" shows "nextMax (set xs) = (if xs = [] then 0 else Suc (last xs))"
zero_sq_mtx_def: shows "0 \<equiv> to_mtx 0"
trancl_rbt_impl: fixes r :: "('b \<times> 'b) list"   and as :: "'b list" shows "rs.\<alpha> (trancl_rbt_impl r as) = {b. \<exists>a\<in>set as. (a, b) \<in> (set r)\<^sup>+}"
sound_nneg: fixes P :: "'b \<Rightarrow> real" assumes "sound P" shows "nneg P"
it_step_insert_iff: fixes it :: "'g set"   and S :: "'g set"   and x :: "'g" assumes "it \<subseteq> S"   and "x \<in> it" shows "S - (it - {x}) = insert x (S - it)"
indexed_const_in_carrier: fixes K :: "'a set"   and k :: "'a"   and m :: "'d multiset" assumes "K \<subseteq> carrier R"   and "k \<in> K" shows "indexed_const k m \<in> carrier R"
encL: fixes Enc :: "'a \<Rightarrow> 'b"   and S :: "'a"   and TRel :: "('b \<times> 'b) set" shows "(TargetTerm (Enc S), SourceTerm S) \<in> encoding.indRelTEQ Enc TRel"
leg0_simps_3: shows "trg r\<^sub>0 = src r"
x_coord_inv_def: fixes x :: "real" shows "x_coord_inv x = 1 / 2 + x"
elims: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and x :: "'a"   and xa :: "'b val"   and y :: "ty option" assumes "heap_base.typeof_h typeof_addr x xa = y" obtains h :: "'a" and a :: "'b" where "x = h__" and "xa = Addr a" and "y = map_option ty_of_htype (typeof_addr h__ a)"   | h :: "'a" where "x = ha__" and "xa = Unit" and "y = typeof Unit"   | h :: "'a" where "x = hb__" and "xa = Null" and "y = typeof Null"   | h :: "'a" and va :: "bool" where "x = hc__" and "xa = Bool va__" and "y = typeof (Bool va__)"   | h :: "'a" and va :: "32 word" where "x = h" and "xa = Intg va" and "y = typeof (Intg va)"
simps: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and g :: "taylor_model option"   and f_exists_on :: "float interval \<Rightarrow> bool" shows "compute_tm_by_comp prec ord I a f g f_exists_on = (case g of None \<Rightarrow> None | Some tg \<Rightarrow> let gI = compute_bound_tm prec I a tg; ga = mid (compute_bound_tm prec a a tg) in if f_exists_on gI then map_option (\<lambda>tf. tm_comp prec ord I a ga tf tg) (tm_floatarith prec ord [gI] [ga] f) else None)"
neqE: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.linorder less_eq less"   and "x \<noteq> y" obtains "less x y"   | "less y x"
assms_2: shows "ipresIGFreshAll h hA MOD MOD'"
stream_all_def: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a stream" shows "stream_all P s = (\<forall>p. P (s !! p))"
lookup_pair_sort_oalist': fixes xs :: "('a \<times> 'b) list" assumes "distinct (map fst xs)" shows "lookup_pair_tc (sort_oalist_tc xs) = lookup_dflt xs"
mset_eq_length: fixes xs :: "'d list"   and ys :: "'d list" assumes "mset xs = mset ys" shows "length xs = length ys"
a1: shows "A = Matrix.mat 2 1 (\<lambda>(i, j). if i = 0 then a0 else a1)"
stutstep_notempty_notempty: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "emptyseq (s |\<^sub>s Suc n)"   and "stutstep s n" shows "emptyseq (s |\<^sub>s n)"
infinite: fixes A :: "'c set" assumes "infinite A" shows "card A = 0"
alw_nxt: fixes \<phi> :: "'d stream \<Rightarrow> bool" shows "alw \<phi> = (\<lambda>xs. \<phi> xs \<and> nxt (alw \<phi>) xs)"
simps_3: fixes y :: "Syntax.var"   and f :: "(Syntax.var, Syntax.term) fmap"   and t :: "Syntax.term" assumes "atom y \<sharp> f" shows "psubst_term (Lam y t) f = Lam y (psubst_term t f)"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_on_right_moving_2 ires rs (l, r) = (\<exists>ml mr rn. l = Bk \<up> ml @ Oc # ires \<and> r = Bk \<up> mr @ Oc \<up> Suc rs @ Bk \<up> rn \<and> Suc 0 < ml + mr)"
ipurge_tr_rev_t_inv_1_def: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list"   and X :: "('d, 'c) ipurge_rec" shows "ipurge_tr_rev_t_inv_1 I D u xs X \<equiv> Out X @ ipurge_tr_rev I D u (In X) = ipurge_tr_rev I D u xs"
sel: fixes x :: "'c \<Rightarrow> ('d \<times> ('c, 'd) resource) spmf" shows "run_resource (Resource x) = x"
induct: fixes x :: "C"   and b :: "nat"   and P :: "C \<Rightarrow> bool" assumes "x \<in> hbase b"   and "P (C [])"   and "\<And>i n ms. \<lbrakk>i \<noteq> 0; i < b; n \<in> hbase b; P n; C ms \<in> hbase b; P (C ms); \<And>m'. m' \<in> set ms \<Longrightarrow> evalC b n < evalC b m'\<rbrakk> \<Longrightarrow> P (C (replicate i n @ ms))" shows "P x"
mult_ac_1: fixes a :: "'e"   and b :: "'e"   and c :: "'e" shows "a \<cdot> b \<cdot> c = a \<cdot> (b \<cdot> c)"
scpp: shows "singular_chain p (subtopology (powertop_real UNIV) (standard_simplex p)) (frag_of (restrict id (standard_simplex p)))"
cringI: fixes R :: "('c, 'd) ring_scheme" assumes "abelian_group R"   and "Group.comm_monoid R"   and "\<And>x y z. \<lbrakk>x \<in> carrier R; y \<in> carrier R; z \<in> carrier R\<rbrakk> \<Longrightarrow> (x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z" shows "cring R"
length_shift: fixes p :: "(action \<times> nat) list"   and n :: "nat" shows "length (shift p n) = length p"
algebraic_int_diff: fixes x :: "'b"   and y :: "'b" assumes "algebraic_int x"   and "algebraic_int y" shows "algebraic_int (x - y)"
mem_eq: shows "subst (to_partial mem |` Y) mem = mem"
diff_eq_diff_less_eq: fixes 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 a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less_eq a b = less_eq c d"
exists_nat_set_suc: fixes n :: "nat"   and P :: "'b \<Rightarrow> nat \<Rightarrow> bool" shows "{x. \<exists>m\<ge>n. P x m} = {x. P x n} \<union> {x. \<exists>m\<ge>Suc n. P x m}"
pred_mapping_alt: fixes Q :: "'a \<Rightarrow> bool"   and P :: "'b \<Rightarrow> 'a option" shows "pred_mapping Q P = (\<forall>p\<in>dom P. Q (the (P p)))"
norm_minus_cancel: fixes x :: "'b" shows "norm (- x) = norm x"
brec_step_abs: fixes rqrm :: "('a, ('a, 'b) ta_rule dlist) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule set"   and Qi :: "('a, unit) HashMap.hashmap"   and \<Sigma> :: "('a, 'b tree) HashMap.hashmap \<times> ('a list \<times> 'a list) \<times> (('a, 'b) ta_rule, nat) HashMap.hashmap \<times> 'a option" assumes "rqrm_invar rqrm"   and "rqrm_prop \<delta> rqrm"   and "hs.invar Qi"   and "\<Sigma> \<in> brec_invar (hs.\<alpha> Qi) \<delta>"   and "brec_cond \<Sigma>" shows "(brec_\<alpha> \<Sigma>, brec_\<alpha> (brec_step rqrm Qi \<Sigma>)) \<in> brw_step \<delta>"
conc_def: fixes A :: "'c list set"   and B :: "'c list set" shows "A @@ B = {xs @ ys |xs ys. xs \<in> A \<and> ys \<in> B}"
cardinal_le_lepoll: fixes A :: "V"   and \<alpha> :: "V" assumes "vcard A \<le> \<alpha>" shows "elts A \<lesssim> elts \<alpha>"
in_set_zipE: fixes x :: "'b"   and y :: "'c"   and xs :: "'b list"   and ys :: "'c list" assumes "(x, y) \<in> set (zip xs ys)" obtains "x \<in> set xs" and "y \<in> set ys"
utos_returntype: fixes w :: "bit list" shows "norm_signed (utos w) = utos w"
elims_2: fixes x :: "'a option"   and xa :: "'a option" assumes "x \<sqsubseteq> xa" obtains x :: "'a" and y :: "'a" where "xa__ = Some x" and "xa = Some y__" and "x \<sqsubseteq> y__"   | y :: "'a option" where "xa__ = None" and "xa = y"
ker_diff_carrier: fixes K :: "'a list set"   and x :: "'a list" assumes "subring K (K [X])" shows "a_kernel (K [X]\<^bsub>K [X]\<^esub>) (K [X]) (\<lambda>p. UP.eval p x) \<noteq> carrier (K [X]\<^bsub>K [X]\<^esub>)"
reachable_mems_def: fixes cms :: "('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) list"   and mem :: "'Var\<^sub>C \<Rightarrow> 'Val" shows "conc.reachable_mems cms mem \<equiv> {mem'. \<exists>sched cms'. conc.meval_sched_abv (cms, mem) sched (cms', mem')}"
SD_preorder: fixes q :: "'b pmf"   and p :: "'b pmf" shows "SD (R i) q p = (p \<in> lotteries_on alts \<and> q \<in> lotteries_on alts \<and> (\<forall>x\<in>alts. measure_pmf.prob q (preferred_alts (R i) x) \<le> measure_pmf.prob p (preferred_alts (R i) x)))"
sharing_correct_def: fixes share :: "'a \<Rightarrow> ('b \<times> 'b) spmf"   and reconstruct :: "'b \<times> 'b \<Rightarrow> 'a spmf"   and input :: "'a" shows "secret_sharing_scheme.sharing_correct share reconstruct input \<equiv> share input \<bind> (\<lambda>(s1, s2). reconstruct (s1, s2)) = return_spmf input"
carrier_stalk_def: shows "stfx.carrier_stalk \<equiv> dlY.carrier_direct_lim"
standard_redundancy_criterion_reductive_axioms: shows "standard_redundancy_criterion_reductive \<Gamma>"
neg_bounds_compare_5: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c \<le>\<^sub>l\<^sub>b b" shows "c >\<^sub>l\<^sub>b b"
F_def: shows "F \<equiv> \<lambda>ltake n xs. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> case n of 0 \<Rightarrow> LNil | eSuc n \<Rightarrow> LCons x (ltake n xs)"
max_step: fixes j :: "nat" assumes "t j \<le> t2" shows "t2.max_stepsize j \<le> \<bar>c\<bar> * a / b / (exp (a * (t2 - t 0) + 1) - 1)"
vsubstEnv_psubstEnv_idEnv: fixes ys :: "'varSort"   and z :: "'var"   and y :: "'var"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" shows "rho &[z // y]_ys = rho &[idEnv [y \<leftarrow> Var ys z]_ys]"
keys_dehomogenizeI: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 nat"   and x :: "'c" assumes "homogeneous p"   and "t \<in> keys p" shows "except t {x} \<in> keys (dehomogenize x p)"
induct: fixes x1 :: "nat"   and x2 :: "'b tree23"   and P :: "nat \<Rightarrow> 'b tree23 \<Rightarrow> bool" assumes "full x1 x2"   and "P 0 \<langle>\<rangle>"   and "\<And>n l r p. \<lbrakk>full n l; P n l; full n r; P n r\<rbrakk> \<Longrightarrow> P (Suc n) \<langle>l, p, r\<rangle>"   and "\<And>n l m r p q. \<lbrakk>full n l; P n l; full n m; P n m; full n r; P n r\<rbrakk> \<Longrightarrow> P (Suc n) \<langle>l, p, m, q, r\<rangle>" shows "P x1 x2"
hyps_4: fixes l :: "char list \<Rightarrow> val option" assumes "P \<turnstile> h \<surd>"   and "P \<turnstile> E \<surd>"   and "\<D>s es \<lfloor>dom l\<rfloor>"   and "\<not> finals es" shows "\<exists>es' s'. P,E \<turnstile> \<langle>es,(h, l)\<rangle> [\<rightarrow>] \<langle>es',s'\<rangle>"
assms_2: shows "mat (rat_poly.row_length m2) (rat_poly.row_length m3) m2"
binomial_le_pow: fixes r :: "nat"   and n :: "nat" assumes "r \<le> n" shows "n choose r \<le> n ^ r"
typing_cond_prot_def: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list set" shows "typing_cond_prot \<P> \<equiv> wf\<^sub>l\<^sub>s\<^sub>t\<^sub>s \<P> \<and> tfr\<^sub>s\<^sub>e\<^sub>t (\<Union> (trms\<^sub>l\<^sub>s\<^sub>t ` \<P>)) \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (\<Union> (trms\<^sub>l\<^sub>s\<^sub>t ` \<P>)) \<and> (\<forall>\<A>\<in>\<P>. list_all tfr\<^sub>s\<^sub>t\<^sub>p (unlabel \<A>)) \<and> Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` unlabel ` \<P>) \<union> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` unlabel ` \<P>))"
square_free_m_cong: fixes f :: "int poly"   and g :: "int poly" assumes "p.square_free_m f"   and "p.eq_m f g" shows "p.square_free_m g"
inj_on_finite: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set" assumes "inj_on f A"   and "f ` A \<subseteq> B"   and "finite B" shows "finite A"
redT_upd_inv_ext: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and I :: "'t \<Rightarrow> 'a option"   and P :: "'a \<Rightarrow> 't \<Rightarrow> 'x \<Rightarrow> 'm \<Rightarrow> bool" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "ts_inv_ok (thr s) I" shows "I \<subseteq>\<^sub>m upd_invs I P \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
simple_conn_matches_def: fixes m :: "'a simple_match"   and c :: "parts_connection" shows "simple_conn_matches m c = (match_iface (iiface m) (pc_iiface c) \<and> match_iface (oiface m) (pc_oiface c) \<and> match_proto (proto m) (pc_proto c) \<and> simple_match_port (sports m) (pc_sport c) \<and> simple_match_port (dports m) (pc_dport c))"
chine_is_identity: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "identity_arrow_of_spans C \<mu>" shows "partial_magma.ide C (Chn \<mu>)"
3: assumes "\<not> (\<forall>i j. x [^] i = x [^] j \<longrightarrow> i = j)" shows "\<exists>n. n \<noteq> 0 \<and> x [^] n = \<one>"
CONSTRAINT_def: fixes P :: "'c \<Rightarrow> 'b"   and x :: "'c" shows "CONSTRAINT P x \<equiv> P x"
prod_algebra_sets_into_space: fixes I :: "'c set"   and M :: "'c \<Rightarrow> 'd measure" shows "prod_algebra I M \<subseteq> Pow (\<Pi>\<^sub>E i\<in>I. space (M i))"
translation_invertible: fixes a :: "'a" assumes "a \<in> G" shows "transformation.sub.invertible (a)\<^sub>L"
binop_app_in_vrange: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>"   and "b \<in>\<^sub>\<circ> \<int>\<^sub>\<circ>" shows "a *\<^sub>\<int> b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> vint_mult"
carrier_subgroup_generated: fixes G :: "('e, 'f) monoid_scheme"   and S :: "'e set" shows "carrier (subgroup_generated G S) = generate G (carrier G \<inter> S)"
respond_Spy_not_see_session_key: fixes PB :: "msg"   and RB :: "msg"   and KAB :: "nat"   and evs :: "event list"   and A :: "agent"   and A' :: "agent"   and K :: "nat"   and N :: "msg" assumes "(PB, RB, KAB) \<in> respond evs"   and "evs \<in> recur"   and "A \<notin> bad \<and> A' \<notin> bad"   and "Crypt (shrK A) \<lbrace>Key K, Agent A', N\<rbrace> \<in> parts {RB}" shows "Key K \<notin> analz (insert RB (knows Spy evs))"
Necessary\<^sub>0_def: shows "Necessary \<equiv> box"
stationary_distribution_def: fixes N :: "'b pmf" shows "K2.stationary_distribution N = (N = N \<bind> K2)"
const_map_quorum_locked: fixes S :: "process set"   and v :: "val" assumes "S \<in> Quorum" shows "locked_in_vf (const_map v S) v"
fls_base_factor_to_fps_one: shows "fls_base_factor_to_fps 1 = 1"
compute_tm_by_comp_None: fixes p :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and x :: "floatarith"   and k :: "float interval \<Rightarrow> bool" shows "compute_tm_by_comp p ord I a x None k = None"
inf_sup_ord_4: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b" assumes "class.lattice (inf::'b \<Rightarrow> 'b \<Rightarrow> 'b) (less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (less::'b \<Rightarrow> 'b \<Rightarrow> bool) (sup::'b \<Rightarrow> 'b \<Rightarrow> 'b)" shows "(less_eq::'b \<Rightarrow> 'b \<Rightarrow> bool) (y::'b) ((sup::'b \<Rightarrow> 'b \<Rightarrow> 'b) (x::'b) y)"
cases: fixes a1 :: "'a"   and a2 :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a1, a2) \<in> r\<^sup>*" obtains a :: "'a" where "a1 = a__" and "a2 = a__"   | a :: "'a" and b :: "'a" and c :: "'a" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>*" and "(b, c) \<in> r"
S_Max_Lt: shows "S_Max_Lt = {d - u c |c d. 0 < v c \<and> v c \<le> n \<and> M (v c) 0 = Lt d}"
coprime_abs_left_iff: fixes k :: "int"   and l :: "int" shows "coprime \<bar>k\<bar> l = coprime k l"
order_bal_nonempty_lasttreebal: fixes k :: "nat"   and t :: "'b btree" assumes "0 < k"   and "root_order k t"   and "bal t" shows "nonempty_lasttreebal t"
Decrypt: fixes K :: "nat"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> analz H"   and "Key (invK K) \<in> analz H" shows "X \<in> analz H"
config'_rand_append: fixes A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and s :: "('a list \<times> 'b) pmf"   and xs :: "'a list"   and ys :: "'a list" shows "Partial_Cost_Model.config'_rand A s (xs @ ys) = Partial_Cost_Model.config'_rand A (Partial_Cost_Model.config'_rand A s xs) ys"
app: fixes A :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm"   and B' :: "'a trm" assumes "substitutes A x M A'"   and "substitutes B x M B'" shows "substitutes (App A B) x M (App A' B')"
exhaust: fixes y :: "facetype" obtains "y = Final"   | "y = Nonfinal"