hcju/isabelleps huggingface dataset:
smcf_0_components_2: fixes \<AA> :: "V" shows "smcf_0 \<AA>\<lparr>ArrMap\<rparr> = []\<^sub>\<circ>"
hb_consistent_append_D2: fixes xs :: "'a list"   and ys :: "'a list" assumes "hb_consistent (xs @ ys)" shows "hb_consistent ys"
Im_rcis: fixes r :: "real"   and a :: "real" shows "Im (rcis r a) = r * sin a"
dec_add_clock: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_dec ck mn env s d res" shows "\<forall>s' r. res = (s', r) \<and> ck = False \<longrightarrow> (\<exists>c. evaluate_dec True mn env (update_clock (\<lambda>_. c) s) d (update_clock (\<lambda>_. 0) s', r))"
NTDYI: shows "NTDom (YMapInv C X F (YMap C X \<eta>)) = YFtor C @@ X"
a: shows "\<forall>k. j + 1 \<le> k \<and> k < length t \<longrightarrow> swap_events i j t ! k = t ! k"
invariant_steps: fixes a :: "'a"   and as :: "'a list" assumes "steps (a # as)"   and "Q a" shows "list_all Q as"
characteristic_iso_1: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>\<a>[f, a, a] : (f \<star> a) \<star> a \<Rightarrow>\<^sub>S f \<star> a \<star> a\<guillemotright>"
qiterg_def: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" shows "prod.qiterg x y = prod.gfp (prod.qiter_fun x y)"
fresh_psubst_I: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho"   and "rho zs z = None \<Longrightarrow> fresh zs z X"   and "\<And>ys y Y. rho ys y = Some Y \<Longrightarrow> fresh ys y X \<or> fresh zs z Y" shows "fresh zs z X #[rho]"
dis_log_def: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "discrete_log.dis_log \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>x. let h = \<^bold>g [^] x in \<A> h \<bind> (\<lambda>x'. return_spmf [x = x'] (mod order \<G>))) ELSE return_spmf False"
umin': fixes t :: "real" assumes "0 \<le> t"   and "t \<le> 1"   and "t < u" shows "g t \<in> S"
Jcc: fixes P\<^sub>1 :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e :: "nat exp"   and h\<^sub>0 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>0 :: "val list"   and ef :: "nat exp"   and h\<^sub>1 :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and ls\<^sub>1 :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and vs :: "val list"   and I :: "nat set"   and v :: "val"   and frs :: "(val list \<times> val list \<times> char list \<times> char list \<times> nat) list"   and xa :: "nat" assumes "P\<^sub>1 \<turnstile>\<^sub>1 \<langle>e,(h\<^sub>0, ls\<^sub>0)\<rangle> \<Rightarrow> \<langle>ef,(h\<^sub>1, ls\<^sub>1)\<rangle>"   and "compP\<^sub>2 P\<^sub>1,C,M,pc \<rhd> compE\<^sub>2 e"   and "compP\<^sub>2 P\<^sub>1,C,M \<rhd> compxE\<^sub>2 e pc (length vs) / I,length vs"   and "{pc..<pc + length (compE\<^sub>2 e)} \<subseteq> I" shows "(ef = Val v \<longrightarrow> compP\<^sub>2 P\<^sub>1 \<turnstile> (None, h\<^sub>0, (vs, ls\<^sub>0, C, M, pc) # frs) -jvm\<rightarrow> (None, h\<^sub>1, (v # vs, ls\<^sub>1, C, M, pc + length (compE\<^sub>2 e)) # frs)) \<and> (ef = Throw xa \<longrightarrow> (\<exists>pc\<^sub>1\<ge>pc. pc\<^sub>1 < pc + length (compE\<^sub>2 e) \<and> \<not> caught (compP\<^sub>2 P\<^sub>1) pc\<^sub>1 h\<^sub>1 xa (compxE\<^sub>2 e pc (length vs)) \<and> compP\<^sub>2 P\<^sub>1 \<turnstile> (None, h\<^sub>0, (vs, ls\<^sub>0, C, M, pc) # frs) -jvm\<rightarrow> handle (compP\<^sub>2 P\<^sub>1) C M xa h\<^sub>1 vs ls\<^sub>1 pc\<^sub>1 frs))"
Invariants_cond_exp_is_integral_fmpt: fixes f :: "'a \<Rightarrow> real" assumes "integrable M f" shows "AE x in M. real_cond_exp M Invariants f x = expectation f / prob (space M)"
simps_2: fixes f :: "'d \<Rightarrow> ('d, 'b, 'c) sp\<^sub>\<mu>"   and b :: "'d"   and sp :: "('a, 'd) sp\<^sub>\<nu>" shows "Get f o\<^sub>\<mu> Put b sp = f b o\<^sub>\<mu> out sp"
app_def: fixes i :: "instr"   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 mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and t :: "(ty list \<times> ty err list) option" shows "app i P mxs T\<^sub>r pc mpc xt t = (case t of None \<Rightarrow> True | \<lfloor>\<tau>\<rfloor> \<Rightarrow> app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt t). pc' < mpc))"
Zp_int_inc_res: fixes n :: "int"   and k :: "nat" shows "([n] \<cdot> \<one>) k = n mod p ^ k"
compile_pr_correct: fixes x :: "nat"   and g :: "recf"   and xs :: "nat list"   and n :: "nat"   and f :: "recf"   and ap :: "abc_inst list"   and arity :: "nat"   and fp :: "nat"   and anything :: "nat list" assumes "\<forall>y<x. terminate g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) \<and> (\<forall>x xa xb. rec_ci g = (x, xa, xb) \<longrightarrow> (\<forall>xc. {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # 0 \<up> (xb - xa) @ xc} x {\<lambda>nl. nl = xs @ y # rec_exec (Pr n f g) (xs @ [y]) # rec_exec g (xs @ [y, rec_exec (Pr n f g) (xs @ [y])]) # 0 \<up> (xb - Suc xa) @ xc}))"   and "terminate f xs"   and "\<And>ap arity fp anything. rec_ci f = (ap, arity, fp) \<Longrightarrow> {\<lambda>nl. nl = xs @ 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ rec_exec f xs # 0 \<up> (fp - Suc arity) @ anything}"   and "length xs = n"   and "rec_ci (Pr n f g) = (ap, arity, fp)" shows "{\<lambda>nl. nl = xs @ x # 0 \<up> (fp - arity) @ anything} ap {\<lambda>nl. nl = xs @ x # rec_exec (Pr n f g) (xs @ [x]) # 0 \<up> (fp - Suc arity) @ anything}"
sel_11: fixes x21 :: "nat"   and x22 :: "'a"   and x23 :: "'a"   and x24 :: "'b"   and x25 :: "'b"   and x26 :: "'c" shows "partner (Send x21 x22 x23 x24 x25 x26) = x23"
set_rev: fixes xs :: "'g list" shows "set (rev xs) = set xs"
f'a_closed: shows "f' \<bullet> a \<in> carrier Zp"
induct: fixes x1 :: "('c \<times> 'd) option"   and x2 :: "('c \<times> 'd) option"   and C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and P :: "('c \<times> 'd) option \<Rightarrow> ('c \<times> 'd) option \<Rightarrow> bool" assumes "(x1, x2) \<in> whileLoop_results C B"   and "\<And>r s. \<not> C r s \<Longrightarrow> P (Some (r, s)) (Some (r, s))"   and "\<And>r s. \<lbrakk>C r s; snd (B r s)\<rbrakk> \<Longrightarrow> P (Some (r, s)) None"   and "\<And>r s r' s' z. \<lbrakk>C r s; (r', s') \<in> fst (B r s); (Some (r', s'), z) \<in> whileLoop_results C B; P (Some (r', s')) z\<rbrakk> \<Longrightarrow> P (Some (r, s)) z" shows "P x1 x2"
frac_integrable: fixes t :: "real" assumes "t \<in> {a..b}" shows "(\<lambda>t. g' t / g t) integrable_on {t..b}"
det3_eq_zeroI: fixes p :: "real \<times> real"   and q :: "real \<times> real"   and x :: "real"   and t :: "real \<times> real" assumes "p = q + x *\<^sub>R (t - q)" shows "coll q t p"
empty: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and t :: "state" assumes "matches \<gamma> m p"   and "\<Gamma>,\<gamma>,p\<turnstile> rs \<Rightarrow>\<^sub>z t" shows "\<Gamma>,\<gamma>,p\<turnstile> Rule m Empty # rs \<Rightarrow>\<^sub>z t"
closure_V: fixes Z :: "('c \<Rightarrow> real) set" shows "Closure\<^sub>\<alpha> Z \<subseteq> V"
n_4: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
inj_freshenLC: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and i :: "nat" assumes "Abstract_Formulas freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP" shows "inj (freshenLC i)"
Cons_in_list_Suc: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat"   and A :: "'b set" shows "(x # xs \<in> list (Suc n) A) = (x \<in> A \<and> xs \<in> list n A)"
assms: shows "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>"
Kleene_iter_gpfp: fixes f :: "'b \<Rightarrow> 'b"   and p :: "'b"   and k :: "nat" assumes "mono f"   and "p \<le> f p" shows "p \<le> (f ^^ k) top"
Eq\<^sub>3_prop: fixes v :: "i"   and F :: "\<Pi>\<^sub>3"   and G :: "\<Pi>\<^sub>3"   and \<phi> :: "\<Pi>\<^sub>3 \<Rightarrow> \<o>" assumes "[F \<^bold>=\<^sub>3 G in v]" shows "[\<phi> F in v] = [\<phi> G in v]"
fls_inverse_subdegree_base: fixes f :: "'a fls" shows "fls_nth (inverse f) (fls_subdegree (inverse f)) = inverse (fls_nth f (fls_subdegree f))"
induct: fixes P :: "'a MFOTL.trm list \<Rightarrow> 'a list \<Rightarrow> bool"   and a0 :: "'a MFOTL.trm list"   and a1 :: "'a list" assumes "P [] []"   and "\<And>x ts y ys. (x = y \<Longrightarrow> P ts ys) \<Longrightarrow> P (trm.Const x # ts) (y # ys)"   and "\<And>x ts y ys. P ts ys \<Longrightarrow> P (trm.Var x # ts) (y # ys)"   and "\<And>vb va. P (trm.Var vb # va) []"   and "\<And>v va. P (v # va) []"   and "\<And>v va. P [] (v # va)" shows "P a0 a1"
sgn_mult_abs: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and 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 a :: "'b" assumes "class.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "times (sgn a) (abs a) = a"
equivp_transp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "equivp R"   and "R x y"   and "R y z" shows "R x z"
asym: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "a \<prec>\<^sub>p b"   and "b \<prec>\<^sub>p a" shows "False"
assms_1: shows "other.s \<delta> \<le> u_max"
LAssThrow: fixes 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 E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "expr"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and V :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>V:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_aft_erase_a (s, l, r) lm n ires = (\<exists>lnl lnr rn ml m. m = Suc (lm ! n) \<and> l = Bk \<up> lnr @ Oc \<up> m @ Bk \<up> lnl @ Bk # Bk # ires \<and> r = <ml> @ Bk \<up> rn)"
neg: fixes s1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and s2 :: "'c \<Rightarrow> 'e \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "module_hom s1 s2 f" shows "f (- x) = - f x"
sublens_put_put: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f"   and \<sigma> :: "'f"   and v :: "'g"   and u :: "'e" assumes "mwb_lens X"   and "Y \<subseteq>\<^sub>L X" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>X\<^esub> \<sigma> u"
GroupHom_restrict0_subgroup: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'a set" assumes "GroupHom G T"   and "Group H \<and> H \<subseteq> G" shows "GroupHom H (T \<down> H)"
simps_2: fixes T :: "ty" shows "class_type_of' (T\<lfloor>\<rceil>) = \<lfloor>Object\<rfloor>"
relhoare_abstract_Un_left: fixes pre :: "('e \<times> 'f) set"   and Ra1 :: "('e \<times> 'e) set"   and Rc :: "('f \<times> 'f) set"   and post :: "('e \<times> 'f) set"   and Ra2 :: "('e \<times> 'e) set" assumes "{pre} Ra1, Rc {> post}" shows "{pre} Ra1 \<union> Ra2, Rc {> post}"
quotient_is_ring: shows "ring (K [X] Quot PIdl\<^bsub>K [X]\<^esub> p)"
map_M_pure_E: fixes h :: "'m"   and g :: "'p \<Rightarrow> ('m, 'n, 'o) prog"   and xs :: "'p list"   and ys :: "'o list"   and x :: "'p" assumes "h \<turnstile> map_M g xs \<rightarrow>\<^sub>r ys"   and "x \<in> set xs"   and "\<And>x h. x \<in> set xs \<Longrightarrow> pure (g x) h" obtains y :: "'o" where "h \<turnstile> g x \<rightarrow>\<^sub>r y" and "y \<in> set ys"
plusU_maybe_strict: fixes ys :: "udom\<cdot>maybe" shows "plusU\<cdot>\<bottom>\<cdot>ys = \<bottom>"
Ex_list_of_length: fixes n :: "nat" shows "\<exists>xs. |xs| = n"
map_iterator_linord_is_it: fixes m :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<times> 'e \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and it :: "'d \<Rightarrow> 'e option" assumes "map_iterator_linord m it" shows "map_iterator m it"
lex_contr: fixes t :: "real \<times> real"   and s :: "real \<times> real"   and q :: "real \<times> real"   and r :: "real \<times> real" assumes "distinct4 t s q r"   and "Counterclockwise_2D_Arbitrary.lex t s"   and "Counterclockwise_2D_Arbitrary.lex s r"   and "coll t s r"   and "ccw' t s q"   and "ccw' t q r" shows "False"
awalk_connected: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" assumes "Digraph_Component.connected G"   and "awalk u p v"   and "set p \<noteq> arcs G" shows "\<exists>e. e \<in> arcs G - set p \<and> (tail G e \<in> set (awalk_verts u p) \<or> head G e \<in> set (awalk_verts u p))"
pair_imageI: fixes a :: "'d"   and b :: "'e"   and A :: "('d \<times> 'e) set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
t12: shows "fst ` set txs \<subseteq> trm"
WTrts_same_length: fixes 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 E :: "char list \<Rightarrow> ty option"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and es :: "expr list"   and Ts :: "ty list" assumes "P,E,h \<turnstile> es [:] Ts" shows "length es = length Ts"
StuckProp: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com" shows "\<Gamma>\<turnstile> \<langle>c,Stuck\<rangle> \<Rightarrow> Stuck"
Red1TryFail: fixes s :: "'heap \<times> 'addr val list"   and a :: "'addr"   and D :: "String.literal"   and P :: "'addr J1_prog"   and C :: "String.literal"   and uf :: "bool"   and t :: "'thread_id"   and V :: "nat"   and e2 :: "'addr expr1" assumes "typeof_addr (hp s) a = \<lfloor>Class_type D\<rfloor>"   and "\<not> P \<turnstile> D \<preceq>\<^sup>* C" shows "uf,P,t \<turnstile>1 \<langle>try Throw a catch(C V) e2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
Q_of_Liminf_state_fin: shows "Q_of_state (Liminf_state Sts') = Q_of_state (Liminf_state Sts)"
bounded_valid_path_image: fixes g :: "real \<Rightarrow> 'a" assumes "valid_path g" shows "bounded (path_image g)"
abs_eq: fixes xa :: "('c, 'd) phi \<Rightarrow> 'd"   and x :: "('c, 'd) phi \<Rightarrow> 'd" assumes "eq_onp (\<lambda>xf. True) xa xa"   and "eq_onp (\<lambda>xf. True) x x" shows "(Abs_lifted_distrib_lattice_top xa < Abs_lifted_distrib_lattice_top x) = ((\<forall>f. xa f \<le> x f) \<and> \<not> (\<forall>f. x f \<le> xa f))"
size_4: fixes x4 :: "fm" shows "size (Neg x4) = size x4 + Suc 0"
sum_rv: fixes K :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> real"   and M :: "'c set set \<times> ('c set \<Rightarrow> real)" assumes "\<forall>k\<in>K. f k \<in> rv M"   and "measure_space M" shows "(\<lambda>t. \<Sum>k\<in>K. f k t) \<in> rv M"
finite_FOV: fixes \<phi> :: "'b formula" shows "finite (FOV \<phi>)"
zero_le: fixes x :: "'b multiset" shows "{#} \<subseteq># x"
rk_abc: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points"   and Q :: "Points"   and a :: "Points"   and b :: "Points"   and c :: "Points"   and R :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "4 \<le> rk {A, B, C, A', B', C', P, Q}"   and "rk {Q, A', a} = 2"   and "rk {Q, B', b} = 2"   and "rk {Q, C', c} = 2"   and "rk {P, Q, R} = 2"   and "rk {P, R} = 2"   and "rk {Q, R} = 2"   and "rk {R, A, a} = 2"   and "rk {R, B, b} = 2"   and "rk {R, C, c} = 2"   and "rk {A', P} = 2"   and "rk {B', P} = 2"   and "rk {C', P} = 2" shows "rk {a, b, c} = 3"
measurable_bind: fixes f :: "'a \<Rightarrow> 'b measure"   and M :: "'a measure"   and N :: "'b measure"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'c measure"   and R :: "'c measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "(\<lambda>x. g (fst x) (snd x)) \<in> M \<Otimes>\<^sub>M N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g x) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
preserves_iso: fixes f :: "'a" assumes "\<phi>.A.iso f" shows "\<phi>.B.iso (\<phi> f)"
backflows_id: fixes E :: "('b \<times> 'b) set" shows "backflows (backflows E) = E"
intensional_simps_11: fixes P :: "'b \<Rightarrow> bool" shows "#False \<longrightarrow> P \<equiv> #True"
intros_4: fixes a :: "name"   and s1 :: "lam"   and s2 :: "lam"   and t1 :: "lam"   and t2 :: "lam" assumes "a \<sharp> (s1, s2)"   and " s1 \<longrightarrow>\<^sub>1 s2"   and " t1 \<longrightarrow>\<^sub>1 t2" shows " App (Lam [a].t1) s1 \<longrightarrow>\<^sub>1 t2[a::=s2]"
NOU1: shows "Oadmit \<sigma> ODE1 (BVO (OProd ODE1 ODE2))"
int_A: fixes i :: "nat" shows "set_integrable M (A i) f"
ik\<^sub>e\<^sub>s\<^sub>t_append: fixes A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "ik\<^sub>e\<^sub>s\<^sub>t (A @ B) = ik\<^sub>e\<^sub>s\<^sub>t A \<union> ik\<^sub>e\<^sub>s\<^sub>t B"
hyps_4: fixes \<xi> :: "'a OCL_Types.type list" assumes "\<Gamma> \<turnstile>\<^sub>L exprs : \<xi>" shows "\<pi> = \<xi>"
step: fixes \<Phi> :: "'c"   and P :: "('a, 'b, 'c) psi"   and xs :: "('c \<times> ('a, 'b, 'c) psi) list" shows "psiCases ((\<Phi>, P) # xs) = \<box> \<Phi> \<Rightarrow> P psiCases xs "
ref_map_soundness: fixes f :: "'e \<Rightarrow> 'f"   and B :: "('e, 'g) ioa"   and A :: "('f, 'g) ioa" assumes "is_ref_map f B A"   and "externals (ioa.asig A) = externals (ioa.asig B)" shows "local.traces B \<subseteq> local.traces A"
set_empty: shows "Tree2.set_tree \<langle>\<rangle> = {}"
no0_1: shows "coeff lp x\<^sub>j \<noteq> 0"
set_id_left: fixes f :: "'b set_arrow"   and U :: "'b set" assumes "f \<in> Ar\<^bsub>set_cat U\<^esub>" shows "set_id U (set_cod f) \<odot> f = f"
openin_prod_topology: fixes X :: "'d topology"   and Y :: "'e topology" shows "openin (prod_topology X Y) = arbitrary union_of (\<lambda>U. U \<in> {S \<times> T |S T. openin X S \<and> openin Y T})"
index_conv_size_if_notin: fixes x :: "'e"   and xs :: "'e list" assumes "x \<notin> set xs" shows "index xs x = length xs"
cont_compose: fixes c :: "'a \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'a" assumes "cont c"   and "cont f" shows "cont (\<lambda>x. c (f x))"
new_tv_nth_nat_A: fixes nat :: "nat"   and A :: "'a list"   and n :: "nat" assumes "nat < length A"   and "new_tv n A" shows "new_tv n (A ! nat)"
c2: assumes "b < 0 \<and> - 1 < b" shows "\<exists>w. \<forall>y<w. b * y < y\<^sup>2"
eq_equiv_class_iff: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "equiv A r"   and "x \<in> A"   and "y \<in> A" shows "(r `` {x} = r `` {y}) = ((x, y) \<in> r)"
app_vrestrictionI: fixes a :: "V"   and A :: "V"   and b :: "V"   and r :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> A"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r \<restriction>\<^sub>\<circ> A"
hyps_4: fixes \<Gamma>\<^sub>2 :: "binding list"   and fs :: "(char list \<times> trm) list"   and \<Gamma>\<^sub>1 :: "binding list"   and t\<^sub>2 :: "trm"   and T\<^sub>2 :: "POPLmarkRecord.type"   and ts :: "trm list" assumes "\<Gamma>\<^sub>2 \<turnstile> fs [:] fTs"   and "\<Gamma>\<^sub>1 @ \<Delta>\<^sub>2 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2 : T\<^sub>2"   and "\<turnstile> fps [\<rhd>] fs \<Rightarrow> ts" shows "\<down>\<^sub>e \<parallel>\<Delta>\<^sub>2\<parallel> 0 \<Gamma>\<^sub>1 @ \<Gamma>\<^sub>2 \<turnstile> t\<^sub>2[\<parallel>\<Gamma>\<^sub>1\<parallel> \<mapsto>\<^sub>s ts] : \<down>\<^sub>\<tau> \<parallel>\<Delta>\<^sub>2\<parallel> \<parallel>\<Gamma>\<^sub>1\<parallel> T\<^sub>2"
sin_periodic_int: fixes x :: "real"   and i :: "int" shows "sin (x + real_of_int i * (2 * pi)) = sin x"
F_Un_D_eq_A: shows "F \<union> D = A"
OV_inv2D: fixes x :: "'a obsv_state_scheme"   and r :: "nat" assumes "x \<in> OV_inv2"   and "v_state.next_round x \<le> r" shows "obsv_state.obs x r = Map.empty"
implmt_subint2: fixes G :: "prog"   and A :: "qtname"   and J :: "qtname"   and K :: "qtname" assumes "G\<turnstile>A\<leadsto>J"   and "G\<turnstile>J\<preceq>I K" shows "G\<turnstile>A\<leadsto>K"
exp_of_nat_mult: fixes n :: "nat"   and x :: "'b" shows "exp (of_nat n * x) = exp x ^ n"
bounded_linearI': fixes f :: "'a \<Rightarrow> 'b" assumes "\<And>x y. f (x + y) = f x + f y"   and "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x" shows "bounded_linear f"
IH_7: assumes "\<forall>X\<langle>\<or>\<noteq>: F \<or>\<notin>: G\<rangle> \<in> set (S \<cdot>\<^sub>s\<^sub>s\<^sub>t \<sigma>)" shows "\<exists>F' G'. F = F' \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<sigma> \<and> G = G' \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s rm_vars (set X) \<sigma> \<and> \<forall>X\<langle>\<or>\<noteq>: F' \<or>\<notin>: G'\<rangle> \<in> set S"
pp_dist_sup: fixes x :: "'a"   and y :: "'a" shows "- - (x \<squnion> y) = - - x \<squnion> - - y"
prime_ex: fixes n :: "nat" shows "\<exists>p>n. p \<le> Suc (n!) \<and> Prime p"
reducedly_statically_complete_reduced_calculus_axioms_def: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "reducedly_statically_complete_reduced_calculus_axioms Bot Inf entails Red_I Red_F \<equiv> \<forall>B N. B \<in> Bot \<longrightarrow> calculus.reduc_saturated Inf Red_I Red_F N \<longrightarrow> entails N {B} \<longrightarrow> (\<exists>B'\<in>Bot. B' \<in> N)"
closedin_subset_trans: fixes U :: "'b set"   and S :: "'b set"   and T :: "'b set" assumes "closedin (top_of_set U) S"   and "S \<subseteq> T"   and "T \<subseteq> U" shows "closedin (top_of_set T) S"
f5: shows "4 \<le> rk {a, b, c, R, A, B}"
dis_log3_def: fixes \<A> :: "'a \<times> 'a \<Rightarrow> nat spmf" shows "local.dis_log_alt.dis_log3 \<A> = TRY sample_uniform (order \<G>) \<bind> (\<lambda>w. let (h, w) = ((\<^bold>g [^] w, local.dis_log_alt.g' [^] w), w) in \<A> h \<bind> (\<lambda>w'. return_spmf [w = w'] (mod order \<G>))) ELSE return_spmf False"
src_simps_2: fixes F :: "('a E.term, 'a) arr" assumes "arr F" shows "Cod (src F) = E.Src (Dom F)"
rel_optionT_simps: fixes M :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and m :: "'g"   and m' :: "'h" shows "rel_optionT M (OptionT m) (OptionT m') = M m m'"