::: ## Lambda calculus environ vocabularies LAMBDA, NUMBERS, NAT_1, XBOOLE_0, SUBSET_1, FINSEQ_1, XXREAL_0, CARD_1, ARYTM_1, ARYTM_3, TARSKI, RELAT_1, ORDINAL4, FUNCOP_1; :: etc... begin reserve D for DecoratedTree, p,q,r for FinSequence of NAT, x for set; definition let D; attr D is LambdaTerm-like means (dom D qua Tree) is finite & ::> *143,306 for r st r in dom D holds r is FinSequence of {0,1} & r^<*0*> in dom D implies D.r = 0; end; registration cluster LambdaTerm-like for DecoratedTree of NAT; existence; ::> *4 end; definition mode LambdaTerm is LambdaTerm-like DecoratedTree of NAT; end; ::: Then we extend this ordinary one-step beta reduction, that is, ::: any subterm is also allowed to reduce. definition let M,N; pred M beta N means ex p st M|p beta_shallow N|p & for q st not p is_a_prefix_of q holds [r,x] in M iff [r,x] in N; end; theorem Th4: ProperPrefixes (v^<*x*>) = ProperPrefixes v \/ {v} proof thus ProperPrefixes (v^<*x*>) c= ProperPrefixes v \/ {v} proof let y; assume y in ProperPrefixes (v^<*x*>); then consider v1 such that A1: y = v1 and A2: v1 is_a_proper_prefix_of v^<*x*> by TREES_1:def 2; v1 is_a_prefix_of v & v1 <> v or v1 = v by A2,TREES_1:9; then v1 is_a_proper_prefix_of v or v1 in {v} by TARSKI:def 1,XBOOLE_0:def 8; then y in ProperPrefixes v or y in {v} by A1,TREES_1:def 2; hence thesis by XBOOLE_0:def 3; end; let y; assume y in ProperPrefixes v \/ {v}; then A3: y in ProperPrefixes v or y in {v} by XBOOLE_0:def 3; A4: now assume y in ProperPrefixes v; then consider v1 such that A5: y = v1 and A6: v1 is_a_proper_prefix_of v by TREES_1:def 2; v is_a_prefix_of v^<*x*> by TREES_1:1; then v1 is_a_proper_prefix_of v^<*x*> by A6,XBOOLE_1:58; hence thesis by A5,TREES_1:def 2; end; v^{} = v by FINSEQ_1:34; then v is_a_prefix_of v^<*x*> & v <> v^<*x*> by FINSEQ_1:33,TREES_1:1; then v is_a_proper_prefix_of v^<*x*> by XBOOLE_0:def 8; then y in ProperPrefixes v or y = v & v in ProperPrefixes (v^<*x*>) by A3,TARSKI:def 1,TREES_1:def 2; hence thesis by A4; end;