Nuprl Lemma : imax-list-subset
∀[L,L':ℤ List]. (imax-list(L) ≤ imax-list(L')) supposing (l_subset(ℤ;L;L') and 0 < ||L||)
Proof
Definitions occuring in Statement :
l_subset: l_subset(T;as;bs),
imax-list: imax-list(L),
length: ||as||,
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
guard: {T},
l_subset: l_subset(T;as;bs)
Lemmas referenced :
and_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
l_exists_iff,
imax-list-ub,
l_member_wf,
le_wf,
l_all_iff,
list_wf,
length_wf,
less_than_wf,
l_subset_wf,
less_than'_wf,
imax-list_wf,
imax-list-lb,
l_subset_pos_length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
productElimination,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
natural_numberEquality,
voidElimination,
setElimination,
rename,
setEquality,
independent_functionElimination,
lambdaFormation,
dependent_pairFormation,
independent_pairFormation,
unionElimination,
int_eqEquality,
voidEquality,
computeAll
Latex:
\mforall{}[L,L':\mBbbZ{} List]. (imax-list(L) \mleq{} imax-list(L')) supposing (l\_subset(\mBbbZ{};L;L') and 0 < ||L||)
Date html generated:
2016_05_14-PM-01_41_56
Last ObjectModification:
2016_01_15-AM-08_23_19
Theory : list_1
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