Nuprl Lemma : iseg-iff-firstn
∀[T:Type]. ∀L1,L2:T List. (L1 ≤ L2 ⇐⇒ (||L1|| ≤ ||L2||) ∧ (L1 = firstn(||L1||;L2) ∈ (T List)))
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
firstn: firstn(n;as),
length: ||as||,
list: T List,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
le: A ≤ B,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T,
subtype_rel: A ⊆r B,
int_iseg: {i...j},
guard: {T},
less_than: a < b,
uiff: uiff(P;Q),
true: True,
sq_type: SQType(T)
Lemmas referenced :
less_than'_wf,
length_wf,
exists_wf,
int_seg_wf,
equal_wf,
list_wf,
firstn_wf,
non_neg_length,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
lelt_wf,
le_wf,
firstn_is_iseg,
iseg_wf,
iff_wf,
squash_wf,
true_wf,
length_firstn_eq,
subtype_rel_sets,
int_seg_properties,
add-is-int-iff,
false_wf,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
length_firstn
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
cumulativity,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
addEquality,
setElimination,
rename,
dependent_pairFormation,
dependent_set_memberEquality,
because_Cache,
unionElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
productEquality,
addLevel,
impliesFunctionality,
independent_functionElimination,
universeEquality,
applyEquality,
imageElimination,
setEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
imageMemberEquality,
hyp_replacement,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. (L1 \mleq{} L2 \mLeftarrow{}{}\mRightarrow{} (||L1|| \mleq{} ||L2||) \mwedge{} (L1 = firstn(||L1||;L2)))
Date html generated:
2017_04_17-AM-07_31_22
Last ObjectModification:
2017_02_27-PM-04_08_34
Theory : list_1
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