Nuprl Lemma : proper-iseg-length
∀[T:Type]. ∀L1,L2:T List. (L1 < L2 ⇐⇒ L1 ≤ L2 ∧ ||L1|| < ||L2||)
Proof
Definitions occuring in Statement :
proper-iseg: L1 < L2,
iseg: l1 ≤ l2,
length: ||as||,
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
proper-iseg: L1 < L2,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
iseg: l1 ≤ l2,
exists: ∃x:A. B[x],
or: P ∨ Q,
cons: [a / b],
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
le: A ≤ B,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
true: True
Lemmas referenced :
iseg_wf,
not_wf,
equal_wf,
list_wf,
less_than_wf,
length_wf,
list-cases,
product_subtype_list,
append_back_nil,
length_wf_nat,
nat_wf,
length-append,
length_of_cons_lemma,
non_neg_length,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
cut,
productEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
independent_functionElimination,
voidElimination,
universeEquality,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
equalitySymmetry,
dependent_set_memberEquality,
isect_memberEquality,
voidEquality,
addEquality,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
computeAll,
hyp_replacement,
Error :applyLambdaEquality,
setElimination,
rename,
applyEquality,
imageElimination,
because_Cache,
imageMemberEquality,
baseClosed,
equalityTransitivity
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. (L1 < L2 \mLeftarrow{}{}\mRightarrow{} L1 \mleq{} L2 \mwedge{} ||L1|| < ||L2||)
Date html generated:
2016_10_21-AM-10_32_28
Last ObjectModification:
2016_07_12-AM-05_45_44
Theory : list_1
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