Nuprl Lemma : iseg_implies_member
∀[T:Type]. ∀l1,l2:T List. (l1 ≤ l2 ⇒ {∀x:T. ((x ∈ l1) ⇒ (x ∈ l2))})
Proof
Definitions occuring in Statement :
iseg: l1 ≤ l2,
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
guard: {T},
l_member: (x ∈ l),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
member: t ∈ T,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
decidable__lt,
length_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
less_than_wf,
select_wf,
decidable__le,
itermConstant_wf,
int_term_value_constant_lemma,
exists_wf,
nat_wf,
le_wf,
all_wf,
isect_wf,
iseg_select,
iseg_wf,
l_member_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
dependent_pairFormation,
hypothesisEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
imageMemberEquality,
baseClosed,
instantiate,
productEquality,
cumulativity,
addLevel,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}l1,l2:T List. (l1 \mleq{} l2 {}\mRightarrow{} \{\mforall{}x:T. ((x \mmember{} l1) {}\mRightarrow{} (x \mmember{} l2))\})
Date html generated:
2019_06_20-PM-01_28_51
Last ObjectModification:
2018_08_24-PM-11_19_51
Theory : list_1
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