Nuprl Lemma : base-is-base-list
∀[T:Type]. ∀[x:Base]. ((x ∈ T List) ⇒ (x ∈ Base List))
Proof
Definitions occuring in Statement :
list: T List,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
nil: [],
it: ⋅,
sq_type: SQType(T),
true: True,
pi2: snd(t),
uiff: uiff(P;Q)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
length_wf,
int_subtype_base,
equal-wf-base,
list_wf,
base_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
equal_wf,
list-cases,
nil_wf,
product_subtype_list,
cons_wf,
length_of_cons_lemma,
le_weakening2,
non_neg_length,
decidable__lt,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
pair-eta,
subtype_rel_product,
top_wf,
exists_wf,
sqequal-wf-base,
subtype_base_sq,
pi2_wf,
pi1_wf,
decidable__equal_int,
add-is-int-iff,
false_wf,
length_wf_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
universeEquality,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
baseClosed,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
baseApply,
closedConclusion,
sqequalIntensionalEquality,
instantiate,
cumulativity,
applyLambdaEquality,
pointwiseFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[x:Base]. ((x \mmember{} T List) {}\mRightarrow{} (x \mmember{} Base List))
Date html generated:
2018_05_21-PM-01_17_08
Last ObjectModification:
2018_05_03-PM-07_20_23
Theory : num_thy_1
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