Nuprl Lemma : coW2natinf_wf
∀[n:ℕ]. ∀[w:coW(ℕ2;x.if (x =z 0) then Void else Unit fi )]. (coW2natinf(w;n) ∈ 𝔹)
Proof
Definitions occuring in Statement :
coW2natinf: coW2natinf(w;n),
coW: coW(A;a.B[a]),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
bool: 𝔹,
uall: ∀[x:A]. B[x],
unit: Unit,
member: t ∈ T,
natural_number: $n,
void: Void
Definitions unfolded in proof :
ext-eq: A ≡ B,
lelt: i ≤ j < k,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
pi2: snd(t),
coW-item: coW-item(w;b),
or: P ∨ Q,
decidable: Dec(P),
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
pi1: fst(t),
coW2natinf: coW2natinf(w;n),
guard: {T},
subtype_rel: A ⊆r B,
so_apply: x[s],
int_seg: {i..j-},
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
nat_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_seg_properties,
subtype_rel-equal,
it_wf,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
bfalse_wf,
btrue_wf,
bool_wf,
subtype_rel_weakening,
coW-ext,
unit_wf2,
eq_int_wf,
ifthenelse_wf,
int_seg_wf,
coW_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties
Rules used in proof :
cumulativity,
promote_hyp,
equalityElimination,
unionElimination,
productElimination,
functionEquality,
productEquality,
applyEquality,
because_Cache,
hypothesis_subsumption,
universeEquality,
instantiate,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[w:coW(\mBbbN{}2;x.if (x =\msubz{} 0) then Void else Unit fi )]. (coW2natinf(w;n) \mmember{} \mBbbB{})
Date html generated:
2018_07_29-AM-09_29_16
Last ObjectModification:
2018_07_27-PM-03_25_40
Theory : basic
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