Nuprl Lemma : equal-nat-inf-infinity2
∀[x:ℕ∞]. uiff(x = ∞ ∈ ℕ∞;∀i:ℕ. (↑(x i)))
Proof
Definitions occuring in Statement :
nat-inf-infinity: ∞,
nat-inf: ℕ∞,
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
assert: ↑b,
ifthenelse: if b then t else f fi ,
nat-inf-infinity: ∞,
btrue: tt,
true: True,
prop: ℙ,
nat-inf: ℕ∞,
implies: P ⇒ Q,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
assert_wf,
nat_wf,
assert_witness,
equal-wf-T-base,
nat-inf_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_wf,
iff_imp_equal_bool,
nat-inf-infinity_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
thin,
natural_numberEquality,
hyp_replacement,
equalitySymmetry,
Error :applyLambdaEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
applyEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
baseClosed,
dependent_set_memberEquality,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionExtensionality,
functionEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
axiomEquality
Latex:
\mforall{}[x:\mBbbN{}\minfty{}]. uiff(x = \minfty{};\mforall{}i:\mBbbN{}. (\muparrow{}(x i)))
Date html generated:
2016_10_25-AM-10_14_14
Last ObjectModification:
2016_07_12-AM-06_25_05
Theory : basic
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