Nuprl Lemma : init0-implies-eq-upto1-zero-seq
∀a:ℕ ⟶ ℕ. (init0(a) ⇒ (a = 0s ∈ (ℕ1 ⟶ ℕ)))
Proof
Definitions occuring in Statement :
init0: init0(a),
zero-seq: 0s,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
subtype_rel: A ⊆r B,
init0: init0(a),
less_than': less_than'(a;b),
le: A ≤ B,
zero-seq: 0s,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
prop: ℙ,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
init0_wf,
int_seg_wf,
equal_wf,
and_wf,
false_wf,
int_subtype_base,
set_subtype_base,
nat_wf,
subtype_base_sq,
le_wf,
decidable__le,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
int_seg_properties
Rules used in proof :
functionEquality,
applyEquality,
applyLambdaEquality,
levelHypothesis,
addLevel,
hyp_replacement,
independent_functionElimination,
cumulativity,
instantiate,
because_Cache,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
productElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
functionExtensionality,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (init0(a) {}\mRightarrow{} (a = 0s))
Date html generated:
2017_04_21-AM-11_23_00
Last ObjectModification:
2017_04_20-PM-04_48_02
Theory : continuity
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