Nuprl Lemma : code-coded-pair
∀n:ℕ. (let a,b = coded-pair(n) in code-pair(a;b) = n ∈ ℤ)
Proof
Definitions occuring in Statement :
code-pair: code-pair(a;b),
coded-pair: coded-pair(m),
nat: ℕ,
all: ∀x:A. B[x],
spread: spread def,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
code-pair: code-pair(a;b),
coded-pair: coded-pair(m),
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
top: Top,
guard: {T},
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,
and: P ∧ Q,
subtract: n - m
Lemmas referenced :
tsqrt_wf,
nat_wf,
value-type-has-value,
set-value-type,
le_wf,
int-value-type,
subtract_wf,
triangular-num_wf,
equal_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
itermMultiply_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_term_value_mul_lemma,
int_formula_prop_wf,
minus-add,
minus-minus,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
callbyvalueReduce,
independent_isectElimination,
intEquality,
lambdaEquality,
natural_numberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
applyEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
addEquality,
multiplyEquality,
minusEquality,
applyLambdaEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}n:\mBbbN{}. (let a,b = coded-pair(n) in code-pair(a;b) = n)
Date html generated:
2019_06_20-PM-02_39_22
Last ObjectModification:
2019_06_12-PM-00_27_32
Theory : num_thy_1
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