Nuprl Lemma : code-pair-bijection
Bij(ℕ;ℕ × ℕ;λx.coded-pair(x))
Proof
Definitions occuring in Statement :
coded-pair: coded-pair(m),
biject: Bij(A;B;f),
nat: ℕ,
lambda: λx.A[x],
product: x:A × B[x]
Definitions unfolded in proof :
biject: Bij(A;B;f),
and: P ∧ Q,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
surject: Surj(A;B;f),
squash: ↓T,
label: ...$L... t,
nat: ℕ,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
equal_wf,
nat_wf,
coded-pair_wf,
code-pair_wf,
squash_wf,
true_wf,
code-coded-pair,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
le_wf,
iff_weakening_equal,
coded-code-pair
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
lambdaFormation,
cut,
sqequalHypSubstitution,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
thin,
productEquality,
applyEquality,
lambdaEquality,
hypothesisEquality,
applyLambdaEquality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
setElimination,
rename,
unionElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
dependent_set_memberEquality,
because_Cache,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairEquality
Latex:
Bij(\mBbbN{};\mBbbN{} \mtimes{} \mBbbN{};\mlambda{}x.coded-pair(x))
Date html generated:
2018_05_21-PM-07_55_03
Last ObjectModification:
2017_07_26-PM-05_32_38
Theory : general
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