Nuprl Lemma : pair-lex_well_fnd
∀[A,B:Type]. ∀[Ra:A ⟶ A ⟶ ℙ]. ∀[Rb:B ⟶ B ⟶ ℙ].
(WellFnd{i}(A;a1,a2.Ra a1 a2) ⇒ WellFnd{i}(B;b1,b2.Rb b1 b2) ⇒ WellFnd{i}(A × B;p1,p2.pair-lex(A;Ra;Rb) p1 p2))
Proof
Definitions occuring in Statement :
pair-lex: pair-lex(A;Ra;Rb),
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_apply: x[s1;s2],
implies: P ⇒ Q,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
all: ∀x:A. B[x],
pair-lex: pair-lex(A;Ra;Rb),
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
or: P ∨ Q,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
product_well_fnd,
pair-lex_wf,
all_wf,
or_wf,
and_wf,
equal_wf,
wellfounded_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
productElimination,
applyEquality,
lambdaEquality,
universeEquality,
productEquality,
functionEquality,
spreadEquality,
cumulativity
Latex:
\mforall{}[A,B:Type]. \mforall{}[Ra:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}[Rb:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}].
(WellFnd\{i\}(A;a1,a2.Ra a1 a2)
{}\mRightarrow{} WellFnd\{i\}(B;b1,b2.Rb b1 b2)
{}\mRightarrow{} WellFnd\{i\}(A \mtimes{} B;p1,p2.pair-lex(A;Ra;Rb) p1 p2))
Date html generated:
2016_05_13-PM-03_18_37
Last ObjectModification:
2015_12_26-AM-09_06_51
Theory : well_fnd
Home
Index