Nuprl Lemma : strong-subtype-product
∀[A,B,C,D:Type]. (strong-subtype(A × B;C × D)) supposing (strong-subtype(B;D) and strong-subtype(A;C))
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
guard: {T},
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
prop: ℙ,
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
strong-subtype-implies,
subtype_rel_product,
exists_wf,
equal_wf,
strong-subtype_witness,
strong-subtype_wf,
pi1_wf,
pi2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
independent_isectElimination,
lambdaFormation,
because_Cache,
independent_pairFormation,
setElimination,
rename,
independent_pairEquality,
setEquality,
productEquality,
applyEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
dependent_pairFormation,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[A,B,C,D:Type].
(strong-subtype(A \mtimes{} B;C \mtimes{} D)) supposing (strong-subtype(B;D) and strong-subtype(A;C))
Date html generated:
2016_05_13-PM-04_11_20
Last ObjectModification:
2015_12_26-AM-11_21_28
Theory : subtype_1
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