Nuprl Lemma : subtype-fpf3
∀[A1,A2:Type]. ∀[B1:A1 ⟶ Type]. ∀[B2:A2 ⟶ Type].
(a:A1 fp-> B1[a] ⊆r a:A2 fp-> B2[a]) supposing ((∀a:A1. (B1[a] ⊆r B2[a])) and strong-subtype(A1;A2))
Proof
Definitions occuring in Statement :
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
fpf: a:A fp-> B[a],
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
all: ∀x:A. B[x]
Lemmas referenced :
strong-subtype-implies,
subtype_rel_list,
l_member_wf,
fpf_wf,
all_wf,
subtype_rel_wf,
strong-subtype_wf,
strong-subtype-l_member-type,
strong-subtype-l_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
dependent_pairEquality,
applyEquality,
independent_isectElimination,
sqequalRule,
functionExtensionality,
setEquality,
functionEquality,
setElimination,
rename,
cumulativity,
universeEquality,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A1,A2:Type]. \mforall{}[B1:A1 {}\mrightarrow{} Type]. \mforall{}[B2:A2 {}\mrightarrow{} Type].
(a:A1 fp-> B1[a] \msubseteq{}r a:A2 fp-> B2[a]) supposing
((\mforall{}a:A1. (B1[a] \msubseteq{}r B2[a])) and
strong-subtype(A1;A2))
Date html generated:
2018_05_21-PM-09_17_07
Last ObjectModification:
2018_02_09-AM-10_16_23
Theory : finite!partial!functions
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