Nuprl Lemma : fpf-compatible-wf2
∀[A:Type]. ∀[B,C:A ⟶ Type]. ∀[eq:EqDecider(A)]. ∀[f:a:A fp-> B[a]]. ∀[g:a:A fp-> C[a]].
f || g ∈ ℙ supposing ∀x:A. ((↑x ∈ dom(f)) ⇒ (↑x ∈ dom(g)) ⇒ (B[x] ⊆r C[x]))
Proof
Definitions occuring in Statement :
fpf-compatible: f || g,
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
deq: EqDecider(T),
assert: ↑b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fpf-compatible: f || g,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
top: Top
Lemmas referenced :
all_wf,
assert_wf,
fpf-dom_wf,
subtype-fpf2,
top_wf,
equal_wf,
fpf-ap_wf,
subtype_rel_wf,
fpf_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
functionEquality,
productEquality,
because_Cache,
applyEquality,
functionExtensionality,
hypothesis,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B,C:A {}\mrightarrow{} Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[g:a:A fp-> C[a]].
f || g \mmember{} \mBbbP{} supposing \mforall{}x:A. ((\muparrow{}x \mmember{} dom(f)) {}\mRightarrow{} (\muparrow{}x \mmember{} dom(g)) {}\mRightarrow{} (B[x] \msubseteq{}r C[x]))
Date html generated:
2018_05_21-PM-09_19_55
Last ObjectModification:
2018_02_09-AM-10_17_47
Theory : finite!partial!functions
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