Nuprl Lemma : compose-fpf_wf
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[f:x:A fp-> B[x]]. ∀[C:Type]. ∀[a:A ⟶ (C?)]. ∀[b:C ⟶ A].
compose-fpf(a;b;f) ∈ y:C fp-> B[b y] supposing ∀y:A. ((↑isl(a y)) ⇒ ((b outl(a y)) = y ∈ A))
Proof
Definitions occuring in Statement :
compose-fpf: compose-fpf(a;b;f),
fpf: a:A fp-> B[a],
outl: outl(x),
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
unit: Unit,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
compose-fpf: compose-fpf(a;b;f),
fpf: a:A fp-> B[a],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
isl: isl(x),
compose: f o g,
fpf-domain: fpf-domain(f),
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
true: True,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
fpf_wf,
equal_wf,
all_wf,
iff_weakening_equal,
list_wf,
true_wf,
squash_wf,
l_member_wf,
member_map_filter,
outl_wf,
assert_wf,
unit_wf2,
isl_wf,
top_wf,
subtype-fpf2,
fpf-domain_wf,
mapfilter_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_pairEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
because_Cache,
applyEquality,
sqequalRule,
lambdaEquality,
hypothesis,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
setEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
functionEquality,
axiomEquality,
unionEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[C:Type]. \mforall{}[a:A {}\mrightarrow{} (C?)]. \mforall{}[b:C {}\mrightarrow{} A].
compose-fpf(a;b;f) \mmember{} y:C fp-> B[b y] supposing \mforall{}y:A. ((\muparrow{}isl(a y)) {}\mRightarrow{} ((b outl(a y)) = y))
Date html generated:
2018_05_21-PM-09_27_54
Last ObjectModification:
2018_02_09-AM-10_23_23
Theory : finite!partial!functions
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