Nuprl Lemma : fpf-compose_wf
∀[A:Type]. ∀[B,C:A ⟶ Type]. ∀[f:a:A fp-> B[a]]. ∀[g:⋂a:A. (B[a] ⟶ C[a])]. (g o f ∈ a:A fp-> C[a])
Proof
Definitions occuring in Statement :
fpf-compose: g o f,
fpf: a:A fp-> B[a],
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
isect: ⋂x:A. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
fpf-compose: g o f,
fpf: a:A fp-> B[a],
pi1: fst(t),
pi2: snd(t),
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
compose: f o g,
subtype_rel: A ⊆r B
Lemmas referenced :
l_member_wf,
fpf_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
functionEquality,
setEquality,
cumulativity,
lemma_by_obid,
isectElimination,
because_Cache,
hypothesis,
lambdaFormation,
setElimination,
rename,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isectEquality,
isect_memberEquality,
lambdaEquality,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B,C:A {}\mrightarrow{} Type]. \mforall{}[f:a:A fp-> B[a]]. \mforall{}[g:\mcap{}a:A. (B[a] {}\mrightarrow{} C[a])]. (g o f \mmember{} a:A fp-> C[a])
Date html generated:
2018_05_21-PM-09_27_33
Last ObjectModification:
2018_02_09-AM-10_23_05
Theory : finite!partial!functions
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