Nuprl Lemma : nh-comp-is-id
∀[I,J:fset(ℕ)].
∀[f:I ⟶ J]. ∀[g:J ⟶ I].
g ⋅ f = 1 ∈ I ⟶ I supposing ∀x:names(I). (((g x) = <x> ∈ Point(dM(J))) ∧ ((f x) = <x> ∈ Point(dM(I))))
supposing I ⊆ J
Proof
Definitions occuring in Statement :
nh-comp: g ⋅ f,
nh-id: 1,
names-hom: I ⟶ J,
dM_inc: <x>,
dM: dM(I),
names: names(I),
lattice-point: Point(l),
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
names-hom: I ⟶ J,
nh-id: 1,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
dM: dM(I),
dM-lift: dM-lift(I;J;f),
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
so_apply: x[s],
nat: ℕ,
squash: ↓T,
dma-hom: dma-hom(dma1;dma2),
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
all: ∀x:A. B[x],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
names_wf,
all_wf,
equal_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM_inc_wf,
names-subtype,
names-hom_wf,
f-subset_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
squash_wf,
true_wf,
dM-lift_wf,
dma-hom_wf,
iff_weakening_equal,
dM-lift-is-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
functionExtensionality,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
productEquality,
applyEquality,
instantiate,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
intEquality,
natural_numberEquality,
imageElimination,
setElimination,
rename,
setEquality,
dependent_functionElimination,
productElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_set_memberEquality,
lambdaFormation
Latex:
\mforall{}[I,J:fset(\mBbbN{})].
\mforall{}[f:I {}\mrightarrow{} J]. \mforall{}[g:J {}\mrightarrow{} I]. g \mcdot{} f = 1 supposing \mforall{}x:names(I). (((g x) = <x>) \mwedge{} ((f x) = <x>))
supposing I \msubseteq{} J
Date html generated:
2017_10_05-AM-01_01_53
Last ObjectModification:
2017_07_28-AM-09_26_00
Theory : cubical!type!theory
Home
Index