Nuprl Lemma : irr-face-morph-property
∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))]. ∀[J:fset(ℕ)]. ∀[g:J ⟶ I].
((irr_face(I;as;bs) g) = 1 ⇒ (g = irr-face-morph(I;as;bs) ⋅ g ∈ J ⟶ I))
Proof
Definitions occuring in Statement :
name-morph-satisfies: (psi f) = 1,
irr-face-morph: irr-face-morph(I;as;bs),
irr_face: irr_face(I;as;bs),
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
names: names(I),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
names-hom: I ⟶ J,
compose: f o g,
irr-face-morph: irr-face-morph(I;as;bs),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
prop: ℙ,
squash: ↓T,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True
Lemmas referenced :
satisfies-irr-face,
nh-comp-sq,
deq-fset-member_wf,
names_wf,
names-deq_wf,
eqtt_to_assert,
assert-deq-fset-member,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
fset-member_wf,
dM-lift-inc,
name-morph-satisfies_wf,
irr_face_wf,
names-hom_wf,
fset_wf,
nat_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
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,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
dM-lift_wf2,
dM0_wf,
subtype_rel_self,
iff_weakening_equal,
dM-lift-0,
dM1_wf,
dM-lift-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
sqequalRule,
functionExtensionality,
Error :memTop,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
universeIsType,
lambdaEquality_alt,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies,
applyEquality,
imageElimination,
universeEquality,
productEquality,
isectEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[as,bs:fset(names(I))]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[g:J {}\mrightarrow{} I].
((irr\_face(I;as;bs) g) = 1 {}\mRightarrow{} (g = irr-face-morph(I;as;bs) \mcdot{} g))
Date html generated:
2020_05_20-PM-01_45_12
Last ObjectModification:
2019_12_27-AM-00_27_42
Theory : cubical!type!theory
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