Nuprl Lemma : satisfies-irr-face
∀[I,J:fset(ℕ)]. ∀[as,bs:fset(names(I))]. ∀[g:J ⟶ I].
uiff((irr_face(I;as;bs) g) = 1;(∀a:names(I). (a ∈ as ⇒ ((g a) = 0 ∈ Point(dM(J)))))
∧ (∀b:names(I). (b ∈ bs ⇒ ((g b) = 1 ∈ Point(dM(J))))))
Proof
Definitions occuring in Statement :
name-morph-satisfies: (psi f) = 1,
irr_face: irr_face(I;as;bs),
names-hom: I ⟶ J,
dM1: 1,
dM0: 0,
dM: dM(I),
names-deq: NamesDeq,
names: names(I),
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
irr_face: irr_face(I;as;bs),
name-morph-satisfies: (psi f) = 1,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
names-hom: I ⟶ J,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
squash: ↓T,
bdd-lattice: BoundedLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
exists: ∃x:A. B[x],
compose: f o g,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
btrue: tt,
cand: A c∧ B,
dM0: 0,
lattice-0: 0,
empty-fset: {},
nil: [],
it: ⋅,
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
fset-union: x ⋃ y,
l-union: as ⋃ bs,
insert: insert(a;L),
eval_list: eval_list(t),
deq-member: x ∈b L,
lattice-join: a ∨ b,
opposite-lattice: opposite-lattice(L),
so_lambda: λ2x y.t[x; y],
lattice-meet: a ∧ b,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
lattice-fset-meet: /\(s),
dM1: 1
Lemmas referenced :
names-hom_wf,
fset_wf,
names_wf,
nat_wf,
lattice-fset-meet-is-1,
fset-member_wf,
names-deq_wf,
lattice-point_wf,
face_lattice_wf,
face_lattice-deq_wf,
fset-image_wf,
fl0_wf,
fl1_wf,
dM_wf,
dM0_wf,
dM1_wf,
iff_weakening_uiff,
equal_wf,
lattice-fset-meet_wf,
decidable__equal_face_lattice,
all_wf,
rev_implies_wf,
lattice-meet_wf,
lattice-1_wf,
lattice-meet-eq-1,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-join_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
fl-morph_wf,
fset-union_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
squash_wf,
true_wf,
istype-universe,
lattice-fset-meet-union,
subtype_rel_self,
iff_weakening_equal,
fl-morph-fset-meet,
decidable_wf,
bdd-lattice_wf,
fset-image-union,
deq_wf,
fset-image-compose,
compose_wf,
exists_wf,
member-fset-image-iff,
fl-morph-fl0,
dM-to-FL_wf,
dm-neg_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
dM-to-FL-eq-1,
dm-neg-neg,
fl-morph-fl1,
dM-to-FL-dM1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
isect_memberEquality_alt,
isectElimination,
hypothesisEquality,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
hypothesis,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
universeIsType,
extract_by_obid,
because_Cache,
independent_pairFormation,
lambdaFormation_alt,
productIsType,
functionIsType,
applyEquality,
equalityIsType1,
independent_isectElimination,
independent_functionElimination,
functionEquality,
promote_hyp,
productEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
dependent_pairFormation_alt,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[as,bs:fset(names(I))]. \mforall{}[g:J {}\mrightarrow{} I].
uiff((irr\_face(I;as;bs) g) = 1;(\mforall{}a:names(I). (a \mmember{} as {}\mRightarrow{} ((g a) = 0)))
\mwedge{} (\mforall{}b:names(I). (b \mmember{} bs {}\mRightarrow{} ((g b) = 1))))
Date html generated:
2019_11_04-PM-05_35_07
Last ObjectModification:
2018_11_08-AM-11_07_40
Theory : cubical!type!theory
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