Nuprl Lemma : fl-morph-fl0-is-1
∀[J,I:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[x:names(I)]. uiff(((x=0))<f> = 1 ∈ Point(face_lattice(J));(f x) = 0 ∈ Point(dM(J)))
Proof
Definitions occuring in Statement :
fl-morph: <f>,
fl0: (x=0),
face_lattice: face_lattice(I),
names-hom: I ⟶ J,
dM0: 0,
dM: dM(I),
names: names(I),
lattice-1: 1,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
rev_implies: P ⇐ Q,
names-hom: I ⟶ J,
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
DeMorgan-algebra: DeMorganAlgebra,
all: ∀x:A. B[x],
dma-neg: ¬(x),
top: Top,
dM0: 0,
dM1: 1
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
face_lattice_wf,
fl-morph-fl0,
lattice-1_wf,
iff_weakening_equal,
fl-morph_wf,
bounded-lattice-hom_wf,
fl0_wf,
dM_wf,
dM0_wf,
uiff_wf,
dM-to-FL_wf,
dm-neg_wf,
names_wf,
names-deq_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
bdd-distributive-lattice_wf,
names-hom_wf,
fset_wf,
nat_wf,
iff_weakening_uiff,
dM-to-FL-eq-1,
rec_select_update_lemma,
dma-neg_wf,
DeMorgan-algebra_wf,
DeMorgan-algebra-laws,
dma-neg-dM0,
dM1_wf
Rules used in proof :
cut,
addLevel,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
independent_pairFormation,
isect_memberFormation,
independent_isectElimination,
applyEquality,
lambdaEquality,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
cumulativity,
setElimination,
rename,
instantiate,
productEquality,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
functionExtensionality,
lambdaFormation,
dependent_functionElimination,
voidElimination,
voidEquality,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[J,I:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[x:names(I)]. uiff(((x=0))<f> = 1;(f x) = 0)
Date html generated:
2017_10_05-AM-01_13_44
Last ObjectModification:
2017_07_28-AM-09_31_07
Theory : cubical!type!theory
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