Nuprl Lemma : dma-neg-eq-1-implies-meet-eq-0
∀[K:fset(ℕ)]. ∀[a2,x1:Point(dM(K))]. ((¬(a2) = 1 ∈ Point(dM(K))) ⇒ (0 = a2 ∧ x1 ∈ Point(dM(K))))
Proof
Definitions occuring in Statement :
dM0: 0,
dM: dM(I),
names-deq: NamesDeq,
names: names(I),
dm-neg: ¬(x),
lattice-1: 1,
lattice-meet: a ∧ b,
lattice-point: Point(l),
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,
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
so_apply: x[s],
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,
rev_implies: P ⇐ Q,
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)
Lemmas referenced :
dM-neg-properties,
equal_wf,
squash_wf,
true_wf,
istype-universe,
lattice-meet_wf,
dM_wf,
subtype_rel_self,
iff_weakening_equal,
dm-neg_wf,
names_wf,
names-deq_wf,
subtype_rel-equal,
lattice-point_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-join_wf,
DeMorgan-algebra-axioms_wf,
free-DeMorgan-lattice_wf,
lattice-1_wf,
fset_wf,
nat_wf,
dM0_wf,
deq_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
lattice-join-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
inhabitedIsType,
instantiate,
universeEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
equalityIsType1,
productEquality,
cumulativity,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[K:fset(\mBbbN{})]. \mforall{}[a2,x1:Point(dM(K))]. ((\mneg{}(a2) = 1) {}\mRightarrow{} (0 = a2 \mwedge{} x1))
Date html generated:
2019_11_04-PM-05_30_39
Last ObjectModification:
2018_11_08-AM-10_18_14
Theory : cubical!type!theory
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