Nuprl Lemma : assert-isdM0
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. uiff(↑isdM0(x);x = 0 ∈ Point(dM(I)))
Proof
Definitions occuring in Statement :
isdM0: isdM0(x),
dM0: 0,
dM: dM(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
guard: {T},
so_apply: x[s],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
fset-antichain: fset-antichain(eq;ac),
fset-pairwise: fset-pairwise(x,y.R[x; y];s),
fset-null: fset-null(s),
null: null(as),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
empty-fset: {},
nil: [],
it: ⋅,
btrue: tt,
true: True,
isdM0: isdM0(x),
dM0: 0,
lattice-0: 0,
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],
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)
Lemmas referenced :
assert_wf,
isdM0_wf,
assert_witness,
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,
dM0_wf,
fset_wf,
nat_wf,
dM-point,
dM0-sq-empty,
fset-antichain_wf,
names_wf,
union-deq_wf,
names-deq_wf,
assert-fset-null
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
unionEquality,
setElimination,
rename,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[x:Point(dM(I))]. uiff(\muparrow{}isdM0(x);x = 0)
Date html generated:
2016_10_26-PM-01_05_13
Last ObjectModification:
2016_07_12-AM-09_26_41
Theory : cubical!type!theory
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