Nuprl Lemma : names-hom-subtype
∀[I1,J1,I2,J2:fset(ℕ)]. (I1 ⟶ J1 ⊆r I2 ⟶ J2) supposing (J2 ⊆ J1 and I1 ⊆ I2)
Proof
Definitions occuring in Statement :
names-hom: I ⟶ J,
f-subset: xs ⊆ ys,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
names-hom: I ⟶ J,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
prop: ℙ,
and: P ∧ Q,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
nat: ℕ
Lemmas referenced :
strong-subtype-self,
le_wf,
strong-subtype-set3,
strong-subtype-deq-subtype,
int-deq_wf,
nat_wf,
f-subset_wf,
dM-point-subtype,
names-subtype,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
dM_wf,
lattice-point_wf,
names_wf,
subtype_rel_dep_function
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
instantiate,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
lambdaFormation,
axiomEquality,
intEquality,
natural_numberEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[I1,J1,I2,J2:fset(\mBbbN{})]. (I1 {}\mrightarrow{} J1 \msubseteq{}r I2 {}\mrightarrow{} J2) supposing (J2 \msubseteq{} J1 and I1 \msubseteq{} I2)
Date html generated:
2016_05_18-AM-11_57_44
Last ObjectModification:
2016_01_26-PM-04_09_25
Theory : cubical!type!theory
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