Nuprl Lemma : K-assignment_subtype
∀[K:mKripkeStruct]. ∀[i,j:World].
  ∀vs1,vs2:ℤ List.  FOAssignment(vs1,Dom(i)) ⊆r FOAssignment(vs2,Dom(j)) supposing vs2 ⊆ vs1 supposing i ≤ j
Proof
Definitions occuring in Statement : 
K-dom: Dom(i)
, 
K-le: i ≤ j
, 
K-world: World
, 
mFO-Kripke-struct: mKripkeStruct
, 
FOAssignment: FOAssignment(vs,Dom)
, 
l_contains: A ⊆ B
, 
list: T List
, 
uimplies: b supposing a
, 
subtype_rel: A ⊆r B
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
int: ℤ
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
l_contains: A ⊆ B
, 
so_lambda: λ2x.t[x]
, 
prop: ℙ
, 
so_apply: x[s]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
FOAssignment: FOAssignment(vs,Dom)
, 
subtype_rel: A ⊆r B
Lemmas referenced : 
l_all_iff, 
l_member_wf, 
istype-int, 
subtype_rel_dep_function, 
K-dom_wf, 
subtype_rel_sets, 
K-dom_subtype, 
l_contains_wf, 
list_wf, 
K-le_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
lambdaFormation_alt, 
sqequalHypSubstitution, 
extract_by_obid, 
isectElimination, 
thin, 
intEquality, 
dependent_functionElimination, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality_alt, 
hypothesis, 
setElimination, 
rename, 
closedConclusion, 
setIsType, 
universeIsType, 
productElimination, 
independent_functionElimination, 
setEquality, 
independent_isectElimination, 
because_Cache, 
axiomEquality, 
inhabitedIsType, 
isect_memberEquality_alt, 
isectIsTypeImplies, 
functionIsTypeImplies
Latex:
\mforall{}[K:mKripkeStruct].  \mforall{}[i,j:World].
    \mforall{}vs1,vs2:\mBbbZ{}  List.    FOAssignment(vs1,Dom(i))  \msubseteq{}r  FOAssignment(vs2,Dom(j))  supposing  vs2  \msubseteq{}  vs1 
    supposing  i  \mleq{}  j
Date html generated:
2019_10_16-AM-11_44_39
Last ObjectModification:
2018_10_13-AM-10_17_46
Theory : minimal-first-order-logic
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