Nuprl Lemma : strong-subtype-set
∀[A,B:Type]. ∀[P:A ⟶ ℙ]. ∀[Q:B ⟶ ℙ]. strong-subtype({x:A| P[x]} ;{x:B| Q[x]} ) supposing ∀x:A. (P[x] ⇒ Q[x]) suppos\000Cing strong-subtype(A;B)
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
guard: {T},
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q
Lemmas referenced :
strong-subtype-implies,
strong-subtype_witness,
all_wf,
strong-subtype_wf,
subtype_rel_sets,
subtype_rel_transitivity,
exists_wf,
equal_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
independent_pairFormation,
setEquality,
cumulativity,
applyEquality,
functionExtensionality,
lambdaEquality,
sqequalRule,
universeEquality,
because_Cache,
functionEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
dependent_set_memberEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
equalityUniverse,
levelHypothesis,
imageElimination,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:B {}\mrightarrow{} \mBbbP{}]. strong-subtype(\{x:A| P[x]\} ;\{x:B| Q[x]\} ) supposing \mforall{}x:A. (P[x] {}\mRightarrow{} Q[x]\000C)
supposing strong-subtype(A;B)
Date html generated:
2016_10_21-AM-09_41_28
Last ObjectModification:
2016_07_12-AM-05_03_33
Theory : subtype_1
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