Nuprl Lemma : type-equating-some
∀T:Type. ∀P:T ⟶ ℙ.
∃T':Type
((T ⊆r T') ∧ (∀x,y:T. (P[x] ⇒ P[y] ⇒ (x = y ∈ T'))) ∧ (∀x,y:T. ((¬P[x]) ⇒ (x = y ∈ T ⇐⇒ x = y ∈ T'))))
Proof
Definitions occuring in Statement :
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
uimplies: b supposing a,
cand: A c∧ B,
implies: P ⇒ Q,
rel_implies: R1 => R2,
infix_ap: x f y,
iff: P ⇐⇒ Q,
guard: {T},
rev_implies: P ⇐ Q,
quotient: x,y:A//B[x; y],
so_lambda: λ2x.t[x],
least-equiv: least-equiv(A;R),
transitive-reflexive-closure: R^*,
or: P ∨ Q,
transitive-closure: TC(R),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
cons: [a / b],
top: Top,
rel_path: rel_path(A;L;x;y),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi2: snd(t),
pi1: fst(t),
not: ¬A
Lemmas referenced :
quotient_wf,
least-equiv_wf,
least-equiv-is-equiv,
subtype_quotient,
quotient-member-eq,
implies-least-equiv,
equal_functionality_wrt_subtype_rel2,
equal_wf,
member_wf,
not_wf,
subtype_rel_wf,
all_wf,
iff_wf,
or_wf,
and_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
productEquality,
functionExtensionality,
hypothesis,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
pertypeElimination,
productElimination,
functionEquality,
universeEquality,
unionElimination,
setElimination,
rename,
imageElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}P:T {}\mrightarrow{} \mBbbP{}.
\mexists{}T':Type
((T \msubseteq{}r T') \mwedge{} (\mforall{}x,y:T. (P[x] {}\mRightarrow{} P[y] {}\mRightarrow{} (x = y))) \mwedge{} (\mforall{}x,y:T. ((\mneg{}P[x]) {}\mRightarrow{} (x = y \mLeftarrow{}{}\mRightarrow{} x = y))))
Date html generated:
2018_05_21-PM-01_12_12
Last ObjectModification:
2018_05_03-PM-02_19_17
Theory : num_thy_1
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