Nuprl Lemma : equal-by-name-cases
∀a,x,y:Name.
∀T:Type. ∀z:T. ∀P,Q:ℙ. ∀t1:⋂p:P. T. ∀t2:⋂q:Q. T.
(((a = x ∈ Name) ∧ P ∧ (z = t1 ∈ T)) ∨ ((a = y ∈ Name) ∧ Q ∧ (z = t2 ∈ T))
⇐⇒ (((a = x ∈ Name) ∧ P) ∨ ((a = y ∈ Name) ∧ Q)) ∧ (z = if name_eq(a;x) then t1 else t2 fi ∈ T))
supposing ¬(x = y ∈ Name)
Proof
Definitions occuring in Statement :
name_eq: name_eq(x;y),
name: Name,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
isect: ⋂x:A. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
iff: P ⇐⇒ Q,
or: P ∨ Q,
cand: A c∧ B,
guard: {T},
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
equal_wf,
name_wf,
name_eq_wf,
bool_wf,
eqtt_to_assert,
assert-name_eq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
voidElimination,
extract_by_obid,
isectElimination,
hypothesis,
rename,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_pairFormation,
inlFormation,
productEquality,
independent_functionElimination,
inrFormation,
because_Cache,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
isectEquality,
universeEquality
Latex:
\mforall{}a,x,y:Name.
\mforall{}T:Type. \mforall{}z:T. \mforall{}P,Q:\mBbbP{}. \mforall{}t1:\mcap{}p:P. T. \mforall{}t2:\mcap{}q:Q. T.
(((a = x) \mwedge{} P \mwedge{} (z = t1)) \mvee{} ((a = y) \mwedge{} Q \mwedge{} (z = t2))
\mLeftarrow{}{}\mRightarrow{} (((a = x) \mwedge{} P) \mvee{} ((a = y) \mwedge{} Q)) \mwedge{} (z = if name\_eq(a;x) then t1 else t2 fi ))
supposing \mneg{}(x = y)
Date html generated:
2017_04_17-AM-09_17_37
Last ObjectModification:
2017_02_27-PM-05_21_54
Theory : decidable!equality
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