Nuprl Lemma : swap_eval_3
∀i,j,k:ℤ. ((¬(k = i ∈ ℤ)) ⇒ (¬(k = j ∈ ℤ)) ⇒ ((swap(i;j) k) = k ∈ ℤ))
Proof
Definitions occuring in Statement :
swap: swap(i;j),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
swap: swap(i;j),
member: t ∈ T,
uall: ∀[x:A]. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
prop: ℙ
Lemmas referenced :
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
int_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not_wf,
equal-wf-base,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
independent_functionElimination,
voidElimination,
dependent_pairFormation_alt,
equalityIsType2,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
because_Cache,
equalityIsType1,
universeIsType,
intEquality
Latex:
\mforall{}i,j,k:\mBbbZ{}. ((\mneg{}(k = i)) {}\mRightarrow{} (\mneg{}(k = j)) {}\mRightarrow{} ((swap(i;j) k) = k))
Date html generated:
2019_10_16-PM-00_59_18
Last ObjectModification:
2018_10_08-AM-09_26_39
Theory : perms_1
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