Nuprl Lemma : mul-distributes
∀[x,y,z:Top]. (x * (y + z) ~ (x * y) + (x * z))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
top: Top,
multiply: n * m,
add: n + m,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
guard: {T},
sq_type: SQType(T),
prop: ℙ,
uimplies: b supposing a,
implies: P ⇒ Q,
and: P ∧ Q,
has-value: (a)↓,
uall: ∀[x:A]. B[x],
false: False
Lemmas referenced :
int-mul-exception,
int-add-exception,
top_wf,
is-exception_wf,
has-value_wf_base,
int_subtype_base,
subtype_base_sq,
equal_wf,
int-value-type,
value-type-has-value,
exception-not-value
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
multiplyDistributive,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
Error :universeIsType,
intEquality,
multiplyEquality,
voidEquality,
voidElimination,
isect_memberEquality,
axiomSqEquality,
because_Cache,
exceptionSqequal,
addExceptionCases,
axiomSqleEquality,
multiplyExceptionCases,
sqleReflexivity,
cumulativity,
instantiate,
callbyvalueAdd,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
isectElimination,
extract_by_obid,
lambdaFormation,
equalitySymmetry,
equalityTransitivity,
productElimination,
baseClosed,
closedConclusion,
baseApply,
sqequalRule,
sqequalHypSubstitution,
callbyvalueMultiply,
divergentSqle,
thin,
sqleRule,
sqequalSqle,
introduction,
isect_memberFormation
Latex:
\mforall{}[x,y,z:Top]. (x * (y + z) \msim{} (x * y) + (x * z))
Date html generated:
2019_06_20-AM-11_22_12
Last ObjectModification:
2018_10_15-AM-11_14_36
Theory : arithmetic
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