Nuprl Lemma : isaxiom-sqequal
∀[C:Base]
∀[A,B,z:Base].
if z = Ax then A z otherwise B z ~ C z
supposing (A Ax ~ C Ax) ∧ ((∀a,b:Base. (if z = Ax then a otherwise b ~ b)) ⇒ (B z ~ C z))
supposing strict(C)
Proof
Definitions occuring in Statement :
strict: strict(F),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
isaxiom: if z = Ax then a otherwise b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
base: Base,
sqequal: s ~ t,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
has-value: (a)↓,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
strict: strict(F)
Lemmas referenced :
strict_wf,
base_wf,
all_wf,
and_wf,
is-exception_wf,
has-value_wf_base,
has-value-implies-dec-isaxiom-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalSqle,
divergentSqle,
callbyvalueIsaxiom,
sqequalHypSubstitution,
hypothesis,
productElimination,
thin,
lemma_by_obid,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
unionElimination,
sqequalRule,
sqleReflexivity,
baseApply,
closedConclusion,
baseClosed,
lambdaFormation,
because_Cache,
isaxiomExceptionCases,
axiomSqleEquality,
isectElimination,
callbyvalueApply,
applyExceptionCases,
sqequalAxiom,
sqequalIntensionalEquality,
functionEquality,
lambdaEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
exceptionSqequal
Latex:
\mforall{}[C:Base]
\mforall{}[A,B,z:Base].
if z = Ax then A z otherwise B z \msim{} C z
supposing (A Ax \msim{} C Ax) \mwedge{} ((\mforall{}a,b:Base. (if z = Ax then a otherwise b \msim{} b)) {}\mRightarrow{} (B z \msim{} C z))
supposing strict(C)
Date html generated:
2016_05_13-PM-03_24_08
Last ObjectModification:
2016_01_14-PM-06_46_29
Theory : call!by!value_1
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