Nuprl Lemma : mrec-induction
∀[L:MutualRectypeSpec]. ∀[P:mobj(L) ⟶ TYPE].
((∀x:mobj(L). ((∀z:{z:mobj(L)| z < x} . P[z]) ⇒ P[x])) ⇒ (∀x:mobj(L). P[x]))
Proof
Definitions occuring in Statement :
mrec-lt: x < y,
mobj: mobj(L),
mrec_spec: MutualRectypeSpec,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
implies: P ⇒ Q,
mrec: mrec(L;i),
all: ∀x:A. B[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
mobj: mobj(L),
mkinds: mKinds,
mrec-lt: x < y,
uimplies: b supposing a,
prop: ℙ
Lemmas referenced :
mobj-ext,
mrec_wf,
istype-atom,
prec-induction-ext,
mrec-spec_wf,
prec_wf,
subtype_rel-equal,
mtype_wf,
mtype-sqequal,
mrec-lt_wf,
prec_sub+_wf,
subtype_rel_self,
mobj_wf,
mrec_spec_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
Error :lambdaFormation_alt,
sqequalRule,
applyEquality,
Error :dependent_pairEquality_alt,
Error :universeIsType,
atomEquality,
Error :lambdaEquality_alt,
Error :inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
setElimination,
rename,
Error :dependent_set_memberEquality_alt,
independent_isectElimination,
Error :setIsType,
Error :functionIsType,
instantiate,
universeEquality,
Error :TYPEMemberIsType,
Error :TYPEIsType
Latex:
\mforall{}[L:MutualRectypeSpec]. \mforall{}[P:mobj(L) {}\mrightarrow{} TYPE].
((\mforall{}x:mobj(L). ((\mforall{}z:\{z:mobj(L)| z < x\} . P[z]) {}\mRightarrow{} P[x])) {}\mRightarrow{} (\mforall{}x:mobj(L). P[x]))
Date html generated:
2019_06_20-PM-02_15_52
Last ObjectModification:
2019_03_12-PM-06_12_37
Theory : tuples
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