Nuprl Lemma : l-ind-sqequal-list_ind
∀[L,x,F:Top]. (l-ind(L;x;h,t,r.F[h;t;r]) ~ rec-case(L) of [] => x | h::t => r.F[h;t;r])
Proof
Definitions occuring in Statement :
l-ind: l-ind(L;nilcase;h,t,r.F[h; t; r]),
list_ind: list_ind,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2;s3],
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
l-ind: l-ind(L;nilcase;h,t,r.F[h; t; r]),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
fun_exp: f^n,
primrec: primrec(n;b;c),
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
list_ind: list_ind,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
sq_type: SQType(T)
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
base_wf,
strictness-apply,
bottom-sqle,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
fun_exp_unroll_1,
cbv_sqle,
int_subtype_base,
has-value-implies-dec-ispair-2,
has-value_wf_base,
is-exception_wf,
top_wf,
has-value-implies-dec-isaxiom-2,
bottom_diverge,
exception-not-bottom,
istype-top,
pair-eta,
subtype_base_sq,
exception-not-value
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
sqequalSqle,
lambdaFormation,
fixpointLeast,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomSqleEquality,
isect_memberEquality,
voidEquality,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
applyEquality,
intEquality,
minusEquality,
because_Cache,
dependent_set_memberEquality,
baseApply,
closedConclusion,
baseClosed,
divergentSqle,
sqleRule,
sqleReflexivity,
axiomSqEquality,
callbyvalueSpread,
equalityTransitivity,
equalitySymmetry,
instantiate,
cumulativity,
spreadExceptionCases
Latex:
\mforall{}[L,x,F:Top]. (l-ind(L;x;h,t,r.F[h;t;r]) \msim{} rec-case(L) of [] => x | h::t => r.F[h;t;r])
Date html generated:
2019_06_20-PM-00_38_48
Last ObjectModification:
2018_09_26-PM-02_46_54
Theory : list_0
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