Nuprl Lemma : term-induction1
∀[opr:Type]. ∀[P:term(opr) ⟶ ℙ].
((∀v:{v:varname()| ¬(v = nullvar() ∈ varname())} . P[varterm(v)])
⇒ (∀f:opr. ∀bts:bound-term(opr) List. ((∀i:ℕ||bts||. P[snd(bts[i])]) ⇒ P[mkterm(f;bts)]))
⇒ {∀t:term(opr). P[t]})
Proof
Definitions occuring in Statement :
bound-term: bound-term(opr),
mkterm: mkterm(opr;bts),
varterm: varterm(v),
term: term(opr),
nullvar: nullvar(),
varname: varname(),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
uimplies: b supposing a,
coterm-fun: coterm-fun(opr;T),
varterm: varterm(v),
mkterm: mkterm(opr;bts),
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
false: False,
le: A ≤ B,
uiff: uiff(P;Q),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bound-term: bound-term(opr),
less_than: a < b,
pi2: snd(t)
Lemmas referenced :
uniform-comp-nat-induction,
term_wf,
le_wf,
term-size_wf,
istype-nat,
term-ext,
sq_stable__le,
istype-le,
subtype_rel_transitivity,
coterm-fun_wf,
subtype_rel_weakening,
ext-eq_inversion,
term_size_var_lemma,
term_size_mkterm_lemma,
term-size-positive,
mkterm_wf,
subtract_wf,
nat_properties,
decidable__le,
add-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
itermAdd_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
false_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
int_seg_wf,
length_wf,
list_wf,
varname_wf,
lsum_wf,
pi2_wf,
l_member_wf,
bound-term_wf,
select_wf,
int_seg_properties,
subtype_rel_self,
nullvar_wf,
istype-void,
varterm_wf,
istype-universe,
summand-le-lsum,
non_neg_length,
select_member
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
lambdaEquality_alt,
functionEquality,
setEquality,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
setElimination,
rename,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
setIsType,
universeIsType,
independent_isectElimination,
inhabitedIsType,
unionElimination,
dependent_functionElimination,
Error :memTop,
natural_numberEquality,
productElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
productIsType,
productEquality,
addEquality,
equalityIstype,
isectIsType,
functionIsType,
instantiate,
universeEquality,
applyLambdaEquality
Latex:
\mforall{}[opr:Type]. \mforall{}[P:term(opr) {}\mrightarrow{} \mBbbP{}].
((\mforall{}v:\{v:varname()| \mneg{}(v = nullvar())\} . P[varterm(v)])
{}\mRightarrow{} (\mforall{}f:opr. \mforall{}bts:bound-term(opr) List. ((\mforall{}i:\mBbbN{}||bts||. P[snd(bts[i])]) {}\mRightarrow{} P[mkterm(f;bts)]))
{}\mRightarrow{} \{\mforall{}t:term(opr). P[t]\})
Date html generated:
2020_05_19-PM-09_54_24
Last ObjectModification:
2020_03_09-PM-04_08_33
Theory : terms
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