Nuprl Lemma : prec-size-induction
∀[P:Type]. ∀[a:Atom ⟶ P ⟶ ((P + P + Type) List)]. ∀[Q:i:P ⟶ prec(lbl,p.a[lbl;p];i) ⟶ TYPE].
((∀i:P. ∀x:prec(lbl,p.a[lbl;p];i). ((∀j:P. ∀z:{z:prec(lbl,p.a[lbl;p];j)| ||j;z|| < ||i;x||} . Q[j;z]) ⇒ Q[i;x]))
⇒ (∀i:P. ∀x:prec(lbl,p.a[lbl;p];i). Q[i;x]))
Proof
Definitions occuring in Statement :
prec-size: ||i;x||,
prec: prec(lbl,p.a[lbl; p];i),
list: T List,
less_than: a < b,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
union: left + right,
atom: Atom,
universe: Type
Definitions unfolded in proof :
guard: {T},
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
lelt: i ≤ j < k,
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
false: False,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
all: ∀x:A. B[x],
nat: ℕ,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
le_witness_for_triv,
istype-universe,
list_wf,
int_term_value_add_lemma,
itermAdd_wf,
decidable__le,
istype-false,
int_seg_subtype_nat,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__lt,
subtract-1-ge-0,
int_seg_properties,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
istype-nat,
prec-size_wf,
istype-le,
istype-atom,
prec_wf,
int_seg_wf
Rules used in proof :
applyLambdaEquality,
functionExtensionality,
imageElimination,
universeEquality,
cumulativity,
unionEquality,
instantiate,
TYPEIsType,
setIsType,
addEquality,
equalityIstype,
equalitySymmetry,
equalityTransitivity,
productIsType,
unionElimination,
dependent_set_memberEquality_alt,
productElimination,
functionIsTypeImplies,
axiomEquality,
independent_pairFormation,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
intWeakElimination,
TYPEMemberIsType,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
because_Cache,
functionIsType,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
universeIsType,
isectIsType,
sqequalRule,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[P:Type]. \mforall{}[a:Atom {}\mrightarrow{} P {}\mrightarrow{} ((P + P + Type) List)]. \mforall{}[Q:i:P {}\mrightarrow{} prec(lbl,p.a[lbl;p];i) {}\mrightarrow{} TYPE].
((\mforall{}i:P. \mforall{}x:prec(lbl,p.a[lbl;p];i).
((\mforall{}j:P. \mforall{}z:\{z:prec(lbl,p.a[lbl;p];j)| ||j;z|| < ||i;x||\} . Q[j;z]) {}\mRightarrow{} Q[i;x]))
{}\mRightarrow{} (\mforall{}i:P. \mforall{}x:prec(lbl,p.a[lbl;p];i). Q[i;x]))
Date html generated:
2019_10_15-AM-10_25_03
Last ObjectModification:
2019_09_26-PM-04_38_54
Theory : tuples
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