Nuprl Lemma : int_formula-induction
∀[P:int_formula() ⟶ ℙ]
((∀left,right:int_term(). P[(left "<" right)])
⇒ (∀left,right:int_term(). P[left "≤" right])
⇒ (∀left,right:int_term(). P[left "=" right])
⇒ (∀left,right:int_formula(). (P[left] ⇒ P[right] ⇒ P[left "∧" right]))
⇒ (∀left,right:int_formula(). (P[left] ⇒ P[right] ⇒ P[left "or" right]))
⇒ (∀left,right:int_formula(). (P[left] ⇒ P[right] ⇒ P[left "=>" right]))
⇒ (∀form:int_formula(). (P[form] ⇒ P["¬"form]))
⇒ {∀v:int_formula(). P[v]})
Proof
Definitions occuring in Statement :
intformnot: "¬"form,
intformimplies: left "=>" right,
intformor: left "or" right,
intformand: left "∧" right,
intformeq: left "=" right,
intformle: left "≤" right,
intformless: (left "<" right),
int_formula: int_formula(),
int_term: int_term(),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
intformless: (left "<" right),
int_formula_size: int_formula_size(p),
pi1: fst(t),
pi2: snd(t),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
intformle: left "≤" right,
intformeq: left "=" right,
intformand: left "∧" right,
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
intformor: left "or" right,
intformimplies: left "=>" right,
intformnot: "¬"form,
ge: i ≥ j ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T
Lemmas referenced :
uniform-comp-nat-induction,
all_wf,
int_formula_wf,
isect_wf,
le_wf,
int_formula_size_wf,
nat_wf,
less_than'_wf,
int_formula-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
set_subtype_base,
int_subtype_base,
add-is-int-iff,
nat_properties,
subtract_wf,
decidable__le,
false_wf,
not-le-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,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
lelt_wf,
uall_wf,
int_seg_wf,
le_reflexive,
intformnot_wf,
intformimplies_wf,
intformor_wf,
intformand_wf,
int_term_wf,
intformeq_wf,
intformle_wf,
intformless_wf,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow,
less_than_wf,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
setElimination,
rename,
functionExtensionality,
independent_functionElimination,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
instantiate,
cumulativity,
atomEquality,
dependent_pairFormation,
independent_pairFormation,
sqequalIntensionalEquality,
intEquality,
natural_numberEquality,
baseClosed,
baseApply,
closedConclusion,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
isect_memberEquality,
voidEquality,
minusEquality,
functionEquality,
universeEquality,
multiplyEquality,
imageMemberEquality
Latex:
\mforall{}[P:int\_formula() {}\mrightarrow{} \mBbbP{}]
((\mforall{}left,right:int\_term(). P[(left "<" right)])
{}\mRightarrow{} (\mforall{}left,right:int\_term(). P[left "\mleq{}" right])
{}\mRightarrow{} (\mforall{}left,right:int\_term(). P[left "=" right])
{}\mRightarrow{} (\mforall{}left,right:int\_formula(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left "\mwedge{}" right]))
{}\mRightarrow{} (\mforall{}left,right:int\_formula(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left "or" right]))
{}\mRightarrow{} (\mforall{}left,right:int\_formula(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left "=>" right]))
{}\mRightarrow{} (\mforall{}form:int\_formula(). (P[form] {}\mRightarrow{} P["\mneg{}"form]))
{}\mRightarrow{} \{\mforall{}v:int\_formula(). P[v]\})
Date html generated:
2017_04_14-AM-09_01_50
Last ObjectModification:
2017_02_27-PM-03_44_24
Theory : omega
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