Nuprl Lemma : rel_exp_add
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀m,n:ℕ. ∀[x,y,z:T]. ((x R^m y) ⇒ (y R^n z) ⇒ (x R^m + n z))
Proof
Definitions occuring in Statement :
rel_exp: R^n,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
nat: ℕ,
all: ∀x:A. B[x],
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
so_apply: x[s],
infix_ap: x f y,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
rel_exp: R^n,
or: P ∨ Q,
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
subtract: n - m,
ge: i ≥ j ,
top: Top,
nat_plus: ℕ+,
less_than: a < b,
true: True,
exists: ∃x:A. B[x],
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
decidable: Dec(P)
Lemmas referenced :
complete_nat_ind_with_y,
all_wf,
nat_wf,
uall_wf,
infix_ap_wf,
rel_exp_wf,
add_nat_wf,
sq_stable__le,
equal_wf,
le_wf,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
eq_int_wf,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_subtype_base,
and_wf,
subtract_wf,
minus-zero,
zero-add,
add-zero,
nat_properties,
add-associates,
add-mul-special,
zero-mul,
not-equal-implies-less,
subtype_rel_self,
less-iff-le,
add_functionality_wrt_le,
le_reflexive,
minus-one-mul-top,
one-mul,
add-commutes,
minus-one-mul,
mul-associates,
omega-shadow,
less_than_wf,
two-mul,
mul-distributes-right,
add-swap,
mul-distributes,
minus-add,
mul-commutes,
mul-swap,
not-le-2,
le-add-cancel,
uiff_transitivity,
decidable__le,
general_arith_equation1,
not-equal-2,
condition-implies-le,
minus-minus,
decidable__lt,
not-lt-2,
le-add-cancel-alt,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
hypothesis,
cumulativity,
hypothesisEquality,
because_Cache,
functionEquality,
universeEquality,
functionExtensionality,
applyEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
lambdaFormation,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
productElimination,
intEquality,
unionElimination,
impliesFunctionality,
addLevel,
hyp_replacement,
applyLambdaEquality,
levelHypothesis,
multiplyEquality,
minusEquality,
isect_memberEquality,
voidElimination,
voidEquality,
promote_hyp,
dependent_pairFormation,
productEquality,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}m,n:\mBbbN{}. \mforall{}[x,y,z:T]. ((x R\^{}m y) {}\mRightarrow{} (y R\^{}n z) {}\mRightarrow{} (x rel\_exp(T; R; m + n) z))
Date html generated:
2017_04_14-AM-07_38_14
Last ObjectModification:
2017_02_27-PM-03_10_26
Theory : relations
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