Nuprl Lemma : sqequal_n_add
∀x,y:Base. ∀n,m:ℕ. ((x ~n + m y) ⇒ (x ~n y))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
add: n + m,
base: Base,
sqequal_n: s ~n t
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
add-zero,
sqequal_n_wf,
decidable__le,
false_wf,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-zero,
add-associates,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
le_wf,
sqequaln_sqlen,
sqle_n_subtype_rel,
sqlen_sqequaln,
subtract_wf,
add_nat_wf,
equal_wf,
subtract-add-cancel,
le_weakening2,
nat_wf,
less-iff-le,
minus-minus,
add-swap,
set_wf,
less_than_wf,
primrec-wf2,
base_wf,
sqequaln_symm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
instantiate
Latex:
\mforall{}x,y:Base. \mforall{}n,m:\mBbbN{}. ((x \msim{}n + m y) {}\mRightarrow{} (x \msim{}n y))
Date html generated:
2017_04_14-AM-07_32_47
Last ObjectModification:
2017_02_27-PM-03_07_20
Theory : int_1
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