Nuprl Lemma : formal-sum-add_functionality
∀S:Type. ∀K:RngSig. ∀x,x',y,y':basic-formal-sum(K;S).
(bfs-equiv(K;S;y;y') ⇒ bfs-equiv(K;S;x;x') ⇒ bfs-equiv(K;S;x + y;x' + y'))
Proof
Definitions occuring in Statement :
formal-sum-add: x + y,
bfs-equiv: bfs-equiv(K;S;fs1;fs2),
basic-formal-sum: basic-formal-sum(K;S),
all: ∀x:A. B[x],
implies: P ⇒ Q,
universe: Type,
rng_sig: RngSig
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
prop: ℙ,
bfs-reduce: bfs-reduce(K;S;as;bs),
or: P ∨ Q,
so_lambda: λ2x.t[x],
and: P ∧ Q,
basic-formal-sum: basic-formal-sum(K;S),
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
exists: ∃x:A. B[x],
formal-sum-add: x + y,
top: Top,
squash: ↓T,
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
rng_sig_wf,
bfs-equiv-implies,
bfs-equiv_wf,
formal-sum-add_wf1,
basic-formal-sum_wf,
bfs-reduce_wf,
implies-bfs-equiv,
exists_wf,
rng_car_wf,
equal_wf,
bag-append_wf,
formal-sum-mul_wf1,
rng_plus_wf,
subtype_rel_self,
bag_wf,
zero-bfs_wf,
bag-append-assoc2,
squash_wf,
true_wf,
bag-append-comm,
bag-append-ac,
iff_weakening_equal,
bfs-equiv-rel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
universeEquality,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
unionElimination,
inlFormation,
productEquality,
applyEquality,
inrFormation,
productElimination,
dependent_pairFormation,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
imageElimination,
equalityTransitivity,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
independent_pairFormation
Latex:
\mforall{}S:Type. \mforall{}K:RngSig. \mforall{}x,x',y,y':basic-formal-sum(K;S).
(bfs-equiv(K;S;y;y') {}\mRightarrow{} bfs-equiv(K;S;x;x') {}\mRightarrow{} bfs-equiv(K;S;x + y;x' + y'))
Date html generated:
2018_05_22-PM-09_45_25
Last ObjectModification:
2018_05_20-PM-10_42_49
Theory : linear!algebra
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