Nuprl Lemma : cons_functionality_wrt_permr_upto
∀T:Type. ∀R:T ⟶ T ⟶ ℙ.
(EquivRel(T;x,y.R[x;y])
⇒ (∀a,b:T. ∀as,bs:T List. (R[a;b] ⇒ as ≡ bs upto x,y.R[x;y] ⇒ [a / as] ≡ [b / bs] upto x,y.R[x;y] )))
Proof
Definitions occuring in Statement :
permr_upto: as ≡ bs upto x,y.R[x; y] ,
cons: [a / b],
list: T List,
equiv_rel: EquivRel(T;x,y.E[x; y]),
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
permr_upto_wf,
subtype_rel_self,
list_wf,
istype-universe,
equiv_rel_wf,
cons_wf,
permr_upto_split,
permr_wf,
lequiv_wf,
cons_functionality_wrt_permr,
cons_functionality_wrt_lequiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
hypothesis,
instantiate,
isectElimination,
universeEquality,
functionIsType,
independent_functionElimination,
productElimination,
dependent_pairFormation_alt,
productIsType,
independent_pairFormation,
because_Cache
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
(EquivRel(T;x,y.R[x;y])
{}\mRightarrow{} (\mforall{}a,b:T. \mforall{}as,bs:T List.
(R[a;b] {}\mRightarrow{} as \mequiv{} bs upto x,y.R[x;y] {}\mRightarrow{} [a / as] \mequiv{} [b / bs] upto x,y.R[x;y] )))
Date html generated:
2019_10_16-PM-01_01_37
Last ObjectModification:
2018_10_08-AM-09_49_03
Theory : perms_2
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