Nuprl Lemma : cons_functionality_wrt_lequiv
∀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 :
lequiv: 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],
lequiv: as = bs upto {x,y.R[x; y]},
cand: A c∧ B,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
int_seg: {i..j-},
sq_type: SQType(T),
guard: {T},
select: L[n],
cons: [a / b],
lelt: i ≤ j < k,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lequiv_wf,
subtype_rel_self,
list_wf,
istype-universe,
equiv_rel_wf,
length_of_cons_lemma,
istype-void,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
int_seg_wf,
length_wf,
cons_wf,
subtype_base_sq,
int_subtype_base,
select_cons_tl,
int_seg_properties,
decidable__lt,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
non_neg_length,
decidable__le,
iff_weakening_equal,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_wf,
less_than_wf
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,
productElimination,
independent_pairFormation,
isect_memberEquality_alt,
voidElimination,
because_Cache,
unionElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
setElimination,
rename,
cumulativity,
intEquality,
dependent_set_memberEquality_alt,
productIsType
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 = bs upto \{x,y.R[x;y]\} {}\mRightarrow{} [a / as] = [b / bs] upto \{x,y.R[x;y]\})))
Date html generated:
2019_10_16-PM-01_01_28
Last ObjectModification:
2018_10_08-AM-09_26_46
Theory : perms_2
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