Nuprl Lemma : map_functionality_wrt_sq
∀[T:Type]. ∀[f,g:Base]. ∀[L:T List]. map(f;L) ~ map(g;L) supposing ∀x:T. ((x ∈ L) ⇒ (f x ~ g x)) supposing T ⊆r Base
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
base: Base,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
list_wf,
base_wf,
subtype_rel_wf,
all_wf,
l_member_wf,
sqequal-wf-base,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
int_subtype_base,
list-cases,
map_nil_lemma,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
decidable__equal_int,
map_cons_lemma,
nil_wf,
cons_wf,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomSqEquality,
hypothesis,
because_Cache,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
extract_by_obid,
Error :inhabitedIsType,
lambdaEquality,
functionEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
universeEquality,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
instantiate,
cumulativity,
imageElimination,
isect_memberFormation,
Error :functionIsType,
sqequalIntensionalEquality,
inrFormation,
inlFormation
Latex:
\mforall{}[T:Type]
\mforall{}[f,g:Base]. \mforall{}[L:T List]. map(f;L) \msim{} map(g;L) supposing \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} (f x \msim{} g x))
supposing T \msubseteq{}r Base
Date html generated:
2019_06_20-PM-01_33_19
Last ObjectModification:
2018_09_26-PM-06_00_39
Theory : list_1
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