Nuprl Lemma : Cn-comb_wf
∀[T:Type]. ∀[n,m:ℕ]. ∀[A:ℕm ⟶ Type].
Cn-comb(n) ∈ funtype(m;A;T) ⟶ funtype(m;λk.if k <z n then A (k + 1)
if (k =z n) then A 0
else A k
fi ;T)
supposing n < m
Proof
Definitions occuring in Statement :
Cn-comb: Cn-comb(n),
funtype: funtype(n;A;T),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
Cn-comb: Cn-comb(n),
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
lt_int: i <z j,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
nequal: a ≠ b ∈ T ,
eq_int: (i =z j),
funtype: funtype(n;A;T),
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
primrec-unroll,
btrue_wf,
uiff_transitivity,
equal-wf-base,
bool_wf,
assert_wf,
lt_int_wf,
less_than_wf,
eqtt_to_assert,
assert_of_lt_int,
le_int_wf,
le_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
int_seg_wf,
subtract-1-ge-0,
int_subtype_base,
istype-nat,
istype-universe,
funtype_wf,
subtype_rel-equal,
add-member-int_seg2,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
eq_int_wf,
assert_of_eq_int,
int_seg_properties,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-le,
neg_assert_of_eq_int,
decidable__equal_int,
istype-false,
C-comb_wf_funtype,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
not_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
set_subtype_base,
primrec_wf,
lelt_wf,
squash_wf,
true_wf,
nat_wf,
itermAdd_wf,
int_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
unionElimination,
equalityElimination,
baseClosed,
productElimination,
equalityIstype,
functionIsType,
baseApply,
closedConclusion,
applyEquality,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
promote_hyp,
cumulativity,
imageElimination,
productIsType,
functionExtensionality_alt,
intEquality,
imageMemberEquality,
equalityIsType1,
equalityIsType4,
functionEquality,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[A:\mBbbN{}m {}\mrightarrow{} Type].
Cn-comb(n) \mmember{} funtype(m;A;T) {}\mrightarrow{} funtype(m;\mlambda{}k.if k <z n then A (k + 1)
if (k =\msubz{} n) then A 0
else A k
fi ;T)
supposing n < m
Date html generated:
2019_10_15-AM-11_15_09
Last ObjectModification:
2019_06_25-PM-01_22_26
Theory : general
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