Nuprl Lemma : assert-init-seg-nat-seq2
∀f,g:finite-nat-seq(). (↑init-seg-nat-seq(f;g) ⇐⇒ ((fst(f)) ≤ (fst(g))) ∧ ((snd(f)) = (snd(g)) ∈ (ℕfst(f) ⟶ ℕ)))
Proof
Definitions occuring in Statement :
init-seg-nat-seq: init-seg-nat-seq(f;g),
finite-nat-seq: finite-nat-seq(),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
pi1: fst(t),
pi2: snd(t),
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
init-seg-nat-seq: init-seg-nat-seq(f;g),
finite-nat-seq: finite-nat-seq(),
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A,
rev_implies: P ⇐ Q
Lemmas referenced :
ble_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
finite-nat-seq_wf,
assert-ble,
int_seg_wf,
nat_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
le_wf,
assert-equal-upto-finite-nat-seq,
assert_wf,
equal-upto-finite-nat-seq_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
rename,
sqequalRule,
cut,
introduction,
extract_by_obid,
isectElimination,
setElimination,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
independent_pairFormation,
functionEquality,
natural_numberEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
productEquality,
addLevel,
impliesFunctionality
Latex:
\mforall{}f,g:finite-nat-seq(). (\muparrow{}init-seg-nat-seq(f;g) \mLeftarrow{}{}\mRightarrow{} ((fst(f)) \mleq{} (fst(g))) \mwedge{} ((snd(f)) = (snd(g))))
Date html generated:
2017_04_20-AM-07_29_44
Last ObjectModification:
2017_02_27-PM-06_00_33
Theory : continuity
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