Nuprl Lemma : assert-eq-seg-nat-seq
∀[n,m:finite-nat-seq()]. (↑eq-seg-nat-seq(n;m) ⇐⇒ n = m ∈ finite-nat-seq())
Proof
Definitions occuring in Statement :
eq-seg-nat-seq: eq-seg-nat-seq(n;m),
finite-nat-seq: finite-nat-seq(),
assert: ↑b,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
eq-seg-nat-seq: eq-seg-nat-seq(n;m),
finite-nat-seq: finite-nat-seq(),
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
sq_type: SQType(T),
uiff: uiff(P;Q),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi
Lemmas referenced :
istype-le,
subtype_rel_function,
int_seg_wf,
nat_wf,
int_seg_subtype,
istype-false,
subtype_rel_self,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
set_subtype_base,
le_wf,
int_subtype_base,
int_seg_properties,
istype-less_than,
intformand_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
subtype_rel_dep_function,
le_weakening,
zero-le-nat,
intformeq_wf,
int_formula_prop_eq_lemma,
subtype_base_sq,
subtype_rel_wf,
assert-init-seg-nat-seq2,
istype-assert,
init-seg-nat-seq_wf,
iff_weakening_uiff,
assert_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
bfalse_wf,
assert_of_band,
assert_witness,
eq-seg-nat-seq_wf,
finite-nat-seq_wf,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
independent_pairFormation,
Error :lambdaFormation_alt,
hypothesis,
Error :productIsType,
extract_by_obid,
isectElimination,
setElimination,
rename,
hypothesisEquality,
Error :equalityIstype,
Error :inhabitedIsType,
applyEquality,
natural_numberEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
hyp_replacement,
equalitySymmetry,
Error :dependent_set_memberEquality_alt,
equalityTransitivity,
applyLambdaEquality,
baseApply,
closedConclusion,
baseClosed,
intEquality,
sqequalBase,
Error :functionExtensionality_alt,
instantiate,
cumulativity,
independent_pairEquality,
Error :functionIsType,
Error :dependent_pairEquality_alt,
promote_hyp,
productEquality,
axiomEquality,
Error :functionIsTypeImplies,
Error :isectIsTypeImplies
Latex:
\mforall{}[n,m:finite-nat-seq()]. (\muparrow{}eq-seg-nat-seq(n;m) \mLeftarrow{}{}\mRightarrow{} n = m)
Date html generated:
2019_06_20-PM-03_04_17
Last ObjectModification:
2018_11_25-PM-05_58_00
Theory : continuity
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