Nuprl Lemma : upto_decomp
∀[m:ℕ]. ∀[n:ℕm + 1]. (upto(m) ~ upto(n) @ map(λx.(x + n);upto(m - n)))
Proof
Definitions occuring in Statement :
upto: upto(n),
map: map(f;as),
append: as @ bs,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
upto: upto(n),
nat: ℕ,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ
Lemmas referenced :
subtract-add-cancel,
zero-add,
subtract_wf,
from-upto-shift,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
int_seg_properties,
from-upto-split,
nat_wf,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalAxiom,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
because_Cache
Latex:
\mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m + 1]. (upto(m) \msim{} upto(n) @ map(\mlambda{}x.(x + n);upto(m - n)))
Date html generated:
2016_05_14-PM-02_03_34
Last ObjectModification:
2016_01_15-AM-08_06_15
Theory : list_1
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