Nuprl Lemma : b-almost-full-intersection
This is the main technical lemma from Veldman & Bezem's proof
of the Intuitionistic Ramsey theorem.
We were able to closely follow their proof except that
before carrying out some of the reasoning steps we have to
"unsquash" some of the hypotheses.
The needed "unsquashing" is usually done using lemmas
implies-quotient-true or all-quotient-true
(making use of the fact that we can prove canonicalizable(StrictInc)).⋅
∀R,T:ℕ ⟶ ℕ ⟶ ℙ. (b-almost-full(n,m.R[n;m]) ⇒ b-almost-full(n,m.T[n;m]) ⇒ ⇃(∃n:ℕ. ∃m:{n + 1...}. (R[n;m] ∧ T[n;m])))
Proof
Definitions occuring in Statement :
b-almost-full: b-almost-full(n,m.R[n; m]),
quotient: x,y:A//B[x; y],
int_upper: {i...},
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
exists: ∃x:A. B[x],
or: P ∨ Q,
ge: i ≥ j ,
decidable: Dec(P),
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
and: P ∧ Q,
strict-inc: StrictInc,
guard: {T},
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
int_seg: {i..j-},
lelt: i ≤ j < k,
strictly-increasing-seq: strictly-increasing-seq(n;s),
seq-add: s.x@n,
nequal: a ≠ b ∈ T ,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
nat_plus: ℕ+,
u-almost-full: u-almost-full(n.A[n]),
compose: f o g,
isr: isr(x),
outr: outr(x),
isl: isl(x),
baf-bar: baf-bar(n,m.R[n; m];n,m.T[n; m];l;a),
pi1: fst(t)
Lemmas referenced :
monotone-bar-induction-strict3,
baf-bar_wf,
int_seg_wf,
strictly-increasing-seq_wf,
istype-nat,
strict-inc_wf,
nat_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
seq-add_wf,
int_upper_wf,
upper_subtype_nat,
istype-false,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
subtype_rel_self,
b-almost-full_wf,
baf-bar-monotone,
le_wf,
trivial-quotient-true,
strict-inc-lower-bound,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nequal-le-implies,
subtract_wf,
int_upper_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
add_nat_wf,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
false_wf,
sq_stable_from_decidable,
decidable__strictly-increasing-seq,
less_than_wf,
int_seg_properties,
set_subtype_base,
lelt_wf,
int_subtype_base,
decidable__equal_int,
less_than_functionality,
le_weakening,
quotient_wf,
all_wf,
exists_wf,
or_wf,
true_wf,
equiv_rel_true,
all-quotient-true,
subtype_rel_function,
int_seg_subtype,
add-mul-special,
zero-mul,
le-add-cancel2,
canonicalizable_wf,
canonicalizable-set,
canonicalizable-base,
Ramsey-n-3,
canonicalizable-nat-to-nat,
primrec-wf2,
nat_plus_properties,
nat_plus_subtype_nat,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_subtype_nat,
implies-quotient-true2,
u-almost-full-finite-intersection,
implies-quotient-true,
compose-strict-inc,
member-less_than,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
decidable__exists_int_seg,
isr_wf,
decidable__assert,
outl_wf,
isl_wf,
le_weakening2,
equal_wf,
le_reflexive,
not_over_exists,
decidable__or,
strictly-increasing-seq-add2-implies,
intformor_wf,
int_formula_prop_or_lemma,
equal-wf-base,
decidable__and2,
int_seg_subtype_special,
int_seg_cases,
lt_int_wf,
le_int_wf,
assert_of_lt_int,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
squash_wf,
nat_plus_wf,
b-almost-full-intersection-lemma,
imax_wf,
subtype_rel_dep_function,
imax_strict_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality_alt,
isectElimination,
applyEquality,
hypothesisEquality,
inhabitedIsType,
setElimination,
rename,
hypothesis,
setIsType,
functionIsType,
universeIsType,
natural_numberEquality,
functionExtensionality,
because_Cache,
closedConclusion,
functionEquality,
productEquality,
unionEquality,
dependent_set_memberEquality_alt,
addEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
instantiate,
universeEquality,
intEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
cumulativity,
hypothesis_subsumption,
productIsType,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
int_eqReduceTrueSq,
equalityIsType1,
int_eqReduceFalseSq,
inlFormation_alt,
unionIsType,
setEquality,
multiplyEquality,
equalityIsType4,
inrFormation_alt,
equalityIsType3,
hyp_replacement
Latex:
\mforall{}R,T:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}.
(b-almost-full(n,m.R[n;m])
{}\mRightarrow{} b-almost-full(n,m.T[n;m])
{}\mRightarrow{} \00D9(\mexists{}n:\mBbbN{}. \mexists{}m:\{n + 1...\}. (R[n;m] \mwedge{} T[n;m])))
Date html generated:
2020_05_19-PM-10_05_50
Last ObjectModification:
2019_10_29-PM-01_46_29
Theory : continuity
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