Nuprl Lemma : type-incr-chain-subtype
∀[X:type-incr-chain{i:l}()]. ∀[n,m:ℕ]. (X n) ⊆r (X m) supposing n ≤ m
Proof
Definitions occuring in Statement :
type-incr-chain: type-incr-chain{i:l}(),
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
type-incr-chain: type-incr-chain{i:l}(),
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
rev_subtype_rel: A ⊇r B,
subtract: n - m
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
subtype_rel-equal,
decidable__le,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
le_wf,
add-zero,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
nat_wf,
type-incr-chain_wf,
sq_stable__subtype_rel,
subtype_rel_functionality_wrt_implies,
subtype_rel_weakening,
ext-eq_inversion,
ext-eq_weakening,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
subtype_rel_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
addEquality,
unionElimination,
productElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality
Latex:
\mforall{}[X:type-incr-chain\{i:l\}()]. \mforall{}[n,m:\mBbbN{}]. (X n) \msubseteq{}r (X m) supposing n \mleq{} m
Date html generated:
2018_05_21-PM-08_43_59
Last ObjectModification:
2018_05_19-PM-05_06_35
Theory : general
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