Nuprl Lemma : isqrt-non-decreasing
∀[x,y:ℕ]. isqrt(x) ≤ isqrt(y) supposing x ≤ y
Proof
Definitions occuring in Statement :
isqrt: isqrt(x),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
nat: ℕ,
ge: i ≥ j ,
prop: ℙ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
decidable__le,
isqrt_wf,
nat_properties,
le_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
less_than'_wf,
nat_wf,
add_nat_wf,
false_wf,
add-is-int-iff,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
isqrt-property,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
addEquality,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
unionElimination,
setElimination,
rename,
dependent_set_memberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
lambdaFormation,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[x,y:\mBbbN{}]. isqrt(x) \mleq{} isqrt(y) supposing x \mleq{} y
Date html generated:
2019_06_20-PM-02_36_58
Last ObjectModification:
2019_06_12-PM-00_25_53
Theory : num_thy_1
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