Nuprl Lemma : isqrt-of-square
∀[z:ℕ]. (isqrt(z * z) = z ∈ ℤ)
Proof
Definitions occuring in Statement :
isqrt: isqrt(x),
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top
Lemmas referenced :
nat_wf,
isqrt-property,
mul_bounds_1a,
le_wf,
isqrt_wf,
less_than_wf,
equal_wf,
decidable__lt,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
mul_preserves_le,
itermMultiply_wf,
int_term_value_mul_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
hypothesis,
Error :universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_set_memberEquality,
multiplyEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
natural_numberEquality,
lambdaFormation,
productElimination,
productEquality,
addEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation
Latex:
\mforall{}[z:\mBbbN{}]. (isqrt(z * z) = z)
Date html generated:
2019_06_20-PM-02_37_12
Last ObjectModification:
2019_06_12-PM-00_26_02
Theory : num_thy_1
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