Nuprl Lemma : isqrt-non-decreasing

[x,y:ℕ].  isqrt(x) ≤ isqrt(y) supposing x ≤ y


Proof




Definitions occuring in Statement :  isqrt: isqrt(x) nat: uimplies: supposing a uall: [x:A]. B[x] le: A ≤ B
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] subtype_rel: A ⊆B decidable: Dec(P) or: P ∨ Q nat: ge: i ≥  prop: satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] false: False implies:  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: 2018_05_21-PM-07_52_55
Last ObjectModification: 2017_07_26-PM-05_30_31

Theory : general


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