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Theorem div-elim-pospos

⊦ 0 < j ∧ 0 ≤ i → (i0 = i / j ∧ j0 = i % j ↔ i = i0 * j + j0 ∧ 0 ≤ j0 ∧ j0 < j ∧ 0 ≤ i0)

Type: A user defined theorem
Validity: The proof is valid.
Features:
Comment:
Used by: add-div-mult, div-add-succ, div-add1-negpos-neg-01, div-add1-negpos-pos, div-add1-posneg-neg, div-add1-posneg-pos-01, div-addthree-rew, div-am-rew, div-eq-same, div-eq-same-01, div-eq-zero, div-eq-zero-01, div-est-denom-pos, div-est-nle-pos, div-est-nls, div-est-num, div-le-modnotzero, div-le-pos, div-le-same, div-ls, div-ls-same, div-mult-le-01, div-mult-ls, div-mult-nle-est, div-mult-nls-est, div-mult-nls-est-01, div-mult-shift-holds, div-mult-three-nls, div-mult-two-nls-01, div-shift-two, div-two-intro, div-zero-nls, div2-nle-02, div2-nls, div2-nls-02, even-not-ex, le-norest, mod-add-zero, mod-add1-negpos-neg, mod-add1-negpos-pos, mod-add1-posneg-neg, mod-add1-posneg-pos, mod-div-ls, mod-ls-same-01, mod-ls-same-02, mod-mult-pos-01, mod-neq, mod-nle, mod-nls-05, mod-pred-02, mod-pred-03, mod-pred-04, mod-shift, mod-succ-01, mod-succ-02, mod-two-06, mod-zero-07, modex-odd, modex-odd-neg, modmod-add-04, modmod-add-zero, modmod-add2-negpos-neg, modmod-add2-negpos-pos, modmod-add2-negpos-pos-01, modmod-ma-negpos-neg, modmod-pos, mult-split-mod, nummult-ls-02, prod-div-less, square-even, square-inject, square-nle, square-nls,
Used by units in proved state: int-pot, int-gcd, intnat,
Proof exists?: Yes

View the proof

Infos about the proof

Proved?: Yes
Interactions: 0
Proof steps: 2
Used lemmas: div-elim-pos, mod-mult-pos, div-mult-pos-02, mod-less-01, mod-zero, divmod-rew,
Used spec theorems:
- Used simplifier rules: