Commit 1cc66522 by MARCHE Claude

### isabelle proof in progress

parent 6af3b6cd
 ... ... @@ -153,40 +153,32 @@ why3_vc Mod_mult using assms by (simp add: emod_def add.commute) why3_vc Mod_bound using assms by (simp_all add: emod_def) why3_vc Div_unique using assms sorry (* proof - have h0: "y \ 0" using H1 by fastforce have h1: "x = y * (x ediv y) + (x emod y)" using h0 Div_mod by blast have h2: "0 \ x emod y \ x emod y < y" using assms H1 h0 Mod_bound zabs_def by (metis abs_sgn monoid_mult_class.mult.right_neutral sgn_pos) have "x - y < y * (x ediv y)" using h1 h2 by linarith have "y * (x ediv y) \ x" using h1 h2 by linarith assume h3: "x ediv y \ q+1" have "y * (x ediv y) >= y * (q + 1)" using h3 assms by auto *) (* (cases "x ediv y \ q") case False "0 < y \ q * y \ x < q * y + y \ x ediv y = q" proof: we have "x = y * (x ediv y) + (x emod y)" by Div_mod we have "0 \ x emod y < y" by Mod_bound hence x - y < y * (x ediv y) \ x case 1 : "x ediv y \ q - 1") then "y * x ediv y <= y * (q - 1) = q * y - y \ x - y" absurd case 2 : "x ediv y \ q + 1") then "y * x ediv y \= y * (q + 1) = q * y + y > x " absurd case 3 : "x ediv y = q") trivial *) proof - have h0: "y \ 0" using assms by auto have h1: "x = y * (x ediv y) + (x emod y)" using h0 Div_mod by blast have h2: "0 \ x emod y \ x emod y < y" using assms H1 h0 Mod_bound zabs_def by (metis abs_sgn monoid_mult_class.mult.right_neutral sgn_pos) have h3: "x - y < y * (x ediv y)" using h1 h2 by linarith have h4: "y * (x ediv y) \ x" using h1 h2 by linarith show ?thesis proof (cases "x ediv y \ q") case False have h5: "x ediv y \ q - 1" using False by linarith have h6: "y * (x ediv y) <= y * (q - 1)" by (metis H1 h5 le_less mult_left_mono) have h7: "y * (x ediv y) <= q * y - y" by (metis Comm1 h6 int_distrib(4) monoid_mult_class.mult.right_neutral) thus "x ediv y = q" using H2 h3 h7 by linarith next show ?thesis proof (cases "x ediv y \ q") case False have h5: "x ediv y \ q + 1" using False by linarith have h6: "y * (x ediv y) >= y * (q + 1)" by (metis H1 h5 le_less mult_left_mono) have h7: "y * (x ediv y) >= q * y + y" by (metis Comm1 Mul_distr_l h6 monoid_mult_class.mult.right_neutral) thus "x ediv y = q" using H3 h1 h2 h7 by linarith next show ?thesis why3_end ... ...
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