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Diffstat (limited to 'hol-light/TacticRecording/examples3.ml')
| -rw-r--r-- | hol-light/TacticRecording/examples3.ml | 110 |
1 files changed, 110 insertions, 0 deletions
diff --git a/hol-light/TacticRecording/examples3.ml b/hol-light/TacticRecording/examples3.ml new file mode 100644 index 00000000..a8fcdb16 --- /dev/null +++ b/hol-light/TacticRecording/examples3.ml @@ -0,0 +1,110 @@ +#use"hol.ml";; +needs "Library/products.ml";; + +#use "TacticRecording/main.ml";; + +prioritize_real();; + + +(* ** LEMMA1 from HOL Light's 100/arithmetic_geometric_mean.ml ** *) + +let LEMMA_1 = prove + (`!x n. x pow (n + 1) - (&n + &1) * x + &n = + (x - &1) pow 2 * sum(1..n) (\k. &k * x pow (n - k))`, + CONV_TAC(ONCE_DEPTH_CONV SYM_CONV) THEN GEN_TAC THEN INDUCT_TAC THEN + REWRITE_TAC[SUM_CLAUSES_NUMSEG; ARITH_EQ; ADD_CLAUSES] THENL + [REAL_ARITH_TAC; REWRITE_TAC[ARITH_RULE `1 <= SUC n`]] THEN + SIMP_TAC[ARITH_RULE `k <= n ==> SUC n - k = SUC(n - k)`; SUB_REFL] THEN + REWRITE_TAC[real_pow; REAL_MUL_RID] THEN + REWRITE_TAC[REAL_ARITH `k * x * x pow n = (k * x pow n) * x`] THEN + ASM_REWRITE_TAC[SUM_RMUL; REAL_MUL_ASSOC; REAL_ADD_LDISTRIB] THEN + REWRITE_TAC[GSYM REAL_OF_NUM_SUC; REAL_POW_ADD] THEN REAL_ARITH_TAC);; + +let LEMMA_1 = theorem_wrap "LEMMA_1" LEMMA_1;; + +top_thm ();; +print_executed_proof true;; +print_flat_proof true;; +print_packaged_proof ();; + +g `!x n. x pow (n + 1) - (&n + &1) * x + &n = + (x - &1) pow 2 * sum(1..n) (\k. &k * x pow (n - k))`;; + +print_executed_proof ();; + +(* LEMMA 2 *) + +let LEMMA_2 = prove + (`!n x. &0 <= x ==> &0 <= x pow (n + 1) - (&n + &1) * x + &n`, + REPEAT STRIP_TAC THEN REWRITE_TAC[LEMMA_1] THEN + MATCH_MP_TAC REAL_LE_MUL THEN REWRITE_TAC[REAL_POW_2; REAL_LE_SQUARE] THEN + MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN + ASM_SIMP_TAC[REAL_LE_MUL; REAL_POS; REAL_POW_LE]);; + +let LEMMA_2 = theorem_wrap "LEMMA_2" LEMMA_2;; + +print_executed_proof true;; +print_flat_proof true;; +print_packaged_proof ();; + +g `!n x. &0 <= x ==> &0 <= x pow (n + 1) - (&n + &1) * x + &n`;; + +(* LEMMA 3 *) + +let LEMMA_3 = prove + (`!n x. 1 <= n /\ (!i. 1 <= i /\ i <= n + 1 ==> &0 <= x i) + ==> x(n + 1) * (sum(1..n) x / &n) pow n + <= (sum(1..n+1) x / (&n + &1)) pow (n + 1)`, + REPEAT STRIP_TAC THEN + ABBREV_TAC `a = sum(1..n+1) x / (&n + &1)` THEN + ABBREV_TAC `b = sum(1..n) x / &n` THEN + SUBGOAL_THEN `x(n + 1) = (&n + &1) * a - &n * b` SUBST1_TAC THENL + [MAP_EVERY EXPAND_TAC ["a"; "b"] THEN + ASM_SIMP_TAC[REAL_DIV_LMUL; REAL_OF_NUM_EQ; LE_1; + REAL_ARITH `~(&n + &1 = &0)`] THEN + SIMP_TAC[SUM_ADD_SPLIT; ARITH_RULE `1 <= n + 1`; SUM_SING_NUMSEG] THEN + REAL_ARITH_TAC; + ALL_TAC] THEN + SUBGOAL_THEN `&0 <= a /\ &0 <= b` STRIP_ASSUME_TAC THENL + [MAP_EVERY EXPAND_TAC ["a"; "b"] THEN CONJ_TAC THEN + MATCH_MP_TAC REAL_LE_DIV THEN + (CONJ_TAC THENL [MATCH_MP_TAC SUM_POS_LE_NUMSEG; REAL_ARITH_TAC]) THEN + ASM_SIMP_TAC[ARITH_RULE `p <= n ==> p <= n + 1`]; + ALL_TAC] THEN + ASM_CASES_TAC `b = &0` THEN + ASM_SIMP_TAC[REAL_POW_ZERO; LE_1; REAL_MUL_RZERO; REAL_POW_LE] THEN + MP_TAC(ISPECL [`n:num`; `a / b`] LEMMA_2) THEN ASM_SIMP_TAC[REAL_LE_DIV] THEN + REWRITE_TAC[REAL_ARITH `&0 <= x - a + b <=> a - b <= x`; REAL_POW_DIV] THEN + SUBGOAL_THEN `&0 < b` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN + ASM_SIMP_TAC[REAL_LE_RDIV_EQ; REAL_POW_LT] THEN + MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN + REWRITE_TAC[REAL_POW_ADD] THEN UNDISCH_TAC `~(b = &0)` THEN + CONV_TAC REAL_FIELD);; + +let LEMMA_3 = theorem_wrap "LEMMA_3" LEMMA_3;; + +print_executed_proof true;; +print_flat_proof true;; +print_thenl_proof ();; +print_packaged_proof ();; + +g `!n x. 1 <= n /\ (!i. 1 <= i /\ i <= n + 1 ==> &0 <= x i) + ==> x(n + 1) * (sum(1..n) x / &n) pow n + <= (sum(1..n+1) x / (&n + &1)) pow (n + 1)`;; + +(* AGM *) + +let AGM = prove + (`!n a. 1 <= n /\ (!i. 1 <= i /\ i <= n ==> &0 <= a(i)) + ==> product(1..n) a <= (sum(1..n) a / &n) pow n`, + INDUCT_TAC THEN REWRITE_TAC[ARITH; PRODUCT_CLAUSES_NUMSEG] THEN + REWRITE_TAC[ARITH_RULE `1 <= SUC n`] THEN X_GEN_TAC `x:num->real` THEN + ASM_CASES_TAC `n = 0` THENL + [ASM_REWRITE_TAC[PRODUCT_CLAUSES_NUMSEG; ARITH; SUM_SING_NUMSEG] THEN + REAL_ARITH_TAC; + REWRITE_TAC[ADD1] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN + EXISTS_TAC `x(n + 1) * (sum(1..n) x / &n) pow n` THEN + ASM_SIMP_TAC[LEMMA_3; GSYM REAL_OF_NUM_ADD; LE_1; + ARITH_RULE `i <= n ==> i <= n + 1`] THEN + GEN_REWRITE_TAC RAND_CONV [REAL_MUL_SYM] THEN MATCH_MP_TAC REAL_LE_RMUL THEN + ASM_SIMP_TAC[LE_REFL; LE_1; ARITH_RULE `i <= n ==> i <= n + 1`]]);; |