(* ========================================================================== *) (* FLYSPECK - BOOK FORMALIZATION *) (* *) (* Chapter: Packing *) (* Lemma: OXLZLEZ *) (* Author: Nguyen Quang Truong *) (* Date: 2012-12-21 *) (* ========================================================================== *) module Oxl_2012 = struct open Oxl_def;; (* needs "/home/user1/flyspeck/working/boot.hl";; ===================== dont worry about this ================ needs "/home/user1/flyspeck/working/update_database_310.ml";; needs "/home/user1/flyspeck/working/asm_search.ml";; needs "/home/user1/flyspeck/working/Oxl_deff.hl";; needs "/home/user1/flyspeck/working/oxl_2012.hl";; *) let asms_search0 sths = let rec immediatesublist l1 l2 = match (l1,l2) with [],_ -> true | _,[] -> false | (h1::t1,h2::t2) -> h1 = h2 & immediatesublist t1 t2 in let rec sublist l1 l2 = match (l1,l2) with [],_ -> true | _,[] -> false | (h1::t1,h2::t2) -> immediatesublist l1 l2 or sublist l1 t2 in let exists_subterm_satisfying p (n,th) = can (find_term p) (concl th) and name_contains s (n,th) = sublist (explode s) (explode n) in let rec filterpred tm = match tm with Comb(Var("",_),t) -> not o filterpred t | Comb(Var("",_),Var(pat,_)) -> name_contains pat | Comb(Var("",_),pat) -> exists_subterm_satisfying (aconv pat) | pat -> exists_subterm_satisfying (can (term_match [] pat)) in fun pats -> let triv,nontriv = partition is_var pats in (if triv <> [] then warn true ("Ignoring plain variables in search: "^ end_itlist (fun s t -> s^", "^t) (map (fst o dest_var) triv)) else ()); (if nontriv = [] & triv <> [] then [] else itlist (filter o filterpred) pats sths);; let asms_search tms = let gstk = !current_goalstack in match gstk with [] -> [] | (meta,gl::_,just)::_ -> let (sths,_) = gl in map snd (asms_search0 sths tms) | _ -> failwith "asm_searchs: Invalid goal state";; let ASMS_SEARCH_TCL (tms:(term)list) (thstac:(thm)list->tactic) : tactic = fun (gl:goal) -> let (sths,tm) = gl in let ths1 = map snd (asms_search0 sths tms) in thstac ths1 gl;; let ASM_SEARCH_TCL (tms:(term)list) (thstac:thm->tactic) : tactic = let foo ths = match ths with [] -> failwith "ASM_SEARCH_TCL: No matching asms found" | _ -> thstac (hd ths) in ASMS_SEARCH_TCL tms foo;; let ATTACH thm = MATCH_MP (MESON[]` ! a b. ( a ==> b ) ==> ( a <=> a /\ b )`) thm;; let NHANH tm = ONCE_REWRITE_TAC[ ATTACH (SPEC_ALL ( tm ))];; let PHA = REWRITE_TAC[ MESON[] ` (a/\b)/\c <=> a/\ b /\ c `; MESON[]` a ==> b ==> c <=> a /\ b ==> c `];; let DOWN_TAC = REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[IMP_IMP] THEN PHA;; let DOWN = FIRST_X_ASSUM MP_TAC;; let DOWNS n = REPLICATE_TAC n DOWN THEN PHA;; let REMOVE_TAC = FIRST_X_ASSUM MP_TAC THEN MATCH_MP_TAC (TAUT` a ==> b ==> a`);; let types_thm th = let cl = concl th in List.map dest_var (frees cl );; let seans_fn () = let (tms,tm) = top_goal () in let vss = map frees (tm::tms) in let vs = setify (flat vss) in map dest_var vs;; let PAT_REWRITE_TAC tm thms = (CONV_TAC (PAT_CONV tm (REWRITE_CONV thms )));; let FOR_ASM th = let th1 = REWRITE_RULE[MESON[]` a /\ b ==> c <=> a ==> b ==> c `] th in let th2 = SPEC_ALL th1 in UNDISCH_ALL th2;; (* change a th having form |- A ==> t to the form A |- t to get ready to some other commands |- A ==> t ----------- FOR_ASM A |- t *) let ASSUME_TAC2 = ASSUME_TAC o FOR_ASM;; let PAT_ONCE_REWRITE_TAC tm thms = (CONV_TAC (PAT_CONV tm (ONCE_REWRITE_CONV thms )));; let ASM_PAT_RW_TAC tm thms = EVERY_ASSUM (fun th -> (CONV_TAC (PAT_CONV tm (ONCE_REWRITE_CONV ( th ::[ thms ] )))));; let PAT_TH_TAC tm th = (CONV_TAC (PAT_CONV tm (REWRITE_CONV[th] )));; let IMP_TO_EQ_RULE th = MATCH_MP (TAUT` (a ==> b ) ==> ( a <=> a /\ b )`) (SPEC_ALL th);; let NHANH_PAT tm th = PAT_ONCE_REWRITE_TAC tm [ IMP_TO_EQ_RULE th ];; let MAKE_FIRST_TAC tm = UNDISCH_TAC tm THEN DISCH_TAC;; (* ---------- BG TEST ------------- *) let rec els L = match L with [] -> p () | (x::l) -> e x; els l;; let rec bls L = match L with [] -> p () | (x::l) -> b (); bls l;; (* ------------- *) let QU_OR_QXY = prove (` ! i. cc_qu_v11 cc i \/ cc_qx_v11 cc i \/ cc_qy_v11 cc i `, REWRITE_TAC[cc_qu_v11; cc_qx_v11; cc_qy_v11] THEN MESON_TAC[]);; let QX_NN0 = REWRITE_RULE[REAL_ARITH` #0.0 = &0`] QX_NN;; let QY_NN0 = REWRITE_RULE[REAL_ARITH` #0.0 = &0`] QY_NN;; let MOD_REFL = REWRITE_RULE[MULT_CLAUSES] (SPECL [`m:num `;`1`] MOD_MULT);; let SUM_POS_LT_NUMSEG = prove_by_refinement ( `!m n f. m <= n /\ (!p. m <= p /\ p <= n ==> &0 < f p) ==> &0 < sum (m..n) f`, [GEN_TAC; INDUCT_TAC; REWRITE_TAC[LE; SUM_CLAUSES_NUMSEG]; SIMP_TAC[]; GEN_TAC; STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` 0`)); ASM_REWRITE_TAC[LE]; GEN_TAC; ASM_CASES_TAC` m = SUC n `; ASM_REWRITE_TAC[SUM_SING_NUMSEG]; STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` SUC n `)); FIRST_X_ASSUM MP_TAC; CONV_TAC TAUT; STRIP_TAC; SUBGOAL_THEN` &0 < sum (m..n) f ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; CONJ_TAC; DOWN_TAC; ARITH_TAC; FIRST_X_ASSUM MP_TAC; MESON_TAC[ARITH_RULE` a <= p ==> a <= SUC p `]; REWRITE_TAC[SUM_CLAUSES_NUMSEG]; ASM_REWRITE_TAC[]; STRIP_TAC; MATCH_MP_TAC REAL_LT_ADD; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[LE_REFL]]);; let periodic_mult = prove_by_refinement ( ` periodic f n <=> !k p. f (k * n + p ) = f p `, [REWRITE_TAC[periodic]; EQ_TAC; STRIP_TAC; INDUCT_TAC; REWRITE_TAC[MULT; ADD]; REWRITE_TAC[ADD1; NUM_RING` (k + 1) * n = k * n + n `]; REWRITE_TAC[ARITH_RULE` (a + b) + c = a + b + (c:num) `]; DOWN_TAC; MESON_TAC[ADD_SYM]; DISCH_THEN (MP_TAC o (SPEC`1 `)); REWRITE_TAC[MULT_CLAUSES]; MESON_TAC[ADD_SYM]]);; (* =================================================== *) CHQSQEY_concl;; let glt = `cc_bool_model_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 /\ indSet = 0..cc_card_v11 cc - 1 ==> (?i j k. ~(i = j \/ j = k \/ k = i) /\ i IN indSet /\ j IN indSet /\ k IN indSet /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\ cc_4cell_v11 cc k)`;; let e1 = (STRIP_TAC THEN ASSUME_TAC2 CC_CARD2 THEN DOWN_TAC THEN NHANH periodic_fn THEN ASM_CASES_TAC` ~ (?i. i IN indSet /\ cc_4cell_v11 cc i)` THENL [ SUBGOAL_THEN` ! i. i IN indSet ==> cc_qy_v11 cc i ` MP_TAC THENL [MP_TAC QU_OR_QXY THEN REWRITE_TAC[cc_qu_v11; cc_qx_v11] THEN FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]; REWRITE_TAC[cc_real_model_v11; cc_bool_model_v11]] THEN REPEAT STRIP_TAC THEN SUBGOAL_THEN` ! i. i IN indSet ==> &0 < cc_gg_v11 cc i ` MP_TAC THENL [ UNDISCH_TAC`!i. i IN indSet ==> cc_qy_v11 cc i` THEN DISCH_THEN NHANH THEN UNDISCH_TAC`!i. cc_qy_v11 cc i ==> #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i` THEN STRIP_TAC THEN FIRST_ASSUM NHANH THEN GEN_TAC THEN UNDISCH_TAC` !i. #0.606 <= cc_azim_v11 cc i ` THEN DISCH_TAC THEN FIRST_ASSUM (MP_TAC o (SPEC` i: num `)) THEN REAL_ARITH_TAC; STRIP_TAC] THEN SUBGOAL_THEN` &0 < sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) ` MP_TAC THENL [MATCH_MP_TAC SUM_POS_LT_NUMSEG THEN CONJ_TAC THENL [UNDISCH_TAC` ~ (cc_card_v11 cc = 0) ` THEN ARITH_TAC; REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG]]; UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ` THEN REAL_ARITH_TAC]; FIRST_X_ASSUM MP_TAC] THEN REWRITE_TAC[] THEN STRIP_TAC THEN STRIP_TAC THEN ASM_CASES_TAC` ~ ( ?j. j IN indSet /\ ~(j = i) /\ cc_4cell_v11 cc j)`);; let e2 = (FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC [ MESON[]` ~(?j. j IN indSet /\ ~(j = i) /\ cc_4cell_v11 cc j) <=> !j. j IN indSet /\ ~(j = i) ==> ~ cc_4cell_v11 cc j `] THEN STRIP_TAC THEN SUBGOAL_THEN`!j. j IN indSet /\ ~(j = i) ==> cc_qy_v11 cc j ` MP_TAC THENL [ FIRST_ASSUM NHANH THEN REPEAT STRIP_TAC THEN MP_TAC (SPEC `j:num` QU_OR_QXY) THEN FIRST_ASSUM MP_TAC THEN SIMP_TAC[cc_qu_v11; cc_qx_v11]; SUBGOAL_THEN` cc_qu_v11 cc i \/ cc_qx_v11 cc i ` MP_TAC] THENL [ MP_TAC (SPEC_ALL QU_OR_QXY) THEN MATCH_MP_TAC (TAUT` ~ c ==> a \/ b \/ c ==> a \/ b `) THEN ASM_REWRITE_TAC[cc_qy_v11]; STRIP_TAC THEN STRIP_TAC THEN SUBGOAL_THEN` cc_qy_v11 cc (i + 1)` MP_TAC]);; let e3 = (ASM_CASES_TAC` i < cc_card_v11 cc - 1 ` THENL [ FIRST_ASSUM (MP_TAC o SPEC` i + 1 `) THEN ANTS_TAC THENL [ ASM_REWRITE_TAC[ARITH_RULE`~( i + 1 = i ) `; IN_NUMSEG] THEN FIRST_ASSUM MP_TAC THEN ARITH_TAC; REWRITE_TAC[]]; UNDISCH_TAC` (i:num) IN indSet ` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASSUME_TAC2 (ARITH_RULE` ~(i < cc_card_v11 cc - 1) /\ i <= cc_card_v11 cc - 1 ==> cc_card_v11 cc - 1 = i `) THEN UNDISCH_TAC` periodic (cc_qy_v11 cc) (cc_card_v11 cc) ` THEN EXPAND_TAC "i" THEN DOWN_TAC THEN REWRITE_TAC[cc_bool_model_v11] THEN NHANH (ARITH_RULE` ~( a = 0) ==> a - 1 + 1 = a `) THEN SIMP_TAC[periodic] THEN STRIP_TAC THEN ONCE_REWRITE_TAC[ARITH_RULE` a = 0 + a `] THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN EXPAND_TAC "i" THEN REWRITE_TAC[LE] THEN MATCH_MP_TAC (ARITH_RULE` 2 <= a ==> ~( 0 = a - 1) `) THEN FIRST_ASSUM ACCEPT_TAC]);; let e4 = ( (* ------------------------------------- *) DOWN_TAC THEN REWRITE_TAC[cc_real_model_v11] THEN STRIP_TAC THEN UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) ` THEN STRIP_TAC THEN FIRST_ASSUM (MP_TAC o SPEC_ALL) THEN ANTS_TAC THENL [ASM_REWRITE_TAC[]; MAKE_FIRST_TAC` !i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `] THEN SUBGOAL_THEN` cc_gg3a_v11 cc ( i + 1) <= cc_gg_v11 cc ( i + 1) ` MP_TAC THENL [ MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC` cc_gg3a_v11 cc (i + 1) + cc_gg3b_v11 cc (i + 1)` THEN CONJ_TAC THENL [ REWRITE_TAC[REAL_LE_ADDR] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]]; PHA] THEN NHANH (REAL_ARITH` a <= b /\ c <= d + a ==> c <= d + b `));; (* `(cc_gg3a_v11 cc (i + 1) <= cc_gg_v11 cc (i + 1) /\ cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) /\ cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ==> (?i j k. ~(i = j \/ j = k \/ k = i) /\ i IN indSet /\ j IN indSet /\ k IN indSet /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\ cc_4cell_v11 cc k)` *) (* ============================================= ================================================ let WKR_COMPTED = prove_by_refinement (glt , g glt;; els [e1 ; e2 ; e3; e4; STRIP_TAC THEN MP_TAC (SPECL [` cc_gg_v11 cc `;` cc_card_v11 cc`] periodic_sum) THEN ANTS_TAC THENL [ ASM_REWRITE_TAC[] THEN UNDISCH_TAC` cc_bool_model_v11 cc ` THEN SIMP_TAC[cc_bool_model_v11]; DISCH_THEN (ASSUME_TAC o (SPEC` i:num`))] THEN SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) ` MP_TAC; ASM_CASES_TAC` i < cc_card_v11 cc - 1 `; SUBGOAL_THEN` 0..cc_card_v11 cc - 1 = ((0..cc_card_v11 cc - 1) DIFF {i, i +1} ) UNION {i, i + 1}` MP_TAC THENL [ MATCH_MP_TAC (SET_RULE` s SUBSET S ==> S = (S DIFF s) UNION s `) THEN UNDISCH_TAC` (i:num) IN indSet` THEN ASM_SIMP_TAC[INSERT_SUBSET; EMPTY_SUBSET] THEN REWRITE_TAC[IN_NUMSEG] THEN FIRST_ASSUM MP_TAC THEN ARITH_TAC; DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[]]; SUBGOAL_THEN` ! f. sum ((0..cc_card_v11 cc - 1) DIFF {i, i + 1} UNION {i, i + 1}) f = sum ((0..cc_card_v11 cc - 1) DIFF {i, i + 1}) f + sum ({i, i + 1}) f ` MP_TAC THENL [ GEN_TAC THEN MATCH_MP_TAC SUM_UNION THEN CONJ_TAC THENL [ MESON_TAC[FINITE_NUMSEG; FINITE_DIFF]; CONJ_TAC] THENL [ MESON_TAC[FINITE_EMPTY; FINITE_INSERT]; SET_TAC[]]; SIMP_TAC[] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC]; MATCH_MP_TAC SUM_POS_LE THEN CONJ_TAC THENL [ MESON_TAC[FINITE_NUMSEG; FINITE_DIFF]; REPEAT STRIP_TAC]; REWRITE_TAC[REAL_ARITH` &0 = #0.0 `] THEN MATCH_MP_TAC QY_NN THEN ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11] THEN FIRST_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_ASSUM MP_TAC THEN REWRITE_TAC[IN_DIFF; IN_INSERT] THEN MESON_TAC[]; MP_TAC (MATCH_MP Geomdetail.SUM_DIS2 (ARITH_RULE` ~( i = i + 1 ) `)) THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC` cc_eps:real` THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[cc_eps] THEN REAL_ARITH_TAC; ASM_CASES_TAC` i = cc_card_v11 cc - 1 ` THENL [ SUBGOAL_THEN` 0..cc_card_v11 cc - 1 = ((0..cc_card_v11 cc - 1) DIFF {i, 0}) UNION {i, 0}` MP_TAC THENL [ MATCH_MP_TAC (SET_RULE` s SUBSET S ==> S = (S DIFF s) UNION s`) THEN REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_NUMSEG] THEN FIRST_X_ASSUM MP_TAC THEN UNDISCH_TAC` (i:num) IN indSet` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN SIMP_TAC[LE; LE_0]; DISCH_THEN SUBST1_TAC] THEN SUBGOAL_THEN` FINITE ((0..cc_card_v11 cc - 1) DIFF {i, 0}) /\ FINITE {i, 0} /\ DISJOINT ((0..cc_card_v11 cc - 1) DIFF {i, 0}) {i, 0} ` MP_TAC THENL [ REWRITE_TAC[SET_RULE` DISJOINT (S DIFF s) s`] THEN MESON_TAC[FINITE_NUMSEG; FINITE_INSERT; FINITE_EMPTY; FINITE_DIFF]; NHANH (ISPECL[` cc_gg_v11 cc`;` (0..cc_card_v11 cc - 1) DIFF {i, 0}`;` {i, 0}`] SUM_UNION) THEN SIMP_TAC[]] THEN STRIP_TAC THEN MATCH_MP_TAC REAL_LE_ADD THEN CONJ_TAC THENL [ MATCH_MP_TAC SUM_POS_LE THEN ASM_REWRITE_TAC[] THEN GEN_TAC THEN REWRITE_TAC[IN_DIFF; IN_INSERT] THEN STRIP_TAC THEN MATCH_MP_TAC QY_NN0 THEN ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MP_TAC THEN CONV_TAC TAUT; SUBGOAL_THEN` ~( i = 0) ` MP_TAC THENL [ ASM_REWRITE_TAC[] THEN UNDISCH_TAC` 2 <= cc_card_v11 cc ` THEN ARITH_TAC; STRIP_TAC]] THEN ASSUME_TAC2 (ISPECL [`i:num `;` 0`; ` cc_gg_v11 cc `] Geomdetail.SUM_DIS2) THEN UNDISCH_TAC` cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` THEN ASSUME_TAC2 (ARITH_RULE` 2 <= cc_card_v11 cc ==> cc_card_v11 cc - 1 + 1 = cc_card_v11 cc `) THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC` periodic (cc_gg_v11 cc) (cc_card_v11 cc) ` THEN REWRITE_TAC[periodic] THEN STRIP_TAC THEN FIRST_ASSUM (MP_TAC o (SPEC` 0 `)) THEN SIMP_TAC[ARITH_RULE` 0 + a = a `] THEN REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC` cc_eps:real ` THEN ASM_REWRITE_TAC[cc_eps] THEN REAL_ARITH_TAC; UNDISCH_TAC` (i:num) IN indSet` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC]; ASM_REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a ) `]; ASSUME_TAC2 QX_NN0 THEN SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc)` MP_TAC THENL [ MATCH_MP_TAC SUM_POS_LE_NUMSEG THEN REPEAT STRIP_TAC THEN ASM_CASES_TAC` p = (i:num) ` THENL [ ASM_REWRITE_TAC[]; MATCH_MP_TAC QY_NN0 THEN ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG]]; ASM_REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a ) `]]; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc)` MP_TAC; MATCH_MP_TAC SUM_POS_LE_NUMSEG; REPEAT STRIP_TAC; ASM_CASES_TAC` p = (i:num) `; ASM_REWRITE_TAC[]; MATCH_MP_TAC QX_NN0; ASM_REWRITE_TAC[]; MATCH_MP_TAC QY_NN0; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG]; ASM_REWRITE_TAC[REAL_ARITH` a <= b <=> ~( b < a ) `]; (* *********** *) FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[] THEN STRIP_TAC THEN ASM_CASES_TAC` ~(? k. (k IN indSet /\ ~(k = i) /\ ~(k = j)) /\ cc_4cell_v11 cc k)`; FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[MESON[]` ~(?x. p x /\ q x) <=> !x. p x ==> ~ q x `] THEN STRIP_TAC THEN SUBGOAL_THEN` !k. k IN indSet /\ ~(k = i) /\ ~(k = j) ==> cc_qy_v11 cc k ` MP_TAC THENL [ FIRST_X_ASSUM NHANH THEN REPEAT STRIP_TAC THEN MP_TAC (SPEC` k:num` QU_OR_QXY) THEN ASM_REWRITE_TAC[cc_qu_v11; cc_qx_v11] ; STRIP_TAC THEN ASM_CASES_TAC`~( ?qy. qy IN indSet /\ cc_qy_v11 cc qy)` THEN FIRST_X_ASSUM MP_TAC THEN FIRST_X_ASSUM MP_TAC THEN PHA THEN NHANH (SET_RULE` (!k. k IN indSet /\ ~(k = i) /\ ~(k = j) ==> cc_qy_v11 cc k) /\ ~(?qy. qy IN indSet /\ cc_qy_v11 cc qy) ==> indSet SUBSET {i, j} `)] THEN DOWN_TAC THEN REWRITE_TAC[cc_real_model_v11] THEN STRIP_TAC THEN UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi ` THEN UNDISCH_TAC` !i. #0.606 <= cc_azim_v11 cc i ` THEN NHANH (REAL_ARITH` #0.606 <= f i ==> &0 <= f i `) THEN STRIP_TAC THEN STRIP_TAC; SUBGOAL_THEN` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) <= sum {i, j} (cc_azim_v11 cc) ` MP_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE THEN UNDISCH_TAC` indSet SUBSET {i, j:num} ` THEN ASM_SIMP_TAC[] THEN MESON_TAC[FINITE_EMPTY; FINITE_INSERT]; ASSUME_TAC2 (ISPECL [`j:num `;` i:num `;` cc_azim_v11 cc `] Geomdetail.SUM_DIS2) THEN ONCE_REWRITE_TAC[INSERT_COMM] THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC` cc_4cell_v11 cc j ` THEN UNDISCH_TAC` cc_4cell_v11 cc i ` THEN UNDISCH_TAC` !i. cc_4cell_v11 cc i ==> cc_azim_v11 cc i < #2.8 ` THEN DISCH_TAC THEN FIRST_ASSUM NHANH THEN REPEAT STRIP_TAC; MP_TAC (MESON[Flyspeck_constants.bounds]` #3.14159 < pi`) THEN STRIP_TAC THEN ASSUME_TAC2 ( REAL_ARITH` &2 * pi <= cc_azim_v11 cc j + cc_azim_v11 cc i /\ cc_azim_v11 cc j < #2.8 /\ cc_azim_v11 cc i < #2.8 ==> pi < #2.8 `) THEN REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC; (* 46 [`qy IN indSet`] 47 [`cc_qy_v11 cc qy`] 48 [`!i. #0.606 <= cc_azim_v11 cc i /\ &0 <= cc_azim_v11 cc i`] 49 [`sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi`] `?i j k. ~(i = j \/ j = k \/ k = i) /\ i IN indSet /\ j IN indSet /\ k IN indSet /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\ cc_4cell_v11 cc k` *) SUBGOAL_THEN` ?ii jj. {ii, jj} = {i:num, j} /\ cc_qy_v11 cc (ii + 1)` MP_TAC; ASM_CASES_TAC` cc_qy_v11 cc ( (i:num) + 1) ` THENL [ EXISTS_TAC` i:num ` THEN EXISTS_TAC` j:num ` THEN FIRST_X_ASSUM MP_TAC THEN SIMP_TAC[]; EXISTS_TAC` j:num ` ] THEN EXISTS_TAC` i:num ` THEN REWRITE_TAC[ INSERT_COMM] THEN SUBGOAL_THEN` ! m. ~cc_qy_v11 cc m ==> (m) MOD (cc_card_v11 cc) = (i:num) \/ m MOD (cc_card_v11 cc) = j ` MP_TAC; GEN_TAC THEN ASSUME_TAC2 (ARITH_RULE` 2 <= cc_card_v11 cc ==> ~( cc_card_v11 cc = 0)`) THEN ABBREV_TAC ` nn = cc_card_v11 cc ` THEN ASSUME_TAC2 (SPECL [`m:num `;` nn:num `] MOD_IN_NUMSEG) THEN ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn: num `;` m:num `] periodic_mod) THEN ABBREV_TAC`m1 = m MOD nn ` THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN SUBGOAL_THEN` m1 = (i:num) \/ m1 = j ` MP_TAC; UNDISCH_TAC` !k. k IN indSet /\ ~(k = i) /\ ~(k = j) ==> cc_qy_v11 cc k` THEN DISCH_THEN (MP_TAC o (SPEC` m1 : num `)) THEN UNDISCH_TAC` ~ cc_qy_v11 cc m1 ` THEN UNDISCH_TAC` m1 IN 0..nn - 1 ` THEN ASM_REWRITE_TAC[] THEN CONV_TAC TAUT; SIMP_TAC[]; (* *) (* 50 [`~cc_qy_v11 cc (i + 1)`] `(!m. ~cc_qy_v11 cc m ==> m MOD cc_card_v11 cc = i \/ m MOD cc_card_v11 cc = j) ==> cc_qy_v11 cc (j + 1)` *) ABBREV_TAC` nn = cc_card_v11 cc ` THEN STRIP_TAC THEN ASM_CASES_TAC` cc_qy_v11 cc (j + 1) ` THENL [ FIRST_X_ASSUM ACCEPT_TAC; SUBGOAL_THEN` ! m. ~(m IN indSet /\ (m + 1) MOD nn = m)` MP_TAC]; GEN_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN STRIP_TAC THEN ASM_CASES_TAC` m = nn - 1 `; ASSUME_TAC2 (ARITH_RULE` 2 <= nn /\ m = nn - 1 ==> m + 1 = 1 * nn + 0 `) THEN MP_TAC (SPECL [` m + 1`;` nn: num `;` 1`; ` 0`] MOD_UNIQ) THEN ANTS_TAC THENL [ ASM_REWRITE_TAC[] THEN UNDISCH_TAC` 2 <= nn ` THEN ARITH_TAC; DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_TAC` 0 = m` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC` 2 <= nn ` THEN ARITH_TAC ]; ASSUME_TAC2 (ARITH_RULE` m <= nn - 1 /\ ~(m = nn - 1) ==> m + 1 < nn`) THEN MP_TAC (SPECL[` m + 1 `;` nn: num `;` 0 `;` m + 1 `] MOD_UNIQ) THEN ANTS_TAC THENL [ ASM_REWRITE_TAC[] THEN ARITH_TAC; DISCH_THEN SUBST_ALL_TAC] THEN UNDISCH_TAC` m + 1 = m` THEN ARITH_TAC; FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 1 `)); FIRST_ASSUM (ASSUME_TAC2 o (SPEC` j + 1 `)); STRIP_TAC; FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i:num`)); FIRST_ASSUM (ASSUME_TAC2 o (SPEC` j:num`)); REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC); UNDISCH_TAC` (i:num) IN indSet `; UNDISCH_TAC` (j:num) IN indSet`; SIMP_TAC[]; REPEAT STRIP_TAC; UNDISCH_TAC` (i + 1) MOD nn = i \/ (i + 1) MOD nn = j `; UNDISCH_TAC` (j + 1) MOD nn = i \/ (j + 1) MOD nn = j`; ASM_REWRITE_TAC[]; ASM_CASES_TAC` j + 1 = nn `; ASM_REWRITE_TAC[]; (* `nn MOD nn = i ==> ~((i + 1) MOD nn = j)` *) ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~( nn = 0) `) THEN ASSUME_TAC2 (SPEC`nn:num` (GEN_ALL MOD_REFL)) THEN ASM_REWRITE_TAC[] THEN DISCH_THEN (SUBST_ALL_TAC o SYM) THEN REWRITE_TAC[ADD] THEN ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> 1 < nn `) THEN FIRST_ASSUM MP_TAC THEN NHANH MOD_LT THEN SIMP_TAC[] THEN REPEAT STRIP_TAC THEN FIRST_X_ASSUM (SUBST_ALL_TAC o SYM) THEN SUBST_ALL_TAC (ARITH_RULE` 1 + 1 = 2 `) THEN UNDISCH_TAC` qy IN (indSet: num -> bool) ` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN UNDISCH_TAC` 2 = nn ` THEN DISCH_THEN (SUBST_ALL_TAC o SYM) THEN REWRITE_TAC[ARITH_RULE` 0 <= x /\ x <= 2 - 1 <=> x = 0 \/ x = 1 `]; STRIP_TAC THENL [ FIRST_X_ASSUM SUBST_ALL_TAC THEN UNDISCH_TAC` cc_4cell_v11 cc 0 ` THEN UNDISCH_TAC` cc_qy_v11 cc 0 ` THEN ASM_REWRITE_TAC[cc_qy_v11]; FIRST_X_ASSUM SUBST_ALL_TAC THEN UNDISCH_TAC` cc_4cell_v11 cc 1 ` THEN UNDISCH_TAC` cc_qy_v11 cc 1 ` THEN ASM_REWRITE_TAC[cc_qy_v11]]; STRIP_TAC THEN SUBGOAL_THEN` j + 1 < (nn: num) ` MP_TAC THENL [ UNDISCH_TAC` j:num IN indSet` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN UNDISCH_TAC` ~( j + 1 = nn ) ` THEN UNDISCH_TAC` 2 <= nn ` THEN ARITH_TAC; NHANH MOD_LT THEN ASM_REWRITE_TAC[] THEN SIMP_TAC[]]; REWRITE_TAC[ARITH_RULE` (j + 1) + 1 = j + 2 `] THEN ASM_CASES_TAC` j + 2 < nn` THENL [ ASSUME_TAC2 (SPECL [`j + 2`;` nn:num `] MOD_LT) THEN FIRST_X_ASSUM SUBST1_TAC THEN ARITH_TAC; STRIP_TAC]; ASSUME_TAC2 (ARITH_RULE` ~(j + 2 < nn) /\ j + 1 < nn ==> j + 2 = nn `) THEN ASM_REWRITE_TAC[] THEN ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~(nn = 0) `) THEN ASSUME_TAC2 (SPEC`nn:num ` (GEN_ALL MOD_REFL)) THEN ASM_REWRITE_TAC[] THEN STRIP_TAC THEN FIRST_X_ASSUM (SUBST_ALL_TAC o SYM) THEN UNDISCH_TAC` qy:num IN indSet ` THEN ASM_REWRITE_TAC[IN_NUMSEG] THEN EXPAND_TAC "nn" THEN REWRITE_TAC[ARITH_RULE` 0 <= qy /\ qy <= (0 + 2) - 1 <=> qy = 0 \/ qy = 1`] THEN STRIP_TAC THENL [ UNDISCH_TAC` cc_qy_v11 cc qy ` THEN UNDISCH_TAC` cc_4cell_v11 cc 0 ` THEN ASM_SIMP_TAC[cc_qy_v11]; UNDISCH_TAC` cc_qy_v11 cc qy ` THEN UNDISCH_TAC` cc_4cell_v11 cc i ` THEN ASM_SIMP_TAC[cc_qy_v11; ADD]]; STRIP_TAC THEN ABBREV_TAC` nn = cc_card_v11 cc ` THEN SUBGOAL_THEN` ? nj k. nj < nn /\ nj + 1 = k * nn + jj /\ cc_qy_v11 cc nj ` MP_TAC; ASM_CASES_TAC` jj = 0 `; EXISTS_TAC` nn - 1 ` THEN EXISTS_TAC` 1 ` THEN FIRST_X_ASSUM SUBST_ALL_TAC THEN ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> nn - 1 < nn /\ nn - 1 + 1 = 1 * nn + 0 `) THEN ASM_SIMP_TAC[] THEN ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~( nn - 1 = 0) `) THEN ASM_CASES_TAC` nn - 1 = ii' ` THENL [ UNDISCH_TAC` cc_qy_v11 cc (ii' + 1)` THEN FIRST_X_ASSUM (SUBST_ALL_TAC o SYM) THEN ASM_REWRITE_TAC[ARITH_RULE` 1 * n + 0 = 0 + n`] THEN UNDISCH_TAC` periodic (cc_qy_v11 cc) nn ` THEN SIMP_TAC[periodic] THEN STRIP_TAC THEN UNDISCH_TAC` {nn - 1, 0} = {i, j} ` THEN UNDISCH_TAC` cc_4cell_v11 cc i ` THEN UNDISCH_TAC` cc_4cell_v11 cc j ` THEN REWRITE_TAC[cc_qy_v11] THEN SET_TAC[]; ASSUME_TAC2 (SET_RULE` ~(nn - 1 = 0) /\ ~(nn - 1 = ii') /\ {ii', 0} = {i, j} ==> ~( nn - 1 = i) /\ ~( nn - 1 = j ) `) THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[IN_NUMSEG; LE_REFL] THEN MATCH_MP_TAC (ARITH_RULE` 2 <= nn ==> 0 <= nn - 1 `) THEN FIRST_X_ASSUM ACCEPT_TAC]; EXISTS_TAC` jj - 1 ` THEN EXISTS_TAC` 0 ` THEN REWRITE_TAC[MULT; ADD] THEN CONJ_TAC THENL [ MATCH_MP_TAC (ARITH_RULE` jj <= nn - 1 /\ ~( jj = 0) ==> jj - 1 < nn `) THEN ASM_REWRITE_TAC[] THEN SUBGOAL_THEN` jj:num IN indSet` MP_TAC THENL [ MP_TAC (SET_RULE` jj IN {ii', jj:num}`) THEN ASM_REWRITE_TAC[] THEN ASM_REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN UNDISCH_TAC` (i:num) IN indSet ` THEN UNDISCH_TAC` (j:num) IN indSet ` THEN ASM_REWRITE_TAC[] THEN MESON_TAC[]; ASM_REWRITE_TAC[IN_NUMSEG] THEN SIMP_TAC[]]; CONJ_TAC THENL [FIRST_X_ASSUM MP_TAC THEN ARITH_TAC; REWRITE_TAC[]]]; ASM_CASES_TAC` jj - 1 = ii' ` THENL [ UNDISCH_TAC` cc_qy_v11 cc (ii' + 1) ` THEN EXPAND_TAC "ii'" THEN ASSUME_TAC2 (ARITH_RULE` ~( jj = 0) ==> jj - 1 + 1 = jj `) THEN FIRST_X_ASSUM SUBST1_TAC THEN MP_TAC (SET_RULE` jj IN {ii', jj:num} `) THEN ASM_REWRITE_TAC[] THEN REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY; cc_qy_v11 ] THEN UNDISCH_TAC` cc_4cell_v11 cc i ` THEN UNDISCH_TAC` cc_4cell_v11 cc j ` THEN MESON_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC]; ASM_REWRITE_TAC[IN_NUMSEG] THEN MATCH_MP_TAC (ARITH_RULE` ~(jj = 0) /\ 0 <= jj /\ jj <= nn - 1 ==> 0 <= jj - 1 /\ jj - 1 <= nn - 1 `) THEN ASM_REWRITE_TAC[] THEN MP_TAC (SET_RULE` jj IN {ii', jj:num} `) THEN ASM_REWRITE_TAC[GSYM IN_NUMSEG] THEN REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY] THEN UNDISCH_TAC` (i:num) IN indSet ` THEN UNDISCH_TAC` (j:num) IN indSet ` THEN ASM_REWRITE_TAC[] THEN MESON_TAC[]; (* 52 [`~(jj = 0)`] 53 [`~(jj - 1 = ii')`] `~(jj - 1 = i) /\ ~(jj - 1 = j)` *) SUBGOAL_THEN` ~( jj - 1 IN {ii', jj:num})` MP_TAC; REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY]; ASM_REWRITE_TAC[]; UNDISCH_TAC` ~( jj = 0) `; ARITH_TAC; ASM_REWRITE_TAC[]; SET_TAC[]; (* ================== *) STRIP_TAC; (* 53 [`nj < nn`] 54 [`nj + 1 = k * nn + jj`] 55 [`cc_qy_v11 cc nj`] `?i j k. ~(i = j \/ j = k \/ k = i) /\ i IN indSet /\ j IN indSet /\ k IN indSet /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\ cc_4cell_v11 cc k` *) ASSUME_TAC2 (ARITH_RULE` 2 <= nn ==> ~( nn = 0) `) THEN ASSUME_TAC2 (SPECL [` ii' + 1 `;` nn:num `] DIVMOD_EXIST) THEN FIRST_X_ASSUM MP_TAC THEN STRIP_TAC THEN SUBGOAL_THEN` {nj, r, ii', jj:num} SUBSET indSet ` MP_TAC THENL [ ONCE_REWRITE_TAC[INSERT_SUBSET] THEN ONCE_REWRITE_TAC[INSERT_SUBSET] THEN ASM_REWRITE_TAC[] THEN CONJ_TAC THENL [ REWRITE_TAC[IN_NUMSEG] THEN UNDISCH_TAC` ~( nn = 0) ` THEN UNDISCH_TAC` nj < nn:num ` THEN ARITH_TAC; CONJ_TAC] THENL [ REWRITE_TAC[IN_NUMSEG] THEN UNDISCH_TAC` ~( nn = 0) ` THEN UNDISCH_TAC` r < nn:num ` THEN ARITH_TAC; UNDISCH_TAC` i:num IN indSet`] THEN UNDISCH_TAC` j:num IN indSet` THEN ASM_REWRITE_TAC[] THEN SET_TAC[]; STRIP_TAC]; SUBGOAL_THEN` &0 <= sum ({nj, r, ii', jj}) (cc_gg_v11 cc)` MP_TAC; UNDISCH_TAC` cc_qy_v11 cc (ii' + 1) ` THEN ASM_REWRITE_TAC[] THEN UNDISCH_TAC` periodic (cc_qy_v11 cc) nn ` THEN REWRITE_TAC[periodic_mult] THEN SIMP_TAC[] THEN STRIP_TAC THEN UNDISCH_TAC` cc_qy_v11 cc nj ` THEN REWRITE_TAC[cc_qy_v11] THEN REPEAT STRIP_TAC THEN ASSUME_TAC2 (SET_RULE` {ii', jj} = {i, j} /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\ ~ (cc_4cell_v11 cc nj) /\ ~( cc_4cell_v11 cc r) ==> DISJOINT {nj, r} {ii', jj}`) THEN REWRITE_TAC[SET_RULE` {a,b,c,d} = {a,b} UNION {c,d} `] THEN ASSUME_TAC (MESON[Geomdetail.FINITE6]` ! a (b:A). FINITE {a,b}`) THEN SUBGOAL_THEN` ! f. sum ({nj, r:num} UNION {i, j}) f = sum {nj, r} f + sum {i, j} f ` MP_TAC THENL [ GEN_TAC THEN MATCH_MP_TAC SUM_UNION THEN UNDISCH_TAC` DISJOINT {nj, r} {ii', jj:num} ` THEN ASM_SIMP_TAC[]; SIMP_TAC[] THEN STRIP_TAC]; (* 61 [`!a b. FINITE {a, b}`] 62 [`!f. sum ({nj, r} UNION {i, j}) f = sum {nj, r} f + sum {i, j} f`] `&0 <= sum {nj, r} (cc_gg_v11 cc) + sum {i, j} (cc_gg_v11 cc)` *) SUBGOAL_THEN` (! i. cc_qy_v11 cc (i + 1) ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) /\ (! i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) ` MP_TAC; CONJ_TAC; (* AHTYRED ================= *) GEN_TAC; ASM_CASES_TAC` cc_qu_v11 cc i' `; FIRST_X_ASSUM MP_TAC; PHA; UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `; DISCH_THEN NHANH; REWRITE_TAC[cc_eps]; REAL_ARITH_TAC; MP_TAC (SPEC` i':num ` QU_OR_QXY); ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [`#0.0 <= cc_gg_v11 cc i `] NHANH; STRIP_TAC; ASM_SEARCH_TCL [`&0 <= cc_gg3a_v11 cc i `] NHANH; FIRST_X_ASSUM MP_TAC; REAL_ARITH_TAC; ASM_SEARCH_TCL [` &0 <= cc_gg3a_v11 cc i`] NHANH; STRIP_TAC; ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i`] (ASSUME_TAC2 o (SPEC` i':num `)); MATCH_MP_TAC REAL_LE_ADD; ASM_REWRITE_TAC[]; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` cc_gg3a_v11 cc i' + cc_gg3b_v11 cc i' `; ASM_REWRITE_TAC[]; MATCH_MP_TAC REAL_LE_ADD; CONJ_TAC; ASM_SEARCH_TCL [` A ==> &0 <= cc_gg3a_v11 cc i' `] (MATCH_MP_TAC o (SPEC` i':num`)); FIRST_X_ASSUM ACCEPT_TAC; ASM_SEARCH_TCL [` A ==> &0 <= cc_gg3b_v11 cc i' `] (MATCH_MP_TAC o (SPEC` i':num`)); FIRST_X_ASSUM ACCEPT_TAC; (* AHTYRED ================= *) (* 60 [`DISJOINT {nj, nj} {ii', jj}`] 61 [`!a b. FINITE {a, b}`] 62 [`!f. sum ({nj, nj} UNION {i, j}) f = sum {nj, nj} f + sum {i, j} f`] `!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)` *) GEN_TAC; ASM_CASES_TAC` cc_qu_v11 cc (i' + 1) `; FIRST_X_ASSUM MP_TAC THEN PHA; ASM_SEARCH_TCL [`cc_eps <= cc_gg3b_v11 cc i' + cc_gg_v11 cc i `] NHANH; REWRITE_TAC[cc_eps]; REAL_ARITH_TAC; SUBST_ALL_TAC (REAL_ARITH` #0.0 = &0 `); ASM_SEARCH_TCL [` &0 <= cc_gg3b_v11 cc i`] NHANH; MP_TAC (SPEC` i' + 1 ` QU_OR_QXY); ASMS_SEARCH_TCL [` ~cc_qu_v11 cc (i' + 1) `] REWRITE_TAC; ASM_SEARCH_TCL [` &0 <= cc_gg_v11 cc i `] NHANH; ASM_SEARCH_TCL [` &0 <= cc_gg_v11 cc i `] NHANH; ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + ccy `] NHANH; REPEAT STRIP_TAC; (* 2 goals *) MATCH_MP_TAC REAL_LE_ADD; ASMS_SEARCH_TCL [` &0 <= x `] REWRITE_TAC; MATCH_MP_TAC REAL_LE_ADD; ASMS_SEARCH_TCL [` &0 <= cc_gg3b_v11 cc i `] REWRITE_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` cc_gg3a_v11 cc (i' + 1) + cc_gg3b_v11 cc (i' + 1)`; ASM_REWRITE_TAC[]; MATCH_MP_TAC REAL_LE_ADD; UNDISCH_TAC` cc_qy_v11 cc (i' + 1)`; ASMS_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= x `] SIMP_TAC; (* end the 2nd goal (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) *) STRIP_TAC; ASM_CASES_TAC` nj = (r:num) `; FIRST_X_ASSUM (SUBST_ALL_TAC o SYM); REWRITE_TAC[INSERT_INSERT; SUM_SING]; ASM_SEARCH_TCL [` {a,b} = {x,y} `] (SUBST1_TAC o SYM); ASSUME_TAC2 (SET_RULE` ~(j = (i:num)) /\ {ii', jj} = {i, j} ==> ~( ii' = jj)`); ASSUME_TAC2 (ISPECL [` ii':num `;` jj:num `;` cc_gg_v11 cc `] Geomdetail.SUM_DIS2); FIRST_X_ASSUM SUBST1_TAC; ASM_SEARCH_TCL [` cc_qy_v11 cc (i + 1) ==> x `] (MP_TAC o (SPEC` ii':num `)); ANTS_TAC; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[cc_qy_v11]; ASM_SEARCH_TCL [` periodic `;` cc_gg3a_v11`] MP_TAC; REWRITE_TAC[periodic_mult]; ASM_REWRITE_TAC[]; SIMP_TAC[]; STRIP_TAC; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> x `;` cc_gg3b_v11 `] (MP_TAC o (SPEC`nj: num`)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; ASM_SEARCH_TCL [` periodic `;` cc_gg_v11`] MP_TAC; REWRITE_TAC[periodic_mult]; ASM_REWRITE_TAC[]; SIMP_TAC[]; STRIP_TAC; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] (MP_TAC o (SPEC` nj: num `)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; PHA; REAL_ARITH_TAC; ASM_SEARCH_TCL [` {a,b} = {c,d} `] (SUBST1_TAC o SYM); ASSUME_TAC2 (ISPECL [`nj: num`;` r:num `;` cc_gg_v11 cc `] Geomdetail.SUM_DIS2); ASSUME_TAC2 (SET_RULE` {ii', jj} = {i,j:num} /\ ~( j = i) ==> ~( ii' = jj) `); ASSUME_TAC2 (ISPECL [`ii': num`;` jj:num `;` cc_gg_v11 cc `] Geomdetail.SUM_DIS2); ASM_REWRITE_TAC[]; FIRST_X_ASSUM (MP_TAC o (SPEC ` nj: num`)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; FIRST_X_ASSUM (MP_TAC o (SPEC ` ii': num`)); ANTS_TAC; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[cc_qy_v11]; ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` periodic `;` cc_gg3a_v11`] MP_TAC; ASM_SEARCH_TCL [` periodic `;` cc_gg_v11`] MP_TAC; SIMP_TAC[periodic_mult]; STRIP_TAC THEN STRIP_TAC; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] (MP_TAC o (SPEC` r:num `)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] (MP_TAC o (SPEC` nj:num `)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= x`;` cc_gg3b_v11 `] (MP_TAC o (SPEC` r:num `)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= x`;` cc_gg3a_v11 `] (MP_TAC o (SPEC` nj:num `)); ANTS_TAC; ASM_REWRITE_TAC[cc_qy_v11]; REAL_ARITH_TAC; (* =============xxxxx=========================== *) FIRST_X_ASSUM MP_TAC; ASM_REWRITE_TAC[]; ABBREV_TAC` sts = {nj, r, i, j:num} `; NHANH (SET_RULE` s SUBSET S ==> (S DIFF s) UNION s = S `); REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum ((0..nn - 1) DIFF sts) (cc_gg_v11 cc) ` MP_TAC; MATCH_MP_TAC SUM_POS_LE; CONJ_TAC; MATCH_MP_TAC FINITE_DIFF; REWRITE_TAC[FINITE_NUMSEG]; GEN_TAC; REWRITE_TAC[IN_DIFF]; STRIP_TAC; MATCH_MP_TAC QY_NN0; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[cc_real_model_v11]; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` ~( (x:num) IN sts ) `; EXPAND_TAC "sts"; REWRITE_TAC[IN_INSERT]; CONV_TAC TAUT; STRIP_TAC; SUBGOAL_THEN` &0 <= sum ((0..nn - 1)) (cc_gg_v11 cc)` MP_TAC; SUBGOAL_THEN` sum (0..nn - 1) (cc_gg_v11 cc) = sum ((0..nn - 1) DIFF sts) (cc_gg_v11 cc) + sum sts (cc_gg_v11 cc)` MP_TAC; ASM_SEARCH_TCL [` a UNION b = c `] (fun x -> PAT_ONCE_REWRITE_TAC`\x. x = y `[SYM x]); MATCH_MP_TAC SUM_UNION; CONJ_TAC; MATCH_MP_TAC FINITE_DIFF; REWRITE_TAC[FINITE_NUMSEG]; CONJ_TAC; EXPAND_TAC "sts"; REWRITE_TAC[Geomdetail.FINITE6]; SET_TAC[]; DISCH_THEN SUBST1_TAC; MATCH_MP_TAC REAL_LE_ADD; ASM_REWRITE_TAC[]; NHANH (REAL_ARITH` &0 <= a ==> ~( a < &0) `); ASM_REWRITE_TAC[]; FIRST_X_ASSUM MP_TAC; REWRITE_TAC[]; STRIP_TAC; EXISTS_TAC`i:num `; EXISTS_TAC`j:num `; EXISTS_TAC`k:num `; ASM_REWRITE_TAC[]]);; ====================================== *) (* xxxxx *) (* =========================== let oxl6142 = prove_by_refinement (CHQSQEY_concl, [GEN_TAC; ABBREV_TAC` indSet = 0..cc_card_v11 cc - 1 `; STRIP_TAC; MP_TAC WKR_COMPTED; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; REWRITE_TAC[cc_size_v11]; SUBGOAL_THEN` {i,j,k} SUBSET {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}` MP_TAC; ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM]; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` FINITE {i | i IN indSet /\ cc_4cell_v11 cc i} ` MP_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` indSet:num -> bool `; EXPAND_TAC "indSet"; REWRITE_TAC[FINITE_NUMSEG]; SET_TAC[]; FIRST_X_ASSUM MP_TAC; PHA; NHANH CARD_SUBSET; STRIP_TAC; ASM_SEARCH_TCL [` a \/ b`] MP_TAC; REWRITE_TAC[GSYM Geomdetail.CARD3]; DISCH_THEN SUBST_ALL_TAC; FIRST_X_ASSUM ACCEPT_TAC]);; =============================== *) (* ================================================ *) (* let CHQSQEY = oxl6142;; *) let SUM_BETA = prove(` sum S (\x. f x) = sum S f `, MATCH_MP_TAC SUM_EQ THEN REWRITE_TAC[BETA_THM]);; let LXDEYBO = prove_by_refinement (LXDEYBO_concl, [ GEN_TAC; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_size_v11]; STRIP_TAC; ABBREV_TAC` cd = CARD {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}`; ASM_CASES_TAC` 4 < cd `; ABBREV_TAC` ss = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}`; ABBREV_TAC` S = 0..cc_card_v11 cc - 1 `; SUBGOAL_THEN` sum ss (cc_gg_v11 cc) <= sum S (cc_gg_v11 cc) ` MP_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE; CONJ_TAC; EXPAND_TAC "S"; REWRITE_TAC[FINITE_NUMSEG]; EXPAND_TAC "ss"; CONJ_TAC; SET_TAC[]; REWRITE_TAC[IN_DIFF; IN_ELIM_THM; DE_MORGAN_THM]; REPEAT STRIP_TAC; FIRST_X_ASSUM MP_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC QY_NN0; ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_qy_v11]; STRIP_TAC; SUBGOAL_THEN` sum ss (\i. a_spine5 + b_spine5 * cc_azim_v11 cc i ) <= sum ss (cc_gg_v11 cc ) ` MP_TAC; MATCH_MP_TAC SUM_LE; CONJ_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` S: num -> bool `; CONJ_TAC; EXPAND_TAC "S"; REWRITE_TAC[FINITE_NUMSEG]; EXPAND_TAC "ss"; SET_TAC[]; EXPAND_TAC "ss"; REWRITE_TAC[IN_ELIM_THM]; ASM_SEARCH_TCL [` cc_4cell_v11 cc i ==> a <= b `] NHANH; SIMP_TAC[]; SUBGOAL_THEN` sum ss (cc_azim_v11 cc ) <= &2 * pi ` MP_TAC; ASM_SEARCH_TCL [`sum S (cc_azim_v11 cc) = &2 * pi `] (SUBST1_TAC o SYM); MATCH_MP_TAC SUM_SUBSET_SIMPLE; CONJ_TAC; EXPAND_TAC "S"; REWRITE_TAC[FINITE_NUMSEG]; CONJ_TAC; EXPAND_TAC "ss"; SET_TAC[]; REPEAT STRIP_TAC; ASM_SEARCH_TCL [` (!i. #0.606 <= cc_azim_v11 cc i) `] MP_TAC; DISCH_THEN (MP_TAC o (SPEC`x:num `)); REAL_ARITH_TAC; ONCE_REWRITE_TAC[MESON[ABS_SIMP]` (\i. a + f i ) = (\i. (\j. a) i + f i)`]; STRIP_TAC THEN STRIP_TAC; SUBGOAL_THEN` sum ss (\i. (\j. a_spine5) i + b_spine5 * cc_azim_v11 cc i) = sum ss (\j. a_spine5) + sum ss (\i. b_spine5 * cc_azim_v11 cc i)` MP_TAC; MATCH_MP_TAC SUM_ADD; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` S:num -> bool`; EXPAND_TAC "S"; REWRITE_TAC[FINITE_NUMSEG]; ASM_REWRITE_TAC[]; EXPAND_TAC "ss"; SET_TAC[]; REWRITE_TAC[SUM_LMUL]; REWRITE_TAC[Sphere.b_spine5; Sphere.a_spine5]; SUBGOAL_THEN` FINITE (S:num -> bool) ` MP_TAC; EXPAND_TAC "S"; REWRITE_TAC[FINITE_NUMSEG]; STRIP_TAC; SUBGOAL_THEN` ss SUBSET (S:num -> bool) ` ASSUME_TAC; EXPAND_TAC "ss"; SET_TAC[]; SUBGOAL_THEN` FINITE (ss:num -> bool) ` MP_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC`S: num -> bool `; ASM_REWRITE_TAC[FINITE_NUMSEG]; STRIP_TAC; ASSUME_TAC2 (ISPECL [` #0.0560305 `;`ss:num -> bool`] SUM_CONST); ASM_REWRITE_TAC[]; ASSUME_TAC2 (ARITH_RULE` 4 < cd ==> 5 <= cd `); FIRST_X_ASSUM MP_TAC; FIRST_X_ASSUM MP_TAC; ASM_REWRITE_TAC[GSYM REAL_OF_NUM_LE]; REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum ss (cc_gg_v11 cc)` ASSUME_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` sum ss (\i. (\j. a_spine5) i + b_spine5 * cc_azim_v11 cc i) `; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5]; REWRITE_TAC[REAL_ARITH` &0 <= a + -- b * x <=> b * x <= a `]; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` #0.0445813 * ( &2 * pi) `; ASM_REWRITE_TAC[REAL_ARITH` #0.0445813 * a <= #0.0445813 * b <=> a <= b `]; ASM_REWRITE_TAC[ISPECL [`ss:num -> bool `;` cc_azim_v11 cc `] (GEN_ALL SUM_BETA)]; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` &5 * #0.0560305`; ASM_REWRITE_TAC[REAL_ARITH` &5 * #0.0560305 <= &cd * #0.0560305 <=> &5 <= &cd`]; MP_TAC (prove(` pi < #3.1416 `, REWRITE_TAC[ Flyspeck_constants.bounds])); REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= sum S (cc_gg_v11 cc)` MP_TAC; UNDISCH_TAC` &0 <= sum ss (cc_gg_v11 cc) `; UNDISCH_TAC` sum ss (cc_gg_v11 cc) <= sum S (cc_gg_v11 cc) `; REAL_ARITH_TAC; UNDISCH_TAC` sum S (cc_gg_v11 cc) < &0 `; REAL_ARITH_TAC; FIRST_X_ASSUM MP_TAC; ARITH_TAC]);; (* ================================== *) let IMP_TAC = ONCE_REWRITE_TAC[TAUT` a /\ b ==> c <=> a ==> b ==> c `];; (* the concls *) let MTMLSRF_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (?i. 0 < i /\ cc_gg_v11 cc i < &0 /\ cc_qu_v11 cc i /\ cc_4cell_v11 cc (i + 1) /\ cc_4cell_v11 cc (i - 1))`;; let UNPNFVW_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> cc_size_v11 cc (cc_qy_v11 cc) <= 1`;; let IPVICGW_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (!i. cc_small_v11 cc i)`;; let RSIWAMP_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (cc_card_v11 cc <= 4)`;; let MTMLSRF = prove_by_refinement (MTMLSRF_concl, [REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; REPEAT STRIP_TAC; ASM_CASES_TAC` ~(? i. 0 < i /\ cc_gg_v11 cc i < &0) `; FIRST_X_ASSUM MP_TAC; REWRITE_TAC[MESON[]` ~(?i. p i /\ q i) <=> ! i. p i ==> ~(q i )`]; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) ` MP_TAC; MATCH_MP_TAC SUM_POS_LE_NUMSEG; REPEAT STRIP_TAC; ASM_CASES_TAC` p = 0 `; ASM_SEARCH_TCL [` periodic `; `cc_gg_v11 `] MP_TAC; FIRST_X_ASSUM SUBST_ALL_TAC; REWRITE_TAC[periodic]; DISCH_THEN (SUBST1_TAC o SYM o (SPEC` 0`)); REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; FIRST_X_ASSUM MATCH_MP_TAC; ASM_SEARCH_TCL [` ~( x = 0) `] MP_TAC; ARITH_TAC; REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM MP_TAC; ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; FIRST_X_ASSUM MP_TAC; REWRITE_TAC[]; STRIP_TAC; ASM_CASES_TAC` ~( ?i. 0 < i /\ cc_gg_v11 cc i < &0 /\ cc_qu_v11 cc i /\ cc_4cell_v11 cc (i + 1) /\ cc_4cell_v11 cc (i - 1) ) `; FIRST_X_ASSUM MP_TAC; REWRITE_TAC[MESON[]` ~(?i. p1 i /\ p2 i /\ p3 i /\ p4 i) <=> (! i. p1 i /\ p2 i /\ p3 i ==> ~( p4 i))`]; STRIP_TAC; ABBREV_TAC` S = 1..cc_card_v11 cc `; ABBREV_TAC` cb i <=> cc_gg_v11 cc i < &0 /\ cc_qu_v11 cc i `; ABBREV_TAC` sa = { i | i IN S /\ cb i /\ cc_qy_v11 cc (i + 1) }`; ABBREV_TAC` sb = { i | i IN S /\ cb i /\ cc_qy_v11 cc ( i - 1)}`; ABBREV_TAC` ss = {i | (i:num) IN S /\ cb i} `; ABBREV_TAC` saa = { i + 1 | i + 1 IN S /\ cc_qy_v11 cc (i + 1) /\ cb i} `; ABBREV_TAC` sbb = { i | i IN S /\ cc_qy_v11 cc i /\ cb ( i + 1)}`; ABBREV_TAC` sab = { i | i IN S /\ cc_qy_v11 cc i /\ (cb (i + 1) \/ (cb ( i - 1)))} `; SUBGOAL_THEN` DISJOINT ss (sab: num -> bool) ` MP_TAC; EXPAND_TAC "sab"; EXPAND_TAC "ss"; ASM_SEARCH_TCL [` a /\ b <=> f i `] ( fun x -> REWRITE_TAC[GSYM x]); REWRITE_TAC[cc_qu_v11; cc_qy_v11]; SET_TAC[]; SUBGOAL_THEN` FINITE (S:num -> bool) ` MP_TAC; EXPAND_TAC "S"; REWRITE_TAC[FINITE_NUMSEG]; STRIP_TAC; SUBGOAL_THEN` ss SUBSET (S:num -> bool) /\ sab SUBSET S ` ASSUME_TAC; EXPAND_TAC "ss"; EXPAND_TAC "sab"; SET_TAC[]; SUBGOAL_THEN` FINITE (ss:num -> bool) ` MP_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC `S:num -> bool `; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` FINITE (sab:num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC `S:num -> bool `; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` sum saa (cc_gg3a_v11 cc) + sum sbb (cc_gg3b_v11 cc) + sum ss (cc_gg_v11 cc) <= sum (ss UNION sab) (cc_gg_v11 cc)` MP_TAC; ASM_SIMP_TAC[SUM_UNION]; REWRITE_TAC[REAL_ARITH` a + b + c <= c + d <=> a + b <= d `]; SUBGOAL_THEN` sbb SUBSET (sab:num -> bool) /\ saa SUBSET (sab:num -> bool) ` ASSUME_TAC; EXPAND_TAC "saa"; EXPAND_TAC "sbb"; EXPAND_TAC "sab"; CONJ_TAC; REWRITE_TAC[SUBSET; IN_ELIM_THM]; MESON_TAC[]; REWRITE_TAC[SUBSET; IN_ELIM_THM]; MESON_TAC[ARITH_RULE` (i + 1) - 1 = i `]; SUBGOAL_THEN` !x. x IN saa ==> cc_qy_v11 cc x ` ASSUME_TAC; EXPAND_TAC "saa"; SET_TAC[]; SUBGOAL_THEN` !x. x IN sbb ==> cc_qy_v11 cc x ` ASSUME_TAC; EXPAND_TAC "sbb"; SET_TAC[]; SUBGOAL_THEN` !x. x IN sab ==> cc_qy_v11 cc x ` ASSUME_TAC; EXPAND_TAC "sab"; SET_TAC[]; SUBGOAL_THEN` sum saa (cc_gg3a_v11 cc) <= sum sab (cc_gg3a_v11 cc) ` MP_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE; ASM_REWRITE_TAC[IN_DIFF]; FIRST_ASSUM NHANH; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> c <= cc_gg3a_v11 cc i `] NHANH; SIMP_TAC[]; SUBGOAL_THEN` sum sbb (cc_gg3b_v11 cc) <= sum sab (cc_gg3b_v11 cc)` MP_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[IN_DIFF]; FIRST_ASSUM NHANH; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> c <= cc_gg3b_v11 cc i `] NHANH THEN SIMP_TAC[]; SUBGOAL_THEN` sum sab (cc_gg3a_v11 cc) + sum sab (cc_gg3b_v11 cc) <= sum sab (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` cc_gg3b_v11 cc `;` sab:num -> bool `] SUM_ADD); FIRST_X_ASSUM (SUBST1_TAC o SYM); MATCH_MP_TAC SUM_LE; ASM_REWRITE_TAC[]; FIRST_ASSUM NHANH; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> a + b <= c`] NHANH; SIMP_TAC[]; REAL_ARITH_TAC; STRIP_TAC; SUBGOAL_THEN`&0 <= sum saa (cc_gg3a_v11 cc) + sum sbb (cc_gg3b_v11 cc) + sum ss (cc_gg_v11 cc) ` MP_TAC; SUBGOAL_THEN` sa SUBSET (ss:num -> bool) ` MP_TAC; EXPAND_TAC "sa"; EXPAND_TAC "ss"; SET_TAC[]; NHANH (SET_RULE` s SUBSET S ==> DISJOINT s (S DIFF s) /\ s UNION ( S DIFF s) = S`); STRIP_TAC; FIRST_ASSUM (SUBST1_TAC o SYM); SUBGOAL_THEN` FINITE (sa:num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` ss: num -> bool `; ASM_REWRITE_TAC[]; SUBGOAL_THEN` FINITE (ss DIFF (sa:num -> bool)) ` ASSUME_TAC; MATCH_MP_TAC FINITE_DIFF; FIRST_X_ASSUM ACCEPT_TAC; ASM_SIMP_TAC[SUM_UNION]; SUBGOAL_THEN` sum saa (cc_gg3a_v11 cc) = sum sa (\i. cc_gg3a_v11 cc (i + 1)) ` MP_TAC; MATCH_MP_TAC (GSYM SUM_EQ_GENERAL); ABBREV_TAC` nn = cc_card_v11 cc `; EXISTS_TAC`\i. if i = nn then 1 else i + 1 `; CONJ_TAC; GEN_TAC; EXPAND_TAC "sa"; EXPAND_TAC "saa"; REWRITE_TAC[IN_ELIM_THM; EXISTS_UNIQUE]; STRIP_TAC; ASM_CASES_TAC` 0 < i' `; EXISTS_TAC` i':num `; ASM_REWRITE_TAC[]; CONJ_TAC; CONJ_TAC; UNDISCH_TAC` i' + 1 IN S `; FIRST_X_ASSUM MP_TAC; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; ARITH_TAC; SUBGOAL_THEN` ~(i' = (nn:num)) ` MP_TAC; UNDISCH_TAC` i' + 1 IN S `; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; ARITH_TAC; SIMP_TAC[]; GEN_TAC; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; ASM_CASES_TAC` y' = (nn:num) `; ASM_REWRITE_TAC[]; FIRST_X_ASSUM SUBST_ALL_TAC; NHANH (ARITH_RULE` a + 1 = 1 ==> ~( 0 < a) `); ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; ARITH_TAC; SUBST_ALL_TAC (ARITH_RULE` ~( 0 < i') <=> i' = 0 `); EXISTS_TAC `nn:num `; ASM_REWRITE_TAC[]; CONJ_TAC; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG; LE_REFL; ADD]; MP_TAC (SPEC_ALL CC_CARD2); ANTS_TAC; ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ASM_SIMP_TAC[ARITH_RULE` 2 <= nn ==> 1 <= nn `]; STRIP_TAC; ASM_SEARCH_TCL [` periodic`; `cc_gg_v11 `] MP_TAC; ASM_SEARCH_TCL [` cc_bool_prep_v11 `] MP_TAC; REWRITE_TAC[cc_bool_prep_v11]; MP_TAC (SPEC_ALL periodic_fn); ANTS_TAC; ASM_REWRITE_TAC[cc_bool_model_v11]; ASM_REWRITE_TAC[]; STRIP_TAC; SIMP_TAC[Oxl_def.periodic]; STRIP_TAC; UNDISCH_TAC`(cb: num -> bool) i'`; ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]); FIRST_X_ASSUM MP_TAC; UNDISCH_TAC` cc_qy_v11 cc (i' + 1) `; ASM_SEARCH_TCL [` periodic `;` cc_qu_v11`] MP_TAC; ASM_SEARCH_TCL [` periodic `;` cc_qy_v11`] MP_TAC; ASM_SEARCH_TCL [` periodic `;` cc_gg_v11`] MP_TAC; SIMP_TAC[Oxl_def.periodic]; UNDISCH_TAC` i' = 0 `; SIMP_TAC[ADD;ADD_SYM]; MESON_TAC[ADD; ADD_SYM]; GEN_TAC; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; ASM_CASES_TAC` y' = (nn:num) `; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; ARITH_TAC; GEN_TAC; EXPAND_TAC "sa"; EXPAND_TAC "saa"; REWRITE_TAC[IN_ELIM_THM]; STRIP_TAC; CONJ_TAC; ASM_CASES_TAC` x = (nn: num) `; EXISTS_TAC` 0`; ASM_REWRITE_TAC[ADD]; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; CONJ_TAC; MP_TAC (SPEC_ALL CC_CARD2); ANTS_TAC; ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ASM_REWRITE_TAC[]; ARITH_TAC; FIRST_X_ASSUM SUBST_ALL_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC); ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]); ASM_SEARCH_TCL [` periodic `;` cc_gg_v11 `] MP_TAC; MP_TAC (SPEC_ALL periodic_fn); ANTS_TAC; ASM_REWRITE_TAC[cc_bool_model_v11]; STRIP_TAC; ASM_SEARCH_TCL [` periodic `;` cc_qy_v11 `] MP_TAC; ASM_SEARCH_TCL [` periodic `;` cc_qu_v11 `] MP_TAC; SIMP_TAC[Oxl_def.periodic]; ASM_SEARCH_TCL [` cc_card_v11 cc = nn `] SUBST1_TAC; MESON_TAC[ADD; ADD_SYM]; EXISTS_TAC `x:num `; ASM_REWRITE_TAC[]; UNDISCH_TAC` (x:num) IN S `; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; FIRST_X_ASSUM MP_TAC; ARITH_TAC; ASM_CASES_TAC` x = (nn:num) `; ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` periodic (cc_gg3a_v11 cc) `] MP_TAC; REWRITE_TAC[Oxl_def.periodic]; MESON_TAC[ADD_SYM]; ASM_REWRITE_TAC[]; MP_TAC (SPEC_ALL periodic_fn); ANTS_TAC; ASM_REWRITE_TAC[cc_bool_model_v11]; ABBREV_TAC` nn = cc_card_v11 cc `; STRIP_TAC; SUBGOAL_THEN ` ss DIFF sa SUBSET (sb: num -> bool) ` MP_TAC; UNDISCH_TAC` {i | (i:num) IN S /\ cb i} = ss `; STRIP_TAC; EXPAND_TAC "ss"; EXPAND_TAC "sa"; EXPAND_TAC "sb"; REWRITE_TAC[SUBSET; IN_DIFF; IN_ELIM_THM]; SUBGOAL_THEN` !x. (x IN S /\ cb x) ==> ~(cc_4cell_v11 cc (x + 1) /\ cc_4cell_v11 cc (x - 1))` MP_TAC; GEN_TAC THEN STRIP_TAC; FIRST_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` (x:num) IN S `; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; ARITH_TAC; REWRITE_TAC[cc_qy_v11]; MESON_TAC[]; SUBGOAL_THEN` sum sbb (cc_gg3b_v11 cc) = sum sb (\i. cc_gg3b_v11 cc ( i - 1))` MP_TAC; MATCH_MP_TAC SUM_EQ_GENERAL; EXISTS_TAC`\i. if i = nn then 1 else i + 1 `; CONJ_TAC; REWRITE_TAC[EXISTS_UNIQUE]; GEN_TAC; EXPAND_TAC "sb"; REWRITE_TAC[IN_ELIM_THM]; EXPAND_TAC "sbb"; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG]; STRIP_TAC; ASM_CASES_TAC` y = 1 `; EXISTS_TAC` nn:num`; REWRITE_TAC[IN_ELIM_THM]; CONJ_TAC; ASM_REWRITE_TAC[]; CONJ_TAC; STRIP_TAC; MP_TAC (SPEC_ALL CC_CARD2); ANTS_TAC; ASM_REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ASM_REWRITE_TAC[]; ARITH_TAC; REWRITE_TAC[LE_REFL]; FIRST_X_ASSUM SUBST_ALL_TAC; FIRST_X_ASSUM MP_TAC; FIRST_X_ASSUM MP_TAC; ASM_SEARCH_TCL [` a /\ b <=> c `] MP_TAC; DISCH_THEN (fun x -> REWRITE_TAC[GSYM x]); ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC; ASM_SEARCH_TCL [` periodic (cc_qy_v11 cc) `] MP_TAC; ASM_SEARCH_TCL [` periodic (cc_qu_v11 cc) `] MP_TAC; REWRITE_TAC[Oxl_def.periodic; SUB_REFL]; MESON_TAC[ADD; ADD_SYM]; GEN_TAC; ASM_CASES_TAC` y' = (nn:num) `; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; ARITH_TAC; EXISTS_TAC` y - 1 `; CONJ_TAC; ASSUME_TAC2 (ARITH_RULE` 1 <= y /\ y <= nn ==> ~( y - 1 = nn) `); ASM_REWRITE_TAC[IN_ELIM_THM]; ASSUME_TAC2 (ARITH_RULE` 1 <= y /\ y <= nn ==> y - 1 + 1 = y /\ y - 1 <= nn`); ASM_REWRITE_TAC[]; UNDISCH_TAC` 1 <= y `; UNDISCH_TAC` ~( y = 1) `; ARITH_TAC; GEN_TAC; ASM_CASES_TAC` y' = (nn:num) `; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; STRIP_TAC; FIRST_X_ASSUM MP_TAC; UNDISCH_TAC` 1 <= y `; ARITH_TAC; GEN_TAC; EXPAND_TAC "sbb"; EXPAND_TAC "sb"; EXPAND_TAC "S"; REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG]; STRIP_TAC; ASM_CASES_TAC` x = (nn: num) `; ASM_REWRITE_TAC[]; FIRST_X_ASSUM SUBST_ALL_TAC; CONJ_TAC; CONJ_TAC; UNDISCH_TAC` 1 <= nn `; ARITH_TAC; UNDISCH_TAC` (cb:num -> bool) (nn + 1) `; UNDISCH_TAC` cc_qy_v11 cc nn `; ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]); ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC; ASM_SEARCH_TCL [` periodic (cc_qu_v11 cc) `] MP_TAC; ASM_SEARCH_TCL [` periodic (cc_qy_v11 cc) `] MP_TAC; REWRITE_TAC[Oxl_def.periodic; SUB_REFL]; MESON_TAC[ADD; ADD_SYM]; REWRITE_TAC[SUB_REFL]; ASM_SEARCH_TCL [` periodic (cc_gg3b_v11 cc) `] MP_TAC; REWRITE_TAC[Oxl_def.periodic]; MESON_TAC[ADD; ADD_SYM]; ASM_REWRITE_TAC[ARITH_RULE` (a + 1) - 1 = a /\ 1 <= a + 1`]; UNDISCH_TAC` x <= (nn:num) `; FIRST_X_ASSUM MP_TAC; ARITH_TAC; SIMP_TAC[]; REPEAT STRIP_TAC; SUBGOAL_THEN` sum (ss DIFF sa) (\i. cc_gg3b_v11 cc (i - 1)) <= sum sb (\i. cc_gg3b_v11 cc (i - 1))` MP_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE; ASM_REWRITE_TAC[]; SUBGOAL_THEN` sb SUBSET (S:num -> bool) ` ASSUME_TAC; EXPAND_TAC "sb"; SET_TAC[]; CONJ_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` S: num -> bool `; ASM_REWRITE_TAC[]; EXPAND_TAC "sb"; REWRITE_TAC[IN_DIFF; IN_ELIM_THM]; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i `] NHANH; SIMP_TAC[]; STRIP_TAC; SUBGOAL_THEN` &0 <= sum sa (\i. cc_gg3a_v11 cc (i + 1)) + sum sa (cc_gg_v11 cc) ` ASSUME_TAC; ASM_SIMP_TAC[GSYM SUM_ADD]; MATCH_MP_TAC SUM_POS_LE; ASM_REWRITE_TAC[]; EXPAND_TAC "sa"; REWRITE_TAC[IN_ELIM_THM]; ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[ GSYM x]); ASM_SEARCH_TCL [` x /\ cc_qy_v11 cc (i + 1) ==> a `] MP_TAC; REWRITE_TAC[cc_eps]; MESON_TAC[REAL_ADD_SYM; REAL_ARITH` &0 <= #0.0057`; REAL_LE_TRANS]; SUBGOAL_THEN` &0 <= sum (ss DIFF sa) (\i. cc_gg3b_v11 cc (i - 1)) + sum (ss DIFF sa) (cc_gg_v11 cc) ` ASSUME_TAC; SUBGOAL_THEN` FINITE (ss DIFF (sa:num -> bool)) ` MP_TAC; MATCH_MP_TAC FINITE_DIFF; FIRST_X_ASSUM ACCEPT_TAC; SIMP_TAC[GSYM SUM_ADD]; STRIP_TAC; MATCH_MP_TAC SUM_POS_LE; ASM_REWRITE_TAC[]; UNDISCH_TAC` ss DIFF sa SUBSET (sb:num -> bool) `; REWRITE_TAC[SUBSET]; DISCH_THEN NHANH; EXPAND_TAC "sb"; EXPAND_TAC "S"; REWRITE_TAC[IN_NUMSEG; IN_ELIM_THM]; NHANH (ARITH_RULE` 1 <= x ==> x - 1 + 1 = x `); ASM_SEARCH_TCL [` a /\ b <=> v`] (fun x -> REWRITE_TAC[GSYM x]); ASM_SEARCH_TCL [` cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> a `] MP_TAC; REWRITE_TAC[cc_eps]; MESON_TAC[REAL_ADD_SYM; REAL_ARITH` &0 <= #0.0057`; REAL_LE_TRANS]; REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC); REAL_ARITH_TAC; SUBGOAL_THEN` ss UNION sab SUBSET (S:num -> bool) ` MP_TAC; EXPAND_TAC "ss"; EXPAND_TAC "sab"; SET_TAC[]; NHANH (SET_RULE` s SUBSET S ==> (S DIFF s ) UNION s = S /\ DISJOINT (S DIFF s) s`); ABBREV_TAC` sg = ss UNION (sab:num -> bool) `; STRIP_TAC; SUBGOAL_THEN` FINITE (sg: num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` S:num -> bool `; ASM_REWRITE_TAC[]; SUBGOAL_THEN` FINITE (S DIFF (sg:num -> bool)) ` ASSUME_TAC; MATCH_MP_TAC FINITE_DIFF; FIRST_X_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= sum S (cc_gg_v11 cc) ` MP_TAC; ASM_SIMP_TAC[SUM_UNION]; EXPAND_TAC "S"; REPLICATE_TAC 4 (FIRST_X_ASSUM MP_TAC); ASM_SIMP_TAC[SUM_UNION]; REPEAT STRIP_TAC; MATCH_MP_TAC REAL_LE_ADD; CONJ_TAC; MATCH_MP_TAC SUM_POS_LE; ASM_REWRITE_TAC[]; GEN_TAC; EXPAND_TAC "sg"; EXPAND_TAC "ss"; REWRITE_TAC[IN_DIFF; IN_ELIM_THM; IN_UNION]; ASM_SEARCH_TCL [` a /\ b <=> c `] (fun x -> REWRITE_TAC[GSYM x]); REWRITE_TAC[DE_MORGAN_THM]; IMP_TAC; SIMP_TAC[]; REPEAT STRIP_TAC; UNDISCH_TAC` ~( cc_gg_v11 cc x < &0 ) `; REAL_ARITH_TAC; SUBGOAL_THEN ` cc_bool_model_v11 cc ` MP_TAC; ASM_REWRITE_TAC[cc_bool_model_v11]; NHANH (SPECL [`cc: cc_v11`;` x:num `] QY_QX_QU); ASM_REWRITE_TAC[]; SUBGOAL_THEN` cc_real_model_v11 cc ` MP_TAC; ASM_REWRITE_TAC[cc_real_model_v11]; MESON_TAC[QY_NN0; QX_NN0]; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` sum saa (cc_gg3a_v11 cc) + sum sbb (cc_gg3b_v11 cc) + sum ss (cc_gg_v11 cc) `; ASM_REWRITE_TAC[]; ABBREV_TAC` nn = cc_card_v11 cc `; STRIP_TAC; UNDISCH_TAC` ~( nn = 0 ) `; ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC; PHA; NHANH periodic_sum; STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC ` 1 `)); ASSUME_TAC2 (ARITH_RULE` ~( nn = 0) ==> nn - 1 + 1 = nn `); ASM_REWRITE_TAC[]; STRIP_TAC; UNDISCH_TAC` &0 <= sum S (cc_gg_v11 cc) `; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0)`]; FIRST_X_ASSUM MP_TAC; SIMP_TAC[]]);; let MOD_INJ1 = prove_by_refinement (` ~( n = 0) /\ k < n /\ ~( k = 0) ==> (! x. ~( x MOD n = (x + k) MOD n)) `, [REPEAT STRIP_TAC; MP_TAC (SPECL [` n: num `;` x:num `;` x:num `;` x + k:num`] MOD_INJ); ANTS_TAC; ASM_REWRITE_TAC[IN_NUMSEG]; DOWN_TAC; SIMP_TAC[LE_REFL; ARITH_RULE` ~( n = 0) ==> x <= n - 1 + x `; ARITH_RULE` a <= a + x:num `]; STRIP_TAC; UNDISCH_TAC` k < (n:num) `; ARITH_TAC; UNDISCH_TAC` ~( k = 0) `; ARITH_TAC]);; (* TTTTTT *) (* ========================================== *) (* WKR_COMPTED *) let WKR_COMPTED = prove_by_refinement (`cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 /\ indSet = 0..cc_card_v11 cc - 1 ==> (?i j k. ~(i = j \/ j = k \/ k = i) /\ i IN indSet /\ j IN indSet /\ k IN indSet /\ cc_4cell_v11 cc i /\ cc_4cell_v11 cc j /\ cc_4cell_v11 cc k)`, [STRIP_TAC; ASSUME_TAC2 (SPEC_ALL MTMLSRF); ASSUME_TAC2 Oxl_def.CC_CARD2; DOWN_TAC; REWRITE_TAC[cc_real_model_v11; cc_qu_v11; cc_bool_model_v11]; STRIP_TAC; ASM_CASES_TAC` cc_card_v11 cc = 2 `; ASM_SEARCH_TCL [` x = &2 * pi `] MP_TAC; FIRST_ASSUM SUBST1_TAC; REWRITE_TAC[ARITH_RULE` 2 - 1 = 1 `]; MP_TAC (ARITH_RULE` 0 < 1 /\ 0 <= 1 `); SIMP_TAC[SUM_CLAUSES_RIGHT; SUB_REFL; SUM_SING_NUMSEG]; STRIP_TAC THEN STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; SUBGOAL_THEN` {i MOD nn, (i + 1) MOD nn} SUBSET 0..nn - 1 ` MP_TAC; REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]; UNDISCH_TAC` ~(nn = 0) `; MESON_TAC[Oxl_def.MOD_IN_NUMSEG]; MP_TAC (SPECL [`1 `;` 2 `] (GEN_ALL MOD_INJ1)); ANTS_TAC; ARITH_TAC; MP_TAC (SPECL [` 0`;` 2 - 1 `] CARD_NUMSEG); ASM_REWRITE_TAC[ARITH_RULE` (2 - 1 + 1) - 0 = 2 `]; REPEAT STRIP_TAC; SUBGOAL_THEN` {i MOD 2, (i + 1) MOD 2} = 0..2 - 1` MP_TAC; MATCH_MP_TAC CARD_SUBSET_EQ; ASM_REWRITE_TAC[FINITE_NUMSEG; Geomdetail.CARD_SET2]; DISCH_THEN (ASSUME_TAC o SYM); ASM_SEARCH_TCL [` sum s f = &2 * pi `] MP_TAC; UNDISCH_TAC` nn = 2 `; SIMP_TAC[]; ASM_SIMP_TAC[]; FIRST_X_ASSUM (MP_TAC o (SPEC` i: num `)); SIMP_TAC[Geomdetail.SUM_DIS2]; REPEAT STRIP_TAC; SUBGOAL_THEN` cc_4cell_v11 cc (i MOD 2) /\ cc_4cell_v11 cc ((i + 1) MOD 2) ` MP_TAC; UNDISCH_TAC ` ~( nn = 0 ) `; UNDISCH_TAC` periodic (cc_4cell_v11 cc) nn `; PHA; UNDISCH_TAC` cc_4cell_v11 cc (i) `; UNDISCH_TAC` cc_4cell_v11 cc (i + 1) `; ASM_REWRITE_TAC[]; MESON_TAC[Oxl_def.periodic_mod]; ASM_SEARCH_TCL [` cc_azim_v11 `;` #2.8`] (NHANH_PAT` \x. x ==> y `); DOWN; MP_TAC (prove(` #3.14159 < pi`, REWRITE_TAC[ Flyspeck_constants.bounds])); REAL_ARITH_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; EXISTS_TAC` (i - 1) MOD nn`; EXISTS_TAC` (i) MOD nn`; EXISTS_TAC` (i + 1) MOD nn`; MP_TAC (ISPECL [` 1 `;`nn: num `] (GEN_ALL MOD_INJ1)); MP_TAC (ISPECL [` 2 `;`nn: num `] (GEN_ALL MOD_INJ1)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` ~( 2 = 0) /\ (2 < nn <=> 2 <= nn /\ ~( nn = 2))`]; STRIP_TAC; ANTS_TAC; ASM_SIMP_TAC[ARITH_RULE` ~( 1 = 0) /\ ( 2 <= x ==> 1 < x)`]; STRIP_TAC; FIRST_ASSUM (MP_TAC o (SPEC` i: num `)); FIRST_X_ASSUM (MP_TAC o (SPEC` i - 1 `)); FIRST_ASSUM (MP_TAC o (SPEC` i - 1 `)); UNDISCH_TAC` 0 < i `; SIMP_TAC[ARITH_RULE` 0 < i ==> i - 1 + 1 = i /\ i - 1 + 2 = i + 1 `]; REPLICATE_TAC 4 STRIP_TAC; ASM_SIMP_TAC[MOD_IN_NUMSEG]; ASM_SEARCH_TCL [` periodic `;` cc_4cell_v11 `] MP_TAC; UNDISCH_TAC` cc_4cell_v11 cc (i - 1) `; UNDISCH_TAC` cc_4cell_v11 cc (i) `; UNDISCH_TAC` cc_4cell_v11 cc (i + 1) `; UNDISCH_TAC` ~( nn = 0) `; MESON_TAC[Oxl_def.periodic_mod]]);; let oxl6142 = prove_by_refinement (CHQSQEY_concl, [GEN_TAC; ABBREV_TAC` indSet = 0..cc_card_v11 cc - 1 `; STRIP_TAC; MP_TAC WKR_COMPTED; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; REWRITE_TAC[cc_size_v11]; SUBGOAL_THEN` {i,j,k} SUBSET {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}` MP_TAC; ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM]; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` FINITE {i | i IN indSet /\ cc_4cell_v11 cc i} ` MP_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` indSet:num -> bool `; EXPAND_TAC "indSet"; REWRITE_TAC[FINITE_NUMSEG]; SET_TAC[]; FIRST_X_ASSUM MP_TAC; PHA; NHANH CARD_SUBSET; STRIP_TAC; ASM_SEARCH_TCL [` a \/ b`] MP_TAC; REWRITE_TAC[GSYM Geomdetail.CARD3]; DISCH_THEN SUBST_ALL_TAC; FIRST_X_ASSUM ACCEPT_TAC]);; let CHQSQEY = oxl6142;; let THREE_LE_CC_CARD = prove_by_refinement (` cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> 3 <= cc_card_v11 cc `, [NHANH CHQSQEY; REWRITE_TAC[cc_size_v11]; ABBREV_TAC` sn = 0..cc_card_v11 cc - 1 `; STRIP_TAC; ASSUME_TAC (SET_RULE` {i | i IN sn /\ cc_4cell_v11 cc i} SUBSET sn`); SUBGOAL_THEN` FINITE (sn:num -> bool) ` MP_TAC; EXPAND_TAC "sn"; REWRITE_TAC[FINITE_NUMSEG]; FIRST_X_ASSUM MP_TAC; PHA; NHANH CARD_SUBSET; STRIP_TAC; ABBREV_TAC` c4 = CARD {i | i IN sn /\ cc_4cell_v11 cc i} `; ASSUME_TAC2 (ARITH_RULE` 3 <= c4 /\ c4 <= CARD (sn:num -> bool) ==> 3 <= CARD sn`); FIRST_X_ASSUM MP_TAC; EXPAND_TAC "sn"; REWRITE_TAC[CARD_NUMSEG]; ARITH_TAC]);; let THREE_MOD_CONSECUTIVES = prove_by_refinement ( ` 0 < i /\ 3 <= n ==> ~( (i - 1) MOD n = i MOD n \/ i MOD n = (i + 1) MOD n \/ (i + 1) MOD n = (i - 1) MOD n )`, [NHANH (ARITH_RULE` 3 <= n ==> 2 < n /\ 1 < n /\ ~(n = 0)`); STRIP_TAC; MP_TAC (SPECL [` 2`;` n:num `] (GEN_ALL MOD_INJ1)); ANTS_TAC; ASM_REWRITE_TAC[]; ARITH_TAC; MP_TAC (SPECL [` 1`;` n:num `] (GEN_ALL MOD_INJ1)); ANTS_TAC; ASM_REWRITE_TAC[]; ARITH_TAC; STRIP_TAC; FIRST_ASSUM (ASSUME_TAC o (SPEC` i - 1 `)); FIRST_X_ASSUM (ASSUME_TAC o (SPEC` i:num `)); STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` i - 1 `)); REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC); PHA; ASSUME_TAC2 (ARITH_RULE` 0 < i ==> i - 1 + 1 = i /\ i - 1 + 2 = i + 1 `); ASM_REWRITE_TAC[]; MESON_TAC[]]);; let MOD_PERIOD_BOUNDED = prove_by_refinement ( ` ~( n = 0) /\ ~( k = 0) ==> (! x. (x + k) MOD n = x MOD n ==> n <= k ) `, [REPEAT STRIP_TAC; ASM_CASES_TAC` n <= (k:num) `; ASM_REWRITE_TAC[]; MP_TAC MOD_INJ1; ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` a < (b:num) <=> ~(b <= a ) `]; ASM_MESON_TAC[]]);; let MOD_PERIOD_BOUNDED2 = prove_by_refinement (` ~(nn = 0) /\ m <= k /\ (i + k ) MOD nn IN { (i + x) MOD nn | x < m } ==> nn <= k `, [REWRITE_TAC[IN_ELIM_THM] ; STRIP_TAC ; MP_TAC (SPECL [` nn:num `;` k - (x:num) `] (GEN_ALL MOD_PERIOD_BOUNDED)) ; ANTS_TAC ; ASM_REWRITE_TAC[] ; ASM_ARITH_TAC ; DISCH_THEN (MP_TAC o (SPEC` i + (x:num) `)) ; ANTS_TAC ; ASSUME_TAC2 (ARITH_RULE` x < m /\ m <= (k:num) ==> (i + x) + k - x = i + k `) ; ASM_REWRITE_TAC[] ; ARITH_TAC]);; let CARD_EMPTY = el 0 (CONJUNCTS CARD_CLAUSES);; let CARD_INSERT = el 1 (CONJUNCTS CARD_CLAUSES);; let SWITCH_TAC tm = UNDISCH_TAC tm THEN DISCH_THEN (ASSUME_TAC o GSYM);; (* ===================================== *) let UNPNFVW = prove_by_refinement ( UNPNFVW_concl, [ NHANH THREE_LE_CC_CARD; NHANH MTMLSRF; NHANH CHQSQEY; REPEAT STRIP_TAC; ABBREV_TAC` indSet = 0..cc_card_v11 cc - 1 `; MP_TAC WKR_COMPTED; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; ASSUME_TAC2 (SPEC_ALL LXDEYBO); ABBREV_TAC` sz = cc_size_v11 cc (cc_4cell_v11 cc) ` ; ASM_CASES_TAC` cc_size_v11 cc (cc_4cell_v11 cc) = 3 `; ABBREV_TAC` nn = cc_card_v11 cc `; UNDISCH_TAC` 3 <= nn `; UNDISCH_TAC` 0 < i `; PHA; NHANH THREE_MOD_CONSECUTIVES; REWRITE_TAC[GSYM Geomdetail.CARD3]; ABBREV_TAC` ia = (i - 1) MOD nn `; ABBREV_TAC` ib = i MOD nn `; ABBREV_TAC ` ic = (i + 1) MOD nn `; SUBGOAL_THEN` cc_4cell_v11 cc ia /\ cc_4cell_v11 cc ib /\ cc_4cell_v11 cc ic ` MP_TAC; UNDISCH_TAC` cc_qu_v11 cc i `; REWRITE_TAC[cc_qu_v11]; ASM_SEARCH_TCL [` cc_bool_model_v11 `] MP_TAC; REWRITE_TAC[cc_bool_model_v11]; ASM_REWRITE_TAC[]; ASM_MESON_TAC[periodic_mod]; REPEAT STRIP_TAC; ASSUME_TAC2 (ARITH_RULE` 3 <= nn ==> ~( nn = 0) `); ASM_SEARCH_TCL [` cc_qu_v11 `] MP_TAC; REWRITE_TAC[cc_qu_v11] THEN STRIP_TAC; SUBGOAL_THEN`! i. cc_4cell_v11 cc (i MOD nn) = cc_4cell_v11 cc i ` MP_TAC; GEN_TAC; MATCH_MP_TAC (GSYM periodic_mod); ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` cc_bool_model_v11 `] MP_TAC; SIMP_TAC[cc_bool_model_v11]; ASM_REWRITE_TAC[]; SIMP_TAC[]; STRIP_TAC; SUBGOAL_THEN` {ia, ib, ic} SUBSET indSet:num -> bool ` MP_TAC; REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]; EXPAND_TAC "indSet"; EXPAND_TAC "ia"; EXPAND_TAC "ib"; EXPAND_TAC "ic"; UNDISCH_TAC` ~(nn = 0) `; SIMP_TAC[MOD_IN_NUMSEG]; STRIP_TAC; SUBGOAL_THEN` {ia, ib, ic} SUBSET {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} ` MP_TAC; REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET; IN_ELIM_THM]; FIRST_X_ASSUM MP_TAC; ASM_SIMP_TAC[INSERT_SUBSET]; STRIP_TAC; EXPAND_TAC "ia"; EXPAND_TAC "ib"; EXPAND_TAC "ic"; ASMS_SEARCH_TCL [` X <=> cc_4cell_v11 cc i `] REWRITE_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MP_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` FINITE (indSet:num -> bool) ` ASSUME_TAC; EXPAND_TAC "indSet"; REWRITE_TAC[FINITE_NUMSEG]; SUBGOAL_THEN`{ia, ib, ic} = {i | i IN indSet /\ cc_4cell_v11 cc i} ` ASSUME_TAC; MATCH_MP_TAC CARD_SUBSET_EQ; ASM_REWRITE_TAC[]; CONJ_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` indSet:num -> bool `; ASM_REWRITE_TAC[]; SET_TAC[]; UNDISCH_TAC` cc_size_v11 cc (cc_4cell_v11 cc) = 3 `; REWRITE_TAC[cc_size_v11]; EXPAND_TAC "indSet"; EXPAND_TAC "nn"; SIMP_TAC[]; ASM_CASES_TAC` cc_4cell_v11 cc ( (i + 2) MOD nn) `; SUBGOAL_THEN` (i + 2) MOD nn IN {ia, ib, ic} ` MP_TAC; ASM_REWRITE_TAC[]; REWRITE_TAC[IN_ELIM_THM]; ASM_REWRITE_TAC[]; EXPAND_TAC "indSet"; MATCH_MP_TAC MOD_IN_NUMSEG; FIRST_X_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` nn = 3 ` ASSUME_TAC; FIRST_X_ASSUM MP_TAC; REWRITE_TAC[IN_INSERT; NOT_IN_EMPTY]; STRIP_TAC; FIRST_X_ASSUM MP_TAC; ASSUME_TAC2 (ARITH_RULE` 0 < i ==> i + 2 = (i - 1) + 3 `); ASM_REWRITE_TAC[]; EXPAND_TAC "ia"; STRIP_TAC; MP_TAC (ARITH_RULE` ~( 3 = 0) `); UNDISCH_TAC` ~( nn = 0) `; PHA; NHANH MOD_PERIOD_BOUNDED; STRIP_TAC; REWRITE_TAC[ARITH_RULE` a = 3 <=> a <= 3 /\ 3 <= a `]; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC; EXISTS_TAC` i - 1 `; ASM_REWRITE_TAC[]; DOWN; EXPAND_TAC "ib"; ASSUME_TAC (ARITH_RULE` ~( 2 = 0) `); ASSUME_TAC2 (SPECL [` nn: num `;` 2 `] (GEN_ALL MOD_PERIOD_BOUNDED)); FIRST_ASSUM NHANH; NHANH (ARITH_RULE` n <= 2 ==> ~( 3 <= n ) `); ASM_REWRITE_TAC[]; DOWN; EXPAND_TAC "ic"; ASSUME_TAC (ARITH_RULE` ~( 1 = 0) `); ASSUME_TAC2 (SPECL [` nn: num `;` 1 `] (GEN_ALL MOD_PERIOD_BOUNDED)); REWRITE_TAC[ARITH_RULE` a + 2 = (a + 1) + 1 `]; FIRST_ASSUM NHANH; NHANH (ARITH_RULE` n <= 1 ==> ~( 3 <= n ) `); ASM_REWRITE_TAC[]; SUBGOAL_THEN` CARD (indSet:num -> bool) = nn ` MP_TAC; EXPAND_TAC "indSet"; REWRITE_TAC[CARD_NUMSEG]; UNDISCH_TAC` ~( nn = 0) `; ARITH_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC (ISPECL [` {ia, ib, ic:num} `;` indSet:num -> bool `] CARD_SUBSET_EQ); ANTS_TAC; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[cc_size_v11]; EXPAND_TAC "indSet"; UNDISCH_TAC` nn = 3 `; SIMP_TAC[cc_qy_v11]; NHANH (SET_RULE` { i | i IN S /\ p i } = S ==> {i | i IN S /\ ~( p i)} = {} `); SIMP_TAC[CARD_CLAUSES]; REPEAT STRIP_TAC; ARITH_TAC; ASM_SEARCH_TCL [` cc_bool_prep_v11 `] MP_TAC; REWRITE_TAC[cc_bool_prep_v11; cc_qy_v11]; ASM_SEARCH_TCL [` p ( x MOD n) <=> p x `] (fun x -> ONCE_REWRITE_TAC[GSYM x]) ; DISCH_THEN (ASSUME_TAC2 o (SPEC` (i + 2)`)) ; SUBGOAL_THEN` { ((i - 1) + x) MOD nn | x < 3} = {ia, ib, ic} ` MP_TAC ; REWRITE_TAC[EXTENSION; IN_ELIM_THM] ; GEN_TAC ; REWRITE_TAC[ARITH_RULE` x < 3 <=> x = 0 \/ x = 1 \/ x = 2 `; IN_ELIM_THM; NOT_IN_EMPTY; IN_INSERT] ; EXPAND_TAC "ia" ; EXPAND_TAC "ib" ; EXPAND_TAC "ic" ; UNDISCH_TAC` 0 < i ` ; NHANH (ARITH_RULE` 0 < i ==> (i - 1 ) + 0 = i - 1 /\ ( i - 1) + 1 = i /\ ( i - 1) + 2 = i + 1 `) ; MESON_TAC[]; STRIP_TAC; SUBGOAL_THEN` (i + 3) MOD nn IN {(i - 1 + x) MOD nn | x < 3} ` MP_TAC; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[IN_ELIM_THM]; EXPAND_TAC "indSet"; CONJ_TAC; MATCH_MP_TAC MOD_IN_NUMSEG; FIRST_X_ASSUM ACCEPT_TAC; DOWN THEN DOWN; ASM_REWRITE_TAC[ARITH_RULE` a + 3 = (a + 2 ) + 1 `]; SIMP_TAC[]; ASSUME_TAC2 (ARITH_RULE` 0 < i ==> i + 3 = (i -1) + 4 `); FIRST_ASSUM SUBST1_TAC; MP_TAC (ARITH_RULE` 3 <= 4 `); UNDISCH_TAC` ~( nn = 0) `; PHA; NHANH MOD_PERIOD_BOUNDED2; STRIP_TAC; ABBREV_TAC` id = ((i - 1) + 4) MOD nn `; SUBGOAL_THEN` (! x. x IN {ia, ib, ic} ==> cc_4cell_v11 cc x)` ASSUME_TAC; UNDISCH_TAC` cc_4cell_v11 cc ia `; UNDISCH_TAC` cc_4cell_v11 cc ib `; UNDISCH_TAC` cc_4cell_v11 cc ic `; SET_TAC[]; SUBGOAL_THEN` ~( ((i + 2) MOD nn) IN {ia, ib, ic})` MP_TAC; UNDISCH_TAC` ~cc_4cell_v11 cc (((i + 2)) MOD nn) `; DOWN; MESON_TAC[]; ABBREV_TAC` ie = (i + 2) MOD nn `; STRIP_TAC; MP_TAC (prove(` FINITE {ia, ib, ic:num}`, REWRITE_TAC[Geomdetail.FINITE6])); NHANH (ISPEC`ie: num ` CARD_INSERT); ABBREV_TAC` cells = {i | i IN indSet /\ cc_4cell_v11 cc i} `; ASM_REWRITE_TAC[ADD1]; STRIP_TAC; SUBGOAL_THEN` CARD (indSet:num -> bool) = nn ` ASSUME_TAC; EXPAND_TAC "indSet"; REWRITE_TAC[CARD_NUMSEG]; MATCH_MP_TAC (ARITH_RULE` ~(nn = 0) ==> (nn - 1 + 1) - 0 = nn `); FIRST_X_ASSUM ACCEPT_TAC; SUBGOAL_THEN` {ie, ia, ib, ic:num } SUBSET indSet ` MP_TAC; ONCE_REWRITE_TAC[INSERT_SUBSET]; ASM_REWRITE_TAC[]; EXPAND_TAC "ie"; EXPAND_TAC "indSet"; MATCH_MP_TAC MOD_IN_NUMSEG; FIRST_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` {ie, ia, ib, ic:num} = indSet` MP_TAC; MATCH_MP_TAC CARD_SUBSET_EQ; ASM_REWRITE_TAC[]; SUBGOAL_THEN` CARD ((ie:num) INSERT cells) <= CARD (indSet:num -> bool) ` MP_TAC; MATCH_MP_TAC CARD_SUBSET; DOWN; ASM_SIMP_TAC[]; ASM_REWRITE_TAC[]; UNDISCH_TAC` nn <= 4 `; ARITH_TAC; ASM_REWRITE_TAC[cc_size_v11; cc_qy_v11]; STRIP_TAC; ASSUME_TAC2 ( SET_RULE` cc_4cell_v11 cc ia /\ cc_4cell_v11 cc ib /\ cc_4cell_v11 cc ic /\ {ia, ib, ic} = cells /\ ~cc_4cell_v11 cc ie /\ ie INSERT cells = indSet ==> {i | i IN indSet /\ ~cc_4cell_v11 cc i} = {ie} `); ASM_REWRITE_TAC[]; REWRITE_TAC[LE_REFL; Geomdetail.CARD_SING]; ASM_CASES_TAC` cc_size_v11 cc (cc_qy_v11 cc) <= 1 `; FIRST_X_ASSUM ACCEPT_TAC; DOWN; REWRITE_TAC[ARITH_RULE` ~( x <= 1) <=> 2 <= x `; cc_size_v11]; ASM_REWRITE_TAC[]; ABBREV_TAC` qyset = {i | i IN indSet /\ cc_qy_v11 cc i} `; SUBGOAL_THEN` qyset SUBSET (indSet:num -> bool) ` MP_TAC; EXPAND_TAC "qyset"; SET_TAC[]; STRIP_TAC; SUBGOAL_THEN` FINITE (indSet:num -> bool) ` ASSUME_TAC; EXPAND_TAC "indSet"; REWRITE_TAC[FINITE_NUMSEG]; SUBGOAL_THEN` FINITE (qyset:num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` indSet:num -> bool `; ASM_REWRITE_TAC[]; DOWN; NHANH CHOOSE_SUBSET; STRIP_TAC; FIRST_X_ASSUM NHANH; REWRITE_TAC[HAS_SIZE_2_EXISTS; SET_RULE` (!z. z IN t <=> z = x \/ z = y) <=> {x,y} = t `]; STRIP_TAC; UNDISCH_TAC` ~(cc_size_v11 cc (cc_4cell_v11 cc) = 3) `; ASM_REWRITE_TAC[]; STRIP_TAC; ASSUME_TAC2 (ARITH_RULE` 3 <= sz /\ ~( sz = 3) /\ sz <= 4 ==> sz = 4 `); DOWN; EXPAND_TAC "sz"; REWRITE_TAC[cc_size_v11]; ASM_REWRITE_TAC[]; ABBREV_TAC` cells = {i | i IN indSet /\ cc_4cell_v11 cc i}`; STRIP_TAC; SUBGOAL_THEN` indSet = qyset UNION cells /\ DISJOINT qyset (cells: num -> bool)` MP_TAC; EXPAND_TAC "qyset"; EXPAND_TAC "cells"; REWRITE_TAC[cc_qy_v11]; SET_TAC[]; STRIP_TAC; MP_TAC (SET_RULE` {i | i IN indSet /\ cc_4cell_v11 cc i} SUBSET indSet`); SWITCH_TAC` indSet = qyset UNION (cells: num -> bool) `; ASM_REWRITE_TAC[]; UNDISCH_TAC` FINITE (indSet:num -> bool)`; PHA; NHANH (FINITE_SUBSET); STRIP_TAC; REPLICATE_TAC 27 DOWN; PHA; ASM_SEARCH_TCL [` cc_real_model_v11 `] MP_TAC; ASM_SEARCH_TCL [` cc_bool_model_v11 `] MP_TAC; REWRITE_TAC[cc_real_model_v11; cc_bool_model_v11]; REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum indSet (cc_gg_v11 cc) ` MP_TAC; EXPAND_TAC "indSet"; SUBGOAL_THEN` sum (qyset UNION cells) (cc_gg_v11 cc) = sum qyset (cc_gg_v11 cc) + sum cells (cc_gg_v11 cc) ` SUBST1_TAC; MATCH_MP_TAC SUM_UNION; ASM_REWRITE_TAC[]; SUBGOAL_THEN` ! j. cc_qy_v11 cc j ==> &0 <= cc_gg_v11 cc j ` ASSUME_TAC; REPEAT STRIP_TAC; MATCH_MP_TAC QY_NN0; ASM_REWRITE_TAC[]; SUBGOAL_THEN` sum t (cc_gg_v11 cc) <= sum qyset (cc_gg_v11 cc)` ASSUME_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE; ASM_REWRITE_TAC[]; EXPAND_TAC "qyset"; REWRITE_TAC[IN_DIFF; IN_ELIM_THM]; FIRST_ASSUM NHANH; SIMP_TAC[]; SUBGOAL_THEN` &2 * #0.606 * #0.008 <= sum qyset (cc_gg_v11 cc) ` ASSUME_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` sum t (cc_gg_v11 cc) `; ASM_REWRITE_TAC[]; EXPAND_TAC "t"; SUBGOAL_THEN` sum {x, y} (cc_gg_v11 cc) = cc_gg_v11 cc x + cc_gg_v11 cc y ` SUBST1_TAC; MATCH_MP_TAC Geomdetail.SUM_DIS2; ASM_REWRITE_TAC[]; SUBGOAL_THEN` cc_qy_v11 cc x /\ cc_qy_v11 cc y ` MP_TAC; UNDISCH_TAC` t SUBSET (qyset: num -> bool) ` ; EXPAND_TAC "t"; EXPAND_TAC "qyset"; SET_TAC[]; STRIP_TAC; ASM_SEARCH_TCL [`#0.008 `] (ASSUME_TAC2 o (SPEC` x:num `)); DOWN; ASM_SEARCH_TCL [`#0.008 `] (ASSUME_TAC2 o (SPEC` y:num `)); DOWN; ASM_SEARCH_TCL [`#0.606`] (MP_TAC o (SPEC` x:num `)); ASM_SEARCH_TCL [`#0.606`] (MP_TAC o (SPEC` y:num `)); REAL_ARITH_TAC; SUBGOAL_THEN` ! i. i IN cells ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i` ASSUME_TAC; EXPAND_TAC "cells"; REWRITE_TAC[IN_ELIM_THM]; ASM_SEARCH_TCL [` cc_4cell_v11 cc i ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i`] NHANH; SIMP_TAC[]; SUBGOAL_THEN` sum cells (\i. (\j. a_spine5) i + b_spine5 * cc_azim_v11 cc i) <= sum cells (cc_gg_v11 cc)` MP_TAC; MATCH_MP_TAC SUM_LE; ASM_REWRITE_TAC[]; ASM_SIMP_TAC[SUM_ADD; SUM_CONST; SUM_LMUL]; REWRITE_TAC[ISPECL [` cells:num -> bool `; `cc_azim_v11 cc `] (GEN_ALL SUM_BETA)]; STRIP_TAC; SUBGOAL_THEN` ! f. sum (indSet:num -> bool) f = sum cells f + sum qyset f ` ASSUME_TAC; GEN_TAC; EXPAND_TAC "indSet"; ONCE_REWRITE_TAC[REAL_ADD_SYM]; MATCH_MP_TAC SUM_UNION; ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` x = &2 * pi`] MP_TAC; ASM_SEARCH_TCL [` 0..m = t `] MP_TAC; SIMP_TAC[]; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` sum (qyset DIFF t) (cc_azim_v11 cc) = sum qyset (cc_azim_v11 cc) - sum t (cc_azim_v11 cc) ` ASSUME_TAC; MATCH_MP_TAC SUM_DIFF; ASM_REWRITE_TAC[]; DOWN; EXPAND_TAC "t"; UNDISCH_TAC` ~( x = (y:num)) `; SIMP_TAC[Geomdetail.SUM_DIS2]; STRIP_TAC; SUBGOAL_THEN` !s. FINITE s ==> &0 <= sum s (cc_azim_v11 cc) ` ASSUME_TAC; GEN_TAC; STRIP_TAC; MATCH_MP_TAC SUM_POS_LE; ASM_REWRITE_TAC[]; REPEAT STRIP_TAC; ASM_SEARCH_TCL [` #0.606 <= x `] (MP_TAC o (SPEC` x':num `)); REAL_ARITH_TAC; FIRST_ASSUM (MP_TAC o (SPEC` qyset DIFF {x, y:num}`)); ANTS_TAC; MATCH_MP_TAC FINITE_DIFF; ASM_REWRITE_TAC[]; PHA; NHANH (REAL_ARITH` &0 <= x /\ x = y - z ==> z <= y `); STRIP_TAC; SUBGOAL_THEN` &2 * #0.606 <= sum qyset (cc_azim_v11 cc) ` MP_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` cc_azim_v11 cc x + cc_azim_v11 cc y `; ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` #0.606 <= x `] (MP_TAC o (SPEC` x:num `)); ASM_SEARCH_TCL [` #0.606 <= x `] (MP_TAC o (SPEC` y:num `)); REAL_ARITH_TAC; STRIP_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` &2 * #0.606 * #0.008 + ( &4 * a_spine5 + b_spine5 * sum cells (cc_azim_v11 cc)) ` ; ONCE_REWRITE_TAC[TAUT` a /\ b <=> b /\ a `]; CONJ_TAC; MATCH_MP_TAC REAL_LE_ADD2; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5]; UNDISCH_TAC` sum cells (cc_azim_v11 cc) + sum qyset (cc_azim_v11 cc) = &2 * pi `; SIMP_TAC[REAL_ARITH` a + b = c <=> a = c - b `]; STRIP_TAC; UNDISCH_TAC` &2 * #0.606 <= sum qyset (cc_azim_v11 cc) `; MP_TAC (prove(` pi < #3.1416 `, REWRITE_TAC[ Flyspeck_constants.bounds])); REAL_ARITH_TAC; UNDISCH_TAC` sum indSet (cc_gg_v11 cc) < &0 `; REAL_ARITH_TAC]);; let CC_QU_SMALL_SITUATION = prove_by_refinement(` cc_bool_model_v11 cc /\ cc_real_model_v11 cc ==> !i. cc_qu_v11 cc i /\ (~cc_small_v11 cc (i + 1) \/ ~cc_small_v11 cc (i)) ==> cc_eps <= cc_gg_v11 cc i `, [REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; STRIP_TAC; ASM_SEARCH_TCL [` x ==> cc_small_v11 cc i`] MP_TAC; UNDISCH_TAC ` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc i ==> cc_eps <= cc_gg_v11 cc i`; UNDISCH_TAC`!i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i`; MESON_TAC[]]);; let CC_4CELL_QUQX = prove(` cc_4cell_v11 cc i <=> cc_qu_v11 cc i \/ cc_qx_v11 cc i `, REWRITE_TAC[cc_qx_v11; cc_qu_v11] THEN CONV_TAC TAUT);; let QX_NN00 = prove(` cc_bool_model_v11 cc /\ cc_real_model_v11 cc ==> (! i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) `, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[QX_NN0]);; let QY_NN00 = prove(` cc_bool_model_v11 cc /\ cc_real_model_v11 cc ==> (! i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i) `, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[QY_NN0]);; let NUMSEG_SET_VER = prove(` 0..m = { i | i <= m } `, REWRITE_TAC[numseg] THEN MP_TAC (ARITH_RULE`! x. (0 <= x /\ x <= m <=> x <= m)`) THEN SET_TAC[]);; let FINITE_INITIAL_SEG = REWRITE_RULE[NUMSEG_SET_VER] (SPEC `0` FINITE_NUMSEG);; let FINITE_INITIAL_SEG2 = prove(` FINITE {i | i < (n:num)} `, MATCH_MP_TAC (ISPECL [` {i | i < n:num } `;` {i | i <= (n:num) }`] FINITE_SUBSET) THEN REWRITE_TAC[FINITE_INITIAL_SEG; SUBSET; IN_ELIM_THM; LT_IMP_LE]);; let CARD_MOD_NUMSEG = prove_by_refinement( ` !m. m <= n ==> CARD {(i + k) MOD n | k | k < m} = m `, [INDUCT_TAC; STRIP_TAC; MP_TAC (ARITH_RULE`(!x. ~( x < 0))`); SIMP_TAC[SET_RULE` (!x. ~(x < 0)) ==> {(i + k) MOD n | k < 0} = {} `; CARD_EMPTY]; NHANH (ARITH_RULE` SUC m <= n ==> m <= n `); FIRST_X_ASSUM NHANH; REWRITE_TAC[ARITH_RULE` k < SUC m <=> k < m \/ k = m `]; REWRITE_TAC[SET_RULE` {(i + k) MOD n | k < m \/ k = m} = (i + m) MOD n INSERT {(i + k) MOD n | k < m }`]; STRIP_TAC; SUBGOAL_THEN` ~( (i + m) MOD n IN {(i + k) MOD n | k < m})` ASSUME_TAC; REWRITE_TAC[IN_ELIM_THM]; STRIP_TAC; MP_TAC (let tt = GEN_ALL MOD_INJ1 in SPECL [` m - (k: num) `;` n: num `] tt); ANTS_TAC; ASM_ARITH_TAC; DISCH_THEN (ASSUME_TAC o (SPEC` i + (k:num) `)); DOWN; ASSUME_TAC2 (ARITH_RULE` k < (m:num) ==> (i + k) + m - k = i + m `); ASM_REWRITE_TAC[]; MP_TAC (ISPECL [`\k. ( i + k) MOD n `; `{ k | k < m:num} `] FINITE_IMAGE); ANTS_TAC; REWRITE_TAC[ FINITE_INITIAL_SEG2; IMAGE; IN_ELIM_THM]; REWRITE_TAC[ IMAGE; IN_ELIM_THM]; ASM_REWRITE_TAC[SET_RULE` {y | ?x. x < m /\ y = (i + x) MOD n} = {(i + k) MOD n | k < m} `]; NHANH (ISPEC ` (i + m) MOD n ` CARD_INSERT); ASM_REWRITE_TAC[]; SIMP_TAC[]]);; let MOD_NUMSEG = prove_by_refinement(` ~( n = 0 ) ==> (! i. 0..(n - 1) = {(i + k) MOD n | k | k < n })`, [STRIP_TAC; GEN_TAC; SUBGOAL_THEN` {(i + k) MOD n | k | k < n} SUBSET 0..(n - 1) ` ASSUME_TAC; REWRITE_TAC[SUBSET]; GEN_TAC; REWRITE_TAC[IN_ELIM_THM]; STRIP_TAC; ASM_REWRITE_TAC[]; ASM_SIMP_TAC[MOD_IN_NUMSEG]; MP_TAC (REWRITE_RULE[LE_REFL] (SPEC`n:num ` CARD_MOD_NUMSEG)); STRIP_TAC; MATCH_MP_TAC (GSYM CARD_SUBSET_EQ); ASM_REWRITE_TAC[FINITE_NUMSEG; CARD_NUMSEG]; UNDISCH_TAC` ~( n = 0) `; ARITH_TAC]);; (* IPVICGW_concl;; *) let IPVICGW_C33 = prove_by_refinement(`! cc. (cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 3 ==> (!i. cc_small_v11 cc i) `, [NHANH LXDEYBO; NHANH CHQSQEY; REWRITE_TAC[cc_size_v11]; GEN_TAC; STRIP_TAC THEN ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; DOWN_TAC; ABBREV_TAC` cells = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} `; ABBREV_TAC` nn = cc_card_v11 cc `; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_bool_prep_v11]; STRIP_TAC; SUBGOAL_THEN` cells SUBSET 0..nn - 1 ` ASSUME_TAC; EXPAND_TAC "cells"; SET_TAC[]; ASSUME_TAC (SPECL [` 0`;` nn - 1 `] FINITE_NUMSEG); MP_TAC (SPECL [` 0`;` nn - 1 `] CARD_NUMSEG); ASM_SIMP_TAC[ARITH_RULE`(3 - 1 + 1) - 0 = 3 `]; ASSUME_TAC2 (ISPECL [` cells:num -> bool `;` 0.. nn - 1 `] CARD_SUBSET); REPLICATE_TAC 3 DOWN THEN PHA; ASM_REWRITE_TAC[]; ASM_SIMP_TAC[ARITH_RULE` 3 <= x ==> ( x <= y /\ y = 3 <=> x = 3 /\ y = 3) `]; STRIP_TAC; SUBGOAL_THEN` 0..3 - 1 = cells ` MP_TAC; MATCH_MP_TAC (GSYM CARD_SUBSET_EQ); ASM_REWRITE_TAC[]; STRIP_TAC; GEN_TAC; ASM_CASES_TAC` ! i. i IN cells ==> cc_qy_v11 cc i `; SUBGOAL_THEN` ! i. i IN cells ==> &0 <= cc_gg_v11 cc i ` ASSUME_TAC; REPEAT STRIP_TAC; ASM_SEARCH_TCL [` !i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] MATCH_MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_ASSUM ACCEPT_TAC; SUBGOAL_THEN` &0 <= sum cells (cc_gg_v11 cc) ` ASSUME_TAC; MATCH_MP_TAC SUM_POS_LE; ASM_REWRITE_TAC[]; EXPAND_TAC "cells"; FIRST_X_ASSUM ACCEPT_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; DOWN; ASM_REWRITE_TAC[]; REAL_ARITH_TAC; ASM_CASES_TAC` (!i. i IN cells ==> cc_qx_v11 cc i) `; SUBGOAL_THEN` (!i. i IN cells ==> &0 <= cc_gg_v11 cc i)` MP_TAC; GEN_TAC THEN STRIP_TAC; ASM_SEARCH_TCL [` (!i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) `] MATCH_MP_TAC; FIRST_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM ACCEPT_TAC; UNDISCH_TAC` FINITE (0..3 - 1 ) `; PHA; ASM_REWRITE_TAC[]; NHANH SUM_POS_LE; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; STRIP_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; ASM_REWRITE_TAC[]; DOWN; REWRITE_TAC[ISPECL [`cells: num -> bool `;` cc_gg_v11 cc `] (GEN_ALL SUM_BETA)]; SIMP_TAC[]; DOWN; REWRITE_TAC[MESON[]` ~(! x. P x ==> q x) <=> (? x. P x /\ ~( q x)) `]; STRIP_TAC; SUBGOAL_THEN` cc_4cell_v11 cc i' ` MP_TAC; REPLICATE_TAC 2 DOWN THEN PHA; UNDISCH_TAC` {i | i IN 0..nn - 1 /\ cc_4cell_v11 cc i} = cells `; DISCH_THEN (SUBST1_TAC o GSYM); SIMP_TAC[IN_ELIM_THM]; ASM_REWRITE_TAC[CC_4CELL_QUQX]; ASM_SEARCH_TCL [` !i. cc_qu_v11 cc i ==> --cc_eps <= cc_gg_v11 cc i `] NHANH; REWRITE_TAC[cc_qu_v11; cc_hassmall_v11]; STRIP_TAC; ASM_CASES_TAC` cc_small_v11 cc (i' + 2) `; UNDISCH_TAC` ~( cc_card_v11 cc = 0) `; ASM_REWRITE_TAC[]; NHANH MOD_NUMSEG; ASM_REWRITE_TAC[]; STRIP_TAC; FIRST_X_ASSUM (ASSUME_TAC o (SPEC` i': num `)); SUBGOAL_THEN` ! j. j IN cells ==> cc_small_v11 cc j ` ASSUME_TAC; ASM_REWRITE_TAC[IN_ELIM_THM]; REPEAT STRIP_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` ~( 3 = 0) `; UNDISCH_TAC ` periodic (cc_small_v11 cc) (cc_card_v11 cc) `; ASM_REWRITE_TAC[]; PHA; SIMP_TAC[GSYM Oxl_def.periodic_mod]; REPLICATE_TAC 9 DOWN; REWRITE_TAC[ARITH_RULE` k < 3 <=> k = 0 \/ k = 1 \/ k = 2 `]; MESON_TAC[ADD_0]; UNDISCH_TAC` ~( 3 = 0) `; UNDISCH_TAC ` periodic (cc_small_v11 cc) (cc_card_v11 cc) `; ASM_REWRITE_TAC[]; PHA; NHANH ( GEN_ALL ( SPEC` i:num` (GEN`m:num ` (SPEC_ALL Oxl_def.periodic_mod)))); SIMP_TAC[]; STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; EXPAND_TAC "cells"; MATCH_MP_TAC MOD_IN_NUMSEG; ASM_REWRITE_TAC[]; SUBGOAL_THEN` ~(cc_qu_v11 cc (i' + 1)) /\ ~(cc_qu_v11 cc (i' + 2)) ` MP_TAC; ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (x + 1) + 1 = x + 2 `]; STRIP_TAC; SUBGOAL_THEN`! i. cc_4cell_v11 cc i ` ASSUME_TAC; GEN_TAC; UNDISCH_TAC` ~( cc_card_v11 cc = 0) `; UNDISCH_TAC ` periodic (cc_4cell_v11 cc) (cc_card_v11 cc) `; ASM_REWRITE_TAC[]; PHA; NHANH ( GEN_ALL ( SPEC` i:num` (GEN`m:num ` (SPEC_ALL Oxl_def.periodic_mod)))); SIMP_TAC[]; STRIP_TAC; SUBGOAL_THEN` i MOD 3 IN cells ` MP_TAC; EXPAND_TAC "cells"; MATCH_MP_TAC MOD_IN_NUMSEG; ARITH_TAC; ASM_SEARCH_TCL [` {i | i IN 0..nn - 1 /\ cc_4cell_v11 cc i} = cells `] (SUBST1_TAC o SYM); SET_TAC[]; FIRST_ASSUM (MP_TAC o (SPEC` i' + 1 `)); FIRST_ASSUM (MP_TAC o (SPEC` i' + 2 `)); DOWN; ASM_REWRITE_TAC[CC_4CELL_QUQX]; REWRITE_TAC[GSYM CC_4CELL_QUQX]; REPEAT STRIP_TAC; SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i' + 1 ) ` ASSUME_TAC; ASM_SEARCH_TCL [` !i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i`] MATCH_MP_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `]; SUBGOAL_THEN` cc_qu_v11 cc i' ` MP_TAC; ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11]; ASM_SEARCH_TCL [`!i. cc_qu_v11 cc i ==> --cc_eps <= cc_gg_v11 cc i`] NHANH; ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= r`] (ASSUME_TAC2 o (SPEC` i' + 2 `)); STRIP_TAC; SUBGOAL_THEN` &0 <= sum (i'.. i' + 2) (cc_gg_v11 cc) ` ASSUME_TAC; MP_TAC (ARITH_RULE` i' <= i' + 2 `); SIMP_TAC[SUM_CLAUSES_LEFT]; STRIP_TAC; MP_TAC (ARITH_RULE` i' + 1 <= i' + 2 `); SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` (i + 1) + 1 = i + 2`; SUM_SING_NUMSEG]; STRIP_TAC; UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc i' `; UNDISCH_TAC` cc_eps <= cc_gg_v11 cc (i' + 1) `; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i' + 2) `; PHA; REAL_ARITH_TAC; UNDISCH_TAC` ~( cc_card_v11 cc = 0) `; ASM_SEARCH_TCL [` periodic (cc_gg_v11 cc) `] MP_TAC; PHA; NHANH Oxl_def.periodic_sum; ASM_REWRITE_TAC[]; STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` i':num `)); REWRITE_TAC[ARITH_RULE` 3 - 1 + i = i + 2 `]; DISCH_THEN (ASSUME_TAC o SYM); UNDISCH_TAC` sum (0.. nn - 1 ) (cc_gg_v11 cc) < &0 `; ASM_REWRITE_TAC[REAL_ARITH` a < &0 <=> ~( &0 <= a) `]]);; let UNIQUE_QY = prove_by_refinement (` !cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (! j. cc_qy_v11 cc j /\ j IN 0..cc_card_v11 cc - 1 ==> (! k. ~( k = j) /\ k IN 0..cc_card_v11 cc - 1 ==> ~(cc_qy_v11 cc k))) `, [NHANH UNPNFVW; REWRITE_TAC[cc_size_v11]; REPEAT STRIP_TAC; ABBREV_TAC ` qys = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_qy_v11 cc i} `; SUBGOAL_THEN` {j,k} SUBSET (qys: num -> bool) ` ASSUME_TAC; EXPAND_TAC "qys"; REWRITE_TAC[SUBSET; EMPTY_SUBSET; IN_ELIM_THM; IN_INSERT; NOT_IN_EMPTY]; REPLICATE_TAC 6 DOWN THEN PHA; MESON_TAC[]; ASSUME_TAC (SPECL [` 0`;` cc_card_v11 cc - 1 `] FINITE_NUMSEG); SUBGOAL_THEN` FINITE (qys: num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` 0..cc_card_v11 cc - 1 `; ASM_REWRITE_TAC[]; EXPAND_TAC "qys"; SET_TAC[]; SUBGOAL_THEN` CARD {j:num, k} <= CARD (qys:num -> bool) ` MP_TAC; MATCH_MP_TAC CARD_SUBSET; ASM_REWRITE_TAC[]; STRIP_TAC; UNDISCH_TAC` ~( k = (j:num)) `; REWRITE_TAC[]; ASM_CASES_TAC` k = (j:num) `; FIRST_X_ASSUM ACCEPT_TAC; DOWN; REWRITE_TAC[GSYM Geomdetail.CARD2]; DOWN; REWRITE_TAC[INSERT_COMM]; PHA; NHANH (ARITH_RULE` a <= b /\ a = 2 ==> ~( b <= 1 )`); ASM_REWRITE_TAC[]]);; let UNIQUE_QY2 = prove(`!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (! j. cc_qy_v11 cc j /\ j IN 0..cc_card_v11 cc - 1 ==> (! k. cc_qy_v11 cc k/\ k IN 0..cc_card_v11 cc - 1 ==> k = j)) `, NHANH UNIQUE_QY THEN GEN_TAC THEN STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN MESON_TAC[]);; let ASM_SEARCH_TC = asms_search;; let IPVICGW_C43_C44 = prove_by_refinement (`!cc. (cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 4 ==> (!i. cc_small_v11 cc i) `, [ NHANH CHQSQEY; NHANH LXDEYBO; GEN_TAC THEN STRIP_TAC; ASSUME_TAC2 QX_NN00; ASSUME_TAC2 QY_NN00; SUBGOAL_THEN` cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ` MP_TAC; ASM_REWRITE_TAC[]; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11; cc_bool_prep_v11]; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; ASM_CASES_TAC` ! i. cc_small_v11 cc i `; FIRST_X_ASSUM ACCEPT_TAC; FIRST_X_ASSUM MP_TAC; REWRITE_TAC[NOT_FORALL_THM]; STRIP_TAC; SUBGOAL_THEN` ~ cc_qu_v11 cc i /\ ~ cc_qu_v11 cc ( i + 3) ` MP_TAC; ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; DE_MORGAN_THM]; DISJ1_TAC; DISJ2_TAC; ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC; REWRITE_TAC[periodic]; ASM_SIMP_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; STRIP_TAC; MP_TAC (SPEC` i:num` QU_OR_QXY); MP_TAC (SPEC` i + 3` QU_OR_QXY); ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[DISJ_SYM]; STRIP_TAC; STRIP_TAC; DOWN THEN DOWN; ASM_SEARCH_TCL [` periodic (cc_qy_v11 cc) `] MP_TAC; ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `; `nn:num `;` i:num `] Oxl_def.periodic_mod); ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `; `nn:num `;` i + 3`] Oxl_def.periodic_mod); REPLICATE_TAC 2 (FIRST_X_ASSUM SUBST1_TAC); ASM_REWRITE_TAC[]; STRIP_TAC THEN STRIP_TAC; MP_TAC (SPEC_ALL UNIQUE_QY2); ANTS_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; ASM_REWRITE_TAC[]; DISCH_THEN (MP_TAC o (SPEC` i MOD 4 `)); ANTS_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC MOD_IN_NUMSEG; ARITH_TAC; DISCH_THEN (MP_TAC o (SPEC` (i + 3) MOD 4 `)); ANTS_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC MOD_IN_NUMSEG; ARITH_TAC; MP_TAC (let th = GEN_ALL MOD_INJ1 in SPECL [` 3 `;` 4 `] th); ANTS_TAC; ARITH_TAC; SIMP_TAC[]; SUBGOAL_THEN` ~ cc_qy_v11 cc ((i + 1) MOD 4) /\ ~( cc_qy_v11 cc ((i + 2) MOD 4)) ` ASSUME_TAC; MP_TAC (SPEC_ALL UNIQUE_QY); ANTS_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; ASM_REWRITE_TAC[]; DISCH_THEN (MP_TAC o (SPEC`(i + 3) MOD 4 `)); ANTS_TAC; ASM_REWRITE_TAC[]; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[GSYM periodic_mod; MOD_IN_NUMSEG]; STRIP_TAC; FIRST_ASSUM (MP_TAC o (SPEC` (i + 1) MOD 4 `)); ANTS_TAC; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[MOD_IN_NUMSEG]; MP_TAC (let t = GEN_ALL MOD_INJ1 in SPECL [`2`; ` 4 `] t); ANTS_TAC; ARITH_TAC; DISCH_THEN (MP_TAC o (SPEC` i + 1 `)); ASM_SIMP_TAC[ARITH_RULE` (i + 1) + 2 = i + 3 `]; FIRST_X_ASSUM (MP_TAC o (SPEC` (i + 2) MOD 4 `)); ANTS_TAC; SWITCH_TAC` nn = 4 `; MP_TAC (let t = GEN_ALL MOD_INJ1 in SPECL [` 1 `;` 4 `] t); ANTS_TAC; ARITH_TAC; ASM_SIMP_TAC[ARITH_RULE` i + 3 = (i + 2) + 1 `; MOD_IN_NUMSEG]; SIMP_TAC[]; MP_TAC (SPEC`(i + 1) MOD 4 ` QU_OR_QXY); MP_TAC (SPEC`(i + 2) MOD 4 ` QU_OR_QXY); ASM_SIMP_TAC[]; ONCE_REWRITE_TAC[DISJ_SYM]; STRIP_TAC THEN STRIP_TAC; REPLICATE_TAC 3 DOWN THEN PHA; SWITCH_TAC` nn = 4 `; ASM_REWRITE_TAC[]; ASM_SIMP_TAC[GSYM periodic_mod]; STRIP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn: num `] periodic_sum); SUBGOAL_THEN` &0 <= sum (i..nn - 1 + i) (cc_gg_v11 cc)` MP_TAC; MATCH_MP_TAC SUM_POS_LE; REWRITE_TAC[FINITE_NUMSEG]; GEN_TAC; EXPAND_TAC "nn"; REWRITE_TAC[IN_NUMSEG; ARITH_RULE` i <= x /\ x <= 4 - 1 + i <=> x = i \/ x = i + 1 \/ x = i + 2 \/ x = i + 3 `]; REPLICATE_TAC 8 DOWN THEN PHA; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] MP_TAC; ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= x `] MP_TAC; MESON_TAC[]; ASM_REWRITE_TAC[REAL_ARITH` &0 <= a <=> ~( a < &0) `]; SUBGOAL_THEN` cc_small_v11 cc (i + 1) /\ cc_small_v11 cc (i + 2) ` MP_TAC; DOWN; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[GSYM periodic_mod]; SIMP_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `; cc_qu_v11; cc_hassmall_v11]; STRIP_TAC; (* =========== *) SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc i ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 1) ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; UNDISCH_TAC` cc_qu_v11 cc ((i + 1) MOD 4 )`; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2)` MP_TAC; ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= x `] MATCH_MP_TAC; UNDISCH_TAC` cc_qx_v11 cc (( i + 2) MOD 4 ) `; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] MATCH_MP_TAC; FIRST_X_ASSUM ACCEPT_TAC; REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (MP_TAC o SYM o (SPEC_ALL)); ASM_SIMP_TAC[]; STRIP_TAC; SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 /\ ((i +1) + 1) + 1 = i + 3 `]; REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 1 = i + 2 `]; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 2) `; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 3) `; UNDISCH_TAC` cc_eps <= cc_gg_v11 cc (i) `; UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc (i + 1) `; PHA; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; REPLICATE_TAC 3 DOWN THEN PHA; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod]; STRIP_TAC; ASM_SEARCH_TCL [` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> xac <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `] (MP_TAC o (SPEC` i + 2 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `]; ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `]; SUBGOAL_THEN` cc_gg3a_v11 cc (i + 3) + cc_gg3b_v11 cc (i + 3) <= cc_gg_v11 cc ( i + 3) ` MP_TAC; FIRST_ASSUM MATCH_MP_TAC; FIRST_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc (i + 3) ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3)` MP_TAC; REPLICATE_TAC 3 DOWN THEN PHA; REAL_ARITH_TAC; ASM_SEARCH_TCL [` cc_qx_v11 cc (i) ==> &0 <= x `] (ASSUME_TAC2 o (SPEC` i + 1 `)); ASM_SEARCH_TCL [` cc_qx_v11 cc (i) ==> &0 <= x `] (ASSUME_TAC2 o (SPEC` i + 1 `)); ASM_SEARCH_TCL [` cc_qx_v11 cc (i) ==> &0 <= x `] (ASSUME_TAC2 o (SPEC` i:num `)); STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1 ) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; `nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i:num `)); EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `]; SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 /\ ((i +1) + 1) + 1 = i + 3 `]; REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 1 = i + 2 `]; DOWN THEN DOWN THEN DOWN THEN PHA; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; REPLICATE_TAC 3 DOWN THEN PHA; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[GSYM Oxl_def.periodic_mod]; STRIP_TAC; (* =================== *) ASM_SEARCH_TCL [` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> xax <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `] (MP_TAC o (SPEC` i + 2 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `]; ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `]; SUBGOAL_THEN` cc_gg3a_v11 cc (i + 3) + cc_gg3b_v11 cc (i + 3) <= cc_gg_v11 cc ( i + 3) ` MP_TAC; FIRST_ASSUM MATCH_MP_TAC; FIRST_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc (i + 3) ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM ACCEPT_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3)` MP_TAC; REPLICATE_TAC 3 DOWN THEN PHA; REAL_ARITH_TAC; STRIP_TAC; SUBGOAL_THEN` cc_small_v11 cc ( i + 1) ` MP_TAC; UNDISCH_TAC` cc_qu_v11 cc (i + 1) `; SIMP_TAC[cc_qu_v11; cc_hassmall_v11]; STRIP_TAC; ASM_SEARCH_TCL [`!i. cc_qx_v11 cc i /\ cc_small_v11 cc (i + 1) /\ ~cc_small_v11 cc i ==> cc_eps <= cc_gg_v11 cc i `] (ASSUME_TAC2 o SPEC_ALL); SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc ( i + 1) ` MP_TAC; FIRST_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM ACCEPT_TAC; REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (MP_TAC o SYM o (SPEC_ALL)); ASM_SIMP_TAC[]; STRIP_TAC; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `]; SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 /\ ((i +1) + 1) + 1 = i + 3 `]; REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 1 = i + 2 `]; REPLICATE_TAC 4 DOWN THEN PHA; DOWN; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= a <=> ~( a < &0) `]; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` MP_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; ASM_SEARCH_TCL [` cc_qu_v11 cc ( i + 1) /\ cc_qy_v11 cc i `] (ASSUME_TAC2 o SPEC_ALL); ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + x <= y `] (ASSUME_TAC2 o SPEC_ALL); ASM_SEARCH_TCL [` x ==> &0 <= cc_gg3a_v11 cc i `] (ASSUME_TAC2 o SPEC_ALL); DOWNS 3; REWRITE_TAC[]; REAL_ARITH_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 2) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 2) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC ` i + 2 ` QU_OR_QXY); ASM_REWRITE_TAC[]; SUBGOAL_THEN`~ cc_small_v11 cc (i + 4) ` MP_TAC; UNDISCH_TAC` periodic (cc_small_v11 cc) nn `; ASM_SIMP_TAC[Oxl_def.periodic]; ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH; STRIP_TAC; REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 2) + 1 = i + 3 `]; STRIP_TAC; ASM_SEARCH_TCL [` !i. cc_qx_v11 cc i /\ cc_small_v11 cc (i) /\ ~cc_small_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i ` ] (MP_TAC o (SPEC` i + 3 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; DOWN; REAL_ARITH_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; SWITCH_TAC ` nn = 4 `; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; ` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (SUBST1_TAC o SYM o SPEC_ALL); FIRST_X_ASSUM (SUBST1_TAC o SYM); SIMP_TAC[ARITH_RULE`4 - 1 + i = i + 3 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + (j:num) /\ i + 1 <= i + 3 /\ (i + 1) + 1 <= i + 3 `; LE_TRANS]; REWRITE_TAC[ARITH_RULE` ((i + 1) + 1) + 1 = i + 3 `; SUM_SING_NUMSEG]; DOWNS 2; REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `]; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; (* pkpk *) SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 1) ` ; SWITCH_TAC` nn = 4 `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `; SWITCH_TAC` nn = 4 `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH; STRIP_TAC; SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc i ` MP_TAC; FIRST_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; DOWNS 2; SIMP_TAC[cc_qu_v11; cc_hassmall_v11]; DOWN; REAL_ARITH_TAC; SUBGOAL_THEN` ~cc_small_v11 cc (i + 4) ` ASSUME_TAC; ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC; SIMP_TAC[Oxl_def.periodic]; SWITCH_TAC` nn = 4 `; ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_CASES_TAC` cc_qy_v11 cc ( i + 2) \/ cc_qx_v11 cc (i + 2)`; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; DOWN THEN STRIP_TAC; ASM_SIMP_TAC[]; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 2 ` QU_OR_QXY); DOWN; SIMP_TAC[DE_MORGAN_THM]; ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH; STRIP_TAC THEN STRIP_TAC; SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_SEARCH_TCL [` !i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i `] MATCH_MP_TAC; UNDISCH_TAC` ~cc_small_v11 cc (i + 4) `; ASM_SIMP_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; UNDISCH_TAC` cc_qu_v11 cc (i + 2) `; SIMP_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 2) + 1 = i + 3 `]; DOWN; REAL_ARITH_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (SUBST1_TAC o SYM o SPEC_ALL); ASM_REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 = i + 2 /\ i + 2 <= i + 3 /\ (i + 2 ) + 1 = i + 3 /\ (i + 1) + 2 = i + 3 `]; REWRITE_TAC[SUM_SING_NUMSEG]; DOWNS 3; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]]);; let EXISTS_QY_CARD5 = prove_by_refinement (` (cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 5 ==> (? i. i < 5 /\ cc_qy_v11 cc i)`, [NHANH UNPNFVW; NHANH LXDEYBO; REWRITE_TAC[cc_size_v11; cc_qy_v11]; MP_TAC (SET_RULE` DISJOINT {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i}`); MP_TAC (SET_RULE` {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} UNION {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} = 0..cc_card_v11 cc - 1`); REPEAT STRIP_TAC; FIRST_X_ASSUM SUBST_ALL_TAC; ABBREV_TAC ` s1 = {i | i IN 0..5 - 1 /\ cc_4cell_v11 cc i}`; ABBREV_TAC ` s2 = {i | i IN 0..5 - 1 /\ ~cc_4cell_v11 cc i}`; SUBGOAL_THEN` CARD (s1 UNION (s2:num -> bool)) = CARD s1 + CARD s2 ` MP_TAC; MATCH_MP_TAC CARD_UNION; ASM_REWRITE_TAC[GSYM DISJOINT]; ASM_REWRITE_TAC[GSYM FINITE_UNION; FINITE_NUMSEG]; ASM_SIMP_TAC[CARD_NUMSEG; ARITH_RULE` (5 - 1 + 1) - 0 = 5 `]; STRIP_TAC; ASSUME_TAC2 (ARITH_RULE` 5 = CARD (s1:num -> bool) + CARD (s2:num -> bool) /\ CARD s1 <= 4 /\ CARD s2 <= 1 ==> CARD s2 = 1 `); DOWN; ASM_CASES_TAC` (s2:num -> bool) = {} `; ASM_REWRITE_TAC[CARD_EMPTY; ARITH_RULE` ~( 0 = 1) `]; DOWN; REWRITE_TAC[GSYM MEMBER_NOT_EMPTY]; FIRST_X_ASSUM (SUBST1_TAC o SYM); EXPAND_TAC "s2"; SIMP_TAC[IN_ELIM_THM; IN_NUMSEG; ARITH_RULE`x <= 5 - 1 <=> x < 5 `; LE_0]]);; let NUMSEG_INTER_22 = prove_by_refinement( ` ~(nn = 0) ==> ( (i <= x /\ x <= i + nn - 1) <=> (? k. k < nn /\ x = i + k )) `, [STRIP_TAC; EQ_TAC; STRIP_TAC; EXISTS_TAC` x - (i:num) `; DOWN_TAC; ARITH_TAC; STRIP_TAC; DOWN_TAC; ARITH_TAC]);; let PERIODIC_IMAGE_EQUAL = prove_by_refinement( ` ~( nn = 0) /\ periodic (f:num -> A) nn ==> IMAGE f (0..nn - 1) = IMAGE f (i..i + nn - 1) `, [NHANH MOD_NUMSEG; STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` i:num `)); SIMP_TAC[]; ASSUME_TAC2 ( REWRITE_RULE[RIGHT_FORALL_IMP_THM] (ISPECL [` f:num -> A `;` nn:num `] Oxl_def.periodic_mod)); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_SIMP_TAC[IMAGE; IN_ELIM_THM; IN_NUMSEG]; ASM_SIMP_TAC[NUMSEG_INTER_22]; DOWN; SET_TAC[]]);; let IMAGE_NUMSEG222 = prove_by_refinement (`~(nn = 0) /\ periodic (f:num -> A) nn ==> ! i. { f k | k IN 0..nn - 1} = { f (i + k) | k IN 0..nn - 1 } `, [STRIP_TAC THEN GEN_TAC THEN DOWN_TAC; NHANH PERIODIC_IMAGE_EQUAL; REWRITE_TAC[IMAGE; IN_NUMSEG; LE_0]; STRIP_TAC; DOWN; ASM_SIMP_TAC[NUMSEG_INTER_22; ARITH_RULE`~(n = 0) ==> (x < (n:num) <=> x <= n - 1) `]; DOWN_TAC; SET_TAC[]]);; let LITTLE_CC_GGB = prove(`(!i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\ ( !i. cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) ==> cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `, STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);; let LITTLE_CC_GGA = prove(` (!i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i) /\ (!i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) ==> cc_qu_v11 cc (i) /\ cc_qy_v11 cc (i + 1) ==> &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `, STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);; let QU_IMP_SMALL = prove(` cc_qu_v11 cc i ==> cc_small_v11 cc i /\ cc_small_v11 cc ( i + 1) `, SIMP_TAC[cc_qu_v11; cc_hassmall_v11]);; let IPVICGW_C5_QX_ALONG_NOT_SMALL = prove_by_refinement( ` !cc. ((cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 5 ) /\ ~cc_small_v11 cc i /\ cc_qx_v11 cc i /\ cc_qx_v11 cc ( i + 4) ==> F `, [REPEAT STRIP_TAC; ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; ASSUME_TAC2 EXISTS_QY_CARD5; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `]; IMP_TAC; IMP_TAC; STRIP_TAC; STRIP_TAC; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; ASSUME_TAC2 (ISPECL [`nn:num `;` cc_qy_v11 cc `] (GEN_ALL IMAGE_NUMSEG222)); SUBGOAL_THEN` cc_qy_v11 cc i' IN {cc_qy_v11 cc k | k IN 0..nn - 1}` MP_TAC; REWRITE_TAC[IN_ELIM_THM]; EXISTS_TAC` i': num `; ASM_REWRITE_TAC[IN_NUMSEG_0; ARITH_RULE` a <= 5 - 1 <=> a < 5 `]; FIRST_ASSUM (SUBST1_TAC o (SPEC` i:num `)); ASM_REWRITE_TAC[IN_ELIM_THM; IN_NUMSEG_0; ARITH_RULE` 5 - 1 = 4 `]; STRIP_TAC; ASM_CASES_TAC` k = 0 `; FIRST_X_ASSUM SUBST_ALL_TAC; DOWN; REWRITE_TAC[ADD_0]; UNDISCH_TAC` cc_qx_v11 cc i `; SIMP_TAC[cc_qx_v11; cc_qy_v11]; ASM_CASES_TAC` k = 4`; FIRST_X_ASSUM SUBST_ALL_TAC; DOWNS 2; UNDISCH_TAC` cc_qx_v11 cc (i + 4) `; SIMP_TAC[cc_qx_v11; cc_qy_v11]; ASM_CASES_TAC` k = 1 `; FIRST_X_ASSUM SUBST_ALL_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) ` MP_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 2) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 2 ) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 2 ` QU_OR_QXY); ASM_SIMP_TAC[]; STRIP_TAC; ASM_SEARCH_TCL [` cc_qu_v11 cc (i + 1) /\ x ==> xpapxa <= cc_gg3b_v11 cc i + cc_gg_v11 cc j `] (MP_TAC o (SPEC ` i + 1 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `]; STRIP_TAC; MP_TAC (SPEC ` i + 1 ` (GEN`i:num ` LITTLE_CC_GGB)); ANTS_TAC; ASM_REWRITE_TAC[]; ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (x + 1) + 1 = x + 2 `]; ASM_SIMP_TAC[ARITH_RULE` (x + 1) + 1 = x + 2 `; Oxl_def.cc_eps]; STRIP_TAC; SWITCH_TAC` nn = 5 `; SUBGOAL_THEN`~ cc_small_v11 cc (i + 5) ` ASSUME_TAC; ASM_SIMP_TAC[ Oxl_def.periodic]; ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC; SIMP_TAC[ Oxl_def.periodic]; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc ( i + 3) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 3 ) ` ; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 3 ` QU_OR_QXY); ASM_REWRITE_TAC[]; NHANH QU_IMP_SMALL; REWRITE_TAC[ARITH_RULE` (i + 3 ) + 1 = i + 4 `]; ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH; STRIP_TAC; ASM_SEARCH_TCL [` a /\ b /\ ~ cc_small_v11 cc (i + 1) ==> cc_eps <= y `] (MP_TAC o (SPEC` i + 4 `)); ANTS_TAC; ASM_SEARCH_TCL [` 5 = n `] (SUBST_ALL_TAC o SYM); ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `]; UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc ( i + 3) `; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i:num `)); EXPAND_TAC "nn"; SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 /\ i <= i + 4 /\ i + 1 <= i + 4 /\ (i +1) + 1 <= i + 4 /\ ((i + 1) + 1) + 1 <= i + 4`; SUM_CLAUSES_LEFT]; REWRITE_TAC[GSYM ADD_ASSOC]; CONV_TAC NUM_REDUCE_CONV; REWRITE_TAC[SUM_SING_NUMSEG]; DOWNS 2; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) `; REAL_ARITH_TAC; ASM_REWRITE_TAC[ARITH_RULE` ~( &0 <= x) <=> x < &0 `]; (* ========== *) ASM_CASES_TAC` k = 2 `; FIRST_X_ASSUM SUBST_ALL_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC (SPEC `i + 1 ` (GEN `i: num ` LITTLE_CC_GGA)); ANTS_TAC; ASM_REWRITE_TAC[]; ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 1) + 1 = i + 2 `]; REWRITE_TAC[Oxl_def.cc_eps; ARITH_RULE` (a + 1) + 1 = a + 2 `]; SUBGOAL_THEN` ~ cc_small_v11 cc ( i + 5) ` ASSUME_TAC; ASM_SEARCH_TCL [` periodic (cc_small_v11 cc) `] MP_TAC; REWRITE_TAC[Oxl_def.periodic]; ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 3 ) ` ; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 3) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 3 ` QU_OR_QXY); ASM_REWRITE_TAC[]; (* ??? *) ASM_SEARCH_TCL [` -- x <= y `] NHANH; REWRITE_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 3) + 1 = i + 4 `]; STRIP_TAC; ASM_SEARCH_TCL [` a /\ b /\ ~ p (i + 1) ==> cc_eps <= x `] (MP_TAC o (SPEC` i + 4 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `]; DOWN; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (MP_TAC o SYM o (SPEC` i :num `)); ASM_SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `]; STRIP_TAC; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; DOWNS 4; DOWN; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `]; ASM_CASES_TAC` k = 3 `; FIRST_X_ASSUM SUBST_ALL_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) + cc_gg_v11 cc (i + 3) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc ( i + 2) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 2) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 2 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC (SPEC` i + 2 ` (GEN`i:num ` LITTLE_CC_GGA)); ANTS_TAC; ASM_REWRITE_TAC[]; ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `]; SIMP_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `; Oxl_def.cc_eps]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_SIMP_TAC[]; ASM_SEARCH_TCL [` x ==> -- y <= z `] NHANH; REWRITE_TAC[cc_qu_v11; cc_hassmall_v11]; STRIP_TAC; ASM_SEARCH_TCL [` a /\ b (i + 1 ) /\ ~ c ==> cc_eps <= x `] (MP_TAC o SPEC_ALL); ANTS_TAC; ASM_REWRITE_TAC[]; DOWN; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 4 ) ` ASSUME_TAC THEN ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] Oxl_def.periodic_sum); FIRST_X_ASSUM (MP_TAC o SYM o (SPEC` i :num `)); ASM_SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `]; STRIP_TAC; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; REMOVE_TAC; DOWNS 4; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `]; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x ) <=> x < &0 `]; DOWNS 5; UNDISCH_TAC` k <= 4 `; ARITH_TAC]);; let CARD5_1074_LE_QYI = prove_by_refinement(` (cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ) /\ (!i. cc_4cell_v11 cc i ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i ) /\ ( !i. cc_qy_v11 cc i ==> #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i ) /\ cc_qy_v11 cc i /\ cc_card_v11 cc = 5 /\ periodic (cc_gg_v11 cc) (cc_card_v11 cc ) /\ (!i. #0.606 <= cc_azim_v11 cc i ) /\ sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi /\ #1.074 <= cc_azim_v11 cc i ==> &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) `, [NHANH UNPNFVW; REWRITE_TAC[TAUT` a /\ b /\ c <=> (a/\b) /\ c `]; IMP_TAC; STRIP_TAC; DISCH_TAC; ABBREV_TAC` S = { i + k | k | k < 5 /\ cc_qy_v11 cc (i + k ) } ` ; SUBGOAL_THEN` i IN (S:num -> bool) ` ASSUME_TAC; EXPAND_TAC "S"; REWRITE_TAC[IN_ELIM_THM]; EXISTS_TAC` 0 `; ASM_REWRITE_TAC[ADD_CLAUSES]; ARITH_TAC; SUBGOAL_THEN` S SUBSET { i + k | k | k < 5 } ` ASSUME_TAC; EXPAND_TAC "S"; CONV_TAC SET_RULE; ASSUME_TAC (SPEC` 5 ` FINITE_NUMSEG_LT ); SUBGOAL_THEN` IMAGE (\k. i + k ) { m | m < 5} = {i + k | k < 5 }` ASSUME_TAC; REWRITE_TAC[EXTENSION; IN_IMAGE; IN_ELIM_THM]; GEN_TAC; MESON_TAC[]; UNDISCH_TAC` FINITE {m | m < 5} `; NHANH (ISPEC `(\x. (i:num) + x ) ` FINITE_IMAGE); ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` sum {i:num} ( cc_azim_v11 cc ) <= sum S ( cc_azim_v11 cc ) ` ASSUME_TAC; MATCH_MP_TAC SUM_SUBSET_SIMPLE; CONJ_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` {i + k | k < 5 } ` ; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]; GEN_TAC THEN STRIP_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` #0.606 `; ASM_REWRITE_TAC[]; REAL_ARITH_TAC; MP_TAC (ISPECL [` cc_gg_v11 cc `;` 5 `] Oxl_def.periodic_sum); ANTS_TAC; UNDISCH_TAC` periodic (cc_gg_v11 cc) (cc_card_v11 cc) `; ASM_SIMP_TAC[]; STRIP_TAC; ARITH_TAC; DISCH_THEN (ASSUME_TAC o SYM o SPEC_ALL); ABBREV_TAC` SS = {i + k | k | k < 5 /\ ~cc_qy_v11 cc (i + k)} `; SUBGOAL_THEN` S INTER (SS: num -> bool) = {} ` ASSUME_TAC; EXPAND_TAC "S"; EXPAND_TAC "SS"; CONV_TAC SET_RULE; SUBGOAL_THEN` { i + k | k | k < 5 } = S UNION (SS: num -> bool)` ASSUME_TAC; EXPAND_TAC "S"; EXPAND_TAC "SS"; CONV_TAC SET_RULE; ASM_REWRITE_TAC[numseg]; SUBGOAL_THEN` {x | i <= x /\ x <= 5 - 1 + i} = {i + k | k < 5 } ` ASSUME_TAC; REWRITE_TAC[EXTENSION; IN_ELIM_THM]; GEN_TAC; EQ_TAC; STRIP_TAC; EXISTS_TAC` x - (i:num) `; DOWN THEN DOWN; ARITH_TAC; STRIP_TAC; DOWNS 2; ARITH_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 UNIQUE_QY; DOWN THEN PHA; DISCH_THEN (MP_TAC o (SPEC` i MOD 5 ` )); ASM_REWRITE_TAC[]; ANTS_TAC; MP_TAC (ARITH_RULE` ~( 5 = 0 ) `); SIMP_TAC[Oxl_def.MOD_IN_NUMSEG]; ASSUME_TAC2 Oxl_def.periodic_fn; DOWN; ASM_REWRITE_TAC[]; STRIP_TAC; DOWN THEN PHA; SIMP_TAC [ GSYM Oxl_def.periodic_mod]; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` cc_qy_v11 cc (i MOD 5 ) ` ASSUME_TAC; ASSUME_TAC2 Oxl_def.periodic_fn; DOWN; ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC (ARITH_RULE` ~( 5 = 0 ) `); DOWN THEN PHA; SIMP_TAC [ GSYM Oxl_def.periodic_mod]; ASM_REWRITE_TAC[]; SUBGOAL_THEN` FINITE (S: num -> bool) ` MP_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` {i + k | k < 5} `; ASM_REWRITE_TAC[]; SUBGOAL_THEN` FINITE (SS: num -> bool) ` MP_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` {i + k | k < 5} `; ASM_REWRITE_TAC[]; CONV_TAC SET_RULE; PHA THEN STRIP_TAC; SUBGOAL_THEN` DISJOINT S (SS: num -> bool) ` MP_TAC; ASM_REWRITE_TAC[DISJOINT]; DOWN THEN DOWN THEN PHA; SIMP_TAC[SUM_UNION]; STRIP_TAC; SUBGOAL_THEN`!i. cc_qy_v11 cc (i MOD 5 ) = cc_qy_v11 cc i ` ASSUME_TAC; MATCH_MP_TAC (* Xwitccn. *) Oxl_def.PERIODIC_PROPERTY; ASSUME_TAC2 Oxl_def.periodic_fn; DOWN; ASM_SIMP_TAC[]; STRIP_TAC; ARITH_TAC; ASSUME_TAC2 Oxl_def.periodic_fn; DOWN THEN ASM_REWRITE_TAC[] THEN STRIP_TAC; SUBGOAL_THEN` S = {i:num } ` ASSUME_TAC; REWRITE_TAC[EXTENSION; IN_INSERT; NOT_IN_EMPTY]; EXPAND_TAC "S"; REWRITE_TAC[IN_ELIM_THM]; GEN_TAC; EQ_TAC; STRIP_TAC; ASM_CASES_TAC` k = 0 `; DOWN THEN DOWN THEN ARITH_TAC; UNDISCH_TAC` !k. ~(k = i MOD 5) /\ k IN 0..5 - 1 ==> ~cc_qy_v11 cc k `; DISCH_THEN (MP_TAC o (SPEC` ( i + k ) MOD 5 `)); ANTS_TAC; CONJ_TAC; MP_TAC (SPECL [` k:num `;` 5 `] Oxl_def.MOD_INJ1_ALT); ANTS_TAC; ASM_REWRITE_TAC[]; ARITH_TAC; SIMP_TAC[]; MATCH_MP_TAC Oxl_def.MOD_IN_NUMSEG; ARITH_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; EXISTS_TAC` 0 `; ASM_REWRITE_TAC[ADD_CLAUSES]; ARITH_TAC; SUBGOAL_THEN` SS = { i + k | k < 5 } DIFF S ` ASSUME_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC (SET_RULE` SS INTER S = {} ==> SS = ( S UNION SS ) DIFF S `); UNDISCH_TAC` (S:num -> bool) INTER SS = {} `; ASM_REWRITE_TAC[INTER_COMM]; SUBGOAL_THEN` SS = { i + k | 0 < k /\ k < 5 } ` ASSUME_TAC; DOWN; FIRST_X_ASSUM SUBST1_TAC; SIMP_TAC[]; DISCH_TAC; REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_DIFF; IN_INSERT; NOT_IN_EMPTY]; GEN_TAC; MESON_TAC[ARITH_RULE` ~( 0 < k ) <=> x + k = x `]; SUBGOAL_THEN` cc_bool_model_v11 cc /\ cc_real_model_v11 cc ` MP_TAC; ASM_REWRITE_TAC[]; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; STRIP_TAC; SUBGOAL_THEN` ! i. i IN SS ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; EXPAND_TAC "SS"; REWRITE_TAC[IN_ELIM_THM]; GEN_TAC THEN STRIP_TAC; ASM_SEARCH_TCL [` !i. cc_4cell_v11 cc i ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `] MATCH_MP_TAC; ASM_REWRITE_TAC[]; DOWN THEN DOWN; SIMP_TAC[cc_qy_v11]; DOWN; UNDISCH_TAC` FINITE (SS: num -> bool) `; PHA; NHANH SUM_LE; UNDISCH_TAC` {i + k | k < 5} = S UNION SS `; DISCH_THEN (ASSUME_TAC o SYM); ASM_REWRITE_TAC[]; UNDISCH_TAC` SS = {i + k | k < 5} DIFF S `; UNDISCH_TAC` SS = {i + k | 0 < k /\ k < 5} `; ASM_REWRITE_TAC[]; DISCH_THEN (ASSUME_TAC o SYM); DISCH_THEN (ASSUME_TAC o SYM); ASM_REWRITE_TAC[SUM_SING; ETA_AX]; SUBGOAL_THEN`! a f. (\(i:num). (a:real) + f i ) = (\i. (\i. a ) i + f i )` ASSUME_TAC; REWRITE_TAC[BETA_THM]; FIRST_X_ASSUM (fun x -> ONCE_REWRITE_TAC[x]); IMP_TAC; ASM_SIMP_TAC[SUM_ADD; Vol1.SUM_CONST2; SUM_LMUL]; SUBGOAL_THEN` CARD {i + k | k < 5 } = 5 ` ASSUME_TAC; MATCH_MP_TAC Topology.CARD_FINITE_SERIES_EQ; GEN_TAC THEN GEN_TAC; ARITH_TAC; SUBGOAL_THEN` CARD ({i + k | k < 5} DIFF {i}) = CARD {i + k | k < 5} - CARD {i} ` ASSUME_TAC; MATCH_MP_TAC CARD_DIFF; ASM_REWRITE_TAC[INSERT_SUBSET]; REWRITE_TAC[EMPTY_SUBSET; IN_ELIM_THM]; EXISTS_TAC` 0 `; CONJ_TAC; ARITH_TAC; ARITH_TAC; DOWN; ASM_REWRITE_TAC[CARD_SING]; STRIP_TAC THEN STRIP_TAC; MP_TAC (GSYM (SPECL [` cc_azim_v11 cc `;` 5`] Oxl_def.periodic_sum)); ANTS_TAC; UNDISCH_TAC` periodic (cc_azim_v11 cc) (cc_card_v11 cc)`; ASM_REWRITE_TAC[]; SIMP_TAC[]; STRIP_TAC; ARITH_TAC; ASM_REWRITE_TAC[ARITH_RULE` 5 - 1 = 4 `]; STRIP_TAC; STRIP_TAC; UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi `; ASM_REWRITE_TAC[ARITH_RULE` 5 - 1 = 4 `]; SUBGOAL_THEN` i.. 4 + i = S UNION SS ` ASSUME_TAC; ASM_REWRITE_TAC[]; REWRITE_TAC[numseg; EXTENSION; IN_ELIM_THM]; GEN_TAC; EQ_TAC; STRIP_TAC; EXISTS_TAC` x - (i:num) `; DOWN THEN DOWN; ARITH_TAC; STRIP_TAC; DOWN THEN DOWN; ARITH_TAC; FIRST_ASSUM SUBST1_TAC; ASSUME_TAC2 (ISPECL [` cc_azim_v11 cc `;` S: num -> bool `;` SS: num -> bool `] SUM_UNION); FIRST_ASSUM SUBST1_TAC; ASM_REWRITE_TAC[SUM_SING; REAL_ARITH` a + b = x <=> b = x - a` ]; STRIP_TAC; UNDISCH_TAC` &4 * a_spine5 + b_spine5 * sum SS (\i. cc_azim_v11 cc i) <= sum SS (cc_gg_v11 cc) `; ASM_REWRITE_TAC[ETA_AX]; UNDISCH_TAC` #1.074 <= cc_azim_v11 cc i `; REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5]; MP_TAC ( REWRITE_RULE[] ( REWRITE_CONV[Flyspeck_constants.bounds]` #3.14159 < pi /\ pi < #3.1416 `)); STRIP_TAC THEN STRIP_TAC; MATCH_MP_TAC (REAL_ARITH` &0 <= x + b ==> b <= c ==> &0 <= x + c `); SUBGOAL_THEN` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; DOWNS 4; REAL_ARITH_TAC]);; let MOD_INJ11 = REWRITE_RULE[GSYM RIGHT_FORALL_IMP_THM] MOD_INJ1;; let PI_BOUNDS = REWRITE_RULE[] (REWRITE_CONV[Flyspeck_constants.bounds]` #3.14159 < pi /\ pi < #3.1416 `);; let CARD5_LT_1074_QYI = prove_by_refinement (` (cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ) /\ cc_qy_v11 cc i /\ cc_card_v11 cc = 5 /\ cc_azim_v11 cc i < #1.074 ==> &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) `, [REPEAT STRIP_TAC; ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; ASSUME_TAC2 EXISTS_QY_CARD5; DOWN_TAC; NHANH Oxl_def.periodic_fn; STRIP_TAC; SUBGOAL_THEN` cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc ` MP_TAC; ASM_REWRITE_TAC[]; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; STRIP_TAC THEN ASSUME_TAC2 UNIQUE_QY; ASM_CASES_TAC` cc_small_eta_v11 cc i \/ cc_small_eta_v11 cc ( i + 1) `; SUBGOAL_THEN` a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; UNDISCH_TAC` !i. cc_qy_v11 cc i /\ cc_small_eta_v11 cc i /\ cc_small_eta_v11 cc (i + 1) ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; UNDISCH_TAC` !i. cc_qy_v11 cc i /\ ~cc_small_eta_v11 cc i /\ cc_small_eta_v11 cc (i + 1) /\ cc_azim_v11 cc i < #1.074 ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; UNDISCH_TAC` !i. cc_qy_v11 cc i /\ cc_small_eta_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1) /\ cc_azim_v11 cc i < #1.074 ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; UNDISCH_TAC` cc_azim_v11 cc i < #1.074 `; UNDISCH_TAC` cc_qy_v11 cc i `; DOWN; MESON_TAC[]; MP_TAC (REWRITE_RULE[RIGHT_FORALL_IMP_THM ] (ISPECL [` cc_qy_v11 cc `;` 5 `] periodic_mod)); ANTS_TAC; UNDISCH_TAC` periodic (cc_qy_v11 cc) (cc_card_v11 cc ) `; ASM_REWRITE_TAC[]; SIMP_TAC[]; STRIP_TAC; ARITH_TAC; DISCH_THEN (ASSUME_TAC o GSYM); UNDISCH_TAC` !j. cc_qy_v11 cc j ==> j IN 0..cc_card_v11 cc - 1 ==> (!k. ~(k = j) ==> k IN 0..cc_card_v11 cc - 1 ==> ~cc_qy_v11 cc k) `; PHA; STRIP_TAC; FIRST_ASSUM (MP_TAC o (SPEC` i MOD 5 ` )); ANTS_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `); ASM_SIMP_TAC[Oxl_def.MOD_IN_NUMSEG]; STRIP_TAC; SUBGOAL_THEN` ! k. 0 < k /\ k < 5 ==> ~cc_qy_v11 cc ((i + k) MOD 5 ) ` ASSUME_TAC; GEN_TAC THEN STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `); ASM_SIMP_TAC[Oxl_def.MOD_IN_NUMSEG]; MATCH_MP_TAC (GSYM MOD_INJ11); ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = x) <=> 0 < x `]; ARITH_TAC; SUBGOAL_THEN` !k. k < 5 ==> a_spine5 + b_spine5 * cc_azim_v11 cc (i + k ) <= cc_gg_v11 cc (i + k ) ` ASSUME_TAC; GEN_TAC; ASM_CASES_TAC` k = 0 `; ASM_REWRITE_TAC[ADD_CLAUSES]; STRIP_TAC; UNDISCH_TAC` !i. cc_4cell_v11 cc i ==> a_spine5 + b_spine5 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; DISCH_THEN MATCH_MP_TAC; UNDISCH_TAC` !k. 0 < k /\ k < 5 ==> ~cc_qy_v11 cc ((i + k) MOD 5) `; ASM_REWRITE_TAC[]; REWRITE_TAC[cc_qy_v11]; DISCH_THEN MATCH_MP_TAC; DOWN THEN DOWN THEN SIMP_TAC[]; ARITH_TAC; MP_TAC (ISPECL [` cc_gg_v11 cc `;` 5 `] Oxl_def.periodic_sum); ANTS_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` periodic (cc_gg_v11 cc) (cc_card_v11 cc) `; ASM_SIMP_TAC[]; ARITH_TAC; DISCH_THEN (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[ARITH_RULE` 5 - 1 = 4 `]; SUBGOAL_THEN` ! j. j IN (i..4 + i) ==> a_spine5 + b_spine5 * cc_azim_v11 cc (j) <= cc_gg_v11 cc (j) ` ASSUME_TAC; GEN_TAC; REWRITE_TAC[IN_NUMSEG]; ABBREV_TAC` ij = j - (i:num) `; NHANH (ARITH_RULE` (i:num) <= j ==> j = i + ( j - i) ` ); ASM_REWRITE_TAC[]; STRIP_TAC; UNDISCH_TAC` j = i + ij:num `; DISCH_THEN SUBST1_TAC; FIRST_X_ASSUM MATCH_MP_TAC; EXPAND_TAC "ij"; UNDISCH_TAC` j <= 4 + i `; ARITH_TAC; UNDISCH_TAC` !j. j IN i..4 + i ==> a_spine5 + b_spine5 * cc_azim_v11 cc j <= cc_gg_v11 cc j ` ; NHANH (REWRITE_RULE[GSYM IN_NUMSEG] SUM_LE_NUMSEG); REWRITE_TAC[SUM_ADD_NUMSEG; ETA_AX;SUM_CONST_NUMSEG; SUM_LMUL]; REWRITE_TAC[ARITH_RULE` ((4 + i) + 1) - i = 5 `]; IMP_TAC; STRIP_TAC; SUBST_ALL_TAC (ARITH_RULE` 5 - 1 = 4 `); ASM_REWRITE_TAC[]; MATCH_MP_TAC (REAL_ARITH` c <= b ==> b <= a ==> c <= a `); UNDISCH_TAC` sum (0..cc_card_v11 cc - 1) (cc_azim_v11 cc) = &2 * pi `; MP_TAC (ISPECL [` cc_azim_v11 cc `;` 5 `] Oxl_def.periodic_sum); ANTS_TAC; UNDISCH_TAC` periodic (cc_azim_v11 cc) (cc_card_v11 cc) ` ; ASM_REWRITE_TAC[]; SIMP_TAC[]; ARITH_TAC; ASM_SIMP_TAC[ARITH_RULE` 5 - 1 + i = 4 + i `]; STRIP_TAC THEN STRIP_TAC; REWRITE_TAC[Sphere.a_spine5; Sphere.b_spine5]; MP_TAC PI_BOUNDS; REAL_ARITH_TAC; UNDISCH_TAC` ~(cc_small_eta_v11 cc i \/ cc_small_eta_v11 cc (i + 1)) `; REWRITE_TAC[DE_MORGAN_THM]; STRIP_TAC; FIRST_ASSUM (MP_TAC o (SPEC` i MOD 5 `)); PHA; MP_TAC ( REWRITE_RULE[RIGHT_FORALL_IMP_THM ] (ISPECL [` cc_qy_v11 cc `;` 5 `] Oxl_def.periodic_mod)); ANTS_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `); ASM_REWRITE_TAC[]; DISCH_THEN (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[]; ANTS_TAC; ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `); ASM_SIMP_TAC[MOD_IN_NUMSEG]; STRIP_TAC; SUBGOAL_THEN` ! k. 0 < k /\ k < 5 ==> ~ cc_qy_v11 cc ( i + k ) ` ASSUME_TAC; GEN_TAC THEN STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` (i + k) MOD 5 `)); ANTS_TAC; ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `); ASM_SIMP_TAC[MOD_IN_NUMSEG]; MATCH_MP_TAC (GSYM MOD_INJ11); ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = k ) <=> 0 < k `]; ASM_SIMP_TAC[]; ARITH_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` ! i. -- cc_eps <= cc_gg_v11 cc i ` ASSUME_TAC; GEN_TAC; ASM_CASES_TAC` cc_qy_v11 cc i'' ` ; DOWN; ASM_SEARCH_TCL [` cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `] NHANH; REWRITE_TAC[Oxl_def.cc_eps]; REAL_ARITH_TAC; ASM_CASES_TAC` cc_qx_v11 cc i'' ` ; DOWN; ASM_SEARCH_TCL [` cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i `] NHANH; REWRITE_TAC[Oxl_def.cc_eps]; REAL_ARITH_TAC; MP_TAC (SPEC` i'':num ` QU_OR_QXY); ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) ` ASSUME_TAC; ASM_CASES_TAC` cc_qu_v11 cc (i + 1 ) ` ; ASM_SEARCH_TCL [`cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc i`;` cc_eps <= x`] (MP_TAC o (SPEC ` i + 1 `)); ANTS_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM (MP_TAC o (SPEC ` i + 2 `)); PHA; MATCH_MP_TAC (REAL_ARITH` &0 <= x ==> -- a <= z /\ a <= y ==> &0 <= x + y + z `); UNDISCH_TAC` cc_qy_v11 cc i `; ASM_REWRITE_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 1 ) `; ASM_CASES_TAC` ~ cc_small_v11 cc ( i + 2 ) ` ; SUBGOAL_THEN` ~ cc_qu_v11 cc ( i + 1) /\ ~ cc_qu_v11 cc ( i + 2) ` ASSUME_TAC; DOWN; SIMP_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 1 ) + 1 = i + 2 `]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); MP_TAC (SPEC` i + 2 ` QU_OR_QXY); ASM_REWRITE_TAC[]; DISCH_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) ` ASSUME_TAC; UNDISCH_TAC` !i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i `; UNDISCH_TAC` !i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `; DOWN; MESON_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 1) ` MP_TAC; UNDISCH_TAC` cc_qx_v11 cc (i + 1) `; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i ` MP_TAC; UNDISCH_TAC` cc_qy_v11 cc i `; ASM_REWRITE_TAC[]; DOWN; REAL_ARITH_TAC; ASM_CASES_TAC` ~ cc_small_v11 cc (i + 1) `; ASM_SEARCH_TCL [` cc_qx_v11 cc i /\ cc_small_v11 cc (i + 1) /\ ~cc_small_v11 cc i ==> cc_eps <= cc_gg_v11 cc i`] (MP_TAC o (SPEC` i + 1 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` ( i + 1) + 1 = i + 2 `]; UNDISCH_TAC` ~ ~cc_small_v11 cc (i + 2) `; SIMP_TAC[]; FIRST_X_ASSUM (MP_TAC o (SPEC ` i + 2 `)); SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; REAL_ARITH_TAC; ASM_SEARCH_TCL [` !i. cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `] (MP_TAC o (SPEC ` i :num ` )); ANTS_TAC; DOWN THEN DOWN; ASM_SIMP_TAC[cc_hassmall_v11; ARITH_RULE` (i + 1) + 1 = i + 2 `]; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC; ASM_REWRITE_TAC[]; REAL_ARITH_TAC; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; SUBGOAL_THEN` ~ cc_qy_v11 cc (i + 1) ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ARITH_TAC; SIMP_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i + cc_gg_v11 cc ( i + 3) + cc_gg_v11 cc ( i + 4 ) ` ASSUME_TAC; ASM_CASES_TAC` ~ cc_small_v11 cc ( i + 4 ) ` ; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) /\ &0 <= cc_gg_v11 cc (i + 4 ) ` ASSUME_TAC; SUBGOAL_THEN` ~cc_qu_v11 cc (i + 3) /\ ~cc_qu_v11 cc ( i + 4) ` MP_TAC; DOWN; SIMP_TAC[cc_qu_v11; cc_hassmall_v11; ARITH_RULE` (i + 3 ) + 1 = i + 4 `]; STRIP_TAC; MP_TAC (SPEC` i + 3 ` QU_OR_QXY); MP_TAC (SPEC` i + 4 ` QU_OR_QXY); ASM_REWRITE_TAC[]; UNDISCH_TAC` !i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i `; UNDISCH_TAC` !i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i `; MESON_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; DOWN; REAL_ARITH_TAC; DOWN; REWRITE_TAC[]; SUBGOAL_THEN` ~cc_small_eta_v11 cc ((i + 5 ) MOD 5 ) ` ASSUME_TAC; MP_TAC ( REWRITE_RULE[RIGHT_FORALL_IMP_THM] ( ISPECL [` cc_small_eta_v11 cc `; ` 5 `] Oxl_def.periodic_mod)); ANTS_TAC; ASM_REWRITE_TAC[]; ABBREV_TAC` nn = cc_card_v11 cc `; SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `); ASM_REWRITE_TAC[]; DISCH_THEN (ASSUME_TAC o GSYM); ONCE_REWRITE_TAC[ARITH_RULE` i + 5 = 1 * 5 + i `]; REWRITE_TAC[MOD_MULT_ADD]; ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC ( REWRITE_RULE[RIGHT_FORALL_IMP_THM] ( ISPECL [` cc_small_eta_v11 cc `; ` 5 `] Oxl_def.periodic_mod)); ANTS_TAC; ASM_REWRITE_TAC[]; ABBREV_TAC` nn = cc_card_v11 cc `; SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `); ASM_REWRITE_TAC[]; DISCH_THEN (ASSUME_TAC o GSYM); UNDISCH_TAC` ~cc_small_eta_v11 cc ((i + 5) MOD 5) `; ASM_REWRITE_TAC[]; STRIP_TAC; ASM_CASES_TAC` cc_qu_v11 cc (i + 4 ) ` ; UNDISCH_TAC` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i `; DISCH_THEN (MP_TAC o (SPEC` i + 4 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 4 ) + 1 = i + 5 `]; SUBGOAL_THEN` --cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; REAL_ARITH_TAC; ASM_CASES_TAC` cc_qx_v11 cc ( i + 4) `; ASM_CASES_TAC`~ cc_small_v11 cc ( i + 5 ) `; UNDISCH_TAC` !i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i `; DISCH_THEN (MP_TAC o (SPEC` i + 4 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 4 ) + 1 = i + 5 ` ]; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc ( i + 3 ) ` MP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; REAL_ARITH_TAC; UNDISCH_TAC` !i. cc_qx_v11 cc i /\ cc_hassmall_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `; DISCH_THEN (MP_TAC o (SPEC` i + 4 `)); ANTS_TAC; DOWN; ASM_SIMP_TAC[cc_hassmall_v11; ARITH_RULE` (i + 4) + 1 = i + 5 `]; STRIP_TAC; SUBGOAL_THEN` cc_qy_v11 cc ((i + 5) MOD 5 ) ` MP_TAC; ONCE_REWRITE_TAC[ARITH_RULE` i + 5 = 1 * 5 + i `]; REWRITE_TAC[MOD_MULT_ADD]; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `]; UNDISCH_TAC` periodic (cc_gg3a_v11 cc) (cc_card_v11 cc) `; ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC (let tt = REWRITE_RULE[RIGHT_FORALL_IMP_THM] Oxl_def.periodic_mod in ISPECL [` cc_gg3a_v11 cc `; ` 5 `] tt); ANTS_TAC; ASM_REWRITE_TAC[]; ARITH_TAC; DISCH_THEN (ASSUME_TAC o GSYM); SUBGOAL_THEN` cc_gg3a_v11 cc ((i + 5) MOD 5) = cc_gg3a_v11 cc (i) ` MP_TAC; ONCE_REWRITE_TAC[ARITH_RULE` i + 5 = 1 * 5 + i `]; REWRITE_TAC[MOD_MULT_ADD]; ASM_REWRITE_TAC[]; ASM_SIMP_TAC[]; STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 3 ) ` MP_TAC; ASM_REWRITE_TAC[]; REAL_ARITH_TAC; SUBGOAL_THEN` ~ cc_qy_v11 cc (( i + 4 ) MOD 5) ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASSUME_TAC (ARITH_RULE` ~( 5 = 0) `); ASM_SIMP_TAC[MOD_IN_NUMSEG]; MATCH_MP_TAC (GSYM MOD_INJ11); ARITH_TAC; MP_TAC (SPEC` i + 4 ` QU_OR_QXY); DOWN; ASM_REWRITE_TAC[]; SIMP_TAC[]; ASM_SEARCH_TCL [` cc_gg3a_v11 cc i + x <= y`] (ASSUME_TAC2 o SPEC_ALL); MP_TAC (ARITH_RULE` ~( 5 = 0) `); UNDISCH_TAC` periodic (cc_gg_v11 cc ) (cc_card_v11 cc) `; ASM_REWRITE_TAC[]; PHA; NHANH Oxl_def.periodic_sum; STRIP_TAC; FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_SIMP_TAC[]; ONCE_REWRITE_TAC[ADD_SYM]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` 5 - 1 = 4 /\ i <= i + 4 /\ i + 1 <= i + 4 /\ i + 2 <= i + 4 /\ i + 3 <= i + 4 /\ i + 4 <= i + 4 /\ (i + 1 ) + 1 = i + 2 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` (i + 1) + 2 <= i + 4 /\ ((i + 1) + 2) + 1 = i + 4`]; REWRITE_TAC[SUM_SING_NUMSEG; ARITH_RULE` (i + 1) + 2 = i + 3 `]; UNDISCH_TAC` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `; UNDISCH_TAC` &0 <= cc_gg3a_v11 cc i + cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) `; UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) `; REAL_ARITH_TAC]);; let PRE_IPVICGW = prove_by_refinement ( ` (cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 5 ==> &0 <= sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) `, [NHANH EXISTS_QY_CARD5; STRIP_TAC; ASM_CASES_TAC` cc_azim_v11 cc i < #1.074 `; ASSUME_TAC2 CARD5_LT_1074_QYI; ASM_REWRITE_TAC[]; MATCH_MP_TAC CARD5_1074_LE_QYI; ASM_REWRITE_TAC[REAL_ARITH` x <= y <=> ~( y < x ) `]; ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; ASSUME_TAC2 EXISTS_QY_CARD5; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `]; IMP_TAC; IMP_TAC; STRIP_TAC; STRIP_TAC; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; ASM_REWRITE_TAC[REAL_ARITH` ~( a < b) <=> b <= a `]; DOWN; SUBST_ALL_TAC (MESON[]` nn = 5 <=> 5 = nn `); ASM_REWRITE_TAC[]]);; let CC_CARD_LESS_THAN_5 = prove_by_refinement(` cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> cc_card_v11 cc <= 5 `, [NHANH LXDEYBO; NHANH UNPNFVW; REWRITE_TAC[cc_size_v11; cc_qy_v11]; MP_TAC (SET_RULE` DISJOINT {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} `); MP_TAC (SET_RULE` {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} UNION {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i} = 0..cc_card_v11 cc - 1 `); REPEAT STRIP_TAC; ABBREV_TAC` s1 = {i | i IN 0..cc_card_v11 cc - 1 /\ ~cc_4cell_v11 cc i} `; ABBREV_TAC` s2 = {i | i IN 0..cc_card_v11 cc - 1 /\ cc_4cell_v11 cc i}`; SUBGOAL_THEN` FINITE (s1:num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` 0.. cc_card_v11 cc - 1 `; REWRITE_TAC[FINITE_NUMSEG]; EXPAND_TAC "s1"; SET_TAC[]; SUBGOAL_THEN` FINITE (s2:num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` 0.. cc_card_v11 cc - 1 `; REWRITE_TAC[FINITE_NUMSEG]; EXPAND_TAC "s2"; SET_TAC[]; MP_TAC (ISPECL [` s1:num -> bool `; ` s2: num -> bool `] CARD_UNION); ANTS_TAC; ASM_REWRITE_TAC[GSYM DISJOINT]; ASM_REWRITE_TAC[CARD_NUMSEG]; ASSUME_TAC2 (prove(` cc_bool_model_v11 cc ==> ~( cc_card_v11 cc = 0) `, SIMP_TAC[cc_bool_model_v11])); ASM_SEARCH_TCL [` x <= 1 `] MP_TAC; ASM_SEARCH_TCL [` x <= 4 `] MP_TAC; ARITH_TAC]);; let IPVICGW = prove_by_refinement(`!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (!i. cc_small_v11 cc i)`, [NHANH CC_CARD_LESS_THAN_5 ; NHANH THREE_LE_CC_CARD; PHA; REWRITE_TAC[ARITH_RULE` 3 <= x /\ x <= 5 <=> x = 3 \/ x = 4 \/ x = 5 `]; MESON_TAC[IPVICGW_C43_C44; IPVICGW_C33; PRE_IPVICGW; REAL_ARITH` a < b ==> ~( b <= a )`]]);; let RSIWAMP = prove_by_refinement ( `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (cc_card_v11 cc <= 4)`, [NHANH CC_CARD_LESS_THAN_5; REWRITE_TAC[ARITH_RULE` a <= 5 <=> a <= 4 \/ a = 5 `]; GEN_TAC THEN STRIP_TAC; ASSUME_TAC2 PRE_IPVICGW; DOWN; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `]]);; let C5_QY_ALONG_NOT_SMALL1 = prove_by_refinement (` (!i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i) /\ (!i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\ (!i. cqu cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) /\ (!i. cc_qx_v11 cc i /\ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i) /\ ( !i. cqu cc i ==> --cc_eps <= cc_gg_v11 cc i ) /\ ~ cc_small_v11 cc i /\ ~cc_small_v11 cc (i + 5) /\ ~cc_small_v11 cc i /\ cc_qy_v11 cc i /\ cqu cc (i + 1) /\ cc_qx_v11 cc (i + 4) /\ ( !cc i. cqu cc i <=> cc_hassmall_v11 cc i /\ cc_4cell_v11 cc i /\ cc_subcrit_v11 cc i ) /\ ( !cc i. cc_hassmall_v11 cc i <=> cc_small_v11 cc i /\ cc_small_v11 cc (i + 1) ) /\ bb = (cqu cc (i + 2) \/ cc_qx_v11 cc (i + 2) \/ cc_qy_v11 cc (i + 2)) /\ bb /\ (! i. cqu cc i \/ cc_qx_v11 cc i \/ cc_qy_v11 cc i ) /\ (((!i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\ (!i. cqu cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) ==> cqu cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1))) ==> &0 <= sum (i..i + 4) (cc_gg_v11 cc) `, [ASM_SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; STRIP_TAC; FIRST_X_ASSUM MP_TAC; ANTS_TAC; ASM_REWRITE_TAC[]; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 2) ` ; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) ` MP_TAC; ASM_SIMP_TAC[]; REWRITE_TAC[Oxl_def.cc_eps]; REAL_ARITH_TAC; ASM_CASES_TAC` cc_qx_v11 cc (i + 2) ` ; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) ` MP_TAC; ASM_SIMP_TAC[]; REWRITE_TAC[Oxl_def.cc_eps]; REAL_ARITH_TAC; UNDISCH_TAC` bb: bool `; FIRST_X_ASSUM SUBST1_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[]; STRIP_TAC THEN STRIP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 3) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 3) ` ; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; FIRST_ASSUM (MP_TAC o (SPEC` i + 3 `)); REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[]; FIRST_X_ASSUM NHANH; ASM_REWRITE_TAC[cc_hassmall_v11; ARITH_RULE` (i + 3) + 1 = i + 4 `]; REPEAT STRIP_TAC; SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i + 4) ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `]; DOWN; REAL_ARITH_TAC; DOWNS 3; REAL_ARITH_TAC]);; let C5_QY_ALONG_NOT_SMALL2 = prove_by_refinement (` (!i. (cqx: cc_v11 -> num -> bool) cc i ==> &0 <= (ggg: cc_v11 -> num -> real) cc i) /\ (!i. cqy cc i ==> &0 <= (ggg: cc_v11 -> num -> real) cc i) /\ (!i. cqy cc i ==> gga cc i + ggb cc i <= (ggg: cc_v11 -> num -> real) cc i) /\ (!i. cqy cc i ==> &0 <= gga cc i) /\ (!i. cqu cc (i + 1) /\ cqy cc i ==> cc_eps <= ggb cc i + (ggg: cc_v11 -> num -> real) cc (i + 1)) /\ (!i. (cqx: cc_v11 -> num -> bool) cc i /\ csmall cc i /\ ~csmall cc (i + 1) ==> cc_eps <= (ggg: cc_v11 -> num -> real) cc i) /\ (!i. cqu cc i ==> --cc_eps <= (ggg: cc_v11 -> num -> real) cc i) /\ ~csmall cc i /\ ~csmall cc (i + 5) /\ ~csmall cc i /\ cqy cc i /\ cqu cc (i + 1) /\ (cqx: cc_v11 -> num -> bool) cc (i + 4) /\ (!cc i. cqu cc i <=> hasmal cc i /\ fcell cc i /\ subcrt cc i) /\ (!cc i. hasmal cc i <=> csmall cc i /\ csmall cc (i + 1)) /\ (bb <=> cqu cc (i + 2) \/ (cqx: cc_v11 -> num -> bool) cc (i + 2) \/ cqy cc (i + 2)) /\ bb /\ (!i. cqu cc i \/ (cqx: cc_v11 -> num -> bool) cc i \/ cqy cc i) /\ ((!i. cqy cc i ==> gga cc i + ggb cc i <= (ggg: cc_v11 -> num -> real) cc i) /\ (!i. cqy cc i ==> &0 <= gga cc i) /\ (!i. cqu cc (i + 1) /\ cqy cc i ==> cc_eps <= ggb cc i + (ggg: cc_v11 -> num -> real) cc (i + 1)) ==> cqu cc (i + 1) /\ cqy cc i ==> cc_eps <= (ggg: cc_v11 -> num -> real) cc i + (ggg: cc_v11 -> num -> real) cc (i + 1)) ==> &0 <= sum (i..i + 4) ((ggg: cc_v11 -> num -> real) cc) `, [ASM_SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; STRIP_TAC; FIRST_X_ASSUM MP_TAC; ANTS_TAC; ASM_REWRITE_TAC[]; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= (ggg: cc_v11 -> num -> real) cc (i + 2) ` ASSUME_TAC; ASM_CASES_TAC` (cqy: cc_v11 -> num -> bool) cc (i + 2) ` ; SUBGOAL_THEN` &0 <= (ggg: cc_v11 -> num -> real) cc ( i + 2) ` MP_TAC; ASM_SIMP_TAC[]; REWRITE_TAC[Oxl_def.cc_eps]; REAL_ARITH_TAC; ASM_CASES_TAC` (cqx: cc_v11 -> num -> bool) cc (i + 2) ` ; SUBGOAL_THEN` &0 <= (ggg: cc_v11 -> num -> real) cc ( i + 2) ` MP_TAC; ASM_SIMP_TAC[]; REWRITE_TAC[Oxl_def.cc_eps]; REAL_ARITH_TAC; UNDISCH_TAC` bb: bool `; FIRST_X_ASSUM SUBST1_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[]; STRIP_TAC THEN STRIP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= (ggg: cc_v11 -> num -> real) cc (i + 3) + (ggg: cc_v11 -> num -> real) cc (i + 4) ` ASSUME_TAC; ASM_CASES_TAC` (cqy: cc_v11 -> num -> bool) cc (i + 3) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` (cqx: cc_v11 -> num -> bool) cc (i + 3) ` ; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; FIRST_ASSUM (MP_TAC o (SPEC` i + 3 `)); REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN SIMP_TAC[]; FIRST_X_ASSUM NHANH; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; REPEAT STRIP_TAC; SUBGOAL_THEN` cc_eps <= (ggg: cc_v11 -> num -> real) cc (i + 4) ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `]; DOWN; REAL_ARITH_TAC; DOWNS 3; REAL_ARITH_TAC]);; let LITTLE_CC_GGAA = prove(` (!i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i) /\ (!i. cc_qx_v11 cc i /\ cc_hassmall_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) ==> cc_qx_v11 cc (i) /\ cc_hassmall_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `, STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);; let LITTLE_CC_GGBB = prove(` (!i. cc_qy_v11 cc i ==> cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\ (!i. cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) ==> cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) `, STRIP_TAC THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN FIRST_X_ASSUM NHANH THEN REAL_ARITH_TAC);; let C5_QX_NEXT_TO_QY = prove_by_refinement(`((cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 5) /\ ~cc_small_v11 cc i /\ cc_qy_v11 cc (i) /\ cc_qx_v11 cc (i + 1) ==> F `, [ REPEAT STRIP_TAC; ASSUME_TAC2 UNIQUE_QY; ASSUME_TAC2 QY_NN00; ASSUME_TAC2 QX_NN00; ASSUME_TAC2 EXISTS_QY_CARD5; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `]; IMP_TAC; IMP_TAC; STRIP_TAC; STRIP_TAC; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc ` ; SUBGOAL_THEN` ~ cc_small_v11 cc (i + nn) ` ASSUME_TAC; ASM_SEARCH_TCL [` periodic ( cc_small_v11 cc) `] MP_TAC; SIMP_TAC[Oxl_def.periodic]; ASM_REWRITE_TAC[]; SUBGOAL_THEN` ~ cc_qu_v11 cc (i + 4 ) ` ASSUME_TAC; SWITCH_TAC` nn = 5 `; ASM_REWRITE_TAC[cc_qu_v11; cc_hassmall_v11 ; ARITH_RULE` (i + 4) + 1 = i + 5 `]; ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num` ;` i:num `] Oxl_def.periodic_mod); ASSUME_TAC2 (ISPECL [` cc_qx_v11 cc `;` nn:num` ;` i + 4 `] Oxl_def.periodic_mod); ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num` ;` i + 4 `] Oxl_def.periodic_mod); SUBGOAL_THEN` ~ cc_qy_v11 cc (i + 4) ` ASSUME_TAC; DOWN THEN DOWN THEN DOWN; REPLICATE_TAC 3 (DISCH_THEN (ASSUME_TAC o SYM)); FIRST_ASSUM (SUBST1_TAC o SYM); SUBGOAL_THEN` cc_qy_v11 cc (i MOD nn) /\ (i MOD nn) IN 0..nn - 1` MP_TAC; SWITCH_TAC` nn = 5 `; ASM_SIMP_TAC[MOD_IN_NUMSEG]; ASM_SEARCH_TCL [` cc_qy_v11 cc x /\ x IN 0..nn - 1 `] NHANH; STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; SWITCH_TAC` nn = 5 `; ASM_SIMP_TAC[MOD_IN_NUMSEG]; MP_TAC (SPECL [` 4 `;` nn:num `] (GEN_ALL MOD_INJ1)); ANTS_TAC; ASM_REWRITE_TAC[]; EXPAND_TAC "nn"; ARITH_TAC; SIMP_TAC[]; MP_TAC (SPEC` i + 4 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; REPLICATE_TAC 5 DOWN; REPLICATE_TAC 3 (DISCH_THEN (ASSUME_TAC o SYM)); STRIP_TAC; SWITCH_TAC` nn = 5 `; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2 ) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 2) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 2) ` ; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 2 ` QU_OR_QXY); ASM_REWRITE_TAC[]; ASM_SEARCH_TCL [` -- cc_eps <= x `] NHANH; NHANH QU_IMP_SMALL; STRIP_TAC; ASM_CASES_TAC` cc_small_v11 cc (i + 1) `; SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC; MP_TAC LITTLE_CC_GGBB; PHA; ANTS_TAC; ASM_REWRITE_TAC[cc_hassmall_v11; ARITH_RULE` (i + 1) + 1 = i + 2 `]; SIMP_TAC[]; DOWNS 3; REAL_ARITH_TAC; SUBGOAL_THEN` cc_eps <= cc_gg_v11 cc (i + 1) ` ASSUME_TAC; ASM_SEARCH_TCL [` a /\ b (i + 1) /\ ~ c ==> cc_eps <= x `;` cc_small_v11 `] MATCH_MP_TAC; ASM_SIMP_TAC[ARITH_RULE` (i + 1 ) + 1 = i + 2 `]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; DOWNS 3; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4 ) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 3) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 3) ` ; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 3 ` QU_OR_QXY); ASM_REWRITE_TAC[]; NHANH QU_IMP_SMALL; REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; STRIP_TAC; ASM_SEARCH_TCL [` a /\ b /\ ~ c (i + 1) ==> cc_eps <= x `] (MP_TAC o (SPEC` i + 4 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 4) + 1 = i + 5 `]; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_SIMP_TAC[]; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num ` ] Oxl_def.periodic_sum); FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC_ALL)); EXPAND_TAC "nn"; SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; DOWNS 2; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `]]);; let LEAST_BOUND_CC_GG = prove(` (!i. cc_qu_v11 cc i ==> --cc_eps <= cc_gg_v11 cc i) /\ (!i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i) /\ (!i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i) ==> (! i. -- cc_eps <= cc_gg_v11 cc i) `, SUBGOAL_THEN` -- cc_eps <= &0 ` MP_TAC THENL [ REWRITE_TAC[Oxl_def.cc_eps] THEN REAL_ARITH_TAC; MESON_TAC[REAL_LE_TRANS; QU_OR_QXY]]);; let C5_QX_NEXT_TO_QY_LEFT = prove_by_refinement(` ((cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 5) /\ ~cc_small_v11 cc i /\ cc_qx_v11 cc i /\ cc_qy_v11 cc (i + 4) /\ cc_qx_v11 cc ( i + 3 ) ==> F `, [ REPEAT STRIP_TAC; ASSUME_TAC2 UNIQUE_QY; ASSUME_TAC2 QY_NN00; ASSUME_TAC2 QX_NN00; ASSUME_TAC2 EXISTS_QY_CARD5; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `]; IMP_TAC; IMP_TAC; STRIP_TAC; STRIP_TAC; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc ` ; SUBGOAL_THEN` ~ cc_small_v11 cc (i + nn) ` ASSUME_TAC; ASM_SEARCH_TCL [` periodic ( cc_small_v11 cc) `] MP_TAC; SIMP_TAC[Oxl_def.periodic]; ASM_REWRITE_TAC[]; SWITCH_TAC` nn = 5 `; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) + cc_gg_v11 cc (i + 4) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc ( i + 2 ) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc ( i + 2 ) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 2 ` QU_OR_QXY); ASM_SIMP_TAC[]; ASM_SEARCH_TCL [` cc_qu_v11 cc i ==> -- x <= y `] NHANH; NHANH QU_IMP_SMALL; REWRITE_TAC[ARITH_RULE` (i + 2) + 1 = i + 3 `]; STRIP_TAC; ASM_CASES_TAC` cc_small_v11 cc (i + 4) `; MP_TAC (SPEC` i + 3 ` (GEN` i:num ` LITTLE_CC_GGAA)); PHA; ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; cc_hassmall_v11]; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; ASSUME_TAC2 LEAST_BOUND_CC_GG; FIRST_X_ASSUM (MP_TAC o (SPEC` i + 2 `)); REAL_ARITH_TAC; ASM_SEARCH_TCL [` a /\ b /\ ~ c (i + 1) ==> cc_eps <= y `] (MP_TAC o (SPEC` i + 3 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 4) ` MP_TAC; ASM_SIMP_TAC[]; UNDISCH_TAC` --cc_eps <= cc_gg_v11 cc (i + 2) ` ; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc ( i + 1) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `; MATCH_MP_TAC REAL_LE_ADD; ASM_SIMP_TAC[]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; NHANH QU_IMP_SMALL; STRIP_TAC; ASM_SEARCH_TCL [` a /\ b (i + 1) /\ ~ c (i) ==> cc_eps <= y `] (ASSUME_TAC2 o (SPEC` i:num `)); ASSUME_TAC2 LEAST_BOUND_CC_GG; FIRST_X_ASSUM (MP_TAC o (SPEC` i + 1 `)); DOWN; REAL_ARITH_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num ` ] Oxl_def.periodic_sum); FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC_ALL)); EXPAND_TAC "nn"; SIMP_TAC[ARITH_RULE` 5 - 1 + i = i + 4 `; SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 4 /\ i + 1 <= i + 4 /\ (i + 1) + 1 = i + 2 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 4 /\ (i + 2) + 1 = i + 3 /\ i + 3 <= i + 4 /\ (i + 3 ) + 1 = i + 4 `; SUM_SING_NUMSEG]; DOWNS 2; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x) <=> x < &0 `]]);; let BOOL_MODEL_PERIODIC_SMALL = prove(` cc_bool_model_v11 cc ==> periodic (cc_small_v11 cc ) (cc_card_v11 cc) `, SIMP_TAC[cc_bool_model_v11]);; let QY_UNIQUE_ANY_SEGMENGT = prove_by_refinement (` cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (!j. cc_qy_v11 cc j ==> (!k. ~(k = j) /\ k IN j..cc_card_v11 cc - 1 + j ==> ~cc_qy_v11 cc k)) `, [NHANH UNIQUE_QY; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11]; REPEAT STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; FIRST_X_ASSUM (MP_TAC o (SPEC` j MOD nn`)); ANTS_TAC; ASM_SIMP_TAC[MOD_IN_NUMSEG]; ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num `;`j:num `] Oxl_def.periodic_mod); FIRST_X_ASSUM (SUBST1_TAC o SYM); FIRST_ASSUM ACCEPT_TAC; MP_TAC (SPECL [` k - (j:num) `;` nn:num `] (GEN_ALL MOD_INJ1)); ANTS_TAC; UNDISCH_TAC` k IN j..nn - 1 + j `; ASM_REWRITE_TAC[IN_NUMSEG; ARITH_RULE` j <= (k:num) <=> j = k \/ j < k `]; UNDISCH_TAC` ~( nn = 0) `; ARITH_TAC; UNDISCH_TAC` k IN j..nn - 1 + j `; ASM_REWRITE_TAC[IN_NUMSEG]; STRIP_TAC; DISCH_THEN (MP_TAC o (SPEC` j:num `)); ASM_SIMP_TAC[ARITH_RULE` j <= k ==> j + k - j = (k:num) `]; STRIP_TAC; STRIP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` k MOD nn `)); ANTS_TAC; ASM_SIMP_TAC[GSYM IN_NUMSEG; MOD_IN_NUMSEG]; ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num `;`k:num `] Oxl_def.periodic_mod); FIRST_X_ASSUM (SUBST1_TAC o SYM); ASM_REWRITE_TAC[]]);; let periodic_mod1 = REWRITE_RULE[RIGHT_FORALL_IMP_THM] periodic_mod;; let LITTLE_CC_GG3BB = prove(` (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i) /\ (!i. cc_qu_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)) /\ (! i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\ (! i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\ cc_qy_v11 cc i ==> &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `, MP_TAC (SPEC` i + 1 ` QU_OR_QXY) THEN MESON_TAC[REAL_ARITH` &0 <= a /\ &0 <= b ==> &0 <= a + b `]);; let MOD_ADD_CANCEL = REWRITE_RULE[ARITH_RULE` 1 * n + p = p + n ` ] ( SPEC` 1 ` MOD_MULT_ADD);; let LITTLE_CC_GG3AA = prove_by_refinement( ` (!i. cc_qy_v11 cc i ==> &0 <= cc_gg3a_v11 cc i) /\ (!i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1)) /\ (! i. cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\ (! i. cc_qx_v11 cc i ==> &0 <= cc_gg_v11 cc i ) /\ periodic (cc_qy_v11 cc ) nn /\ periodic (cc_gg3a_v11 cc ) nn /\ ~( nn = 0) /\ cc_qy_v11 cc i ==> &0 <= cc_gg_v11 cc ( i + nn - 1 ) + cc_gg3a_v11 cc (i) `, [ASM_CASES_TAC` cc_qx_v11 cc (i + nn - 1) \/ cc_qy_v11 cc ( i + nn - 1 ) `; DOWN; MESON_TAC[REAL_ARITH` &0 <= a /\ &0 <= b ==> &0 <= a + b `]; MP_TAC (SPEC` i + nn - 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; ABBREV_TAC` nii = i + nn - 1 `; STRIP_TAC; STRIP_TAC; UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `; DISCH_THEN (MP_TAC o (SPEC` nii: num `)); ANTS_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ARITH_RULE ` ~( nn = 0) ==> (i + nn - 1 ) + 1 = i + nn `); DOWN; ASM_REWRITE_TAC[]; SIMP_TAC[]; ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `;` nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); SUBGOAL_THEN` cc_qy_v11 cc ((i + nn) MOD nn) ` ASSUME_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; DOWN; ASM_SIMP_TAC[]; SUBGOAL_THEN` nii + 1 = i + nn ` ASSUME_TAC; EXPAND_TAC "nii"; UNDISCH_TAC` ~( nn = 0) `; ARITH_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `; `nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); SUBGOAL_THEN` cc_gg3a_v11 cc ((i + nn) MOD nn) = cc_gg3a_v11 cc i ` MP_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; ASM_SIMP_TAC[]]);; let SUM_NUMSEG2 = SIMP_RULE[ARITH_RULE`m <= m + 1`; SUM_SING_NUMSEG] (SPEC` m + 1` (GEN` n:num ` (SPEC_ALL SUM_CLAUSES_LEFT)));; let SUM_NUMSEG33 = SIMP_RULE[ARITH_RULE`m <= m + 2 /\ m + 2 = (m + 1 ) + 1`; SUM_NUMSEG2] (SPEC` m + 2` (GEN` n:num ` (SPEC_ALL SUM_CLAUSES_LEFT)));; let SUM_NUMSEG3 = REWRITE_RULE[ARITH_RULE` (a + 1) + 1 = a + 2 `] SUM_NUMSEG33;; let SUM_NUMSEG44 = SIMP_RULE[ARITH_RULE`m <= m + 3 /\ m + 3 = (m + 1 ) + 2 `; SUM_NUMSEG3] (SPEC` m + 3` (GEN` n:num ` (SPEC_ALL SUM_CLAUSES_LEFT)));; let SUM_NUMSEG4 = REWRITE_RULE[ARITH_RULE` (m + 1) + 2 = m + 3 /\ (m + 1) + 1 = m + 2 `] SUM_NUMSEG44;; let UTEOITF_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (!i. cc_4cell_v11 cc i)`;; let UTEOITF = prove_by_refinement(UTEOITF_concl, [GEN_TAC; NHANH RSIWAMP; STRIP_TAC THEN ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; ASM_CASES_TAC` !i. cc_4cell_v11 cc i `; ASM_REWRITE_TAC[]; DOWN; REWRITE_TAC[NOT_FORALL_THM; GSYM cc_qy_v11]; STRIP_TAC; ASSUME_TAC2 RSIWAMP; ASSUME_TAC2 THREE_LE_CC_CARD; ASSUME_TAC2 IPVICGW; ASSUME_TAC2 UNIQUE_QY; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ABBREV_TAC` nn = cc_card_v11 cc `; STRIP_TAC; ASM_CASES_TAC` nn = 3 `; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i /\ &0 <= cc_gg3b_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) ` ASSUME_TAC; ASM_CASES_TAC` cc_qy_v11 cc (i + 1) `; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 1) ` MP_TAC; ASM_SIMP_TAC[]; UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i `; REAL_ARITH_TAC; ASM_CASES_TAC` cc_qx_v11 cc (i + 1) `; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 1) ` MP_TAC; ASM_SIMP_TAC[]; UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i `; REAL_ARITH_TAC; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; MP_TAC (GSYM ( ISPECL [` cc_qy_v11 cc `;` nn:num `] periodic_mod1)); ANTS_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` nn = 3 `; DISCH_THEN (ASSUME_TAC o SYM); ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` cc_qy_v11 cc ((1 * 3 + i ) MOD 3) ` MP_TAC; REWRITE_TAC[MOD_MULT_ADD]; ASM_REWRITE_TAC[]; MP_TAC LITTLE_CC_GG3AA; ANTS_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc (i) <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; STRIP_TAC THEN STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASSUME_TAC2 ( ISPECL [` cc_gg_v11 cc `;` nn:num `] periodic_sum); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[]; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 /\ 2 + i = i + 2`; SUM_NUMSEG3]; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + nn - 1) + cc_gg3a_v11 cc i `; UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `; UNDISCH_TAC` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `]; PHA; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `]; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;` nn:num `] periodic_sum); FIRST_X_ASSUM (ASSUME_TAC o GSYM); DOWN; FIRST_X_ASSUM (MP_TAC o (SPEC` i MOD nn`)); ASSUME_TAC2 (ISPECL [` cc_qy_v11 cc `; `nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ANTS_TAC; ASM_SIMP_TAC[MOD_IN_NUMSEG]; STRIP_TAC; FIRST_ASSUM (MP_TAC o (SPEC` (i + 1) MOD nn `)); ANTS_TAC; ASM_SIMP_TAC[MOD_IN_NUMSEG]; MATCH_MP_TAC (GSYM MOD_INJ11); ASM_REWRITE_TAC[]; UNDISCH_TAC` 3 <= nn `; ARITH_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; ASM_CASES_TAC` cc_qx_v11 cc (i + 1 ) `; ASM_SEARCH_TCL [` !i. cc_qx_v11 cc (i + 1) /\ cc_hassmall_v11 cc (i + 1) /\ cc_qy_v11 cc i ==> cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1)`] (MP_TAC o (SPEC` i:num `)); ANTS_TAC; ASM_REWRITE_TAC[cc_hassmall_v11]; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg_v11 cc (i + 3 ) + cc_gg3a_v11 cc ((i + 3) + 1 ) ` ASSUME_TAC; ASM_CASES_TAC` cc_qu_v11 cc ( i + 3) `; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ARITH_RULE` ~( nn = 3) /\ 3 <= nn /\ nn <= 4 ==> 4 = nn`); SUBGOAL_THEN` cc_qy_v11 cc ((i + 4 ) MOD 4 ) ` MP_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; ASSUME_TAC2 (ARITH_RULE` ~( nn = 3) /\ 3 <= nn /\ nn <= 4 ==> 4 = nn`); SUBGOAL_THEN` cc_qy_v11 cc ((i + 4) MOD 4 ) ` ASSUME_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; DOWN; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc (i + nn) ` MP_TAC; ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 3) ` MP_TAC; MP_TAC (SPEC` i + 3 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; ASM_SIMP_TAC[]; ASM_SIMP_TAC[]; REAL_ARITH_TAC; ASSUME_TAC2 LEAST_BOUND_CC_GG; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ARITH_RULE` ~( nn = 3) /\ 3 <= nn /\ nn <= 4 ==> 4 = nn`); EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_NUMSEG4]; ASSUME_TAC2 ( ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); SUBGOAL_THEN` cc_gg3a_v11 cc (((i + 3) + 1) MOD 4) = cc_gg3a_v11 cc i ` ASSUME_TAC; REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; DOWN; ASM_REWRITE_TAC[]; DISCH_THEN SUBST_ALL_TAC; UNDISCH_TAC` cc_eps <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 3) + cc_gg3a_v11 cc i`; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc ( i + 2) ` MP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; DOWN; PHA; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `]; MP_TAC (SPEC` i + 1 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; (* ========================== *) ASSUME_TAC2 LEAST_BOUND_CC_GG; DOWN; FIRST_ASSUM (MP_TAC o (SPEC ` (i + 2) MOD nn `)); ANTS_TAC; ASM_SIMP_TAC[MOD_IN_NUMSEG]; MATCH_MP_TAC (GSYM MOD_INJ11); ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = 2) `]; UNDISCH_TAC` 3 <= nn `; ARITH_TAC; STRIP_TAC; FIRST_ASSUM (MP_TAC o (SPEC ` (i + 3) MOD nn `)); ANTS_TAC; ASM_SIMP_TAC[MOD_IN_NUMSEG]; MATCH_MP_TAC (GSYM MOD_INJ11); ASM_REWRITE_TAC[ARITH_RULE` ~( 0 = 3) `]; UNDISCH_TAC` 3 <= nn `; UNDISCH_TAC` ~( nn = 3)`; ARITH_TAC; STRIP_TAC; DOWN THEN DOWN; ASM_REWRITE_TAC[]; STRIP_TAC THEN STRIP_TAC; ASM_CASES_TAC` cc_qx_v11 cc ( i + 2 ) `; SUBGOAL_THEN` &0 <= cc_gg_v11 cc ( i + 2) ` ASSUME_TAC; ASM_SIMP_TAC[]; MP_TAC LITTLE_CC_GG3BB; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; MP_TAC LITTLE_CC_GG3AA; ANTS_TAC; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ARITH_RULE` 3 <= nn /\ nn <= 4 /\ ~( nn = 3) ==> 4 = nn `); EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1 ) (cc_gg_v11 cc) ` MP_TAC; EXPAND_TAC "nn"; ASM_REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 /\ 3 + i = i + 3`]; REWRITE_TAC[SUM_NUMSEG4]; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 3) + cc_gg3a_v11 cc i `; UNDISCH_TAC` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i `; UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `; UNDISCH_TAC` &0 <= cc_gg_v11 cc (i + 2) `; PHA; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0 ) `]; ASM_CASES_TAC` cc_qx_v11 cc ( i + 3) `; ASSUME_TAC2 (ARITH_RULE` 3 <= nn /\ nn <= 4 /\ ~( nn = 3) ==> 4 = nn `); SUBGOAL_THEN` cc_qy_v11 cc ((i + 4) MOD 4) ` ASSUME_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; DOWN; ASM_REWRITE_TAC[]; STRIP_TAC; ASM_SEARCH_TCL [` cc_eps <= x + cc_gg3a_v11 cc i `] (MP_TAC o (SPEC` i + 3 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; cc_hassmall_v11]; STRIP_TAC; MP_TAC LITTLE_CC_GG3BB; ANTS_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC; ASM_SIMP_TAC[]; STRIP_TAC; SUBGOAL_THEN` cc_gg3a_v11 cc ((i + 4) MOD 4) = cc_gg3a_v11 cc i ` MP_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[]; STRIP_TAC; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC; ASM_REWRITE_TAC[]; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `; SUM_NUMSEG4]; SUBGOAL_THEN` cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; UNDISCH_TAC` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `; UNDISCH_TAC` cc_eps <= cc_gg_v11 cc (i + 3) + cc_gg3a_v11 cc ((i + 3) + 1) `; PHA; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; SUBGOAL_THEN` --cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC; ASM_REWRITE_TAC[]; REAL_ARITH_TAC; DOWN; ASM_REWRITE_TAC[ARITH_RULE` &0 <= x <=> ~( x < &0 ) `]; MP_TAC (SPEC` i + 2 ` QU_OR_QXY); MP_TAC (SPEC` i + 3 ` QU_OR_QXY); ASM_REWRITE_TAC[]; STRIP_TAC; STRIP_TAC; SUBGOAL_THEN` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; ASSUME_TAC2 (ISPECL [` cc_azim_v11 cc `;` nn:num `] periodic_sum); ASM_SEARCH_TCL [` &2 * pi `] MP_TAC; FIRST_X_ASSUM (ASSUME_TAC o GSYM); DOWN; SIMP_TAC[]; ASSUME_TAC2 (ARITH_RULE` 3 <= nn /\ nn <= 4 /\ ~( nn = 3) ==> 4 = nn `); EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 /\ 4 - 1 = 3 `]; REPEAT STRIP_TAC; UNDISCH_TAC` sum (i..i + 3) (cc_azim_v11 cc) = &2 * pi `; REWRITE_TAC[SUM_NUMSEG4]; STRIP_TAC; ASM_CASES_TAC` #2.089 <= cc_azim_v11 cc i ` ; ASM_SEARCH_TCL [`#0.161517`] (ASSUME_TAC2 o (SPEC` i + 1 `)); ASM_SEARCH_TCL [`!i. cc_qu_v11 cc i ==> #0.161517 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i`] (ASSUME_TAC2 o (SPEC` i + 2 `)); ASM_SEARCH_TCL [`!i. cc_qu_v11 cc i ==> #0.161517 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i`] (ASSUME_TAC2 o (SPEC` i + 3 `)); SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; ASM_REWRITE_TAC[SUM_NUMSEG4]; DOWNS 4; DOWN; UNDISCH_TAC` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; MP_TAC (PI_BOUNDS); REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; ASM_CASES_TAC` cc_azim_v11 cc i <= #1.946 `; UNDISCH_TAC` !i. cc_qu_v11 cc i ==> -- #0.0659 + #0.042 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; STRIP_TAC; FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 1 `)); FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 2 `)); FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i + 3 `)); SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; ASM_REWRITE_TAC[SUM_NUMSEG4]; DOWNS 3; UNDISCH_TAC` cc_azim_v11 cc i <= #1.946 `; UNDISCH_TAC` cc_azim_v11 cc i + cc_azim_v11 cc (i + 1) + cc_azim_v11 cc (i + 2) + cc_azim_v11 cc (i + 3) = &2 * pi `; MP_TAC PI_BOUNDS; UNDISCH_TAC` #0.008 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; ASM_CASES_TAC` cc_small_eta_v11 cc i /\ cc_small_eta_v11 cc (i + 1) `; UNDISCH_TAC` !i. cc_qy_v11 cc i /\ cc_small_eta_v11 cc i /\ cc_small_eta_v11 cc (i + 1) /\ #1.946 <= cc_azim_v11 cc i /\ cc_azim_v11 cc i <= #2.089 ==> #3.0 * cc_eps <= cc_gg_v11 cc i `; DISCH_THEN (MP_TAC o (SPEC` i:num `)); ANTS_TAC; ASM_REWRITE_TAC[]; DOWNS 3; REAL_ARITH_TAC; STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 1) /\ -- cc_eps <= cc_gg_v11 cc (i + 2) /\ -- cc_eps <= cc_gg_v11 cc (i + 3) ` MP_TAC; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..nn - 1) (cc_gg_v11 cc ) ` MP_TAC; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; ASM_REWRITE_TAC[SUM_NUMSEG4]; DOWNS 4; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` &0 <= x <=> ~( x < &0) `]; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; ASM_REWRITE_TAC[]; REWRITE_TAC[REAL_ARITH` a < &0 <=> ~( &0 <= a ) `; SUM_NUMSEG4]; STRIP_TAC; UNDISCH_TAC`~(cc_small_eta_v11 cc i /\ cc_small_eta_v11 cc (i + 1)) `; REWRITE_TAC[DE_MORGAN_THM]; STRIP_TAC; ASSUME_TAC2 (ISPECL [` cc_small_eta_v11 cc `; `nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); SUBGOAL_THEN` ~cc_small_eta_v11 cc ((i + 4) MOD 4)` ASSUME_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; DOWN; ASM_REWRITE_TAC[]; STRIP_TAC; UNDISCH_TAC` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc (i + 1) ==> cc_eps <= cc_gg_v11 cc i `; DISCH_THEN (MP_TAC o (SPEC` i + 3 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2 ) ` ASSUME_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i + cc_gg_v11 cc (i + 1) `ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= cc_gg3a_v11 cc i /\ cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; ASM_SIMP_TAC[]; SUBGOAL_THEN` &0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3) ` MP_TAC; DOWNS 4; REAL_ARITH_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` !i. cc_qu_v11 cc i /\ ~cc_small_eta_v11 cc i ==> cc_eps <= cc_gg_v11 cc i `; DISCH_THEN (MP_TAC o (SPEC` i + 1 `)); ANTS_TAC; ASM_REWRITE_TAC[]; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2 ) ` ASSUME_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` !i. cc_qu_v11 cc i /\ cc_qy_v11 cc (i + 1) ==> &0 <= cc_gg_v11 cc i + cc_gg3a_v11 cc (i + 1) `; DISCH_THEN (MP_TAC o (SPEC` i + 3 `)); ANTS_TAC; ASM_REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `]; SUBGOAL_THEN` cc_qy_v11 cc ((i + nn) MOD nn) ` MP_TAC; REWRITE_TAC[MOD_ADD_CANCEL]; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[]; DOWN; SUBGOAL_THEN` &0 <= cc_gg3b_v11 cc i /\ cc_gg3a_v11 cc i + cc_gg3b_v11 cc i <= cc_gg_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; SUBGOAL_THEN` cc_gg3a_v11 cc (((i + 3) + 1) MOD 4) = cc_gg3a_v11 cc i ` MP_TAC; REWRITE_TAC[ARITH_RULE` (i + 3) + 1 = i + 4 `; MOD_ADD_CANCEL]; ASM_SIMP_TAC[]; MATCH_MP_TAC (GSYM periodic_mod); ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg3a_v11 cc `;` nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[]; SIMP_TAC[]; STRIP_TAC; UNDISCH_TAC` ~(&0 <= cc_gg_v11 cc i + cc_gg_v11 cc (i + 1) + cc_gg_v11 cc (i + 2) + cc_gg_v11 cc (i + 3)) `; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc (i + 2) ` MP_TAC; ASM_REWRITE_TAC[]; REAL_ARITH_TAC]);; let GRHIDFA_ALT = prove_by_refinement(`!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> ~(cc_card_v11 cc = 4) `, [NHANH UTEOITF; NHANH IPVICGW; REPEAT STRIP_TAC; ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `]; ABBREV_TAC` nn = cc_card_v11 cc `; STRIP_TAC; ABBREV_TAC ` squ = { i | i <= 3 /\ cc_qu_v11 cc i } `; SWITCH_TAC` nn = 4 `; ASSUME_TAC (SPECL [` 0 `;` 3 `] FINITE_NUMSEG); SUBGOAL_THEN` squ SUBSET 0..3 ` ASSUME_TAC; EXPAND_TAC "squ"; REWRITE_TAC[numseg; LE_0]; CONV_TAC SET_RULE; SUBGOAL_THEN` FINITE (squ: num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` 0..3 `; ASM_REWRITE_TAC[]; SUBGOAL_THEN` ! i. cc_qu_v11 cc i \/ cc_qx_v11 cc i ` ASSUME_TAC; ASM_REWRITE_TAC[cc_qx_v11]; MESON_TAC[]; ABBREV_TAC` sqx = { i | i <= 3 /\ cc_qx_v11 cc i } ` ; SUBGOAL_THEN` sqx UNION squ = 0..3` ASSUME_TAC; EXPAND_TAC "squ"; EXPAND_TAC "sqx"; REWRITE_TAC[numseg; LE_0]; UNDISCH_TAC` !i. cc_qu_v11 cc i \/ cc_qx_v11 cc i `; REWRITE_TAC[EXTENSION; IN_ELIM_THM; IN_UNION]; MESON_TAC[]; SUBGOAL_THEN` DISJOINT sqx (squ:num -> bool) ` ASSUME_TAC; EXPAND_TAC "squ" THEN EXPAND_TAC "sqx"; REWRITE_TAC[cc_qx_v11]; CONV_TAC SET_RULE; SUBGOAL_THEN` sqx SUBSET 0..3 ` ASSUME_TAC; EXPAND_TAC "sqx"; REWRITE_TAC[numseg; LE_0]; CONV_TAC SET_RULE; SUBGOAL_THEN` FINITE (sqx:num -> bool) ` ASSUME_TAC; MATCH_MP_TAC FINITE_SUBSET; EXISTS_TAC` 0..3 `; ASM_REWRITE_TAC[]; SUBGOAL_THEN` DISJOINT squ (sqx: num -> bool) ` ASSUME_TAC; ONCE_REWRITE_TAC[DISJOINT_SYM]; ASM_REWRITE_TAC[]; SUBGOAL_THEN`! f. sum (squ UNION sqx) f = sum squ f + sum (sqx:num -> bool) f ` ASSUME_TAC; GEN_TAC; MATCH_MP_TAC SUM_UNION; ASM_REWRITE_TAC[]; MP_TAC (ISPECL [` squ: num -> bool `;` sqx: num -> bool `] CARD_UNION); ANTS_TAC; ASM_REWRITE_TAC[GSYM DISJOINT]; UNDISCH_TAC` sqx UNION squ = 0..3 `; REWRITE_TAC[UNION_COMM]; STRIP_TAC; ASM_REWRITE_TAC[CARD_NUMSEG; ARITH_RULE` (3 + 1) - 0 = 4 `]; STRIP_TAC; SUBGOAL_THEN` !i. i IN squ ==> #0.161517 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; EXPAND_TAC "squ"; REWRITE_TAC[IN_ELIM_THM]; GEN_TAC THEN STRIP_TAC; ASM_SIMP_TAC[]; SUBGOAL_THEN` ! i. i IN sqx ==> #0.213849 - #0.119482 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; EXPAND_TAC "sqx"; REWRITE_TAC[IN_ELIM_THM]; GEN_TAC THEN STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[cc_hassmall_v11]; ASM_CASES_TAC` CARD (squ: num -> bool) <= 2 ` ; SWITCH_TAC` squ UNION sqx = 0..3 `; SUBGOAL_THEN` sum squ (\i. #0.161517 - #0.119482 * cc_azim_v11 cc i ) <= sum squ (cc_gg_v11 cc) ` ASSUME_TAC; MATCH_MP_TAC SUM_LE; ASM_REWRITE_TAC[]; SUBGOAL_THEN` sum sqx (\i. #0.213849 - #0.119482 * cc_azim_v11 cc i ) <= sum sqx (cc_gg_v11 cc) ` ASSUME_TAC; MATCH_MP_TAC SUM_LE; ASM_REWRITE_TAC[]; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; ASM_REWRITE_TAC[ARITH_RULE` ~( x < &0 ) <=> &0 <= x `]; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` sum squ (\i. #0.161517 - #0.119482 * cc_azim_v11 cc i) + sum sqx (\i. #0.213849 - #0.119482 * cc_azim_v11 cc i ) `; CONJ_TAC; ASM_SIMP_TAC[SUM_SUB; SUM_LMUL; SUM_CONST]; UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; ASM_REWRITE_TAC[ETA_AX]; UNDISCH_TAC` nn = CARD (squ: num -> bool) + CARD (sqx: num -> bool) `; UNDISCH_TAC` CARD (squ: num -> bool) <= 2 `; REWRITE_TAC[GSYM REAL_OF_NUM_LE; GSYM REAL_OF_NUM_EQ; GSYM REAL_OF_NUM_ADD]; EXPAND_TAC "nn"; MP_TAC PI_BOUNDS; REAL_ARITH_TAC; DOWNS 2; REAL_ARITH_TAC; ASM_CASES_TAC` ! i. i < 4 ==> cc_qu_v11 cc i ` ; SUBGOAL_THEN` ! i. 0 <= i /\ i <= 3 ==> -- #0.0659 + #0.042 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` ASSUME_TAC; REWRITE_TAC[ARITH_RULE` 0 <= x /\ x <= 3 <=> x < 4 `]; FIRST_X_ASSUM NHANH; ASM_SIMP_TAC[]; DOWN; PHA; NHANH SUM_LE_NUMSEG; REWRITE_TAC[SUM_ADD_NUMSEG; SUM_LMUL; ETA_AX; SUM_CONST_NUMSEG]; REWRITE_TAC[ARITH_RULE` ( 3 + 1 ) - 0 = 4`]; UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 = 3 `]; SIMP_TAC[]; REPEAT STRIP_TAC; SUBGOAL_THEN` &0 <= sum (0..3) (cc_gg_v11 cc) ` MP_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` &4 * -- #0.0659 + #0.042 * &2 * pi `; DOWN THEN SIMP_TAC[]; MP_TAC PI_BOUNDS; REAL_ARITH_TAC; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x ) <=> x < &0 `]; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; EXPAND_TAC "nn"; SIMP_TAC[ARITH_RULE` 4 - 1 = 3 `]; SUBGOAL_THEN` ? i. i < 4 /\ ~ cc_qu_v11 cc i ` ASSUME_TAC; DOWN; MESON_TAC[]; DOWN THEN STRIP_TAC; SUBGOAL_THEN` cc_qx_v11 cc i ` ASSUME_TAC; ASM_REWRITE_TAC[cc_qx_v11]; SUBGOAL_THEN` CARD (sqx:num -> bool) <= 1 ` ASSUME_TAC; UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx:num -> bool) `; UNDISCH_TAC` ~(CARD (squ:num -> bool) <= 2) `; EXPAND_TAC "nn"; ARITH_TAC; SUBGOAL_THEN` ! j. j <= 3 /\ ~(j = i) ==> cc_qu_v11 cc j ` ASSUME_TAC; GEN_TAC THEN STRIP_TAC; ASM_CASES_TAC` cc_qx_v11 cc j `; SUBGOAL_THEN` {i, j:num} SUBSET sqx ` ASSUME_TAC; EXPAND_TAC "sqx"; REWRITE_TAC[IN_ELIM_THM; INSERT_SUBSET; EMPTY_SUBSET]; ASM_REWRITE_TAC[]; UNDISCH_TAC` i < 4 `; ARITH_TAC; UNDISCH_TAC` ~( j = (i:num)) ` ; REWRITE_TAC[GSYM Geomdetail.CARD2; INSERT_COMM]; STRIP_TAC; SUBGOAL_THEN` CARD {i, j:num} <= CARD (sqx:num -> bool)` ASSUME_TAC; MATCH_MP_TAC CARD_SUBSET; ASM_REWRITE_TAC[]; DOWN; ASM_REWRITE_TAC[]; NHANH (ARITH_RULE` 2 <= a ==> ~( a <= 1 ) `); ASM_REWRITE_TAC[]; UNDISCH_TAC` !i. cc_qu_v11 cc i \/ cc_qx_v11 cc i `; DISCH_THEN (MP_TAC o (SPEC` j:num `)); ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_qu_v11 cc `;` nn:num `] periodic_mod1); FIRST_X_ASSUM (ASSUME_TAC o GSYM); MP_TAC (ISPECL [` 3`] Hypermap.LE_MOD_SUC); REWRITE_TAC[ARITH_RULE` SUC 3 = 4 `]; STRIP_TAC; SUBGOAL_THEN` i MOD nn = i ` ASSUME_TAC; MATCH_MP_TAC MOD_LT; UNDISCH_TAC` i < 4 `; ASM_REWRITE_TAC[]; UNDISCH_TAC` !j. j <= 3 /\ ~(j = i) ==> cc_qu_v11 cc j `; EXPAND_TAC "i"; STRIP_TAC; SUBGOAL_THEN` cc_qu_v11 cc (( i + 1) MOD 4 ) ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `; DISCH_THEN (ASSUME_TAC o SYM); CONJ_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` 4 = nn `; SIMP_TAC[]; STRIP_TAC; MATCH_MP_TAC (GSYM MOD_INJ11); EXPAND_TAC "nn"; ARITH_TAC; SUBGOAL_THEN` cc_qu_v11 cc (( i + 2) MOD 4 ) ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `; DISCH_THEN (ASSUME_TAC o SYM); CONJ_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` 4 = nn `; SIMP_TAC[]; STRIP_TAC; MATCH_MP_TAC (GSYM MOD_INJ11); EXPAND_TAC "nn"; ARITH_TAC; SUBGOAL_THEN` cc_qu_v11 cc (( i + 3) MOD 4 ) ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `; DISCH_THEN (ASSUME_TAC o SYM); CONJ_TAC; ASM_REWRITE_TAC[]; UNDISCH_TAC` 4 = nn `; SIMP_TAC[]; STRIP_TAC; MATCH_MP_TAC (GSYM MOD_INJ11); EXPAND_TAC "nn"; ARITH_TAC; DOWNS 3; UNDISCH_TAC` nn = CARD (squ:num -> bool) + CARD (sqx: num -> bool) `; DISCH_THEN (ASSUME_TAC o SYM); ASM_REWRITE_TAC[]; ASM_CASES_TAC` cc_crit_v11 cc i `; STRIP_TAC; UNDISCH_TAC` nn = 4 /\ (?i. cc_4cell_v11 cc i /\ cc_crit_v11 cc i /\ cc_qu_v11 cc (i + 1) /\ cc_qu_v11 cc (i + 2) /\ cc_qu_v11 cc (i + 3)) ==> &0 <= sum (0..nn - 1) (cc_gg_v11 cc) `; ANTS_TAC; ASM_REWRITE_TAC[]; EXISTS_TAC` i:num `; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[REAL_ARITH` ~( &0 <= x ) <=> x < &0 `]; ASM_CASES_TAC` cc_subcrit_v11 cc i `; UNDISCH_TAC` cc_qx_v11 cc i `; REWRITE_TAC[cc_qx_v11; cc_qu_v11]; ASM_REWRITE_TAC[cc_hassmall_v11]; SUBGOAL_THEN` cc_crit_v11 cc i \/ cc_subcrit_v11 cc i \/ cc_supercrit_v11 cc i ` ASSUME_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; DOWN; ASM_REWRITE_TAC[]; STRIP_TAC; SUBGOAL_THEN` #0.00457511 + #0.00609451 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[cc_hassmall_v11]; STRIP_TAC; SUBGOAL_THEN` ! i. cc_hassmall_v11 cc i ` ASSUME_TAC; ASM_REWRITE_TAC[cc_hassmall_v11]; UNDISCH_TAC` !i. cc_qu_v11 cc i ==> -- #0.0142852 + #0.00609451 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; DISCH_THEN NHANH; STRIP_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0 `; UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `;`nn:num `] periodic_sum); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASSUME_TAC2 (ISPECL [` cc_azim_v11 cc `;`nn:num `] periodic_sum); FIRST_X_ASSUM (ASSUME_TAC o GSYM); ASM_REWRITE_TAC[]; UNDISCH_TAC` 4 = nn `; STRIP_TAC; EXPAND_TAC "nn"; REWRITE_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `]; REWRITE_TAC[SUM_NUMSEG4]; UNDISCH_TAC ` -- #0.0142852 + #0.00609451 * cc_azim_v11 cc (i + 3) <= cc_gg_v11 cc (i + 3) `; UNDISCH_TAC ` -- #0.0142852 + #0.00609451 * cc_azim_v11 cc (i + 2) <= cc_gg_v11 cc (i + 2) `; UNDISCH_TAC ` -- #0.0142852 + #0.00609451 * cc_azim_v11 cc (i + 1) <= cc_gg_v11 cc (i + 1) `; UNDISCH_TAC` #0.00457511 + #0.00609451 * cc_azim_v11 cc i <= cc_gg_v11 cc i `; PHA; MP_TAC PI_BOUNDS; REAL_ARITH_TAC]);; let LUIKGMH_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (4 <= cc_card_v11 cc)`;; let GRHIDFA_concl = `!cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ (sum (0..cc_card_v11 cc -1) (cc_gg_v11 cc) < &0) ==> (F)`;; let NOT_QY_LEM = prove(` !cc. cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0 ==> (!i. ~ cc_qy_v11 cc i) `, NHANH UTEOITF THEN SIMP_TAC[cc_qy_v11]);; let LUIKGMH = prove_by_refinement(LUIKGMH_concl, [NHANH NOT_QY_LEM; REPEAT STRIP_TAC; ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; ASSUME_TAC2 THREE_LE_CC_CARD; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `; ASM_CASES_TAC` nn = 3 `; SUBGOAL_THEN` ! i. cc_qu_v11 cc i /\ cc_gg_v11 cc i < &0 ==> cc_azim_v11 cc i < #1.65 ` ASSUME_TAC; GEN_TAC THEN STRIP_TAC; SUBGOAL_THEN` -- #0.0659 + #0.042 * cc_azim_v11 cc i <= cc_gg_v11 cc i ` MP_TAC; ASM_SIMP_TAC[]; DOWN THEN PHA; NHANH (REAL_ARITH` x < &0 /\ a <= x ==> a < &0 `); STRIP_TAC; DOWN; CONV_TAC REAL_FIELD; ABBREV_TAC` sxx = { i | i <= 2 /\ cc_qu_v11 cc i /\ cc_gg_v11 cc i < &0 }`; SUBGOAL_THEN` ! i. cc_qu_v11 cc i \/ cc_qx_v11 cc i ` ASSUME_TAC; GEN_TAC; MP_TAC (SPEC` i:num ` QU_OR_QXY); ASM_REWRITE_TAC[]; ASM_CASES_TAC` (sxx:num -> bool) = {} `; SUBGOAL_THEN` ! i. i <= 2 ==> &0 <= cc_gg_v11 cc i ` ASSUME_TAC; GEN_TAC THEN STRIP_TAC; ASM_CASES_TAC` cc_qx_v11 cc i `; ASM_SIMP_TAC[]; FIRST_X_ASSUM (MP_TAC o (SPEC`i:num `)); ASM_REWRITE_TAC[]; STRIP_TAC; ASM_CASES_TAC` cc_gg_v11 cc i < &0 `; SUBGOAL_THEN` i IN (sxx:num -> bool) ` MP_TAC; EXPAND_TAC "sxx"; REWRITE_TAC[IN_ELIM_THM]; ASM_REWRITE_TAC[]; ASM_REWRITE_TAC[NOT_IN_EMPTY]; DOWN; REAL_ARITH_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc ) < &0 `; ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 0 + 2 `; SUM_NUMSEG3; ADD]; REWRITE_TAC[REAL_ARITH` x < &0 <=> ~( &0 <= x) `]; DOWN; MESON_TAC[LE_0; SUM_POS_LE_NUMSEG]; DOWN; REWRITE_TAC[Local_lemmas.EMPTY_NOT_EXISTS_IN]; STRIP_TAC; ASM_CASES_TAC` ? y. y IN sxx /\ ~( y = x:num ) `; DOWN THEN STRIP_TAC; ASSUME_TAC (SET_RULE` DISJOINT {x,y:num} ( (0..2) DIFF {x,y}) `); ABBREV_TAC` sc = (0..2) DIFF {x,y} `; SUBGOAL_THEN` x <= 2 /\ y <= 2 /\ 0 <= x /\ 0 <= y` MP_TAC; UNDISCH_TAC` x IN (sxx: num -> bool) `; UNDISCH_TAC` y IN (sxx: num -> bool) `; EXPAND_TAC "sxx"; SIMP_TAC[IN_ELIM_THM; LE_0]; STRIP_TAC; SUBGOAL_THEN` 0..2 = {x,y} UNION sc ` ASSUME_TAC; EXPAND_TAC "sc"; SUBGOAL_THEN` x IN 0..2 /\ y IN 0..2 ` MP_TAC; ASM_REWRITE_TAC[IN_NUMSEG]; CONV_TAC SET_RULE; SUBGOAL_THEN` CARD {x,y:num} = 2 ` ASSUME_TAC; ASM_REWRITE_TAC[Geomdetail.CARD2]; UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc ) < &0 `; ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `]; ASSUME_TAC (SPECL [` 0 `;` 2 `] FINITE_NUMSEG); ASSUME_TAC (ISPECL [` x:num `;` y:num `] Hypermap.FINITE_TWO_ELEMENTS); SUBGOAL_THEN` FINITE (sc:num -> bool) ` ASSUME_TAC; EXPAND_TAC "sc"; MATCH_MP_TAC FINITE_DIFF; FIRST_ASSUM ACCEPT_TAC; STRIP_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `; ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `]; ASM_SIMP_TAC[SUM_UNION]; ASSUME_TAC2 ( ISPECL [` {x:num, y} `;` sc:num -> bool `] ( REWRITE_RULE[GSYM DISJOINT] CARD_UNION)); DOWN; ASM_REWRITE_TAC[]; MP_TAC (SPECL [` 0 `;` 2 `] CARD_NUMSEG); ASM_SIMP_TAC[ARITH_RULE` ( 2 + 1) - 0 = 3 `]; PHA THEN STRIP_TAC; SUBGOAL_THEN` sum {x,y} (cc_azim_v11 cc) <= &(CARD {x,y}) * ( #1.65) ` ASSUME_TAC; MATCH_MP_TAC SUM_BOUND; ASM_REWRITE_TAC[]; SUBGOAL_THEN` {x,y} SUBSET (sxx: num -> bool) ` MP_TAC; ASM_REWRITE_TAC[INSERT_SUBSET; EMPTY_SUBSET]; REWRITE_TAC[SUBSET]; DISCH_THEN NHANH; GEN_TAC THEN STRIP_TAC; MATCH_MP_TAC REAL_LT_IMP_LE; FIRST_X_ASSUM MATCH_MP_TAC; DOWN; EXPAND_TAC "sxx"; REWRITE_TAC[IN_ELIM_THM]; SIMP_TAC[]; SUBGOAL_THEN` sum sc (cc_azim_v11 cc) < &(CARD sc) * ( #2.8) ` ASSUME_TAC; MATCH_MP_TAC SUM_BOUND_LT_ALL; ASM_REWRITE_TAC[]; CONJ_TAC; STRIP_TAC; FIRST_X_ASSUM SUBST_ALL_TAC; UNDISCH_TAC` 3 = 2 + (CARD ({}:num -> bool))` ; REWRITE_TAC[CARD_EMPTY]; ARITH_TAC; GEN_TAC THEN STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` x':num `)); SIMP_TAC[cc_qu_v11; cc_qx_v11]; MESON_TAC[]; DOWNS 2; NHANH (REAL_ARITH` a <= aa /\ b < bb ==> a + b < aa + bb `); ASM_REWRITE_TAC[]; STRIP_TAC; UNDISCH_TAC` 3 = 2 + CARD (sc:num -> bool) `; REWRITE_TAC[ARITH_RULE` 3 = 2 + x <=> x = 1 `]; DISCH_THEN (SUBST_ALL_TAC); DOWN; MP_TAC PI_BOUNDS; REAL_ARITH_TAC; SUBGOAL_THEN` ! i. #2.3 < cc_azim_v11 cc i ==> cc_eps <= cc_gg_v11 cc i ` ASSUME_TAC; GEN_TAC; FIRST_X_ASSUM (MP_TAC o (SPEC` i:num `)); STRIP_TAC; STRIP_TAC; MATCH_MP_TAC REAL_LE_TRANS; EXISTS_TAC` -- #0.0659 + #0.042 * cc_azim_v11 cc i `; ASM_SIMP_TAC[]; DOWN; REWRITE_TAC[cc_eps]; REAL_ARITH_TAC; ASM_SIMP_TAC[]; ASSUME_TAC2 ( SET_RULE` x IN sxx /\ ~(?y. y IN sxx /\ ~(y = x)) ==> sxx = {x:num} `); ABBREV_TAC` sc = (0..2) DIFF sxx `; ASSUME_TAC (SPECL [` 0 `;` 2 `] FINITE_NUMSEG); SUBGOAL_THEN` FINITE ((0..2) DIFF sxx ) ` ASSUME_TAC; DOWN; REWRITE_TAC[FINITE_DIFF]; DOWNS 3; ASM_REWRITE_TAC[]; STRIP_TAC; UNDISCH_TAC` x IN (sxx:num -> bool) `; EXPAND_TAC "sxx"; REWRITE_TAC[IN_ELIM_THM]; STRIP_TAC; SUBGOAL_THEN` x IN (0..2) ` ASSUME_TAC; ASM_REWRITE_TAC[IN_NUMSEG; LE_0]; DOWN; NHANH (SET_RULE` x IN S ==> DISJOINT {x} (S DIFF {x}) `); ASM_REWRITE_TAC[]; STRIP_TAC; ASSUME_TAC2 (SET_RULE` x IN (0..2) ==> {x} UNION ((0..2) DIFF {x}) = (0..2) `); MP_TAC (ISPECL [` {x:num } `;`sc: num -> bool `] CARD_UNION); ANTS_TAC; ASM_REWRITE_TAC[GSYM DISJOINT]; UNDISCH_TAC` FINITE ((0..2) DIFF {x})`; ASM_REWRITE_TAC[FINITE_SING]; DOWN; ASM_SIMP_TAC[CARD_SING; CARD_NUMSEG; ARITH_RULE` (2 + 1) - 0 = 3 `]; REWRITE_TAC[ARITH_RULE` 3 = 1 + x <=> x = 2 `]; PHA THEN STRIP_TAC; SUBGOAL_THEN` FINITE (sc:num -> bool) ` ASSUME_TAC; EXPAND_TAC "sc"; FIRST_X_ASSUM ACCEPT_TAC; ASM_CASES_TAC` ! i. i IN sc ==> cc_azim_v11 cc i <= #2.3 ` ; DOWNS 2; NHANH SUM_LE; REWRITE_TAC[ETA_AX]; IMP_TAC; ASM_SIMP_TAC[SUM_CONST]; PHA THEN STRIP_TAC; UNDISCH_TAC` sum (0..nn - 1) (cc_azim_v11 cc) = &2 * pi `; ASM_REWRITE_TAC[ARITH_RULE` 3 - 1 = 2 `]; UNDISCH_TAC` {x} UNION sc = 0..2 `; DISCH_THEN (SUBST1_TAC o SYM); ASM_SIMP_TAC[SUM_UNION; FINITE_SING; SUM_SING]; SUBGOAL_THEN` cc_azim_v11 cc x < #1.65 ` MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ASM_REWRITE_TAC[]; DOWN; MP_TAC PI_BOUNDS; REAL_ARITH_TAC; DOWN; REWRITE_TAC[MESON[]` (~(! i. p i ==> q i)) <=> (? i. p i /\ ~ q i ) `]; REWRITE_TAC[REAL_ARITH` ~( x <= y ) <=> y < x `]; STRIP_TAC; FIRST_ASSUM (ASSUME_TAC2 o (SPEC` i:num `)); ASSUME_TAC2 (SET_RULE` (i:num) IN sc ==> sc = {i} UNION (sc DIFF {i})`); ABBREV_TAC` scc = sc DIFF {i:num} `; SUBGOAL_THEN` FINITE ( sc DIFF {i:num}) ` ASSUME_TAC; MATCH_MP_TAC FINITE_DIFF; ASM_REWRITE_TAC[]; SUBGOAL_THEN` ! j. j IN sc ==> &0 <= cc_gg_v11 cc j ` ASSUME_TAC; EXPAND_TAC "sc"; UNDISCH_TAC` sxx = {x:num} `; DISCH_THEN (SUBST1_TAC o SYM); EXPAND_TAC "sxx"; REWRITE_TAC[IN_DIFF; IN_NUMSEG; IN_ELIM_THM; LE_0; DE_MORGAN_THM]; GEN_TAC THEN IMP_TAC; SIMP_TAC[]; PHA THEN STRIP_TAC; MP_TAC (SPEC` j:num` QU_OR_QXY); ASM_REWRITE_TAC[]; DOWN; REAL_ARITH_TAC; UNDISCH_TAC` sum (0.. nn - 1) (cc_gg_v11 cc ) < &0 `; ASM_SIMP_TAC[ARITH_RULE` 3 - 1 = 2 `]; UNDISCH_TAC` {x} UNION sc = 0..2 `; DISCH_THEN (SUBST1_TAC o SYM); UNDISCH_TAC ` sc = {i:num} UNION scc `; DISCH_THEN (ASSUME_TAC o SYM); ASM_SIMP_TAC[SUM_UNION; FINITE_SING; SUM_SING]; FIRST_X_ASSUM (ASSUME_TAC o SYM); UNDISCH_TAC` FINITE (sc DIFF {i:num}) `; MP_TAC (SET_RULE` DISJOINT {i} (sc DIFF {i:num} ) `); ASM_SIMP_TAC[SUM_UNION; FINITE_SING; SUM_SING]; PHA THEN STRIP_TAC; SUBGOAL_THEN` -- cc_eps <= cc_gg_v11 cc x ` ASSUME_TAC; ASSUME_TAC2 (LEAST_BOUND_CC_GG); ASM_REWRITE_TAC[]; SUBGOAL_THEN` &0 <= sum scc (cc_gg_v11 cc) ` ASSUME_TAC; MATCH_MP_TAC SUM_POS_LE; ASM_REWRITE_TAC[]; EXPAND_TAC "scc"; REWRITE_TAC[IN_DIFF]; FIRST_X_ASSUM NHANH; SIMP_TAC[]; DOWNS 3; UNDISCH_TAC` cc_eps <= cc_gg_v11 cc i `; REAL_ARITH_TAC; DOWN; UNDISCH_TAC` 3 <= nn `; ARITH_TAC]);; let GRHIDFA = prove(GRHIDFA_concl, PHA THEN NHANH LUIKGMH THEN NHANH GRHIDFA_ALT THEN NHANH RSIWAMP THEN GEN_TAC THEN STRIP_TAC THEN DOWNS 3 THEN ARITH_TAC);; (* =================================================== *) (* ================= search[name "RSI"];; g ` !cc. ((cc_bool_model_v11 cc /\ cc_bool_prep_v11 cc /\ cc_real_model_v11 cc /\ sum (0..cc_card_v11 cc - 1) (cc_gg_v11 cc) < &0) /\ cc_card_v11 cc = 5 ) /\ ~cc_small_v11 cc i /\ cc_qx_v11 cc i /\ cc_qx_v11 cc ( i + 4) ==> F `;; els [REPEAT STRIP_TAC; ASSUME_TAC2 QY_NN00 THEN ASSUME_TAC2 QX_NN00; ASSUME_TAC2 EXISTS_QY_CARD5; DOWN_TAC; NHANH Oxl_def.periodic_fn; REWRITE_TAC[cc_bool_model_v11; cc_real_model_v11]; ONCE_REWRITE_TAC[TAUT` ~ a <=> a ==> F `]; IMP_TAC; IMP_TAC; STRIP_TAC; STRIP_TAC; STRIP_TAC; ABBREV_TAC` nn = cc_card_v11 cc `];; ASM_REWRITE_TAC[]; ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; ` nn:num `] Oxl_def.periodic_sum); SUBGOAL_THEN` cc_gg_v11 cc i' + cc_gg_v11 cc (i' + 1) + cc_gg_v11 cc (i' + 3 - 1) = sum (0..nn - 1) (cc_gg_v11 cc) ` MP_TAC; FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i':num `)); ASM_SIMP_TAC[ARITH_RULE` 3 - 1 + i = i + 2 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 2 /\ i + 1 <= i + 2 /\ (i + 1) + 1 = i + 2 `; SUM_SING_NUMSEG]; SIMP_TAC[ARITH_RULE` 3 - 1 = 2 `]; SIMP_TAC[Oxl_def.cc_eps]; NHANH (REAL_ARITH` &2 * #0.0057 <= x ==> ~( x < &0 ) `); ASM_REWRITE_TAC[]; ASSUME_TAC2 (ARITH_RULE` nn <= 4 /\ 3 <= nn /\ ~( nn = 3) ==> nn = 4 `); ASSUME_TAC2 (ISPECL [` cc_gg_v11 cc `; ` nn:num `] Oxl_def.periodic_sum); UNDISCH_TAC` sum (0..nn - 1) (cc_gg_v11 cc) < &0`; FIRST_X_ASSUM (SUBST1_TAC o SYM o (SPEC` i':num `)); ASM_SIMP_TAC[ARITH_RULE` 4 - 1 + i = i + 3 `]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i <= i + 3 /\ i + 1 <= i + 3 /\ (i + 1) + 1 = i + 2 `; SUM_SING_NUMSEG]; SIMP_TAC[SUM_CLAUSES_LEFT; ARITH_RULE` i + 2 <= i + 3 /\ (i + 2) + 1 = i + 3`; SUM_SING_NUMSEG]; ASSUME_TAC2 LEAST_BOUND_CC_GG; FIRST_X_ASSUM (ASSUME_TAC o (SPEC `i' + 2 `)); DOWN; UNDISCH_TAC` &2 * cc_eps <= cc_gg_v11 cc i' + cc_gg_v11 cc (i' + 1) + cc_gg_v11 cc (i' + nn - 1) `; ASM_SIMP_TAC[ARITH_RULE` 4 - 1 = 3 `; Oxl_def.cc_eps]];; *) end;;