(* ========================================================================== *) (* FLYSPECK - BOOK FORMALIZATION *) (* Section: Appendix, Main Estimate, check_completeness *) (* Chapter: Local Fan *) (* Lemma: IUNBUIG *) (* Author: Thomas Hales *) (* Date: 2013-07-29 *) (* ========================================================================== *) module Iunbuig = struct open Hales_tactic;; let LET_THM = CONJ LET_DEF LET_END_DEF;; let quadratic_at_most_2_roots = prove_by_refinement( `!a b c. ?x1 x2. !x. ~(a = &0) /\ a * x pow 2 + b * x + c = &0 ==> x = x1 \/ x = x2`, (* {{{ proof *) [ REPEAT GEN_TAC; ASM_CASES_TAC `~(?x1. a * x1 pow 2 + b * x1 + c = &0)`; BY(ASM_MESON_TAC[]); FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[] THEN REPEAT STRIP_TAC; ASM_CASES_TAC `!x2. a * x2 pow 2 + b * x2 + c = &0 ==> x2 = x1`; BY(ASM_MESON_TAC[]); FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[NOT_FORALL_THM] THEN REPEAT STRIP_TAC; GEXISTL_TAC [`x1`;`x2`]; BY(GEN_TAC THEN ASM_TAC THEN CONV_TAC REAL_RING) (* String.length above = 414. String.length below = 233. 56% the length. *) (* g/r asmcs `~(?x1. a * x1 pow 2 + b * x1 + c = &0)` amt[] fx mp then rt[] then str/r asmcs `!x2. a * x2 pow 2 + b * x2 + c = &0 ==> x2 = x1` amt[] fx mp then rt[NOT_FORALL_THM] then str/r exl [`x1`;`x2`] g then asm then cvc REAL_RING *) (* If there are no roots, it is obvious. Otherwise, let x1 be a root. If every root is x1 it is obvious. Otherwise, let x2 be a second root. Instantiate the existential quantifiers with these two roots, and solve the resulting problem by Groebner basis algorithm. *) ]);; (* }}} *) let ARCV_ADD_AFF_GE = prove_by_refinement( `!(u:real^N) v w x. ~(v = u) /\ ~(w = u) /\ ~(x = u) /\ x IN aff_ge {u} {v, w} ==> arcV u v x + arcV u x w = arcV u v w`, (* {{{ proof *) [ BY(REWRITE_TAC[ARCV_ANGLE;ANGLES_ADD_AFF_GE]) ]);; (* }}} *) let SUM_NUMSEG4 = prove_by_refinement( `!t. sum (0..4) t = t 0 + t 1 + t 2 + t 3 + t 4`, (* {{{ proof *) [ REWRITE_TAC[arith `4 = SUC 3`;SUM_CLAUSES_NUMSEG;Qknvmlb.SUM_NUMSEG3;arith `0 <= SUC 3`]; BY(REAL_ARITH_TAC) ]);; (* }}} *) let epsilon_pent = prove_by_refinement( `!e e' e'' e''' e''''. &0 < e /\ &0 < e' /\ &0 < e'' /\ &0 < e''' /\ &0 < e'''' ==> (?e'''''. &0 < e''''' /\ (!t. abs t < e''''' ==> abs t < e) /\ (!t. abs t < e''''' ==> abs t < e') /\ (!t. abs t < e''''' ==> abs t < e'') /\ (!t. abs t < e''''' ==> abs t < e''') /\ (!t. abs t < e''''' ==> abs t < e''''))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_quad [`e`;`e'`;`e''`;`e'''`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_pair [`e''''`;`e'''''`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''''` EXISTS_TAC; BY(ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN REPEAT WEAKER_STRIP_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) let epsilon_hex = prove_by_refinement( `!e e' e'' e''' e'''' e5. &0 < e /\ &0 < e' /\ &0 < e'' /\ &0 < e''' /\ &0 < e'''' /\ &0 < e5 ==> (?e6. &0 < e6 /\ (!t. abs t < e6 ==> abs t < e) /\ (!t. abs t < e6 ==> abs t < e') /\ (!t. abs t < e6 ==> abs t < e'') /\ (!t. abs t < e6 ==> abs t < e''') /\ (!t. abs t < e6 ==> abs t < e'''') /\ (!t. abs t < e6 ==> abs t < e5))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; INTRO_TAC epsilon_pent [`e`;`e'`;`e''`;`e'''`;`e''''`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_pair [`e'''''`;`e5`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''''` EXISTS_TAC; BY(ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN REPEAT WEAKER_STRIP_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) let epsilon_hept = prove_by_refinement( `!e e' e'' e''' e'''' e5 e6. &0 < e /\ &0 < e' /\ &0 < e'' /\ &0 < e''' /\ &0 < e'''' /\ &0 < e5 /\ &0 < e6 ==> (?e7. &0 < e7 /\ (!t. abs t < e7 ==> abs t < e) /\ (!t. abs t < e7 ==> abs t < e') /\ (!t. abs t < e7 ==> abs t < e'') /\ (!t. abs t < e7 ==> abs t < e''') /\ (!t. abs t < e7 ==> abs t < e'''') /\ (!t. abs t < e7 ==> abs t < e5) /\ (!t. abs t < e7 ==> abs t < e6))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; INTRO_TAC epsilon_hex [`e`;`e'`;`e''`;`e'''`;`e''''`;`e5`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_pair [`e6`;`e6'`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''''` EXISTS_TAC; BY(ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THEN REPEAT WEAKER_STRIP_TAC THEN (REPEAT (FIRST_X_ASSUM MATCH_MP_TAC)) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) let re_eqvl_imp_le = prove_by_refinement( `!a b. re_eqvl a b ==> (&0 <= a <=> &0 <= b)`, (* {{{ proof *) [ TYPIFY `!a b. re_eqvl a b /\ &0 <= b ==> &0 <= a` ENOUGH_TO_SHOW_TAC; BY(MESON_TAC[Leaf_cell.RE_EQVL_SYM]); REWRITE_TAC[Trigonometry2.re_eqvl]; REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; GMATCH_SIMP_TAC REAL_LE_MUL; BY(ASM_TAC THEN REAL_ARITH_TAC) ]);; (* }}} *) let re_eqvl_real_sgn = prove_by_refinement( `!x y. re_eqvl x y <=> real_sgn x = real_sgn y`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC (TAUT `((a ==>b) /\ (b ==> a)) ==> (a = b)`); REWRITE_TAC[Trigonometry2.re_eqvl]; CONJ_TAC THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]; REPEAT (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[REAL_SGN_MUL;GSYM REAL_SGN_EQ;arith `&0 < t <=> t > &0`]; BY(DISCH_THEN (unlist REWRITE_TAC) THEN REAL_ARITH_TAC); FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[REAL_SGN]; TYPIFY `x = &0` ASM_CASES_TAC; ASM_REWRITE_TAC[arith `abs(&0) = &0`;arith `&0 / &0 = &0`;arith `&0 = x <=> x = &0`;REAL_DIV_EQ_0;REAL_ENTIRE;arith `abs y = &0 <=> y = &0`]; BY(MESON_TAC[arith `&0 < &1`]); DISCH_TAC; TYPIFY `~(y = &0)` (C SUBGOAL_THEN ASSUME_TAC); DISCH_TAC; FIRST_X_ASSUM_ST `abs` MP_TAC; BY(ASM_REWRITE_TAC[arith `&0 / t = &0`;REAL_DIV_EQ_0] THEN ASM_TAC THEN REAL_ARITH_TAC); TYPIFY `x / y` EXISTS_TAC; CONJ2_TAC; Calc_derivative.CALC_ID_TAC; BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); REWRITE_TAC[Calc_derivative.invert_den_lt]; REWRITE_TAC[REAL_MUL_POS_LT]; FIRST_X_ASSUM_ST `abs` MP_TAC THEN REWRITE_TAC[GSYM REAL_SGN;real_sgn]; RULE_ASSUM_TAC (REWRITE_RULE[arith `~(x = &0) <=> (&0 < x \/ x < &0) /\ ~(&0 < x /\ x < &0)`]); ASSUME_TAC (arith ` ~(&1 = -- &1)`); BY(REPEAT COND_CASES_TAC THEN ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]) ]);; (* }}} *) let coplanar_delta_y_zero = prove_by_refinement( `!(u0:real^3) u1 u2 u3. coplanar {u0, u1, u2, u3} <=> delta_y (dist (u0,u1)) (dist (u0,u2)) (dist (u0,u3)) (dist (u2,u3)) (dist (u1,u3)) (dist (u1,u2)) = &0`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; BY(ASM_MESON_TAC[Terminal.DELTA_Y_POS_4POINTS;Oxlzlez.coplanar_delta_y;arith `&0 <= x ==> (~(&0 < x) <=> (x = &0))`]) ]);; (* }}} *) let real_continuous_finite_range_constant = prove_by_refinement( `!f a b. f real_continuous_on (real_interval (a,b)) /\ FINITE (IMAGE f (real_interval (a,b))) ==> (?c. !x. x IN real_interval (a,b) ==> f x = c)`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; TYPIFY `(?c. !x. x IN IMAGE lift (real_interval(a,b)) ==> ((lift o f o drop) x = c))` ENOUGH_TO_SHOW_TAC; REPEAT WEAKER_STRIP_TAC; TYPIFY `drop c` EXISTS_TAC; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `IMAGE` (C INTRO_TAC [`lift x`]); REWRITE_TAC[o_THM;LIFT_DROP]; ANTS_TAC; MATCH_MP_TAC FUN_IN_IMAGE; BY(ASM_REWRITE_TAC[]); DISCH_THEN (SUBST1_TAC o GSYM); BY(REWRITE_TAC[LIFT_DROP]); MATCH_MP_TAC CONTINUOUS_FINITE_RANGE_CONSTANT; CONJ_TAC; ONCE_REWRITE_TAC[GSYM LIFT_DROP]; REWRITE_TAC[GSYM INTERVAL_REAL_INTERVAL]; BY(REWRITE_TAC[CONNECTED_INTERVAL]); CONJ_TAC; REWRITE_TAC[ETA_AX]; BY(ASM_REWRITE_TAC[GSYM REAL_CONTINUOUS_ON]); REWRITE_TAC[GSYM IMAGE_o;ETA_AX;GSYM o_ASSOC]; TYPIFY_GOAL_THEN `!X. IMAGE (lift o f o drop o lift) X = IMAGE lift (IMAGE f (IMAGE (drop o lift) X))` (unlist REWRITE_TAC); BY(REWRITE_TAC[GSYM IMAGE_o;GSYM o_ASSOC]); REWRITE_TAC[IMAGE_LIFT_DROP]; MATCH_MP_TAC FINITE_IMAGE; BY(ASM_REWRITE_TAC[]) ]);; (* }}} *) let planar_deform_dist = prove_by_refinement( `!(v0:real^3) v1 v2 f V e'. v1 IN V /\ v2 IN V /\ ~(v0 = vec 0) /\ &0 < e' /\ deformation f V (--e',e') /\ (!v t. ~(v =v1) /\ ~(v = v2) ==> f v t = v) /\ (!t. abs t < e' ==> coplanar {vec 0, v0, f v1 t, f v2 t} /\ dist (v0,f v1 t) = dist (v0,v1) /\ norm (f v1 t) = norm v1 /\ dist (f v2 t,v0) = dist (v2,v0) /\ norm (f v2 t) = norm v2) ==> (!t. abs t < e' ==> dist(f v1 t,f v2 t) = dist(v1,v2)) `, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; TYPIFY `f v1 (&0) = v1 /\ f v2 (&0) = v2` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation]; BY(ASM_MESON_TAC[]); TYPED_ABBREV_TAC `b = ((norm v0 * norm v0) * norm v0 * norm v0 + (norm v2 * norm v2 - dist (v0,v2) * dist (v0,v2)) * (norm v1 * norm v1 - dist (v0,v1) * dist (v0,v1)) - (norm v0 * norm v0) * (norm v1 * norm v1 + norm v2 * norm v2 + dist (v0,v2) * dist (v0,v2) + dist (v0,v1) * dist (v0,v1)))`; TYPED_ABBREV_TAC `c = (norm v0 * norm v0) * (norm v2 * norm v2) * dist (v0,v2) * dist (v0,v2) + (norm v0 * norm v0) * (norm v1 * norm v1) * dist (v0,v1) * dist (v0,v1) - (norm v2 * norm v2) * (norm v0 * norm v0 + norm v1 * norm v1 - norm v2 * norm v2 + dist (v0,v2) * dist (v0,v2) - dist (v0,v1) * dist (v0,v1)) * dist (v0,v1) * dist (v0,v1) - (norm v1 * norm v1) * (dist (v0,v2) * dist (v0,v2)) * (norm v0 * norm v0 - norm v1 * norm v1 + norm v2 * norm v2 - dist (v0,v2) * dist (v0,v2) + dist (v0,v1) * dist (v0,v1))`; TYPED_ABBREV_TAC `a = (norm v0 * norm v0)`; INTRO_TAC quadratic_at_most_2_roots [`a`;`b`;`c`]; REPEAT WEAKER_STRIP_TAC; INTRO_TAC real_continuous_finite_range_constant [`(\t. dist(f v1 t, f v2 t))`;`-- e'`;`e'`]; ANTS_TAC; CONJ_TAC; GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT; REWRITE_TAC[Ocbicby.REAL_OPEN_REAL_INTERVAL]; REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC Local_lemmas1.CON_ATREAL_REAL_CON2_REDO; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation;ETA_AX]; BY(ASM_MESON_TAC[]); TYPIFY `IMAGE (\t. dist (f v1 t,f v2 t) pow 2) (real_interval (--e',e')) SUBSET {x1,x2}` ENOUGH_TO_SHOW_TAC; TYPIFY `FINITE {x1,x2}` ENOUGH_TO_SHOW_TAC; TYPIFY_GOAL_THEN `!X. IMAGE (\t. dist (f v1 t,f v2 t) pow 2) X = IMAGE (\t. t pow 2) (IMAGE (\t. dist (f v1 t,f v2 t)) X)` (unlist REWRITE_TAC); REWRITE_TAC[GSYM IMAGE_o]; BY(GEN_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM;o_THM]); TYPED_ABBREV_TAC `s = (IMAGE (\t. dist (f v1 t,f v2 t)) (real_interval (--e',e')))`; INTRO_TAC HAS_SIZE_IMAGE_INJ_EQ [`(\t. t pow 2)`;`s`;`CARD (IMAGE (\t. t pow 2) s)`]; ANTS_TAC; EXPAND_TAC "s"; REWRITE_TAC[IN_IMAGE] THEN REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; ASM_REWRITE_TAC[]; REWRITE_TAC[GSYM REAL_EQ_SQUARE_ABS]; REPEAT (GMATCH_SIMP_TAC (arith `&0 <= x ==> abs x = x`)); BY(REWRITE_TAC[DIST_POS_LE]); REWRITE_TAC[HAS_SIZE]; BY(MESON_TAC[FINITE_SUBSET;FINITE_IMAGE]); BY(REWRITE_TAC[FINITE_INSERT;FINITE_EMPTY]); REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY;IN_IMAGE]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; CONJ_TAC; EXPAND_TAC "a"; REWRITE_TAC[REAL_ENTIRE]; REWRITE_TAC[NORM_EQ_0]; BY(ASM_REWRITE_TAC[]); FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`x'`]); ANTS_TAC; BY(FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[IN_REAL_INTERVAL;arith `-- e < t /\ t < e <=> abs t < e`]); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `coplanar` MP_TAC THEN REWRITE_TAC[coplanar_delta_y_zero]; REPEAT (FIRST_X_ASSUM_ST `dist` MP_TAC) THEN REWRITE_TAC[DIST_SYM;DIST_0] THEN REPEAT WEAKER_STRIP_TAC THEN (FIRST_X_ASSUM MP_TAC) THEN ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[arith `x = &0 <=> -- x = &0`]; ASM_REWRITE_TAC[Sphere.delta_y;Nonlinear_lemma.delta_quadratic]; BY(REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; FIRST_ASSUM (C INTRO_TAC [`&0`]); FIRST_X_ASSUM (C INTRO_TAC [`t`]); REWRITE_TAC[]; REPEAT (DISCH_THEN GMATCH_SIMP_TAC); ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN ASM_REWRITE_TAC[Localization.deformation;IN_REAL_INTERVAL;arith `--e < t /\ t < e <=> abs t < e`;arith `abs(&0) = &0`]; BY(ASM_MESON_TAC[]) ]);; (* }}} *) (* let deform_pent_exists = prove_by_refinement( `!V g23 v0 v1 v2 v3 e. ?f. ( ( v1 IN V /\ v2 IN V /\ (coplanar {vec 0,v0,v1,v2} ==> (v1 cross v0) dot (v2 cross v0) > &0) /\ ~coplanar {vec 0,v0,v2,v3} /\ ~(collinear {vec 0,v1,v2}) /\ &0 < azim (vec 0) v1 v2 v0 /\ azim (vec 0) v1 v2 v0 <= pi /\ v2 dot (v3 cross v0) > &0 /\ &0 < e /\ g23 real_continuous_on (real_interval (--e,e)) /\ g23 (&0) = &0 ==> (?e'. &0 < e' /\ deformation f V (-- e', e') /\ (!v t. ~(v = v1) /\ ~(v = v2) ==> f v t = v) /\ (!t. abs t < e' ==> dist(v0,f v1 t) = dist(v0,v1) /\ dist(f v2 t,f v1 t) = dist(v2,v1) /\ norm(f v1 t) = norm(v1) /\ dist (f v2 t,v0) = dist(v2,v0) /\ dist (f v2 t,v3) = dist(v2,v3)+ g23 t /\ norm(f v2 t) = norm (v2) ))))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[MESON [] `(?f. a ==> P f) <=> (a ==> (?f. P f))`]; REPEAT WEAKER_STRIP_TAC; TYPIFY `~coplanar {vec 0,v0,v1,v2}` ASM_CASES_TAC; INTRO_TAC Cuxvzoz.deform_684_pent_exists [`V`;`g23`;`v0`;`v1`;`v2`;`v3`;`e`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[]; REWRITE_TAC[arith `x > &0 <=> &0 <= x /\ ~(x = &0)`]; ANTS_TAC; ASM_REWRITE_TAC[]; CONJ2_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `coplanar` MP_TAC) THEN REWRITE_TAC[GSYM Local_lemmas.CROSS_DOT_COPLANAR] THEN VEC3_TAC); INTRO_TAC Trigonometry2.JBDNJJB [`v1`;`v2`;`v0`]; TYPIFY `((v1 cross v2) dot v0) = v1 dot (v2 cross v0)` (C SUBGOAL_THEN SUBST1_TAC); BY(VEC3_TAC); ONCE_REWRITE_TAC[Leaf_cell.RE_EQVL_SYM]; DISCH_THEN ((unlist REWRITE_TAC) o (MATCH_MP re_eqvl_imp_le)); BY(ASM_MESON_TAC[Local_lemmas.SIN_AZIM_POS_PI_LT]); REPEAT WEAKER_STRIP_TAC; BY(GEXISTL_TAC [`f`;`e'`] THEN ASM_MESON_TAC[]); COMMENT "planar case"; INTRO_TAC Cuxvzoz.deform_planar_exists [`V`;`(\ (t:real). &0)`;`(\ (t:real). &0)`;`g23`;`v0`;`v1`;`v2`;`v3`;`e`]; RULE_ASSUM_TAC(REWRITE_RULE[]); REPEAT WEAKER_STRIP_TAC THEN FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[arith `x + &0 = x`]; GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT THEN REWRITE_TAC[REAL_CONTINUOUS_CONST;Ocbicby.REAL_OPEN_REAL_INTERVAL]; ANTS_TAC; CONJ_TAC; BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]); BY(FIRST_X_ASSUM_ST `collinear` MP_TAC THEN MESON_TAC[COLLINEAR_2;SET_RULE `{a,b,b} = {a,b}`]); REPEAT WEAKER_STRIP_TAC; GEXISTL_TAC [`f`;`e'`] THEN ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; TYPIFY `~(v0 = vec 0)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[COPLANAR_3;SET_RULE `{a,a,b,c} = {a,b,c}`]); INTRO_TAC planar_deform_dist [`v0`;`v1`;`v2`;`f`;`V`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; BY(ASM_MESON_TAC[]); DISCH_TAC; BY(REPLICATE_TAC 2 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) *) (* let deform_pent_exists = prove_by_refinement( `!V g23 v0 v1 v2 v3 e. ?f. ( ( v1 IN V /\ v2 IN V /\ (coplanar {vec 0,v0,v1,v2} ==> (v1 cross v0) dot (v2 cross v0) > &0) /\ ~coplanar {vec 0,v0,v2,v3} /\ ~(collinear {vec 0,v1,v2}) /\ azim (vec 0) v1 v2 v0 <= pi /\ v2 dot (v3 cross v0) > &0 /\ &0 < e /\ g23 real_continuous_on (real_interval (--e,e)) /\ g23 (&0) = &0 ==> (?e'. &0 < e' /\ deformation f V (-- e', e') /\ (!v t. ~(v = v1) /\ ~(v = v2) ==> f v t = v) /\ (!t. abs t < e' ==> dist(v0,f v1 t) = dist(v0,v1) /\ dist(f v2 t,f v1 t) = dist(v2,v1) /\ norm(f v1 t) = norm(v1) /\ dist (f v2 t,v0) = dist(v2,v0) /\ dist (f v2 t,v3) = dist(v2,v3)+ g23 t /\ norm(f v2 t) = norm (v2) ))))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[MESON [] `(?f. a ==> P f) <=> (a ==> (?f. P f))`]; REPEAT WEAKER_STRIP_TAC; TYPIFY `~coplanar {vec 0,v0,v1,v2}` ASM_CASES_TAC; INTRO_TAC Cuxvzoz.deform_684_pent_exists [`V`;`g23`;`v0`;`v1`;`v2`;`v3`;`e`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[]; REWRITE_TAC[arith `x > &0 <=> &0 <= x /\ ~(x = &0)`]; ANTS_TAC; ASM_REWRITE_TAC[]; CONJ2_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `coplanar` MP_TAC) THEN REWRITE_TAC[GSYM Local_lemmas.CROSS_DOT_COPLANAR] THEN VEC3_TAC); INTRO_TAC Trigonometry2.JBDNJJB [`v1`;`v2`;`v0`]; TYPIFY `((v1 cross v2) dot v0) = v1 dot (v2 cross v0)` (C SUBGOAL_THEN SUBST1_TAC); BY(VEC3_TAC); ONCE_REWRITE_TAC[Leaf_cell.RE_EQVL_SYM]; DISCH_THEN ((unlist REWRITE_TAC) o (MATCH_MP re_eqvl_imp_le)); BY(ASM_MESON_TAC[Local_lemmas.SIN_AZIM_POS_PI_LT]); REPEAT WEAKER_STRIP_TAC; BY(GEXISTL_TAC [`f`;`e'`] THEN ASM_MESON_TAC[]); COMMENT "planar case"; INTRO_TAC Cuxvzoz.deform_planar_exists [`V`;`(\ (t:real). &0)`;`(\ (t:real). &0)`;`g23`;`v0`;`v1`;`v2`;`v3`;`e`]; RULE_ASSUM_TAC(REWRITE_RULE[]); REPEAT WEAKER_STRIP_TAC THEN FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[arith `x + &0 = x`]; GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT THEN REWRITE_TAC[REAL_CONTINUOUS_CONST;Ocbicby.REAL_OPEN_REAL_INTERVAL]; ANTS_TAC; CONJ_TAC; BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]); BY(FIRST_X_ASSUM_ST `collinear` MP_TAC THEN MESON_TAC[COLLINEAR_2;SET_RULE `{a,b,b} = {a,b}`]); REPEAT WEAKER_STRIP_TAC; GEXISTL_TAC [`f`;`e'`] THEN ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; TYPIFY `~(v0 = vec 0)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[COPLANAR_3;SET_RULE `{a,a,b,c} = {a,b,c}`]); INTRO_TAC planar_deform_dist [`v0`;`v1`;`v2`;`f`;`V`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; BY(ASM_MESON_TAC[]); DISCH_TAC; BY(REPLICATE_TAC 2 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) *) let deform_pent_exists = prove_by_refinement( `!V g23 v0 v1 v2 v3 e. ?f. v1 IN V /\ v2 IN V /\ (coplanar {vec 0,v0,v1,v2} ==> (v1 cross v0) dot (v2 cross v0) > &0) /\ ~coplanar {vec 0,v0,v2,v3} /\ ~(collinear {vec 0,v1,v2}) /\ azim (vec 0) v1 v2 v0 <= pi /\ v2 dot (v3 cross v0) > &0 /\ &0 < e /\ g23 real_continuous_on (real_interval (--e,e)) /\ g23 (&0) = &0 ==> (?e'. &0 < e' /\ deformation f V (-- e', e') /\ (!v t. ~(v = v1) /\ ~(v = v2) ==> f v t = v) /\ (!t. abs t < e' ==> dist(v0,f v1 t) = dist(v0,v1) /\ dist(f v2 t,f v1 t) = dist(v2,v1) /\ norm(f v1 t) = norm(v1) /\ dist (f v2 t,v0) = dist(v2,v0) /\ dist (f v2 t,v3) = dist(v2,v3)+ g23 t /\ norm(f v2 t) = norm (v2) /\ (coplanar {vec 0,v0,v1,v2} ==> coplanar {vec 0,v0,f v1 t,f v2 t}) ))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[MESON [] `(?f. a ==> P f) <=> (a ==> (?f. P f))`]; REPEAT WEAKER_STRIP_TAC; TYPIFY `~coplanar {vec 0,v0,v1,v2}` ASM_CASES_TAC; INTRO_TAC Cuxvzoz.deform_684_pent_exists [`V`;`g23`;`v0`;`v1`;`v2`;`v3`;`e`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[]; REWRITE_TAC[arith `x > &0 <=> &0 <= x /\ ~(x = &0)`]; ANTS_TAC; ASM_REWRITE_TAC[]; CONJ2_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `coplanar` MP_TAC) THEN REWRITE_TAC[GSYM Local_lemmas.CROSS_DOT_COPLANAR] THEN VEC3_TAC); INTRO_TAC Trigonometry2.JBDNJJB [`v1`;`v2`;`v0`]; TYPIFY `((v1 cross v2) dot v0) = v1 dot (v2 cross v0)` (C SUBGOAL_THEN SUBST1_TAC); BY(VEC3_TAC); ONCE_REWRITE_TAC[Leaf_cell.RE_EQVL_SYM]; DISCH_THEN ((unlist REWRITE_TAC) o (MATCH_MP re_eqvl_imp_le)); BY(ASM_MESON_TAC[Local_lemmas.SIN_AZIM_POS_PI_LT]); REPEAT WEAKER_STRIP_TAC; BY(GEXISTL_TAC [`f`;`e'`] THEN ASM_MESON_TAC[]); COMMENT "planar case"; INTRO_TAC Cuxvzoz.deform_planar_exists [`V`;`(\ (t:real). &0)`;`(\ (t:real). &0)`;`g23`;`v0`;`v1`;`v2`;`v3`;`e`]; RULE_ASSUM_TAC(REWRITE_RULE[]); REPEAT WEAKER_STRIP_TAC THEN FIRST_X_ASSUM MP_TAC THEN ASM_REWRITE_TAC[arith `x + &0 = x`]; GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT THEN REWRITE_TAC[REAL_CONTINUOUS_CONST;Ocbicby.REAL_OPEN_REAL_INTERVAL]; ANTS_TAC; CONJ_TAC; BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]); BY(FIRST_X_ASSUM_ST `collinear` MP_TAC THEN MESON_TAC[COLLINEAR_2;SET_RULE `{a,b,b} = {a,b}`]); REPEAT WEAKER_STRIP_TAC; GEXISTL_TAC [`f`;`e'`] THEN ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; TYPIFY `~(v0 = vec 0)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[COPLANAR_3;SET_RULE `{a,a,b,c} = {a,b,c}`]); INTRO_TAC planar_deform_dist [`v0`;`v1`;`v2`;`f`;`V`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; BY(ASM_MESON_TAC[]); DISCH_TAC; BY(REPLICATE_TAC 2 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[DIST_SYM] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) (* let EGHNAVX_SCS = prove_by_refinement( `!s k v p. (let bta = (\j. azim (vec 0) (v p) (v (p+1)) (v (p+j))) in ( is_scs_v39 s /\ scs_k_v39 s = k /\ 3 < k /\ BBs_v39 s v /\ scs_basic_v39 s /\ scs_generic v ==> bta 1 = &0 /\ bta (k - 1) <= pi /\ (!i j. i < j /\ j < k ==> bta i <= bta j) /\ (!q j. bta q = &0 /\ 0 < j /\ j <= q /\ q < k ==> (j azim (vec 0) (v (p+j)) (v (p+j+1)) (v (p+j+k-1)) = pi) /\ v (p+j) IN aff_gt {vec 0,v p} {v (p+1)}) /\ (!q j. bta q = bta (k - 1) /\ 0 < q /\ q < k - 1 /\ q <= j /\ j < k ==> (q < j ==> azim (vec 0) (v (p+j)) (v (p+j+1)) (v(p+j+k-1)) = pi) /\ v(p+j) IN aff_gt {vec 0, v p} {v (p + k - 1)})))`, (* {{{ proof *) [ REPEAT GEN_TAC; LET_TAC; REPEAT WEAKER_STRIP_TAC; TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`; TYPED_ABBREV_TAC `V = IMAGE v (:num)`; TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`; INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `periodic v k /\ convex_local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN ASM_MESON_TAC[arith `3 < k ==> ~(k<= 3)`]); INTRO_TAC Terminal.ITER_vv_rho_node1 [`v`;`k`]; ASM_REWRITE_TAC[LET_THM]; TYPIFY_GOAL_THEN `3 <= k` (unlist REWRITE_TAC); BY(ASM_MESON_TAC[arith `3 < k ==> 3 <= k`]); DISCH_TAC; INTRO_TAC (GEN_ALL Odxlstcv2.CARD_V_EQ_SCS_K1) [`s`;`v`;`V`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; EXPAND_TAC "V"; MATCH_MP_TAC FUN_IN_IMAGE; BY(REWRITE_TAC[IN_UNIV]); TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0, v i, v j}` (C SUBGOAL_THEN ASSUME_TAC); INTRO_TAC Nkezbfc_local.PROPERTIES_GENERIC_LOCAL_FAN [`V`;`E`;`FF`]; BY(ASM_MESON_TAC[Appendix.scs_generic]); INTRO_TAC (GENL [`v0:real^3`;`vv:num->real^3`;`i:num`] Local_lemmas.EGHNAVX) [`v p`;`\ (i:num). v (p + i)`]; REWRITE_TAC[RIGHT_FORALL_IMP_THM;GSYM RIGHT_AND_FORALL_THM]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; REPEAT WEAKER_STRIP_TAC; TYPIFY `?i'. v' = v i'` (C SUBGOAL_THEN MP_TAC); FIRST_X_ASSUM_ST `IN` MP_TAC; EXPAND_TAC "V"; BY(REWRITE_TAC[IN_IMAGE;IN_UNIV]); REPEAT WEAKER_STRIP_TAC; BY(ASM_MESON_TAC[]); EXPAND_TAC "bta"; BY(REWRITE_TAC[]); REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; INTRO_TAC (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM) [`s`;`v`]; ASM_REWRITE_TAC[LET_THM]; TYPIFY_GOAL_THEN `v IN BBs_v39 s` (unlist REWRITE_TAC); BY(ASM_REWRITE_TAC[IN]); DISCH_TAC; COMMENT "first conjunct"; CONJ_TAC; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `SUBSET` (C INTRO_TAC [`q`]); ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; CONJ_TAC; DISCH_TAC; FIRST_X_ASSUM (C INTRO_TAC [`j`]); ASM_REWRITE_TAC[]; BY(REWRITE_TAC[arith `SUC (p+j) = p+j+1 /\ ((p+j) + k - 1 = p+j+k-1)`]); FIRST_X_ASSUM kill; FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[SUBSET;IN_ELIM_THM]; DISCH_THEN (C INTRO_TAC [`v (p+(j:num))`]); DISCH_THEN MATCH_MP_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `SUBSET` (C INTRO_TAC [`q`]); ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; CONJ_TAC; DISCH_TAC; FIRST_X_ASSUM (C INTRO_TAC [`j`]); ASM_REWRITE_TAC[]; BY(REWRITE_TAC[arith `SUC (p+j) = p+j+1 /\ ((p+j) + k - 1 = p+j+k-1)`]); FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[SUBSET;IN_ELIM_THM]; DISCH_THEN (C INTRO_TAC [`v (p+(j:num))`]); DISCH_THEN MATCH_MP_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]) ]);; (* }}} *) *) let EGHNAVX_COLL_SCS = prove_by_refinement( `!s k v p. (let bta = (\j. azim (vec 0) (v p) (v (p+1)) (v (p+j))) in ( is_scs_v39 s /\ scs_k_v39 s = k /\ 3 < k /\ BBs_v39 s v /\ scs_basic_v39 s /\ (!i. ~(v i = v p) ==> ~collinear {vec 0,v p,v i}) ==> bta 1 = &0 /\ bta (k - 1) <= pi /\ (!i j. i < j /\ j < k ==> bta i <= bta j) /\ (!q j. bta q = &0 /\ 0 < j /\ j <= q /\ q < k ==> (j azim (vec 0) (v (p+j)) (v (p+j+1)) (v (p+j+k-1)) = pi) /\ v (p+j) IN aff_gt {vec 0,v p} {v (p+1)}) /\ (!q j. bta q = bta (k - 1) /\ 0 < q /\ q < k - 1 /\ q <= j /\ j < k ==> (q < j ==> azim (vec 0) (v (p+j)) (v (p+j+1)) (v(p+j+k-1)) = pi) /\ v(p+j) IN aff_gt {vec 0, v p} {v (p + k - 1)})))`, (* {{{ proof *) [ REPEAT GEN_TAC; LET_TAC; REPEAT WEAKER_STRIP_TAC; TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`; TYPED_ABBREV_TAC `V = IMAGE v (:num)`; TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`; INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `periodic v k /\ convex_local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN ASM_MESON_TAC[arith `3 < k ==> ~(k<= 3)`]); INTRO_TAC Terminal.ITER_vv_rho_node1 [`v`;`k`]; ASM_REWRITE_TAC[LET_THM]; TYPIFY_GOAL_THEN `3 <= k` (unlist REWRITE_TAC); BY(ASM_MESON_TAC[arith `3 < k ==> 3 <= k`]); DISCH_TAC; INTRO_TAC (GEN_ALL Odxlstcv2.CARD_V_EQ_SCS_K1) [`s`;`v`;`V`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; EXPAND_TAC "V"; MATCH_MP_TAC FUN_IN_IMAGE; BY(REWRITE_TAC[IN_UNIV]); INTRO_TAC (GENL [`v0:real^3`;`vv:num->real^3`;`i:num`] Local_lemmas.EGHNAVX) [`v p`;`\ (i:num). v (p + i)`]; REWRITE_TAC[RIGHT_FORALL_IMP_THM;GSYM RIGHT_AND_FORALL_THM]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; EXPAND_TAC "V"; (REWRITE_TAC[IN_IMAGE;IN_UNIV]); BY(ASM_MESON_TAC[]); EXPAND_TAC "bta"; BY(REWRITE_TAC[]); REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; INTRO_TAC (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM) [`s`;`v`]; ASM_REWRITE_TAC[LET_THM]; TYPIFY_GOAL_THEN `v IN BBs_v39 s` (unlist REWRITE_TAC); BY(ASM_REWRITE_TAC[IN]); DISCH_TAC; COMMENT "first conjunct"; CONJ_TAC; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `SUBSET` (C INTRO_TAC [`q`]); ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; CONJ_TAC; DISCH_TAC; FIRST_X_ASSUM (C INTRO_TAC [`j`]); ASM_REWRITE_TAC[]; BY(REWRITE_TAC[arith `SUC (p+j) = p+j+1 /\ ((p+j) + k - 1 = p+j+k-1)`]); FIRST_X_ASSUM kill; FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[SUBSET;IN_ELIM_THM]; DISCH_THEN (C INTRO_TAC [`v (p+(j:num))`]); DISCH_THEN MATCH_MP_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `SUBSET` (C INTRO_TAC [`q`]); ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; CONJ_TAC; DISCH_TAC; FIRST_X_ASSUM (C INTRO_TAC [`j`]); ASM_REWRITE_TAC[]; BY(REWRITE_TAC[arith `SUC (p+j) = p+j+1 /\ ((p+j) + k - 1 = p+j+k-1)`]); FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[SUBSET;IN_ELIM_THM]; DISCH_THEN (C INTRO_TAC [`v (p+(j:num))`]); DISCH_THEN MATCH_MP_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]) ]);; (* }}} *) let EGHNAVX_SCS = prove_by_refinement( `!s k v p. (let bta = (\j. azim (vec 0) (v p) (v (p+1)) (v (p+j))) in ( is_scs_v39 s /\ scs_k_v39 s = k /\ 3 < k /\ BBs_v39 s v /\ scs_basic_v39 s /\ scs_generic v ==> bta 1 = &0 /\ bta (k - 1) <= pi /\ (!i j. i < j /\ j < k ==> bta i <= bta j) /\ (!q j. bta q = &0 /\ 0 < j /\ j <= q /\ q < k ==> (j azim (vec 0) (v (p+j)) (v (p+j+1)) (v (p+j+k-1)) = pi) /\ v (p+j) IN aff_gt {vec 0,v p} {v (p+1)}) /\ (!q j. bta q = bta (k - 1) /\ 0 < q /\ q < k - 1 /\ q <= j /\ j < k ==> (q < j ==> azim (vec 0) (v (p+j)) (v (p+j+1)) (v(p+j+k-1)) = pi) /\ v(p+j) IN aff_gt {vec 0, v p} {v (p + k - 1)})))`, (* {{{ proof *) [ REWRITE_TAC[LET_THM]; REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC (REWRITE_RULE[LET_THM] EGHNAVX_COLL_SCS); TYPIFY `s` EXISTS_TAC; ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `scs_generic` MP_TAC; REWRITE_TAC[Appendix.scs_generic]; DISCH_TAC; TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`; TYPED_ABBREV_TAC `V = IMAGE v (:num)`; TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`; TYPIFY `convex_local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN ASM_MESON_TAC[arith `3 < k ==> ~(k<= 3)`]); TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; EXPAND_TAC "V"; MATCH_MP_TAC FUN_IN_IMAGE; BY(REWRITE_TAC[IN_UNIV]); BY(ASM_MESON_TAC[Nkezbfc_local.PROPERTIES_GENERIC_LOCAL_FAN]) ]);; (* }}} *) let coplanar_cross_reduction = prove_by_refinement( `!s k v i. is_scs_v39 s /\ scs_k_v39 s = k /\ 3 < k /\ MMs_v39 s v /\ scs_basic_v39 s /\ scs_generic v /\ coplanar {vec 0,v i, v(i+1),v(i+2)} ==> (v(i+1) cross v i) dot (v(i+2) cross v i) > &0`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; TYPIFY `~(k=0)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`; TYPED_ABBREV_TAC `V = IMAGE v (:num)`; TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`; FIRST_ASSUM_ST `coplanar` MP_TAC; REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; DISCH_TAC; TYPIFY `~coplanar ({vec 0} UNION V)` (C SUBGOAL_THEN ASSUME_TAC); EXPAND_TAC "V"; MATCH_MP_TAC Fektyiy.FEKTYIY; TYPIFY `s` EXISTS_TAC; BY(ASM_REWRITE_TAC[IN]); TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs]); TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.BBprime2_v39]); TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.BBprime_v39]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `periodic v k /\ convex_local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN ASM_MESON_TAC[arith `3 < k ==> ~(k<= 3)`]); INTRO_TAC Terminal.ITER_vv_rho_node1 [`v`;`k`]; ASM_REWRITE_TAC[LET_THM]; ASM_SIMP_TAC[arith `3 < k ==> 3 <= k`]; DISCH_TAC; INTRO_TAC (GEN_ALL Odxlstcv2.CARD_V_EQ_SCS_K1) [`s`;`v`;`V`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; EXPAND_TAC "V"; MATCH_MP_TAC FUN_IN_IMAGE; BY(REWRITE_TAC[IN_UNIV]); TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0, v i, v j}` (C SUBGOAL_THEN ASSUME_TAC); INTRO_TAC Nkezbfc_local.PROPERTIES_GENERIC_LOCAL_FAN [`V`;`E`;`FF`]; BY(ASM_MESON_TAC[Appendix.scs_generic]); INTRO_TAC EGHNAVX_SCS [`s`;`k`;`v`;`i`]; REWRITE_TAC[LET_THM]; ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; COMMENT "insert here"; TYPIFY `!a. 0 < a /\ (a:num) < k ==> ~(v i = v (i+a))` (C SUBGOAL_THEN ASSUME_TAC); TYPIFY `!i. v (i MOD k) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_MESON_TAC[arith `3 < k ==> ~(0=k)`]); FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM); GEN_TAC THEN DISCH_TAC; TYPIFY_GOAL_THEN `!i j. i < k /\ j < (k:num) /\ ~(i = j) ==> ~(v i = v j)` MATCH_MP_TAC; BY(ASM_MESON_TAC[]); REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC; BY(ASM_MESON_TAC[DIVISION]); TYPIFY `i MOD k = (i+0) MOD k` (C SUBGOAL_THEN SUBST1_TAC); BY(REWRITE_TAC[arith `i + 0 = i`]); GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT; REPEAT (GMATCH_SIMP_TAC MOD_LT); BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN FIRST_X_ASSUM MP_TAC THEN ARITH_TAC); FIRST_X_ASSUM DISJ_CASES_TAC; FIRST_X_ASSUM_ST `&0` (C INTRO_TAC [`2`;`2`]); ASM_REWRITE_TAC[arith `0 < 2 /\ 2 <= 2 /\ ~(2 < 2)`]; ANTS_TAC; BY(MATCH_MP_TAC (arith `3 < k ==> 2 < k`) THEN ASM_REWRITE_TAC[]); GMATCH_SIMP_TAC Local_lemmas1.COLL_AFF_GT_2_1; SUBCONJ_TAC; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); DISCH_TAC; REWRITE_TAC[IN_ELIM_THM]; REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[CROSS_LMUL;CROSS_LADD;DOT_RMUL;DOT_RADD;CROSS_LZERO;DOT_RZERO;CROSS_REFL]; REWRITE_TAC[arith `t1 * &0 + x = x`;arith `x > &0 <=> &0 < x`]; GMATCH_SIMP_TAC REAL_LT_MUL_EQ; ASM_REWRITE_TAC[]; REWRITE_TAC[DOT_POS_LT]; REWRITE_TAC[CROSS_EQ_0]; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`]; BY(ASM_REWRITE_TAC[]); COMMENT "second conj"; FIRST_X_ASSUM_ST `azim` (C INTRO_TAC [`2`]); FIRST_X_ASSUM_ST `aff_gt` kill; FIRST_X_ASSUM_ST `0 < a` MP_TAC THEN BURY_MP_TAC; FIRST_X_ASSUM (C INTRO_TAC [`2`;`k-1`]); ANTS_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); DISCH_TAC; TYPIFY_GOAL_THEN `azim (vec 0) (v i) (v (i + 1)) (v (i + 2)) = azim (vec 0) (v i) (v (i + 1)) (v (i + k - 1))` (unlist REWRITE_TAC); BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC); DISCH_TAC; PROOF_BY_CONTR_TAC; TYPIFY `V SUBSET affine hull {vec 0,v i,v (i+k-1)}` ENOUGH_TO_SHOW_TAC; DISCH_TAC THEN FIRST_X_ASSUM_ST `coplanar` MP_TAC THEN REWRITE_TAC[coplanar]; GEXISTL_TAC [`(vec 0):real^3`;`v i`;`v (i+k-1)`]; FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[SUBSET;IN_UNION;IN_SING]; TYPIFY `vec 0 IN affine hull {vec 0,v i ,v(i+k-1)}` ENOUGH_TO_SHOW_TAC; BY(MESON_TAC[]); MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); TYPIFY `!w. w IN IMAGE v (i..i + k - 1) ==> w IN affine hull {vec 0,v i , v(i+k-1)}` ENOUGH_TO_SHOW_TAC; GMATCH_SIMP_TAC (GSYM Oxl_2012.PERIODIC_IMAGE_EQUAL); ASM_REWRITE_TAC[]; DISCH_TAC; INTRO_TAC Oxl_def.periodic_numseg [`(\j. v j IN affine hull {vec 0,v i,v (i+k-1)})`;`k`]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; FIRST_X_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic]; BY(MESON_TAC[]); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; MATCH_MP_TAC FUN_IN_IMAGE; BY(ASM_REWRITE_TAC[]); DISCH_TAC; EXPAND_TAC "V"; REWRITE_TAC[IMAGE;IN_UNIV;SUBSET;IN_ELIM_THM]; BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]); REWRITE_TAC[IN_IMAGE;IN_NUMSEG]; REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; TYPIFY `?j. 0 <= j /\ j < k /\ x = i + j` (C SUBGOAL_THEN MP_TAC); TYPIFY `x - (i:num)` EXISTS_TAC; BY(REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; TYPIFY `!j. 2 <= j /\ j < k==> v (i + j) IN affine hull {vec 0,v i, v(i+k-1)}` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM (C INTRO_TAC [`j'`]); ANTS_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC); TYPIFY `aff_gt {vec 0,v i} {v (i+k-1)} SUBSET affine hull {vec 0,v i,v (i+k-1)}` ENOUGH_TO_SHOW_TAC; BY(SET_TAC[]); TYPIFY `{vec 0,v i,v (i+k-1)} = {vec 0,v i} UNION {v (i+k-1)}` (C SUBGOAL_THEN SUBST1_TAC); BY(SET_TAC[]); BY(REWRITE_TAC[ AFF_GT_SUBSET_AFFINE_HULL]); TYPIFY `{vec 0,v i, v (i+2)} SUBSET affine hull {vec 0, v i, v (i+k-1)}` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]; REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); INTRO_TAC AFFINE_HULL_3_GENERATED [`{vec 0, v i, v (i + 2)}`;`(vec 0):real^3`;`v i`;`v (i+k-1)`]; ANTS_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); DISCH_TAC; ASSUME_TAC (arith ` 2 <= j \/ j = 0 \/ j = 1` ); REPEAT (POP_ASSUM DISJ_CASES_TAC); BY(ASM_MESON_TAC[]); ASM_REWRITE_TAC[arith `i + 0 = i`] THEN MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); ASM_REWRITE_TAC[]; (FIRST_X_ASSUM_ST `coplanar (a INSERT X)` MP_TAC); REWRITE_TAC[coplanar] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC AFFINE_HULL_3_GENERATED [`{vec 0, v i, v (i + 2)}`;`u`;`v'`;`w'`]; ANTS_TAC; CONJ_TAC; BY(FIRST_X_ASSUM MP_TAC THEN SET_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); BY(FIRST_X_ASSUM MP_TAC THEN SET_TAC[]) ]);; (* }}} *) let coplanar_aff_gt_simple = prove_by_refinement( `!s k v i. is_scs_v39 s /\ scs_k_v39 s = k /\ 3 < k /\ MMs_v39 s v /\ scs_basic_v39 s /\ (!j. ~(v j = v i) ==> ~collinear {vec 0,v i,v j}) /\ coplanar {vec 0,v i, v(i+1),v(i+2)} ==> v (i+1) IN aff_gt {vec 0} {v i, v(i+2)}`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; TYPIFY `~(k=0)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); TYPED_ABBREV_TAC `FF = IMAGE (\i. (v i,v (SUC i))) (:num)`; TYPED_ABBREV_TAC `V = IMAGE v (:num)`; TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`; FIRST_ASSUM_ST `coplanar` MP_TAC; REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; DISCH_TAC; TYPIFY `~coplanar ({vec 0} UNION V)` (C SUBGOAL_THEN ASSUME_TAC); EXPAND_TAC "V"; MATCH_MP_TAC Fektyiy.FEKTYIY; TYPIFY `s` EXISTS_TAC; BY(ASM_REWRITE_TAC[IN]); TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs]); TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.BBprime2_v39]); TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.BBprime_v39]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `periodic v k /\ convex_local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN ASM_MESON_TAC[arith `3 < k ==> ~(k<= 3)`]); INTRO_TAC Terminal.ITER_vv_rho_node1 [`v`;`k`]; ASM_REWRITE_TAC[LET_THM]; ASM_SIMP_TAC[arith `3 < k ==> 3 <= k`]; DISCH_TAC; INTRO_TAC (GEN_ALL Odxlstcv2.CARD_V_EQ_SCS_K1) [`s`;`v`;`V`;`k`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; EXPAND_TAC "V"; MATCH_MP_TAC FUN_IN_IMAGE; BY(REWRITE_TAC[IN_UNIV]); INTRO_TAC EGHNAVX_COLL_SCS [`s`;`k`;`v`;`i`]; REWRITE_TAC[LET_THM]; ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; TYPIFY `!a. 0 < a /\ (a:num) < k ==> ~(v i = v (i+a))` (C SUBGOAL_THEN ASSUME_TAC); TYPIFY `!i. v (i MOD k) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_MESON_TAC[arith `3 < k ==> ~(0=k)`]); FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM); GEN_TAC THEN DISCH_TAC; TYPIFY_GOAL_THEN `!i j. i < k /\ j < (k:num) /\ ~(i = j) ==> ~(v i = v j)` MATCH_MP_TAC; BY(ASM_MESON_TAC[]); REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC; BY(ASM_MESON_TAC[DIVISION]); TYPIFY `i MOD k = (i+0) MOD k` (C SUBGOAL_THEN SUBST1_TAC); BY(REWRITE_TAC[arith `i + 0 = i`]); GMATCH_SIMP_TAC Ocbicby.MOD_EQ_MOD_SHIFT; REPEAT (GMATCH_SIMP_TAC MOD_LT); BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN FIRST_X_ASSUM MP_TAC THEN ARITH_TAC); FIRST_X_ASSUM DISJ_CASES_TAC; FIRST_X_ASSUM_ST `&0` (C INTRO_TAC [`2`;`2`]); NUM_REDUCE_TAC; ANTS_TAC; ASM_REWRITE_TAC[]; BY(MATCH_MP_TAC (arith `3 < k ==> 2 < k`) THEN ASM_REWRITE_TAC[]); GMATCH_SIMP_TAC Local_lemmas1.COLL_AFF_GT_2_1; GMATCH_SIMP_TAC AFF_GT_1_2; REWRITE_TAC[IN_ELIM_THM;arith `t % vec 0 + a = a`]; SUBCONJ_TAC; ONCE_REWRITE_TAC[DISJOINT_SYM]; MATCH_MP_TAC Fan.th3a; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,a,b}`] THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC); ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); DISCH_TAC; CONJ_TAC; FIRST_X_ASSUM MATCH_MP_TAC; ONCE_REWRITE_TAC[EQ_SYM_EQ]; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); REPEAT WEAKER_STRIP_TAC; GEXISTL_TAC [`&1 + t2 / t3 - &1 / t3`;`-- t2 / t3`;`&1 / t3`]; ASM_REWRITE_TAC[]; GMATCH_SIMP_TAC REAL_LT_DIV; GMATCH_SIMP_TAC REAL_LT_DIV; ASM_REWRITE_TAC[arith `&0 < &1`]; nCONJ_TAC 2; TYPIFY `--t2 / t3 % v i + &1 / t3 % (t2 % v i + t3 % v (i + 1)) = (-- t2 / t3 + (&1 / t3) * t2) % v i + ((&1 / t3)*t3) % v (i+1) ` (C SUBGOAL_THEN SUBST1_TAC); BY(VECTOR_ARITH_TAC); TYPIFY_GOAL_THEN `(--t2 / t3 + &1 / t3 * t2) = &0 /\ (&1 / t3 * t3 = &1)` (unlist REWRITE_TAC); BY(CONJ_TAC THEN Calc_derivative.CALC_ID_TAC THEN (REPLICATE_TAC 3 (FIRST_X_ASSUM MP_TAC)) THEN REAL_ARITH_TAC); BY(VECTOR_ARITH_TAC); nCONJ_TAC 1; BY(Calc_derivative.CALC_ID_TAC THEN REPLICATE_TAC 5 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC); TYPIFY `~(&0 <= t2)` ENOUGH_TO_SHOW_TAC; BY(REPLICATE_TAC 2 (POP_ASSUM MP_TAC) THEN MESON_TAC[arith `~(&0 <= t2) ==> &0 < -- t2`]); DISCH_TAC; INTRO_TAC (GEN_ALL Local_lemmas.FAN_IN_AFF_GE_IMP_EQ) [`V`;`E`;`(vec 0):real^3`;`v i`;`v (i+2)`;`v (i+1)`]; ANTS_TAC; CONJ_TAC; MATCH_MP_TAC Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN; BY(ASM_REWRITE_TAC[]); CONJ_TAC; EXPAND_TAC "V"; MATCH_MP_TAC FUN_IN_IMAGE; BY(REWRITE_TAC[IN_UNIV]); CONJ_TAC; EXPAND_TAC "E"; BY(REWRITE_TAC[arith `i+1 = SUC i`;IN_IMAGE;IN_UNIV] THEN MESON_TAC[]); GMATCH_SIMP_TAC AFF_GE_1_2; REWRITE_TAC[IN_ELIM_THM]; CONJ_TAC; ONCE_REWRITE_TAC[DISJOINT_SYM]; MATCH_MP_TAC Fan.th3a; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {c,a,b}`] THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC); ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); GEXISTL_TAC [`t1`;`t2`;`t3`]; FIRST_X_ASSUM_ST `i+2` SUBST1_TAC; REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC; BY(VECTOR_ARITH_TAC); BY(REPLICATE_TAC 6 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC); REWRITE_TAC[DE_MORGAN_THM]; CONJ_TAC; ONCE_REWRITE_TAC[EQ_SYM_EQ] THEN FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); ONCE_REWRITE_TAC[EQ_SYM_EQ]; MATCH_MP_TAC Hypermap_iso.fan_in_e_imp_neq; GEXISTL_TAC [`V`;`E`]; CONJ_TAC; MATCH_MP_TAC Wrgcvdr_cizmrrh.LOCAL_FAN_IMP_FAN; BY(ASM_REWRITE_TAC[]); EXPAND_TAC "E"; BY(REWRITE_TAC[arith `i+2 = SUC (i+1)`;IN_IMAGE;IN_UNIV] THEN MESON_TAC[]); COMMENT "second disj"; FIRST_X_ASSUM_ST `azim` (C INTRO_TAC [`2`]); FIRST_X_ASSUM_ST `aff_gt` kill; FIRST_X_ASSUM_ST `0 < a` MP_TAC THEN BURY_MP_TAC; FIRST_X_ASSUM (C INTRO_TAC [`2`;`k-1`]); ANTS_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); DISCH_TAC; TYPIFY_GOAL_THEN `azim (vec 0) (v i) (v (i + 1)) (v (i + 2)) = azim (vec 0) (v i) (v (i + 1)) (v (i + k - 1))` (unlist REWRITE_TAC); BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC); DISCH_TAC; PROOF_BY_CONTR_TAC; TYPIFY `V SUBSET affine hull {vec 0,v i,v (i+k-1)}` ENOUGH_TO_SHOW_TAC; DISCH_TAC THEN FIRST_X_ASSUM_ST `coplanar` MP_TAC THEN REWRITE_TAC[coplanar]; GEXISTL_TAC [`(vec 0):real^3`;`v i`;`v (i+k-1)`]; FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[SUBSET;IN_UNION;IN_SING]; TYPIFY `vec 0 IN affine hull {vec 0,v i ,v(i+k-1)}` ENOUGH_TO_SHOW_TAC; BY(MESON_TAC[]); MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); TYPIFY `!w. w IN IMAGE v (i..i + k - 1) ==> w IN affine hull {vec 0,v i , v(i+k-1)}` ENOUGH_TO_SHOW_TAC; GMATCH_SIMP_TAC (GSYM Oxl_2012.PERIODIC_IMAGE_EQUAL); ASM_REWRITE_TAC[]; DISCH_TAC; INTRO_TAC Oxl_def.periodic_numseg [`(\j. v j IN affine hull {vec 0,v i,v (i+k-1)})`;`k`]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; FIRST_X_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic]; BY(MESON_TAC[]); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; MATCH_MP_TAC FUN_IN_IMAGE; BY(ASM_REWRITE_TAC[]); DISCH_TAC; EXPAND_TAC "V"; REWRITE_TAC[IMAGE;IN_UNIV;SUBSET;IN_ELIM_THM]; BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]); REWRITE_TAC[IN_IMAGE;IN_NUMSEG]; REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; TYPIFY `?j. 0 <= j /\ j < k /\ x = i + j` (C SUBGOAL_THEN MP_TAC); TYPIFY `x - (i:num)` EXISTS_TAC; BY(REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; TYPIFY `!j. 2 <= j /\ j < k==> v (i + j) IN affine hull {vec 0,v i, v(i+k-1)}` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM (C INTRO_TAC [`j'`]); ANTS_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN ARITH_TAC); TYPIFY `aff_gt {vec 0,v i} {v (i+k-1)} SUBSET affine hull {vec 0,v i,v (i+k-1)}` ENOUGH_TO_SHOW_TAC; BY(SET_TAC[]); TYPIFY `{vec 0,v i,v (i+k-1)} = {vec 0,v i} UNION {v (i+k-1)}` (C SUBGOAL_THEN SUBST1_TAC); BY(SET_TAC[]); BY(REWRITE_TAC[ AFF_GT_SUBSET_AFFINE_HULL]); TYPIFY `{vec 0,v i, v (i+2)} SUBSET affine hull {vec 0, v i, v (i+k-1)}` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]; REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); INTRO_TAC AFFINE_HULL_3_GENERATED [`{vec 0, v i, v (i + 2)}`;`(vec 0):real^3`;`v i`;`v (i+k-1)`]; ANTS_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC; ONCE_REWRITE_TAC[EQ_SYM_EQ]; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); INTRO_TAC AFFINE_HULL_3_GENERATED [`{vec 0, v i, v (i + 2)}`;`(vec 0):real^3`;`v i`;`v (i+k-1)`]; ANTS_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM MATCH_MP_TAC; ONCE_REWRITE_TAC[EQ_SYM_EQ]; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); DISCH_TAC; ASSUME_TAC (arith ` 2 <= j \/ j = 0 \/ j = 1` ); REPEAT (POP_ASSUM DISJ_CASES_TAC); BY(ASM_MESON_TAC[]); ASM_REWRITE_TAC[arith `i + 0 = i`] THEN MATCH_MP_TAC Marchal_cells_2_new.IN_AFFINE_KY_LEMMA1; BY(SET_TAC[]); ASM_REWRITE_TAC[]; (FIRST_X_ASSUM_ST `coplanar (a INSERT X)` MP_TAC); REWRITE_TAC[coplanar] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC AFFINE_HULL_3_GENERATED [`{vec 0, v i, v (i + 2)}`;`u`;`v'`;`w'`]; ANTS_TAC; CONJ_TAC; BY(FIRST_X_ASSUM MP_TAC THEN SET_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; ONCE_REWRITE_TAC[EQ_SYM_EQ]; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM_ST `3 < k` MP_TAC THEN ARITH_TAC); BY(FIRST_X_ASSUM MP_TAC THEN SET_TAC[]) ]);; (* }}} *) (* let a5_assumption_reduction = prove_by_refinement( `!s f v e. is_scs_v39 s /\ scs_k_v39 s = 5 /\ BBs_v39 s v /\ (!i j. scs_diag 5 i j ==> scs_a_v39 s i j < dist(v i,v j)) /\ deformation f (IMAGE v (:num)) (--e,e) /\ (!w t. ~(w = v 0) /\ ~(w = v 1) ==> (f w t = w)) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist(f (v 1) t,v 2) = dist(v 1,v 2) /\ dist(f (v 0) t,v 4) = dist(v 0,v 4))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> scs_a_v39 s 0 1 <= dist(f (v 0) t,f (v 1) t))) ==> (!i j. ?e1. &0 < e1 /\ (scs_a_v39 s i j = dist(v i,v j) ==> (!t. abs t < e1 ==> scs_a_v39 s i j <= dist(f (v i) t,f (v j) t)))) `, (* {{{ proof *) [ REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC Cuxvzoz.I_IMP THEN ASM_TAC THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `!i j.scs_a_v39 s (i MOD 5) (j MOD 5) = scs_a_v39 s i j /\ scs_b_v39 s (i MOD 5) (j MOD 5) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC; ASM_REWRITE_TAC[Appendix.is_scs_v39]; REPEAT WEAKER_STRIP_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `periodic2` MP_TAC) THEN MESON_TAC[Cuxvzoz.periodic2_MOD;arith `~(5 = 0)`]); TYPIFY `periodic v 5` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]); TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `!j. j < 5 ==> (j MOD 5 = j)` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[MOD_LT]); COMMENT "scs_a_v39"; REWRITE_TAC[I_THM]; MATCH_MP_TAC Terminal.periodic2_mod_reduce; TYPIFY `5` EXISTS_TAC; ASM_REWRITE_TAC[arith `~(5=0)`]; CONJ_TAC; MATCH_MP_TAC Cuxvzoz.MOD_periodic2; BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[arith `~(5=0)`]); TYPIFY_GOAL_THEN `!P. (!i j. P i j ==> P j i) /\ (!i j. (i <= (j:num)) ==> P i j) ==> (!i j. P i j)` MATCH_MP_TAC; BY(MESON_TAC[arith `i <= j \/ j <= (i:num)`]); CONJ_TAC; REPEAT WEAKER_STRIP_TAC; ONCE_REWRITE_TAC[DIST_SYM]; TYPIFY `scs_a_v39 s j i = scs_a_v39 s i j` (C SUBGOAL_THEN SUBST1_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN REPEAT WEAKER_STRIP_TAC; BY(ASM_MESON_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); TYPIFY `!t. f (v 2) t = v 2 /\ f (v 3) t = v 3 /\ f (v 4) t = v 4` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[arith `0 < 5 /\ 1 < 5 /\ 2 < 5 /\ 3 < 5 /\ 4 < 5 /\ ~(2 = 0) /\ ~(2 = 1) /\ ~(3 = 0) /\ ~(3 = 1) /\ ~(4 =0) /\ ~(4=1)`]); REWRITE_TAC[arith `i < 5 <=> (i = 0) \/ (i = 1) \/ (i=2) \/ (i =3) \/ (i=4)`]; TYPIFY `scs_diag 5 0 2 /\ scs_diag 5 0 3 /\ scs_diag 5 1 3 /\ scs_diag 5 1 4 /\ scs_diag 5 2 4` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[Appendix.scs_diag;Uxckfpe.ARITH_5_TAC]); ONCE_REWRITE_TAC[TAUT `(a ==> b ==> c) <=> (b ==> a ==> c)`]; TYPIFY `!i j. scs_diag 5 i j ==> (?e1. &0 < e1 /\ (scs_a_v39 s i j = dist (v i,v j) ==> (!t. abs t < e1 ==> scs_a_v39 s i j <= dist (f (v i) t,f (v j) t))))` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]); TYPIFY `!i. T ==> (?e1. &0 < e1 /\ (scs_a_v39 s i i = dist (v i,v i) ==> (!t. abs t < e1 ==> scs_a_v39 s i i <= dist (f (v i) t,f (v i) t))))` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[DIST_REFL]; BY(MESON_TAC[arith `x = &0 ==> x <= &0`;arith `&0 < &1`]); REPEAT GEN_TAC; TYPIFY `(?e1. &0 < e1 /\ (scs_a_v39 s 0 1 = dist (v 0,v 1) ==> (!t. abs t < e1 ==> scs_a_v39 s 0 1 <= dist (f (v 0) t,f (v 1) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 1 2 = dist (v 1,v 2) ==> (!t. abs t < e1 ==> scs_a_v39 s 1 2 <= dist (f (v 1) t,f (v 2) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 2 3 = dist (v 2,v 3) ==> (!t. abs t < e1 ==> scs_a_v39 s 2 3 <= dist (f (v 2) t,f (v 3) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 3 4 = dist (v 3,v 4) ==> (!t. abs t < e1 ==> scs_a_v39 s 3 4 <= dist (f (v 3) t,f (v 4) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 0 4 = dist (v 0,v 4) ==> (!t. abs t < e1 ==> scs_a_v39 s 0 4 <= dist (f (v 0) t,f (v 4) t))))` (C SUBGOAL_THEN MP_TAC); ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `!a b. (?e1. &0 < e1 /\ (a = b ==> (!t. abs t < e1 ==> a <= b)))` (unlist REWRITE_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `(a = b) ==> (a <= b)`]); CONJ_TAC; BY(ASM_MESON_TAC[]); BY(ASM_MESON_TAC[arith `x = y ==> x <= y`]); REWRITE_TAC[LEFT_OR_DISTRIB;RIGHT_OR_DISTRIB;GSYM DISJ_ASSOC]; ONCE_REWRITE_TAC[TAUT `(a1 /\ a2 /\ a3 /\ a4 /\ a5) <=> ((T ==> a1) /\ (T ==> a2) /\ (T ==> a3) /\ (T ==> a4) /\ (T ==> a5))`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN FIRST_X_ASSUM (unlist REWRITE_TAC) THEN NUM_REDUCE_TAC THEN TRY (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) *) let a5_assumption_reduction = prove_by_refinement( `!s f v e. is_scs_v39 s /\ scs_k_v39 s = 5 /\ BBs_v39 s v /\ (!i j. scs_diag 5 i j ==> scs_a_v39 s i j < dist(v i,v j)) /\ deformation f (IMAGE v (:num)) (--e,e) /\ (!w t. ~(w = v 4) /\ ~(w = v 0) ==> (f w t = w)) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist(f (v 4) t,f (v 0) t) = dist(v 4,v 0) /\ dist(f (v 4) t,v 3) = dist(v 4,v 3))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> scs_a_v39 s 0 1 <= dist(f (v 0) t,v 1))) ==> (!i j. ?e1. &0 < e1 /\ (scs_a_v39 s i j = dist(v i,v j) ==> (!t. abs t < e1 ==> scs_a_v39 s i j <= dist(f (v i) t,f (v j) t)))) `, (* {{{ proof *) [ REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC Cuxvzoz.I_IMP THEN ASM_TAC THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `!i j.scs_a_v39 s (i MOD 5) (j MOD 5) = scs_a_v39 s i j /\ scs_b_v39 s (i MOD 5) (j MOD 5) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC; ASM_REWRITE_TAC[Appendix.is_scs_v39]; REPEAT WEAKER_STRIP_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `periodic2` MP_TAC) THEN MESON_TAC[Cuxvzoz.periodic2_MOD;arith `~(5 = 0)`]); TYPIFY `periodic v 5` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]); TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `!j. j < 5 ==> (j MOD 5 = j)` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[MOD_LT]); COMMENT "scs_a_v39"; REWRITE_TAC[I_THM]; MATCH_MP_TAC Terminal.periodic2_mod_reduce; TYPIFY `5` EXISTS_TAC; ASM_REWRITE_TAC[arith `~(5=0)`]; CONJ_TAC; MATCH_MP_TAC Cuxvzoz.MOD_periodic2; BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[arith `~(5=0)`]); TYPIFY_GOAL_THEN `!P. (!i j. P i j ==> P j i) /\ (!i j. (i <= (j:num)) ==> P i j) ==> (!i j. P i j)` MATCH_MP_TAC; BY(MESON_TAC[arith `i <= j \/ j <= (i:num)`]); CONJ_TAC; REPEAT WEAKER_STRIP_TAC; ONCE_REWRITE_TAC[DIST_SYM]; TYPIFY `scs_a_v39 s j i = scs_a_v39 s i j` (C SUBGOAL_THEN SUBST1_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN REPEAT WEAKER_STRIP_TAC; BY(ASM_MESON_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); TYPIFY `!i j. i < 5 /\ j < 5 /\ ~(i=j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[]); TYPIFY `!t. f (v 2) t = v 2 /\ f (v 3) t = v 3 /\ f (v 1) t = v 1` (C SUBGOAL_THEN ASSUME_TAC); BY(GEN_TAC THEN REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC THEN REPEAT CONJ_TAC) THEN NUM_REDUCE_TAC); REWRITE_TAC[arith `i < 5 <=> (i = 0) \/ (i = 1) \/ (i=2) \/ (i =3) \/ (i=4)`]; TYPIFY `scs_diag 5 0 2 /\ scs_diag 5 0 3 /\ scs_diag 5 1 3 /\ scs_diag 5 1 4 /\ scs_diag 5 2 4` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[Appendix.scs_diag;Uxckfpe.ARITH_5_TAC]); ONCE_REWRITE_TAC[TAUT `(a ==> b ==> c) <=> (b ==> a ==> c)`]; TYPIFY `!i j. scs_diag 5 i j ==> (?e1. &0 < e1 /\ (scs_a_v39 s i j = dist (v i,v j) ==> (!t. abs t < e1 ==> scs_a_v39 s i j <= dist (f (v i) t,f (v j) t))))` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[arith `x < y ==> ~(x = y)`]); TYPIFY `!i. T ==> (?e1. &0 < e1 /\ (scs_a_v39 s i i = dist (v i,v i) ==> (!t. abs t < e1 ==> scs_a_v39 s i i <= dist (f (v i) t,f (v i) t))))` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[DIST_REFL]; BY(MESON_TAC[arith `x = &0 ==> x <= &0`;arith `&0 < &1`]); REPEAT GEN_TAC; TYPIFY `(?e1. &0 < e1 /\ (scs_a_v39 s 0 1 = dist (v 0,v 1) ==> (!t. abs t < e1 ==> scs_a_v39 s 0 1 <= dist (f (v 0) t,f (v 1) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 1 2 = dist (v 1,v 2) ==> (!t. abs t < e1 ==> scs_a_v39 s 1 2 <= dist (f (v 1) t,f (v 2) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 2 3 = dist (v 2,v 3) ==> (!t. abs t < e1 ==> scs_a_v39 s 2 3 <= dist (f (v 2) t,f (v 3) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 3 4 = dist (v 3,v 4) ==> (!t. abs t < e1 ==> scs_a_v39 s 3 4 <= dist (f (v 3) t,f (v 4) t)))) /\ (?e1. &0 < e1 /\ (scs_a_v39 s 0 4 = dist (v 0,v 4) ==> (!t. abs t < e1 ==> scs_a_v39 s 0 4 <= dist (f (v 0) t,f (v 4) t))))` (C SUBGOAL_THEN MP_TAC); ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `!a b. (?e1. &0 < e1 /\ (a = b ==> (!t. abs t < e1 ==> a <= b)))` (unlist REWRITE_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `(a = b) ==> (a <= b)`]); CONJ_TAC; BY(ASM_MESON_TAC[]); ONCE_REWRITE_TAC[DIST_SYM]; BY(ASM_MESON_TAC[arith `x = y ==> x <= y`]); REWRITE_TAC[LEFT_OR_DISTRIB;RIGHT_OR_DISTRIB;GSYM DISJ_ASSOC]; ONCE_REWRITE_TAC[TAUT `(a1 /\ a2 /\ a3 /\ a4 /\ a5) <=> ((T ==> a1) /\ (T ==> a2) /\ (T ==> a3) /\ (T ==> a4) /\ (T ==> a5))`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN FIRST_X_ASSUM (unlist REWRITE_TAC) THEN NUM_REDUCE_TAC THEN TRY (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) (* let b5_assumption_reduction = prove_by_refinement( `!s f v e. is_scs_v39 s /\ scs_k_v39 s = 5 /\ BBs_v39 s v /\ (!i j. scs_diag 5 i j ==> &4 * h0 < scs_b_v39 s i j) /\ deformation f (IMAGE v (:num)) (--e,e) /\ (!w t. ~(w = v 0) /\ ~(w = v 1) ==> (f w t = w)) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist(f (v 1) t,v 2) = dist(v 1,v 2) /\ dist(f (v 0) t,v 4) = dist(v 0,v 4))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist(f (v 0) t,f (v 1) t) <= dist(v 0, v 1))) ==> (!i j. ?e1. &0 < e1 /\ (dist(v i,v j) = scs_b_v39 s i j ==> (!t. abs t < e1 ==> dist(f (v i) t,f (v j) t) <= scs_b_v39 s i j))) `, (* {{{ proof *) [ REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC Cuxvzoz.I_IMP THEN ASM_TAC THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `!i j.scs_a_v39 s (i MOD 5) (j MOD 5) = scs_a_v39 s i j /\ scs_b_v39 s (i MOD 5) (j MOD 5) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC; ASM_REWRITE_TAC[Appendix.is_scs_v39]; REPEAT WEAKER_STRIP_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `periodic2` MP_TAC) THEN MESON_TAC[Cuxvzoz.periodic2_MOD;arith `~(5 = 0)`]); TYPIFY `periodic v 5` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]); TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `!j. j < 5 ==> (j MOD 5 = j)` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[MOD_LT]); COMMENT "scs_b_v39"; REWRITE_TAC[I_THM]; MATCH_MP_TAC Terminal.periodic2_mod_reduce; TYPIFY `5` EXISTS_TAC; ASM_REWRITE_TAC[arith `~(5=0)`]; CONJ_TAC; MATCH_MP_TAC Cuxvzoz.MOD_periodic2; BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[arith `~(5=0)`]); TYPIFY_GOAL_THEN `!P. (!i j. P i j ==> P j i) /\ (!i j. (i <= (j:num)) ==> P i j) ==> (!i j. P i j)` MATCH_MP_TAC; BY(MESON_TAC[arith `i <= j \/ j <= (i:num)`]); CONJ_TAC; REPEAT WEAKER_STRIP_TAC; ONCE_REWRITE_TAC[DIST_SYM]; TYPIFY `scs_b_v39 s j i = scs_b_v39 s i j` (C SUBGOAL_THEN SUBST1_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN REPEAT WEAKER_STRIP_TAC; BY(ASM_MESON_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); TYPIFY `!t. f (v 2) t = v 2 /\ f (v 3) t = v 3 /\ f (v 4) t = v 4` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[arith `0 < 5 /\ 1 < 5 /\ 2 < 5 /\ 3 < 5 /\ 4 < 5 /\ ~(2 = 0) /\ ~(2 = 1) /\ ~(3 = 0) /\ ~(3 = 1) /\ ~(4 =0) /\ ~(4=1)`]); TYPIFY `scs_diag 5 0 2 /\ scs_diag 5 0 3 /\ scs_diag 5 1 3 /\ scs_diag 5 1 4 /\ scs_diag 5 2 4` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[Appendix.scs_diag;Uxckfpe.ARITH_5_TAC]); TYPIFY `!i j. scs_diag 5 i j ==> (?e1. &0 < e1 /\ (dist (v i,v j) = scs_b_v39 s i j ==> (!t. abs t < e1 ==> dist (f (v i) t,f (v j) t) <= scs_b_v39 s i j)))` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; TYPIFY `&1` EXISTS_TAC; (REWRITE_TAC[arith `&0 < &1`]); TYPIFY `dist(v i,v j) <= &4 * h0` ENOUGH_TO_SHOW_TAC; FIRST_X_ASSUM_ST `&4 * h0` (C INTRO_TAC [`i`;`j`]); ASM_REWRITE_TAC[]; BY(REAL_ARITH_TAC); INTRO_TAC DIST_TRIANGLE [`v i`;`(vec 0):real^3`;`v j`]; REWRITE_TAC[DIST_0]; TYPIFY `!i. norm (v i) <= &2 * h0` ENOUGH_TO_SHOW_TAC; BY(DISCH_TAC THEN FIRST_ASSUM (C INTRO_TAC [`i`]) THEN FIRST_X_ASSUM (C INTRO_TAC [`j`]) THEN REAL_ARITH_TAC); FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC; BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN REWRITE_TAC[Terminal.IMAGE_SUBSET_IN;IN_UNIV] THEN MESON_TAC[Fnjlbxs.in_ball_annulus]); TYPIFY `!i. T ==> (?e1. &0 < e1 /\ (dist (v i,v i) = scs_b_v39 s i i ==> (!t. abs t < e1 ==> dist (f (v i) t,f (v i) t) <= scs_b_v39 s i i)))` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[DIST_REFL]; BY(MESON_TAC[arith `x = &0 ==> x <= &0`;arith `&0 < &1`]); REPEAT GEN_TAC; TYPIFY `(?e1. &0 < e1 /\ (dist (v 0,v 1) = scs_b_v39 s 0 1 ==> (!t. abs t < e1 ==> dist (f (v 0) t,f (v 1) t) <= scs_b_v39 s 0 1))) /\(?e1. &0 < e1 /\ (dist (v 1,v 2) = scs_b_v39 s 1 2 ==> (!t. abs t < e1 ==> dist (f (v 1) t,f (v 2) t) <= scs_b_v39 s 1 2))) /\(?e1. &0 < e1 /\ (dist (v 2,v 3) = scs_b_v39 s 2 3 ==> (!t. abs t < e1 ==> dist (f (v 2) t,f (v 3) t) <= scs_b_v39 s 2 3))) /\(?e1. &0 < e1 /\ (dist (v 3,v 4) = scs_b_v39 s 3 4 ==> (!t. abs t < e1 ==> dist (f (v 3) t,f (v 4) t) <= scs_b_v39 s 3 4))) /\(?e1. &0 < e1 /\ (dist (v 0,v 4) = scs_b_v39 s 0 4 ==> (!t. abs t < e1 ==> dist (f (v 0) t,f (v 4) t) <= scs_b_v39 s 0 4)))` (C SUBGOAL_THEN MP_TAC); ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `!a b. (?e1. &0 < e1 /\ (a = b ==> (!t. abs t < e1 ==> a <= b)))` (unlist REWRITE_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `(a = b) ==> (a <= b)`]); CONJ_TAC; BY(ASM_MESON_TAC[]); BY(ASM_MESON_TAC[arith `x = y ==> x <= y`]); DISCH_TAC; REWRITE_TAC[arith `i < 5 <=> (i = 0) \/ (i = 1) \/ (i=2) \/ (i =3) \/ (i=4)`]; ONCE_REWRITE_TAC[TAUT `(a ==> b ==> c) <=> (b ==> a ==> c)`]; REWRITE_TAC[LEFT_OR_DISTRIB;RIGHT_OR_DISTRIB;GSYM DISJ_ASSOC]; FIRST_X_ASSUM MP_TAC; ONCE_REWRITE_TAC[TAUT `(a1 /\ a2 /\ a3 /\ a4 /\ a5) <=> ((T ==> a1) /\ (T ==> a2) /\ (T ==> a3) /\ (T ==> a4) /\ (T ==> a5))`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN FIRST_X_ASSUM (unlist REWRITE_TAC) THEN NUM_REDUCE_TAC THEN TRY (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) *) let b5_assumption_reduction = prove_by_refinement( `!s f v e. is_scs_v39 s /\ scs_k_v39 s = 5 /\ BBs_v39 s v /\ (!i j. scs_diag 5 i j ==> &4 * h0 < scs_b_v39 s i j) /\ deformation f (IMAGE v (:num)) (--e,e) /\ (!w t. ~(w = v 4) /\ ~(w = v 0) ==> (f w t = w)) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist(f (v 4) t,f (v 0) t) = dist(v 4,v 0) /\ dist(f (v 4) t,v 3) = dist(v 4,v 3))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist(f (v 0) t, (v 1)) <= dist(v 0, v 1))) ==> (!i j. ?e1. &0 < e1 /\ (dist(v i,v j) = scs_b_v39 s i j ==> (!t. abs t < e1 ==> dist(f (v i) t,f (v j) t) <= scs_b_v39 s i j))) `, (* {{{ proof *) [ REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC Cuxvzoz.I_IMP THEN ASM_TAC THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `!i j.scs_a_v39 s (i MOD 5) (j MOD 5) = scs_a_v39 s i j /\ scs_b_v39 s (i MOD 5) (j MOD 5) = scs_b_v39 s i j ` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC; ASM_REWRITE_TAC[Appendix.is_scs_v39]; REPEAT WEAKER_STRIP_TAC; BY(REPEAT (FIRST_X_ASSUM_ST `periodic2` MP_TAC) THEN MESON_TAC[Cuxvzoz.periodic2_MOD;arith `~(5 = 0)`]); TYPIFY `periodic v 5` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]); TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `!j. j < 5 ==> (j MOD 5 = j)` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[MOD_LT]); COMMENT "scs_b_v39"; REWRITE_TAC[I_THM]; MATCH_MP_TAC Terminal.periodic2_mod_reduce; TYPIFY `5` EXISTS_TAC; ASM_REWRITE_TAC[arith `~(5=0)`]; CONJ_TAC; MATCH_MP_TAC Cuxvzoz.MOD_periodic2; BY(REPLICATE_TAC 7 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[] THEN MESON_TAC[arith `~(5=0)`]); TYPIFY_GOAL_THEN `!P. (!i j. P i j ==> P j i) /\ (!i j. (i <= (j:num)) ==> P i j) ==> (!i j. P i j)` MATCH_MP_TAC; BY(MESON_TAC[arith `i <= j \/ j <= (i:num)`]); CONJ_TAC; REPEAT WEAKER_STRIP_TAC; ONCE_REWRITE_TAC[DIST_SYM]; TYPIFY `scs_b_v39 s j i = scs_b_v39 s i j` (C SUBGOAL_THEN SUBST1_TAC); FIRST_X_ASSUM_ST `is_scs_v39` MP_TAC THEN REWRITE_TAC[Appendix.is_scs_v39] THEN REPEAT WEAKER_STRIP_TAC; BY(ASM_MESON_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); TYPIFY `!i j. i < 5 /\ j < 5 /\ ~(i=j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[]); TYPIFY `!t. f (v 2) t = v 2 /\ f (v 3) t = v 3 /\ f (v 1) t = v 1` (C SUBGOAL_THEN ASSUME_TAC); BY(GEN_TAC THEN REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC THEN REPEAT CONJ_TAC) THEN NUM_REDUCE_TAC); TYPIFY `scs_diag 5 0 2 /\ scs_diag 5 0 3 /\ scs_diag 5 1 3 /\ scs_diag 5 1 4 /\ scs_diag 5 2 4` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[Appendix.scs_diag;Uxckfpe.ARITH_5_TAC]); TYPIFY `!i j. scs_diag 5 i j ==> (?e1. &0 < e1 /\ (dist (v i,v j) = scs_b_v39 s i j ==> (!t. abs t < e1 ==> dist (f (v i) t,f (v j) t) <= scs_b_v39 s i j)))` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; TYPIFY `&1` EXISTS_TAC; (REWRITE_TAC[arith `&0 < &1`]); TYPIFY `dist(v i,v j) <= &4 * h0` ENOUGH_TO_SHOW_TAC; FIRST_X_ASSUM_ST `&4 * h0` (C INTRO_TAC [`i`;`j`]); ASM_REWRITE_TAC[]; BY(REAL_ARITH_TAC); INTRO_TAC DIST_TRIANGLE [`v i`;`(vec 0):real^3`;`v j`]; REWRITE_TAC[DIST_0]; TYPIFY `!i. norm (v i) <= &2 * h0` ENOUGH_TO_SHOW_TAC; BY(DISCH_TAC THEN FIRST_ASSUM (C INTRO_TAC [`i`]) THEN FIRST_X_ASSUM (C INTRO_TAC [`j`]) THEN REAL_ARITH_TAC); FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN REPEAT WEAKER_STRIP_TAC; BY(FIRST_X_ASSUM_ST `ball_annulus` MP_TAC THEN REWRITE_TAC[Terminal.IMAGE_SUBSET_IN;IN_UNIV] THEN MESON_TAC[Fnjlbxs.in_ball_annulus]); TYPIFY `!i. T ==> (?e1. &0 < e1 /\ (dist (v i,v i) = scs_b_v39 s i i ==> (!t. abs t < e1 ==> dist (f (v i) t,f (v i) t) <= scs_b_v39 s i i)))` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[DIST_REFL]; BY(MESON_TAC[arith `x = &0 ==> x <= &0`;arith `&0 < &1`]); REPEAT GEN_TAC; TYPIFY `(?e1. &0 < e1 /\ (dist (v 0,v 1) = scs_b_v39 s 0 1 ==> (!t. abs t < e1 ==> dist (f (v 0) t,f (v 1) t) <= scs_b_v39 s 0 1))) /\(?e1. &0 < e1 /\ (dist (v 1,v 2) = scs_b_v39 s 1 2 ==> (!t. abs t < e1 ==> dist (f (v 1) t,f (v 2) t) <= scs_b_v39 s 1 2))) /\(?e1. &0 < e1 /\ (dist (v 2,v 3) = scs_b_v39 s 2 3 ==> (!t. abs t < e1 ==> dist (f (v 2) t,f (v 3) t) <= scs_b_v39 s 2 3))) /\(?e1. &0 < e1 /\ (dist (v 3,v 4) = scs_b_v39 s 3 4 ==> (!t. abs t < e1 ==> dist (f (v 3) t,f (v 4) t) <= scs_b_v39 s 3 4))) /\(?e1. &0 < e1 /\ (dist (v 0,v 4) = scs_b_v39 s 0 4 ==> (!t. abs t < e1 ==> dist (f (v 0) t,f (v 4) t) <= scs_b_v39 s 0 4)))` (C SUBGOAL_THEN MP_TAC); ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `!a b. (?e1. &0 < e1 /\ (a = b ==> (!t. abs t < e1 ==> a <= b)))` (unlist REWRITE_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `(a = b) ==> (a <= b)`]); CONJ_TAC; BY(ASM_MESON_TAC[]); REPEAT (FIRST_X_ASSUM_ST `dist(v 4,v 0)` MP_TAC); REWRITE_TAC[DIST_SYM]; BY(ASM_MESON_TAC[arith `x = y ==> x <= y`]); DISCH_TAC; REWRITE_TAC[arith `i < 5 <=> (i = 0) \/ (i = 1) \/ (i=2) \/ (i =3) \/ (i=4)`]; ONCE_REWRITE_TAC[TAUT `(a ==> b ==> c) <=> (b ==> a ==> c)`]; REWRITE_TAC[LEFT_OR_DISTRIB;RIGHT_OR_DISTRIB;GSYM DISJ_ASSOC]; FIRST_X_ASSUM MP_TAC; ONCE_REWRITE_TAC[TAUT `(a1 /\ a2 /\ a3 /\ a4 /\ a5) <=> ((T ==> a1) /\ (T ==> a2) /\ (T ==> a3) /\ (T ==> a4) /\ (T ==> a5))`]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; BY(REPEAT (FIRST_X_ASSUM DISJ_CASES_TAC) THEN FIRST_X_ASSUM (unlist REWRITE_TAC) THEN NUM_REDUCE_TAC THEN TRY (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]) ]);; (* }}} *) (* was dihV_dih_y *) let azim_dih_y = prove_by_refinement( `!v0 v1 v2 v3. ~collinear {v0,v1,v2} /\ ~collinear{v0,v1,v3} /\ azim v0 v1 v2 v3 <= pi ==> azim v0 v1 v2 v3 = dih_y (dist(v0,v1)) (dist(v0,v2)) (dist(v0,v3)) (dist(v2,v3)) (dist(v1,v3)) (dist(v1,v2))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; INTRO_TAC AZIM_DIHV_SAME_STRONG [`v0`;`v1`;`v2`;`v3`]; ASM_REWRITE_TAC[]; DISCH_THEN SUBST1_TAC; GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] Merge_ineq.DIHV_DIH_X); REWRITE_TAC[Sphere.dih_y;LET_THM;arith `x*x = x pow 2`]; BY(ASM_REWRITE_TAC[GSYM Collect_geom2.NOT_COL_EQ_UPS_X_POS]) ]);; (* }}} *) (* next azim reduction *) (* let azm5_reduction = prove_by_refinement( `!s f v e. is_scs_v39 s /\ scs_k_v39 s = 5 /\ BBs_v39 s v /\ scs_generic v /\ deformation f (IMAGE v (:num)) (--e,e) /\ (!w t. ~(w = v 0) /\ ~(w = v 4) ==> f w t = w) /\ azim (vec 0) (v 0) (v 1) (v 4) < pi /\ azim (vec 0) (v 1) (v 2) (v 0) < pi /\ &0 < azim (vec 0) (v 0) (v 1) (v 3) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 0) t) (v 1) <= azim (vec 0) (v 3) (v 0) (v 1))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 // make this a dih_y condition?? ==> azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) = azim (vec 0) (v 3) (v 4) (v 0) /\ azim (vec 0) (f (v 4) t) (f (v 0) t) (v 3) = azim (vec 0) (v 4) (v 0) (v 3) /\ azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) = azim (vec 0) (v 0) (v 3) (v 4))) ==> azim (vec 0) (v 3) (v 4) (v 2) = azim (vec 0) (v 3) (v 4) (v 0) + azim (vec 0) (v 3) (v 0) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) /\ azim (vec 0) (v 0) (v 1) (v 4) = azim (vec 0) (v 0) (v 1) (v 3) + azim (vec 0) (v 0) (v 3) (v 4) /\ azim (vec 0) (v 1) (v 2) (v 0) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (v 0) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (v 2) = azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) + azim (vec 0) (v 3) (f (v 0) t) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) /\ azim (vec 0) (f (v 0) t) (v 1) (f (v 4) t) = azim (vec 0) (f (v 0) t) (v 1) (v 3) + azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) /\ azim (vec 0) (v 1) (v 2) (f (v 0) t) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (f (v 0) t) /\ &0 < azim (vec 0) (f (v 0) t) (v 1) (v 3) /\ azim (vec 0) (f (v 0) t) (v 1) (v 3) < pi)) /\ (!i. ?e0. &0 < e0 /\ (azim (vec 0) (v i) (v (i + 1)) (v (i + 4)) = pi ==> (!t. abs t < e0 ==> azim (vec 0) (f (v i) t) (f (v (i + 1)) t) (f (v (i + 4)) t) <= pi)))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; ASSUME_TAC (arith `~(0 = 5)`); TYPIFY `!i. i MOD 5 < 5` (C SUBGOAL_THEN ASSUME_TAC); BY((FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION]); TYPIFY `periodic v 5` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]); TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); TYPIFY `v 5 = v 0 /\ v 6 = v 1 /\ v 7 = v 2 /\ v 8 = v 3 /\ v 9 = v 4` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM); BY(REWRITE_TAC[Ocbicby.MOD_5_EXPLICIT;arith `8 = 1*5 + 3`;arith `9 = 1*5 + 4`;MOD_MULT_ADD]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY ` generic (IMAGE v (:num)) (IMAGE (\i. {v i, v (SUC i)}) (:num)) /\ convex_local_fan (IMAGE v (:num), IMAGE (\i. {v i, v (SUC i)}) (:num), IMAGE (\i. v i,v (SUC i)) (:num))` (C SUBGOAL_THEN ASSUME_TAC); CONJ2_TAC; FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[LET_THM;Appendix.BBs_v39]; BY(MESON_TAC[arith `~(5 <= 3)`]); BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN MESON_TAC[Appendix.scs_generic]); FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC; COMMENT "first addition"; SUBCONJ_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`3`;`4`]; ASM_REWRITE_TAC[LET_THM;arith `SUC 3 = 4 /\ 3 + 4 = 7 /\ 3 + 3 = 6 /\ 3 <= 5 /\ 0 < 3 /\ 3 < 4 /\ 4 < 5`]; DISCH_THEN SUBST1_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`2`;`3`]; ASM_REWRITE_TAC[LET_THM;arith `SUC 3 = 4 /\ 3 + 3 = 6 /\ 3 + 2 = 5 /\ 3 <= 5 /\ 0 < 2 /\ 2 < 3 /\ 3 < 5`]; DISCH_THEN SUBST1_TAC; BY(REAL_ARITH_TAC); DISCH_TAC; COMMENT "second addition"; SUBCONJ_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`0`;`3`;`4`]; NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[LET_THM]); COMMENT "third addition"; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`0`;`3`;`4`]; NUM_REDUCE_TAC; ASM_REWRITE_TAC[LET_THM]; DISCH_TAC; DISCH_TAC; SUBCONJ_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`1`;`2`;`4`]; NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[LET_THM]); DISCH_TAC; TYPIFY `!i j. i < 5 /\ j < 5 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[]); TYPIFY `!t. f (v 1) t = v 1 /\ f (v 2) t = v 2 /\ f (v 3) t = v 3` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC); COMMENT "colinearity"; TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC); REPEAT GEN_TAC THEN DISCH_TAC; MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT; GEXISTL_TAC [`IMAGE v (:num)`;`IMAGE (\i. {v i, v (SUC i)}) (:num)`;`IMAGE (\i. (v i, v (SUC i))) (:num)`]; ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]; CONJ_TAC; MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); BY(MESON_TAC[]); COMMENT "deform collinear"; TYPIFY `!i j. ~collinear {vec 0,v i,v j} ==> (?e1. &0 < e1 /\ (!t. abs t < e1 ==> ~collinear {vec 0, f (v i) t, f (v j) t}))` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC (GEN_ALL Local_lemmas1.CONTINUOUS_PRESERVE_COLLINEAR) [`&0`;`(vec 0):real^3`;`f (v i)`;`f (v j)`]; ANTS_TAC; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation;arith `abs(&0 - x) = abs x`]; REWRITE_TAC[IN_IMAGE;IN_UNIV]; BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]); BY(REWRITE_TAC[arith `abs(&0 - r') = abs r'`]); TYPIFY `!i. azim (vec 0) (v i) (v (i+1)) (v(i+4)) <= pi` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[arith `i+1 = SUC i`;arith `4 = 5 - 1`]; GEN_TAC; MATCH_MP_TAC (REWRITE_RULE[LET_THM] Terminal.convex_local_fan_azim_le_pi); ASM_REWRITE_TAC[arith `3 <= 5`]; FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM]; BY(DISCH_THEN (unlist REWRITE_TAC)); COMMENT "merge e"; TYPIFY_GOAL_THEN `(?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (v 2) = azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) + azim (vec 0) (v 3) (f (v 0) t) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) )) /\(?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (f (v 0) t) (v 1) (f (v 4) t) = azim (vec 0) (f (v 0) t) (v 1) (v 3) + azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) )) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 1) (v 2) (f (v 0) t) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (f (v 0) t))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> &0 < azim (vec 0) (f (v 0) t) (v 1) (v 3) /\ azim (vec 0) (f (v 0) t) (v 1) (v 3) < pi)) ==> (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (v 2) = azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) + azim (vec 0) (v 3) (f (v 0) t) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) /\ azim (vec 0) (f (v 0) t) (v 1) (f (v 4) t) = azim (vec 0) (f (v 0) t) (v 1) (v 3) + azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) /\ azim (vec 0) (v 1) (v 2) (f (v 0) t) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (f (v 0) t) /\ &0 < azim (vec 0) (f (v 0) t) (v 1) (v 3) /\ azim (vec 0) (f (v 0) t) (v 1) (v 3) < pi))` (GMATCH_SIMP_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_quad [`e1''`;`e1'''`;`e1''''`;`e1'''''`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC THEN REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]; BY(ASM_MESON_TAC[]); BY(ASM_MESON_TAC[]); REWRITE_TAC[GSYM CONJ_ASSOC]; COMMENT "azim bounds"; TYPIFY `azim (vec 0) (v 0) (v 1) (v 3) < pi` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `azim (vec 0) (v 0) (v 1) (v 4) < pi` MP_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC (arith `&0 <= y ==> (x + y < pi ==> x < pi)`); BY(REWRITE_TAC[Counting_spheres.AZIM_NN]); TYPIFY `~coplanar {vec 0,v 0,v 1,v 3}` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; BY(REPEAT (FIRST_X_ASSUM_ST `azim (vec 0) (v 0) (v 1) (v 3)` MP_TAC) THEN REAL_ARITH_TAC); TYPIFY `&0 < azim (vec 0) (v 3) (v 0) (v 1) /\ &0 < azim (vec 0) (v 1) (v 3) (v 0)` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[Zlzthic.azim_pos_iff_nz]; FIRST_X_ASSUM_ST `coplanar` MP_TAC; BY(MESON_TAC[Local_lemmas1.AZIM_COND_FOR_COPLANAR;SET_RULE `{a,b,c,d} = {a,d,b,c}`;SET_RULE `{a,b,c,d}={a,c,d,b}`]); INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v 3`;`v 0`;`v 1`;`f`;`-- e`;`e`;`pi - azim (vec 0) (v 0) (v 3) (v 4)`]; ANTS_TAC; ASM_REWRITE_TAC[arith `a < pi - b <=> a + b < pi`]; CONJ_TAC; MATCH_MP_TAC Zlzthic.deformation_subset; TYPIFY `IMAGE v (:num)` EXISTS_TAC; ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY;IN_UNIV]; BY(MESON_TAC[FUN_IN_IMAGE;IN_UNIV]); REWRITE_TAC[CONJ_ASSOC]; CONJ_TAC; FIRST_X_ASSUM_ST `coplanar` MP_TAC; BY(MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan;SET_RULE `{a,b,c} = {a,c,b}`]); BY(REPEAT (FIRST_X_ASSUM_ST `x < pi` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v 0`;`v 1`;`v 3`;`f`;`-- e`;`e`;`pi - azim (vec 0) (v 1) (v 2) (v 3)`]; ANTS_TAC; ASM_REWRITE_TAC[arith `a < pi - b <=> a + b < pi`]; CONJ_TAC; MATCH_MP_TAC Zlzthic.deformation_subset; TYPIFY `IMAGE v (:num)` EXISTS_TAC; ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY;IN_UNIV]; BY(MESON_TAC[FUN_IN_IMAGE;IN_UNIV]); REWRITE_TAC[CONJ_ASSOC]; CONJ_TAC; FIRST_X_ASSUM_ST `coplanar` MP_TAC; BY(MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan;SET_RULE `{a,b,c} = {a,c,b}`]); BY(REPEAT (FIRST_X_ASSUM_ST `x < pi` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; COMMENT "azim additivity"; TYPIFY `?e3. &0 < e3 /\ (!t. abs t < e3 ==> ~collinear {vec 0,f (v 0) t,f (v 4) t} /\ ~collinear {vec 0,f (v 0) t,f (v 3) t} /\ ~collinear {vec 0,f (v 0) t,f (v 1) t} /\ ~collinear {vec 0,f (v 4) t,f (v 3) t})` (C SUBGOAL_THEN ASSUME_TAC); FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`0`;`4`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`0`;`3`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`0`;`1`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; FIRST_X_ASSUM_ST `collinear` (C INTRO_TAC [`4`;`3`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; INTRO_TAC Cuxvzoz.epsilon_quad [`e1''`;`e1'''`;`e1''''`;`e1'''''`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; TYPIFY `e''''` EXISTS_TAC THEN CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]); FIRST_X_ASSUM MP_TAC THEN REPEAT WEAKER_STRIP_TAC; SUBCONJ_TAC; INTRO_TAC Cuxvzoz.epsilon_triple [`e1`;`e1'`;`e3`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 3 (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.azim_dominated_split [`(vec 0):real^3`;`v 3`;`v 4`;`v 1`;`v 2`;`(vec 0):real^3`;`v 3`;`f (v 4) t`;`v 1`;`v 2`]; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`3`;`4`]; ASM_REWRITE_TAC[LET_THM;arith `SUC 3 = 4 /\ 3 + 4 = 7 /\ 3 + 3 = 6 /\ 3 <= 5 /\ 0 < 3 /\ 3 < 4 /\ 4 < 5`]; DISCH_THEN (SUBST1_TAC o GSYM) THEN ASM_REWRITE_TAC[arith `x <= x`]; INTRO_TAC Cuxvzoz.azim_dominated_split [`(vec 0):real^3`;`v 3`;`v 4`;`v 0`;`v 1`;`(vec 0):real^3`;`v 3`;`f (v 4) t`;`f (v 0) t`;`v 1`]; ASM_REWRITE_TAC[arith `x <= x`]; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`2`;`3`]; NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[LET_THM]; ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `~collinear {vec 0, v 1, v 3} /\ ~collinear {vec 0, v 1, v 3} /\ ~collinear {vec 0, v 2, v 3}` (unlist REWRITE_TAC); BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); BY(REPLICATE_TAC 15 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC); (DISCH_TAC); COMMENT "azim 0 additive"; FIRST_X_ASSUM MP_TAC THEN REPEAT WEAKER_STRIP_TAC; SUBCONJ_TAC; INTRO_TAC Cuxvzoz.epsilon_triple [`e'`;`e1'`;`e3`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC Fan.sum3_azim_fan; ASM_REWRITE_TAC[]; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; COMMENT "azim 1 additive"; SUBCONJ_TAC; INTRO_TAC Cuxvzoz.epsilon_pair [`e''`;`e3`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC Fan.sum3_azim_fan; ASM_REWRITE_TAC[]; CONJ_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC); ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`]; ASM_REWRITE_TAC[]; BY(CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); REPEAT WEAKER_STRIP_TAC; COMMENT "pi bounds"; CONJ_TAC; TYPIFY `e'` EXISTS_TAC; CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; TYPIFY ` &0 < azim (vec 0) (f (v 0) t) (f (v 1) t) (f (v 3) t) /\ azim (vec 0) (f (v 0) t) (f (v 1) t) (f (v 3) t) < pi - azim (vec 0) (v 0) (v 3) (v 4)` ENOUGH_TO_SHOW_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC (arith `&0 <= b ==> (&0 < a /\ a < pi - b ==> &0 < a /\ a < pi)`); BY(REWRITE_TAC[Counting_spheres.AZIM_NN]); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); MATCH_MP_TAC Oxl_def.periodic_numseg; TYPIFY `5` EXISTS_TAC; REWRITE_TAC[arith `~(5 = 0)`]; CONJ_TAC; FIRST_X_ASSUM_ST `periodic` MP_TAC; REWRITE_TAC[Oxl_def.periodic]; REWRITE_TAC[arith `(i+5)+j = (i+j)+5`]; BY(DISCH_THEN (unlist REWRITE_TAC)); REWRITE_TAC[IN_NUMSEG;arith `(0 <= i /\ i <= 5-1) <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3 \/ i = 4)`]; REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC; EXISTS_TAC `&1` THEN REPEAT (FIRST_X_ASSUM_ST `azim (vec 0) (v 0) (v 1) (v 4) < pi` MP_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `x < pi ==> ~(x=pi)`]); REPEAT (FIRST_X_ASSUM_ST `+` kill); ASM_REWRITE_TAC[] THEN EXISTS_TAC `&1` THEN REPEAT (FIRST_X_ASSUM_ST `azim (vec 0) (v 1) (v 2) (v 0) < pi` MP_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `x < pi ==> ~(x=pi)`]); BY(ASM_REWRITE_TAC[] THEN EXISTS_TAC `&1` THEN MESON_TAC[arith `&0 < &1`;arith `x <= x`]); INTRO_TAC Cuxvzoz.epsilon_triple [`e1''`;`e1'`;`e1`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; TYPIFY `e'''` EXISTS_TAC THEN CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; REPLICATE_TAC 3(FIRST_X_ASSUM (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC); REPEAT (FIRST_X_ASSUM_ST `azim` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `a = pi` (SUBST1_TAC o GSYM); ASM_REWRITE_TAC[]; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); TYPIFY `e1'` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `4` (C INTRO_TAC [`t`])); ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC) ]);; (* }}} *) *) let azim5_reduction = prove_by_refinement( `!s f v e. is_scs_v39 s /\ scs_k_v39 s = 5 /\ BBs_v39 s v /\ scs_generic v /\ deformation f (IMAGE v (:num)) (--e,e) /\ (!w t. ~(w = v 4) /\ ~(w = v 0) ==> f w t = w) /\ azim (vec 0) (v 0) (v 1) (v 4) < pi /\ azim (vec 0) (v 1) (v 2) (v 0) < pi /\ &0 < azim (vec 0) (v 0) (v 1) (v 3) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 0) t) (v 1) <= azim (vec 0) (v 3) (v 0) (v 1))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> dist (v 3,f (v 4) t) = dist (v 3,v 4) /\ dist (f (v 0) t,f (v 4) t) = dist (v 0,v 4) /\ norm (f (v 4) t) = norm (v 4) /\ dist (f (v 0) t,v 3) = dist (v 0,v 3) /\ dist (f (v 0) t,v 1) = dist (v 0,v 1) - abs t /\ norm (f (v 0) t) = norm (v 0) /\ (coplanar {vec 0, v 3, v 4, v 0} ==> coplanar {vec 0, v 3, f (v 4) t, f (v 0) t}))) ==> azim (vec 0) (v 3) (v 4) (v 2) = azim (vec 0) (v 3) (v 4) (v 0) + azim (vec 0) (v 3) (v 0) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) /\ azim (vec 0) (v 0) (v 1) (v 4) = azim (vec 0) (v 0) (v 1) (v 3) + azim (vec 0) (v 0) (v 3) (v 4) /\ azim (vec 0) (v 1) (v 2) (v 0) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (v 0) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (v 2) = azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) + azim (vec 0) (v 3) (f (v 0) t) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) /\ azim (vec 0) (f (v 0) t) (v 1) (f (v 4) t) = azim (vec 0) (f (v 0) t) (v 1) (v 3) + azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) /\ azim (vec 0) (v 1) (v 2) (f (v 0) t) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (f (v 0) t) /\ &0 < azim (vec 0) (f (v 0) t) (v 1) (v 3) /\ azim (vec 0) (f (v 0) t) (v 1) (v 3) < pi)) /\ (!i. ?e0. &0 < e0 /\ (azim (vec 0) (v i) (v (i + 1)) (v (i + 4)) = pi ==> (!t. abs t < e0 ==> azim (vec 0) (f (v i) t) (f (v (i + 1)) t) (f (v (i + 4)) t) <= pi))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) = azim (vec 0) (v 3) (v 4) (v 0) /\ azim (vec 0) (f (v 4) t) (f (v 0) t) (v 3) = azim (vec 0) (v 4) (v 0) (v 3) /\ azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) = azim (vec 0) (v 0) (v 3) (v 4)))`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; ASSUME_TAC (arith `~(0 = 5)`); TYPIFY `!i. i MOD 5 < 5` (C SUBGOAL_THEN ASSUME_TAC); BY((FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[DIVISION]); TYPIFY `periodic v 5` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN MESON_TAC[]); TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); TYPIFY `v 5 = v 0 /\ v 6 = v 1 /\ v 7 = v 2 /\ v 8 = v 3 /\ v 9 = v 4` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM); BY(REWRITE_TAC[Ocbicby.MOD_5_EXPLICIT;arith `8 = 1*5 + 3`;arith `9 = 1*5 + 4`;MOD_MULT_ADD]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY ` generic (IMAGE v (:num)) (IMAGE (\i. {v i, v (SUC i)}) (:num)) /\ convex_local_fan (IMAGE v (:num), IMAGE (\i. {v i, v (SUC i)}) (:num), IMAGE (\i. v i,v (SUC i)) (:num))` (C SUBGOAL_THEN ASSUME_TAC); CONJ2_TAC; FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN ASM_REWRITE_TAC[LET_THM;Appendix.BBs_v39]; BY(MESON_TAC[arith `~(5 <= 3)`]); BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN MESON_TAC[Appendix.scs_generic]); FIRST_X_ASSUM MP_TAC THEN BURY_MP_TAC; COMMENT "first addition"; SUBCONJ_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`3`;`4`]; ASM_REWRITE_TAC[LET_THM;arith `SUC 3 = 4 /\ 3 + 4 = 7 /\ 3 + 3 = 6 /\ 3 <= 5 /\ 0 < 3 /\ 3 < 4 /\ 4 < 5`]; DISCH_THEN SUBST1_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`2`;`3`]; ASM_REWRITE_TAC[LET_THM;arith `SUC 3 = 4 /\ 3 + 3 = 6 /\ 3 + 2 = 5 /\ 3 <= 5 /\ 0 < 2 /\ 2 < 3 /\ 3 < 5`]; DISCH_THEN SUBST1_TAC; BY(REAL_ARITH_TAC); DISCH_TAC; COMMENT "second addition"; SUBCONJ_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`0`;`3`;`4`]; NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[LET_THM]); COMMENT "third addition"; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`0`;`3`;`4`]; NUM_REDUCE_TAC; ASM_REWRITE_TAC[LET_THM]; DISCH_TAC; DISCH_TAC; SUBCONJ_TAC; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`1`;`2`;`4`]; NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[LET_THM]); DISCH_TAC; TYPIFY `!i j. i < 5 /\ j < 5 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[]); TYPIFY `!t. f (v 1) t = v 1 /\ f (v 2) t = v 2 /\ f (v 3) t = v 3` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC); COMMENT "colinearity"; TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC); REPEAT GEN_TAC THEN DISCH_TAC; MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT; GEXISTL_TAC [`IMAGE v (:num)`;`IMAGE (\i. {v i, v (SUC i)}) (:num)`;`IMAGE (\i. (v i, v (SUC i))) (:num)`]; ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]; CONJ_TAC; MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); BY(MESON_TAC[]); COMMENT "deform collinear"; TYPIFY `!i j. ~collinear {vec 0,v i,v j} ==> (?e1. &0 < e1 /\ (!t. abs t < e1 ==> ~collinear {vec 0, f (v i) t, f (v j) t}))` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC (GEN_ALL Local_lemmas1.CONTINUOUS_PRESERVE_COLLINEAR) [`&0`;`(vec 0):real^3`;`f (v i)`;`f (v j)`]; ANTS_TAC; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation;arith `abs(&0 - x) = abs x`]; REWRITE_TAC[IN_IMAGE;IN_UNIV]; BY(FIRST_X_ASSUM MP_TAC THEN MESON_TAC[]); BY(REWRITE_TAC[arith `abs(&0 - r') = abs r'`]); TYPIFY `!i. azim (vec 0) (v i) (v (i+1)) (v(i+4)) <= pi` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[arith `i+1 = SUC i`;arith `4 = 5 - 1`]; GEN_TAC; MATCH_MP_TAC (REWRITE_RULE[LET_THM] Terminal.convex_local_fan_azim_le_pi); ASM_REWRITE_TAC[arith `3 <= 5`]; FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM]; BY(DISCH_THEN (unlist REWRITE_TAC)); COMMENT "merge e"; TYPIFY_GOAL_THEN `(?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (v 2) = azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) + azim (vec 0) (v 3) (f (v 0) t) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) )) /\(?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (f (v 0) t) (v 1) (f (v 4) t) = azim (vec 0) (f (v 0) t) (v 1) (v 3) + azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) )) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 1) (v 2) (f (v 0) t) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (f (v 0) t))) /\ (?e1. &0 < e1 /\ (!t. abs t < e1 ==> &0 < azim (vec 0) (f (v 0) t) (v 1) (v 3) /\ azim (vec 0) (f (v 0) t) (v 1) (v 3) < pi)) ==> (?e1. &0 < e1 /\ (!t. abs t < e1 ==> azim (vec 0) (v 3) (f (v 4) t) (v 2) = azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) + azim (vec 0) (v 3) (f (v 0) t) (v 1) + azim (vec 0) (v 3) (v 1) (v 2) /\ azim (vec 0) (f (v 0) t) (v 1) (f (v 4) t) = azim (vec 0) (f (v 0) t) (v 1) (v 3) + azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) /\ azim (vec 0) (v 1) (v 2) (f (v 0) t) = azim (vec 0) (v 1) (v 2) (v 3) + azim (vec 0) (v 1) (v 3) (f (v 0) t) /\ &0 < azim (vec 0) (f (v 0) t) (v 1) (v 3) /\ azim (vec 0) (f (v 0) t) (v 1) (v 3) < pi))` (GMATCH_SIMP_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_quad [`e1''`;`e1'''`;`e1''''`;`e1'''''`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC THEN REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]; BY(ASM_MESON_TAC[]); BY(ASM_MESON_TAC[]); REWRITE_TAC[GSYM CONJ_ASSOC]; COMMENT "azim bounds"; TYPIFY `azim (vec 0) (v 0) (v 1) (v 3) < pi` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `azim (vec 0) (v 0) (v 1) (v 4) < pi` MP_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC (arith `&0 <= y ==> (x + y < pi ==> x < pi)`); BY(REWRITE_TAC[Counting_spheres.AZIM_NN]); TYPIFY `~coplanar {vec 0,v 0,v 1,v 3}` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; BY(REPEAT (FIRST_X_ASSUM_ST `azim (vec 0) (v 0) (v 1) (v 3)` MP_TAC) THEN REAL_ARITH_TAC); TYPIFY `&0 < azim (vec 0) (v 3) (v 0) (v 1) /\ &0 < azim (vec 0) (v 1) (v 3) (v 0)` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[Zlzthic.azim_pos_iff_nz]; FIRST_X_ASSUM_ST `coplanar` MP_TAC; BY(MESON_TAC[Local_lemmas1.AZIM_COND_FOR_COPLANAR;SET_RULE `{a,b,c,d} = {a,d,b,c}`;SET_RULE `{a,b,c,d}={a,c,d,b}`]); INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v 3`;`v 0`;`v 1`;`f`;`-- e`;`e`;`pi - azim (vec 0) (v 0) (v 3) (v 4)`]; ANTS_TAC; ASM_REWRITE_TAC[arith `a < pi - b <=> a + b < pi`]; CONJ_TAC; MATCH_MP_TAC Zlzthic.deformation_subset; TYPIFY `IMAGE v (:num)` EXISTS_TAC; ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY;IN_UNIV]; BY(MESON_TAC[FUN_IN_IMAGE;IN_UNIV]); REWRITE_TAC[CONJ_ASSOC]; CONJ_TAC; FIRST_X_ASSUM_ST `coplanar` MP_TAC; BY(MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan;SET_RULE `{a,b,c} = {a,c,b}`]); BY(REPEAT (FIRST_X_ASSUM_ST `x < pi` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.WNWSHJT_ALT [`v 0`;`v 1`;`v 3`;`f`;`-- e`;`e`;`pi - azim (vec 0) (v 1) (v 2) (v 3)`]; ANTS_TAC; ASM_REWRITE_TAC[arith `a < pi - b <=> a + b < pi`]; CONJ_TAC; MATCH_MP_TAC Zlzthic.deformation_subset; TYPIFY `IMAGE v (:num)` EXISTS_TAC; ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY;IN_UNIV]; BY(MESON_TAC[FUN_IN_IMAGE;IN_UNIV]); REWRITE_TAC[CONJ_ASSOC]; CONJ_TAC; FIRST_X_ASSUM_ST `coplanar` MP_TAC; BY(MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan;SET_RULE `{a,b,c} = {a,c,b}`]); BY(REPEAT (FIRST_X_ASSUM_ST `x < pi` MP_TAC) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; COMMENT "034 triangle"; TYPIFY `?e3. &0 < e3 /\ (!t. abs t < e3 ==> ~collinear {vec 0,f (v 0) t,f (v 4) t} /\ ~collinear {vec 0,f (v 0) t,f (v 3) t} /\ ~collinear {vec 0,f (v 0) t,f (v 1) t} /\ ~collinear {vec 0,f (v 4) t,f (v 3) t})` (C SUBGOAL_THEN ASSUME_TAC); FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`0`;`4`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`0`;`3`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; FIRST_ASSUM_ST `collinear` (C INTRO_TAC [`0`;`1`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; FIRST_X_ASSUM_ST `collinear` (C INTRO_TAC [`4`;`3`]) THEN ANTS_TAC THENL [REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC;REPEAT WEAKER_STRIP_TAC]; INTRO_TAC Cuxvzoz.epsilon_quad [`e1''`;`e1'''`;`e1''''`;`e1'''''`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; TYPIFY `e''''` EXISTS_TAC THEN CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN ASM_REWRITE_TAC[]); FIRST_X_ASSUM MP_TAC THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `!i. f (v i) (&0) = v i` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation]; REPEAT WEAKER_STRIP_TAC; BY(FIRST_X_ASSUM MATCH_MP_TAC THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_UNIV]); REWRITE_TAC[CONJ_ASSOC] THEN SUBCONJ2_TAC; TYPIFY `?e2. &0 < e2 /\ (!t. abs t < e2 ==> azim (vec 0) (f (v 4) t) (f (v 0) t) (v 3) <= pi)` ASM_CASES_TAC; FIRST_X_ASSUM MP_TAC THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.epsilon_triple [`e1'`;`e3`;`e2`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 3 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `azim (vec 0) (f (v 0) t) (v 3) (f (v 4) t) <= pi /\ azim (vec 0) (v 3) (f (v 4) t) (f (v 0) t) <= pi /\ azim (vec 0) (v 0) (v 3) (v 4) <= pi /\ azim (vec 0) (v 3) (v 4) (v 0) <= pi /\ azim (vec 0) (v 4) (v 0) (v 3) <= pi` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM MP_TAC THEN (FIRST_X_ASSUM_ST `x <= pi` (C INTRO_TAC [`4`])); NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]; BY(MESON_TAC[Xivphks.FSQKWKK]); FIRST_X_ASSUM MP_TAC THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `a <= pi` (ASSUME_TAC o (MATCH_MP (REWRITE_RULE[TAUT `(a /\ b /\ c ==> d) <=> (c ==> (a /\ b ==> d))`] azim_dih_y)))); REPLICATE_TAC 6 (POP_ASSUM GMATCH_SIMP_TAC); ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `(~collinear {vec 0, v 0, v 4} /\ ~collinear {vec 0, v 3, v 4}) /\ (~collinear {vec 0, v 4, v 3} /\ ~collinear {vec 0, v 0, v 3}) /\ (~collinear {vec 0, v 3, v 0} /\ ~collinear {vec 0, v 4, v 0})` (unlist REWRITE_TAC); BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); ASM_REWRITE_TAC[DIST_SYM]; BY(REPEAT (FIRST_X_ASSUM_ST `dist` MP_TAC) THEN REWRITE_TAC[DIST_SYM;DIST_0] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]); PROOF_BY_CONTR_TAC; FIRST_X_ASSUM kill; FIRST_X_ASSUM MP_TAC THEN REWRITE_TAC[]; COMMENT "second half of assumption. All angles 034 <=pi"; TYPIFY `coplanar {vec 0, v 3, v 4, v 0}` ASM_CASES_TAC; TYPIFY `e1'` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; TYPIFY `{vec 0, v 3, f (v 4) t, f (v 0) t} = {vec 0, f (v 4) t, f (v 0) t, v 3}` (C SUBGOAL_THEN SUBST1_TAC); BY(SET_TAC[]); REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; BY(MP_TAC PI_POS THEN REAL_ARITH_TAC); POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`] THEN REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR;DE_MORGAN_THM] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `&0 < azim (vec 0) (v 4) (v 0) (v 3) /\ azim (vec 0) (v 4) (v 0) (v 3) < pi` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[arith `&0 < x <=> &0 <= x /\ ~(x = &0) `;arith `x < pi <=> x <= pi /\ ~(x = pi)`]; REWRITE_TAC[Counting_spheres.AZIM_NN]; ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `a <= pi` (C INTRO_TAC [`4`]) THEN NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[]); TYPIFY `&0 < e` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation;IN_REAL_INTERVAL]; BY(MESON_TAC[]); INTRO_TAC Cuxvzoz.epsilon_pair [`e3`;`e`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Zlzthic.WNWSHJT [`v 3`;`v 4`;`v 0`;`f`;`-- e'''`;`e'''`] THEN ASM_REWRITE_TAC[]; ANTS_TAC; TYPIFY_GOAL_THEN `~collinear {vec 0, v 4, v 0} /\ ~collinear {vec 0, v 4, v 3}` (unlist REWRITE_TAC); BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); MATCH_MP_TAC Cuxvzoz.deformation_restrict; TYPIFY `e` EXISTS_TAC THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC Zlzthic.deformation_subset; TYPIFY `(IMAGE v (:num))` EXISTS_TAC; ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]; BY(GEN_TAC THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_UNIV]); BY(MESON_TAC[arith `x < pi ==> x <= pi`]); (REPEAT WEAKER_STRIP_TAC THEN REWRITE_TAC[GSYM CONJ_ASSOC]); COMMENT "azim additivity"; SUBCONJ_TAC; INTRO_TAC Cuxvzoz.epsilon_triple [`e1`;`e1''`;`e3`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 3 (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC Cuxvzoz.azim_dominated_split [`(vec 0):real^3`;`v 3`;`v 4`;`v 1`;`v 2`;`(vec 0):real^3`;`v 3`;`f (v 4) t`;`v 1`;`v 2`]; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`3`;`4`]; ASM_REWRITE_TAC[LET_THM;arith `SUC 3 = 4 /\ 3 + 4 = 7 /\ 3 + 3 = 6 /\ 3 <= 5 /\ 0 < 3 /\ 3 < 4 /\ 4 < 5`]; DISCH_THEN (SUBST1_TAC o GSYM) THEN ASM_REWRITE_TAC[arith `x <= x`]; INTRO_TAC Cuxvzoz.azim_dominated_split [`(vec 0):real^3`;`v 3`;`v 4`;`v 0`;`v 1`;`(vec 0):real^3`;`v 3`;`f (v 4) t`;`f (v 0) t`;`v 1`]; ASM_REWRITE_TAC[arith `x <= x`]; INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`3`;`2`;`3`]; NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[LET_THM]; ASM_REWRITE_TAC[]; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`] THEN ASM_REWRITE_TAC[]; TYPIFY_GOAL_THEN `~collinear {vec 0, v 1, v 3} /\ ~collinear {vec 0, v 1, v 3} /\ ~collinear {vec 0, v 2, v 3}` (unlist REWRITE_TAC); BY(REPEAT CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); BY(REPLICATE_TAC 15 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC); COMMENT "azim 0 additive"; REPEAT WEAKER_STRIP_TAC; SUBCONJ_TAC; INTRO_TAC Cuxvzoz.epsilon_triple [`e'`;`e1''`;`e3`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC Fan.sum3_azim_fan; ASM_REWRITE_TAC[]; BY(REPLICATE_TAC 15 (FIRST_X_ASSUM MP_TAC) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; COMMENT "azim 1 additive"; SUBCONJ_TAC; INTRO_TAC Cuxvzoz.epsilon_pair [`e''`;`e3`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''` EXISTS_TAC; ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 2 (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPEAT (FIRST_X_ASSUM_ST `abs t < e` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC Fan.sum3_azim_fan; ASM_REWRITE_TAC[]; CONJ_TAC; BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MP_TAC PI_POS THEN REAL_ARITH_TAC); ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`]; ASM_REWRITE_TAC[]; BY(CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); REPEAT WEAKER_STRIP_TAC; COMMENT "pi bounds"; CONJ_TAC; TYPIFY `e'` EXISTS_TAC; CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; TYPIFY ` &0 < azim (vec 0) (f (v 0) t) (f (v 1) t) (f (v 3) t) /\ azim (vec 0) (f (v 0) t) (f (v 1) t) (f (v 3) t) < pi - azim (vec 0) (v 0) (v 3) (v 4)` ENOUGH_TO_SHOW_TAC; ASM_REWRITE_TAC[]; MATCH_MP_TAC (arith `&0 <= b ==> (&0 < a /\ a < pi - b ==> &0 < a /\ a < pi)`); BY(REWRITE_TAC[Counting_spheres.AZIM_NN]); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); MATCH_MP_TAC Oxl_def.periodic_numseg; TYPIFY `5` EXISTS_TAC; REWRITE_TAC[arith `~(5 = 0)`]; CONJ_TAC; FIRST_X_ASSUM_ST `periodic` MP_TAC; REWRITE_TAC[Oxl_def.periodic]; REWRITE_TAC[arith `(i+5)+j = (i+j)+5`]; BY(DISCH_THEN (unlist REWRITE_TAC)); REWRITE_TAC[IN_NUMSEG;arith `(0 <= i /\ i <= 5-1) <=> (i = 0 \/ i = 1 \/ i = 2 \/ i = 3 \/ i = 4)`]; REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC; EXISTS_TAC `&1` THEN REPEAT (FIRST_X_ASSUM_ST `azim (vec 0) (v 0) (v 1) (v 4) < pi` MP_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `x < pi ==> ~(x=pi)`]); REPEAT (FIRST_X_ASSUM_ST `+` kill); ASM_REWRITE_TAC[] THEN EXISTS_TAC `&1` THEN REPEAT (FIRST_X_ASSUM_ST `azim (vec 0) (v 1) (v 2) (v 0) < pi` MP_TAC); BY(MESON_TAC[arith `&0 < &1`;arith `x < pi ==> ~(x=pi)`]); BY(ASM_REWRITE_TAC[] THEN EXISTS_TAC `&1` THEN MESON_TAC[arith `&0 < &1`;arith `x <= x`]); INTRO_TAC Cuxvzoz.epsilon_triple [`e1'''`;`e1''`;`e1`] THEN ANTS_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; TYPIFY `e'''` EXISTS_TAC THEN CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; REPLICATE_TAC 3(FIRST_X_ASSUM (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC); REPEAT (FIRST_X_ASSUM_ST `azim` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `a = pi` (SUBST1_TAC o GSYM); ASM_REWRITE_TAC[]; BY(REPLICATE_TAC 15 (FIRST_X_ASSUM MP_TAC) THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); TYPIFY `e1''` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `azim a b c d = azim a' b' c' d'` (C INTRO_TAC [`t`]); BY(ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[arith `x <= x`]) ]);; (* }}} *) let L13_LEMMA = prove_by_refinement( `main_nonlinear_terminal_v11 ==> (!V E FF s v. IMAGE (\i. v i,v (SUC i)) (:num) = FF /\ is_scs_v39 s /\ scs_k_v39 s = 5 /\ scs_basic_v39 s /\ scs_generic v /\ BBs_v39 s v /\ IMAGE v (:num) = V /\ IMAGE (\i. {v i, v (SUC i)}) (:num) = E /\ ~coplanar {vec 0, v 1, v 2, v 0} ==> &0 < azim (vec 0) (v 0) (v 1) (v 3) )`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[arith `&0 < a <=> ~(a = &0) /\ (&0 <= a)`]; REWRITE_TAC[Counting_spheres.AZIM_NN]; DISCH_TAC; FIRST_X_ASSUM_ST `coplanar {vec 0,v 1 ,v 2,v 0}` MP_TAC; REWRITE_TAC[]; INTRO_TAC EGHNAVX_SCS [`s`;`5`;`v`;`0`]; ASM_REWRITE_TAC[LET_THM]; NUM_REDUCE_TAC; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM kill; FIRST_X_ASSUM (C INTRO_TAC [`3`;`2`]); NUM_REDUCE_TAC; ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[coplanar]; GEXISTL_TAC [`(vec 0):real^3`;`v 0`;`v 1`]; TYPIFY `{vec 0, v 0, v 1} SUBSET affine hull {vec 0, v 0, v 1} /\ v 2 IN affine hull {vec 0, v 0, v 1}` ENOUGH_TO_SHOW_TAC; BY(SET_TAC[]); REWRITE_TAC[HULL_SUBSET]; INTRO_TAC AFF_GT_SUBSET_AFFINE_HULL [`{vec 0,v 0}`;`{v 1}`]; TYPIFY `{vec 0, v 0} UNION {v 1} = {vec 0, v 0, v 1}` (C SUBGOAL_THEN SUBST1_TAC); BY(SET_TAC[]); BY(FIRST_X_ASSUM MP_TAC THEN SET_TAC[]) ]);; (* }}} *) let IUNBUIG = prove_by_refinement( `main_nonlinear_terminal_v11 ==> (!s FF v. IMAGE (\i. (v i,v (SUC i))) (:num) = FF /\ is_scs_v39 s /\ scs_k_v39 s = 5 /\ MMs_v39 s v /\ scs_basic_v39 s /\ scs_generic v /\ (!i j. scs_diag 5 i j ==> scs_a_v39 s i j <= cstab /\ cstab < dist(v i,v j) /\ &4 * h0 < scs_b_v39 s i j) /\ scs_a_v39 s 0 1 = &2 /\ scs_b_v39 s 0 1 <= cstab /\ interior_angle1 (vec 0) FF (v 0) < pi /\ interior_angle1 (vec 0) FF (v 1) < pi /\ (!i. ~(i MOD 5 = 0) ==> dist(v i,v (i+1)) = &2) /\ xrr (norm (v 0)) (norm (v 3)) (dist(v 0,v 3)) <= #15.53 /\ xrr (norm (v 1)) (norm (v 3)) (dist(v 1,v 3)) <= #15.53 ==> dist(v 0,v 1) = &2)`, (* {{{ proof *) [ REPEAT WEAKER_STRIP_TAC; TYPIFY `BBprime2_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.scs_basic;Ayqjtmd.unadorned_MMs]); TYPIFY `BBprime_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.BBprime2_v39]); TYPIFY `BBs_v39 s v` (C SUBGOAL_THEN ASSUME_TAC); BY(ASM_MESON_TAC[Appendix.BBprime_v39]); INTRO_TAC Appendix.BBs_v39 [`s`;`v`]; ASM_REWRITE_TAC[LET_DEF;LET_END_DEF;Terminal.IMAGE_SUBSET_IN;IN_UNIV;arith `~(5 <= 3)`]; REPEAT WEAKER_STRIP_TAC; TYPED_ABBREV_TAC `V = IMAGE v (:num)`; TYPED_ABBREV_TAC `E = IMAGE (\i. {v i, v (SUC i)}) (:num)`; TYPIFY `local_fan (V,E,FF)` (C SUBGOAL_THEN ASSUME_TAC); MATCH_MP_TAC Local_lemmas.CVX_LO_IMP_LO; BY(ASM_REWRITE_TAC[]); TYPIFY `!i. v i IN V` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; EXPAND_TAC "V"; REWRITE_TAC[IN_IMAGE;IN_UNIV]; BY(MESON_TAC[]); INTRO_TAC Cuxvzoz.BBs_inj [`s`;`v`;`5`]; ASM_REWRITE_TAC[]; DISCH_TAC; TYPIFY `!i. interior_angle1 (vec 0) FF (v i) = azim (vec 0) (v i) (v (i+1)) (v (i+4))` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; EXPAND_TAC "FF"; GMATCH_SIMP_TAC (REWRITE_RULE[LET_THM] (GSYM Ocbicby.INTERIOR_ANGLE1_AZIM)); TYPIFY `s` EXISTS_TAC; BY(ASM_REWRITE_TAC[IN;arith `5 - 1 = 4`;arith `3 < 5`;arith `SUC i = i+1`]); INTRO_TAC Ocbicby.LOCAL_FAN_AZIM_POS [`s`;`v`]; ASM_REWRITE_TAC[LET_THM;IN;arith `5-1 = 4`;arith `SUC i = i+1`]; DISCH_TAC; TYPIFY `!i. v (i MOD 5) = v i` (C SUBGOAL_THEN ASSUME_TAC); GEN_TAC; MATCH_MP_TAC (GSYM Oxl_def.periodic_mod); BY(ASM_REWRITE_TAC[arith `~(0=5)`]); TYPIFY `v 5 = v 0 /\ v 6 = v 1 /\ v 7 = v 2 /\ v 8 = v 3 /\ v 9 = v 4` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM ((unlist ONCE_REWRITE_TAC) o GSYM); BY(REWRITE_TAC[Ocbicby.MOD_5_EXPLICIT;arith `8 = 1*5 + 3`;arith `9 = 1*5 + 4`;MOD_MULT_ADD]); TYPIFY `~coplanar {vec 0, v 0,v 1,v 4} /\ ~coplanar {vec 0,v 1,v 2,v 0}` (C SUBGOAL_THEN ASSUME_TAC); ASM_REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; BY(ASM_MESON_TAC[arith `x < pi ==> ~(x = pi)`;arith `&0 < a ==> ~(a = &0)`;arith `0+i=i`;arith `1+1=2 /\ 1+4=5`]); TYPIFY `generic V E` (C SUBGOAL_THEN ASSUME_TAC); BY(FIRST_X_ASSUM_ST `scs_generic` MP_TAC THEN ASM_REWRITE_TAC[Appendix.scs_generic]); INTRO_TAC Terminal.vv_split_azim_generic [`v`;`5`;`0`;`3`;`4`]; NUM_REDUCE_TAC; ASM_REWRITE_TAC[LET_THM]; DISCH_TAC; COMMENT "collinearity"; TYPIFY `!i j. ~(v i = v j) ==> ~collinear {vec 0,v i, v j}` (C SUBGOAL_THEN ASSUME_TAC); REPEAT GEN_TAC THEN DISCH_TAC; MATCH_MP_TAC Zlzthic.PROPERTIES_GENERIC_LOCAL_FAN_ALT; GEXISTL_TAC [`IMAGE v (:num)`;`IMAGE (\i. {v i, v (SUC i)}) (:num)`;`IMAGE (\i. (v i, v (SUC i))) (:num)`]; ASM_REWRITE_TAC[IN_IMAGE;IN_UNIV]; BY(MESON_TAC[]); TYPIFY `!i j. i < 5 /\ j < 5 /\ ~(i = j) ==> ~(v i = v j)` (C SUBGOAL_THEN ASSUME_TAC); BY(REPLICATE_TAC 10 (FIRST_X_ASSUM MP_TAC) THEN MESON_TAC[]); COMMENT "non coplanar 013 digression"; INTRO_TAC (UNDISCH L13_LEMMA) [`V`;`E`;`FF`;`s`;`v`]; ASM_REWRITE_TAC[]; DISCH_TAC; REPEAT (FIRST_X_ASSUM_ST `interior_angle1 a b c < pi` MP_TAC) THEN ASM_REWRITE_TAC[] THEN NUM_REDUCE_TAC; REPEAT WEAKER_STRIP_TAC; TYPIFY `azim (vec 0) (v 0) (v 1) (v 3) < pi` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `+` MP_TAC; MATCH_MP_TAC (arith `&0 <= a2 /\ a < pi ==> (a = a1 + a2 ==> a1 < pi)`); REWRITE_TAC[Counting_spheres.AZIM_NN]; BY(ASM_REWRITE_TAC[]); TYPIFY `~coplanar {vec 0,v 0, v 1,v 3}` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; BY(REPLICATE_TAC 4 (POP_ASSUM MP_TAC) THEN REAL_ARITH_TAC); COMMENT "introduce deformation"; INTRO_TAC deform_pent_exists [`V`;`(\t. -- abs t)`;`v 3`;`v 4`;`v 0`;`v 1`;`&1`]; ASM_REWRITE_TAC[]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM MP_TAC; ANTS_TAC; REWRITE_TAC[arith `&0 < &1`;arith `--abs (&0) = &0`]; CONJ_TAC; DISCH_TAC; INTRO_TAC coplanar_cross_reduction [`s`;`5`;`v`;`3`]; NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[]); CONJ_TAC; FIRST_X_ASSUM MP_TAC; BY(MESON_TAC[SET_RULE `{a,b,c,d} = {a,d,b,c}`]); CONJ_TAC; BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); CONJ_TAC; INTRO_TAC (GEN_ALL Local_lemmas.CONVEX_LOFA_IMP_INANGLE_LE_PI) [`E`;`V`;`FF`;`v 4`]; ASM_REWRITE_TAC[]; NUM_REDUCE_TAC; BY(ASM_REWRITE_TAC[]); CONJ_TAC; INTRO_TAC Trigonometry.JBDNJJB [`v 0`;`v 1`;`v 3`]; TYPIFY `v 0 dot (v 1 cross v 3) = ((v 0 cross v 1) dot v 3)` (C SUBGOAL_THEN SUBST1_TAC); BY(VEC3_TAC); REWRITE_TAC[arith `x > &0 <=> &0 < x`]; ONCE_REWRITE_TAC[Leaf_cell.RE_EQVL_SYM]; REWRITE_TAC[Trigonometry2.re_eqvl]; REPEAT WEAKER_STRIP_TAC; ASM_REWRITE_TAC[]; GMATCH_SIMP_TAC REAL_LT_MUL_EQ; BY(ASM_REWRITE_TAC[Cuxvzoz.sin_azim_pos]); GMATCH_SIMP_TAC REAL_CONTINUOUS_ON_EQ_REAL_CONTINUOUS_AT; REWRITE_TAC[Ocbicby.REAL_OPEN_REAL_INTERVAL]; REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC REAL_CONTINUOUS_NEG; BY(REWRITE_TAC[Cuxvzoz.real_continuous_abs]); REPEAT WEAKER_STRIP_TAC; (COMMENT "reduction lemmas"); PROOF_BY_CONTR_TAC; TYPIFY `&2 < dist(v 0,v 1)` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `scs_a_v39` (C INTRO_TAC [`0`;`1`]); BY(ASM_REWRITE_TAC[] THEN FIRST_X_ASSUM MP_TAC THEN REAL_ARITH_TAC); FIRST_X_ASSUM_ST `~` MP_TAC THEN REWRITE_TAC[]; TYPIFY `!i. f (v i) (&0) = v i` (C SUBGOAL_THEN ASSUME_TAC); FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation]; BY(ASM_MESON_TAC[]); INTRO_TAC a5_assumption_reduction [`s`;`f`;`v`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `scs_diag` (C INTRO_TAC [`i`;`j`]); BY(ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC); CONJ_TAC; TYPIFY `e'` EXISTS_TAC THEN CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; FIRST_X_ASSUM (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[DIST_SYM]; BY(MESON_TAC[]); INTRO_TAC Cuxvzoz.continuous_nbd_pos [`\ t. dist(f (v 0) t,v 1) - &2`;`&0`]; ASM_REWRITE_TAC[arith `abs (t' - &0) = abs t'`;arith `&0 < a - b <=> b < a`]; ANTS_TAC; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation]; REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `atreal` (C INTRO_TAC [`v 0`;`&0`]); ASM_REWRITE_TAC[]; DISCH_TAC; BY((REPEAT (REPEAT CONJ_TAC THEN (MATCH_MP_TAC REAL_CONTINUOUS_ADD ORELSE MATCH_MP_TAC Local_lemmas1.CON_ATREAL_REAL_CON ORELSE MATCH_MP_TAC Local_lemmas1.CON_ATREAL_REAL_CON2 ORELSE MATCH_MP_TAC REAL_CONTINUOUS_MUL ORELSE MATCH_MP_TAC REAL_CONTINUOUS_POW ORELSE MATCH_MP_TAC REAL_CONTINUOUS_SUB THEN ASM_REWRITE_TAC[REAL_CONTINUOUS_CONST;ETA_AX])))); BY(MESON_TAC[arith `x < y ==> x <= y`]); DISCH_TAC; (COMMENT "b5 reduction"); INTRO_TAC b5_assumption_reduction [`s`;`f`;`v`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; BY(ASM_MESON_TAC[]); CONJ_TAC; TYPIFY `e'` EXISTS_TAC; FIRST_X_ASSUM_ST `norm` MP_TAC THEN ASM_REWRITE_TAC[DIST_SYM]; BY(MESON_TAC[]); TYPIFY `e'` EXISTS_TAC; CONJ_TAC THENL [ASM_REWRITE_TAC[];REPEAT WEAKER_STRIP_TAC]; FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[DIST_SYM]; DISCH_THEN (unlist REWRITE_TAC); BY(REAL_ARITH_TAC); DISCH_TAC; COMMENT "deform collinear"; TYPIFY `!t. f (v 1) t = v 1 /\ f (v 2) t = v 2 /\ f (v 3) t = v 3` (C SUBGOAL_THEN ASSUME_TAC); BY(GEN_TAC THEN REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN NUM_REDUCE_TAC); TYPIFY `!i j. i < 5 /\ j < 5 /\ ~(v i = v j) ==> (?e. &0 < e /\ (!t. abs t < e==> ~collinear {vec 0, f (v i) t, f (v j) t}))` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; INTRO_TAC (GEN_ALL Local_lemmas1.CONTINUOUS_PRESERVE_COLLINEAR) [`&0`;`(vec 0):real^3`;`f (v i)`;`f (v j)`]; REWRITE_TAC[arith `abs(&0 - r') = abs r'`] THEN DISCH_THEN MATCH_MP_TAC; ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation]; REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC; BY(REPEAT CONJ_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]); FIRST_X_ASSUM MATCH_MP_TAC; FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_MESON_TAC[]); TYPIFY `!i. azim (vec 0) (v i) (v (i+1)) (v(i+4)) <= pi` (C SUBGOAL_THEN ASSUME_TAC); REWRITE_TAC[arith `i+1 = SUC i`;arith `4 = 5 - 1`]; GEN_TAC; MATCH_MP_TAC (REWRITE_RULE[LET_THM] Terminal.convex_local_fan_azim_le_pi); ASM_REWRITE_TAC[arith `3 <= 5`]; FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM]; BY(ASM_REWRITE_TAC[]); COMMENT "coplanar"; INTRO_TAC Zlzthic.NONPLANAR_OPEN [`(\ (t:real). (vec 0):real^3)`;`f (v 0)`;`\ (t:real). v 1`;`\ (t:real). v 3`;`&0`]; ASM_REWRITE_TAC[CONTINUOUS_CONST]; ANTS_TAC; FIRST_X_ASSUM_ST `deformation` MP_TAC THEN REWRITE_TAC[Localization.deformation] THEN REPEAT WEAKER_STRIP_TAC; BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]); REWRITE_TAC[arith `abs (&0 - t') = abs t'`] THEN REPEAT WEAKER_STRIP_TAC; COMMENT "azim reduction"; INTRO_TAC azim5_reduction [`s`;`f`;`v`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; BY(REPLICATE_TAC 3 (FIRST_X_ASSUM_ST `a < pi` MP_TAC) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[]); CONJ2_TAC; TYPIFY `e'` EXISTS_TAC; BY(ASM_REWRITE_TAC[arith `x - abs t = x + --abs t`]); INTRO_TAC Zlzthic.WNWSHJT [`v 1`;`v 3`;`v 0`;`f`;`-- e'`;`e'`]; ASM_REWRITE_TAC[]; ANTS_TAC; CONJ_TAC; MATCH_MP_TAC Zlzthic.deformation_subset; TYPIFY `V` EXISTS_TAC; ASM_REWRITE_TAC[SUBSET;IN_INSERT;NOT_IN_EMPTY]; BY(EXPAND_TAC "V" THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[] THEN MATCH_MP_TAC FUN_IN_IMAGE THEN REWRITE_TAC[IN_UNIV]); TYPIFY_GOAL_THEN `~collinear {vec 0, v 3, v 0} /\ ~collinear {vec 0, v 3, v 1}` (unlist REWRITE_TAC); BY(CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); FIRST_X_ASSUM_ST `coplanar` kill; FIRST_X_ASSUM_ST `~coplanar s` MP_TAC THEN ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,d,b,c}`] THEN REWRITE_TAC[GSYM Local_lemmas1.AZIM_COND_FOR_COPLANAR]; REWRITE_TAC[arith `&0 < a <=> &0 <= a /\ ~(a = &0)`]; REWRITE_TAC[Counting_spheres.AZIM_NN]; REWRITE_TAC[arith `a < pi <=> a <= pi /\ ~(a = pi)`;DE_MORGAN_THM]; DISCH_THEN (unlist REWRITE_TAC); REPLICATE_TAC 2(MATCH_MP_TAC Xivphks.FSQKWKK); FIRST_X_ASSUM_ST `azim` (C INTRO_TAC [`0`]) THEN NUM_REDUCE_TAC THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC (arith `&0 <= a2 ==> (a1 + a2 <= pi ==> a1 <= pi)`); BY(REWRITE_TAC[Counting_spheres.AZIM_NN]); (REPEAT WEAKER_STRIP_TAC); INTRO_TAC Cuxvzoz.epsilon_triple [`e`;`e''`;`e'`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `e''''` EXISTS_TAC THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 3 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; INTRO_TAC azim_dih_y [`(vec 0):real^3`;`v 3`;`f (v 0) t`;`v 1`]; INTRO_TAC azim_dih_y [`(vec 0):real^3`;`v 3`;`(v 0)`;`v 1`]; DISCH_THEN GMATCH_SIMP_TAC; CONJ_TAC; FIRST_X_ASSUM_ST `x < pi` (C INTRO_TAC [`&0`]) THEN ASM_REWRITE_TAC[arith `abs(&0) = &0`]; SIMP_TAC[arith `x < pi ==> x <= pi`]; DISCH_TAC; BY(CONJ_TAC THEN REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); DISCH_THEN GMATCH_SIMP_TAC; CONJ_TAC; REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC; MATCH_MP_TAC (arith `x < pi ==> x <= pi`); BY(FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[]); CONJ_TAC; ONCE_REWRITE_TAC[SET_RULE `{a,b,c} = {a,c,b}`]; BY(FIRST_X_ASSUM_ST `coplanar` (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[] THEN MESON_TAC[Planarity.notcoplanar_imp_notcollinear_fan]); BY(REPEAT (FIRST_X_ASSUM MATCH_MP_TAC) THEN NUM_REDUCE_TAC); FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`]); ASM_REWRITE_TAC[DIST_SYM;DIST_0]; REPEAT WEAKER_STRIP_TAC THEN ASM_REWRITE_TAC[]; MATCH_MP_TAC Ocbicby.dih_y_mono; FIRST_X_ASSUM_ST `~coplanar s` (C INTRO_TAC [`t`]) THEN ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `~coplanar s` MP_TAC; REPEAT WEAKER_STRIP_TAC; REWRITE_TAC[arith `d + -- abs t <= d`]; CONJ_TAC; REWRITE_TAC[NORM_POS_LT]; POP_ASSUM (MP_TAC o (MATCH_MP Planarity.notcoplanar_disjoint)); BY(MESON_TAC[]); CONJ_TAC; FIRST_X_ASSUM_ST `abs` (SUBST1_TAC o GSYM); REWRITE_TAC[GSYM DIST_NZ]; POP_ASSUM (MP_TAC o (MATCH_MP Planarity.notcoplanar_disjoint)); BY(MESON_TAC[]); CONJ_TAC; POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[Oxlzlez.coplanar_delta_y;DIST_SYM;DIST_0]; BY(MESON_TAC[Merge_ineq.delta_y_sym]); INTRO_TAC Terminal.DELTA_Y_POS_4POINTS [`(vec 0):real^3`;`v 3`;`v 0`;`v 1`]; BY(REWRITE_TAC[DIST_0;DIST_SYM]); REPEAT WEAKER_STRIP_TAC; (COMMENT "end of azim reduction"); COMMENT "deformation in BBs"; INTRO_TAC Cuxvzoz.deformation_BBs_ALT [`s`;`5`;`f`;`v`;`e'`]; NUM_REDUCE_TAC; ASM_REWRITE_TAC[]; ANTS_TAC; MATCH_MP_TAC Terminal.periodic_mod_reduce; TYPIFY `5` EXISTS_TAC THEN REWRITE_TAC[arith `~(5 = 0)`]; CONJ_TAC; FIRST_X_ASSUM_ST `periodic` MP_TAC THEN REWRITE_TAC[Oxl_def.periodic]; BY(MESON_TAC[]); REWRITE_TAC[arith `i < 5 <=> i = 0 \/ i = 1 \/ i = 2 \/ i = 3 \/ i = 4`]; REPEAT WEAKER_STRIP_TAC THEN (FIRST_X_ASSUM_ST `norm` (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; BY(FIRST_X_ASSUM_ST ` \/ ` MP_TAC THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[]); REPEAT WEAKER_STRIP_TAC; COMMENT "back to root, next tau"; TYPIFY `!t. abs t < e'' ==> sum {i | i < 5} (\i. rho_fun (norm (v i)) * azim (vec 0) (v i) (v (i + 1)) (v (i + 5 - 1))) <= sum {i | i < 5} (\i. rho_fun (norm (f (v i) t)) * azim (vec 0) (f (v i) t) (f(v(i+1)) t) (f (v (i + 5 - 1)) t))` (C SUBGOAL_THEN ASSUME_TAC); REPEAT WEAKER_STRIP_TAC; MATCH_MP_TAC Cuxvzoz.MMs_minimize_tau_fun; TYPIFY `s` EXISTS_TAC; ASM_REWRITE_TAC[arith `3 < 5`]; FIRST_X_ASSUM MATCH_MP_TAC; BY(FIRST_X_ASSUM (unlist REWRITE_TAC)); POP_ASSUM MP_TAC; TYPIFY `{i | i < 5} = 0..4` (C SUBGOAL_THEN SUBST1_TAC); BY(REWRITE_TAC[EXTENSION;IN_ELIM_THM;IN_NUMSEG] THEN ARITH_TAC); REWRITE_TAC[SUM_NUMSEG4]; NUM_REDUCE_TAC; DISCH_TAC; COMMENT "introduce nonlinear 684"; TYPIFY `scs_diag 5 0 2 /\ scs_diag 5 0 3 /\ scs_diag 5 1 3 /\ scs_diag 5 1 4 /\ scs_diag 5 2 4` (C SUBGOAL_THEN ASSUME_TAC); BY(REWRITE_TAC[Appendix.scs_diag;Uxckfpe.ARITH_5_TAC]); INTRO_TAC (UNDISCH Ocbicby.LEMMA_6843920790) [`norm (v 3)`;`norm (v 0)`;`norm (v 1)`;`dist(v 0,v 1)`;`dist (v 1,v 3)`;`dist(v 0,v 3)`]; ASM_REWRITE_TAC[]; ANTS_TAC; ONCE_REWRITE_TAC[Ocbicby.xrr_sym]; ASM_REWRITE_TAC[GSYM Sphere.cstab]; TYPIFY_GOAL_THEN `(&2 <= norm (v 3) /\ norm (v 3) <= &2 * h0) /\ (&2 <= norm (v 0) /\ norm (v 0) <= &2 * h0) /\ (&2 <= norm (v 1) /\ norm (v 1) <= &2 * h0)` (unlist REWRITE_TAC); REWRITE_TAC[GSYM Fnjlbxs.in_ball_annulus]; FIRST_X_ASSUM_ST `BBs_v39` kill THEN FIRST_X_ASSUM_ST `BBs_v39` MP_TAC THEN REWRITE_TAC[Appendix.BBs_v39;LET_THM] THEN EXPAND_TAC "V" THEN REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `ball_annulus` MP_TAC; REWRITE_TAC[SUBSET;IN_IMAGE;IN_UNIV]; BY(MESON_TAC[]); REWRITE_TAC[ CONJ_ASSOC] THEN CONJ2_TAC; INTRO_TAC Oxlzlez.coplanar_delta_y [`(vec 0):real^3`;`v 3`;`v 0`;`v 1`]; REWRITE_TAC[DIST_0;DIST_SYM] THEN DISCH_THEN (SUBST1_TAC o GSYM); BY(ONCE_REWRITE_TAC[SET_RULE `{a,b,c,d} = {a,c,d,b}`] THEN ASM_REWRITE_TAC[]); TYPIFY_GOAL_THEN `(&2 <= dist (v 0,v 1) /\ dist (v 0,v 1) <= cstab)` (unlist REWRITE_TAC); REPEAT (FIRST_X_ASSUM_ST `scs_a_v39` (C INTRO_TAC [`0`;`1`])) THEN FIRST_X_ASSUM_ST `scs_a_v39 s 0 1 = &2` MP_TAC THEN FIRST_X_ASSUM_ST `scs_b_v39 s 0 1 <= cstab` MP_TAC; BY(REAL_ARITH_TAC); CONJ_TAC; REPEAT (FIRST_X_ASSUM_ST `scs_a_v39` (C INTRO_TAC [`1`;`3`]) ); BY(ASM_REWRITE_TAC[Sphere.cstab] THEN REAL_ARITH_TAC); REPEAT (FIRST_X_ASSUM_ST `scs_a_v39` (C INTRO_TAC [`0`;`3`]) ); BY(ASM_REWRITE_TAC[Sphere.cstab] THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; PROOF_BY_CONTR_TAC; FIRST_X_ASSUM_ST `IN` MP_TAC THEN REWRITE_TAC[NOT_FORALL_THM;TAUT `~(a ==> b) <=> (a /\ ~b)`;arith `~(a < b) <=> b <= a`]; TYPIFY `?e2. &0 < e2 /\ (!t. abs t < e2 ==> dist(v 0,v 1) + t IN real_interval(a,b))` (C SUBGOAL_THEN MP_TAC); INTRO_TAC Zlzthic.real_interval_contains_0_ball [`a-dist(v 0,v 1)`;`b - dist(v 0,v 1)`;`&1`]; REWRITE_TAC[arith `&0 < &1`;arith `a - d < &0 <=> a < d`;arith `&0 < b - d <=> d < b`]; ANTS_TAC; FIRST_X_ASSUM_ST `dist s IN real_interval (a,b)` MP_TAC; BY(REWRITE_TAC[IN_REAL_INTERVAL]); REPEAT WEAKER_STRIP_TAC; TYPIFY `e'''` EXISTS_TAC; (ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC THEN (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPLICATE_TAC 2 (FIRST_X_ASSUM MP_TAC) THEN REWRITE_TAC[IN_REAL_INTERVAL]); BY(REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; TYPIFY `?t. (t < &0) /\ abs t < e2 /\ taum (norm (v 3)) (norm (v 0)) (norm (v 1)) (dist(v 0, v 1)) (dist (v 1,v 3)) (dist (v 0,v 3)) <= taum (norm (v 3)) (norm (v 0)) (norm (v 1)) (dist (v 0,v 1) + t) (dist (v 1,v 3)) (dist (v 0,v 3))` ENOUGH_TO_SHOW_TAC; REPEAT WEAKER_STRIP_TAC; GEXISTL_TAC [`dist(v 0,v 1) + t`;`dist(v 0,v 1)`]; ASM_REWRITE_TAC[]; CONJ2_TAC; BY(REPLICATE_TAC 5 (FIRST_X_ASSUM MP_TAC) THEN REAL_ARITH_TAC); FIRST_X_ASSUM MATCH_MP_TAC; BY(ASM_REWRITE_TAC[]); COMMENT "choose epsilon and t"; INTRO_TAC epsilon_hex [`e`;`e'`;`e''`;`e1`;`e1'`;`e2`] THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `?t. t < &0 /\ abs t < e6` (C SUBGOAL_THEN MP_TAC); TYPIFY `-- e6 / &2` EXISTS_TAC; BY(FIRST_X_ASSUM_ST `&0 < e6` MP_TAC THEN REAL_ARITH_TAC); REPEAT WEAKER_STRIP_TAC; REPLICATE_TAC 6 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; TYPIFY `t` EXISTS_TAC THEN ASM_REWRITE_TAC[]; REPLICATE_TAC 8 (FIRST_X_ASSUM (C INTRO_TAC [`t`])) THEN ASM_REWRITE_TAC[] THEN REPEAT WEAKER_STRIP_TAC; FIRST_X_ASSUM_ST `rho_fun` MP_TAC THEN ASM_REWRITE_TAC[Appendix.rho_rho_fun]; REWRITE_TAC[arith `r0 * (a013 + a034) + r1 * (a123 + a130) + r2 * a231 + r3 * (a340 + a301+a312) + r4 * a403 <= r0 * (f013 + a034) + r1 * (a123 + f130) + r2 * a231 + r3 *(a340+f301 +a312) + r4*a403 <=> r0 * a013 + r1*a130 + r3* a301 <= r0 * f013 + r1 * f130 + r3 * f301`]; MATCH_MP_TAC (arith `a - (pi+sol0) = a' /\ b - (pi+sol0) = b' ==> (a <= b ==> a' <= b')`); REWRITE_TAC[arith `(a + b+c) - (pi + sol0) = a + b + c - (pi + sol0)`]; CONJ_TAC; GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim); GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar; ASM_REWRITE_TAC[]; REWRITE_TAC[DIST_SYM]; CONJ2_TAC; BY(MESON_TAC[Terminal.taum_sym]); BY(REPEAT (FIRST_X_ASSUM_ST `a < pi` MP_TAC) THEN REAL_ARITH_TAC); FIRST_X_ASSUM_ST `norm a = norm b` (SUBST1_TAC o GSYM); TYPIFY `dist(v 0,v 1) + t = dist(v 0,v 1) + -- abs t` (C SUBGOAL_THEN SUBST1_TAC); BY(FIRST_X_ASSUM_ST `t < &0` MP_TAC THEN REAL_ARITH_TAC); FIRST_X_ASSUM_ST `d = d' + -- abs t` (SUBST1_TAC o GSYM); GMATCH_SIMP_TAC (GSYM Cuxvzoz.tau3_azim); GMATCH_SIMP_TAC Cuxvzoz.tau3_taum_nonplanar; REWRITE_TAC[DIST_SYM]; REWRITE_TAC[CONJ_ASSOC] THEN CONJ2_TAC; ASM_REWRITE_TAC[DIST_SYM]; BY(MESON_TAC[Terminal.taum_sym]); ASM_REWRITE_TAC[]; FIRST_X_ASSUM_ST `a < pi` MP_TAC; BY(REAL_ARITH_TAC) ]);; (* }}} *) end;;