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| (* ========================================================================= *) | |
| (* *) | |
| (* Quantum optics library: single mode electromagnetic field. *) | |
| (* *) | |
| (* (c) Copyright, Mohamed Yousri Mahmoud , 2012-2014 *) | |
| (* Hardware Verification Group, *) | |
| (* Concordia University *) | |
| (* *) | |
| (* Contact: <mo_solim@ece.concordia.ca>, *) | |
| (* *) | |
| (* Last update: April 18, 2016 *) | |
| (* *) | |
| (* ========================================================================= *) | |
| needs "Functionspaces/cfunspace.ml";; | |
| (*****************************************************************************) | |
| (* SQUARE INTEGRABLE FUNCTIONS (L2) *) | |
| (*****************************************************************************) | |
| parse_as_infix("complex_measurable_on",(12,"right"));; | |
| let complex_measurable = new_definition | |
| `f complex_measurable_on s <=> (\x. Re (f x)) real_measurable_on s /\ | |
| (\x. Im (f x)) real_measurable_on s`;; | |
| let sq_integrable = new_specification ["sq_integrable"] | |
| (prove(`?s. !f. f IN s <=> f complex_measurable_on (:real) /\ (\x. norm (f x) pow 2) | |
| real_integrable_on (:real)`, EXISTS_TAC `{f| f complex_measurable_on (:real) /\ | |
| (\x. norm (f x) pow 2) real_integrable_on (:real)}` THEN SIMP_TAC[IN_ELIM_THM]));; | |
| let r_inprod = new_definition | |
| `r_inprod f g = complex(real_integral (:real) (\x:real. Re (cnj (f x) * (g x))), | |
| real_integral (:real) (\x. Im (cnj (f x) * (g x))) )`;; | |
| (*****************************************************************************) | |
| (*We will prove each property of the inner space in the following *) | |
| (*theorems. We will conclude all properties in one theorem at the very end *) | |
| (*****************************************************************************) | |
| let FRECHET_REAL_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL = prove | |
| (`!f f' f'' x a b. | |
| a < b /\ x IN real_interval[a,b] /\ | |
| (f has_real_derivative f') (atreal x within (real_interval[a,b])) /\ | |
| (f has_real_derivative f'') (atreal x within (real_interval[a,b])) | |
| ==> f' = f''`, | |
| let tem = REWRITE_RULE[MESON[] `A/\B/\C ==> Q <=> C ==> A /\ B ==> Q `] | |
| FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL in | |
| REPEAT GEN_TAC THEN REWRITE_TAC[HAS_REAL_FRECHET_DERIVATIVE_WITHIN ] | |
| THEN REWRITE_TAC[MESON[] `A/\B/\C ==> Q <=> C ==> A /\ B ==> Q `; | |
| IMAGE_LIFT_REAL_INTERVAL ] THEN | |
| DISCH_THEN (ASSUME_TAC o (MATCH_MP tem)) THEN | |
| POP_ASSUM(ASSUME_TAC o( SIMP_RULE[LIFT_IN_INTERVAL ;DIMINDEX_1;LIFT_DROP; | |
| ARITH_RULE`x <= i /\ i <= x <=> i=(x:num)`;lift;LAMBDA_BETA])) THEN | |
| DISCH_THEN (fun th1 -> POP_ASSUM (MP_TAC o (SIMP_RULE[GSYM LIFT_EQ_CMUL; | |
| LIFT_EQ])o(Pa.SPEC `vec 1:`)o (SIMP_RULE[th1;FUN_EQ_THM]))) | |
| THEN REWRITE_TAC[]);; | |
| let cfun_almost_zero = new_specification ["cfun_almost_zero"] | |
| (prove(`?f.(?k. real_negligible k /\ !x. ~(x IN k) ==> f x = Cx(&0))`, | |
| Pa.EXISTS_TAC `cfun_zero:` THEN REWRITE_TAC[cfun_zero;K_THM]THEN Pa.EXISTS_TAC `{}:` | |
| THEN REWRITE_TAC[REAL_NEGLIGIBLE_EMPTY]));; | |
| let is_almost_zero = new_definition | |
| `is_almost_zero1 f = !a b. (?k. real_negligible k /\ !x. x IN real_interval[a,b] | |
| DIFF k ==> f x = Cx(&0))`;; | |
| let REAL_INTEGRA_ZERO_SUBINTERVALS = prove | |
| (`!f. (!x. &0 <= f x) /\ (f has_real_integral &0) (:real) ==> | |
| !a b. (f has_real_integral &0) (real_interval[a,b])`, | |
| REPEAT STRIP_TAC THEN Pa.ASM_CASES_TAC `b<=a:` | |
| THENL[ASM_SIMP_TAC[HAS_REAL_INTEGRAL_NULL];ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `!a b. f real_integrable_on (real_interval[a,b]):` | |
| ASSUME_TAC THENL[ | |
| RULE_ASSUM_TAC(REWRITE_RULE[HAS_REAL_INTEGRAL_ALT;SET_RULE `x IN (:real)`;ETA_AX]) | |
| THEN ASM_REWRITE_TAC[];ALL_TAC] THEN | |
| MP_TAC (Pa.SPECL [`f:`;`real_interval[a,b]:`] REAL_INTEGRAL_POS) | |
| THEN ASM_SIMP_TAC[] THEN | |
| MP_TAC (Pa.SPECL [`f:`;`real_interval[a,b]:`;`(:real):`] REAL_INTEGRAL_SUBSET_LE) | |
| THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_REAL_INTEGRAL_INTEGRABLE_INTEGRAL]) | |
| THEN IMP_REWRITE_TAC[SET_RULE `!s. ~(s={}) ==> s SUBSET (:real)`; | |
| REAL_INTERVAL_NE_EMPTY;GSYM REAL_LE_ANTISYM; | |
| HAS_REAL_INTEGRAL_INTEGRABLE_INTEGRAL] THEN | |
| ASM_SIMP_TAC[REAL_ARITH `~(b <= a) ==> a <= b`]);; | |
| let REAL_POW2_0 = REWRITE_RULE[REAL_ADD_LID;REAL_POW_ZERO; ARITH] | |
| (SPEC `&0` REAL_SOS_EQ_0);; | |
| let RINPROD_ALMOST_ZERO = prove( | |
| `!f. f IN sq_integrable ==> (r_inprod f f = Cx (&0) <=> is_almost_zero1 f)`, | |
| REWRITE_TAC[sq_integrable;r_inprod;r_inprod;RE_CX;IM_CX;GSYM CX_DEF; | |
| COMPLEX_MUL_CNJ;GSYM CX_POW;REAL_INTEGRAL_0; CX_INJ] THEN | |
| REPEAT STRIP_TAC THEN EQ_TAC THENL[ | |
| POP_ASSUM MP_TAC THEN REWRITE_TAC[MESON[] `P==>Q==>A <=> P/\Q ==>A`] THEN | |
| DISCH_THEN (fun thm -> ASSUME_TAC(REWRITE_RULE | |
| [GSYM HAS_REAL_INTEGRAL_INTEGRABLE_INTEGRAL] thm) THEN MP_TAC thm) THEN | |
| Pa.SUBGOAL_THEN `!a b. (\x. norm (f x) pow 2) real_integrable_on (real_interval[a,b]):` | |
| ASSUME_TAC THENL[ | |
| RULE_ASSUM_TAC(REWRITE_RULE[HAS_REAL_INTEGRAL_ALT;SET_RULE `x IN (:real)`;ETA_AX]) | |
| THEN ASM_REWRITE_TAC[];ALL_TAC] | |
| THEN MP_TAC (Pa.SPEC `(\x. norm ((f:real->complex) x) pow 2):` | |
| HAS_REAL_DERIVATIVE_INDEFINITE_INTEGRAL) THEN ASM_REWRITE_TAC[] THEN | |
| MP_TAC (Pa.SPEC `(\x. norm ((f:real->complex) x) pow 2):` | |
| REAL_INTEGRA_ZERO_SUBINTERVALS) THEN | |
| ASM_SIMP_TAC[REAL_LE_POW_2;HAS_REAL_INTEGRAL_INTEGRABLE_INTEGRAL] | |
| THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN | |
| SUBGOAL_THEN `(!a b. ?k. real_negligible k /\ | |
| (!x. x IN real_interval [a,b] DIFF k | |
| ==> ((\x. &0) has_real_derivative norm ((f:real->complex) x) pow 2) | |
| (atreal x within real_interval [a,b]))) ==> | |
| (!a b. | |
| ?k. real_negligible k /\ | |
| (!x. x IN real_interval [a,b] DIFF k ==> norm (f x) pow 2 = &0))` | |
| ASSUME_TAC | |
| THENL[REPEAT STRIP_TAC THEN | |
| POP_ASSUM (MP_TAC o SPECL [`a:real`;`b:real`]) THEN REPEAT STRIP_TAC | |
| THEN Pa.ASM_CASES_TAC `a < b:` THENL[ | |
| Pa.EXISTS_TAC `k:` THEN ASM_SIMP_TAC[] THEN | |
| ASSUME_TAC (Pa.SPECL [`&0:`;`atreal x within real_interval [a,b]:`] HAS_REAL_DERIVATIVE_CONST) | |
| THEN REPEAT STRIP_TAC THEN | |
| MATCH_MP_TAC FRECHET_REAL_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL | |
| THEN MAP_EVERY Pa.EXISTS_TAC [`(\x. &0):`;`x':`;`a:`;`b:`] THEN | |
| ASM_SIMP_TAC[] THEN ASM_MESON_TAC[IN_DIFF ]; | |
| Pa.EXISTS_TAC `{a}:` | |
| THEN ASM_SIMP_TAC[REAL_NEGLIGIBLE_FINITE ;FINITE_SING;real_interval;IN_ELIM_THM;IN_DIFF;IN_SING] | |
| THEN ASM_MESON_TAC[REAL_FIELD `~(a < b) /\ (a <= x /\ x <= b) /\ | |
| ~(x = a) <=> F`]];ALL_TAC] THEN | |
| DISCH_THEN (fun th -> POP_ASSUM (fun th1 -> ASSUME_TAC | |
| (SIMP_RULE[REAL_POW2_0;COMPLEX_NORM_ZERO ](MATCH_MP th1 th)))) | |
| THEN ASM_SIMP_TAC[is_almost_zero];ALL_TAC] THEN | |
| REWRITE_TAC[is_almost_zero] THEN REPEAT STRIP_TAC | |
| THEN MATCH_MP_TAC REAL_INTEGRAL_UNIQUE THEN | |
| REWRITE_TAC[HAS_REAL_INTEGRAL_ALT;SET_RULE `x IN (:real)`] THEN | |
| Pa.SUBGOAL_THEN `!a b. ((\x. norm (f x) pow 2) has_real_integral &0) | |
| (real_interval [a,b]):` ASSUME_TAC | |
| THENL[ | |
| IMP_REWRITE_TAC[HAS_REAL_INTEGRAL_NEGLIGIBLE;REAL_POW2_0;COMPLEX_NORM_ZERO ]; | |
| ALL_TAC] THEN REPEAT STRIP_TAC THENL | |
| [ASM_MESON_TAC[HAS_REAL_INTEGRAL_INTEGRABLE]; | |
| EXISTS_TAC `&1` THEN ASM_SIMP_TAC [REAL_ARITH `&0 < &1`] THEN | |
| REPEAT STRIP_TAC THEN | |
| IMP_REWRITE_TAC[REAL_INTEGRAL_UNIQUE] THEN EXISTS_TAC `&0` THEN | |
| ASM_SIMP_TAC [REAL_ARITH `&0 - &0 = &0`;REAL_ABS_NUM]]);; | |
| let ALOMST_ZERO_ZERO = prove | |
| (`!f g. is_almost_zero1 f ==> r_inprod g f = Cx(&0)`, | |
| REWRITE_TAC[r_inprod;COMPLEX_EQ;CX_DEF;RE;IM] THEN | |
| REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_INTEGRAL_UNIQUE THEN | |
| REWRITE_TAC[HAS_REAL_INTEGRAL_ALT;SET_RULE `x IN (:real)`] | |
| THEN RULE_ASSUM_TAC(REWRITE_RULE[is_almost_zero]) | |
| THENL[Pa.SUBGOAL_THEN `!a b. ((\x. Re (cnj (g x) * f x)) has_real_integral &0) | |
| (real_interval [a,b]):` ASSUME_TAC THENL[ | |
| REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_REAL_INTEGRAL_NEGLIGIBLE THEN | |
| POP_ASSUM ( (X_CHOOSE_TAC `s:real->bool`) o SPEC_ALL) THEN | |
| Pa.EXISTS_TAC `s:` THEN ASM_SIMP_TAC[COMPLEX_MUL_RZERO;RE_CX;IM_CX];ALL_TAC]; | |
| Pa.SUBGOAL_THEN `!a b. ((\x. Im (cnj (g x) * f x)) has_real_integral &0) | |
| (real_interval [a,b]):` ASSUME_TAC | |
| THENL [REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_REAL_INTEGRAL_NEGLIGIBLE THEN | |
| POP_ASSUM ( (X_CHOOSE_TAC `s:real->bool`) o SPEC_ALL) | |
| THEN Pa.EXISTS_TAC `s:` THEN ASM_SIMP_TAC[COMPLEX_MUL_RZERO;RE_CX;IM_CX];ALL_TAC]] | |
| THEN ASM_MESON_TAC[REAL_SUB_RZERO;HAS_REAL_INTEGRAL_INTEGRABLE_INTEGRAL;REAL_ABS_0]);; | |
| let RINPROD_ZERO_EQ = prove | |
| (`!x y. x IN sq_integrable /\ r_inprod x x = Cx(&0) | |
| ==> r_inprod y x = Cx(&0)`, | |
| MESON_TAC[ALOMST_ZERO_ZERO;RINPROD_ALMOST_ZERO]);; | |
| let SQ_RULE = REAL_FIELD `(a+b) pow 2 = a pow 2 + b pow 2 + &2 * a * b`;; | |
| let SQ_RULE_SUB = REAL_FIELD `(a-b) pow 2 = a pow 2 + b pow 2 - &2 * a * b`;; | |
| let ABS_POW_2 = MESON[REAL_ABS_REFL;REAL_LE_POW_2] `!x. abs (x pow 2) = x pow 2`;; | |
| let SQ_INTEGRABLE_SUBSPACE = prove( | |
| `is_cfun_subspace sq_integrable`, | |
| REWRITE_TAC[is_cfun_subspace;sq_integrable;complex_measurable;cfun_zero; | |
| K_THM;RE_CX;IM_CX;REAL_MEASURABLE_ON_0;COMPLEX_NORM_0;REAL_POW_ZERO; | |
| ARITH;REAL_INTEGRABLE_0] THEN REPEAT STRIP_TAC | |
| THENL[ | |
| ASM_SIMP_TAC[CFUN_SMUL;REAL_MEASURABLE_ON_LMUL;RE;complex_mul; | |
| REAL_MEASURABLE_ON_SUB] | |
| ;ASM_SIMP_TAC[CFUN_SMUL;REAL_MEASURABLE_ON_LMUL;IM;complex_mul; | |
| REAL_MEASURABLE_ON_ADD] | |
| ;ASM_SIMP_TAC[CFUN_SMUL;COMPLEX_NORM_MUL;REAL_POW_MUL;REAL_INTEGRABLE_LMUL] | |
| ;ASM_SIMP_TAC[CFUN_ADD_THM;RE_ADD;REAL_MEASURABLE_ON_ADD] | |
| ;ASM_SIMP_TAC[CFUN_ADD_THM;IM_ADD;REAL_MEASURABLE_ON_ADD] | |
| ;RULE_ASSUM_TAC(REWRITE_RULE[SQ_RULE;COMPLEX_SQNORM]) | |
| THEN ASM_SIMP_TAC[CFUN_ADD_THM;complex_add;COMPLEX_SQNORM;RE;IM;SQ_RULE; | |
| REAL_FIELD `((a1:real)+b1+c1) + a2 + b2 + c2 = (a1+a2) + (b1+b2) + c1+ c2`] | |
| THEN MATCH_MP_TAC REAL_INTEGRABLE_ADD THEN ASM_REWRITE_TAC[] | |
| THEN MATCH_MP_TAC REAL_INTEGRABLE_ADD THEN ASM_REWRITE_TAC[] | |
| THEN MATCH_MP_TAC REAL_INTEGRABLE_ADD THEN REPEAT STRIP_TAC | |
| THENL [MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.( Re (x x') pow 2 + Im (x x') pow 2) + | |
| (Re (y x') pow 2 + Im (y x') pow 2):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_ABS_MUL;REAL_ABS_NUM] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+a2-c1)+b1+b2`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2]; | |
| ALL_TAC] THEN | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.( Re (x x') pow 2 + Im (x x') pow 2) + | |
| (Re (y x') pow 2 + Im (y x') pow 2):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_ABS_MUL;REAL_ABS_NUM] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+a2)+(b1+b2-c1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2]]);; | |
| let RINPROD_SELF_POS = prove( | |
| `!f. f IN sq_integrable | |
| ==> real (r_inprod f f) /\ | |
| &0 <= real_of_complex (r_inprod f f)`, | |
| REWRITE_TAC[sq_integrable;REAL;r_inprod;COMPLEX_MUL_CNJ;RE_CX;IM_CX;GSYM CX_POW | |
| ;RE;IM;REAL_INTEGRAL_0;GSYM CX_DEF;REAL_OF_COMPLEX_CX] THEN | |
| REPEAT STRIP_TAC THEN MATCH_MP_TAC REAL_INTEGRAL_POS | |
| THEN ASM_REWRITE_TAC[REAL_LE_POW_2]);; | |
| let RINPROD_CNJ = prove( | |
| `!f g. f IN sq_integrable /\ g IN sq_integrable | |
| ==> cnj (r_inprod f g) = r_inprod g f`, | |
| REWRITE_TAC[sq_integrable;complex_measurable;RE;IM;cnj;COMPLEX_SQNORM; | |
| r_inprod;complex_mul] | |
| THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN | |
| REWRITE_TAC[PAIR_EQ] THEN REPEAT STRIP_TAC | |
| THENL[AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN | |
| REAL_ARITH_TAC;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (f x) * Im (g x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (g x') pow 2 + Im (g x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+b2-c1)+(b1+a2)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (f x) * Re (g x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (g x') pow 2 + Im (g x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+a2-c1)+(b2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| ASM_SIMP_TAC[GSYM REAL_NEG_LMUL;REAL_INTEGRABLE_SUB;REAL_INTEGRABLE_NEG;REAL_INTEGRABLE_ADD; | |
| GSYM REAL_INTEGRAL_NEG] THEN | |
| AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REAL_ARITH_TAC);; | |
| let RINPROD_RSMUL = prove( | |
| `!f g a. f IN sq_integrable /\ g IN sq_integrable | |
| ==> r_inprod f (a%g) = a * r_inprod f g`, | |
| REWRITE_TAC[sq_integrable;complex_measurable;CFUN_SMUL;RE;IM;cnj;COMPLEX_SQNORM; | |
| r_inprod;complex_mul] | |
| THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN | |
| REWRITE_TAC[PAIR_EQ] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (f x) * Im (g x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (g x') pow 2 + Im (g x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+b2-c1)+(b1+a2)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (f x) * Re (g x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (g x') pow 2 + Im (g x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+a2-c1)+(b2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (f x) * Re (g x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (g x') pow 2 + Im (g x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+a2-c1)+(b2+b1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (f x) * Im (g x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (g x') pow 2 + Im (g x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+b2-c1)+(a2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| IMP_REWRITE_TAC[GSYM REAL_NEG_LMUL;REAL_INTEGRABLE_SUB; | |
| REAL_INTEGRABLE_NEG;REAL_INTEGRABLE_ADD; | |
| GSYM REAL_INTEGRAL_LMUL;REAL_INTEGRABLE_LMUL; | |
| GSYM REAL_INTEGRAL_SUB;GSYM REAL_INTEGRAL_ADD] | |
| THEN REPEAT STRIP_TAC THEN | |
| ((AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REAL_ARITH_TAC) | |
| ORELSE (MATCH_MP_TAC REAL_INTEGRABLE_LMUL ORELSE ALL_TAC)) THEN | |
| (MATCH_MP_TAC REAL_INTEGRABLE_SUB ORELSE MATCH_MP_TAC REAL_INTEGRABLE_ADD) | |
| THEN ASM_SIMP_TAC[REAL_INTEGRABLE_NEG]);; | |
| let RINPROD_LADD = prove | |
| (`!f g z. f IN sq_integrable /\ g IN sq_integrable /\ z IN sq_integrable | |
| ==> r_inprod (f+g) z= r_inprod f z + r_inprod g z`, | |
| REWRITE_TAC[sq_integrable;complex_measurable;CFUN_ADD_THM;RE;IM;cnj;COMPLEX_SQNORM; | |
| r_inprod;complex_mul;RE_ADD;IM_ADD;GSYM REAL_NEG_LMUL;REAL_SUB_RNEG; | |
| REAL_ADD_RDISTRIB;GSYM real_sub;complex_add] | |
| THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN | |
| REWRITE_TAC[PAIR_EQ] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (f x) * Im (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+b2-c1)+(b1+a2)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (f x) * Re (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+a2-c1)+(b2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (g x) * Im (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (g x') pow 2 + Im (g x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+b2-c1)+(b1+a2)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (g x) * Re (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (g x') pow 2 + Im (g x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+a2-c1)+(b2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (f x) * Re (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+a2-c1)+(b2+b1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (f x) * Im (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (f x') pow 2 + Im (f x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+b2-c1)+(a2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Re (g x) * Re (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (g x') pow 2 + Im (g x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (a1+a2-c1)+(b2+b1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| Pa.SUBGOAL_THEN `(\x. Im (g x) * Im (z x)) real_integrable_on (:real):` | |
| ASSUME_TAC THENL[ | |
| MATCH_MP_TAC REAL_MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_INTEGRABLE | |
| THEN Pa.EXISTS_TAC `\x'.inv (&2) * (( Re (g x') pow 2 + Im (g x') pow 2) + | |
| (Re (z x') pow 2 + Im (z x') pow 2)):` | |
| THEN ASM_SIMP_TAC[REAL_MEASURABLE_ON_LMUL;REAL_MEASURABLE_ON_MUL; | |
| REAL_INTEGRABLE_ADD;REAL_INTEGRABLE_LMUL;REAL_ABS_MUL;REAL_ABS_NUM | |
| ;REAL_FIELD `x <= inv (&2) * y <=> &2 * x <= y`] THEN | |
| ONCE_REWRITE_TAC[GSYM REAL_SUB_LE] THEN | |
| ONCE_REWRITE_TAC[GSYM ABS_POW_2] THEN | |
| REWRITE_TAC[REAL_ARITH `(((a1:real)+b1)+a2+b2)-c1 = (b1+b2-c1)+(a2+a1)`; | |
| GSYM SQ_RULE_SUB;REAL_ABS_POW] THEN MESON_TAC[REAL_LE_ADD;REAL_LE_POW_2] | |
| ;ALL_TAC] THEN | |
| IMP_REWRITE_TAC[GSYM REAL_NEG_LMUL;REAL_INTEGRABLE_SUB; | |
| REAL_INTEGRABLE_NEG;REAL_INTEGRABLE_ADD; | |
| GSYM REAL_INTEGRAL_LMUL;REAL_INTEGRABLE_LMUL; | |
| GSYM REAL_INTEGRAL_SUB;GSYM REAL_INTEGRAL_ADD] | |
| THEN REPEAT STRIP_TAC THEN | |
| ((AP_TERM_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REAL_ARITH_TAC) | |
| ORELSE (MATCH_MP_TAC REAL_INTEGRABLE_LMUL ORELSE ALL_TAC)) THEN | |
| (MATCH_MP_TAC REAL_INTEGRABLE_SUB ORELSE MATCH_MP_TAC REAL_INTEGRABLE_ADD) | |
| THEN ASM_SIMP_TAC[REAL_INTEGRABLE_NEG]);; | |
| let SQ_INTEGRABLE_INNER_SPACE = prove | |
| (`is_inner_space (sq_integrable, r_inprod)`, | |
| REWRITE_TAC[is_inner_space] THEN | |
| REPEAT STRIP_TAC THEN | |
| ASM_SIMP_TAC[RINPROD_LADD;RINPROD_RSMUL;RINPROD_RSMUL;RINPROD_ZERO_EQ;RINPROD_CNJ; | |
| RINPROD_SELF_POS;SQ_INTEGRABLE_SUBSPACE] | |
| );; | |