(* ========================================================================= *) (* A library for vectors of complex numbers. *) (* Much inspired from HOL-Light real vector library <"vectors.ml">. *) (* *) (* (c) Copyright, Sanaz Khan Afshar & Vincent Aravantinos 2011-13 *) (* Hardware Verification Group, *) (* Concordia University *) (* *) (* Contact: *) (* *) (* *) (* Acknowledgements: *) (* - Harsh Singhal: n-dimensional dot product, utility theorems *) (* *) (* Updated for the latest version of HOL Light (JULY 2014) *) (* *) (* Distributed under the same license as HOL Light. *) (* ========================================================================= *) needs "Multivariate/complexes.ml";; needs "Multivariate/cross.ml";; (* ========================================================================= *) (* ADDITIONS TO THE BASE LIBRARY *) (* ========================================================================= *) (* ----------------------------------------------------------------------- *) (* Additional tacticals *) (* ----------------------------------------------------------------------- *) let SINGLE f x = f [x];; let distrib fs x = map (fun f -> f x) fs;; let DISTRIB ttacs x = EVERY (distrib ttacs x);; let REWRITE_TACS = MAP_EVERY (SINGLE REWRITE_TAC);; let GCONJUNCTS thm = map GEN_ALL (CONJUNCTS (SPEC_ALL thm));; (* ----------------------------------------------------------------------- *) (* Additions to the vectors library *) (* ----------------------------------------------------------------------- *) let COMPONENT_LE_NORM_ALT = prove (`!x:real^N i. 1 <= i /\ i <= dimindex (:N) ==> x$i <= norm x`, MESON_TAC [REAL_ABS_LE;COMPONENT_LE_NORM;REAL_LE_TRANS]);; (* ----------------------------------------------------------------------- *) (* Additions to the library of complex numbers *) (* ----------------------------------------------------------------------- *) (* Lemmas *) let RE_IM_NORM = prove (`!x. Re x <= norm x /\ Im x <= norm x /\ abs(Re x) <= norm x /\ abs(Im x) <= norm x`, REWRITE_TAC[RE_DEF;IM_DEF] THEN GEN_TAC THEN REPEAT CONJ_TAC THEN ((MATCH_MP_TAC COMPONENT_LE_NORM_ALT THEN REWRITE_TAC[DIMINDEX_2] THEN ARITH_TAC) ORELSE SIMP_TAC [COMPONENT_LE_NORM]));; let [RE_NORM;IM_NORM;ABS_RE_NORM;ABS_IM_NORM] = GCONJUNCTS RE_IM_NORM;; let NORM_RE = prove (`!x. &0 <= norm x + Re x /\ &0 <= norm x - Re x`, GEN_TAC THEN MP_TAC (SPEC_ALL ABS_RE_NORM) THEN REAL_ARITH_TAC);; let [NORM_RE_ADD;NORM_RE_SUB] = GCONJUNCTS NORM_RE;; let NORM2_ADD_REAL = prove (`!x y. real x /\ real y ==> norm (x + ii * y) pow 2 = norm x pow 2 + norm y pow 2`, SIMP_TAC[real;complex_norm;RE_ADD;IM_ADD;RE_MUL_II;IM_MUL_II;REAL_NEG_0; REAL_ADD_LID;REAL_ADD_RID;REAL_POW_ZERO;ARITH_RULE `~(2=0)`;REAL_LE_POW_2; SQRT_POW_2;REAL_LE_ADD]);; let COMPLEX_EQ_RCANCEL_IMP = GEN_ALL (MATCH_MP (MESON [] `(p <=> r \/ q) ==> (p /\ ~r ==> q) `) (SPEC_ALL COMPLEX_EQ_MUL_RCANCEL));; let COMPLEX_BALANCE_DIV_MUL = prove (`!x y z t. ~(z=Cx(&0)) ==> (x = y/z * t <=> x*z = y * t)`, REPEAT STRIP_TAC THEN POP_ASSUM (fun x -> ASSUME_TAC (REWRITE_RULE[x] (SPEC_ALL COMPLEX_EQ_MUL_RCANCEL)) THEN ASSUME_TAC (REWRITE_RULE[x] (SPECL [`x:complex`;`z:complex`] COMPLEX_DIV_RMUL))) THEN SUBGOAL_THEN `x=y/z*t <=> x*z=(y/z*t)*z:complex` (SINGLE REWRITE_TAC) THENL [ASM_REWRITE_TAC[]; REWRITE_TAC[SIMPLE_COMPLEX_ARITH `(y/z*t)*z=(y/z*z)*t:complex`] THEN ASM_REWRITE_TAC[]]);; let CSQRT_MUL_LCX_LT = prove (`!x y. &0 < x ==> csqrt(Cx x * y) = Cx(sqrt x) * csqrt y`, REWRITE_TAC[csqrt;complex_mul;IM;RE;IM_CX;REAL_MUL_LZERO;REAL_ADD_RID;RE_CX; REAL_SUB_RZERO] THEN REPEAT STRIP_TAC THEN REPEAT COND_CASES_TAC THEN FIRST_ASSUM (ASSUME_TAC o MATCH_MP REAL_LT_IMP_LE) THEN ASM_SIMP_TAC[IM;RE;REAL_MUL_RZERO;SQRT_MUL] THENL [ REPEAT (POP_ASSUM MP_TAC) THEN REWRITE_TAC[REAL_ENTIRE;REAL_MUL_POS_LE] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[SQRT_0;REAL_MUL_LZERO;REAL_MUL_RZERO]; REPEAT (POP_ASSUM MP_TAC) THEN SIMP_TAC [REAL_ENTIRE] THEN MESON_TAC [REAL_LT_IMP_NZ]; ASM_MESON_TAC [REAL_LE_MUL_EQ;REAL_ARITH `~(&0 <= y) = &0 > y`]; SIMP_TAC [REAL_NEG_RMUL] THEN REPEAT (POP_ASSUM MP_TAC) THEN SIMP_TAC [REAL_ARITH `~(&0 <= y) = y < &0`] THEN SIMP_TAC [GSYM REAL_NEG_GT0] THEN MESON_TAC[REAL_LT_IMP_LE;SQRT_MUL]; REPEAT (POP_ASSUM MP_TAC) THEN SIMP_TAC [REAL_ENTIRE] THEN MESON_TAC [REAL_LT_IMP_NZ]; REPEAT (POP_ASSUM MP_TAC) THEN SIMP_TAC [REAL_ENTIRE] THEN SIMP_TAC [DE_MORGAN_THM]; REPEAT (POP_ASSUM MP_TAC) THEN SIMP_TAC [REAL_ENTIRE] THEN SIMP_TAC [DE_MORGAN_THM]; ALL_TAC] THENL [ SIMP_TAC [REAL_NEG_0;SQRT_0;REAL_MUL_RZERO]; ASM_MESON_TAC[REAL_ARITH `~(x csqrt(Cx x * y) = Cx(sqrt x) * csqrt y`, REWRITE_TAC[REAL_LE_LT] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CSQRT_MUL_LCX_LT] THEN EXPAND_TAC "x" THEN REWRITE_TAC[COMPLEX_MUL_LZERO;SQRT_0;CSQRT_0]);; let REAL_ADD_POW_2 = prove (`!x y:real. (x+y) pow 2 = x pow 2 + y pow 2 + &2 * x * y`, REAL_ARITH_TAC);; let COMPLEX_ADD_POW_2 = prove (`!x y:complex. (x+y) pow 2 = x pow 2 + y pow 2 + Cx(&2) * x * y`, REWRITE_TAC[COMPLEX_POW_2] THEN SIMPLE_COMPLEX_ARITH_TAC);; (* ----------------------------------------------------------------------- *) (* Additions to the topology library *) (* ----------------------------------------------------------------------- *) prioritize_vector ();; (* Lemmas *) let FINITE_INTER_ENUM = prove (`!s n. FINITE(s INTER (0..n))`, MESON_TAC[FINITE_INTER;FINITE_NUMSEG]);; let NORM_PASTECART_GE1 = prove (`!x y. norm x <= norm (pastecart x y)`, MESON_TAC[FSTCART_PASTECART;NORM_FSTCART]);; let NORM_PASTECART_GE2 = prove (`!x y. norm y <= norm (pastecart x y)`, MESON_TAC[SNDCART_PASTECART;NORM_SNDCART]);; let SUMS_PASTECART = prove (`!s f1:num->real^N f2:num->real^M l1 l2. (f1 sums l1) s /\ (f2 sums l2) s <=> ((\x. pastecart (f1 x) (f2 x)) sums (pastecart l1 l2)) s`, SIMP_TAC[sums;FINITE_INTER_ENUM;GSYM PASTECART_VSUM; GSYM LIM_PASTECART_EQ]);; let LINEAR_SUMS = prove( `!s f l g. linear g ==> ((f sums l) s ==> ((g o f) sums (g l)) s)`, SIMP_TAC[sums;FINITE_INTER_ENUM;GSYM LINEAR_VSUM; REWRITE_RULE[o_DEF;CONTINUOUS_AT_SEQUENTIALLY] LINEAR_CONTINUOUS_AT]);; (* ----------------------------------------------------------------------- *) (* Embedding of reals in complex numbers *) (* ----------------------------------------------------------------------- *) let real_of_complex = new_definition `real_of_complex c = @r. c = Cx r`;; let REAL_OF_COMPLEX = prove (`!c. real c ==> Cx(real_of_complex c) = c`, MESON_TAC[REAL;real_of_complex]);; let REAL_OF_COMPLEX_RE = prove (`!c. real c ==> real_of_complex c = Re c`, MESON_TAC[RE_CX;REAL_OF_COMPLEX]);; let REAL_OF_COMPLEX_CX = prove (`!r. real_of_complex (Cx r) = r`, SIMP_TAC[REAL_CX;REAL_OF_COMPLEX_RE;RE_CX]);; let REAL_OF_COMPLEX_NORM = prove (`!c. real c ==> norm c = abs (real_of_complex c)`, MESON_TAC[REAL_NORM;REAL_OF_COMPLEX_RE]);; let REAL_OF_COMPLEX_ADD = prove (`!x y. real x /\ real y ==> real_of_complex (x+y) = real_of_complex x + real_of_complex y`, MESON_TAC[REAL_ADD;REAL_OF_COMPLEX_RE;RE_ADD]);; let REAL_MUL = prove (`!x y. real x /\ real y ==> real (x*y)`, REWRITE_TAC[real] THEN SIMPLE_COMPLEX_ARITH_TAC);; let REAL_OF_COMPLEX_MUL = prove( `!x y. real x /\ real y ==> real_of_complex (x*y) = real_of_complex x * real_of_complex y`, MESON_TAC[REAL_MUL;REAL_OF_COMPLEX;CX_MUL;REAL_OF_COMPLEX_CX]);; let REAL_OF_COMPLEX_0 = prove( `!x. real x ==> (real_of_complex x = &0 <=> x = Cx(&0))`, REWRITE_TAC[REAL_EXISTS] THEN REPEAT STRIP_TAC THEN ASM_SIMP_TAC[REAL_OF_COMPLEX_CX;CX_INJ]);; let REAL_COMPLEX_ADD_CNJ = prove( `!x. real(cnj x + x) /\ real(x + cnj x)`, REWRITE_TAC[COMPLEX_ADD_CNJ;REAL_CX]);; (* TODO *let RE_EQ_NORM = prove(`!x. Re x = norm x <=> real x /\ &0 <= real_of_complex x`, *) (* ----------------------------------------------------------------------- *) (* Additions to the vectors library *) (* ----------------------------------------------------------------------- *) let vector_const = new_definition `vector_const (k:A) :A^N = lambda i. k`;; let vector_map = new_definition `vector_map (f:A->B) (v:A^N) :B^N = lambda i. f(v$i)`;; let vector_map2 = new_definition `vector_map2 (f:A->B->C) (v1:A^N) (v2:B^N) :C^N = lambda i. f (v1$i) (v2$i)`;; let vector_map3 = new_definition `vector_map3 (f:A->B->C->D) (v1:A^N) (v2:B^N) (v3:C^N) :D^N = lambda i. f (v1$i) (v2$i) (v3$i)`;; let FINITE_INDEX_INRANGE_2 = prove (`!i. ?k. 1 <= k /\ k <= dimindex(:N) /\ (!x:A^N. x$i = x$k) /\ (!x:B^N. x$i = x$k) /\ (!x:C^N. x$i = x$k) /\ (!x:D^N. x$i = x$k)`, REWRITE_TAC[finite_index] THEN MESON_TAC[FINITE_INDEX_WORKS]);; let COMPONENT_TAC x = REPEAT GEN_TAC THEN CHOOSE_TAC (SPEC_ALL FINITE_INDEX_INRANGE_2) THEN ASM_SIMP_TAC[x;LAMBDA_BETA];; let VECTOR_CONST_COMPONENT = prove (`!i k. ((vector_const k):A^N)$i = k`, COMPONENT_TAC vector_const);; let VECTOR_MAP_COMPONENT = prove (`!i f:A->B v:A^N. (vector_map f v)$i = f (v$i)`, COMPONENT_TAC vector_map);; let VECTOR_MAP2_COMPONENT = prove (`!i f:A->B->C v1:A^N v2. (vector_map2 f v1 v2)$i = f (v1$i) (v2$i)`, COMPONENT_TAC vector_map2);; let VECTOR_MAP3_COMPONENT = prove( `!i f:A->B->C->D v1:A^N v2 v3. (vector_map3 f v1 v2 v3)$i = f (v1$i) (v2$i) (v3$i)`, COMPONENT_TAC vector_map3);; let COMMON_TAC = REWRITE_TAC[vector_const;vector_map;vector_map2;vector_map3] THEN ONCE_REWRITE_TAC[CART_EQ] THEN SIMP_TAC[LAMBDA_BETA;o_DEF];; let VECTOR_MAP_VECTOR_CONST = prove (`!f:A->B k. vector_map f ((vector_const k):A^N) = vector_const (f k)`, COMMON_TAC);; let VECTOR_MAP_VECTOR_MAP = prove (`!f:A->B g:C->A v:C^N. vector_map f (vector_map g v) = vector_map (f o g) v`, COMMON_TAC);; let VECTOR_MAP_VECTOR_MAP2 = prove (`!f:A->B g:C->D->A u v:D^N. vector_map f (vector_map2 g u v) = vector_map2 (\x y. f (g x y)) u v`, COMMON_TAC);; let VECTOR_MAP2_LVECTOR_CONST = prove (`!f:A->B->C k v:B^N. vector_map2 f (vector_const k) v = vector_map (f k) v`, COMMON_TAC);; let VECTOR_MAP2_RVECTOR_CONST = prove (`!f:A->B->C k v:A^N. vector_map2 f v (vector_const k) = vector_map (\x. f x k) v`, COMMON_TAC);; let VECTOR_MAP2_LVECTOR_MAP = prove (`!f:A->B->C g:D->A v1 v2:B^N. vector_map2 f (vector_map g v1) v2 = vector_map2 (f o g) v1 v2`, COMMON_TAC);; let VECTOR_MAP2_RVECTOR_MAP = prove (`!f:A->B->C g:D->B v1 v2:D^N. vector_map2 f v1 (vector_map g v2) = vector_map2 (\x y. f x (g y)) v1 v2`, COMMON_TAC);; let VECTOR_MAP2_LVECTOR_MAP2 = prove (`!f:A->B->C g:D->E->A v1 v2 v3:B^N. vector_map2 f (vector_map2 g v1 v2) v3 = vector_map3 (\x y. f (g x y)) v1 v2 v3`, COMMON_TAC);; let VECTOR_MAP2_RVECTOR_MAP2 = prove( `!f:A->B->C g:D->E->B v1 v2 v3:E^N. vector_map2 f v1 (vector_map2 g v2 v3) = vector_map3 (\x y z. f x (g y z)) v1 v2 v3`, COMMON_TAC);; let VECTOR_MAP_SIMP_TAC = REWRITE_TAC[ VECTOR_MAP_VECTOR_CONST;VECTOR_MAP2_LVECTOR_CONST; VECTOR_MAP2_RVECTOR_CONST;VECTOR_MAP_VECTOR_MAP;VECTOR_MAP2_RVECTOR_MAP; VECTOR_MAP2_LVECTOR_MAP;VECTOR_MAP2_RVECTOR_MAP2;VECTOR_MAP2_LVECTOR_MAP2; VECTOR_MAP_VECTOR_MAP2];; let VECTOR_MAP_PROPERTY_TAC fs prop = REWRITE_TAC fs THEN VECTOR_MAP_SIMP_TAC THEN ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[VECTOR_MAP_COMPONENT;VECTOR_MAP2_COMPONENT; VECTOR_MAP3_COMPONENT;VECTOR_CONST_COMPONENT;o_DEF;prop];; let VECTOR_MAP_PROPERTY thm f prop = prove(thm,VECTOR_MAP_PROPERTY_TAC f prop);; let COMPLEX_VECTOR_MAP = prove (`!f:complex->complex g. f = vector_map g <=> !z. f z = complex (g (Re z), g (Im z))`, REWRITE_TAC[vector_map;FUN_EQ_THM;complex] THEN REPEAT (GEN_TAC ORELSE EQ_TAC) THEN ASM_SIMP_TAC[CART_EQ;DIMINDEX_2;FORALL_2;LAMBDA_BETA;VECTOR_2;RE_DEF;IM_DEF]);; let COMPLEX_NEG_IS_A_MAP = prove (`(--):complex->complex = vector_map ((--):real->real)`, REWRITE_TAC[COMPLEX_VECTOR_MAP;complex_neg]);; let VECTOR_NEG_IS_A_MAP = prove (`(--):real^N->real^N = vector_map ((--):real->real)`, REWRITE_TAC[FUN_EQ_THM;CART_EQ;VECTOR_NEG_COMPONENT;VECTOR_MAP_COMPONENT]);; let VECTOR_MAP_VECTOR_MAP_ALT = prove (`!f:A^N->B^N g:C^N->A^N f' g'. f = vector_map f' /\ g = vector_map g' ==> f o g = vector_map (f' o g')`, SIMP_TAC[o_DEF;FUN_EQ_THM;VECTOR_MAP_VECTOR_MAP]);; let COMPLEX_VECTOR_MAP2 = prove (`!f:complex->complex->complex g. f = vector_map2 g <=> !z1 z2. f z1 z2 = complex (g (Re z1) (Re z2), g (Im z1) (Im z2))`, REWRITE_TAC[vector_map2;FUN_EQ_THM;complex] THEN REPEAT (GEN_TAC ORELSE EQ_TAC) THEN ASM_SIMP_TAC[CART_EQ;DIMINDEX_2;FORALL_2;LAMBDA_BETA;VECTOR_2;RE_DEF; IM_DEF]);; let VECTOR_MAP2_RVECTOR_MAP_ALT = prove( `!f:complex->complex->complex g:complex->complex f' g'. f = vector_map2 f' /\ g = vector_map g' ==> (\x y. f x (g y)) = vector_map2 (\x y. f' x (g' y))`, SIMP_TAC[FUN_EQ_THM;VECTOR_MAP2_RVECTOR_MAP]);; let COMPLEX_ADD_IS_A_MAP = prove (`(+):complex->complex->complex = vector_map2 ((+):real->real->real)`, REWRITE_TAC[COMPLEX_VECTOR_MAP2;complex_add]);; let VECTOR_ADD_IS_A_MAP = prove (`(+):real^N->real^N->real^N = vector_map2 ((+):real->real->real)`, REWRITE_TAC[FUN_EQ_THM;CART_EQ;VECTOR_ADD_COMPONENT;VECTOR_MAP2_COMPONENT]);; let COMPLEX_SUB_IS_A_MAP = prove (`(-):complex->complex->complex = vector_map2 ((-):real->real->real)`, ONCE_REWRITE_TAC[prove(`(-) = \x y:complex. x-y`,REWRITE_TAC[FUN_EQ_THM])] THEN ONCE_REWRITE_TAC[prove(`(-) = \x y:real. x-y`,REWRITE_TAC[FUN_EQ_THM])] THEN REWRITE_TAC[complex_sub;real_sub] THEN MATCH_MP_TAC VECTOR_MAP2_RVECTOR_MAP_ALT THEN REWRITE_TAC[COMPLEX_NEG_IS_A_MAP;COMPLEX_ADD_IS_A_MAP]);; let VECTOR_SUB_IS_A_MAP = prove (`(-):real^N->real^N->real^N = vector_map2 ((-):real->real->real)`, REWRITE_TAC[FUN_EQ_THM;CART_EQ;VECTOR_SUB_COMPONENT;VECTOR_MAP2_COMPONENT]);; let COMMON_TAC x = SIMP_TAC[CART_EQ;pastecart;x;LAMBDA_BETA] THEN REPEAT STRIP_TAC THEN REPEAT COND_CASES_TAC THEN POP_ASSUM MP_TAC THEN REWRITE_TAC[] THEN SUBGOAL_THEN `1<= i-dimindex(:N) /\ i-dimindex(:N) <= dimindex(:M)` ASSUME_TAC THEN ASM_SIMP_TAC[LAMBDA_BETA] THEN REPEAT (POP_ASSUM (MP_TAC o REWRITE_RULE[DIMINDEX_FINITE_SUM])) THEN ARITH_TAC;; let PASTECART_VECTOR_MAP = prove (`!f:A->B x:A^N y:A^M. pastecart (vector_map f x) (vector_map f y) = vector_map f (pastecart x y)`, COMMON_TAC vector_map);; let PASTECART_VECTOR_MAP2 = prove (`!f:A->B->C x1:A^N x2 y1:A^M y2. pastecart (vector_map2 f x1 x2) (vector_map2 f y1 y2) = vector_map2 f (pastecart x1 y1) (pastecart x2 y2)`, COMMON_TAC vector_map2);; let vector_zip = new_definition `vector_zip (v1:A^N) (v2:B^N) : (A#B)^N = lambda i. (v1$i,v2$i)`;; let VECTOR_ZIP_COMPONENT = prove (`!i v1:A^N v2:B^N. (vector_zip v1 v2)$i = (v1$i,v2$i)`, REPEAT GEN_TAC THEN CHOOSE_TAC (INST_TYPE [`:A#B`,`:C`] (SPEC_ALL FINITE_INDEX_INRANGE_2)) THEN ASM_SIMP_TAC[vector_zip;LAMBDA_BETA]);; let vector_unzip = new_definition `vector_unzip (v:(A#B)^N):(A^N)#(B^N) = vector_map FST v,vector_map SND v`;; let VECTOR_UNZIP_COMPONENT = prove (`!i v:(A#B)^N. (FST (vector_unzip v))$i = FST (v$i) /\ (SND (vector_unzip v))$i = SND (v$i)`, REWRITE_TAC[vector_unzip;VECTOR_MAP_COMPONENT]);; let VECTOR_MAP2_AS_VECTOR_MAP = prove (`!f:A->B->C v1:A^N v2:B^N. vector_map2 f v1 v2 = vector_map (UNCURRY f) (vector_zip v1 v2)`, REWRITE_TAC[CART_EQ;VECTOR_MAP2_COMPONENT;VECTOR_MAP_COMPONENT; VECTOR_ZIP_COMPONENT;UNCURRY_DEF]);; (* ========================================================================= *) (* BASIC ARITHMETIC *) (* ========================================================================= *) make_overloadable "%" `:A-> B-> B`;; let prioritize_cvector () = overload_interface("--",`(cvector_neg):complex^N->complex^N`); overload_interface("+",`(cvector_add):complex^N->complex^N->complex^N`); overload_interface("-",`(cvector_sub):complex^N->complex^N->complex^N`); overload_interface("%",`(cvector_mul):complex->complex^N->complex^N`);; let cvector_zero = new_definition `cvector_zero:complex^N = vector_const (Cx(&0))`;; let cvector_neg = new_definition `cvector_neg :complex^N->complex^N = vector_map (--)`;; let cvector_add = new_definition `cvector_add :complex^N->complex^N->complex^N = vector_map2 (+)`;; let cvector_sub = new_definition `cvector_sub :complex^N->complex^N->complex^N = vector_map2 (-)`;; let cvector_mul = new_definition `(cvector_mul:complex->complex^N->complex^N) a = vector_map (( * ) a)`;; overload_interface("%",`(%):real->real^N->real^N`);; prioritize_cvector ();; let CVECTOR_ZERO_COMPONENT = prove (`!i. (cvector_zero:complex^N)$i = Cx(&0)`, REWRITE_TAC[cvector_zero;VECTOR_CONST_COMPONENT]);; let CVECTOR_NON_ZERO = prove (`!x:complex^N. ~(x=cvector_zero) <=> ?i. 1 <= i /\ i <= dimindex(:N) /\ ~(x$i = Cx(&0))`, GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o ONCE_DEPTH_CONV) [CART_EQ] THEN REWRITE_TAC[CVECTOR_ZERO_COMPONENT] THEN MESON_TAC[]);; let CVECTOR_ADD_COMPONENT = prove (`!X Y:complex^N i. ((X + Y)$i = X$i + Y$i)`, REWRITE_TAC[cvector_add;VECTOR_MAP2_COMPONENT]);; let CVECTOR_SUB_COMPONENT = prove (`!X:complex^N Y i. ((X - Y)$i = X$i - Y$i)`, REWRITE_TAC[cvector_sub;VECTOR_MAP2_COMPONENT]);; let CVECTOR_NEG_COMPONENT = prove (`!X:complex^N i. ((--X)$i = --(X$i))`, REWRITE_TAC[cvector_neg;VECTOR_MAP_COMPONENT]);; let CVECTOR_MUL_COMPONENT = prove (`!c:complex X:complex^N i. ((c % X)$i = c * X$i)`, REWRITE_TAC[cvector_mul;VECTOR_MAP_COMPONENT]);; (* Simple generic tactic adapted from VECTOR_ARITH_TAC *) let CVECTOR_ARITH_TAC = let RENAMED_LAMBDA_BETA th = if fst(dest_fun_ty(type_of(funpow 3 rand (concl th)))) = aty then INST_TYPE [aty,bty; bty,aty] LAMBDA_BETA else LAMBDA_BETA in POP_ASSUM_LIST(K ALL_TAC) THEN REPEAT(GEN_TAC ORELSE CONJ_TAC ORELSE DISCH_TAC ORELSE EQ_TAC) THEN REPEAT(POP_ASSUM MP_TAC) THEN REWRITE_TAC[IMP_IMP; GSYM CONJ_ASSOC] THEN GEN_REWRITE_TAC ONCE_DEPTH_CONV [CART_EQ] THEN REWRITE_TAC[AND_FORALL_THM] THEN TRY EQ_TAC THEN TRY(MATCH_MP_TAC MONO_FORALL) THEN GEN_TAC THEN REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> a ==> b /\ c`; TAUT `(a ==> b) \/ (a ==> c) <=> a ==> b \/ c`] THEN TRY(MATCH_MP_TAC(TAUT `(a ==> b ==> c) ==> (a ==> b) ==> (a ==> c)`)) THEN REWRITE_TAC[cvector_zero;cvector_add; cvector_sub; cvector_neg; cvector_mul; vector_map;vector_map2;vector_const] THEN DISCH_THEN(fun th -> REWRITE_TAC[MATCH_MP(RENAMED_LAMBDA_BETA th) th]) THEN SIMPLE_COMPLEX_ARITH_TAC;; let CVECTOR_ARITH tm = prove(tm,CVECTOR_ARITH_TAC);; (* ========================================================================= *) (* VECTOR SPACE AXIOMS AND ADDITIONAL BASIC RESULTS *) (* ========================================================================= *) let CVECTOR_MAP_PROPERTY thm = VECTOR_MAP_PROPERTY thm [cvector_zero;cvector_add;cvector_sub;cvector_neg; cvector_mul];; let CVECTOR_ADD_SYM = CVECTOR_MAP_PROPERTY `!x y:complex^N. x + y = y + x` COMPLEX_ADD_SYM;; let CVECTOR_ADD_ASSOC = CVECTOR_MAP_PROPERTY `!x y z:complex^N. x + (y + z) = (x + y) + z` COMPLEX_ADD_ASSOC;; let CVECTOR_ADD_ID = CVECTOR_MAP_PROPERTY `!x:complex^N. x + cvector_zero = x /\ cvector_zero + x = x` (CONJ COMPLEX_ADD_RID COMPLEX_ADD_LID);; let [CVECTOR_ADD_RID;CVECTOR_ADD_LID] = GCONJUNCTS CVECTOR_ADD_ID;; let CVECTOR_ADD_INV = CVECTOR_MAP_PROPERTY `!x:complex^N. x + -- x = cvector_zero /\ --x + x = cvector_zero` (CONJ COMPLEX_ADD_RINV COMPLEX_ADD_LINV);; let CVECTOR_MUL_ASSOC = CVECTOR_MAP_PROPERTY `!a b x:complex^N. a % (b % x) = (a * b) % x` COMPLEX_MUL_ASSOC;; let CVECTOR_SUB_LDISTRIB = CVECTOR_MAP_PROPERTY `!c x y:complex^N. c % (x - y) = c % x - c % y` COMPLEX_SUB_LDISTRIB;; let CVECTOR_SCALAR_RDIST = CVECTOR_MAP_PROPERTY `!a b x:complex^N. (a + b) % x = a % x + b % x` COMPLEX_ADD_RDISTRIB;; let CVECTOR_MUL_ID = CVECTOR_MAP_PROPERTY `!x:complex^N. Cx(&1) % x = x` COMPLEX_MUL_LID;; let CVECTOR_SUB_REFL = CVECTOR_MAP_PROPERTY `!x:complex^N. x - x = cvector_zero` COMPLEX_SUB_REFL;; let CVECTOR_SUB_RADD = CVECTOR_MAP_PROPERTY `!x y:complex^N. x - (x + y) = --y` COMPLEX_ADD_SUB2;; let CVECTOR_NEG_SUB = CVECTOR_MAP_PROPERTY `!x y:complex^N. --(x - y) = y - x` COMPLEX_NEG_SUB;; let CVECTOR_SUB_EQ = CVECTOR_MAP_PROPERTY `!x y:complex^N. (x - y = cvector_zero) <=> (x = y)` COMPLEX_SUB_0;; let CVECTOR_MUL_LZERO = CVECTOR_MAP_PROPERTY `!x. Cx(&0) % x = cvector_zero` COMPLEX_MUL_LZERO;; let CVECTOR_SUB_ADD = CVECTOR_MAP_PROPERTY `!x y:complex^N. (x - y) + y = x` COMPLEX_SUB_ADD;; let CVECTOR_SUB_ADD2 = CVECTOR_MAP_PROPERTY `!x y:complex^N. y + (x - y) = x` COMPLEX_SUB_ADD2;; let CVECTOR_ADD_LDISTRIB = CVECTOR_MAP_PROPERTY `!c x y:complex^N. c % (x + y) = c % x + c % y` COMPLEX_ADD_LDISTRIB;; let CVECTOR_ADD_RDISTRIB = CVECTOR_MAP_PROPERTY `!a b x:complex^N. (a + b) % x = a % x + b % x` COMPLEX_ADD_RDISTRIB;; let CVECTOR_SUB_RDISTRIB = CVECTOR_MAP_PROPERTY `!a b x:complex^N. (a - b) % x = a % x - b % x` COMPLEX_SUB_RDISTRIB;; let CVECTOR_ADD_SUB = CVECTOR_MAP_PROPERTY `!x y:complex^N. (x + y:complex^N) - x = y` COMPLEX_ADD_SUB;; let CVECTOR_EQ_ADDR = CVECTOR_MAP_PROPERTY `!x y:complex^N. (x + y = x) <=> (y = cvector_zero)` COMPLEX_EQ_ADD_LCANCEL_0;; let CVECTOR_SUB = CVECTOR_MAP_PROPERTY `!x y:complex^N. x - y = x + --(y:complex^N)` complex_sub;; let CVECTOR_SUB_RZERO = CVECTOR_MAP_PROPERTY `!x:complex^N. x - cvector_zero = x` COMPLEX_SUB_RZERO;; let CVECTOR_MUL_RZERO = CVECTOR_MAP_PROPERTY `!c:complex. c % cvector_zero = cvector_zero` COMPLEX_MUL_RZERO;; let CVECTOR_MUL_LZERO = CVECTOR_MAP_PROPERTY `!x:complex^N. Cx(&0) % x = cvector_zero` COMPLEX_MUL_LZERO;; let CVECTOR_MUL_EQ_0 = prove (`!a:complex x:complex^N. (a % x = cvector_zero <=> a = Cx(&0) \/ x = cvector_zero)`, REPEAT STRIP_TAC THEN EQ_TAC THENL [ ASM_CASES_TAC `a=Cx(&0)` THENL [ ASM_REWRITE_TAC[]; GEN_REWRITE_TAC (RATOR_CONV o DEPTH_CONV) [CART_EQ] THEN ASM_REWRITE_TAC[CVECTOR_MUL_COMPONENT;CVECTOR_ZERO_COMPONENT; COMPLEX_ENTIRE] THEN GEN_REWRITE_TAC (RAND_CONV o DEPTH_CONV) [CART_EQ] THEN REWRITE_TAC[CVECTOR_ZERO_COMPONENT]; ]; REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[CVECTOR_MUL_RZERO;CVECTOR_MUL_LZERO]; ]);; let CVECTOR_NEG_MINUS1 = CVECTOR_MAP_PROPERTY `!x:complex^N. --x = (--(Cx(&1))) % x` (GSYM COMPLEX_NEG_MINUS1);; let CVECTOR_SUB_LZERO = CVECTOR_MAP_PROPERTY `!x:complex^N. cvector_zero - x = --x` COMPLEX_SUB_LZERO;; let CVECTOR_NEG_NEG = CVECTOR_MAP_PROPERTY `!x:complex^N. --(--(x:complex^N)) = x` COMPLEX_NEG_NEG;; let CVECTOR_MUL_LNEG = CVECTOR_MAP_PROPERTY `!c x:complex^N. --c % x = --(c % x)` COMPLEX_MUL_LNEG;; let CVECTOR_MUL_RNEG = CVECTOR_MAP_PROPERTY `!c x:complex^N. c % --x = --(c % x)` COMPLEX_MUL_RNEG;; let CVECTOR_NEG_0 = CVECTOR_MAP_PROPERTY `--cvector_zero = cvector_zero` COMPLEX_NEG_0;; let CVECTOR_NEG_EQ_0 = CVECTOR_MAP_PROPERTY `!x:complex^N. --x = cvector_zero <=> x = cvector_zero` COMPLEX_NEG_EQ_0;; let CVECTOR_ADD_AC = prove (`!x y z:complex^N. (x + y = y + x) /\ ((x + y) + z = x + y + z) /\ (x + y + z = y + x + z)`, MESON_TAC[CVECTOR_ADD_SYM;CVECTOR_ADD_ASSOC]);; let CVECTOR_MUL_LCANCEL = prove (`!a x y:complex^N. a % x = a % y <=> a = Cx(&0) \/ x = y`, MESON_TAC[CVECTOR_MUL_EQ_0;CVECTOR_SUB_LDISTRIB;CVECTOR_SUB_EQ]);; let CVECTOR_MUL_RCANCEL = prove (`!a b x:complex^N. a % x = b % x <=> a = b \/ x = cvector_zero`, MESON_TAC[CVECTOR_MUL_EQ_0;CVECTOR_SUB_RDISTRIB;COMPLEX_SUB_0;CVECTOR_SUB_EQ]);; (* ========================================================================= *) (* LINEARITY *) (* ========================================================================= *) let clinear = new_definition `clinear (f:complex^M->complex^N) <=> (!x y. f(x + y) = f(x) + f(y)) /\ (!c x. f(c % x) = c % f(x))`;; let COMMON_TAC additional_thms = SIMP_TAC[clinear] THEN REPEAT STRIP_TAC THEN ONCE_REWRITE_TAC[CART_EQ] THEN SIMP_TAC(CVECTOR_ADD_COMPONENT::CVECTOR_MUL_COMPONENT::additional_thms) THEN SIMPLE_COMPLEX_ARITH_TAC;; let CLINEAR_COMPOSE_CMUL = prove (`!f:complex^M->complex^N c. clinear f ==> clinear (\x. c % f x)`, COMMON_TAC[]);; let CLINEAR_COMPOSE_NEG = prove (`!f:complex^M->complex^N. clinear f ==> clinear (\x. --(f x))`, COMMON_TAC[CVECTOR_NEG_COMPONENT]);; let CLINEAR_COMPOSE_ADD = prove (`!f:complex^M->complex^N g. clinear f /\ clinear g ==> clinear (\x. f x + g x)`, COMMON_TAC[]);; let CLINEAR_COMPOSE_SUB = prove (`!f:complex^M->complex^N g. clinear f /\ clinear g ==> clinear (\x. f x - g x)`, COMMON_TAC[CVECTOR_SUB_COMPONENT]);; let CLINEAR_COMPOSE = prove (`!f:complex^M->complex^N g. clinear f /\ clinear g ==> clinear (g o f)`, SIMP_TAC[clinear;o_THM]);; let CLINEAR_ID = prove (`clinear (\x:complex^N. x)`, REWRITE_TAC[clinear]);; let CLINEAR_I = prove (`clinear (I:complex^N->complex^N)`, REWRITE_TAC[I_DEF;CLINEAR_ID]);; let CLINEAR_ZERO = prove (`clinear ((\x. cvector_zero):complex^M->complex^N)`, COMMON_TAC[CVECTOR_ZERO_COMPONENT]);; let CLINEAR_NEGATION = prove (`clinear ((--):complex^N->complex^N)`, COMMON_TAC[CVECTOR_NEG_COMPONENT]);; let CLINEAR_VMUL_COMPONENT = prove (`!f:complex^M->complex^N v:complex^P k. clinear f /\ 1 <= k /\ k <= dimindex(:N) ==> clinear (\x. (f x)$k % v)`, COMMON_TAC[]);; let CLINEAR_0 = prove (`!f:complex^M->complex^N. clinear f ==> (f cvector_zero = cvector_zero)`, MESON_TAC[CVECTOR_MUL_LZERO;clinear]);; let CLINEAR_CMUL = prove (`!f:complex^M->complex^N c x. clinear f ==> (f (c % x) = c % f x)`, SIMP_TAC[clinear]);; let CLINEAR_NEG = prove (`!f:complex^M->complex^N x. clinear f ==> (f (--x) = --(f x))`, ONCE_REWRITE_TAC[CVECTOR_NEG_MINUS1] THEN SIMP_TAC[CLINEAR_CMUL]);; let CLINEAR_ADD = prove (`!f:complex^M->complex^N x y. clinear f ==> (f (x + y) = f x + f y)`, SIMP_TAC[clinear]);; let CLINEAR_SUB = prove (`!f:complex^M->complex^N x y. clinear f ==> (f(x - y) = f x - f y)`, SIMP_TAC[CVECTOR_SUB;CLINEAR_ADD;CLINEAR_NEG]);; let CLINEAR_INJECTIVE_0 = prove (`!f:complex^M->complex^N. clinear f ==> ((!x y. f x = f y ==> x = y) <=> (!x. f x = cvector_zero ==> x = cvector_zero))`, ONCE_REWRITE_TAC[GSYM CVECTOR_SUB_EQ] THEN SIMP_TAC[CVECTOR_SUB_RZERO;GSYM CLINEAR_SUB] THEN MESON_TAC[CVECTOR_SUB_RZERO]);; (* ========================================================================= *) (* PASTING COMPLEX VECTORS *) (* ========================================================================= *) let CLINEAR_FSTCART_SNDCART = prove (`clinear fstcart /\ clinear sndcart`, SIMP_TAC[clinear;fstcart;sndcart;CART_EQ;LAMBDA_BETA;CVECTOR_ADD_COMPONENT; CVECTOR_MUL_COMPONENT; DIMINDEX_FINITE_SUM; ARITH_RULE `x <= a ==> x <= a + b:num`; ARITH_RULE `x <= b ==> x + a <= a + b:num`]);; let FSTCART_CLINEAR = CONJUNCT1 CLINEAR_FSTCART_SNDCART;; let SNDCART_CLINEAR = CONJUNCT2 CLINEAR_FSTCART_SNDCART;; let FSTCART_SNDCART_CVECTOR_ZERO = prove (`fstcart cvector_zero = cvector_zero /\ sndcart cvector_zero = cvector_zero`, SIMP_TAC[CVECTOR_ZERO_COMPONENT;fstcart;sndcart;LAMBDA_BETA;CART_EQ; DIMINDEX_FINITE_SUM;ARITH_RULE `x <= a ==> x <= a + b:num`; ARITH_RULE `x <= b ==> x + a <= a + b:num`]);; let FSTCART_CVECTOR_ZERO = CONJUNCT1 FSTCART_SNDCART_CVECTOR_ZERO;; let SNDCART_CVECTOR_ZERO = CONJUNCT2 FSTCART_SNDCART_CVECTOR_ZERO;; let FSTCART_SNDCART_CVECTOR_ADD = prove (`!x:complex^(M,N)finite_sum y. fstcart(x + y) = fstcart(x) + fstcart(y) /\ sndcart(x + y) = sndcart(x) + sndcart(y)`, REWRITE_TAC[REWRITE_RULE[clinear] CLINEAR_FSTCART_SNDCART]);; let FSTCART_SNDCART_CVECTOR_MUL = prove (`!x:complex^(M,N)finite_sum c. fstcart(c % x) = c % fstcart(x) /\ sndcart(c % x) = c % sndcart(x)`, REWRITE_TAC[REWRITE_RULE[clinear] CLINEAR_FSTCART_SNDCART]);; let PASTECART_TAC xs = REWRITE_TAC(PASTECART_EQ::FSTCART_PASTECART::SNDCART_PASTECART::xs);; let PASTECART_CVECTOR_ZERO = prove (`pastecart (cvector_zero:complex^N) (cvector_zero:complex^M) = cvector_zero`, PASTECART_TAC[FSTCART_SNDCART_CVECTOR_ZERO]);; let PASTECART_EQ_CVECTOR_ZERO = prove (`!x:complex^N y:complex^M. pastecart x y = cvector_zero <=> x = cvector_zero /\ y = cvector_zero`, PASTECART_TAC [FSTCART_SNDCART_CVECTOR_ZERO]);; let PASTECART_CVECTOR_ADD = prove (`!x1 y2 x2:complex^N y2:complex^M. pastecart x1 y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)`, PASTECART_TAC [FSTCART_SNDCART_CVECTOR_ADD]);; let PASTECART_CVECTOR_MUL = prove (`!x1 x2 c:complex. pastecart (c % x1) (c % y1) = c % pastecart x1 y1`, PASTECART_TAC [FSTCART_SNDCART_CVECTOR_MUL]);; (* ========================================================================= *) (* REAL AND IMAGINARY VECTORS *) (* ========================================================================= *) let cvector_re = new_definition `cvector_re :complex^N -> real^N = vector_map Re`;; let cvector_im = new_definition `cvector_im :complex^N -> real^N = vector_map Im`;; let vector_to_cvector = new_definition `vector_to_cvector :real^N -> complex^N = vector_map Cx`;; let CVECTOR_RE_COMPONENT = prove (`!x:complex^N i. (cvector_re x)$i = Re (x$i)`, REWRITE_TAC[cvector_re;VECTOR_MAP_COMPONENT]);; let CVECTOR_IM_COMPONENT = prove (`!x:complex^N i. (cvector_im x)$i = Im (x$i)`, REWRITE_TAC[cvector_im;VECTOR_MAP_COMPONENT]);; let VECTOR_TO_CVECTOR_COMPONENT = prove (`!x:real^N i. (vector_to_cvector x)$i = Cx(x$i)`, REWRITE_TAC[vector_to_cvector;VECTOR_MAP_COMPONENT]);; let VECTOR_TO_CVECTOR_ZERO = prove (`vector_to_cvector (vec 0) = cvector_zero:complex^N`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[VECTOR_TO_CVECTOR_COMPONENT;CVECTOR_ZERO_COMPONENT; VEC_COMPONENT]);; let VECTOR_TO_CVECTOR_ZERO_EQ = prove (`!x:real^N. vector_to_cvector x = cvector_zero <=> x = vec 0`, GEN_TAC THEN EQ_TAC THEN SIMP_TAC[VECTOR_TO_CVECTOR_ZERO] THEN ONCE_REWRITE_TAC[CART_EQ] THEN SIMP_TAC[VECTOR_TO_CVECTOR_COMPONENT;CVECTOR_ZERO_COMPONENT; VEC_COMPONENT;CX_INJ]);; let CVECTOR_ZERO_VEC0 = prove (`!x:complex^N. x = cvector_zero <=> cvector_re x = vec 0 /\ cvector_im x = vec 0`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_ZERO_COMPONENT;CVECTOR_RE_COMPONENT; CVECTOR_IM_COMPONENT;VEC_COMPONENT;COMPLEX_EQ;RE_CX;IM_CX] THEN MESON_TAC[]);; let VECTOR_TO_CVECTOR_MUL = prove (`!a x:real^N. vector_to_cvector (a % x) = Cx a % vector_to_cvector x`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[VECTOR_TO_CVECTOR_COMPONENT;CVECTOR_MUL_COMPONENT;VECTOR_MUL_COMPONENT;CX_MUL]);; let CVECTOR_EQ = prove (`!x:complex^N y z. x = vector_to_cvector y + ii % vector_to_cvector z <=> cvector_re x = y /\ cvector_im x = z`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_ADD_COMPONENT;CVECTOR_MUL_COMPONENT; CVECTOR_RE_COMPONENT;CVECTOR_IM_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT] THEN REWRITE_TAC[COMPLEX_EQ;RE_CX;IM_CX;RE_ADD;IM_ADD;RE_MUL_II;REAL_NEG_0; REAL_ADD_RID;REAL_ADD_LID;IM_MUL_II] THEN MESON_TAC[]);; let CVECTOR_RE_VECTOR_TO_CVECTOR = prove (`!x:real^N. cvector_re (vector_to_cvector x) = x`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_RE_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT;RE_CX]);; let CVECTOR_IM_VECTOR_TO_CVECTOR = prove (`!x:real^N. cvector_im (vector_to_cvector x) = vec 0`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_IM_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT;IM_CX; VEC_COMPONENT]);; let CVECTOR_IM_VECTOR_TO_CVECTOR_IM = prove (`!x:real^N. cvector_im (ii % vector_to_cvector x) = x`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_IM_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT;IM_CX; VEC_COMPONENT;CVECTOR_MUL_COMPONENT;IM_MUL_II;RE_CX]);; let VECTOR_TO_CVECTOR_CVECTOR_RE_IM = prove (`!x:complex^N. vector_to_cvector (cvector_re x) + ii % vector_to_cvector (cvector_im x) = x`, GEN_TAC THEN MATCH_MP_TAC EQ_SYM THEN REWRITE_TAC[CVECTOR_EQ]);; let CVECTOR_IM_VECTOR_TO_CVECTOR_RE_IM = prove (`!x y:real^N. cvector_im (vector_to_cvector x + ii % vector_to_cvector y) = y`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_IM_COMPONENT;CVECTOR_ADD_COMPONENT; CVECTOR_MUL_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT;IM_ADD;IM_CX;IM_MUL_II; RE_CX;REAL_ADD_LID]);; let CVECTOR_RE_VECTOR_TO_CVECTOR_RE_IM = prove (`!x y:real^N. cvector_re (vector_to_cvector x + ii % vector_to_cvector y)= x`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_RE_COMPONENT;CVECTOR_ADD_COMPONENT; CVECTOR_MUL_COMPONENT;RE_ADD;VECTOR_TO_CVECTOR_COMPONENT;RE_CX;RE_MUL_CX; RE_II;REAL_MUL_LZERO;REAL_ADD_RID]);; let CVECTOR_RE_ADD = prove (`!x y:complex^N. cvector_re (x+y) = cvector_re x + cvector_re y`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_RE_COMPONENT; VECTOR_ADD_COMPONENT;CVECTOR_ADD_COMPONENT;RE_ADD]);; let CVECTOR_IM_ADD = prove (`!x y:complex^N. cvector_im (x+y) = cvector_im x + cvector_im y`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_IM_COMPONENT;VECTOR_ADD_COMPONENT; CVECTOR_ADD_COMPONENT;IM_ADD]);; let CVECTOR_RE_MUL = prove (`!a x:complex^N. cvector_re (Cx a % x) = a % cvector_re x`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_RE_COMPONENT;VECTOR_MUL_COMPONENT; CVECTOR_MUL_COMPONENT;RE_MUL_CX]);; let CVECTOR_IM_MUL = prove (`!a x:complex^N. cvector_im (Cx a % x) = a % cvector_im x`, ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_IM_COMPONENT;VECTOR_MUL_COMPONENT; CVECTOR_MUL_COMPONENT;IM_MUL_CX]);; let CVECTOR_RE_VECTOR_MAP = prove (`!f v:A^N. cvector_re (vector_map f v) = vector_map (Re o f) v`, REWRITE_TAC[cvector_re;VECTOR_MAP_VECTOR_MAP]);; let CVECTOR_IM_VECTOR_MAP = prove (`!f v:A^N. cvector_im (vector_map f v) = vector_map (Im o f) v`, REWRITE_TAC[cvector_im;VECTOR_MAP_VECTOR_MAP]);; let VECTOR_MAP_CVECTOR_RE = prove (`!f:real->A v:complex^N. vector_map f (cvector_re v) = vector_map (f o Re) v`, REWRITE_TAC[cvector_re;VECTOR_MAP_VECTOR_MAP]);; let VECTOR_MAP_CVECTOR_IM = prove (`!f:real->A v:complex^N. vector_map f (cvector_im v) = vector_map (f o Im) v`, REWRITE_TAC[cvector_im;VECTOR_MAP_VECTOR_MAP]);; let CVECTOR_RE_VECTOR_MAP2 = prove (`!f v1:A^N v2:B^N. cvector_re (vector_map2 f v1 v2) = vector_map2 (\x y. Re (f x y)) v1 v2`, REWRITE_TAC[cvector_re;VECTOR_MAP_VECTOR_MAP2]);; let CVECTOR_IM_VECTOR_MAP2 = prove (`!f v1:A^N v2:B^N. cvector_im (vector_map2 f v1 v2) = vector_map2 (\x y. Im (f x y)) v1 v2`, REWRITE_TAC[cvector_im;VECTOR_MAP_VECTOR_MAP2]);; (* ========================================================================= *) (* FLATTENING COMPLEX VECTORS INTO REAL VECTORS *) (* *) (* Note: *) (* Theoretically, the following could be defined more generally for matrices *) (* instead of complex vectors, but this would require a "finite_prod" type *) (* constructor, which is not available right now, and which, at first sight, *) (* would probably require dependent types. *) (* ========================================================================= *) let cvector_flatten = new_definition `cvector_flatten (v:complex^N) :real^(N,N) finite_sum = pastecart (cvector_re v) (cvector_im v)`;; let FLATTEN_RE_IM_COMPONENT = prove (`!v:complex^N i. 1 <= i /\ i <= 2 * dimindex(:N) ==> (cvector_flatten v)$i = if i <= dimindex(:N) then (cvector_re v)$i else (cvector_im v)$(i-dimindex(:N))`, SIMP_TAC[MULT_2;GSYM DIMINDEX_FINITE_SUM;cvector_flatten;pastecart; LAMBDA_BETA]);; let complex_vector = new_definition `complex_vector (v1,v2) :complex^N = vector_map2 (\x y. Cx x + ii * Cx y) v1 v2`;; let COMPLEX_VECTOR_TRANSPOSE = prove( `!v1 v2:real^N. complex_vector (v1,v2) = vector_to_cvector v1 + ii % vector_to_cvector v2`, ONCE_REWRITE_TAC[CART_EQ] THEN SIMP_TAC[complex_vector;CVECTOR_ADD_COMPONENT;CVECTOR_MUL_COMPONENT; VECTOR_TO_CVECTOR_COMPONENT;VECTOR_MAP2_COMPONENT]);; let cvector_unflatten = new_definition `cvector_unflatten (v:real^(N,N) finite_sum) :complex^N = complex_vector (fstcart v, sndcart v)`;; let UNFLATTEN_FLATTEN = prove (`cvector_unflatten o cvector_flatten = I :complex^N -> complex^N`, REWRITE_TAC[FUN_EQ_THM;o_DEF;I_DEF;cvector_flatten;cvector_unflatten; FSTCART_PASTECART;SNDCART_PASTECART;COMPLEX_VECTOR_TRANSPOSE; VECTOR_TO_CVECTOR_CVECTOR_RE_IM]);; let FLATTEN_UNFLATTEN = prove (`cvector_flatten o cvector_unflatten = I :real^(N,N) finite_sum -> real^(N,N) finite_sum`, REWRITE_TAC[FUN_EQ_THM;o_DEF;I_DEF;cvector_flatten;cvector_unflatten; PASTECART_FST_SND;COMPLEX_VECTOR_TRANSPOSE; CVECTOR_RE_VECTOR_TO_CVECTOR_RE_IM;CVECTOR_IM_VECTOR_TO_CVECTOR_RE_IM]);; let FLATTEN_CLINEAR = prove (`!f:complex^N->complex^M. clinear f ==> linear (cvector_flatten o f o cvector_unflatten)`, REWRITE_TAC[clinear;linear;cvector_flatten;cvector_unflatten;o_DEF; FSTCART_ADD;SNDCART_ADD;PASTECART_ADD;complex_vector;GSYM PASTECART_CMUL] THEN REPEAT STRIP_TAC THEN REPEAT (AP_TERM_TAC ORELSE MK_COMB_TAC) THEN REWRITE_TAC(map GSYM [CVECTOR_RE_ADD;CVECTOR_IM_ADD;CVECTOR_RE_MUL; CVECTOR_IM_MUL]) THEN AP_TERM_TAC THEN ASSUM_LIST (REWRITE_TAC o map GSYM) THEN AP_TERM_TAC THEN ONCE_REWRITE_TAC[CART_EQ] THEN SIMP_TAC[VECTOR_MAP2_COMPONENT;VECTOR_ADD_COMPONENT; CVECTOR_ADD_COMPONENT;CX_ADD;VECTOR_MUL_COMPONENT;CVECTOR_MUL_COMPONENT; FSTCART_CMUL;SNDCART_CMUL;CX_MUL] THEN SIMPLE_COMPLEX_ARITH_TAC);; let FLATTEN_MAP = prove (`!f g. f = vector_map g ==> !x:complex^N. cvector_flatten (vector_map f x) = vector_map g (cvector_flatten x)`, SIMP_TAC[cvector_flatten;CVECTOR_RE_VECTOR_MAP;CVECTOR_IM_VECTOR_MAP; GSYM PASTECART_VECTOR_MAP;VECTOR_MAP_CVECTOR_RE;VECTOR_MAP_CVECTOR_IM; o_DEF;IM_DEF;RE_DEF;VECTOR_MAP_COMPONENT]);; let FLATTEN_NEG = prove (`!x:complex^N. cvector_flatten (--x) = --(cvector_flatten x)`, REWRITE_TAC[cvector_neg;MATCH_MP FLATTEN_MAP COMPLEX_NEG_IS_A_MAP] THEN REWRITE_TAC[VECTOR_NEG_IS_A_MAP]);; let CVECTOR_NEG_ALT = prove (`!x:complex^N. --x = cvector_unflatten (--(cvector_flatten x))`, REWRITE_TAC[GSYM FLATTEN_NEG; REWRITE_RULE[o_DEF;FUN_EQ_THM;I_DEF] UNFLATTEN_FLATTEN]);; let FLATTEN_MAP2 = prove( `!f g. f = vector_map2 g ==> !x y:complex^N. cvector_flatten (vector_map2 f x y) = vector_map2 g (cvector_flatten x) (cvector_flatten y)`, SIMP_TAC[cvector_flatten;CVECTOR_RE_VECTOR_MAP2;CVECTOR_IM_VECTOR_MAP2; CVECTOR_RE_VECTOR_MAP2;GSYM PASTECART_VECTOR_MAP2] THEN REWRITE_TAC[cvector_re;cvector_im;VECTOR_MAP2_LVECTOR_MAP; VECTOR_MAP2_RVECTOR_MAP] THEN REPEAT MK_COMB_TAC THEN REWRITE_TAC[FUN_EQ_THM;IM_DEF;RE_DEF;VECTOR_MAP2_COMPONENT;o_DEF]);; let FLATTEN_ADD = prove (`!x y:complex^N. cvector_flatten (x+y) = cvector_flatten x + cvector_flatten y`, REWRITE_TAC[cvector_add;MATCH_MP FLATTEN_MAP2 COMPLEX_ADD_IS_A_MAP] THEN REWRITE_TAC[VECTOR_ADD_IS_A_MAP]);; let CVECTOR_ADD_ALT = prove (`!x y:complex^N. x+y = cvector_unflatten (cvector_flatten x + cvector_flatten y)`, REWRITE_TAC[GSYM FLATTEN_ADD; REWRITE_RULE[o_DEF;FUN_EQ_THM;I_DEF] UNFLATTEN_FLATTEN]);; let FLATTEN_SUB = prove (`!x y:complex^N. cvector_flatten (x-y) = cvector_flatten x - cvector_flatten y`, REWRITE_TAC[cvector_sub;MATCH_MP FLATTEN_MAP2 COMPLEX_SUB_IS_A_MAP] THEN REWRITE_TAC[VECTOR_SUB_IS_A_MAP]);; let CVECTOR_SUB_ALT = prove (`!x y:complex^N. x-y = cvector_unflatten (cvector_flatten x - cvector_flatten y)`, REWRITE_TAC[GSYM FLATTEN_SUB; REWRITE_RULE[o_DEF;FUN_EQ_THM;I_DEF] UNFLATTEN_FLATTEN]);; (* ========================================================================= *) (* CONJUGATE VECTOR. *) (* ========================================================================= *) let cvector_cnj = new_definition `cvector_cnj : complex^N->complex^N = vector_map cnj`;; let CVECTOR_MAP_PROPERTY thm = VECTOR_MAP_PROPERTY thm [cvector_zero;cvector_add;cvector_sub;cvector_neg; cvector_mul;cvector_cnj;cvector_re;cvector_im];; let CVECTOR_CNJ_ADD = CVECTOR_MAP_PROPERTY `!x y:complex^N. cvector_cnj (x+y) = cvector_cnj x + cvector_cnj y` CNJ_ADD;; let CVECTOR_CNJ_SUB = CVECTOR_MAP_PROPERTY `!x y:complex^N. cvector_cnj (x-y) = cvector_cnj x - cvector_cnj y` CNJ_SUB;; let CVECTOR_CNJ_NEG = CVECTOR_MAP_PROPERTY `!x:complex^N. cvector_cnj (--x) = --(cvector_cnj x)` CNJ_NEG;; let CVECTOR_RE_CNJ = CVECTOR_MAP_PROPERTY `!x:complex^N. cvector_re (cvector_cnj x) = cvector_re x` RE_CNJ;; let CVECTOR_IM_CNJ = prove (`!x:complex^N. cvector_im (cvector_cnj x) = --(cvector_im x)`, VECTOR_MAP_PROPERTY_TAC[cvector_im;cvector_cnj;VECTOR_NEG_IS_A_MAP] IM_CNJ);; let CVECTOR_CNJ_CNJ = CVECTOR_MAP_PROPERTY `!x:complex^N. cvector_cnj (cvector_cnj x) = x` CNJ_CNJ;; (* ========================================================================= *) (* CROSS PRODUCTS IN COMPLEX^3. *) (* ========================================================================= *) prioritize_vector();; parse_as_infix("ccross",(20,"right"));; let ccross = new_definition `((ccross):complex^3 -> complex^3 -> complex^3) x y = vector [ x$2 * y$3 - x$3 * y$2; x$3 * y$1 - x$1 * y$3; x$1 * y$2 - x$2 * y$1 ]`;; let CCROSS_COMPONENT = prove (`!x y:complex^3. (x ccross y)$1 = x$2 * y$3 - x$3 * y$2 /\ (x ccross y)$2 = x$3 * y$1 - x$1 * y$3 /\ (x ccross y)$3 = x$1 * y$2 - x$2 * y$1`, REWRITE_TAC[ccross;VECTOR_3]);; (* simple handy instantiation of CART_EQ for dimension 3*) let CART_EQ3 = prove (`!x y:complex^3. x = y <=> x$1 = y$1 /\ x$2 = y$2 /\ x$3 = y$3`, GEN_REWRITE_TAC (PATH_CONV "rbrblr") [CART_EQ] THEN REWRITE_TAC[DIMINDEX_3;FORALL_3]);; let CCROSS_TAC lemmas = REWRITE_TAC(CART_EQ3::CCROSS_COMPONENT::lemmas) THEN SIMPLE_COMPLEX_ARITH_TAC;; let CCROSS_LZERO = prove (`!x:complex^3. cvector_zero ccross x = cvector_zero`, CCROSS_TAC[CVECTOR_ZERO_COMPONENT]);; let CCROSS_RZERO = prove (`!x:complex^3. x ccross cvector_zero = cvector_zero`, CCROSS_TAC[CVECTOR_ZERO_COMPONENT]);; let CCROSS_SKEW = prove (`!x y:complex^3. (x ccross y) = --(y ccross x)`, CCROSS_TAC[CVECTOR_NEG_COMPONENT]);; let CCROSS_REFL = prove (`!x:complex^3. x ccross x = cvector_zero`, CCROSS_TAC[CVECTOR_ZERO_COMPONENT]);; let CCROSS_LADD = prove (`!x y z:complex^3. (x + y) ccross z = (x ccross z) + (y ccross z)`, CCROSS_TAC[CVECTOR_ADD_COMPONENT]);; let CCROSS_RADD = prove (`!x y z:complex^3. x ccross(y + z) = (x ccross y) + (x ccross z)`, CCROSS_TAC[CVECTOR_ADD_COMPONENT]);; let CCROSS_LMUL = prove (`!c x y:complex^3. (c % x) ccross y = c % (x ccross y)`, CCROSS_TAC[CVECTOR_MUL_COMPONENT]);; let CCROSS_RMUL = prove (`!c x y:complex^3. x ccross (c % y) = c % (x ccross y)`, CCROSS_TAC[CVECTOR_MUL_COMPONENT]);; let CCROSS_LNEG = prove (`!x y:complex^3. (--x) ccross y = --(x ccross y)`, CCROSS_TAC[CVECTOR_NEG_COMPONENT]);; let CCROSS_RNEG = prove (`!x y:complex^3. x ccross (--y) = --(x ccross y)`, CCROSS_TAC[CVECTOR_NEG_COMPONENT]);; let CCROSS_JACOBI = prove (`!(x:complex^3) y z. x ccross (y ccross z) + y ccross (z ccross x) + z ccross (x ccross y) = cvector_zero`, REWRITE_TAC[CART_EQ3] THEN REWRITE_TAC[CVECTOR_ADD_COMPONENT;CCROSS_COMPONENT; CVECTOR_ZERO_COMPONENT] THEN SIMPLE_COMPLEX_ARITH_TAC);; (* ========================================================================= *) (* DOT PRODUCTS IN COMPLEX^N *) (* *) (* Only difference with the real case: *) (* we take the conjugate of the 2nd argument *) (* ========================================================================= *) prioritize_complex();; parse_as_infix("cdot",(20,"right"));; let cdot = new_definition `(cdot) (x:complex^N) (y:complex^N) = vsum (1..dimindex(:N)) (\i. x$i * cnj(y$i))`;; (* The dot product is symmetric MODULO the conjugate *) let CDOT_SYM = prove (`!x:complex^N y. x cdot y = cnj (y cdot x)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (SPEC_ALL CNJ_VSUM) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[CNJ_MUL;CNJ_CNJ;COMPLEX_MUL_SYM]);; let REAL_CDOT_SELF = prove (`!x:complex^N. real(x cdot x)`, REWRITE_TAC[REAL_CNJ;GSYM CDOT_SYM]);; (* The following theorems are usual axioms of the hermitian dot product, they are proved later on. * let CDOT_SELF_POS = prove(`!x:complex^N. &0 <= real_of_complex (x cdot x)`, ... * let CDOT_EQ_0 = prove(`!x:complex^N. x cdot x = Cx(&0) <=> x = cvector_zero` *) let CDOT_LADD = prove (`!x:complex^N y z. (x + y) cdot z = (x cdot z) + (y cdot z)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_ADD) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[SPECL [`(x:real^2^N)$(x':num)`;`(y:real^2^N)$(x':num)`; `cnj ((z:real^2^N)$(x':num))`] (GSYM COMPLEX_ADD_RDISTRIB)] THEN REWRITE_TAC[CVECTOR_ADD_COMPONENT]);; let CDOT_RADD = prove (`!x:complex^N y z. x cdot (y + z) = (x cdot y) + (x cdot z)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_ADD) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[SPECL [`(x:real^2^N)$(x':num)`; `cnj((y:real^2^N)$(x':num))`; `cnj ((z:real^2^N)$(x':num))`] (GSYM COMPLEX_ADD_LDISTRIB)] THEN REWRITE_TAC[CNJ_ADD; CVECTOR_ADD_COMPONENT]);; let CDOT_LSUB = prove (`!x:complex^N y z. (x - y) cdot z = (x cdot z) - (y cdot z)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_SUB) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[SPECL [`(x:real^2^N)$(x':num)`; `(y:real^2^N)$(x':num)`; `cnj ((z:real^2^N)$(x':num))`] (GSYM COMPLEX_SUB_RDISTRIB)] THEN REWRITE_TAC[CVECTOR_SUB_COMPONENT]);; let CDOT_RSUB = prove (`!x:complex^N y z. x cdot (y - z) = (x cdot y) - (x cdot z)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_SUB) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REPEAT GEN_TAC THEN MATCH_MP_TAC VSUM_EQ THEN GEN_TAC THEN DISCH_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN REWRITE_TAC[SPECL [`(x:real^2^N)$(x':num)`; `cnj((y:real^2^N)$(x':num))`; `cnj ((z:real^2^N)$(x':num))`] (GSYM COMPLEX_SUB_LDISTRIB)] THEN REWRITE_TAC[CNJ_SUB; CVECTOR_SUB_COMPONENT]);; let CDOT_LMUL = prove (`!c:complex x:complex^N y. (c % x) cdot y = c * (x cdot y)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_COMPLEX_LMUL) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[CVECTOR_MUL_COMPONENT; GSYM COMPLEX_MUL_ASSOC]);; let CDOT_RMUL = prove (`!c:complex x:complex^N x y. x cdot (c % y) = cnj c * (x cdot y)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_COMPLEX_LMUL) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[CVECTOR_MUL_COMPONENT; CNJ_MUL; COMPLEX_MUL_AC]);; let CDOT_LNEG = prove (`!x:complex^N y. (--x) cdot y = --(x cdot y)`, REWRITE_TAC[cdot] THEN ONCE_REWRITE_TAC[COMPLEX_NEG_MINUS1] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_COMPLEX_LMUL) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[CVECTOR_NEG_COMPONENT] THEN ONCE_REWRITE_TAC[GSYM COMPLEX_NEG_MINUS1] THEN REWRITE_TAC[COMPLEX_NEG_LMUL]);; let CDOT_RNEG = prove (`!x:complex^N y. x cdot (--y) = --(x cdot y)`, REWRITE_TAC[cdot] THEN ONCE_REWRITE_TAC[COMPLEX_NEG_MINUS1] THEN REWRITE_TAC[MATCH_MP (GSYM VSUM_COMPLEX_LMUL) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN ONCE_REWRITE_TAC[GSYM COMPLEX_NEG_MINUS1] THEN REWRITE_TAC[CVECTOR_NEG_COMPONENT; CNJ_NEG; COMPLEX_NEG_RMUL]);; let CDOT_LZERO = prove (`!x:complex^N. cvector_zero cdot x = Cx (&0)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[CVECTOR_ZERO_COMPONENT] THEN REWRITE_TAC[COMPLEX_MUL_LZERO; GSYM COMPLEX_VEC_0; VSUM_0]);; let CNJ_ZERO = prove( `cnj (Cx(&0)) = Cx(&0)`, REWRITE_TAC[cnj;RE_CX;IM_CX;CX_DEF;REAL_NEG_0]);; let CDOT_RZERO = prove( `!x:complex^N. x cdot cvector_zero = Cx (&0)`, REWRITE_TAC[cdot] THEN REWRITE_TAC[CVECTOR_ZERO_COMPONENT] THEN REWRITE_TAC[CNJ_ZERO] THEN REWRITE_TAC[COMPLEX_MUL_RZERO;GSYM COMPLEX_VEC_0;VSUM_0]);; (* Cauchy Schwarz inequality: proved later on * let CDOT_CAUCHY_SCHWARZ = prove (`!x y:complex^N. norm (x cdot y) pow 2 <= cnorm2 x * cnorm2 y`, * let CDOT_CAUCHY_SCHWARZ_EQUAL = prove(`!x y:complex^N. norm (x cdot y) pow 2 = cnorm2 x * cnorm2 y <=> collinear_cvectors x y`, *) let CDOT3 = prove (`!x y:complex^3. x cdot y = (x$1 * cnj (y$1) + x$2 * cnj (y$2) + x$3 * cnj (y$3))`, REWRITE_TAC[cdot] THEN SIMP_TAC [DIMINDEX_3] THEN REWRITE_TAC[VSUM_3]);; let ADD_CDOT_SYM = prove( `!x y:complex^N. x cdot y + y cdot x = Cx(&2 * Re(x cdot y))`, MESON_TAC[CDOT_SYM;COMPLEX_ADD_CNJ]);; (* ========================================================================= *) (* RELATION WITH REAL DOT AND CROSS PRODUCTS *) (* ========================================================================= *) let CCROSS_LREAL = prove (`!r c. (vector_to_cvector r) ccross c = vector_to_cvector (r cross (cvector_re c)) + ii % (vector_to_cvector (r cross (cvector_im c)))`, REWRITE_TAC[CART_EQ3;CVECTOR_ADD_COMPONENT;CVECTOR_MUL_COMPONENT; VECTOR_TO_CVECTOR_COMPONENT;CCROSS_COMPONENT;CROSS_COMPONENTS; CVECTOR_RE_COMPONENT;CVECTOR_IM_COMPONENT;complex_mul;RE_CX;IM_CX;CX_DEF; complex_sub;complex_neg;complex_add;RE;IM;RE_II;IM_II] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[PAIR_EQ] THEN ARITH_TAC);; let CCROSS_RREAL = prove (`!r c. c ccross (vector_to_cvector r) = vector_to_cvector ((cvector_re c) cross r) + ii % (vector_to_cvector ((cvector_im c) cross r))`, REWRITE_TAC[CART_EQ3;CVECTOR_ADD_COMPONENT;CVECTOR_MUL_COMPONENT; VECTOR_TO_CVECTOR_COMPONENT;CCROSS_COMPONENT;CROSS_COMPONENTS; CVECTOR_RE_COMPONENT;CVECTOR_IM_COMPONENT;complex_mul;RE_CX;IM_CX;CX_DEF; complex_sub;complex_neg;complex_add;RE;IM;RE_II;IM_II] THEN REPEAT STRIP_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[PAIR_EQ] THEN ARITH_TAC);; let CDOT_LREAL = prove (`!r c. (vector_to_cvector r) cdot c = Cx (r dot (cvector_re c)) - ii * Cx (r dot (cvector_im c))`, REWRITE_TAC[cdot; dot; VECTOR_TO_CVECTOR_COMPONENT;CVECTOR_RE_COMPONENT; CVECTOR_IM_COMPONENT] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o ONCE_DEPTH_CONV) [COMPLEX_EXPAND] THEN REWRITE_TAC[MATCH_MP RE_VSUM (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[MATCH_MP (IM_VSUM) (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[RE_MUL_CX;RE_CNJ;IM_MUL_CX;IM_CNJ] THEN REWRITE_TAC[COMPLEX_POLY_NEG_CLAUSES] THEN REWRITE_TAC[COMPLEX_MUL_AC] THEN REWRITE_TAC[COMPLEX_MUL_ASSOC] THEN REWRITE_TAC[GSYM CX_MUL] THEN REWRITE_TAC[GSYM SUM_LMUL] THEN REWRITE_TAC[GSYM REAL_NEG_MINUS1;GSYM REAL_MUL_RNEG]);; let CDOT_RREAL = prove (`!r c. c cdot (vector_to_cvector r) = Cx ((cvector_re c) dot r) + ii * Cx ((cvector_im c) dot r)`, REWRITE_TAC[cdot; dot; VECTOR_TO_CVECTOR_COMPONENT;CVECTOR_RE_COMPONENT; CVECTOR_IM_COMPONENT] THEN REPEAT GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o ONCE_DEPTH_CONV) [COMPLEX_EXPAND] THEN REWRITE_TAC[MATCH_MP RE_VSUM (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[MATCH_MP IM_VSUM (SPEC `dimindex (:N)` (GEN_ALL (CONJUNCT1 (SPEC_ALL (REWRITE_RULE[HAS_SIZE] HAS_SIZE_NUMSEG_1)))))] THEN REWRITE_TAC[CNJ_CX] THEN REWRITE_TAC[RE_MUL_CX;RE_CNJ;IM_MUL_CX;IM_CNJ]);; (* ========================================================================= *) (* NORM, UNIT VECTORS. *) (* ========================================================================= *) let cnorm2 = new_definition `cnorm2 (v:complex^N) = real_of_complex (v cdot v)`;; let CX_CNORM2 = prove (`!v:complex^N. Cx(cnorm2 v) = v cdot v`, SIMP_TAC[cnorm2;REAL_CDOT_SELF;REAL_OF_COMPLEX]);; let CNORM2_CVECTOR_ZERO = prove (`cnorm2 (cvector_zero:complex^N) = &0`, REWRITE_TAC[cnorm2;CDOT_RZERO;REAL_OF_COMPLEX_CX]);; let CNORM2_MODULUS = prove (`!x:complex^N. cnorm2 x = (vector_map norm x) dot (vector_map norm x)`, REWRITE_TAC[cnorm2;cdot;COMPLEX_MUL_CNJ;COMPLEX_POW_2;GSYM CX_MUL; VSUM_CX_NUMSEG;dot;VECTOR_MAP_COMPONENT;REAL_OF_COMPLEX_CX]);; let CNORM2_EQ_0 = prove (`!x:complex^N. cnorm2 x = &0 <=> x = cvector_zero`, REWRITE_TAC[CNORM2_MODULUS;CX_INJ;DOT_EQ_0] THEN GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o DEPTH_CONV) [CART_EQ] THEN REWRITE_TAC[VEC_COMPONENT;VECTOR_MAP_COMPONENT;COMPLEX_NORM_ZERO] THEN GEN_REWRITE_TAC (RAND_CONV o DEPTH_CONV) [CART_EQ] THEN REWRITE_TAC[CVECTOR_ZERO_COMPONENT]);; let CDOT_EQ_0 = prove (`!x:complex^N. x cdot x = Cx(&0) <=> x = cvector_zero`, SIMP_TAC[TAUT `(p<=>q) <=> ((p==>q) /\ (q==>p))`;CDOT_LZERO] THEN GEN_TAC THEN DISCH_THEN (MP_TAC o MATCH_MP (MESON[REAL_OF_COMPLEX_CX] `x = Cx y ==> real_of_complex x = y`)) THEN REWRITE_TAC[GSYM cnorm2;CNORM2_EQ_0]);; let CNORM2_POS = prove (`!x:complex^N. &0 <= cnorm2 x`, REWRITE_TAC[CNORM2_MODULUS;DOT_POS_LE]);; let CDOT_SELF_POS = prove (`!x:complex^N. &0 <= real_of_complex (x cdot x)`, REWRITE_TAC[GSYM cnorm2;CNORM2_POS]);; let CNORM2_MUL = prove (`!a x:complex^N. cnorm2 (a % x) = (norm a) pow 2 * cnorm2 x`, SIMP_TAC[cnorm2;CDOT_LMUL;CDOT_RMUL; SIMPLE_COMPLEX_ARITH `x * cnj x * y = (x * cnj x) * y`;COMPLEX_MUL_CNJ; REAL_OF_COMPLEX_CX;REAL_OF_COMPLEX_MUL;REAL_CX;REAL_CDOT_SELF; GSYM CX_POW]);; let CNORM2_NORM2_2 = prove (`!x y:real^N. cnorm2 (vector_to_cvector x + ii % vector_to_cvector y) = norm x pow 2 + norm y pow 2`, REWRITE_TAC[cnorm2;vector_norm;cdot;CVECTOR_ADD_COMPONENT; CVECTOR_MUL_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT;CNJ_ADD;CNJ_CX;CNJ_MUL; CNJ_II;COMPLEX_ADD_RDISTRIB;COMPLEX_ADD_LDISTRIB; SIMPLE_COMPLEX_ARITH `(x*x+x*(--ii)*y)+(ii*y)*x+(ii*y)*(--ii)*y = x*x-(ii*ii)*y*y`] THEN REWRITE_TAC[GSYM COMPLEX_POW_2;COMPLEX_POW_II_2; SIMPLE_COMPLEX_ARITH `x-(--Cx(&1))*y = x+y`] THEN SIMP_TAC[MESON[CARD_NUMSEG_1;HAS_SIZE_NUMSEG_1;FINITE_HAS_SIZE] `FINITE (1..dimindex(:N))`;VSUM_ADD;GSYM CX_POW;VSUM_CX;GSYM dot; REAL_POW_2;GSYM dot] THEN SIMP_TAC[GSYM CX_ADD;REAL_OF_COMPLEX_CX;GSYM REAL_POW_2;DOT_POS_LE; SQRT_POW_2]);; let CNORM2_NORM2 = prove (`!v:complex^N. cnorm2 v = norm (cvector_re v) pow 2 + norm (cvector_im v) pow 2`, GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o ONCE_DEPTH_CONV) [GSYM VECTOR_TO_CVECTOR_CVECTOR_RE_IM] THEN REWRITE_TAC[CNORM2_NORM2_2]);; let CNORM2_ALT = prove (`!x:complex^N. cnorm2 x = norm (x cdot x)`, SIMP_TAC[cnorm2;REAL_OF_COMPLEX_NORM;REAL_CDOT_SELF;EQ_SYM_EQ;REAL_ABS_REFL; REWRITE_RULE[cnorm2] CNORM2_POS]);; let CNORM2_SUB = prove (`!x y:complex^N. cnorm2 (x-y) = cnorm2 (y-x)`, REWRITE_TAC[cnorm2;CDOT_LSUB;CDOT_RSUB] THEN REPEAT GEN_TAC THEN AP_TERM_TAC THEN SIMPLE_COMPLEX_ARITH_TAC);; let CNORM2_VECTOR_TO_CVECTOR = prove (`!x:real^N. cnorm2 (vector_to_cvector x) = norm x pow 2`, REWRITE_TAC[CNORM2_ALT;CDOT_RREAL;CVECTOR_RE_VECTOR_TO_CVECTOR; CVECTOR_IM_VECTOR_TO_CVECTOR;DOT_LZERO;COMPLEX_MUL_RZERO;COMPLEX_ADD_RID; DOT_SQUARE_NORM;CX_POW;COMPLEX_NORM_POW;COMPLEX_NORM_CX;REAL_POW2_ABS]);; let cnorm = new_definition `cnorm :complex^N->real = sqrt o cnorm2`;; overload_interface ("norm",`cnorm:complex^N->real`);; let CNORM_CVECTOR_ZERO = prove (`norm (cvector_zero:complex^N) = &0`, REWRITE_TAC[cnorm;CNORM2_CVECTOR_ZERO;o_DEF;SQRT_0]);; let CNORM_POW_2 = prove (`!x:complex^N. norm x pow 2 = cnorm2 x`, SIMP_TAC[cnorm;o_DEF;SQRT_POW_2;CNORM2_POS]);; let CNORM_NORM_2 = prove (`!x y:real^N. norm (vector_to_cvector x + ii % vector_to_cvector y) = sqrt(norm x pow 2 + norm y pow 2)`, REWRITE_TAC[cnorm;o_DEF;CNORM2_NORM2_2]);; let CNORM_NORM = prove( `!v:complex^N. norm v = sqrt(norm (cvector_re v) pow 2 + norm (cvector_im v) pow 2)`, GEN_TAC THEN GEN_REWRITE_TAC (RATOR_CONV o ONCE_DEPTH_CONV) [GSYM VECTOR_TO_CVECTOR_CVECTOR_RE_IM] THEN REWRITE_TAC[CNORM_NORM_2]);; let CNORM_MUL = prove (`!a x:complex^N. norm (a % x) = norm a * norm x`, SIMP_TAC[cnorm;o_DEF;CNORM2_MUL;REAL_LE_POW_2;SQRT_MUL;CNORM2_POS; NORM_POS_LE;POW_2_SQRT]);; let CNORM_EQ_0 = prove (`!x:complex^N. norm x = &0 <=> x = cvector_zero`, SIMP_TAC[cnorm;o_DEF;SQRT_EQ_0;CNORM2_POS;CNORM2_EQ_0]);; let CNORM_POS = prove (`!x:complex^N. &0 <= norm x`, SIMP_TAC[cnorm;o_DEF;SQRT_POS_LE;CNORM2_POS]);; let CNORM_SUB = prove (`!x y:complex^N. norm (x-y) = norm (y-x)`, REWRITE_TAC[cnorm;o_DEF;CNORM2_SUB]);; let CNORM_VECTOR_TO_CVECTOR = prove (`!x:real^N. norm (vector_to_cvector x) = norm x`, SIMP_TAC[cnorm;o_DEF;CNORM2_VECTOR_TO_CVECTOR;POW_2_SQRT;NORM_POS_LE]);; let CNORM_BASIS = prove (`!k. 1 <= k /\ k <= dimindex(:N) ==> norm (vector_to_cvector (basis k :real^N)) = &1`, SIMP_TAC[NORM_BASIS;CNORM_VECTOR_TO_CVECTOR]);; let CNORM_BASIS_1 = prove (`norm(basis 1:real^N) = &1`, SIMP_TAC[NORM_BASIS_1;CNORM_VECTOR_TO_CVECTOR]);; let CVECTOR_CHOOSE_SIZE = prove (`!c. &0 <= c ==> ?x:complex^N. norm(x) = c`, MESON_TAC[VECTOR_CHOOSE_SIZE;CNORM_VECTOR_TO_CVECTOR]);; (* Triangle inequality. Proved later on using Cauchy Schwarz inequality. * let CNORM_TRIANGLE = prove(`!x y:complex^N. norm (x+y) <= norm x + norm y`, ... *) let cunit = new_definition `cunit (X:complex^N) = inv(Cx(norm X))% X`;; let CUNIT_CVECTOR_ZERO = prove (`cunit cvector_zero = cvector_zero:complex^N`, REWRITE_TAC[cunit;CNORM_CVECTOR_ZERO;COMPLEX_INV_0;CVECTOR_MUL_LZERO]);; let CDOT_CUNIT_MUL_CUNIT = prove (`!x:complex^N. (cunit x cdot x) % cunit x = x`, GEN_TAC THEN ASM_CASES_TAC `x = cvector_zero:complex^N` THEN ASM_REWRITE_TAC[CUNIT_CVECTOR_ZERO;CDOT_LZERO;CVECTOR_MUL_LZERO] THEN SIMP_TAC[cunit;CVECTOR_MUL_ASSOC;CDOT_LMUL; SIMPLE_COMPLEX_ARITH `(x*y)*x=(x*x)*y`;GSYM COMPLEX_INV_MUL;GSYM CX_MUL; GSYM REAL_POW_2;cnorm;o_DEF;CNORM2_POS;SQRT_POW_2] THEN ASM_SIMP_TAC[cnorm2;REAL_OF_COMPLEX;REAL_CDOT_SELF;CDOT_EQ_0; CNORM2_CVECTOR_ZERO;CVECTOR_MUL_RZERO;CNORM2_EQ_0;COMPLEX_MUL_LINV; CVECTOR_MUL_ID]);; (* ========================================================================= *) (* COLLINEARITY *) (* ========================================================================= *) (* Definition of collinearity between complex vectors. * Note: This is different from collinearity between points (which is the one defined in HOL-Light library) *) let collinear_cvectors = new_definition `collinear_cvectors x (y:complex^N) <=> ?a. y = a % x \/ x = a % y`;; let COLLINEAR_CVECTORS_SYM = prove (`!x y:complex^N. collinear_cvectors x y <=> collinear_cvectors y x`, REWRITE_TAC[collinear_cvectors] THEN MESON_TAC[]);; let COLLINEAR_CVECTORS_0 = prove (`!x:complex^N. collinear_cvectors x cvector_zero`, REWRITE_TAC[collinear_cvectors] THEN GEN_TAC THEN EXISTS_TAC `Cx(&0)` THEN REWRITE_TAC[CVECTOR_MUL_LZERO]);; let NON_NULL_COLLINEARS = prove (`!x y:complex^N. collinear_cvectors x y /\ ~(x=cvector_zero) /\ ~(y=cvector_zero) ==> ?a. ~(a=Cx(&0)) /\ y = a % x`, REWRITE_TAC[collinear_cvectors] THEN REPEAT STRIP_TAC THENL [ ASM_MESON_TAC[CVECTOR_MUL_LZERO]; SUBGOAL_THEN `~(a=Cx(&0))` ASSUME_TAC THENL [ ASM_MESON_TAC[CVECTOR_MUL_LZERO]; EXISTS_TAC `inv a :complex` THEN ASM_REWRITE_TAC[COMPLEX_INV_EQ_0;CVECTOR_MUL_ASSOC] THEN ASM_SIMP_TAC[COMPLEX_MUL_LINV;CVECTOR_MUL_ID]]]);; let COLLINEAR_LNONNULL = prove( `!x y:complex^N. collinear_cvectors x y /\ ~(x=cvector_zero) ==> ?a. y = a % x`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `y=cvector_zero:complex^N` THENL [ ASM_REWRITE_TAC[] THEN EXISTS_TAC `Cx(&0)` THEN ASM_MESON_TAC[CVECTOR_MUL_LZERO]; ASM_MESON_TAC[NON_NULL_COLLINEARS] ]);; let COLLINEAR_RNONNULL = prove( `!x y:complex^N. collinear_cvectors x y /\ ~(y=cvector_zero) ==> ?a. x = a % y`, MESON_TAC[COLLINEAR_LNONNULL;COLLINEAR_CVECTORS_SYM]);; let COLLINEAR_RUNITREAL = prove( `!x y:real^N. collinear_cvectors x (vector_to_cvector y) /\ norm y = &1 ==> x = (x cdot (vector_to_cvector y)) % vector_to_cvector y`, REPEAT STRIP_TAC THEN POP_ASSUM (DISTRIB [ASSUME_TAC; ASSUME_TAC o REWRITE_RULE[NORM_EQ_0; GSYM VECTOR_TO_CVECTOR_ZERO_EQ] o MATCH_MP (REAL_ARITH `!x. x= &1 ==> ~(x= &0)`)]) THEN FIRST_X_ASSUM (fun x -> FIRST_X_ASSUM (fun y -> CHOOSE_THEN (SINGLE ONCE_REWRITE_TAC) (MATCH_MP COLLINEAR_RNONNULL (CONJ y x)))) THEN REWRITE_TAC[CDOT_LMUL;CDOT_LREAL;CVECTOR_RE_VECTOR_TO_CVECTOR; CVECTOR_IM_VECTOR_TO_CVECTOR;DOT_RZERO;COMPLEX_MUL_RZERO;COMPLEX_SUB_RZERO] THEN POP_ASSUM ((fun x -> REWRITE_TAC[x;COMPLEX_MUL_RID]) o REWRITE_RULE[NORM_EQ_1]));; let CCROSS_COLLINEAR_CVECTORS = prove (`!x y:complex^3. x ccross y = cvector_zero <=> collinear_cvectors x y`, REWRITE_TAC[ccross;collinear_cvectors;CART_EQ3;VECTOR_3; CVECTOR_ZERO_COMPONENT;COMPLEX_SUB_0;CVECTOR_MUL_COMPONENT] THEN REPEAT GEN_TAC THEN EQ_TAC THENL [ REPEAT (POP_ASSUM MP_TAC) THEN ASM_CASES_TAC `(x:complex^3)$1 = Cx(&0)` THENL [ ASM_CASES_TAC `(x:complex^3)$2 = Cx(&0)` THENL [ ASM_CASES_TAC `(x:complex^3)$3 = Cx(&0)` THENL [ REPEAT DISCH_TAC THEN EXISTS_TAC `Cx(&0)` THEN ASM_REWRITE_TAC[COMPLEX_POLY_CLAUSES]; REPEAT STRIP_TAC THEN EXISTS_TAC `(y:complex^3)$3/(x:complex^3)$3` THEN ASM_SIMP_TAC[COMPLEX_BALANCE_DIV_MUL] THEN ASM_MESON_TAC[COMPLEX_MUL_AC];]; REPEAT STRIP_TAC THEN EXISTS_TAC `(y:complex^3)$2/(x:complex^3)$2` THEN ASM_SIMP_TAC[COMPLEX_BALANCE_DIV_MUL] THEN ASM_MESON_TAC[COMPLEX_MUL_AC]; ]; REPEAT STRIP_TAC THEN EXISTS_TAC `(y:complex^3)$1/(x:complex^3)$1` THEN ASM_SIMP_TAC[COMPLEX_BALANCE_DIV_MUL] THEN ASM_MESON_TAC[COMPLEX_MUL_AC];]; SIMPLE_COMPLEX_ARITH_TAC ]);; let CVECTOR_MUL_INV = prove (`!a x y:complex^N. ~(a = Cx(&0)) /\ x = a % y ==> y = inv a % x`, REPEAT STRIP_TAC THEN ASM_SIMP_TAC[CVECTOR_MUL_ASSOC; MESON[] `(p\/q) <=> (~p ==> q)`;COMPLEX_MUL_LINV;CVECTOR_MUL_ID]);; let CVECTOR_MUL_INV2 = prove (`!a x y:complex^N. ~(x = cvector_zero) /\ x = a % y ==> y = inv a % x`, REPEAT STRIP_TAC THEN ASM_CASES_TAC `a=Cx(&0)` THEN ASM_MESON_TAC[CVECTOR_MUL_LZERO;CVECTOR_MUL_INV]);; let COLLINEAR_CVECTORS_VECTOR_TO_CVECTOR = prove( `!x y:real^N. collinear_cvectors (vector_to_cvector x) (vector_to_cvector y) <=> collinear {vec 0,x,y}`, REWRITE_TAC[COLLINEAR_LEMMA_ALT;collinear_cvectors] THEN REPEAT (STRIP_TAC ORELSE EQ_TAC) THENL [ POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_MUL_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT; VECTOR_MUL_COMPONENT;COMPLEX_EQ;RE_CX;RE_MUL_CX] THEN REPEAT STRIP_TAC THEN DISJ2_TAC THEN EXISTS_TAC `Re a` THEN ASM_SIMP_TAC[]; REWRITE_TAC[MESON[]`(p\/q) <=> (~p ==> q)`] THEN REWRITE_TAC[GSYM VECTOR_TO_CVECTOR_ZERO_EQ] THEN DISCH_TAC THEN SUBGOAL_TAC "" `vector_to_cvector (y:real^N) = inv a % vector_to_cvector x` [ASM_MESON_TAC[CVECTOR_MUL_INV2]] THEN POP_ASSUM MP_TAC THEN ONCE_REWRITE_TAC[CART_EQ] THEN REWRITE_TAC[CVECTOR_MUL_COMPONENT;VECTOR_TO_CVECTOR_COMPONENT; VECTOR_MUL_COMPONENT;COMPLEX_EQ;RE_CX;RE_MUL_CX] THEN REPEAT STRIP_TAC THEN EXISTS_TAC `Re(inv a)` THEN ASM_SIMP_TAC[]; EXISTS_TAC `Cx(&0)` THEN ASM_REWRITE_TAC[VECTOR_TO_CVECTOR_ZERO; CVECTOR_MUL_LZERO]; ASM_REWRITE_TAC[VECTOR_TO_CVECTOR_MUL] THEN EXISTS_TAC `Cx c` THEN REWRITE_TAC[]; ]);; (* ========================================================================= *) (* ORTHOGONALITY *) (* ========================================================================= *) let corthogonal = new_definition `corthogonal (x:complex^N) y <=> x cdot y = Cx(&0)`;; let CORTHOGONAL_SYM = prove( `!x y:complex^N. corthogonal x y <=> corthogonal y x`, MESON_TAC[corthogonal;CDOT_SYM;CNJ_ZERO]);; let CORTHOGONAL_0 = prove( `!x:complex^N. corthogonal cvector_zero x /\ corthogonal x cvector_zero`, REWRITE_TAC[corthogonal;CDOT_LZERO;CDOT_RZERO]);; let [CORTHOGONAL_LZERO;CORTHOGONAL_RZERO] = GCONJUNCTS CORTHOGONAL_0;; let CORTHOGONAL_COLLINEAR_CVECTORS = prove (`!x y:complex^N. collinear_cvectors x y /\ ~(x=cvector_zero) /\ ~(y=cvector_zero) ==> ~(corthogonal x y)`, REWRITE_TAC[collinear_cvectors;corthogonal] THEN REPEAT STRIP_TAC THEN POP_ASSUM MP_TAC THEN ASM_REWRITE_TAC[CDOT_RMUL;CDOT_LMUL;COMPLEX_ENTIRE;GSYM cnorm2; CDOT_EQ_0;CNJ_EQ_0] THEN ASM_MESON_TAC[CVECTOR_MUL_LZERO]);; let CORTHOGONAL_MUL_CLAUSES = prove (`!x y a. (corthogonal x y ==> corthogonal x (a%y)) /\ (corthogonal x y \/ a = Cx(&0) <=> corthogonal x (a%y)) /\ (corthogonal x y ==> corthogonal (a%x) y) /\ (corthogonal x y \/ a = Cx(&0) <=> corthogonal (a%x) y)`, SIMP_TAC[corthogonal;CDOT_RMUL;CDOT_LMUL;COMPLEX_ENTIRE;CNJ_EQ_0] THEN MESON_TAC[]);; let [CORTHOGONAL_RMUL;CORTHOGONAL_RMUL_EQ;CORTHOGONAL_LMUL; CORTHOGONAL_LMUL_EQ] = GCONJUNCTS CORTHOGONAL_MUL_CLAUSES;; let CORTHOGONAL_LRMUL_CLAUSES = prove (`!x y a b. (corthogonal x y ==> corthogonal (a%x) (b%y)) /\ (corthogonal x y \/ a = Cx(&0) \/ b = Cx(&0) <=> corthogonal (a%x) (b%y))`, MESON_TAC[CORTHOGONAL_MUL_CLAUSES]);; let [CORTHOGONAL_LRMUL;CORTHOGONAL_LRMUL_EQ] = GCONJUNCTS CORTHOGONAL_LRMUL_CLAUSES;; let CORTHOGONAL_REAL_CLAUSES = prove (`!r c. (corthogonal c (vector_to_cvector r) <=> orthogonal (cvector_re c) r /\ orthogonal (cvector_im c) r) /\ (corthogonal (vector_to_cvector r) c <=> orthogonal r (cvector_re c) /\ orthogonal r (cvector_im c))`, REWRITE_TAC[corthogonal;orthogonal;CDOT_LREAL;CDOT_RREAL;COMPLEX_SUB_0; COMPLEX_EQ;RE_CX;IM_CX;RE_SUB;IM_SUB;RE_ADD;IM_ADD] THEN REWRITE_TAC[RE_DEF;CX_DEF;IM_DEF;complex;complex_mul;VECTOR_2;ii] THEN CONV_TAC REAL_FIELD);; let [CORTHOGONAL_RREAL;CORTHOGONAL_LREAL] = GCONJUNCTS CORTHOGONAL_REAL_CLAUSES;; let CORTHOGONAL_UNIT = prove (`!x y:complex^N. (corthogonal x (cunit y) <=> corthogonal x y) /\ (corthogonal (cunit x) y <=> corthogonal x y)`, REWRITE_TAC[cunit;GSYM CORTHOGONAL_MUL_CLAUSES;COMPLEX_INV_EQ_0;CX_INJ; CNORM_EQ_0] THEN MESON_TAC[CORTHOGONAL_0]);; let [CORTHOGONAL_RUNIT;CORTHOGONAL_LUNIT] = GCONJUNCTS CORTHOGONAL_UNIT;; let CORTHOGONAL_PROJECTION = prove( `!x y:complex^N. corthogonal (x - (x cdot cunit y) % cunit y) y`, REPEAT GEN_TAC THEN ASM_CASES_TAC `y=cvector_zero:complex^N` THEN ASM_REWRITE_TAC[corthogonal;CDOT_RZERO] THEN REWRITE_TAC[CDOT_LSUB;cunit;CVECTOR_MUL_ASSOC;GSYM cnorm2;CDOT_LMUL; CDOT_RMUL;REWRITE_RULE[REAL_CNJ] (MATCH_MP REAL_INV (SPEC_ALL REAL_CX))] THEN REWRITE_TAC[COMPLEX_MUL_AC;GSYM COMPLEX_INV_MUL;GSYM COMPLEX_POW_2; cnorm;o_DEF;CSQRT] THEN SIMP_TAC[CNORM2_POS;CX_SQRT;cnorm2;REAL_CDOT_SELF;REAL_OF_COMPLEX;CSQRT] THEN ASM_SIMP_TAC[CDOT_EQ_0;COMPLEX_MUL_RINV;COMPLEX_MUL_RID; COMPLEX_SUB_REFL]);; let CDOT_PYTHAGOREAN = prove (`!x y:complex^N. corthogonal x y ==> cnorm2 (x+y) = cnorm2 x + cnorm2 y`, SIMP_TAC[corthogonal;cnorm2;CDOT_LADD;CDOT_RADD;COMPLEX_ADD_RID; COMPLEX_ADD_LID;REAL_OF_COMPLEX_ADD;REAL_CDOT_SELF; MESON[CDOT_SYM;CNJ_ZERO] `x cdot y = Cx (&0) ==> y cdot x = Cx(&0)`]);; let CDOT_CAUCHY_SCHWARZ_POW_2 = prove (`!x y:complex^N. norm (x cdot y) pow 2 <= cnorm2 x * cnorm2 y`, REPEAT GEN_TAC THEN ASM_CASES_TAC `y = cvector_zero:complex^N` THEN ASM_REWRITE_TAC[CNORM2_CVECTOR_ZERO;CDOT_RZERO;COMPLEX_NORM_0; REAL_POW_2;REAL_MUL_RZERO;REAL_OF_COMPLEX_CX;REAL_LE_REFL] THEN ONCE_REWRITE_TAC[MATCH_MP (MESON[CVECTOR_SUB_ADD] `(!x:complex^N y. p (x - f x y) y) ==> cnorm2 x * z = cnorm2 (x - f x y + f x y) * z`) CORTHOGONAL_PROJECTION] THEN MATCH_MP_TAC (GEN_ALL (MATCH_MP (MESON[] `(!x y. P x y ==> f x y = (g x y:real)) ==> P x y /\ a <= g x y * b ==> a <= f x y * b`) CDOT_PYTHAGOREAN)) THEN REWRITE_TAC[GSYM CORTHOGONAL_MUL_CLAUSES;CORTHOGONAL_RUNIT; CORTHOGONAL_PROJECTION] THEN SIMP_TAC[cnorm2;GSYM REAL_OF_COMPLEX_ADD;REAL_CDOT_SELF;REAL_ADD; GSYM REAL_OF_COMPLEX_MUL] THEN REWRITE_TACS[cunit;CDOT_RMUL;CVECTOR_MUL_ASSOC;REWRITE_RULE[REAL_CNJ] (MATCH_MP REAL_INV (SPEC_ALL REAL_CX));COMPLEX_MUL_AC;GSYM COMPLEX_INV_MUL; GSYM COMPLEX_POW_2;cnorm;o_DEF;CSQRT;COMPLEX_ADD_LDISTRIB;cnorm2;CDOT_RMUL; CNJ_MUL;CDOT_LMUL;GSYM cnorm2; REWRITE_RULE[REAL_CNJ] (MATCH_MP REAL_INV (SPEC_ALL REAL_CX))] THEN SIMP_TAC[CX_SQRT;CNORM2_POS;CSQRT;CX_CNORM2] THEN REWRITE_TAC[SIMPLE_COMPLEX_ARITH `x * ((y * inv x) * x) * (z * inv x') * inv x' = (y * z) * (x * inv x) * (x * inv x' * inv x'):complex`] THEN ASM_SIMP_TAC[CDOT_EQ_0;COMPLEX_MUL_RINV;COMPLEX_MUL_LID;COMPLEX_MUL_CNJ; GSYM COMPLEX_INV_MUL] THEN ONCE_REWRITE_TAC[ GSYM (MATCH_MP REAL_OF_COMPLEX (SPEC_ALL REAL_CDOT_SELF))] THEN SIMP_TAC[GSYM cnorm2;GSYM CX_SQRT;CNORM2_POS;GSYM CX_MUL; GSYM COMPLEX_POW_2;GSYM CX_POW;SQRT_POW_2;GSYM CX_INV] THEN ASM_SIMP_TAC[REAL_MUL_RINV;CNORM2_EQ_0;REAL_MUL_RID;GSYM CX_ADD; REAL_OF_COMPLEX_CX;GSYM REAL_POW_2;REAL_LE_ADDL;REAL_LE_MUL;CNORM2_POS]);; let CDOT_CAUCHY_SCHWARZ = prove (`!x y:complex^N. norm (x cdot y) <= norm x * norm y`, REPEAT GEN_TAC THEN MATCH_MP_TAC (REWRITE_RULE[REAL_LE_SQUARE_ABS] (REAL_ARITH `&0 <= x /\ &0 <= y /\ abs x <= abs y ==> x <= y`)) THEN SIMP_TAC[NORM_POS_LE;CNORM_POS;REAL_LE_MUL;REAL_POW_MUL;CNORM_POW_2; CDOT_CAUCHY_SCHWARZ_POW_2]);; let CDOT_CAUCHY_SCHWARZ_POW_2_EQUAL = prove (`!x y:complex^N. norm (x cdot y) pow 2 = cnorm2 x * cnorm2 y <=> collinear_cvectors x y`, REPEAT GEN_TAC THEN ASM_CASES_TAC `y = cvector_zero:complex^N` THEN ASM_REWRITE_TAC[CNORM2_CVECTOR_ZERO;CDOT_RZERO;COMPLEX_NORM_0; REAL_POW_2;REAL_MUL_RZERO;REAL_OF_COMPLEX_CX;COLLINEAR_CVECTORS_0] THEN EQ_TAC THENL [ ONCE_REWRITE_TAC[MATCH_MP (MESON[CVECTOR_SUB_ADD] `(!x:complex^N y. p (x - f x y) y) ==> cnorm2 x * z = cnorm2 (x - f x y + f x y) * z`) CORTHOGONAL_PROJECTION] THEN MATCH_MP_TAC (GEN_ALL (MATCH_MP (MESON[] `(!x y. P x y ==> g x y = (f x y:real)) ==> P x y /\ (a = f x y * z ==> R) ==> (a = g x y * z ==> R)`) CDOT_PYTHAGOREAN)) THEN REWRITE_TAC[GSYM CORTHOGONAL_MUL_CLAUSES;CORTHOGONAL_RUNIT; CORTHOGONAL_PROJECTION] THEN SIMP_TAC[cnorm2;GSYM REAL_OF_COMPLEX_ADD;REAL_CDOT_SELF;REAL_ADD; GSYM REAL_OF_COMPLEX_MUL] THEN REWRITE_TACS[cunit;CDOT_RMUL;CVECTOR_MUL_ASSOC;REWRITE_RULE[REAL_CNJ] (MATCH_MP REAL_INV (SPEC_ALL REAL_CX));COMPLEX_MUL_AC; GSYM COMPLEX_INV_MUL;GSYM COMPLEX_POW_2;cnorm;o_DEF;CSQRT; COMPLEX_ADD_LDISTRIB;cnorm2;CDOT_RMUL;CNJ_MUL;CDOT_LMUL;GSYM cnorm2; REWRITE_RULE[REAL_CNJ] (MATCH_MP REAL_INV (SPEC_ALL REAL_CX))] THEN SIMP_TAC[CX_SQRT;CNORM2_POS;CSQRT;CX_CNORM2] THEN REWRITE_TAC[SIMPLE_COMPLEX_ARITH `x * ((y * inv x) * x) * (z * inv x') * inv x' = (y * z) * (x * inv x) * (x * inv x' * inv x'):complex`] THEN ONCE_REWRITE_TAC[GSYM (MATCH_MP REAL_OF_COMPLEX (SPEC_ALL REAL_CDOT_SELF))] THEN SIMP_TAC[GSYM cnorm2;GSYM CX_SQRT;CNORM2_POS;GSYM CX_MUL; GSYM COMPLEX_POW_2;GSYM CX_POW;SQRT_POW_2;GSYM CX_INV;REAL_POW_INV] THEN ASM_SIMP_TAC[REAL_MUL_RINV;CNORM2_EQ_0;REAL_MUL_RID;GSYM CX_ADD; REAL_OF_COMPLEX_CX;GSYM REAL_POW_2;REAL_LE_ADDL;REAL_LE_MUL;CNORM2_POS; GSYM CX_POW;REAL_POW_ONE;COMPLEX_MUL_RID;COMPLEX_MUL_CNJ; REAL_ARITH `x = y + x <=> y = &0`;REAL_ENTIRE;CNORM2_EQ_0; CVECTOR_SUB_EQ;collinear_cvectors] THEN MESON_TAC[]; REWRITE_TAC[collinear_cvectors] THEN REPEAT STRIP_TAC THEN ASM_REWRITE_TAC[cnorm2;CDOT_LMUL;CDOT_RMUL;COMPLEX_NORM_MUL; COMPLEX_MUL_ASSOC] THEN SIMP_TAC[COMPLEX_MUL_CNJ;GSYM cnorm2;COMPLEX_NORM_CNJ;GSYM CX_POW; REAL_OF_COMPLEX_MUL;REAL_CX;REAL_CDOT_SELF;REAL_OF_COMPLEX_CX; GSYM CNORM2_ALT] THEN SIMPLE_COMPLEX_ARITH_TAC ]);; let CDOT_CAUCHY_SCHWARZ_EQUAL = prove (`!x y:complex^N. norm (x cdot y) = norm x * norm y <=> collinear_cvectors x y`, ONCE_REWRITE_TAC[REWRITE_RULE[REAL_EQ_SQUARE_ABS] (REAL_ARITH `x=y <=> abs x = abs y /\ (&0 <= x /\ &0 <= y \/ x < &0 /\ y < &0)`)] THEN SIMP_TAC[NORM_POS_LE;CNORM_POS;REAL_LE_MUL;REAL_POW_MUL;CNORM_POW_2; CDOT_CAUCHY_SCHWARZ_POW_2_EQUAL]);; let CNORM_TRIANGLE = prove (`!x y:complex^N. norm (x+y) <= norm x + norm y`, REPEAT GEN_TAC THEN MATCH_MP_TAC (REWRITE_RULE[REAL_LE_SQUARE_ABS] (REAL_ARITH `abs x <= abs y /\ &0 <= x /\ &0 <= y ==> x <= y`)) THEN SIMP_TAC[CNORM_POS;REAL_LE_ADD;REAL_ADD_POW_2;CNORM_POW_2;cnorm2; CDOT_LADD;CDOT_RADD;SIMPLE_COMPLEX_ARITH `(x+y)+z+t = x+(y+z)+t:complex`; ADD_CDOT_SYM;REAL_OF_COMPLEX_ADD;REAL_CDOT_SELF;REAL_CX;REAL_ADD; REAL_OF_COMPLEX_CX;REAL_ARITH `x+ &2*y+z<=x+z+ &2*t <=> y<=t:real`] THEN MESON_TAC[CDOT_CAUCHY_SCHWARZ;RE_NORM;REAL_LE_TRANS]);; let REAL_ABS_SUB_CNORM = prove (`!x y:complex^N. abs (norm x - norm y) <= norm (x-y)`, let lemma = REWRITE_RULE[CVECTOR_SUB_ADD2;REAL_ARITH `x<=y+z <=> x-y<=z:real`] (SPECL [`x:complex^N`;`y-x:complex^N`] CNORM_TRIANGLE) in REPEAT GEN_TAC THEN MATCH_MP_TAC (MATCH_MP (MESON[] `(!x y. P x y <=> Q x y) ==> Q x y ==> P x y`) REAL_ABS_BOUNDS) THEN ONCE_REWRITE_TAC[REAL_ARITH `--x <= y <=> --y <= x`] THEN REWRITE_TAC[REAL_NEG_SUB] THEN REWRITE_TAC[lemma;ONCE_REWRITE_RULE[CNORM_SUB] lemma]);; (* ========================================================================= *) (* VSUM *) (* ========================================================================= *) let cvsum = new_definition `(cvsum:(A->bool)->(A->complex^N)->complex^N) s f = lambda i. vsum s (\x. (f x)$i)`;; (* ========================================================================= *) (* INFINITE SUM *) (* ========================================================================= *) let csummable = new_definition `csummable (s:num->bool) (f:num->complex^N) <=> summable s (cvector_re o f) /\ summable s (cvector_im o f)`;; let cinfsum = new_definition `cinfsum (s:num->bool) (f:num->complex^N) :complex^N = vector_to_cvector (infsum s (\x. cvector_re (f x))) + ii % vector_to_cvector (infsum s (\x.cvector_im (f x)))`;; let CSUMMABLE_FLATTEN_CVECTOR = prove (`!s (f:num->complex^N). csummable s f <=> summable s (cvector_flatten o f)`, REWRITE_TAC[csummable;summable;cvector_flatten;o_DEF] THEN REPEAT (STRIP_TAC ORELSE EQ_TAC) THENL [ EXISTS_TAC `pastecart (l:real^N) (l':real^N)` THEN ASM_SIMP_TAC[GSYM SUMS_PASTECART]; EXISTS_TAC `fstcart (l:real^(N,N) finite_sum)` THEN MATCH_MP_TAC (GEN_ALL (MATCH_MP (TAUT `(p /\ q <=> r) ==> (r ==> p)`) (INST_TYPE [`:N`,`:M`] (SPEC_ALL SUMS_PASTECART)))) THEN EXISTS_TAC `(cvector_im o f) :num->real^N` THEN EXISTS_TAC `sndcart (l:real^(N,N) finite_sum)` THEN ASM_REWRITE_TAC[ETA_AX;o_DEF;PASTECART_FST_SND]; EXISTS_TAC `sndcart (l:real^(N,N) finite_sum)` THEN MATCH_MP_TAC (GEN_ALL (MATCH_MP (TAUT `(p /\ q <=> r) ==> (r ==> q)`) (INST_TYPE [`:N`,`:M`] (SPEC_ALL SUMS_PASTECART)))) THEN EXISTS_TAC `(cvector_re o f) :num->real^N` THEN EXISTS_TAC `fstcart (l:real^(N,N) finite_sum)` THEN ASM_REWRITE_TAC[ETA_AX;o_DEF;PASTECART_FST_SND]; ]);; let FLATTEN_CINFSUM = prove (`!s f. csummable s f ==> ((cinfsum s f):complex^N) = cvector_unflatten (infsum s (cvector_flatten o f))`, SIMP_TAC[cinfsum;cvector_unflatten;COMPLEX_VECTOR_TRANSPOSE;LINEAR_FSTCART; LINEAR_SNDCART;CSUMMABLE_FLATTEN_CVECTOR;GSYM INFSUM_LINEAR;o_DEF; cvector_flatten;FSTCART_PASTECART;SNDCART_PASTECART]);; let CSUMMABLE_LINEAR = prove (`!f h:complex^N->complex^M s. csummable s f /\ clinear h ==> csummable s (h o f)`, REWRITE_TAC[CSUMMABLE_FLATTEN_CVECTOR] THEN REPEAT STRIP_TAC THEN POP_ASSUM (ASSUME_TAC o MATCH_MP FLATTEN_CLINEAR) THEN SUBGOAL_THEN `cvector_flatten o (h:complex^N -> complex^M) o (f:num -> complex^N) = \n. (cvector_flatten o h o cvector_unflatten) (cvector_flatten (f n))` (SINGLE REWRITE_TAC) THENL [ REWRITE_TAC[o_DEF;FUN_EQ_THM] THEN GEN_TAC THEN REPEAT AP_TERM_TAC THEN REWRITE_TAC[REWRITE_RULE[o_DEF;I_DEF;FUN_EQ_THM] UNFLATTEN_FLATTEN]; MATCH_MP_TAC SUMMABLE_LINEAR THEN ASM_SIMP_TAC[GSYM o_DEF] ]);; let CINFSUM_LINEAR = prove (`!f (h:complex^M->complex^N) s. csummable s f /\ clinear h ==> cinfsum s (h o f) = h (cinfsum s f)`, REPEAT GEN_TAC THEN DISCH_THEN (fun x -> MP_TAC (CONJ (MATCH_MP CSUMMABLE_LINEAR x) x)) THEN SIMP_TAC[FLATTEN_CINFSUM;CSUMMABLE_FLATTEN_CVECTOR] THEN REPEAT STRIP_TAC THEN POP_ASSUM (ASSUME_TAC o MATCH_MP FLATTEN_CLINEAR) THEN SUBGOAL_THEN `cvector_flatten o (h:complex^M->complex^N) o (f:num->complex^M) = \n. (cvector_flatten o h o cvector_unflatten) ((cvector_flatten o f) n)` (SINGLE REWRITE_TAC) THENL [ REWRITE_TAC[o_DEF;FUN_EQ_THM] THEN GEN_TAC THEN REPEAT AP_TERM_TAC THEN REWRITE_TAC[REWRITE_RULE[o_DEF;I_DEF;FUN_EQ_THM] UNFLATTEN_FLATTEN]; FIRST_ASSUM (fun x -> FIRST_ASSUM (fun y -> REWRITE_TAC[MATCH_MP (MATCH_MP (REWRITE_RULE[IMP_CONJ] INFSUM_LINEAR) x) y])) THEN REWRITE_TAC[o_DEF;REWRITE_RULE[o_DEF;I_DEF;FUN_EQ_THM] UNFLATTEN_FLATTEN] ]);;