(* ========================================================================= *)
(* Multivariate calculus in Euclidean space. *)
(* *)
(* (c) Copyright, John Harrison 1998-2008 *)
(* ========================================================================= *)
needs "Multivariate/dimension.ml";;
(* ------------------------------------------------------------------------- *)
(* Derivatives. The definition is slightly tricky since we make it work over *)
(* nets of a particular form. This lets us prove theorems generally and use *)
(* "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *)
(* ------------------------------------------------------------------------- *)
parse_as_infix ("has_derivative",(12,"right"));;
let has_derivative = new_definition
`(f has_derivative f') net <=>
linear f' /\
((\y. inv(norm(y - netlimit net)) %
(f(y) -
(f(netlimit net) + f'(y - netlimit net)))) --> vec 0) net`;;
(* ------------------------------------------------------------------------- *)
(* These are the only cases we'll care about, probably. *)
(* ------------------------------------------------------------------------- *)
let has_derivative_within = prove
(`!f:real^M->real^N f' x s.
(f has_derivative f') (at x within s) <=>
linear f' /\
((\y. inv(norm(y - x)) % (f(y) - (f(x) + f'(y - x)))) --> vec 0)
(at x within s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_derivative] THEN AP_TERM_TAC THEN
ASM_CASES_TAC `trivial_limit(at (x:real^M) within s)` THENL
[ASM_REWRITE_TAC[LIM]; ASM_SIMP_TAC[NETLIMIT_WITHIN]]);;
let has_derivative_at = prove
(`!f:real^M->real^N f' x.
(f has_derivative f') (at x) <=>
linear f' /\
((\y. inv(norm(y - x)) % (f(y) - (f(x) + f'(y - x)))) --> vec 0)
(at x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[has_derivative_within]);;
(* ------------------------------------------------------------------------- *)
(* More explicit epsilon-delta forms. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_WITHIN = prove
(`(f has_derivative f')(at x within s) <=>
linear f' /\
!e. &0 < e
==> ?d. &0 < d /\
!x'. x' IN s /\
&0 < norm(x' - x) /\ norm(x' - x) < d
==> norm(f(x') - f(x) - f'(x' - x)) /
norm(x' - x) < e`,
SIMP_TAC[has_derivative_within; LIM_WITHIN] THEN AP_TERM_TAC THEN
REWRITE_TAC[dist; VECTOR_ARITH `(x' - (x + d)) = x' - x - d:real^N`] THEN
REWRITE_TAC[real_div; VECTOR_SUB_RZERO; NORM_MUL] THEN
REWRITE_TAC[REAL_MUL_AC; REAL_ABS_INV; REAL_ABS_NORM]);;
let HAS_DERIVATIVE_AT = prove
(`(f has_derivative f')(at x) <=>
linear f' /\
!e. &0 < e
==> ?d. &0 < d /\
!x'. &0 < norm(x' - x) /\ norm(x' - x) < d
==> norm(f(x') - f(x) - f'(x' - x)) /
norm(x' - x) < e`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[HAS_DERIVATIVE_WITHIN; IN_UNIV]);;
let HAS_DERIVATIVE_AT_WITHIN = prove
(`!f x s. (f has_derivative f') (at x)
==> (f has_derivative f') (at x within s)`,
REWRITE_TAC[HAS_DERIVATIVE_WITHIN; HAS_DERIVATIVE_AT] THEN MESON_TAC[]);;
let HAS_DERIVATIVE_WITHIN_OPEN = prove
(`!f f' a s.
a IN s /\ open s
==> ((f has_derivative f') (at a within s) <=>
(f has_derivative f') (at a))`,
SIMP_TAC[has_derivative_within; has_derivative_at; LIM_WITHIN_OPEN]);;
(* ------------------------------------------------------------------------- *)
(* Combining theorems. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_LINEAR = prove
(`!f net. linear f ==> (f has_derivative f) net`,
REWRITE_TAC[has_derivative; linear] THEN
SIMP_TAC[VECTOR_ARITH `x - y = x + --(&1) % y`] THEN
REWRITE_TAC[VECTOR_ARITH `x + --(&1) % (y + x + --(&1) % y) = vec 0`] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; LIM_CONST]);;
let HAS_DERIVATIVE_ID = prove
(`!net. ((\x. x) has_derivative (\h. h)) net`,
SIMP_TAC[HAS_DERIVATIVE_LINEAR; LINEAR_ID]);;
let HAS_DERIVATIVE_CONST = prove
(`!c net. ((\x. c) has_derivative (\h. vec 0)) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_derivative; linear] THEN
REWRITE_TAC[VECTOR_ADD_RID; VECTOR_SUB_REFL; VECTOR_MUL_RZERO; LIM_CONST]);;
let HAS_DERIVATIVE_LIFT_COMPONENT = prove
(`!net:(real^N)net. ((\x. lift(x$i)) has_derivative (\x. lift(x$i))) net`,
GEN_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_LINEAR THEN
REWRITE_TAC[linear; VECTOR_MUL_COMPONENT; VECTOR_ADD_COMPONENT] THEN
REWRITE_TAC[LIFT_ADD; LIFT_CMUL]);;
let HAS_DERIVATIVE_CMUL = prove
(`!f f' net c. (f has_derivative f') net
==> ((\x. c % f(x)) has_derivative (\h. c % f'(h))) net`,
REPEAT GEN_TAC THEN SIMP_TAC[has_derivative; LINEAR_COMPOSE_CMUL] THEN
DISCH_THEN(MP_TAC o SPEC `c:real` o MATCH_MP LIM_CMUL o CONJUNCT2) THEN
REWRITE_TAC[VECTOR_MUL_RZERO] THEN
MATCH_MP_TAC EQ_IMP THEN
AP_THM_TAC THEN AP_THM_TAC THEN
AP_TERM_TAC THEN ABS_TAC THEN VECTOR_ARITH_TAC);;
let HAS_DERIVATIVE_CMUL_EQ = prove
(`!f f' net c.
~(c = &0)
==> (((\x. c % f(x)) has_derivative (\h. c % f'(h))) net <=>
(f has_derivative f') net)`,
REPEAT STRIP_TAC THEN EQ_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_DERIVATIVE_CMUL) THENL
[DISCH_THEN(MP_TAC o SPEC `inv(c):real`);
DISCH_THEN(MP_TAC o SPEC `c:real`)] THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID; ETA_AX]);;
let HAS_DERIVATIVE_NEG = prove
(`!f f' net. (f has_derivative f') net
==> ((\x. --(f(x))) has_derivative (\h. --(f'(h)))) net`,
ONCE_REWRITE_TAC[VECTOR_NEG_MINUS1] THEN
SIMP_TAC[HAS_DERIVATIVE_CMUL]);;
let HAS_DERIVATIVE_NEG_EQ = prove
(`!f f' net. ((\x. --(f(x))) has_derivative (\h. --(f'(h)))) net <=>
(f has_derivative f') net`,
REPEAT GEN_TAC THEN EQ_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_DERIVATIVE_NEG) THEN
REWRITE_TAC[VECTOR_NEG_NEG; ETA_AX]);;
let HAS_DERIVATIVE_ADD = prove
(`!f f' g g' net.
(f has_derivative f') net /\ (g has_derivative g') net
==> ((\x. f(x) + g(x)) has_derivative (\h. f'(h) + g'(h))) net`,
REPEAT GEN_TAC THEN SIMP_TAC[has_derivative; LINEAR_COMPOSE_ADD] THEN
DISCH_THEN(MP_TAC o MATCH_MP (TAUT `(a /\ b) /\ (c /\ d) ==> b /\ d`)) THEN
DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD) THEN REWRITE_TAC[VECTOR_ADD_LID] THEN
MATCH_MP_TAC EQ_IMP THEN
AP_THM_TAC THEN AP_THM_TAC THEN
AP_TERM_TAC THEN ABS_TAC THEN VECTOR_ARITH_TAC);;
let HAS_DERIVATIVE_SUB = prove
(`!f f' g g' net.
(f has_derivative f') net /\ (g has_derivative g') net
==> ((\x. f(x) - g(x)) has_derivative (\h. f'(h) - g'(h))) net`,
SIMP_TAC[VECTOR_SUB; HAS_DERIVATIVE_ADD; HAS_DERIVATIVE_NEG]);;
let HAS_DERIVATIVE_VSUM = prove
(`!f net s.
FINITE s /\
(!a. a IN s ==> ((f a) has_derivative (f' a)) net)
==> ((\x. vsum s (\a. f a x)) has_derivative (\h. vsum s (\a. f' a h)))
net`,
GEN_TAC THEN GEN_TAC THEN REWRITE_TAC[IMP_CONJ] THEN
MATCH_MP_TAC FINITE_INDUCT_STRONG THEN
SIMP_TAC[VSUM_CLAUSES; HAS_DERIVATIVE_CONST] THEN
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_ADD THEN
REWRITE_TAC[ETA_AX] THEN ASM_SIMP_TAC[IN_INSERT]);;
let HAS_DERIVATIVE_VSUM_NUMSEG = prove
(`!f net m n.
(!i. m <= i /\ i <= n ==> ((f i) has_derivative (f' i)) net)
==> ((\x. vsum (m..n) (\i. f i x)) has_derivative
(\h. vsum (m..n) (\i. f' i h))) net`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_VSUM THEN
ASM_REWRITE_TAC[IN_NUMSEG; FINITE_NUMSEG]);;
let HAS_DERIVATIVE_COMPONENTWISE_WITHIN = prove
(`!f:real^M->real^N f' a s.
(f has_derivative f') (at a within s) <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> ((\x. lift(f(x)$i)) has_derivative (\x. lift(f'(x)$i)))
(at a within s)`,
REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[has_derivative_within] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[LINEAR_COMPONENTWISE; LIM_COMPONENTWISE_LIFT] THEN
SIMP_TAC[AND_FORALL_THM; TAUT `(p ==> q) /\ (p ==> r) <=> p ==> q /\ r`] THEN
REWRITE_TAC[GSYM LIFT_ADD; GSYM LIFT_CMUL; GSYM LIFT_SUB] THEN
REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT;
VECTOR_SUB_COMPONENT; LIFT_NUM]);;
let HAS_DERIVATIVE_COMPONENTWISE_AT = prove
(`!f:real^M->real^N f' a.
(f has_derivative f') (at a) <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> ((\x. lift(f(x)$i)) has_derivative (\x. lift(f'(x)$i))) (at a)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
MATCH_ACCEPT_TAC HAS_DERIVATIVE_COMPONENTWISE_WITHIN);;
(* ------------------------------------------------------------------------- *)
(* Somewhat different results for derivative of scalar multiplier. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_VMUL_COMPONENT = prove
(`!c:real^M->real^N c' k v:real^P.
1 <= k /\ k <= dimindex(:N) /\ (c has_derivative c') net
==> ((\x. c(x)$k % v) has_derivative (\x. c'(x)$k % v)) net`,
SIMP_TAC[has_derivative; LINEAR_VMUL_COMPONENT] THEN
REPEAT STRIP_TAC THEN
REWRITE_TAC[GSYM VECTOR_ADD_RDISTRIB; GSYM VECTOR_SUB_RDISTRIB] THEN
SUBST1_TAC(VECTOR_ARITH `vec 0 = &0 % (v:real^P)`) THEN
REWRITE_TAC[VECTOR_MUL_ASSOC] THEN MATCH_MP_TAC LIM_VMUL THEN
ASM_SIMP_TAC[GSYM VECTOR_SUB_COMPONENT; GSYM VECTOR_ADD_COMPONENT] THEN
ASM_SIMP_TAC[GSYM VECTOR_MUL_COMPONENT] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM]) THEN REWRITE_TAC[LIM] THEN
REWRITE_TAC[dist; LIFT_NUM; VECTOR_SUB_RZERO; o_THM; NORM_LIFT] THEN
ASM_SIMP_TAC[VECTOR_MUL_COMPONENT; REAL_ABS_MUL; NORM_MUL] THEN
ASM_MESON_TAC[REAL_LET_TRANS; COMPONENT_LE_NORM;
REAL_LE_LMUL; REAL_ABS_POS]);;
let HAS_DERIVATIVE_VMUL_DROP = prove
(`!c c' v. (c has_derivative c') net
==> ((\x. drop(c(x)) % v) has_derivative (\x. drop(c'(x)) % v)) net`,
SIMP_TAC[drop; LE_REFL; DIMINDEX_1; HAS_DERIVATIVE_VMUL_COMPONENT]);;
let HAS_DERIVATIVE_LIFT_DOT = prove
(`!f:real^M->real^N f'.
(f has_derivative f') net
==> ((\x. lift(v dot f(x))) has_derivative (\t. lift(v dot (f' t)))) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_derivative] THEN
REWRITE_TAC[GSYM LIFT_SUB; GSYM LIFT_ADD; GSYM LIFT_CMUL] THEN
REWRITE_TAC[GSYM DOT_RADD; GSYM DOT_RSUB; GSYM DOT_RMUL] THEN
SUBGOAL_THEN
`(\t. lift (v dot (f':real^M->real^N) t)) = (\y. lift(v dot y)) o f'`
SUBST1_TAC THENL [REWRITE_TAC[o_DEF]; ALL_TAC] THEN
SIMP_TAC[LINEAR_COMPOSE; LINEAR_LIFT_DOT] THEN
DISCH_THEN(MP_TAC o MATCH_MP LIM_LIFT_DOT o CONJUNCT2) THEN
SIMP_TAC[o_DEF; DOT_RZERO; LIFT_NUM]);;
(* ------------------------------------------------------------------------- *)
(* Limit transformation for derivatives. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_TRANSFORM_WITHIN = prove
(`!f f' g x s d.
&0 < d /\ x IN s /\
(!x'. x' IN s /\ dist (x',x) < d ==> f x' = g x') /\
(f has_derivative f') (at x within s)
==> (g has_derivative f') (at x within s)`,
REPEAT GEN_TAC THEN SIMP_TAC[has_derivative_within; IMP_CONJ] THEN
REPLICATE_TAC 4 DISCH_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[TAUT `a /\ b /\ c ==> d <=> a /\ b ==> c ==> d`]
LIM_TRANSFORM_WITHIN) THEN
EXISTS_TAC `d:real` THEN ASM_SIMP_TAC[DIST_REFL]);;
let HAS_DERIVATIVE_TRANSFORM_AT = prove
(`!f f' g x d.
&0 < d /\ (!x'. dist (x',x) < d ==> f x' = g x') /\
(f has_derivative f') (at x)
==> (g has_derivative f') (at x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
MESON_TAC[HAS_DERIVATIVE_TRANSFORM_WITHIN; IN_UNIV]);;
let HAS_DERIVATIVE_TRANSFORM_WITHIN_OPEN = prove
(`!f g:real^M->real^N s x.
open s /\ x IN s /\
(!y. y IN s ==> f y = g y) /\
(f has_derivative f') (at x)
==> (g has_derivative f') (at x)`,
REPEAT GEN_TAC THEN SIMP_TAC[has_derivative_at; IMP_CONJ] THEN
REPLICATE_TAC 4 DISCH_TAC THEN
MATCH_MP_TAC(REWRITE_RULE
[TAUT `a /\ b /\ c /\ d ==> e <=> a /\ b /\ c ==> d ==> e`]
LIM_TRANSFORM_WITHIN_OPEN) THEN
EXISTS_TAC `s:real^M->bool` THEN ASM_SIMP_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* Differentiability. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix ("differentiable",(12,"right"));;
parse_as_infix ("differentiable_on",(12,"right"));;
let differentiable = new_definition
`f differentiable net <=> ?f'. (f has_derivative f') net`;;
let differentiable_on = new_definition
`f differentiable_on s <=> !x. x IN s ==> f differentiable (at x within s)`;;
let HAS_DERIVATIVE_IMP_DIFFERENTIABLE = prove
(`!f f' net. (f has_derivative f') net ==> f differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[]);;
let DIFFERENTIABLE_AT_WITHIN = prove
(`!f s x. f differentiable (at x)
==> f differentiable (at x within s)`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_AT_WITHIN]);;
let DIFFERENTIABLE_WITHIN_OPEN = prove
(`!f a s.
a IN s /\ open s
==> (f differentiable (at a within s) <=> (f differentiable (at a)))`,
SIMP_TAC[differentiable; HAS_DERIVATIVE_WITHIN_OPEN]);;
let DIFFERENTIABLE_AT_IMP_DIFFERENTIABLE_ON = prove
(`!f s. (!x. x IN s ==> f differentiable at x) ==> f differentiable_on s`,
REWRITE_TAC[differentiable_on] THEN MESON_TAC[DIFFERENTIABLE_AT_WITHIN]);;
let DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT = prove
(`!f s. open s ==> (f differentiable_on s <=>
!x. x IN s ==> f differentiable at x)`,
SIMP_TAC[differentiable_on; DIFFERENTIABLE_WITHIN_OPEN]);;
let DIFFERENTIABLE_TRANSFORM_WITHIN = prove
(`!f g x s d.
&0 < d /\ x IN s /\
(!x'. x' IN s /\ dist (x',x) < d ==> f x' = g x') /\
f differentiable (at x within s)
==> g differentiable (at x within s)`,
REWRITE_TAC[differentiable] THEN
MESON_TAC[HAS_DERIVATIVE_TRANSFORM_WITHIN]);;
let DIFFERENTIABLE_TRANSFORM_AT = prove
(`!f g x d.
&0 < d /\
(!x'. dist (x',x) < d ==> f x' = g x') /\
f differentiable at x
==> g differentiable at x`,
REWRITE_TAC[differentiable] THEN
MESON_TAC[HAS_DERIVATIVE_TRANSFORM_AT]);;
(* ------------------------------------------------------------------------- *)
(* Frechet derivative and Jacobian matrix. *)
(* ------------------------------------------------------------------------- *)
let frechet_derivative = new_definition
`frechet_derivative f net = @f'. (f has_derivative f') net`;;
let FRECHET_DERIVATIVE_WORKS = prove
(`!f net. f differentiable net <=>
(f has_derivative (frechet_derivative f net)) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[frechet_derivative] THEN
CONV_TAC(RAND_CONV SELECT_CONV) THEN REWRITE_TAC[differentiable]);;
let LINEAR_FRECHET_DERIVATIVE = prove
(`!f net. f differentiable net ==> linear(frechet_derivative f net)`,
SIMP_TAC[FRECHET_DERIVATIVE_WORKS; has_derivative]);;
let jacobian = new_definition
`jacobian f net = matrix(frechet_derivative f net)`;;
let JACOBIAN_WORKS = prove
(`!f net. f differentiable net <=>
(f has_derivative (\h. jacobian f net ** h)) net`,
REPEAT GEN_TAC THEN EQ_TAC THENL [ALL_TAC; MESON_TAC[differentiable]] THEN
REWRITE_TAC[FRECHET_DERIVATIVE_WORKS] THEN
SIMP_TAC[jacobian; MATRIX_WORKS; has_derivative] THEN SIMP_TAC[ETA_AX]);;
(* ------------------------------------------------------------------------- *)
(* Differentiability implies continuity. *)
(* ------------------------------------------------------------------------- *)
let LIM_MUL_NORM_WITHIN = prove
(`!f a s. (f --> vec 0) (at a within s)
==> ((\x. norm(x - a) % f(x)) --> vec 0) (at a within s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[LIM_WITHIN] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `e:real` THEN
ASM_CASES_TAC `&0 < e` THEN ASM_REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `min d (&1)` THEN ASM_REWRITE_TAC[REAL_LT_MIN; REAL_LT_01] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[NORM_MUL; REAL_ABS_NORM] THEN
GEN_REWRITE_TAC RAND_CONV [GSYM REAL_MUL_LID] THEN
ASM_SIMP_TAC[REAL_LT_MUL2; NORM_POS_LE]);;
let DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN = prove
(`!f:real^M->real^N s.
f differentiable (at x within s) ==> f continuous (at x within s)`,
REWRITE_TAC[differentiable; has_derivative_within; CONTINUOUS_WITHIN] THEN
REPEAT GEN_TAC THEN
DISCH_THEN(X_CHOOSE_THEN `f':real^M->real^N` MP_TAC) THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o MATCH_MP LIM_MUL_NORM_WITHIN) THEN
SUBGOAL_THEN
`((f':real^M->real^N) o (\y. y - x)) continuous (at x within s)`
MP_TAC THENL
[MATCH_MP_TAC CONTINUOUS_WITHIN_COMPOSE THEN
ASM_SIMP_TAC[LINEAR_CONTINUOUS_WITHIN] THEN
SIMP_TAC[CONTINUOUS_SUB; CONTINUOUS_CONST; CONTINUOUS_WITHIN_ID];
ALL_TAC] THEN
REWRITE_TAC[CONTINUOUS_WITHIN; o_DEF] THEN
ASM_REWRITE_TAC[VECTOR_MUL_ASSOC; IMP_IMP; IN_UNIV] THEN
DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD) THEN
SIMP_TAC[LIM_WITHIN; GSYM DIST_NZ; REAL_MUL_RINV; NORM_EQ_0;
VECTOR_ARITH `(x - y = vec 0) <=> (x = y)`;
VECTOR_MUL_LID; VECTOR_SUB_REFL] THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP LINEAR_0) THEN
REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN
REWRITE_TAC[VECTOR_ARITH `(a + b - (c + a)) - (vec 0 + vec 0) = b - c`]);;
let DIFFERENTIABLE_IMP_CONTINUOUS_AT = prove
(`!f:real^M->real^N x. f differentiable (at x) ==> f continuous (at x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN]);;
let DIFFERENTIABLE_IMP_CONTINUOUS_ON = prove
(`!f:real^M->real^N s. f differentiable_on s ==> f continuous_on s`,
SIMP_TAC[differentiable_on; CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN;
DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN]);;
let HAS_DERIVATIVE_WITHIN_SUBSET = prove
(`!f s t x. (f has_derivative f') (at x within s) /\ t SUBSET s
==> (f has_derivative f') (at x within t)`,
REWRITE_TAC[has_derivative_within] THEN MESON_TAC[LIM_WITHIN_SUBSET]);;
let DIFFERENTIABLE_WITHIN_SUBSET = prove
(`!f:real^M->real^N s t.
f differentiable (at x within t) /\ s SUBSET t
==> f differentiable (at x within s)`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_WITHIN_SUBSET]);;
let DIFFERENTIABLE_ON_SUBSET = prove
(`!f:real^M->real^N s t.
f differentiable_on t /\ s SUBSET t ==> f differentiable_on s`,
REWRITE_TAC[differentiable_on] THEN
MESON_TAC[SUBSET; DIFFERENTIABLE_WITHIN_SUBSET]);;
let DIFFERENTIABLE_ON_EMPTY = prove
(`!f. f differentiable_on {}`,
REWRITE_TAC[differentiable_on; NOT_IN_EMPTY]);;
(* ------------------------------------------------------------------------- *)
(* Several results are easier using a "multiplied-out" variant. *)
(* (I got this idea from Dieudonne's proof of the chain rule). *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_WITHIN_ALT = prove
(`!f:real^M->real^N f' s x.
(f has_derivative f') (at x within s) <=>
linear f' /\
!e. &0 < e
==> ?d. &0 < d /\
!y. y IN s /\ norm(y - x) < d
==> norm(f(y) - f(x) - f'(y - x)) <=
e * norm(y - x)`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_derivative_within; LIM_WITHIN] THEN
ASM_REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN
ASM_CASES_TAC `linear(f':real^M->real^N)` THEN
ASM_REWRITE_TAC[NORM_MUL; REAL_ABS_INV; REAL_ABS_NORM] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [REAL_MUL_SYM] THEN
SIMP_TAC[GSYM real_div; REAL_LT_LDIV_EQ] THEN
REWRITE_TAC[VECTOR_ARITH `a - (b + c) = a - b - c :real^M`] THEN
EQ_TAC THEN DISCH_TAC THEN X_GEN_TAC `e:real` THEN DISCH_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN DISCH_TAC THEN
ASM_CASES_TAC `&0 < norm(y - x :real^M)` THENL
[ASM_SIMP_TAC[REAL_LT_IMP_LE]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE RAND_CONV [NORM_POS_LT]) THEN
ASM_SIMP_TAC[VECTOR_SUB_EQ; VECTOR_SUB_REFL; NORM_0; REAL_MUL_RZERO;
VECTOR_ARITH `vec 0 - x = --x`; NORM_NEG] THEN
ASM_MESON_TAC[LINEAR_0; NORM_0; REAL_LE_REFL];
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN STRIP_TAC THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `e / &2 * norm(y - x :real^M)` THEN
ASM_SIMP_TAC[REAL_LT_RMUL_EQ; REAL_LT_LDIV_EQ; REAL_OF_NUM_LT; ARITH] THEN
UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC]);;
let HAS_DERIVATIVE_AT_ALT = prove
(`!f:real^M->real^N f' x.
(f has_derivative f') (at x) <=>
linear f' /\
!e. &0 < e
==> ?d. &0 < d /\
!y. norm(y - x) < d
==> norm(f(y) - f(x) - f'(y - x)) <= e * norm(y - x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[HAS_DERIVATIVE_WITHIN_ALT; IN_UNIV]);;
(* ------------------------------------------------------------------------- *)
(* The chain rule. *)
(* ------------------------------------------------------------------------- *)
let DIFF_CHAIN_WITHIN = prove
(`!f:real^M->real^N g:real^N->real^P f' g' x s.
(f has_derivative f') (at x within s) /\
(g has_derivative g') (at (f x) within (IMAGE f s))
==> ((g o f) has_derivative (g' o f'))(at x within s)`,
REPEAT GEN_TAC THEN SIMP_TAC[HAS_DERIVATIVE_WITHIN_ALT; LINEAR_COMPOSE] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
FIRST_ASSUM(X_CHOOSE_TAC `B1:real` o MATCH_MP LINEAR_BOUNDED_POS) THEN
DISCH_THEN(fun th -> X_GEN_TAC `e:real` THEN DISCH_TAC THEN MP_TAC th) THEN
DISCH_THEN(CONJUNCTS_THEN2
(fun th -> ASSUME_TAC th THEN X_CHOOSE_TAC `B2:real` (MATCH_MP
LINEAR_BOUNDED_POS th)) MP_TAC) THEN
FIRST_X_ASSUM(fun th -> MP_TAC th THEN MP_TAC(SPEC `e / &2 / B2` th)) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN DISCH_TAC THEN
DISCH_THEN(MP_TAC o SPEC `e / &2 / (&1 + B1)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; REAL_LT_ADD] THEN
DISCH_THEN(X_CHOOSE_THEN `de:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `&1`) THEN
REWRITE_TAC[REAL_LT_01; REAL_MUL_LID] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`d1:real`; `d2:real`] REAL_DOWN2) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_ADD; REAL_LT_01] THEN
DISCH_THEN(X_CHOOSE_THEN `d0:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`d0:real`; `de / (B1 + &1)`] REAL_DOWN2) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_ADD; REAL_LT_01] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN
DISCH_TAC THEN UNDISCH_TAC
`!y. y IN s /\ norm(y - x) < d2
==> norm ((f:real^M->real^N) y - f x - f'(y - x)) <= norm(y - x)` THEN
DISCH_THEN(MP_TAC o SPEC `y:real^M`) THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_LT_TRANS]; DISCH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_LT_TRANS]; DISCH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `(f:real^M->real^N) y`) THEN ANTS_TAC THENL
[CONJ_TAC THENL [ASM_MESON_TAC[IN_IMAGE]; ALL_TAC] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN EXISTS_TAC
`norm(f'(y - x)) + norm((f:real^M->real^N) y - f x - f'(y - x))` THEN
REWRITE_TAC[NORM_TRIANGLE_SUB] THEN
MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `B1 * norm(y - x) + norm(y - x :real^M)` THEN
ASM_SIMP_TAC[REAL_LE_ADD2] THEN
REWRITE_TAC[REAL_ARITH `a * x + x = x * (a + &1)`] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_ADD; REAL_LT_01] THEN
ASM_MESON_TAC[REAL_LT_TRANS];
DISCH_TAC] THEN
REWRITE_TAC[o_THM] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `norm((g:real^N->real^P)(f(y:real^M)) - g(f x) - g'(f y - f x)) +
norm((g(f y) - g(f x) - g'(f'(y - x))) -
(g(f y) - g(f x) - g'(f y - f x)))` THEN
REWRITE_TAC[NORM_TRIANGLE_SUB] THEN
REWRITE_TAC[VECTOR_ARITH `(a - b - c1) - (a - b - c2) = c2 - c1:real^M`] THEN
ASM_SIMP_TAC[GSYM LINEAR_SUB] THEN
FIRST_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`a <= d ==> b <= ee - d ==> a + b <= ee`)) THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `B2 * norm((f:real^M->real^N) y - f x - f'(y - x))` THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `B2 * e / &2 / B2 * norm(y - x :real^M)` THEN
ASM_SIMP_TAC[REAL_LE_LMUL; REAL_LT_IMP_LE] THEN REWRITE_TAC[real_div] THEN
ONCE_REWRITE_TAC[REAL_ARITH
`b * ((e * h) * b') * x <= e * x - d <=>
d <= e * (&1 - h * b' * b) * x`] THEN
ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_LE_LMUL_EQ; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN REWRITE_TAC[GSYM real_div] THEN
ASM_SIMP_TAC[REAL_LE_LDIV_EQ; REAL_LT_ADD; REAL_LT_01] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
`norm(f'(y - x)) + norm((f:real^M->real^N) y - f x - f'(y - x))` THEN
REWRITE_TAC[NORM_TRIANGLE_SUB] THEN MATCH_MP_TAC(REAL_ARITH
`u <= x * b /\ v <= b ==> u + v <= b * (&1 + x)`) THEN
ASM_REWRITE_TAC[]);;
let DIFF_CHAIN_AT = prove
(`!f:real^M->real^N g:real^N->real^P f' g' x.
(f has_derivative f') (at x) /\
(g has_derivative g') (at (f x))
==> ((g o f) has_derivative (g' o f')) (at x)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
ASM_MESON_TAC[DIFF_CHAIN_WITHIN; LIM_WITHIN_SUBSET; SUBSET_UNIV;
HAS_DERIVATIVE_WITHIN_SUBSET]);;
(* ------------------------------------------------------------------------- *)
(* Composition rules stated just for differentiability. *)
(* ------------------------------------------------------------------------- *)
let DIFFERENTIABLE_LINEAR = prove
(`!net f:real^M->real^N. linear f ==> f differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_LINEAR]);;
let DIFFERENTIABLE_CONST = prove
(`!c net. (\z. c) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_CONST]);;
let DIFFERENTIABLE_ID = prove
(`!net. (\z. z) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_ID]);;
let DIFFERENTIABLE_LIFT_COMPONENT = prove
(`!net:(real^N)net. (\x. lift(x$i)) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_LIFT_COMPONENT]);;
let DIFFERENTIABLE_CMUL = prove
(`!net f c. f differentiable net ==> (\x. c % f(x)) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_CMUL]);;
let DIFFERENTIABLE_NEG = prove
(`!f net. f differentiable net ==> (\z. --(f z)) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_NEG]);;
let DIFFERENTIABLE_ADD = prove
(`!f g net.
f differentiable net /\
g differentiable net
==> (\z. f z + g z) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_ADD]);;
let DIFFERENTIABLE_SUB = prove
(`!f g net.
f differentiable net /\
g differentiable net
==> (\z. f z - g z) differentiable net`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_SUB]);;
let DIFFERENTIABLE_VSUM = prove
(`!f net s.
FINITE s /\
(!a. a IN s ==> (f a) differentiable net)
==> (\x. vsum s (\a. f a x)) differentiable net`,
REPEAT GEN_TAC THEN REWRITE_TAC[differentiable] THEN
GEN_REWRITE_TAC (LAND_CONV o TOP_DEPTH_CONV)
[RIGHT_IMP_EXISTS_THM; SKOLEM_THM; RIGHT_AND_EXISTS_THM] THEN
DISCH_THEN(CHOOSE_THEN (MP_TAC o MATCH_MP HAS_DERIVATIVE_VSUM)) THEN
MESON_TAC[]);;
let DIFFERENTIABLE_VSUM_NUMSEG = prove
(`!f net m n.
FINITE s /\
(!i. m <= i /\ i <= n ==> (f i) differentiable net)
==> (\x. vsum (m..n) (\a. f a x)) differentiable net`,
SIMP_TAC[DIFFERENTIABLE_VSUM; FINITE_NUMSEG; IN_NUMSEG]);;
let DIFFERENTIABLE_CHAIN_AT = prove
(`!f g x.
f differentiable (at x) /\
g differentiable (at(f x))
==> (g o f) differentiable (at x)`,
REWRITE_TAC[differentiable] THEN MESON_TAC[DIFF_CHAIN_AT]);;
let DIFFERENTIABLE_CHAIN_WITHIN = prove
(`!f g x s.
f differentiable (at x within s) /\
g differentiable (at(f x) within IMAGE f s)
==> (g o f) differentiable (at x within s)`,
REWRITE_TAC[differentiable] THEN MESON_TAC[DIFF_CHAIN_WITHIN]);;
let DIFFERENTIABLE_COMPONENTWISE_WITHIN = prove
(`!f:real^M->real^N a s.
f differentiable (at a within s) <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> (\x. lift(f(x)$i)) differentiable (at a within s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[differentiable] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV)
[HAS_DERIVATIVE_COMPONENTWISE_WITHIN] THEN
REWRITE_TAC[RIGHT_IMP_EXISTS_THM; SKOLEM_THM] THEN EQ_TAC THENL
[DISCH_THEN(X_CHOOSE_TAC `g':real^M->real^N`) THEN
EXISTS_TAC `\i x. lift((g':real^M->real^N) x$i)` THEN ASM_REWRITE_TAC[];
DISCH_THEN(X_CHOOSE_TAC `g':num->real^M->real^1`) THEN
EXISTS_TAC `(\x. lambda i. drop((g':num->real^M->real^1) i x))
:real^M->real^N` THEN
ASM_SIMP_TAC[LAMBDA_BETA; LIFT_DROP; ETA_AX]]);;
let DIFFERENTIABLE_COMPONENTWISE_AT = prove
(`!f:real^M->real^N a.
f differentiable (at a) <=>
!i. 1 <= i /\ i <= dimindex(:N)
==> (\x. lift(f(x)$i)) differentiable (at a)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
MATCH_ACCEPT_TAC DIFFERENTIABLE_COMPONENTWISE_WITHIN);;
(* ------------------------------------------------------------------------- *)
(* Similarly for "differentiable_on". *)
(* ------------------------------------------------------------------------- *)
let DIFFERENTIABLE_ON_LINEAR = prove
(`!f:real^M->real^N s. linear f ==> f differentiable_on s`,
SIMP_TAC[differentiable_on; DIFFERENTIABLE_LINEAR]);;
let DIFFERENTIABLE_ON_CONST = prove
(`!s c. (\z. c) differentiable_on s`,
REWRITE_TAC[differentiable_on; DIFFERENTIABLE_CONST]);;
let DIFFERENTIABLE_ON_ID = prove
(`!s. (\z. z) differentiable_on s`,
REWRITE_TAC[differentiable_on; DIFFERENTIABLE_ID]);;
let DIFFERENTIABLE_ON_COMPOSE = prove
(`!f g s. f differentiable_on s /\ g differentiable_on (IMAGE f s)
==> (g o f) differentiable_on s`,
SIMP_TAC[differentiable_on; FORALL_IN_IMAGE] THEN
MESON_TAC[DIFFERENTIABLE_CHAIN_WITHIN]);;
let DIFFERENTIABLE_ON_NEG = prove
(`!f s. f differentiable_on s ==> (\z. --(f z)) differentiable_on s`,
SIMP_TAC[differentiable_on; DIFFERENTIABLE_NEG]);;
let DIFFERENTIABLE_ON_ADD = prove
(`!f g s.
f differentiable_on s /\ g differentiable_on s
==> (\z. f z + g z) differentiable_on s`,
SIMP_TAC[differentiable_on; DIFFERENTIABLE_ADD]);;
let DIFFERENTIABLE_ON_SUB = prove
(`!f g s.
f differentiable_on s /\ g differentiable_on s
==> (\z. f z - g z) differentiable_on s`,
SIMP_TAC[differentiable_on; DIFFERENTIABLE_SUB]);;
(* ------------------------------------------------------------------------- *)
(* Uniqueness of derivative. *)
(* *)
(* The general result is a bit messy because we need approachability of the *)
(* limit point from any direction. But OK for nontrivial intervals etc. *)
(* ------------------------------------------------------------------------- *)
let FRECHET_DERIVATIVE_UNIQUE_WITHIN = prove
(`!f:real^M->real^N f' f'' x s.
(f has_derivative f') (at x within s) /\
(f has_derivative f'') (at x within s) /\
(!i e. 1 <= i /\ i <= dimindex(:M) /\ &0 < e
==> ?d. &0 < abs(d) /\ abs(d) < e /\ (x + d % basis i) IN s)
==> f' = f''`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_derivative] THEN
ONCE_REWRITE_TAC[CONJ_ASSOC] THEN
DISCH_THEN(CONJUNCTS_THEN2 MP_TAC STRIP_ASSUME_TAC) THEN
SUBGOAL_THEN `(x:real^M) limit_point_of s` ASSUME_TAC THENL
[REWRITE_TAC[LIMPT_APPROACHABLE] THEN X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`1`; `e:real`]) THEN
ASM_REWRITE_TAC[DIMINDEX_GE_1; LE_REFL] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `(x:real^M) + d % basis 1` THEN
ASM_REWRITE_TAC[dist] THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
ASM_SIMP_TAC[VECTOR_ADD_SUB; NORM_MUL; NORM_BASIS; DIMINDEX_GE_1; LE_REFL;
VECTOR_MUL_EQ_0; REAL_MUL_RID; DE_MORGAN_THM; REAL_ABS_NZ;
BASIS_NONZERO];
ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_SUB) THEN
SUBGOAL_THEN `netlimit(at x within s) = x:real^M` SUBST_ALL_TAC THENL
[ASM_MESON_TAC[NETLIMIT_WITHIN; TRIVIAL_LIMIT_WITHIN]; ALL_TAC] THEN
REWRITE_TAC[GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN
REWRITE_TAC[VECTOR_ARITH
`fx - (fa + f'') - (fx - (fa + f')):real^M = f' - f''`] THEN
DISCH_TAC THEN MATCH_MP_TAC LINEAR_EQ_STDBASIS THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `i:num` THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
GEN_REWRITE_TAC I [TAUT `p = ~ ~p`] THEN
PURE_REWRITE_TAC[GSYM NORM_POS_LT] THEN DISCH_TAC THEN ABBREV_TAC
`e = norm((f':real^M->real^N) (basis i) - f''(basis i :real^M))` THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN]) THEN
DISCH_THEN(MP_TAC o SPEC `e:real`) THEN
ASM_REWRITE_TAC[dist; VECTOR_SUB_RZERO] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`i:num`; `d:real`]) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `c:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `(x:real^M) + c % basis i`) THEN
ASM_REWRITE_TAC[VECTOR_ADD_SUB; NORM_MUL] THEN
ASM_SIMP_TAC[NORM_BASIS; REAL_MUL_RID] THEN
ASM_SIMP_TAC[LINEAR_CMUL; GSYM VECTOR_SUB_LDISTRIB; NORM_MUL] THEN
REWRITE_TAC[REAL_ABS_INV; REAL_ABS_ABS] THEN
ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; REAL_MUL_ASSOC] THEN
REWRITE_TAC[REAL_MUL_LID; REAL_LT_REFL]);;
let FRECHET_DERIVATIVE_UNIQUE_AT = prove
(`!f:real^M->real^N f' f'' x.
(f has_derivative f') (at x) /\ (f has_derivative f'') (at x)
==> f' = f''`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FRECHET_DERIVATIVE_UNIQUE_WITHIN THEN
MAP_EVERY EXISTS_TAC
[`f:real^M->real^N`; `x:real^M`; `(:real^M)`] THEN
ASM_REWRITE_TAC[IN_UNIV; WITHIN_UNIV] THEN
MESON_TAC[REAL_ARITH `&0 < e ==> &0 < abs(e / &2) /\ abs(e / &2) < e`]);;
let HAS_FRECHET_DERIVATIVE_UNIQUE_AT = prove
(`!f:real^M->real^N f' x.
(f has_derivative f') (at x)
==> frechet_derivative f (at x) = f'`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FRECHET_DERIVATIVE_UNIQUE_AT THEN
MAP_EVERY EXISTS_TAC [`f:real^M->real^N`; `x:real^M`] THEN
ASM_REWRITE_TAC[frechet_derivative] THEN CONV_TAC SELECT_CONV THEN
ASM_MESON_TAC[]);;
let FRECHET_DERIVATIVE_CONST_AT = prove
(`!c:real^N a:real^M. frechet_derivative (\x. c) (at a) = \h. vec 0`,
REPEAT GEN_TAC THEN MATCH_MP_TAC HAS_FRECHET_DERIVATIVE_UNIQUE_AT THEN
REWRITE_TAC[HAS_DERIVATIVE_CONST]);;
let FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL = prove
(`!f:real^M->real^N f' f'' x a b.
(!i. 1 <= i /\ i <= dimindex(:M) ==> a$i < b$i) /\
x IN interval[a,b] /\
(f has_derivative f') (at x within interval[a,b]) /\
(f has_derivative f'') (at x within interval[a,b])
==> f' = f''`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FRECHET_DERIVATIVE_UNIQUE_WITHIN THEN
MAP_EVERY EXISTS_TAC
[`f:real^M->real^N`; `x:real^M`; `interval[a:real^M,b]`] THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`i:num`; `e:real`] THEN STRIP_TAC THEN
MATCH_MP_TAC(MESON[] `(?a. P a \/ P(--a)) ==> (?a:real. P a)`) THEN
EXISTS_TAC `(min ((b:real^M)$i - (a:real^M)$i) e) / &2` THEN
REWRITE_TAC[REAL_ABS_NEG; GSYM LEFT_OR_DISTRIB] THEN
REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
[UNDISCH_TAC `&0 < e` THEN FIRST_X_ASSUM(MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
UNDISCH_TAC `(x:real^M) IN interval[a,b]` THEN REWRITE_TAC[IN_INTERVAL] THEN
DISCH_TAC THEN
ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
BASIS_COMPONENT] THEN
SUBGOAL_THEN
`!P. (!j. 1 <= j /\ j <= dimindex(:M) ==> P j) <=>
P i /\
(!j. 1 <= j /\ j <= dimindex(:M) /\ ~(j = i) ==> P j)`
(fun th -> ONCE_REWRITE_TAC[th])
THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_ADD_RID; REAL_MUL_RID] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN
UNDISCH_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let FRECHET_DERIVATIVE_UNIQUE_WITHIN_OPEN_INTERVAL = prove
(`!f:real^M->real^N f' f'' x a b.
x IN interval(a,b) /\
(f has_derivative f') (at x within interval(a,b)) /\
(f has_derivative f'') (at x within interval(a,b))
==> f' = f''`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC FRECHET_DERIVATIVE_UNIQUE_WITHIN THEN
MAP_EVERY EXISTS_TAC
[`f:real^M->real^N`; `x:real^M`; `interval(a:real^M,b)`] THEN
ASM_REWRITE_TAC[] THEN
MAP_EVERY X_GEN_TAC [`i:num`; `e:real`] THEN STRIP_TAC THEN
MATCH_MP_TAC(MESON[] `(?a. P a \/ P(--a)) ==> (?a:real. P a)`) THEN
EXISTS_TAC `(min ((b:real^M)$i - (a:real^M)$i) e) / &3` THEN
REWRITE_TAC[REAL_ABS_NEG; GSYM LEFT_OR_DISTRIB] THEN
REWRITE_TAC[CONJ_ASSOC] THEN CONJ_TAC THENL
[UNDISCH_TAC `&0 < e` THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERVAL]) THEN
DISCH_THEN(MP_TAC o SPEC `i:num`) THEN
ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
UNDISCH_TAC `(x:real^M) IN interval(a,b)` THEN REWRITE_TAC[IN_INTERVAL] THEN
DISCH_TAC THEN
ASM_SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_MUL_COMPONENT;
BASIS_COMPONENT] THEN
SUBGOAL_THEN
`!P. (!j. 1 <= j /\ j <= dimindex(:M) ==> P j) <=>
P i /\
(!j. 1 <= j /\ j <= dimindex(:M) /\ ~(j = i) ==> P j)`
(fun th -> ONCE_REWRITE_TAC[th])
THENL [ASM_MESON_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[REAL_MUL_RZERO; REAL_ADD_RID; REAL_MUL_RID] THEN
REPEAT(FIRST_X_ASSUM(MP_TAC o SPEC `i:num`)) THEN
UNDISCH_TAC `&0 < e` THEN ASM_REWRITE_TAC[] THEN REAL_ARITH_TAC);;
let FRECHET_DERIVATIVE_AT = prove
(`!f:real^M->real^N f' x.
(f has_derivative f') (at x) ==> (f' = frechet_derivative f (at x))`,
MESON_TAC[has_derivative; FRECHET_DERIVATIVE_WORKS;
differentiable; FRECHET_DERIVATIVE_UNIQUE_AT]);;
let FRECHET_DERIVATIVE_WITHIN_CLOSED_INTERVAL = prove
(`!f:real^M->real^N f' x a b.
(!i. 1 <= i /\ i <= dimindex(:M) ==> a$i < b$i) /\
x IN interval[a,b] /\
(f has_derivative f') (at x within interval[a,b])
==> frechet_derivative f (at x within interval[a,b]) = f'`,
ASM_MESON_TAC[has_derivative; FRECHET_DERIVATIVE_WORKS;
differentiable; FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL]);;
(* ------------------------------------------------------------------------- *)
(* Component of the differential must be zero if it exists at a local *)
(* maximum or minimum for that corresponding component. Start with slightly *)
(* sharper forms that fix the sign of the derivative on the boundary. *)
(* ------------------------------------------------------------------------- *)
let DIFFERENTIAL_COMPONENT_POS_AT_MINIMUM = prove
(`!f:real^M->real^N f' x s k e.
1 <= k /\ k <= dimindex(:N) /\
x IN s /\ convex s /\ (f has_derivative f') (at x within s) /\
&0 < e /\ (!w. w IN s INTER ball(x,e) ==> (f x)$k <= (f w)$k)
==> !y. y IN s ==> &0 <= (f'(y - x))$k`,
REWRITE_TAC[has_derivative_within] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `y:real^M = x` THENL
[ASM_MESON_TAC[VECTOR_SUB_REFL; LINEAR_0; VEC_COMPONENT; REAL_LE_REFL];
ALL_TAC] THEN
ONCE_REWRITE_TAC[GSYM REAL_NOT_LT] THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [LIM_WITHIN]) THEN
DISCH_THEN(MP_TAC o SPEC
`--((f':real^M->real^N)(y - x)$k) / norm(y - x)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; VECTOR_SUB_EQ;
NOT_EXISTS_THM; REAL_ARITH `&0 < --x <=> x < &0`] THEN
X_GEN_TAC `d:real` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ABBREV_TAC `de = min (&1) ((min d e) / &2 / norm(y - x:real^M))` THEN
DISCH_THEN(MP_TAC o SPEC `x + de % (y - x):real^M`) THEN
REWRITE_TAC[dist; VECTOR_ADD_SUB; NOT_IMP; GSYM CONJ_ASSOC] THEN
SUBGOAL_THEN `norm(de % (y - x):real^M) < min d e` MP_TAC THENL
[ASM_SIMP_TAC[NORM_MUL; GSYM REAL_LT_RDIV_EQ;
NORM_POS_LT; VECTOR_SUB_EQ] THEN
EXPAND_TAC "de" THEN MATCH_MP_TAC(REAL_ARITH
`&0 < de / x ==> abs(min (&1) (de / &2 / x)) < de / x`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_LT_MIN; NORM_POS_LT; VECTOR_SUB_EQ];
REWRITE_TAC[REAL_LT_MIN] THEN STRIP_TAC] THEN
SUBGOAL_THEN `&0 < de /\ de <= &1` STRIP_ASSUME_TAC THENL
[EXPAND_TAC "de" THEN CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN
ASM_SIMP_TAC[REAL_LT_MIN; REAL_LT_01; REAL_HALF; REAL_LT_DIV;
NORM_POS_LT; VECTOR_SUB_EQ];
ALL_TAC] THEN
MATCH_MP_TAC(TAUT `a /\ (a ==> b) ==> a /\ b`) THEN CONJ_TAC THENL
[REWRITE_TAC[VECTOR_ARITH
`x + a % (y - x):real^N = (&1 - a) % x + a % y`] THEN
MATCH_MP_TAC IN_CONVEX_SET THEN ASM_SIMP_TAC[REAL_LT_IMP_LE];
DISCH_TAC] THEN
ASM_REWRITE_TAC[] THEN REWRITE_TAC[NORM_MUL] THEN
ASM_SIMP_TAC[REAL_LT_MUL; REAL_ARITH `&0 < x ==> &0 < abs x`;
NORM_POS_LT; VECTOR_SUB_EQ; VECTOR_SUB_RZERO] THEN
MATCH_MP_TAC(NORM_ARITH
`abs(y$k) <= norm(y) /\ ~(abs(y$k) < e) ==> ~(norm y < e)`) THEN
ASM_SIMP_TAC[COMPONENT_LE_NORM] THEN REWRITE_TAC[VECTOR_MUL_COMPONENT] THEN
REWRITE_TAC[REAL_ABS_INV; REAL_ABS_MUL; REAL_ABS_NORM; REAL_ABS_ABS] THEN
REWRITE_TAC[REAL_NOT_LT; REAL_INV_MUL; REAL_ARITH
`d <= (a * inv b) * c <=> d <= (c * a) / b`] THEN
ASM_SIMP_TAC[REAL_LE_DIV2_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_RDIV_EQ; VECTOR_SUB_COMPONENT;
VECTOR_ADD_COMPONENT; REAL_ARITH `&0 < x ==> &0 < abs x`] THEN
MATCH_MP_TAC(REAL_ARITH
`fx <= fy /\ a = --b /\ b < &0 ==> a <= abs(fy - (fx + b))`) THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_CMUL th]) THEN
ASM_SIMP_TAC[real_abs; VECTOR_MUL_COMPONENT; REAL_LT_IMP_LE] THEN
ONCE_REWRITE_TAC[REAL_ARITH `x * y < &0 <=> &0 < x * --y`] THEN
ASM_SIMP_TAC[REAL_LT_MUL_EQ] THEN
CONJ_TAC THENL [FIRST_X_ASSUM MATCH_MP_TAC; ASM_REAL_ARITH_TAC] THEN
ASM_REWRITE_TAC[IN_INTER; IN_BALL; NORM_ARITH
`dist(x:real^M,x + e) = norm e`]);;
let DIFFERENTIAL_COMPONENT_NEG_AT_MAXIMUM = prove
(`!f:real^M->real^N f' x s k e.
1 <= k /\ k <= dimindex(:N) /\
x IN s /\ convex s /\ (f has_derivative f') (at x within s) /\
&0 < e /\ (!w. w IN s INTER ball(x,e) ==> (f w)$k <= (f x)$k)
==> !y. y IN s ==> (f'(y - x))$k <= &0`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`\x. --((f:real^M->real^N) x)`; `\x. --((f':real^M->real^N) x)`;
`x:real^M`; `s:real^M->bool`; `k:num`; `e:real`]
DIFFERENTIAL_COMPONENT_POS_AT_MINIMUM) THEN
ASM_SIMP_TAC[HAS_DERIVATIVE_NEG] THEN
ASM_SIMP_TAC[REAL_LE_NEG2; VECTOR_NEG_COMPONENT; REAL_NEG_GE0]);;
let DROP_DIFFERENTIAL_POS_AT_MINIMUM = prove
(`!f:real^N->real^1 f' x s e.
x IN s /\ convex s /\ (f has_derivative f') (at x within s) /\
&0 < e /\ (!w. w IN s INTER ball(x,e) ==> drop(f x) <= drop(f w))
==> !y. y IN s ==> &0 <= drop(f'(y - x))`,
REPEAT GEN_TAC THEN REWRITE_TAC[drop] THEN STRIP_TAC THEN
MATCH_MP_TAC DIFFERENTIAL_COMPONENT_POS_AT_MINIMUM THEN
MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `e:real`] THEN
ASM_REWRITE_TAC[DIMINDEX_1; LE_REFL]);;
let DROP_DIFFERENTIAL_NEG_AT_MAXIMUM = prove
(`!f:real^N->real^1 f' x s e.
x IN s /\ convex s /\ (f has_derivative f') (at x within s) /\
&0 < e /\ (!w. w IN s INTER ball(x,e) ==> drop(f w) <= drop(f x))
==> !y. y IN s ==> drop(f'(y - x)) <= &0`,
REPEAT GEN_TAC THEN REWRITE_TAC[drop] THEN STRIP_TAC THEN
MATCH_MP_TAC DIFFERENTIAL_COMPONENT_NEG_AT_MAXIMUM THEN
MAP_EVERY EXISTS_TAC [`f:real^N->real^1`; `e:real`] THEN
ASM_REWRITE_TAC[DIMINDEX_1; LE_REFL]);;
let DIFFERENTIAL_COMPONENT_ZERO_AT_MAXMIN = prove
(`!f:real^M->real^N f' x s k.
1 <= k /\ k <= dimindex(:N) /\
x IN s /\ open s /\ (f has_derivative f') (at x) /\
((!w. w IN s ==> (f w)$k <= (f x)$k) \/
(!w. w IN s ==> (f x)$k <= (f w)$k))
==> !h. (f' h)$k = &0`,
REPEAT GEN_TAC THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_CBALL]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[SUBSET] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM DISJ_CASES_TAC THENL
[MP_TAC(ISPECL [`f:real^M->real^N`; `f':real^M->real^N`;
`x:real^M`; `cball(x:real^M,e)`; `k:num`; `e:real`]
DIFFERENTIAL_COMPONENT_NEG_AT_MAXIMUM);
MP_TAC(ISPECL [`f:real^M->real^N`; `f':real^M->real^N`;
`x:real^M`; `cball(x:real^M,e)`; `k:num`; `e:real`]
DIFFERENTIAL_COMPONENT_POS_AT_MINIMUM)] THEN
ASM_SIMP_TAC[HAS_DERIVATIVE_AT_WITHIN; CENTRE_IN_CBALL;
CONVEX_CBALL; REAL_LT_IMP_LE; IN_INTER] THEN
DISCH_THEN(LABEL_TAC "*") THEN X_GEN_TAC `h:real^M` THEN
FIRST_X_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [has_derivative_at]) THEN
(ASM_CASES_TAC `h:real^M = vec 0` THENL
[ASM_MESON_TAC[LINEAR_0; VEC_COMPONENT]; ALL_TAC]) THEN
REMOVE_THEN "*" (fun th ->
MP_TAC(SPEC `x + e / norm h % h:real^M` th) THEN
MP_TAC(SPEC `x - e / norm h % h:real^M` th)) THEN
REWRITE_TAC[IN_CBALL; NORM_ARITH
`dist(x:real^N,x - e) = norm e /\ dist(x:real^N,x + e) = norm e`] THEN
REWRITE_TAC[NORM_MUL; REAL_ABS_DIV; REAL_ABS_NORM] THEN
ASM_SIMP_TAC[real_abs; REAL_DIV_RMUL; NORM_EQ_0; REAL_LT_IMP_LE;
REAL_LE_REFL] THEN
REWRITE_TAC[VECTOR_ARITH `x - e - x:real^N = --e /\ (x + e) - x = e`] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_NEG th]) THEN
REWRITE_TAC[IMP_IMP; REAL_ARITH `&0 <= --x /\ &0 <= x <=> x = &0`;
VECTOR_NEG_COMPONENT; REAL_ARITH `--x <= &0 /\ x <= &0 <=> x = &0`] THEN
DISCH_THEN(MP_TAC o AP_TERM `(*) (norm(h:real^M) / e)`) THEN
REWRITE_TAC[GSYM VECTOR_MUL_COMPONENT] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[GSYM(MATCH_MP LINEAR_CMUL th)]) THEN
REWRITE_TAC[REAL_MUL_RZERO; VECTOR_MUL_ASSOC] THEN
ASM_SIMP_TAC[REAL_FIELD `~(x = &0) /\ ~(y = &0) ==> x / y * y / x = &1`;
NORM_EQ_0; REAL_LT_IMP_NZ; VECTOR_MUL_LID]);;
let DIFFERENTIAL_ZERO_MAXMIN_COMPONENT = prove
(`!f:real^M->real^N x e k.
1 <= k /\ k <= dimindex(:N) /\ &0 < e /\
((!y. y IN ball(x,e) ==> (f y)$k <= (f x)$k) \/
(!y. y IN ball(x,e) ==> (f x)$k <= (f y)$k)) /\
f differentiable (at x)
==> (jacobian f (at x) $ k = vec 0)`,
REWRITE_TAC[JACOBIAN_WORKS] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`f:real^M->real^N`; `\h. jacobian (f:real^M->real^N) (at x) ** h`;
`x:real^M`; `ball(x:real^M,e)`; `k:num`]
DIFFERENTIAL_COMPONENT_ZERO_AT_MAXMIN) THEN
ASM_REWRITE_TAC[CENTRE_IN_BALL; OPEN_BALL] THEN
ASM_SIMP_TAC[MATRIX_VECTOR_MUL_COMPONENT; FORALL_DOT_EQ_0]);;
let DIFFERENTIAL_ZERO_MAXMIN = prove
(`!f:real^N->real^1 f' x s.
x IN s /\ open s /\ (f has_derivative f') (at x) /\
((!y. y IN s ==> drop(f y) <= drop(f x)) \/
(!y. y IN s ==> drop(f x) <= drop(f y)))
==> (f' = \v. vec 0)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`f:real^N->real^1`; `f':real^N->real^1`;
`x:real^N`; `s:real^N->bool`; `1:num`]
DIFFERENTIAL_COMPONENT_ZERO_AT_MAXMIN) THEN
ASM_REWRITE_TAC[GSYM drop; DIMINDEX_1; LE_REFL] THEN
REWRITE_TAC[GSYM LIFT_EQ; LIFT_NUM; FUN_EQ_THM; LIFT_DROP]);;
(* ------------------------------------------------------------------------- *)
(* The traditional Rolle theorem in one dimension. *)
(* ------------------------------------------------------------------------- *)
let ROLLE = prove
(`!f:real^1->real^1 f' a b.
drop a < drop b /\ (f a = f b) /\
f continuous_on interval[a,b] /\
(!x. x IN interval(a,b) ==> (f has_derivative f'(x)) (at x))
==> ?x. x IN interval(a,b) /\ (f'(x) = \v. vec 0)`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`f:real^1->real^1`; `a:real^1`; `b:real^1`]
CONTINUOUS_IVT_LOCAL_EXTREMUM) THEN
ASM_SIMP_TAC[SEGMENT_1; REAL_LT_IMP_LE] THEN
ANTS_TAC THENL [ASM_MESON_TAC[REAL_LT_REFL]; MATCH_MP_TAC MONO_EXISTS] THEN
X_GEN_TAC `c:real^1` THEN DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC DIFFERENTIAL_ZERO_MAXMIN THEN MAP_EVERY EXISTS_TAC
[`f:real^1->real^1`; `c:real^1`; `interval(a:real^1,b)`] THEN
ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET; OPEN_INTERVAL]);;
(* ------------------------------------------------------------------------- *)
(* One-dimensional mean value theorem. *)
(* ------------------------------------------------------------------------- *)
let MVT = prove
(`!f:real^1->real^1 f' a b.
drop a < drop b /\
f continuous_on interval[a,b] /\
(!x. x IN interval(a,b) ==> (f has_derivative f'(x)) (at x))
==> ?x. x IN interval(a,b) /\ (f(b) - f(a) = f'(x) (b - a))`,
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`\x. f(x) - (drop(f b - f a) / drop(b - a)) % x`;
`\k:real^1 x:real^1.
f'(k)(x) - (drop(f b - f a) / drop(b - a)) % x`;
`a:real^1`; `b:real^1`]
ROLLE) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CMUL; CONTINUOUS_ON_ID] THEN
CONJ_TAC THENL
[REWRITE_TAC[VECTOR_ARITH
`(fa - k % a = fb - k % b) <=> (fb - fa = k % (b - a))`];
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_SUB THEN
ASM_SIMP_TAC[HAS_DERIVATIVE_CMUL; HAS_DERIVATIVE_ID; ETA_AX]];
MATCH_MP_TAC MONO_EXISTS THEN SIMP_TAC[FUN_EQ_THM] THEN
X_GEN_TAC `x:real^1` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `b - a:real^1`))] THEN
SIMP_TAC[VECTOR_SUB_EQ; GSYM DROP_EQ; DROP_SUB; DROP_CMUL] THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; REAL_SUB_LT; REAL_LT_IMP_NZ]);;
let MVT_SIMPLE = prove
(`!f:real^1->real^1 f' a b.
drop a < drop b /\
(!x. x IN interval[a,b]
==> (f has_derivative f'(x)) (at x within interval[a,b]))
==> ?x. x IN interval(a,b) /\ (f(b) - f(a) = f'(x) (b - a))`,
MP_TAC MVT THEN
REPEAT(MATCH_MP_TAC MONO_FORALL THEN GEN_TAC) THEN
REPEAT STRIP_TAC THEN FIRST_X_ASSUM MATCH_MP_TAC THEN ASM_REWRITE_TAC[] THEN
CONJ_TAC THENL
[MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN
ASM_MESON_TAC[differentiable_on; differentiable];
ASM_MESON_TAC[HAS_DERIVATIVE_WITHIN_OPEN; OPEN_INTERVAL;
HAS_DERIVATIVE_WITHIN_SUBSET; INTERVAL_OPEN_SUBSET_CLOSED;
SUBSET]]);;
let MVT_VERY_SIMPLE = prove
(`!f:real^1->real^1 f' a b.
drop a <= drop b /\
(!x. x IN interval[a,b]
==> (f has_derivative f'(x)) (at x within interval[a,b]))
==> ?x. x IN interval[a,b] /\ (f(b) - f(a) = f'(x) (b - a))`,
REPEAT GEN_TAC THEN ASM_CASES_TAC `b:real^1 = a` THENL
[ASM_REWRITE_TAC[VECTOR_SUB_REFL] THEN REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `a:real^1`) THEN
REWRITE_TAC[INTERVAL_SING; IN_SING; has_derivative; UNWIND_THM2] THEN
MESON_TAC[LINEAR_0];
ASM_REWRITE_TAC[REAL_LE_LT; DROP_EQ] THEN
DISCH_THEN(MP_TAC o MATCH_MP MVT_SIMPLE) THEN
MATCH_MP_TAC MONO_EXISTS THEN
SIMP_TAC[REWRITE_RULE[SUBSET] INTERVAL_OPEN_SUBSET_CLOSED]]);;
(* ------------------------------------------------------------------------- *)
(* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *)
(* ------------------------------------------------------------------------- *)
let MVT_GENERAL = prove
(`!f:real^1->real^N f' a b.
drop a < drop b /\
f continuous_on interval[a,b] /\
(!x. x IN interval(a,b) ==> (f has_derivative f'(x)) (at x))
==> ?x. x IN interval(a,b) /\
norm(f(b) - f(a)) <= norm(f'(x) (b - a))`,
REPEAT STRIP_TAC THEN
MP_TAC(SPECL [`(lift o (\y. (f(b) - f(a)) dot y)) o (f:real^1->real^N)`;
`\x t. lift((f(b:real^1) - f(a)) dot
((f':real^1->real^1->real^N) x t))`;
`a:real^1`; `b:real^1`] MVT) THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[CONTINUOUS_ON_LIFT_DOT; CONTINUOUS_ON_COMPOSE] THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[o_DEF] THEN
MATCH_MP_TAC HAS_DERIVATIVE_LIFT_DOT THEN ASM_SIMP_TAC[ETA_AX];
ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `x:real^1` THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
ASM_REWRITE_TAC[o_THM; GSYM LIFT_SUB; GSYM DOT_RSUB; LIFT_EQ] THEN
DISCH_TAC THEN ASM_CASES_TAC `(f:real^1->real^N) b = f a` THENL
[ASM_REWRITE_TAC[VECTOR_SUB_REFL; NORM_0; NORM_POS_LE]; ALL_TAC] THEN
MATCH_MP_TAC REAL_LE_LCANCEL_IMP THEN
EXISTS_TAC `norm((f:real^1->real^N) b - f a)` THEN
ASM_REWRITE_TAC[NORM_POS_LT; VECTOR_SUB_EQ; GSYM REAL_POW_2] THEN
ASM_REWRITE_TAC[NORM_POW_2; NORM_CAUCHY_SCHWARZ]);;
(* ------------------------------------------------------------------------- *)
(* Still more general bound theorem. *)
(* ------------------------------------------------------------------------- *)
let DIFFERENTIABLE_BOUND = prove
(`!f:real^M->real^N f' s B.
convex s /\
(!x. x IN s ==> (f has_derivative f'(x)) (at x within s)) /\
(!x. x IN s ==> onorm(f'(x)) <= B)
==> !x y. x IN s /\ y IN s ==> norm(f(x) - f(y)) <= B * norm(x - y)`,
ONCE_REWRITE_TAC[NORM_SUB] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`!x y. x IN s ==> norm((f':real^M->real^M->real^N)(x) y) <= B * norm(y)`
ASSUME_TAC THENL
[ASM_MESON_TAC[ONORM; has_derivative; REAL_LE_TRANS; NORM_POS_LE;
REAL_LE_RMUL]; ALL_TAC] THEN
SUBGOAL_THEN
`!u. u IN interval[vec 0,vec 1] ==> (x + drop u % (y - x) :real^M) IN s`
ASSUME_TAC THENL
[REWRITE_TAC[IN_INTERVAL; FORALL_DIMINDEX_1; drop] THEN
SIMP_TAC[VEC_COMPONENT; LE_REFL; DIMINDEX_1] THEN
REWRITE_TAC[VECTOR_ARITH `x + u % (y - x) = (&1 - u) % x + u % y`] THEN
ASM_MESON_TAC[CONVEX_ALT];
ALL_TAC] THEN
SUBGOAL_THEN
`!u. u IN interval(vec 0,vec 1) ==> (x + drop u % (y - x) :real^M) IN s`
ASSUME_TAC THENL
[ASM_MESON_TAC[SUBSET; INTERVAL_OPEN_SUBSET_CLOSED]; ALL_TAC] THEN
MP_TAC(SPECL
[`(f:real^M->real^N) o (\u. x + drop u % (y - x))`;
`\u. (f':real^M->real^M->real^N) (x + drop u % (y - x)) o
(\u. vec 0 + drop u % (y - x))`;
`vec 0:real^1`; `vec 1:real^1`] MVT_GENERAL) THEN
REWRITE_TAC[o_THM; DROP_VEC; VECTOR_ARITH `x + &1 % (y - x) = y`;
VECTOR_MUL_LZERO; VECTOR_SUB_RZERO; VECTOR_ADD_RID] THEN
REWRITE_TAC[VECTOR_MUL_LID] THEN ANTS_TAC THENL
[ALL_TAC; ASM_MESON_TAC[VECTOR_ADD_LID; REAL_LE_TRANS]] THEN
REWRITE_TAC[REAL_LT_01] THEN CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_VMUL;
o_DEF; LIFT_DROP; CONTINUOUS_ON_ID] THEN
MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN EXISTS_TAC `s:real^M->bool` THEN
ASM_REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
ASM_MESON_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN; differentiable;
CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN];
ALL_TAC] THEN
X_GEN_TAC `a:real^1` THEN DISCH_TAC THEN
SUBGOAL_THEN `a IN interval(vec 0:real^1,vec 1) /\
open(interval(vec 0:real^1,vec 1))`
MP_TAC THENL [ASM_MESON_TAC[OPEN_INTERVAL]; ALL_TAC] THEN
DISCH_THEN(fun th -> ONCE_REWRITE_TAC[GSYM(MATCH_MP
HAS_DERIVATIVE_WITHIN_OPEN th)]) THEN
MATCH_MP_TAC DIFF_CHAIN_WITHIN THEN
ASM_SIMP_TAC[HAS_DERIVATIVE_ADD; HAS_DERIVATIVE_CONST;
HAS_DERIVATIVE_VMUL_DROP; HAS_DERIVATIVE_ID] THEN
MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN
EXISTS_TAC `s:real^M->bool` THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN
ASM_MESON_TAC[INTERVAL_OPEN_SUBSET_CLOSED; SUBSET]);;
(* ------------------------------------------------------------------------- *)
(* In particular. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_ZERO_CONSTANT = prove
(`!f:real^M->real^N s.
convex s /\
(!x. x IN s ==> (f has_derivative (\h. vec 0)) (at x within s))
==> ?c. !x. x IN s ==> f(x) = c`,
REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`f:real^M->real^N`; `(\x h. vec 0):real^M->real^M->real^N`;
`s:real^M->bool`; `&0`] DIFFERENTIABLE_BOUND) THEN
ASM_REWRITE_TAC[REAL_MUL_LZERO; ONORM_CONST; NORM_0; REAL_LE_REFL] THEN
SIMP_TAC[NORM_LE_0; VECTOR_SUB_EQ] THEN MESON_TAC[]);;
let HAS_DERIVATIVE_ZERO_UNIQUE = prove
(`!f s a c. convex s /\ a IN s /\ f a = c /\
(!x. x IN s ==> (f has_derivative (\h. vec 0)) (at x within s))
==> !x. x IN s ==> f x = c`,
MESON_TAC[HAS_DERIVATIVE_ZERO_CONSTANT]);;
let HAS_DERIVATIVE_ZERO_CONNECTED_CONSTANT = prove
(`!f:real^M->real^N s.
open s /\ connected s /\
(!x. x IN s ==> (f has_derivative (\h. vec 0)) (at x))
==> ?c. !x. x IN s ==> f(x) = c`,
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `s:real^M->bool = {}` THEN ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `a:real^M`) THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONNECTED_CLOPEN]) THEN
DISCH_THEN(MP_TAC o SPEC `{x | x IN s /\ (f:real^M->real^N) x = f a}`) THEN
ANTS_TAC THENL [ALL_TAC; ASM SET_TAC[]] THEN CONJ_TAC THENL
[SIMP_TAC[open_in; SUBSET; IN_ELIM_THM] THEN
X_GEN_TAC `x:real^M` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
DISCH_THEN(MP_TAC o SPEC `x:real^M`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `e:real` THEN
REWRITE_TAC[SUBSET; IN_BALL] THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
MP_TAC(ISPECL [`f:real^M->real^N`; `ball(x:real^M,e)`]
HAS_DERIVATIVE_ZERO_CONSTANT) THEN
REWRITE_TAC[IN_BALL; CONVEX_BALL] THEN
ASM_MESON_TAC[HAS_DERIVATIVE_AT_WITHIN; DIST_SYM; DIST_REFL];
MATCH_MP_TAC CONTINUOUS_CLOSED_IN_PREIMAGE_CONSTANT THEN
MATCH_MP_TAC DIFFERENTIABLE_IMP_CONTINUOUS_ON THEN
ASM_SIMP_TAC[DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN
ASM_MESON_TAC[differentiable]]);;
let HAS_DERIVATIVE_ZERO_CONNECTED_UNIQUE = prove
(`!f s a c. open s /\ connected s /\ a IN s /\ f a = c /\
(!x. x IN s ==> (f has_derivative (\h. vec 0)) (at x))
==> !x. x IN s ==> f x = c`,
MESON_TAC[HAS_DERIVATIVE_ZERO_CONNECTED_CONSTANT]);;
(* ------------------------------------------------------------------------- *)
(* Differentiability of inverse function (most basic form). *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE_BASIC = prove
(`!f:real^M->real^N g f' g' t y.
(f has_derivative f') (at (g y)) /\ linear g' /\ (g' o f' = I) /\
g continuous (at y) /\
open t /\ y IN t /\ (!z. z IN t ==> (f(g(z)) = z))
==> (g has_derivative g') (at y)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REPEAT STRIP_TAC THEN
FIRST_ASSUM(X_CHOOSE_TAC `C:real` o MATCH_MP LINEAR_BOUNDED_POS) THEN
SUBGOAL_THEN
`!e. &0 < e ==> ?d. &0 < d /\
!z. norm(z - y) < d
==> norm((g:real^N->real^M)(z) - g(y) - g'(z - y))
<= e * norm(g(z) - g(y))`
ASSUME_TAC THENL
[X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [HAS_DERIVATIVE_AT_ALT]) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (MP_TAC o SPEC `e / C`)) THEN
ASM_SIMP_TAC[REAL_LT_DIV] THEN
DISCH_THEN(X_CHOOSE_THEN `d0:real`
(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(ASSUME_TAC o GEN `z:real^N` o SPEC `(g:real^N->real^M) z`) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_at]) THEN
DISCH_THEN(MP_TAC o SPEC `d0:real`) THEN ASM_REWRITE_TAC[dist] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^N` o
GEN_REWRITE_RULE I [open_def]) THEN
ASM_REWRITE_TAC[dist] THEN
DISCH_THEN(X_CHOOSE_THEN `d2:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`d1:real`; `d2:real`] REAL_DOWN2) THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN STRIP_TAC THEN
ASM_REWRITE_TAC[] THEN
X_GEN_TAC `z:real^N` THEN DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `C * (e / C) * norm((g:real^N->real^M) z - g y)` THEN
CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[REAL_MUL_ASSOC; REAL_LE_RMUL; REAL_DIV_LMUL;
REAL_EQ_IMP_LE; REAL_LT_IMP_NZ; NORM_POS_LE]] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `C * norm(f((g:real^N->real^M) z) - y - f'(g z - g y))` THEN
CONJ_TAC THENL [ALL_TAC; ASM_MESON_TAC[REAL_LT_TRANS; REAL_LE_LMUL_EQ]] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC
`norm(g'(f((g:real^N->real^M) z) - y - f'(g z - g y)):real^M)` THEN
ASM_REWRITE_TAC[] THEN ASM_SIMP_TAC[LINEAR_SUB] THEN
GEN_REWRITE_TAC LAND_CONV [GSYM NORM_NEG] THEN
REWRITE_TAC[VECTOR_ARITH
`--(gz:real^N - gy - (z - y)) = z - y - (gz - gy)`] THEN
ASM_MESON_TAC[REAL_LE_REFL; REAL_LT_TRANS];
ALL_TAC] THEN
SUBGOAL_THEN
`?B d. &0 < B /\ &0 < d /\
!z. norm(z - y) < d
==> norm((g:real^N->real^M)(z) - g(y)) <= B * norm(z - y)`
STRIP_ASSUME_TAC THENL
[EXISTS_TAC `&2 * C` THEN
FIRST_X_ASSUM(MP_TAC o SPEC `&1 / &2`) THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
ASM_SIMP_TAC[REAL_LT_MUL; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC) THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `z:real^N` THEN
MATCH_MP_TAC MONO_IMP THEN REWRITE_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH
`norm(dg) <= norm(dg') + norm(dg - dg') /\
((&2 * (&1 - h)) * norm(dg) = &1 * norm(dg)) /\
norm(dg') <= c * norm(d)
==> norm(dg - dg') <= h * norm(dg)
==> norm(dg) <= (&2 * c) * norm(d)`) THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN ASM_REWRITE_TAC[NORM_TRIANGLE_SUB];
ALL_TAC] THEN
REWRITE_TAC[HAS_DERIVATIVE_AT_ALT] THEN ASM_REWRITE_TAC[] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / B`) THEN
ASM_SIMP_TAC[REAL_LT_DIV] THEN
DISCH_THEN(X_CHOOSE_THEN `d':real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`d:real`; `d':real`] REAL_DOWN2) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `k:real` THEN
STRIP_TAC THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `z:real^N` THEN
DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `e / B * norm ((g:real^N->real^M) z - g y)` THEN
CONJ_TAC THENL [ASM_MESON_TAC[REAL_LT_TRANS]; ALL_TAC] THEN
ASM_SIMP_TAC[real_div; GSYM REAL_MUL_ASSOC; REAL_LE_LMUL_EQ] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ] THEN
ASM_MESON_TAC[REAL_MUL_SYM; REAL_LT_TRANS]);;
(* ------------------------------------------------------------------------- *)
(* Simply rewrite that based on the domain point x. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE_BASIC_X = prove
(`!f:real^M->real^N g f' g' t x.
(f has_derivative f') (at x) /\ linear g' /\ (g' o f' = I) /\
g continuous (at (f(x))) /\ (g(f(x)) = x) /\
open t /\ f(x) IN t /\ (!y. y IN t ==> (f(g(y)) = y))
==> (g has_derivative g') (at (f(x)))`,
MESON_TAC[HAS_DERIVATIVE_INVERSE_BASIC]);;
(* ------------------------------------------------------------------------- *)
(* This is the version in Dieudonne', assuming continuity of f and g. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE_DIEUDONNE = prove
(`!f:real^M->real^N g s.
open s /\ open (IMAGE f s) /\
f continuous_on s /\ g continuous_on (IMAGE f s) /\
(!x. x IN s ==> (g(f(x)) = x))
==> !f' g' x. x IN s /\ (f has_derivative f') (at x) /\
linear g' /\ (g' o f' = I)
==> (g has_derivative g') (at (f(x)))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_BASIC_X THEN
EXISTS_TAC `f':real^M->real^N` THEN
EXISTS_TAC `IMAGE (f:real^M->real^N) s` THEN
ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT; IN_IMAGE]);;
(* ------------------------------------------------------------------------- *)
(* Here's the simplest way of not assuming much about g. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE = prove
(`!f:real^M->real^N g f' g' s x.
compact s /\ x IN s /\ f(x) IN interior(IMAGE f s) /\
f continuous_on s /\ (!x. x IN s ==> (g(f(x)) = x)) /\
(f has_derivative f') (at x) /\ linear g' /\ (g' o f' = I)
==> (g has_derivative g') (at (f(x)))`,
REPEAT STRIP_TAC THEN
MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_BASIC_X THEN
EXISTS_TAC `f':real^M->real^N` THEN
EXISTS_TAC `interior(IMAGE (f:real^M->real^N) s)` THEN
ASM_MESON_TAC[CONTINUOUS_ON_INTERIOR; CONTINUOUS_ON_INVERSE;
OPEN_INTERIOR; IN_IMAGE; INTERIOR_SUBSET; SUBSET]);;
(* ------------------------------------------------------------------------- *)
(* Proving surjectivity via Brouwer fixpoint theorem. *)
(* ------------------------------------------------------------------------- *)
let BROUWER_SURJECTIVE = prove
(`!f:real^N->real^N s t.
compact t /\ convex t /\ ~(t = {}) /\ f continuous_on t /\
(!x y. x IN s /\ y IN t ==> x + (y - f(y)) IN t)
==> !x. x IN s ==> ?y. y IN t /\ (f(y) = x)`,
REPEAT STRIP_TAC THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`((f:real^N->real^N)(y) = x) <=> (x + (y - f(y)) = y)`] THEN
ASM_SIMP_TAC[CONTINUOUS_ON_ADD; CONTINUOUS_ON_CONST; CONTINUOUS_ON_SUB;
BROUWER; SUBSET; FORALL_IN_IMAGE; CONTINUOUS_ON_ID]);;
let BROUWER_SURJECTIVE_CBALL = prove
(`!f:real^N->real^N s a e.
&0 < e /\
f continuous_on cball(a,e) /\
(!x y. x IN s /\ y IN cball(a,e) ==> x + (y - f(y)) IN cball(a,e))
==> !x. x IN s ==> ?y. y IN cball(a,e) /\ (f(y) = x)`,
REPEAT GEN_TAC THEN STRIP_TAC THEN MATCH_MP_TAC BROUWER_SURJECTIVE THEN
ASM_REWRITE_TAC[COMPACT_CBALL; CONVEX_CBALL] THEN
ASM_SIMP_TAC[CBALL_EQ_EMPTY; REAL_LT_IMP_LE; REAL_NOT_LT]);;
(* ------------------------------------------------------------------------- *)
(* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *)
(* ------------------------------------------------------------------------- *)
let SUSSMANN_OPEN_MAPPING = prove
(`!f:real^M->real^N f' g' s x.
open s /\ f continuous_on s /\
x IN s /\ (f has_derivative f') (at x) /\ linear g' /\ (f' o g' = I)
==> !t. t SUBSET s /\ x IN interior(t)
==> f(x) IN interior(IMAGE f t)`,
REWRITE_TAC[HAS_DERIVATIVE_AT_ALT] THEN
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REPEAT STRIP_TAC THEN
MP_TAC(MATCH_MP LINEAR_BOUNDED_POS (ASSUME `linear(g':real^N->real^M)`)) THEN
DISCH_THEN(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `&1 / (&2 * B)`) THEN
ASM_SIMP_TAC[REAL_LT_MUL; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `e0:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [IN_INTERIOR_CBALL]) THEN
DISCH_THEN(X_CHOOSE_THEN `e1:real` STRIP_ASSUME_TAC) THEN
MP_TAC(SPECL [`e0 / B`; `e1 / B`] REAL_DOWN2) THEN
ASM_SIMP_TAC[REAL_LT_DIV] THEN
DISCH_THEN(X_CHOOSE_THEN `e:real` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL
[`\y. (f:real^M->real^N)(x + g'(y - f(x)))`;
`cball((f:real^M->real^N) x,e / &2)`; `(f:real^M->real^N) x`; `e:real`]
BROUWER_SURJECTIVE_CBALL) THEN
REWRITE_TAC[] THEN ANTS_TAC THENL
[ASM_REWRITE_TAC[] THEN CONJ_TAC THENL
[ONCE_REWRITE_TAC[GSYM o_DEF] THEN
MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN CONJ_TAC THENL
[MATCH_MP_TAC CONTINUOUS_ON_ADD THEN
REWRITE_TAC[CONTINUOUS_ON_CONST] THEN
ONCE_REWRITE_TAC[GSYM o_DEF] THEN
MATCH_MP_TAC CONTINUOUS_ON_COMPOSE THEN
ASM_SIMP_TAC[CONTINUOUS_ON_SUB; CONTINUOUS_ON_CONST;
CONTINUOUS_ON_ID; LINEAR_CONTINUOUS_ON];
ALL_TAC] THEN
MATCH_MP_TAC CONTINUOUS_ON_SUBSET THEN
EXISTS_TAC `cball(x:real^M,e1)` THEN CONJ_TAC THENL
[ASM_MESON_TAC[CONTINUOUS_ON_SUBSET; SUBSET_TRANS]; ALL_TAC] THEN
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN X_GEN_TAC `y:real^N` THEN
REWRITE_TAC[IN_CBALL; dist] THEN
REWRITE_TAC[VECTOR_ARITH `x - (x + y) = --y:real^N`] THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [NORM_SUB] THEN
DISCH_TAC THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `B * norm(y - (f:real^M->real^N) x)` THEN
ASM_REWRITE_TAC[NORM_NEG] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
ASM_MESON_TAC[REAL_LE_TRANS; REAL_LT_IMP_LE];
ALL_TAC] THEN
MAP_EVERY X_GEN_TAC [`y:real^N`; `z:real^N`] THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x + g'(z - (f:real^M->real^N) x)`) THEN
ASM_REWRITE_TAC[VECTOR_ADD_SUB] THEN ANTS_TAC THENL
[MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `B * norm(z - (f:real^M->real^N) x)` THEN
ASM_REWRITE_TAC[NORM_NEG] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ] THEN
ASM_MESON_TAC[IN_CBALL; dist; NORM_SUB; REAL_LET_TRANS];
ALL_TAC] THEN
REWRITE_TAC[VECTOR_ARITH `a - b - (c - b) = a - c:real^N`] THEN
DISCH_TAC THEN REWRITE_TAC[IN_CBALL; dist] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`f0 - (y + z - f1) = (f1 - z) + (f0 - y):real^N`] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC
`norm(f(x + g'(z - (f:real^M->real^N) x)) - z) + norm(f x - y)` THEN
REWRITE_TAC[NORM_TRIANGLE] THEN
FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REAL_ARITH
`x <= a ==> y <= b - a ==> x + y <= b`)) THEN
MATCH_MP_TAC REAL_LE_TRANS THEN EXISTS_TAC `e / &2` THEN CONJ_TAC THENL
[ASM_MESON_TAC[IN_CBALL; dist]; ALL_TAC] THEN
REWRITE_TAC[REAL_ARITH `e / &2 <= e - x <=> x <= e / &2`] THEN
REWRITE_TAC[real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN
CONV_TAC REAL_RAT_REDUCE_CONV THEN
SIMP_TAC[REAL_ARITH `(&1 / &2 * b) * x <= e * &1 / &2 <=> x * b <= e`] THEN
ASM_SIMP_TAC[GSYM real_div; REAL_LE_LDIV_EQ] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `B * norm(z - (f:real^M->real^N) x)` THEN
ASM_REWRITE_TAC[] THEN
ASM_MESON_TAC[REAL_LE_LMUL_EQ; REAL_MUL_SYM; IN_CBALL; dist; DIST_SYM];
ALL_TAC] THEN
REWRITE_TAC[IN_INTERIOR] THEN
DISCH_THEN(fun th -> EXISTS_TAC `e / &2` THEN MP_TAC th) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH; SUBSET] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^N` THEN
MATCH_MP_TAC MONO_IMP THEN
REWRITE_TAC[REWRITE_RULE[SUBSET] BALL_SUBSET_CBALL] THEN
DISCH_THEN(X_CHOOSE_THEN `z:real^N` (STRIP_ASSUME_TAC o GSYM)) THEN
ASM_REWRITE_TAC[IN_IMAGE] THEN
EXISTS_TAC `x + g'(z - (f:real^M->real^N) x)` THEN REWRITE_TAC[] THEN
FIRST_ASSUM(MATCH_MP_TAC o REWRITE_RULE[SUBSET]) THEN
REWRITE_TAC[IN_CBALL; dist; VECTOR_ARITH `x - (x + y) = --y:real^N`] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `B * norm(z - (f:real^M->real^N) x)` THEN
ASM_REWRITE_TAC[NORM_NEG] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
ASM_MESON_TAC[IN_CBALL; dist; NORM_SUB; REAL_LT_IMP_LE; REAL_LE_TRANS]);;
(* ------------------------------------------------------------------------- *)
(* Hence the following eccentric variant of the inverse function theorem. *)
(* This has no continuity assumptions, but we do need the inverse function. *)
(* We could put f' o g = I but this happens to fit with the minimal linear *)
(* algebra theory I've set up so far. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE_STRONG = prove
(`!f:real^N->real^N g f' g' s x.
open s /\ x IN s /\ f continuous_on s /\
(!x. x IN s ==> (g(f(x)) = x)) /\
(f has_derivative f') (at x) /\ (f' o g' = I)
==> (g has_derivative g') (at (f(x)))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_BASIC_X THEN
SUBGOAL_THEN `linear (g':real^N->real^N) /\ (g' o f' = I)`
STRIP_ASSUME_TAC THENL
[ASM_MESON_TAC[has_derivative; RIGHT_INVERSE_LINEAR; LINEAR_INVERSE_LEFT];
ALL_TAC] THEN
EXISTS_TAC `f':real^N->real^N` THEN
EXISTS_TAC `interior (IMAGE (f:real^N->real^N) s)` THEN
ASM_REWRITE_TAC[] THEN REPEAT CONJ_TAC THENL
[ALL_TAC;
ASM_SIMP_TAC[];
REWRITE_TAC[OPEN_INTERIOR];
ASM_MESON_TAC[INTERIOR_OPEN; SUSSMANN_OPEN_MAPPING; LINEAR_INVERSE_LEFT;
SUBSET_REFL; has_derivative];
ASM_MESON_TAC[IN_IMAGE; SUBSET; INTERIOR_SUBSET]] THEN
REWRITE_TAC[continuous_at] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
SUBGOAL_THEN
`!t. t SUBSET s /\ x IN interior(t)
==> (f:real^N->real^N)(x) IN interior(IMAGE f t)`
MP_TAC THENL
[ASM_MESON_TAC[SUSSMANN_OPEN_MAPPING; LINEAR_INVERSE_LEFT; has_derivative];
ALL_TAC] THEN
DISCH_THEN(MP_TAC o SPEC `ball(x:real^N,e) INTER s`) THEN ANTS_TAC THENL
[ASM_SIMP_TAC[IN_INTER; OPEN_BALL; INTERIOR_OPEN; OPEN_INTER;
INTER_SUBSET; CENTRE_IN_BALL];
ALL_TAC] THEN
REWRITE_TAC[IN_INTERIOR] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `d:real` THEN
ASM_CASES_TAC `&0 < d` THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[SUBSET; IN_BALL; IN_IMAGE; IN_INTER] THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `y:real^N` THEN
REWRITE_TAC[DIST_SYM] THEN MATCH_MP_TAC MONO_IMP THEN
ASM_MESON_TAC[DIST_SYM]);;
(* ------------------------------------------------------------------------- *)
(* A rewrite based on the other domain. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE_STRONG_X = prove
(`!f:real^N->real^N g f' g' s y.
open s /\ (g y) IN s /\ f continuous_on s /\
(!x. x IN s ==> (g(f(x)) = x)) /\
(f has_derivative f') (at (g y)) /\ (f' o g' = I) /\
f(g y) = y
==> (g has_derivative g') (at y)`,
REPEAT STRIP_TAC THEN
FIRST_ASSUM(fun th -> GEN_REWRITE_TAC (RAND_CONV o RAND_CONV) [SYM th]) THEN
MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_STRONG THEN
MAP_EVERY EXISTS_TAC [`f':real^N->real^N`; `s:real^N->bool`] THEN
ASM_REWRITE_TAC[]);;
(* ------------------------------------------------------------------------- *)
(* On a region. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_INVERSE_ON = prove
(`!f:real^N->real^N s.
open s /\
(!x. x IN s ==> (f has_derivative f'(x)) (at x) /\ (g(f(x)) = x) /\
(f'(x) o g'(x) = I))
==> !x. x IN s ==> (g has_derivative g'(x)) (at (f(x)))`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_STRONG THEN
EXISTS_TAC `(f':real^N->real^N->real^N) x` THEN
EXISTS_TAC `s:real^N->bool` THEN
ASM_MESON_TAC[CONTINUOUS_ON_EQ_CONTINUOUS_AT;
DIFFERENTIABLE_IMP_CONTINUOUS_AT; differentiable]);;
(* ------------------------------------------------------------------------- *)
(* Invertible derivative continous at a point implies local injectivity. *)
(* It's only for this we need continuity of the derivative, except of course *)
(* if we want the fact that the inverse derivative is also continuous. So if *)
(* we know for some other reason that the inverse function exists, it's OK. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_LOCALLY_INJECTIVE = prove
(`!f:real^M->real^N f' g' s a.
a IN s /\ open s /\ linear g' /\ (g' o f'(a) = I) /\
(!x. x IN s ==> (f has_derivative f'(x)) (at x)) /\
(!e. &0 < e
==> ?d. &0 < d /\
!x. dist(a,x) < d ==> onorm(\v. f'(x) v - f'(a) v) < e)
==> ?t. a IN t /\ open t /\
!x x'. x IN t /\ x' IN t /\ (f x' = f x) ==> (x' = x)`,
REWRITE_TAC[FUN_EQ_THM; o_THM; I_THM] THEN REPEAT STRIP_TAC THEN
SUBGOAL_THEN `&0 < onorm(g':real^N->real^M)` ASSUME_TAC THENL
[ASM_SIMP_TAC[ONORM_POS_LT] THEN ASM_MESON_TAC[VEC_EQ; ARITH_EQ];
ALL_TAC] THEN
ABBREV_TAC `k = &1 / onorm(g':real^N->real^M) / &2` THEN
SUBGOAL_THEN
`?d. &0 < d /\ ball(a,d) SUBSET s /\
!x. x IN ball(a,d)
==> onorm(\v. (f':real^M->real^M->real^N)(x) v - f'(a) v) < k`
STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o SPEC `k:real`) THEN EXPAND_TAC "k" THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [OPEN_CONTAINS_BALL]) THEN
DISCH_THEN(MP_TAC o SPEC `a:real^M`) THEN ASM_REWRITE_TAC[] THEN
REWRITE_TAC[SUBSET; IN_BALL] THEN DISCH_THEN(X_CHOOSE_TAC `d2:real`) THEN
EXISTS_TAC `min d1 d2` THEN ASM_REWRITE_TAC[REAL_LT_MIN; IN_BALL] THEN
ASM_MESON_TAC[REAL_LT_TRANS];
ALL_TAC] THEN
EXISTS_TAC `ball(a:real^M,d)` THEN
ASM_SIMP_TAC[OPEN_BALL; CENTRE_IN_BALL] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `x':real^M`] THEN STRIP_TAC THEN
ABBREV_TAC `ph = \w. w - g'(f(w) - (f:real^M->real^N)(x))` THEN
SUBGOAL_THEN `norm((ph:real^M->real^M) x' - ph x) <= norm(x' - x) / &2`
MP_TAC THENL
[ALL_TAC;
EXPAND_TAC "ph" THEN ASM_REWRITE_TAC[VECTOR_SUB_REFL] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_0 th]) THEN
ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
REWRITE_TAC[VECTOR_SUB_RZERO; GSYM NORM_LE_0] THEN REAL_ARITH_TAC] THEN
SUBGOAL_THEN
`!u v:real^M. u IN ball(a,d) /\ v IN ball(a,d)
==> norm(ph u - ph v :real^M) <= norm(u - v) / &2`
(fun th -> ASM_SIMP_TAC[th]) THEN
REWRITE_TAC[real_div] THEN ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN
MATCH_MP_TAC DIFFERENTIABLE_BOUND THEN
REWRITE_TAC[CONVEX_BALL; OPEN_BALL] THEN
EXISTS_TAC `\x v. v - g'((f':real^M->real^M->real^N) x v)` THEN
CONJ_TAC THEN X_GEN_TAC `u:real^M` THEN DISCH_TAC THEN REWRITE_TAC[] THENL
[EXPAND_TAC "ph" THEN
MATCH_MP_TAC HAS_DERIVATIVE_SUB THEN REWRITE_TAC[HAS_DERIVATIVE_ID] THEN
FIRST_ASSUM(fun th -> REWRITE_TAC[MATCH_MP LINEAR_SUB th]) THEN
GEN_REWRITE_TAC (RATOR_CONV o BINDER_CONV) [GSYM VECTOR_SUB_RZERO] THEN
MATCH_MP_TAC HAS_DERIVATIVE_SUB THEN REWRITE_TAC[HAS_DERIVATIVE_CONST] THEN
ONCE_REWRITE_TAC[GSYM o_DEF] THEN MATCH_MP_TAC DIFF_CHAIN_WITHIN THEN
ONCE_REWRITE_TAC[ETA_AX] THEN
ASM_MESON_TAC[HAS_DERIVATIVE_LINEAR; SUBSET; HAS_DERIVATIVE_AT_WITHIN];
ALL_TAC] THEN
SUBGOAL_THEN
`(\w. w - g'((f':real^M->real^M->real^N) u w)) =
g' o (\w. f' a w - f' u w)`
SUBST1_TAC THENL
[REWRITE_TAC[FUN_EQ_THM; o_THM] THEN ASM_MESON_TAC[LINEAR_SUB];
ALL_TAC] THEN
SUBGOAL_THEN `linear(\w. f' a w - (f':real^M->real^M->real^N) u w)`
ASSUME_TAC THENL
[MATCH_MP_TAC LINEAR_COMPOSE_SUB THEN ONCE_REWRITE_TAC[ETA_AX] THEN
ASM_MESON_TAC[has_derivative; SUBSET; CENTRE_IN_BALL];
ALL_TAC] THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC
`onorm(g':real^N->real^M) *
onorm(\w. f' a w - (f':real^M->real^M->real^N) u w)` THEN
ASM_SIMP_TAC[ONORM_COMPOSE] THEN
ONCE_REWRITE_TAC[REAL_MUL_SYM] THEN ASM_SIMP_TAC[GSYM REAL_LE_RDIV_EQ] THEN
REWRITE_TAC[real_div; REAL_ARITH `inv(&2) * x = (&1 * x) * inv(&2)`] THEN
ASM_REWRITE_TAC[GSYM real_div] THEN
SUBGOAL_THEN `onorm(\w. (f':real^M->real^M->real^N) a w - f' u w) =
onorm(\w. f' u w - f' a w)`
(fun th -> ASM_SIMP_TAC[th; REAL_LT_IMP_LE]) THEN
GEN_REWRITE_TAC (LAND_CONV o ONCE_DEPTH_CONV) [GSYM VECTOR_NEG_SUB] THEN
MATCH_MP_TAC ONORM_NEG THEN ONCE_REWRITE_TAC[GSYM VECTOR_NEG_SUB] THEN
ASM_SIMP_TAC[LINEAR_COMPOSE_NEG]);;
(* ------------------------------------------------------------------------- *)
(* More conventional "C1" version of inverse function theorem. *)
(* ------------------------------------------------------------------------- *)
let INVERSE_FUNCTION_C1 = prove
(`!f:real^N->real^N f' a s.
a IN s /\ open s /\
(!x. x IN s ==> (f has_derivative f'(x)) (at x)) /\
(!e. &0 < e
==> ?d. &0 < d /\
!x. dist(a,x) < d ==> onorm(\v. f'(x) v - f'(a) v) < e) /\
~(det(matrix(f' a)) = &0)
==> ?t u g. open t /\ a IN t /\ open u /\ f(a) IN u /\
(!x. x IN t ==> g(f(x)) = x) /\
(!y. y IN u ==> f(g(y)) = y) /\
g differentiable_on u`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN
`(\x:real^N. lift(det(matrix(f' x:real^N->real^N)))) continuous at a`
ASSUME_TAC THENL
[MATCH_MP_TAC CONTINUOUS_LIFT_DET THEN
REPEAT STRIP_TAC THEN REWRITE_TAC[continuous_at; DIST_LIFT] THEN
X_GEN_TAC `e:real` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [open_def]) THEN
DISCH_THEN(MP_TAC o SPEC `a:real^N`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `k:real` STRIP_ASSUME_TAC) THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` STRIP_ASSUME_TAC) THEN
EXISTS_TAC `min d k:real` THEN ASM_REWRITE_TAC[REAL_LT_MIN] THEN
X_GEN_TAC `x:real^N` THEN STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o SPEC `x:real^N`) THEN
ANTS_TAC THENL [ASM_MESON_TAC[DIST_SYM]; ALL_TAC] THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ] REAL_LET_TRANS) THEN
REWRITE_TAC[GSYM MATRIX_SUB_COMPONENT] THEN
W(MP_TAC o PART_MATCH lhand MATRIX_COMPONENT_LE_ONORM o lhand o snd) THEN
MATCH_MP_TAC(REWRITE_RULE[IMP_CONJ_ALT] REAL_LE_TRANS) THEN
MATCH_MP_TAC REAL_EQ_IMP_LE THEN AP_TERM_TAC THEN ABS_TAC THEN
REWRITE_TAC[MATRIX_VECTOR_MUL_SUB_RDISTRIB] THEN BINOP_TAC THEN
MATCH_MP_TAC(REWRITE_RULE[RIGHT_IMP_FORALL_THM] MATRIX_WORKS) THEN
RULE_ASSUM_TAC(REWRITE_RULE[has_derivative]) THEN
ASM_MESON_TAC[];
ALL_TAC] THEN
SUBGOAL_THEN
`?u. a IN u /\ open u /\
!x:real^N. x IN u ==> ~(det(matrix(f' x:real^N->real^N)) = &0)`
STRIP_ASSUME_TAC THENL
[FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [continuous_at]) THEN
DISCH_THEN(MP_TAC o SPEC
`abs(det(matrix((f':real^N->real^N->real^N) a)))`) THEN
ASM_REWRITE_TAC[REAL_ARITH `&0 < abs x <=> ~(x = &0)`] THEN
REWRITE_TAC[DIST_LIFT; LEFT_IMP_EXISTS_THM] THEN
X_GEN_TAC `d:real` THEN STRIP_TAC THEN
EXISTS_TAC `ball(a:real^N,d)` THEN
ASM_REWRITE_TAC[DIST_REFL; OPEN_BALL; IN_BALL] THEN
ASM_MESON_TAC[DIST_SYM; REAL_ARITH `abs(x - y) < abs y ==> ~(x = &0)`];
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM INVERTIBLE_DET_NZ]) THEN
SUBGOAL_THEN `!x. x IN s ==> linear((f':real^N->real^N->real^N) x)`
ASSUME_TAC THENL [ASM_MESON_TAC[has_derivative]; ALL_TAC] THEN
ASM_SIMP_TAC[MATRIX_INVERTIBLE] THEN
DISCH_THEN(X_CHOOSE_THEN `g'a:real^N->real^N` STRIP_ASSUME_TAC) THEN
MP_TAC(ISPECL [`f:real^N->real^N`; `f':real^N->real^N->real^N`;
`g'a:real^N->real^N`; `s:real^N->bool`; `a:real^N`]
HAS_DERIVATIVE_LOCALLY_INJECTIVE) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `t:real^N->bool` THEN STRIP_TAC THEN
ONCE_REWRITE_TAC[SWAP_EXISTS_THM] THEN
MP_TAC(ISPECL [`f:real^N->real^N`; `t:real^N->bool`]
INJECTIVE_ON_LEFT_INVERSE) THEN
MATCH_MP_TAC(TAUT `p /\ (q ==> r) ==> (p <=> q) ==> r`) THEN
CONJ_TAC THENL [ASM_MESON_TAC[]; MATCH_MP_TAC MONO_EXISTS] THEN
X_GEN_TAC `g:real^N->real^N` THEN STRIP_TAC THEN ASM_REWRITE_TAC[] THEN
SUBGOAL_THEN
`!x. x IN s INTER u
==> ?g. linear g /\ (f':real^N->real^N->real^N) x o g = I /\
g o f' x = I`
MP_TAC THENL
[ASM_SIMP_TAC[IN_INTER; GSYM MATRIX_INVERTIBLE; INVERTIBLE_DET_NZ];
GEN_REWRITE_TAC (LAND_CONV o BINDER_CONV) [RIGHT_IMP_EXISTS_THM] THEN
REWRITE_TAC[SKOLEM_THM; LEFT_IMP_EXISTS_THM]] THEN
X_GEN_TAC `g':real^N->real^N->real^N` THEN STRIP_TAC THEN
EXISTS_TAC `interior (IMAGE (f:real^N->real^N) (s INTER t INTER u))` THEN
SIMP_TAC[OPEN_INTERIOR; DIFFERENTIABLE_ON_EQ_DIFFERENTIABLE_AT] THEN
CONJ_TAC THENL
[MP_TAC(ISPECL [`f:real^N->real^N`; `(f':real^N->real^N->real^N) a`;
`g'a:real^N->real^N`; `s INTER t INTER u:real^N->bool`;
`a:real^N`]
SUSSMANN_OPEN_MAPPING) THEN
ASM_SIMP_TAC[OPEN_INTER; IN_INTER] THEN ANTS_TAC THENL
[ASM_MESON_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_AT;
CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_INTER;
differentiable; SUBSET; IN_INTER];
DISCH_THEN MATCH_MP_TAC THEN CONJ_TAC THENL [SET_TAC[]; ALL_TAC] THEN
ASM_SIMP_TAC[INTERIOR_OPEN; OPEN_INTER; IN_INTER]];
ALL_TAC] THEN
ONCE_REWRITE_TAC[MESON[SUBSET; INTERIOR_SUBSET]
`(!x. x IN interior s ==> P x) <=>
(!x. x IN s ==> x IN interior s ==> P x)`] THEN
REWRITE_TAC[FORALL_IN_IMAGE] THEN
CONJ_TAC THENL [ASM_MESON_TAC[IN_INTER]; ALL_TAC] THEN
X_GEN_TAC `x:real^N` THEN REWRITE_TAC[IN_INTER] THEN REPEAT STRIP_TAC THEN
REWRITE_TAC[differentiable] THEN
EXISTS_TAC `(g':real^N->real^N->real^N) x` THEN
MATCH_MP_TAC HAS_DERIVATIVE_INVERSE_STRONG THEN
EXISTS_TAC `(f':real^N->real^N->real^N) x` THEN
EXISTS_TAC `s INTER t INTER u:real^N->bool` THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN_INTER]) THEN
ASM_SIMP_TAC[IN_INTER; OPEN_INTER] THEN
ASM_MESON_TAC[DIFFERENTIABLE_IMP_CONTINUOUS_AT;
CONTINUOUS_ON_EQ_CONTINUOUS_AT; OPEN_INTER;
differentiable; SUBSET; IN_INTER]);;
(* ------------------------------------------------------------------------- *)
(* Uniformly convergent sequence of derivatives. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_SEQUENCE_LIPSCHITZ = prove
(`!s f:num->real^M->real^N f' g'.
convex s /\
(!n x. x IN s ==> ((f n) has_derivative (f' n x)) (at x within s)) /\
(!e. &0 < e
==> ?N. !n x h. n >= N /\ x IN s
==> norm(f' n x h - g' x h) <= e * norm(h))
==> !e. &0 < e
==> ?N. !m n x y. m >= N /\ n >= N /\ x IN s /\ y IN s
==> norm((f m x - f n x) - (f m y - f n y))
<= e * norm(x - y)`,
REPEAT STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[REAL_HALF] THEN MATCH_MP_TAC MONO_EXISTS THEN
X_GEN_TAC `N:num` THEN DISCH_TAC THEN
MAP_EVERY X_GEN_TAC [`m:num`; `n:num`] THEN
ASM_CASES_TAC `m:num >= N` THEN ASM_REWRITE_TAC[] THEN
ASM_CASES_TAC `n:num >= N` THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC DIFFERENTIABLE_BOUND THEN
EXISTS_TAC `\x h. (f':num->real^M->real^M->real^N) m x h - f' n x h` THEN
ASM_SIMP_TAC[HAS_DERIVATIVE_SUB; ETA_AX] THEN
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
SUBGOAL_THEN
`!h. norm((f':num->real^M->real^M->real^N) m x h - f' n x h) <= e * norm(h)`
MP_TAC THEN RULE_ASSUM_TAC(REWRITE_RULE[HAS_DERIVATIVE_WITHIN_ALT]) THEN
ASM_SIMP_TAC[ONORM; LINEAR_COMPOSE_SUB; ETA_AX] THEN
X_GEN_TAC `h:real^M` THEN SUBST1_TAC(VECTOR_ARITH
`(f':num->real^M->real^M->real^N) m x h - f' n x h =
(f' m x h - g' x h) + --(f' n x h - g' x h)`) THEN
MATCH_MP_TAC NORM_TRIANGLE_LE THEN
ASM_SIMP_TAC[NORM_NEG; REAL_ARITH
`a <= e / &2 * h /\ b <= e / &2 * h ==> a + b <= e * h`]);;
let HAS_DERIVATIVE_SEQUENCE = prove
(`!s f:num->real^M->real^N f' g'.
convex s /\
(!n x. x IN s ==> ((f n) has_derivative (f' n x)) (at x within s)) /\
(!e. &0 < e
==> ?N. !n x h. n >= N /\ x IN s
==> norm(f' n x h - g' x h) <= e * norm(h)) /\
(?x l. x IN s /\ ((\n. f n x) --> l) sequentially)
==> ?g. !x. x IN s
==> ((\n. f n x) --> g x) sequentially /\
(g has_derivative g'(x)) (at x within s)`,
REPEAT GEN_TAC THEN
REPLICATE_TAC 2 (DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
DISCH_THEN(CONJUNCTS_THEN2 (LABEL_TAC "O") MP_TAC) THEN
DISCH_THEN(X_CHOOSE_THEN `x0:real^M` STRIP_ASSUME_TAC) THEN
SUBGOAL_TAC "A"
`!e. &0 < e
==> ?N. !m n x y. m >= N /\ n >= N /\ x IN s /\ y IN s
==> norm(((f:num->real^M->real^N) m x - f n x) -
(f m y - f n y))
<= e * norm(x - y)`
[MATCH_MP_TAC HAS_DERIVATIVE_SEQUENCE_LIPSCHITZ THEN
ASM_REWRITE_TAC[] THEN ASM_MESON_TAC[]] THEN
SUBGOAL_THEN
`?g:real^M->real^N. !x. x IN s ==> ((\n. f n x) --> g x) sequentially`
MP_TAC THENL
[REWRITE_TAC[GSYM SKOLEM_THM; RIGHT_EXISTS_IMP_THM] THEN
X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
GEN_REWRITE_TAC I [CONVERGENT_EQ_CAUCHY] THEN
FIRST_X_ASSUM(MP_TAC o MATCH_MP CONVERGENT_IMP_CAUCHY) THEN
REWRITE_TAC[cauchy; dist] THEN DISCH_THEN(LABEL_TAC "B") THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
ASM_CASES_TAC `x:real^M = x0` THEN ASM_SIMP_TAC[] THEN
REMOVE_THEN "B" (MP_TAC o SPEC `e / &2`) THEN
ASM_REWRITE_TAC[REAL_HALF] THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` STRIP_ASSUME_TAC) THEN
REMOVE_THEN "A" (MP_TAC o SPEC `e / &2 / norm(x - x0:real^M)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_HALF; VECTOR_SUB_EQ] THEN
DISCH_THEN(X_CHOOSE_TAC `N2:num`) THEN
EXISTS_TAC `N1 + N2:num` THEN REPEAT GEN_TAC THEN
DISCH_THEN(CONJUNCTS_THEN (STRIP_ASSUME_TAC o MATCH_MP
(ARITH_RULE `m >= N1 + N2:num ==> m >= N1 /\ m >= N2`))) THEN
SUBST1_TAC(VECTOR_ARITH
`(f:num->real^M->real^N) m x - f n x =
(f m x - f n x - (f m x0 - f n x0)) + (f m x0 - f n x0)`) THEN
MATCH_MP_TAC NORM_TRIANGLE_LT THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`m:num`; `n:num`; `x:real^M`; `x0:real^M`]) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`m:num`; `n:num`]) THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0; VECTOR_SUB_EQ] THEN REAL_ARITH_TAC;
ALL_TAC] THEN
MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN SIMP_TAC[] THEN
DISCH_THEN(LABEL_TAC "B") THEN X_GEN_TAC `x:real^M` THEN DISCH_TAC THEN
REWRITE_TAC[HAS_DERIVATIVE_WITHIN_ALT] THEN
SUBGOAL_TAC "C"
`!e. &0 < e
==> ?N. !n x y. n >= N /\ x IN s /\ y IN s
==> norm(((f:num->real^M->real^N) n x - f n y) -
(g x - g y))
<= e * norm(x - y)`
[X_GEN_TAC `e:real` THEN DISCH_TAC THEN
REMOVE_THEN "A" (MP_TAC o SPEC `e:real`) THEN ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN
MATCH_MP_TAC MONO_FORALL THEN X_GEN_TAC `n:num` THEN
DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`] THEN
STRIP_TAC THEN FIRST_X_ASSUM(MP_TAC o GEN `m:num` o SPECL
[`m:num`; `u:real^M`; `v:real^M`]) THEN
DISCH_TAC THEN MATCH_MP_TAC(ISPEC `sequentially` LIM_NORM_UBOUND) THEN
EXISTS_TAC
`\m. ((f:num->real^M->real^N) n u - f n v) - (f m u - f m v)` THEN
REWRITE_TAC[eventually; TRIVIAL_LIMIT_SEQUENTIALLY] THEN
ASM_SIMP_TAC[SEQUENTIALLY; LIM_SUB; LIM_CONST] THEN EXISTS_TAC `N:num` THEN
ONCE_REWRITE_TAC[VECTOR_ARITH
`(x - y) - (u - v) = (x - u) - (y - v):real^N`] THEN
ASM_MESON_TAC[GE_REFL]] THEN
CONJ_TAC THENL
[SUBGOAL_TAC "D"
`!u. ((\n. (f':num->real^M->real^M->real^N) n x u) --> g' x u) sequentially`
[REWRITE_TAC[LIM_SEQUENTIALLY; dist] THEN REPEAT STRIP_TAC THEN
ASM_CASES_TAC `u = vec 0:real^M` THENL
[REMOVE_THEN "O" (MP_TAC o SPEC `e:real`);
REMOVE_THEN "O" (MP_TAC o SPEC `e / &2 / norm(u:real^M)`)] THEN
ASM_SIMP_TAC[NORM_POS_LT; REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
MATCH_MP_TAC MONO_EXISTS THEN GEN_TAC THEN
MATCH_MP_TAC MONO_FORALL THEN GEN_TAC THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^M`; `u:real^M`]) THEN
DISCH_THEN(fun th -> DISCH_TAC THEN MP_TAC th) THEN
ASM_SIMP_TAC[GE; NORM_0; REAL_MUL_RZERO; NORM_LE_0] THEN
ASM_SIMP_TAC[REAL_DIV_RMUL; NORM_EQ_0] THEN
UNDISCH_TAC `&0 < e` THEN REAL_ARITH_TAC] THEN
REWRITE_TAC[linear] THEN ONCE_REWRITE_TAC[GSYM VECTOR_SUB_EQ] THEN
CONJ_TAC THENL
[MAP_EVERY X_GEN_TAC [`u:real^M`; `v:real^M`];
MAP_EVERY X_GEN_TAC [`c:real`; `u:real^M`]] THEN
MATCH_MP_TAC(ISPEC `sequentially` LIM_UNIQUE) THENL
[EXISTS_TAC
`\n. (f':num->real^M->real^M->real^N) n x (u + v) -
(f' n x u + f' n x v)`;
EXISTS_TAC
`\n. (f':num->real^M->real^M->real^N) n x (c % u) -
c % f' n x u`] THEN
ASM_SIMP_TAC[TRIVIAL_LIMIT_SEQUENTIALLY; LIM_SUB; LIM_ADD; LIM_CMUL] THEN
RULE_ASSUM_TAC(REWRITE_RULE[has_derivative_within; linear]) THEN
ASM_SIMP_TAC[VECTOR_SUB_REFL; LIM_CONST];
ALL_TAC] THEN
X_GEN_TAC `e:real` THEN DISCH_TAC THEN
MAP_EVERY (fun s -> REMOVE_THEN s (MP_TAC o SPEC `e / &3`)) ["C"; "O"] THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `N1:num` (LABEL_TAC "C")) THEN
DISCH_THEN(X_CHOOSE_THEN `N2:num` (LABEL_TAC "A")) THEN
REMOVE_THEN "C" (MP_TAC o GEN `y:real^M` o
SPECL [`N1 + N2:num`; `x:real^M`; `y - x:real^M`]) THEN
REMOVE_THEN "A" (MP_TAC o GEN `y:real^M` o
SPECL [`N1 + N2:num`; `y:real^M`; `x:real^M`]) THEN
FIRST_X_ASSUM(MP_TAC o SPECL [`N1 + N2:num`; `x:real^M`]) THEN
ASM_REWRITE_TAC[ARITH_RULE `m + n >= m:num /\ m + n >= n`] THEN
REWRITE_TAC[HAS_DERIVATIVE_WITHIN_ALT] THEN
DISCH_THEN(MP_TAC o SPEC `e / &3` o CONJUNCT2) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_OF_NUM_LT; ARITH] THEN
DISCH_THEN(X_CHOOSE_THEN `d1:real` STRIP_ASSUME_TAC) THEN
DISCH_THEN(LABEL_TAC "D1") THEN DISCH_THEN(LABEL_TAC "D2") THEN
EXISTS_TAC `d1:real` THEN ASM_REWRITE_TAC[] THEN X_GEN_TAC `y:real^M` THEN
DISCH_TAC THEN REMOVE_THEN "D2" (MP_TAC o SPEC `y:real^M`) THEN
REMOVE_THEN "D1" (MP_TAC o SPEC `y:real^M`) THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_LT_TRANS; NORM_SUB]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `y:real^M`) THEN ANTS_TAC THENL
[ASM_MESON_TAC[REAL_LT_TRANS; NORM_SUB]; ALL_TAC] THEN
MATCH_MP_TAC(REAL_ARITH
`d <= a + b + c
==> a <= e / &3 * n ==> b <= e / &3 * n ==> c <= e / &3 * n
==> d <= e * n`) THEN
GEN_REWRITE_TAC (funpow 2 RAND_CONV o LAND_CONV) [NORM_SUB] THEN
MATCH_MP_TAC(REAL_ARITH
`(norm(x + y + z) = norm(a)) /\
norm(x + y + z) <= norm(x) + norm(y + z) /\
norm(y + z) <= norm(y) + norm(z)
==> norm(a) <= norm(x) + norm(y) + norm(z)`) THEN
REWRITE_TAC[NORM_TRIANGLE] THEN AP_TERM_TAC THEN VECTOR_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Can choose to line up antiderivatives if we want. *)
(* ------------------------------------------------------------------------- *)
let HAS_ANTIDERIVATIVE_SEQUENCE = prove
(`!s f:num->real^M->real^N f' g'.
convex s /\
(!n x. x IN s ==> ((f n) has_derivative (f' n x)) (at x within s)) /\
(!e. &0 < e
==> ?N. !n x h. n >= N /\ x IN s
==> norm(f' n x h - g' x h) <= e * norm(h))
==> ?g. !x. x IN s ==> (g has_derivative g'(x)) (at x within s)`,
REPEAT STRIP_TAC THEN ASM_CASES_TAC `(s:real^M->bool) = {}` THEN
ASM_REWRITE_TAC[NOT_IN_EMPTY] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [GSYM MEMBER_NOT_EMPTY]) THEN
DISCH_THEN(X_CHOOSE_TAC `a:real^M`) THEN
MP_TAC(ISPECL
[`s:real^M->bool`;
`\n x. (f:num->real^M->real^N) n x + (f 0 a - f n a)`;
`f':num->real^M->real^M->real^N`;
`g':real^M->real^M->real^N`]
HAS_DERIVATIVE_SEQUENCE) THEN
ASM_REWRITE_TAC[] THEN ANTS_TAC THENL [ALL_TAC; MESON_TAC[]] THEN
CONJ_TAC THENL
[REPEAT STRIP_TAC THEN
SUBGOAL_THEN `(f':num->real^M->real^M->real^N) n x =
\h. f' n x h + vec 0`
SUBST1_TAC THENL [SIMP_TAC[FUN_EQ_THM] THEN VECTOR_ARITH_TAC; ALL_TAC] THEN
MATCH_MP_TAC HAS_DERIVATIVE_ADD THEN
ASM_SIMP_TAC[HAS_DERIVATIVE_CONST; ETA_AX];
MAP_EVERY EXISTS_TAC [`a:real^M`; `f 0 (a:real^M) :real^N`] THEN
ASM_REWRITE_TAC[VECTOR_ARITH `a + b - a = b:real^N`; LIM_CONST]]);;
let HAS_ANTIDERIVATIVE_LIMIT = prove
(`!s g':real^M->real^M->real^N.
convex s /\
(!e. &0 < e
==> ?f f'. !x. x IN s
==> (f has_derivative (f' x)) (at x within s) /\
(!h. norm(f' x h - g' x h) <= e * norm(h)))
==> ?g. !x. x IN s ==> (g has_derivative g'(x)) (at x within s)`,
REPEAT STRIP_TAC THEN
FIRST_X_ASSUM(MP_TAC o GEN `n:num` o SPEC `inv(&n + &1)`) THEN
REWRITE_TAC[REAL_LT_INV_EQ; REAL_ARITH `&0 < &n + &1`] THEN
REWRITE_TAC[SKOLEM_THM] THEN DISCH_TAC THEN
MATCH_MP_TAC HAS_ANTIDERIVATIVE_SEQUENCE THEN
UNDISCH_TAC `convex(s:real^M->bool)` THEN SIMP_TAC[] THEN
DISCH_THEN(K ALL_TAC) THEN POP_ASSUM MP_TAC THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f:num->real^M->real^N` THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `f':num->real^M->real^M->real^N` THEN
REPEAT STRIP_TAC THEN ASM_SIMP_TAC[] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [REAL_ARCH_INV]) THEN
MATCH_MP_TAC MONO_EXISTS THEN X_GEN_TAC `N:num` THEN STRIP_TAC THEN
X_GEN_TAC `n:num` THEN REWRITE_TAC[GE] THEN
MAP_EVERY X_GEN_TAC [`x:real^M`; `h:real^M`] THEN STRIP_TAC THEN
MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `inv(&n + &1) * norm(h:real^M)` THEN
ASM_SIMP_TAC[] THEN MATCH_MP_TAC REAL_LE_RMUL THEN
REWRITE_TAC[NORM_POS_LE] THEN MATCH_MP_TAC REAL_LE_TRANS THEN
EXISTS_TAC `inv(&N)` THEN ASM_SIMP_TAC[REAL_LT_IMP_LE] THEN
MATCH_MP_TAC REAL_LE_INV2 THEN
REWRITE_TAC[REAL_OF_NUM_ADD; REAL_OF_NUM_LE; REAL_OF_NUM_LT] THEN
ASM_ARITH_TAC);;
(* ------------------------------------------------------------------------- *)
(* Differentiation of a series. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_SERIES = prove
(`!s f:num->real^M->real^N f' g' k.
convex s /\
(!n x. x IN s ==> ((f n) has_derivative (f' n x)) (at x within s)) /\
(!e. &0 < e
==> ?N. !n x h. n >= N /\ x IN s
==> norm(vsum(k INTER (0..n)) (\i. f' i x h) -
g' x h) <= e * norm(h)) /\
(?x l. x IN s /\ ((\n. f n x) sums l) k)
==> ?g. !x. x IN s ==> ((\n. f n x) sums (g x)) k /\
(g has_derivative g'(x)) (at x within s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[sums] THEN
DISCH_THEN(REPEAT_TCL CONJUNCTS_THEN ASSUME_TAC) THEN
MATCH_MP_TAC HAS_DERIVATIVE_SEQUENCE THEN EXISTS_TAC
`\n:num x:real^M h:real^M. vsum(k INTER (0..n)) (\n. f' n x h):real^N` THEN
ASM_SIMP_TAC[ETA_AX; FINITE_INTER_NUMSEG; HAS_DERIVATIVE_VSUM]);;
(* ------------------------------------------------------------------------- *)
(* Derivative with composed bilinear function. *)
(* ------------------------------------------------------------------------- *)
let HAS_DERIVATIVE_BILINEAR_WITHIN = prove
(`!h:real^M->real^N->real^P f g f' g' x:real^Q s.
(f has_derivative f') (at x within s) /\
(g has_derivative g') (at x within s) /\
bilinear h
==> ((\x. h (f x) (g x)) has_derivative
(\d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)`,
REPEAT STRIP_TAC THEN
SUBGOAL_TAC "contg" `((g:real^Q->real^N) --> g(x)) (at x within s)`
[REWRITE_TAC[GSYM CONTINUOUS_WITHIN] THEN
ASM_MESON_TAC[differentiable; DIFFERENTIABLE_IMP_CONTINUOUS_WITHIN]] THEN
UNDISCH_TAC `((f:real^Q->real^M) has_derivative f') (at x within s)` THEN
REWRITE_TAC[has_derivative_within] THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "df")) THEN
SUBGOAL_TAC "contf"
`((\y. (f:real^Q->real^M)(x) + f'(y - x)) --> f(x)) (at x within s)`
[GEN_REWRITE_TAC LAND_CONV [GSYM VECTOR_ADD_RID] THEN
MATCH_MP_TAC LIM_ADD THEN REWRITE_TAC[LIM_CONST] THEN
SUBGOAL_THEN `vec 0 = (f':real^Q->real^M)(x - x)` SUBST1_TAC THENL
[ASM_MESON_TAC[LINEAR_0; VECTOR_SUB_REFL]; ALL_TAC] THEN
ASM_SIMP_TAC[LIM_LINEAR; LIM_SUB; LIM_CONST; LIM_WITHIN_ID]] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [has_derivative_within]) THEN
DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC (LABEL_TAC "dg")) THEN
CONJ_TAC THENL
[FIRST_ASSUM(STRIP_ASSUME_TAC o GEN_REWRITE_RULE I [bilinear]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[linear]) THEN ASM_REWRITE_TAC[linear] THEN
REPEAT STRIP_TAC THEN VECTOR_ARITH_TAC;
ALL_TAC] THEN
MP_TAC(ISPECL [`at (x:real^Q) within s`; `h:real^M->real^N->real^P`]
LIM_BILINEAR) THEN
ASM_REWRITE_TAC[] THEN STRIP_TAC THEN
REMOVE_THEN "contg" MP_TAC THEN REMOVE_THEN "df" MP_TAC THEN
REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
REMOVE_THEN "dg" MP_TAC THEN REMOVE_THEN "contf" MP_TAC THEN
ONCE_REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(ANTE_RES_THEN MP_TAC) THEN
REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD) THEN
SUBGOAL_THEN
`((\y:real^Q. inv(norm(y - x)) %
(h:real^M->real^N->real^P) (f'(y - x)) (g'(y - x)))
--> vec 0) (at x within s)`
MP_TAC THENL
[FIRST_ASSUM(X_CHOOSE_THEN `B:real` STRIP_ASSUME_TAC o MATCH_MP
BILINEAR_BOUNDED_POS) THEN
X_CHOOSE_THEN `C:real` STRIP_ASSUME_TAC
(MATCH_MP LINEAR_BOUNDED_POS (ASSUME `linear (f':real^Q->real^M)`)) THEN
X_CHOOSE_THEN `D:real` STRIP_ASSUME_TAC
(MATCH_MP LINEAR_BOUNDED_POS (ASSUME `linear (g':real^Q->real^N)`)) THEN
REWRITE_TAC[LIM_WITHIN; dist; VECTOR_SUB_RZERO] THEN
X_GEN_TAC `e:real` THEN STRIP_TAC THEN EXISTS_TAC `e / (B * C * D)` THEN
ASM_SIMP_TAC[REAL_LT_DIV; NORM_MUL; REAL_LT_MUL] THEN
X_GEN_TAC `x':real^Q` THEN
REWRITE_TAC[REAL_ABS_MUL; REAL_ABS_NORM; REAL_ABS_INV] THEN
STRIP_TAC THEN MATCH_MP_TAC REAL_LET_TRANS THEN
EXISTS_TAC `inv(norm(x' - x :real^Q)) *
B * (C * norm(x' - x)) * (D * norm(x' - x))` THEN
CONJ_TAC THENL
[MATCH_MP_TAC REAL_LE_LMUL THEN SIMP_TAC[REAL_LE_INV_EQ; NORM_POS_LE] THEN
ASM_MESON_TAC[REAL_LE_LMUL; REAL_LT_IMP_LE; REAL_LE_MUL2; NORM_POS_LE;
REAL_LE_TRANS];
ONCE_REWRITE_TAC[AC REAL_MUL_AC
`i * b * (c * x) * (d * x) = (i * x) * x * (b * c * d)`] THEN
ASM_SIMP_TAC[REAL_MUL_LINV; REAL_LT_IMP_NZ; REAL_MUL_LID] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; REAL_LT_MUL]];
REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP LIM_ADD) THEN
REWRITE_TAC (map (C MATCH_MP (ASSUME `bilinear(h:real^M->real^N->real^P)`))
[BILINEAR_RZERO; BILINEAR_LZERO; BILINEAR_LADD; BILINEAR_RADD;
BILINEAR_LMUL; BILINEAR_RMUL; BILINEAR_LSUB; BILINEAR_RSUB]) THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN
BINOP_TAC THEN REWRITE_TAC[FUN_EQ_THM] THEN VECTOR_ARITH_TAC]);;
let HAS_DERIVATIVE_BILINEAR_AT = prove
(`!h:real^M->real^N->real^P f g f' g' x:real^Q.
(f has_derivative f') (at x) /\
(g has_derivative g') (at x) /\
bilinear h
==> ((\x. h (f x) (g x)) has_derivative
(\d. h (f x) (g' d) + h (f' d) (g x))) (at x)`,
REWRITE_TAC[has_derivative_at] THEN
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[GSYM has_derivative_within; HAS_DERIVATIVE_BILINEAR_WITHIN]);;
let BILINEAR_DIFFERENTIABLE_AT_COMPOSE = prove
(`!f:real^M->real^N g:real^M->real^P h:real^N->real^P->real^Q a.
f differentiable at a /\ g differentiable at a /\ bilinear h
==> (\x. h (f x) (g x)) differentiable at a`,
REPEAT GEN_TAC THEN REWRITE_TAC[FRECHET_DERIVATIVE_WORKS] THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_DERIVATIVE_BILINEAR_AT) THEN
REWRITE_TAC[GSYM FRECHET_DERIVATIVE_WORKS; differentiable] THEN
MESON_TAC[]);;
let BILINEAR_DIFFERENTIABLE_WITHIN_COMPOSE = prove
(`!f:real^M->real^N g:real^M->real^P h:real^N->real^P->real^Q x s.
f differentiable at x within s /\ g differentiable at x within s /\
bilinear h
==> (\x. h (f x) (g x)) differentiable at x within s`,
REPEAT GEN_TAC THEN REWRITE_TAC[FRECHET_DERIVATIVE_WORKS] THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_DERIVATIVE_BILINEAR_WITHIN) THEN
REWRITE_TAC[GSYM FRECHET_DERIVATIVE_WORKS; differentiable] THEN
MESON_TAC[]);;
let BILINEAR_DIFFERENTIABLE_ON_COMPOSE = prove
(`!f:real^M->real^N g:real^M->real^P h:real^N->real^P->real^Q s.
f differentiable_on s /\ g differentiable_on s /\ bilinear h
==> (\x. h (f x) (g x)) differentiable_on s`,
REWRITE_TAC[differentiable_on] THEN
MESON_TAC[BILINEAR_DIFFERENTIABLE_WITHIN_COMPOSE]);;
let DIFFERENTIABLE_AT_LIFT_DOT2 = prove
(`!f:real^M->real^N g x.
f differentiable at x /\ g differentiable at x
==> (\x. lift(f x dot g x)) differentiable at x`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (MATCH_MP (REWRITE_RULE
[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`]
BILINEAR_DIFFERENTIABLE_AT_COMPOSE) BILINEAR_DOT)) THEN REWRITE_TAC[]);;
let DIFFERENTIABLE_WITHIN_LIFT_DOT2 = prove
(`!f:real^M->real^N g x s.
f differentiable (at x within s) /\ g differentiable (at x within s)
==> (\x. lift(f x dot g x)) differentiable (at x within s)`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (MATCH_MP (REWRITE_RULE
[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`]
BILINEAR_DIFFERENTIABLE_WITHIN_COMPOSE) BILINEAR_DOT)) THEN REWRITE_TAC[]);;
let DIFFERENTIABLE_ON_LIFT_DOT2 = prove
(`!f:real^M->real^N g s.
f differentiable_on s /\ g differentiable_on s
==> (\x. lift(f x dot g x)) differentiable_on s`,
REPEAT GEN_TAC THEN DISCH_THEN(MP_TAC o MATCH_MP (MATCH_MP (REWRITE_RULE
[TAUT `p /\ q /\ r ==> s <=> r ==> p /\ q ==> s`]
BILINEAR_DIFFERENTIABLE_ON_COMPOSE) BILINEAR_DOT)) THEN REWRITE_TAC[]);;
let HAS_DERIVATIVE_MUL_WITHIN = prove
(`!f f' g:real^M->real^N g' a s.
((lift o f) has_derivative (lift o f')) (at a within s) /\
(g has_derivative g') (at a within s)
==> ((\x. f x % g x) has_derivative
(\h. f a % g' h + f' h % g a)) (at a within s)`,
REPEAT GEN_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP (REWRITE_RULE[BILINEAR_DROP_MUL]
(ISPEC `\x y:real^M. drop x % y` HAS_DERIVATIVE_BILINEAR_WITHIN))) THEN
REWRITE_TAC[o_DEF; DROP_CMUL; LIFT_DROP]);;
let HAS_DERIVATIVE_MUL_AT = prove
(`!f f' g:real^M->real^N g' a.
((lift o f) has_derivative (lift o f')) (at a) /\
(g has_derivative g') (at a)
==> ((\x. f x % g x) has_derivative
(\h. f a % g' h + f' h % g a)) (at a)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[HAS_DERIVATIVE_MUL_WITHIN]);;
let HAS_DERIVATIVE_SQNORM_AT = prove
(`!a:real^N.
((\x. lift(norm x pow 2)) has_derivative (\x. &2 % lift(a dot x))) (at a)`,
GEN_TAC THEN MP_TAC(ISPECL
[`\x y:real^N. lift(x dot y)`;
`\x:real^N. x`; `\x:real^N. x`; `\x:real^N. x`; `\x:real^N. x`;
`a:real^N`] HAS_DERIVATIVE_BILINEAR_AT) THEN
REWRITE_TAC[HAS_DERIVATIVE_ID; BILINEAR_DOT; NORM_POW_2] THEN
MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN AP_TERM_TAC THEN
REWRITE_TAC[FUN_EQ_THM; DOT_SYM] THEN VECTOR_ARITH_TAC);;
let DIFFERENTIABLE_MUL_WITHIN = prove
(`!f g:real^M->real^N a s.
(lift o f) differentiable (at a within s) /\
g differentiable (at a within s)
==> (\x. f x % g x) differentiable (at a within s)`,
REPEAT GEN_TAC THEN MP_TAC(ISPECL
[`lift o (f:real^M->real)`; `g:real^M->real^N`; `\x y:real^N. drop x % y`;
`a:real^M`; `s:real^M->bool`] BILINEAR_DIFFERENTIABLE_WITHIN_COMPOSE) THEN
REWRITE_TAC[o_DEF; LIFT_DROP; BILINEAR_DROP_MUL]);;
let DIFFERENTIABLE_MUL_AT = prove
(`!f g:real^M->real^N a.
(lift o f) differentiable (at a) /\ g differentiable (at a)
==> (\x. f x % g x) differentiable (at a)`,
ONCE_REWRITE_TAC[GSYM WITHIN_UNIV] THEN
REWRITE_TAC[DIFFERENTIABLE_MUL_WITHIN]);;
let DIFFERENTIABLE_SQNORM_AT = prove
(`!a:real^N. (\x. lift(norm x pow 2)) differentiable (at a)`,
REWRITE_TAC[differentiable] THEN MESON_TAC[HAS_DERIVATIVE_SQNORM_AT]);;
let DIFFERENTIABLE_ON_MUL = prove
(`!f g:real^M->real^N s.
(lift o f) differentiable_on s /\ g differentiable_on s
==> (\x. f x % g x) differentiable_on s`,
REPEAT GEN_TAC THEN MP_TAC(ISPECL
[`lift o (f:real^M->real)`; `g:real^M->real^N`; `\x y:real^N. drop x % y`;
`s:real^M->bool`] BILINEAR_DIFFERENTIABLE_ON_COMPOSE) THEN
REWRITE_TAC[o_DEF; LIFT_DROP; BILINEAR_DROP_MUL]);;
let DIFFERENTIABLE_ON_SQNORM = prove
(`!s:real^N->bool. (\x. lift(norm x pow 2)) differentiable_on s`,
SIMP_TAC[DIFFERENTIABLE_AT_IMP_DIFFERENTIABLE_ON;
DIFFERENTIABLE_SQNORM_AT]);;
(* ------------------------------------------------------------------------- *)
(* Considering derivative R(^1)->R^n as a vector. *)
(* ------------------------------------------------------------------------- *)
parse_as_infix ("has_vector_derivative",(12,"right"));;
let has_vector_derivative = new_definition
`(f has_vector_derivative f') net <=>
(f has_derivative (\x. drop(x) % f')) net`;;
let vector_derivative = new_definition
`vector_derivative (f:real^1->real^N) net =
@f'. (f has_vector_derivative f') net`;;
let VECTOR_DERIVATIVE_WORKS = prove
(`!net f:real^1->real^N.
f differentiable net <=>
(f has_vector_derivative (vector_derivative f net)) net`,
REPEAT GEN_TAC THEN REWRITE_TAC[vector_derivative] THEN
CONV_TAC(RAND_CONV SELECT_CONV) THEN
SIMP_TAC[FRECHET_DERIVATIVE_WORKS; has_vector_derivative] THEN EQ_TAC THENL
[ALL_TAC; MESON_TAC[FRECHET_DERIVATIVE_WORKS; differentiable]] THEN
DISCH_TAC THEN EXISTS_TAC `column 1 (jacobian (f:real^1->real^N) net)` THEN
FIRST_ASSUM MP_TAC THEN MATCH_MP_TAC EQ_IMP THEN AP_THM_TAC THEN
AP_TERM_TAC THEN REWRITE_TAC[jacobian] THEN
MATCH_MP_TAC LINEAR_FROM_REALS THEN
RULE_ASSUM_TAC(REWRITE_RULE[has_derivative]) THEN ASM_REWRITE_TAC[]);;
let VECTOR_DERIVATIVE_UNIQUE_AT = prove
(`!f:real^1->real^N x f' f''.
(f has_vector_derivative f') (at x) /\
(f has_vector_derivative f'') (at x)
==> f' = f''`,
REWRITE_TAC[has_vector_derivative; drop] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`f:real^1->real^N`;
`\x. drop x % (f':real^N)`; `\x. drop x % (f'':real^N)`;
`x:real^1`] FRECHET_DERIVATIVE_UNIQUE_AT) THEN
ASM_SIMP_TAC[DIMINDEX_1; LE_ANTISYM; drop] THEN
REWRITE_TAC[FUN_EQ_THM] THEN DISCH_THEN(MP_TAC o SPEC `vec 1:real^1`) THEN
SIMP_TAC[VEC_COMPONENT; DIMINDEX_1; ARITH; VECTOR_MUL_LID]);;
let HAS_VECTOR_DERIVATIVE_UNIQUE_AT = prove
(`!f:real^1->real^N f' x.
(f has_vector_derivative f') (at x)
==> vector_derivative f (at x) = f'`,
REPEAT STRIP_TAC THEN MATCH_MP_TAC VECTOR_DERIVATIVE_UNIQUE_AT THEN
MAP_EVERY EXISTS_TAC [`f:real^1->real^N`; `x:real^1`] THEN
ASM_REWRITE_TAC[vector_derivative] THEN CONV_TAC SELECT_CONV THEN
ASM_MESON_TAC[]);;
let VECTOR_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL = prove
(`!f:real^1->real^N a b x f' f''.
drop a < drop b /\
x IN interval [a,b] /\
(f has_vector_derivative f') (at x within interval [a,b]) /\
(f has_vector_derivative f'') (at x within interval [a,b])
==> f' = f''`,
REWRITE_TAC[has_vector_derivative; drop] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL [`f:real^1->real^N`;
`\x. drop x % (f':real^N)`; `\x. drop x % (f'':real^N)`;
`x:real^1`; `a:real^1`; `b:real^1`]
FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL) THEN
ASM_SIMP_TAC[DIMINDEX_1; LE_ANTISYM; drop] THEN
ANTS_TAC THENL [ASM_MESON_TAC[]; ALL_TAC] THEN REWRITE_TAC[FUN_EQ_THM] THEN
DISCH_THEN(MP_TAC o SPEC `vec 1:real^1`) THEN
SIMP_TAC[VEC_COMPONENT; DIMINDEX_1; ARITH; VECTOR_MUL_LID]);;
let VECTOR_DERIVATIVE_AT = prove
(`(f has_vector_derivative f') (at x) ==> vector_derivative f (at x) = f'`,
ASM_MESON_TAC[VECTOR_DERIVATIVE_UNIQUE_AT;
VECTOR_DERIVATIVE_WORKS; differentiable; has_vector_derivative]);;
let VECTOR_DERIVATIVE_WITHIN_CLOSED_INTERVAL = prove
(`!f:real^1->real^N f' x a b.
drop a < drop b /\ x IN interval[a,b] /\
(f has_vector_derivative f') (at x within interval [a,b])
==> vector_derivative f (at x within interval [a,b]) = f'`,
ASM_MESON_TAC[VECTOR_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL;
VECTOR_DERIVATIVE_WORKS; differentiable; has_vector_derivative]);;
let HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET = prove
(`!f s t x. (f has_vector_derivative f') (at x within s) /\ t SUBSET s
==> (f has_vector_derivative f') (at x within t)`,
REWRITE_TAC[has_vector_derivative; HAS_DERIVATIVE_WITHIN_SUBSET]);;
let HAS_VECTOR_DERIVATIVE_CONST = prove
(`!c net. ((\x. c) has_vector_derivative vec 0) net`,
REWRITE_TAC[has_vector_derivative] THEN
REWRITE_TAC[VECTOR_MUL_RZERO; HAS_DERIVATIVE_CONST]);;
let VECTOR_DERIVATIVE_CONST_AT = prove
(`!c:real^N a. vector_derivative (\x. c) (at a) = vec 0`,
REPEAT GEN_TAC THEN MATCH_MP_TAC HAS_VECTOR_DERIVATIVE_UNIQUE_AT THEN
REWRITE_TAC[HAS_VECTOR_DERIVATIVE_CONST]);;
let HAS_VECTOR_DERIVATIVE_ID = prove
(`!net. ((\x. x) has_vector_derivative (vec 1)) net`,
REWRITE_TAC[has_vector_derivative] THEN
SUBGOAL_THEN `(\x. drop x % vec 1) = (\x. x)`
(fun th -> REWRITE_TAC[HAS_DERIVATIVE_ID; th]) THEN
REWRITE_TAC[FUN_EQ_THM; GSYM DROP_EQ; DROP_CMUL; DROP_VEC] THEN
REAL_ARITH_TAC);;
let HAS_VECTOR_DERIVATIVE_CMUL = prove
(`!f f' net c. (f has_vector_derivative f') net
==> ((\x. c % f(x)) has_vector_derivative (c % f')) net`,
SIMP_TAC[has_vector_derivative] THEN
ONCE_REWRITE_TAC[VECTOR_ARITH `a % b % x = b % a % x`] THEN
SIMP_TAC[HAS_DERIVATIVE_CMUL]);;
let HAS_VECTOR_DERIVATIVE_CMUL_EQ = prove
(`!f f' net c.
~(c = &0)
==> (((\x. c % f(x)) has_vector_derivative (c % f')) net <=>
(f has_vector_derivative f') net)`,
REPEAT STRIP_TAC THEN EQ_TAC THEN
DISCH_THEN(MP_TAC o MATCH_MP HAS_VECTOR_DERIVATIVE_CMUL) THENL
[DISCH_THEN(MP_TAC o SPEC `inv(c):real`);
DISCH_THEN(MP_TAC o SPEC `c:real`)] THEN
ASM_SIMP_TAC[VECTOR_MUL_ASSOC; REAL_MUL_LINV; VECTOR_MUL_LID; ETA_AX]);;
let HAS_VECTOR_DERIVATIVE_NEG = prove
(`!f f' net. (f has_vector_derivative f') net
==> ((\x. --(f(x))) has_vector_derivative (--f')) net`,
SIMP_TAC[has_vector_derivative; VECTOR_MUL_RNEG; HAS_DERIVATIVE_NEG]);;
let HAS_VECTOR_DERIVATIVE_NEG_EQ = prove
(`!f f' net. ((\x. --(f(x))) has_vector_derivative --f') net <=>
(f has_vector_derivative f') net`,
SIMP_TAC[has_vector_derivative; HAS_DERIVATIVE_NEG_EQ; VECTOR_MUL_RNEG]);;
let HAS_VECTOR_DERIVATIVE_ADD = prove
(`!f f' g g' net.
(f has_vector_derivative f') net /\ (g has_vector_derivative g') net
==> ((\x. f(x) + g(x)) has_vector_derivative (f' + g')) net`,
SIMP_TAC[has_vector_derivative; VECTOR_ADD_LDISTRIB; HAS_DERIVATIVE_ADD]);;
let HAS_VECTOR_DERIVATIVE_SUB = prove
(`!f f' g g' net.
(f has_vector_derivative f') net /\ (g has_vector_derivative g') net
==> ((\x. f(x) - g(x)) has_vector_derivative (f' - g')) net`,
SIMP_TAC[has_vector_derivative; VECTOR_SUB_LDISTRIB; HAS_DERIVATIVE_SUB]);;
let HAS_VECTOR_DERIVATIVE_BILINEAR_WITHIN = prove
(`!h:real^M->real^N->real^P f g f' g' x s.
(f has_vector_derivative f') (at x within s) /\
(g has_vector_derivative g') (at x within s) /\
bilinear h
==> ((\x. h (f x) (g x)) has_vector_derivative
(h (f x) g' + h f' (g x))) (at x within s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_vector_derivative] THEN
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP HAS_DERIVATIVE_BILINEAR_WITHIN) THEN
RULE_ASSUM_TAC(REWRITE_RULE[bilinear; linear]) THEN
ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB]);;
let HAS_VECTOR_DERIVATIVE_BILINEAR_AT = prove
(`!h:real^M->real^N->real^P f g f' g' x.
(f has_vector_derivative f') (at x) /\
(g has_vector_derivative g') (at x) /\
bilinear h
==> ((\x. h (f x) (g x)) has_vector_derivative
(h (f x) g' + h f' (g x))) (at x)`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_vector_derivative] THEN
DISCH_TAC THEN
FIRST_ASSUM(MP_TAC o MATCH_MP HAS_DERIVATIVE_BILINEAR_AT) THEN
RULE_ASSUM_TAC(REWRITE_RULE[bilinear; linear]) THEN
ASM_REWRITE_TAC[VECTOR_ADD_LDISTRIB]);;
let HAS_VECTOR_DERIVATIVE_AT_WITHIN = prove
(`!f x s. (f has_vector_derivative f') (at x)
==> (f has_vector_derivative f') (at x within s)`,
SIMP_TAC[has_vector_derivative; HAS_DERIVATIVE_AT_WITHIN]);;
let HAS_VECTOR_DERIVATIVE_TRANSFORM_WITHIN = prove
(`!f f' g x s d.
&0 < d /\ x IN s /\
(!x'. x' IN s /\ dist (x',x) < d ==> f x' = g x') /\
(f has_vector_derivative f') (at x within s)
==> (g has_vector_derivative f') (at x within s)`,
REWRITE_TAC[has_vector_derivative; HAS_DERIVATIVE_TRANSFORM_WITHIN]);;
let HAS_VECTOR_DERIVATIVE_TRANSFORM_AT = prove
(`!f f' g x d.
&0 < d /\ (!x'. dist (x',x) < d ==> f x' = g x') /\
(f has_vector_derivative f') (at x)
==> (g has_vector_derivative f') (at x)`,
REWRITE_TAC[has_vector_derivative; HAS_DERIVATIVE_TRANSFORM_AT]);;
let HAS_VECTOR_DERIVATIVE_TRANSFORM_WITHIN_OPEN = prove
(`!f g s x.
open s /\ x IN s /\
(!y. y IN s ==> f y = g y) /\
(f has_vector_derivative f') (at x)
==> (g has_vector_derivative f') (at x)`,
REWRITE_TAC[has_vector_derivative; HAS_DERIVATIVE_TRANSFORM_WITHIN_OPEN]);;
let VECTOR_DIFF_CHAIN_AT = prove
(`!f g f' g' x.
(f has_vector_derivative f') (at x) /\
(g has_vector_derivative g') (at (f x))
==> ((g o f) has_vector_derivative (drop f' % g')) (at x)`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_vector_derivative] THEN
DISCH_THEN(MP_TAC o MATCH_MP DIFF_CHAIN_AT) THEN
REWRITE_TAC[o_DEF; DROP_CMUL; GSYM VECTOR_MUL_ASSOC]);;
let VECTOR_DIFF_CHAIN_WITHIN = prove
(`!f g f' g' s x.
(f has_vector_derivative f') (at x within s) /\
(g has_vector_derivative g') (at (f x) within IMAGE f s)
==> ((g o f) has_vector_derivative (drop f' % g')) (at x within s)`,
REPEAT GEN_TAC THEN REWRITE_TAC[has_vector_derivative] THEN
DISCH_THEN(MP_TAC o MATCH_MP DIFF_CHAIN_WITHIN) THEN
REWRITE_TAC[o_DEF; DROP_CMUL; GSYM VECTOR_MUL_ASSOC]);;
(* ------------------------------------------------------------------------- *)
(* Various versions of Kachurovskii's theorem. *)
(* ------------------------------------------------------------------------- *)
let CONVEX_ON_DERIVATIVE_SECANT_IMP = prove
(`!f f' s x y:real^N.
f convex_on s /\ segment[x,y] SUBSET s /\
((lift o f) has_derivative (lift o f')) (at x within s)
==> f'(y - x) <= f y - f x`,
REPEAT STRIP_TAC THEN
SUBGOAL_THEN `(x:real^N) IN s /\ (y:real^N) IN s` ASSUME_TAC THENL
[ASM_MESON_TAC[SUBSET; ENDS_IN_SEGMENT]; ALL_TAC] THEN
FIRST_X_ASSUM(MP_TAC o GEN_REWRITE_RULE I [has_derivative_within]) THEN
REWRITE_TAC[LIM_WITHIN; DIST_0; o_THM] THEN
REWRITE_TAC[GSYM LIFT_ADD; GSYM LIFT_SUB; GSYM LIFT_CMUL; NORM_LIFT] THEN
STRIP_TAC THEN ASM_CASES_TAC `y:real^N = x` THENL
[FIRST_X_ASSUM(MP_TAC o MATCH_MP LINEAR_0) THEN
REWRITE_TAC[o_THM; VECTOR_SUB_REFL; GSYM DROP_EQ; DROP_VEC; LIFT_DROP] THEN
ASM_SIMP_TAC[REAL_SUB_REFL; REAL_LE_REFL; VECTOR_SUB_REFL];
ALL_TAC] THEN
ABBREV_TAC `e = (f':real^N->real)(y - x) - (f y - f x)` THEN
ASM_CASES_TAC `&0 < e` THENL [ALL_TAC; ASM_REAL_ARITH_TAC] THEN
FIRST_X_ASSUM(MP_TAC o SPEC `e / &2 / norm(y - x:real^N)`) THEN
ASM_SIMP_TAC[REAL_LT_DIV; REAL_HALF; NORM_POS_LT; VECTOR_SUB_EQ] THEN
DISCH_THEN(X_CHOOSE_THEN `d:real` (CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN
ABBREV_TAC `u = min (&1 / &2) (d / &2 / norm (y - x:real^N))` THEN
SUBGOAL_THEN `&0 < u /\ u < &1` STRIP_ASSUME_TAC THENL
[EXPAND_TAC "u" THEN REWRITE_TAC[REAL_LT_MIN; REAL_MIN_LT] THEN
ASM_SIMP_TAC[REAL_LT_DIV; NORM_POS_LT; REAL_HALF; VECTOR_SUB_EQ] THEN
CONV_TAC REAL_RAT_REDUCE_CONV;
ALL_TAC] THEN
ABBREV_TAC `z:real^N = (&1 - u) % x + u % y` THEN
SUBGOAL_THEN `(z:real^N) IN segment(x,y)` MP_TAC THENL
[ASM_MESON_TAC[IN_SEGMENT]; ALL_TAC] THEN
SIMP_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY; DE_MORGAN_THM] THEN
STRIP_TAC THEN DISCH_THEN(MP_TAC o SPEC `z:real^N`) THEN
SUBGOAL_THEN `(z:real^N) IN s` ASSUME_TAC THENL [ASM SET_TAC[]; ALL_TAC] THEN
ANTS_TAC THENL
[ASM_SIMP_TAC[DIST_POS_LT] THEN
EXPAND_TAC "z" THEN REWRITE_TAC[dist; NORM_MUL; VECTOR_ARITH
`((&1 - u) % x + u % y) - x:real^N = u % (y - x)`] THEN
ASM_SIMP_TAC[GSYM REAL_LT_RDIV_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
FIRST_ASSUM(MP_TAC o GEN_REWRITE_RULE I [CONVEX_ON_LEFT_SECANT]) THEN
DISCH_THEN(MP_TAC o SPECL [`x:real^N`; `y:real^N`; `z:real^N`]) THEN
ASM_REWRITE_TAC[open_segment; IN_DIFF; IN_INSERT; NOT_IN_EMPTY] THEN
SIMP_TAC[REAL_ARITH `inv y * (z - (x + d)):real = (z - x) / y - d / y`] THEN
REWRITE_TAC[IMP_IMP] THEN DISCH_THEN(MP_TAC o MATCH_MP (REAL_ARITH
`z <= y / n /\ abs(z - d) < e / n ==> d <= (y + e) / n`)) THEN
SUBGOAL_THEN
`(f':real^N->real)(z - x) / norm(z - x) = f'(y - x) / norm(y - x)`
SUBST1_TAC THENL
[EXPAND_TAC "z" THEN
REWRITE_TAC[VECTOR_ARITH
`((&1 - u) % x + u % y) - x:real^N = u % (y - x)`] THEN
FIRST_ASSUM(MP_TAC o MATCH_MP LINEAR_CMUL) THEN
DISCH_THEN(MP_TAC o SPECL [`u:real`; `y - x:real^N`]) THEN
ASM_REWRITE_TAC[GSYM LIFT_CMUL; o_THM; LIFT_EQ] THEN
DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[NORM_MUL] THEN
ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE] THEN
REWRITE_TAC[real_div; REAL_INV_MUL; REAL_MUL_ASSOC] THEN
AP_THM_TAC THEN AP_TERM_TAC THEN REWRITE_TAC[GSYM REAL_MUL_ASSOC] THEN
REWRITE_TAC[GSYM real_div] THEN MATCH_MP_TAC REAL_DIV_LMUL THEN
ASM_REAL_ARITH_TAC;
ASM_SIMP_TAC[REAL_LE_DIV2_EQ; NORM_POS_LT; VECTOR_SUB_EQ] THEN
ASM_REAL_ARITH_TAC]);;
let CONVEX_ON_SECANT_DERIVATIVE_IMP = prove
(`!f f' s x y:real^N.
f convex_on s /\ segment[x,y] SUBSET s /\
((lift o f) has_derivative (lift o f')) (at y within s)
==> f y - f x <= f'(y - x)`,
ONCE_REWRITE_TAC[SEGMENT_SYM] THEN REPEAT STRIP_TAC THEN
MP_TAC(ISPECL
[`f:real^N->real`; `f':real^N->real`; `s:real^N->bool`;
`y:real^N`; `x:real^N`] CONVEX_ON_DERIVATIVE_SECANT_IMP) THEN
ASM_REWRITE_TAC[] THEN ONCE_REWRITE_TAC[SEGMENT_SYM] THEN
MATCH_MP_TAC(REAL_ARITH
`f' = --f'' ==> f' <= x - y ==> y - x <= f''`) THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM VECTOR_NEG_SUB] THEN
GEN_REWRITE_TAC I [GSYM LIFT_EQ] THEN REWRITE_TAC[LIFT_NEG] THEN
SPEC_TAC(`y - x:real^N`,`z:real^N`) THEN
MATCH_MP_TAC(REWRITE_RULE[RIGHT_FORALL_IMP_THM] LINEAR_NEG) THEN
REWRITE_TAC[GSYM o_DEF] THEN ASM_MESON_TAC[has_derivative]);;
let CONVEX_ON_DERIVATIVES_IMP = prove
(`!f f'x f'y s x y:real^N.
f convex_on s /\ segment[x,y] SUBSET s /\
((lift o f) has_derivative (lift o f'x)) (at x within s) /\
((lift o f) has_derivative (lift o f'y)) (at y within s)
==> f'x(y - x) <= f'y(y - x)`,
ASM_MESON_TAC[CONVEX_ON_DERIVATIVE_SECANT_IMP;
CONVEX_ON_SECANT_DERIVATIVE_IMP;
SEGMENT_SYM; REAL_LE_TRANS]);;
let CONVEX_ON_DERIVATIVE_SECANT,CONVEX_ON_DERIVATIVES =
(CONJ_PAIR o prove)
(`(!f f' s:real^N->bool.
convex s /\
(!x. x IN s ==> ((lift o f) has_derivative (lift o f'(x)))
(at x within s))
==> (f convex_on s <=>
!x y. x IN s /\ y IN s ==> f'(x)(y - x) <= f y - f x)) /\
(!f f' s:real^N->bool.
convex s /\
(!x. x IN s ==> ((lift o f) has_derivative (lift o f'(x)))
(at x within s))
==> (f convex_on s <=>
!x y. x IN s /\ y IN s ==> f'(x)(y - x) <= f'(y)(y - x)))`,
REWRITE_TAC[AND_FORALL_THM] THEN REPEAT GEN_TAC THEN
REWRITE_TAC[TAUT `(a ==> b) /\ (a ==> c) <=> a ==> b /\ c`] THEN
STRIP_TAC THEN MATCH_MP_TAC(TAUT
`(a ==> b) /\ (b ==> c) /\ (c ==> a) ==> (a <=> b) /\ (a <=> c)`) THEN
REPEAT CONJ_TAC THENL
[REPEAT STRIP_TAC THEN MATCH_MP_TAC CONVEX_ON_DERIVATIVE_SECANT_IMP THEN
EXISTS_TAC `s:real^N->bool` THEN ASM_SIMP_TAC[ETA_AX] THEN
ASM_MESON_TAC[CONVEX_CONTAINS_SEGMENT];
DISCH_TAC THEN MAP_EVERY X_GEN_TAC [`x:real^N`; `y:real^N`] THEN
STRIP_TAC THEN FIRST_X_ASSUM(fun th ->
MP_TAC(ISPECL [`x:real^N`; `y:real^N`] th) THEN
MP_TAC(ISPECL [`y:real^N`; `x:real^N`] th)) THEN
ASM_REWRITE_TAC[] THEN MATCH_MP_TAC(REAL_ARITH
`f''' = --f'' ==> f''' <= x - y ==> f' <= y - x ==> f' <= f''`) THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM VECTOR_NEG_SUB] THEN
GEN_REWRITE_TAC I [GSYM LIFT_EQ] THEN REWRITE_TAC[LIFT_NEG] THEN
SPEC_TAC(`y - x:real^N`,`z:real^N`) THEN
MATCH_MP_TAC(REWRITE_RULE[RIGHT_FORALL_IMP_THM] LINEAR_NEG) THEN
REWRITE_TAC[GSYM o_DEF] THEN REWRITE_TAC[GSYM I_DEF; I_O_ID] THEN
ASM_MESON_TAC[has_derivative];
ALL_TAC] THEN
DISCH_TAC THEN REWRITE_TAC[convex_on] THEN
MAP_EVERY X_GEN_TAC [`a:real^N`; `b:real^N`] THEN
ONCE_REWRITE_TAC[SWAP_FORALL_THM] THEN
REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e <=> e /\ a /\ b /\ c /\ d`] THEN
REWRITE_TAC[IMP_CONJ; REAL_ARITH `u + v = &1 <=> u = &1 - v`] THEN
REWRITE_TAC[FORALL_UNWIND_THM2; REAL_SUB_LE] THEN X_GEN_TAC `u:real` THEN
REPEAT STRIP_TAC THEN
ASM_CASES_TAC `u = &0` THEN
ASM_SIMP_TAC[REAL_SUB_RZERO; VECTOR_MUL_LZERO; VECTOR_MUL_LID; REAL_LE_REFL;
REAL_MUL_LZERO; REAL_MUL_LID; VECTOR_ADD_RID; REAL_ADD_RID] THEN
ASM_CASES_TAC `u = &1` THEN
ASM_SIMP_TAC[REAL_SUB_REFL; VECTOR_MUL_LZERO; VECTOR_MUL_LID; REAL_LE_REFL;
REAL_MUL_LZERO; REAL_MUL_LID; VECTOR_ADD_LID; REAL_ADD_LID] THEN
SUBGOAL_THEN `&0 < u /\ u < &1` STRIP_ASSUME_TAC THENL
[ASM_REWRITE_TAC[REAL_LT_LE]; ALL_TAC] THEN
MP_TAC(ISPECL
[`lift o (f:real^N->real) o (\u. (&1 - drop u) % a + drop u % b)`;
`\x:real^1. lift o f'((&1 - drop x) % a + drop x % b) o
(\u. --(drop u) % a + drop u % b:real^N)`] MVT_VERY_SIMPLE) THEN
DISCH_THEN(fun th ->
MP_TAC(ISPECL [`vec 0:real^1`; `lift u`] th) THEN
MP_TAC(ISPECL [`lift u`; `vec 1:real^1`] th)) THEN
ASM_REWRITE_TAC[LIFT_DROP; o_THM] THEN
ASM_SIMP_TAC[DROP_VEC; VECTOR_MUL_LZERO; REAL_SUB_RZERO; REAL_LT_IMP_LE;
VECTOR_ADD_RID; VECTOR_MUL_LID; VECTOR_SUB_RZERO] THEN
MATCH_MP_TAC(TAUT
`(a1 /\ a2) /\ (b1 ==> b2 ==> c) ==> (a1 ==> b1) ==> (a2 ==> b2) ==> c`) THEN
CONJ_TAC THENL
[CONJ_TAC THEN X_GEN_TAC `v:real^1` THEN DISCH_TAC THEN
(REWRITE_TAC[o_ASSOC] THEN MATCH_MP_TAC DIFF_CHAIN_WITHIN THEN
REWRITE_TAC[] THEN CONJ_TAC THENL
[MATCH_MP_TAC HAS_DERIVATIVE_ADD THEN CONJ_TAC THENL
[ONCE_REWRITE_TAC[VECTOR_ARITH `(&1 - a) % x:real^N = x + --a % x`;
VECTOR_ARITH `--u % a:real^N = vec 0 + --u % a`] THEN
MATCH_MP_TAC HAS_DERIVATIVE_ADD THEN
REWRITE_TAC[HAS_DERIVATIVE_CONST];
ALL_TAC] THEN
MATCH_MP_TAC HAS_DERIVATIVE_LINEAR THEN
REWRITE_TAC[linear; DROP_ADD; DROP_CMUL] THEN VECTOR_ARITH_TAC;
MATCH_MP_TAC HAS_DERIVATIVE_WITHIN_SUBSET THEN
EXISTS_TAC `s:real^N->bool` THEN CONJ_TAC THENL
[FIRST_X_ASSUM MATCH_MP_TAC;
REWRITE_TAC[SUBSET; FORALL_IN_IMAGE] THEN GEN_TAC THEN DISCH_TAC] THEN
FIRST_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
RULE_ASSUM_TAC(REWRITE_RULE[IN_INTERVAL_1; LIFT_DROP; DROP_VEC]) THEN
ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC]);
REWRITE_TAC[REAL_SUB_REFL; VECTOR_MUL_LZERO; VECTOR_ADD_LID] THEN
REWRITE_TAC[EXISTS_LIFT; LIFT_DROP; IN_INTERVAL_1; DROP_VEC] THEN
REWRITE_TAC[GSYM LIFT_SUB; LIFT_EQ] THEN
REWRITE_TAC[DROP_SUB; DROP_VEC; LIFT_DROP] THEN
REWRITE_TAC[VECTOR_ARITH `--u % a + u % b:real^N = u % (b - a)`] THEN
REWRITE_TAC[LEFT_IMP_EXISTS_THM; RIGHT_IMP_FORALL_THM] THEN
MAP_EVERY X_GEN_TAC [`w:real`; `v:real`] THEN
DISCH_THEN(CONJUNCTS_THEN2 STRIP_ASSUME_TAC MP_TAC) THEN
ONCE_REWRITE_TAC[TAUT `a ==> b /\ c ==> d <=> b ==> a ==> c ==> d`] THEN
STRIP_TAC THEN REWRITE_TAC[IMP_IMP] THEN
DISCH_THEN(CONJUNCTS_THEN2 (MP_TAC o AP_TERM `(*) (u:real)`)
(MP_TAC o AP_TERM `(*) (&1 - u:real)`)) THEN
MATCH_MP_TAC(REAL_ARITH
`f1 <= f2 /\ (xa <= xb ==> a <= b)
==> xa = f1 ==> xb = f2 ==> a <= b`) THEN
CONJ_TAC THENL [ALL_TAC; REAL_ARITH_TAC] THEN
SUBGOAL_THEN
`((&1 - v) % a + v % b:real^N) IN s /\
((&1 - w) % a + w % b:real^N) IN s`
STRIP_ASSUME_TAC THENL
[CONJ_TAC THEN
FIRST_X_ASSUM(MATCH_MP_TAC o GEN_REWRITE_RULE I [CONVEX_ALT]) THEN
ASM_REWRITE_TAC[] THEN ASM_REAL_ARITH_TAC;
ALL_TAC] THEN
SUBGOAL_THEN
`linear(lift o (f'((&1 - v) % a + v % b:real^N):real^N->real)) /\
linear(lift o (f'((&1 - w) % a + w % b:real^N):real^N->real))`
MP_TAC THENL [ASM_MESON_TAC[has_derivative]; ALL_TAC] THEN
DISCH_THEN(CONJUNCTS_THEN(MP_TAC o MATCH_MP LINEAR_CMUL)) THEN
ASM_REWRITE_TAC[o_THM; GSYM LIFT_NEG; GSYM LIFT_CMUL; LIFT_EQ] THEN
REPEAT DISCH_TAC THEN ASM_REWRITE_TAC[] THEN
ONCE_REWRITE_TAC[REAL_ARITH `(&1 - u) * u * x = u * (&1 - u) * x`] THEN
REPEAT(MATCH_MP_TAC REAL_LE_LMUL THEN
CONJ_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC]) THEN
FIRST_X_ASSUM(MP_TAC o SPECL
[`(&1 - v) % a + v % b:real^N`; `(&1 - w) % a + w % b:real^N`]) THEN
ASM_REWRITE_TAC[VECTOR_ARITH
`((&1 - v) % a + v % b) - ((&1 - w) % a + w % b):real^N =
(v - w) % (b - a)`] THEN
ASM_CASES_TAC `v:real = w` THEN ASM_SIMP_TAC[REAL_LE_REFL] THEN
SUBGOAL_THEN `&0 < w - v` (fun th -> SIMP_TAC[th; REAL_LE_LMUL_EQ]) THEN
ASM_REAL_ARITH_TAC]);;
let CONVEX_ON_SECANT_DERIVATIVE = prove
(`!f f' s:real^N->bool.
convex s /\
(!x. x IN s ==> ((lift o f) has_derivative (lift o f'(x)))
(at x within s))
==> (f convex_on s <=>
!x y. x IN s /\ y IN s ==> f y - f x <= f'(y)(y - x))`,
REPEAT GEN_TAC THEN DISCH_TAC THEN
FIRST_ASSUM(SUBST1_TAC o MATCH_MP CONVEX_ON_DERIVATIVE_SECANT) THEN
GEN_REWRITE_TAC RAND_CONV [SWAP_FORALL_THM] THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `x:real^N` THEN REWRITE_TAC[] THEN
AP_TERM_TAC THEN GEN_REWRITE_TAC I [FUN_EQ_THM] THEN
X_GEN_TAC `y:real^N` THEN REWRITE_TAC[] THEN
MAP_EVERY ASM_CASES_TAC [`(x:real^N) IN s`; `(y:real^N) IN s`] THEN
ASM_REWRITE_TAC[] THEN
MATCH_MP_TAC(REAL_ARITH
`f' = --f'' ==> (f' <= y - x <=> x - y <= f'')`) THEN
GEN_REWRITE_TAC (LAND_CONV o RAND_CONV) [GSYM VECTOR_NEG_SUB] THEN
GEN_REWRITE_TAC I [GSYM LIFT_EQ] THEN REWRITE_TAC[LIFT_NEG] THEN
SPEC_TAC(`x - y:real^N`,`z:real^N`) THEN
MATCH_MP_TAC(REWRITE_RULE[RIGHT_FORALL_IMP_THM] LINEAR_NEG) THEN
REWRITE_TAC[GSYM o_DEF] THEN
REWRITE_TAC[GSYM I_DEF; I_O_ID] THEN ASM_MESON_TAC[has_derivative]);;