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divisionRework.sml
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Theorem divisionErrorBounded:
!(e1lo e1hi nR1 e2lo e2hi nR2 nF1 nF2 err1 err2:real).
(e1lo <= nR1) /\ (nR1 <= e1hi) /\
(e2lo <= nR2) /\ (nR2 <= e2hi) /\
abs (nR1 − nF1) <= err1 /\
abs (nR2 − nF2) <= err2 /\
0 <= err1 /\
0 <= err2 /\
contained nF1 (widenInterval (e1lo,e1hi) err1) /\
contained nF2 (widenInterval (e2lo,e2hi) err2) /\
noDivzero e2hi e2lo /\
noDivzero (SND (widenInterval (e2lo,e2hi) err2))
(FST (widenInterval (e2lo,e2hi) err2)) ==>
abs (nR1 / nR2 − nF1 / nF2) ≤
maxAbs (e1lo,e1hi) *
(1 /
(minAbsFun (widenInterval (e2lo,e2hi) err2) *
minAbsFun (widenInterval (e2lo,e2hi) err2)) * err2) +
maxAbs (invertInterval (e2lo,e2hi)) * err1 +
err1 *
(1 /
(minAbsFun (widenInterval (e2lo,e2hi) err2) *
minAbsFun (widenInterval (e2lo,e2hi) err2)) * err2)
Proof
rpt strip_tac
\\ `contained (inv nR2) (invertInterval (e2lo, e2hi))`
by (irule interval_inversion_valid \\ conj_tac
\\ fs[contained_def, IVlo_def, IVhi_def, noDivzero_def])
\\ `contained (inv nF2) (invertInterval (widenInterval (e2lo, e2hi) err2))`
by (irule interval_inversion_valid \\ conj_tac
\\ fs[contained_def, IVlo_def, IVhi_def, noDivzero_def])
\\ `nR1 <= maxAbs (e1lo, e1hi)`
by (irule contained_leq_maxAbs_val
\\ fs[contained_def, IVlo_def, IVhi_def])
\\ `inv nR2 <= maxAbs (invertInterval(e2lo, e2hi))`
by (irule contained_leq_maxAbs_val
\\ fs[contained_def, IVlo_def, IVhi_def])
\\ `-nR1 <= maxAbs (e1lo, e1hi)`
by (irule contained_leq_maxAbs_neg_val
\\ fs[contained_def, IVlo_def, IVhi_def])
\\ `- inv nR2 <= maxAbs (invertInterval (e2lo, e2hi))`
by (irule contained_leq_maxAbs_neg_val
\\ fs[contained_def, IVlo_def, IVhi_def])
\\ `nR1 * err2 <= maxAbs (e1lo, e1hi) * err2`
by (irule REAL_LE_RMUL_IMP \\ fs[])
\\ `-nR1 * err2 <= maxAbs (e1lo, e1hi) * err2`
by (irule REAL_LE_RMUL_IMP \\ fs[])
\\ `inv nR2 * err1 <= maxAbs (invertInterval(e2lo, e2hi)) * err1`
by (irule REAL_LE_RMUL_IMP \\ fs[])
\\ `- inv nR2 * err1 <= maxAbs (invertInterval(e2lo, e2hi)) * err1`
by (irule REAL_LE_RMUL_IMP \\ fs[])
\\ `- (err1 * err2) <= err1 * err2`
by (rewrite_tac[REAL_NEG_LMUL] \\ irule REAL_LE_RMUL_IMP
\\ REAL_ASM_ARITH_TAC)
\\ `0 <= maxAbs (e1lo, e1hi) * err2` by REAL_ASM_ARITH_TAC
\\ `0 <= maxAbs (invertInterval (e2lo, e2hi)) * err1` by REAL_ASM_ARITH_TAC
\\ `maxAbs (e1lo, e1hi) * err2 <=
maxAbs (e1lo, e1hi) * err2 + maxAbs (invertInterval (e2lo, e2hi)) * err1`
by (REAL_ASM_ARITH_TAC)
\\ `maxAbs (e1lo, e1hi) * err2 + maxAbs (invertInterval(e2lo, e2hi)) * err1 <=
maxAbs (e1lo, e1hi) * err2 + maxAbs (invertInterval (e2lo, e2hi)) * err1 + err1 * err2`
by REAL_ASM_ARITH_TAC
\\ ntac 2 (first_x_assum (fn thm1 => (mp_then Any assume_tac REAL_LE_ABS_TRANS thm1 \\ mp_tac thm1)))
\\ rpt strip_tac
(* Case distinction for divisor range
positive or negative in float and real valued execution *)
\\ fs [IVlo_def, IVhi_def, widenInterval_def, contained_def, noDivzero_def]
(* Solve trivial, bogus cases *)
\\ TRY (`e2hi < e2lo` by REAL_ASM_ARITH_TAC \\ REAL_ASM_ARITH_TAC)
(* e2hi + err2 < 0, e2hi < 0;
The range of the divisor lies in the range from -infinity until 0 *)
>- (
`abs (inv nR2 - inv nF2) <= err2 * inv ((e2hi + err2) * (e2hi + err2))`
by (irule err_prop_inversion_neg \\ fs[] \\ qexists_tac `e2lo` \\ simp[])
\\ fs [widenInterval_def, IVlo_def, IVhi_def]
\\ `minAbsFun (e2lo - err2, e2hi + err2) = - (e2hi + err2)`
by (irule minAbs_negative_iv_is_hi \\ REAL_ASM_ARITH_TAC)
\\ simp[]
\\ qpat_x_assum `minAbsFun _ = _ ` kall_tac
\\ `nF1 <= nR1 + err1` by REAL_ASM_ARITH_TAC
\\ `nR1 - err1 <= nF1` by REAL_ASM_ARITH_TAC
\\ `(0 < nR2 - nF2 /\ nR2 - nF2 <= err2) \/ (nR2 - nF2 <= 0 /\ - (nR2 - nF2) <= err2)`
by REAL_ASM_ARITH_TAC
(* 0 < nR2 - nF2; nR2 - nF2 ≤ err2 *)
>- (
`nF2 < nR2` by REAL_ASM_ARITH_TAC
\\ qpat_x_assum `nF2 < nR2` (fn thm => assume_tac (ONCE_REWRITE_RULE [GSYM REAL_LT_NEG] thm))
\\ `inv (- nF2) < inv (- nR2)` by (irule REAL_LT_INV \\ REAL_ASM_ARITH_TAC)
\\ `inv (- nF2) = - (inv nF2)` by (irule (GSYM REAL_NEG_INV) \\ REAL_ASM_ARITH_TAC)
\\ `inv (- nR2) = - (inv nR2)` by (irule (GSYM REAL_NEG_INV) \\ REAL_ASM_ARITH_TAC)
\\ rpt (qpat_x_assum `inv (- _) = - (inv _)`
(fn thm => rule_assum_tac (fn hyp => REWRITE_RULE [thm] hyp)))
\\ `inv nR2 < inv nF2` by REAL_ASM_ARITH_TAC
\\ qpat_x_assum `- _ < - _` kall_tac
\\ `inv nR2 - inv nF2 < 0` by REAL_ASM_ARITH_TAC
\\ `- (nR2⁻¹ − nF2⁻¹) ≤ err2 * ((e2hi + err2) * (e2hi + err2))⁻¹`
by (fs[] \\ REAL_ASM_ARITH_TAC)
\\ `inv nF2 <= inv nR2 + err2 * inv ((e2hi + err2) * (e2hi + err2))`
by REAL_ASM_ARITH_TAC
\\ `inv nR2 - err2 * inv ((e2hi + err2) * (e2hi + err2)) <= inv nF2`
by REAL_ASM_ARITH_TAC
(* Next do a case distinction for the absolute value *)
\\ `! (x:real). ((abs x = x) /\ 0 <= x) \/ ((abs x = - x) /\ x < 0)` by REAL_ASM_ARITH_TAC
\\ qpat_x_assum `!x. A /\ B \/ C`
(qspec_then `(nR1:real / nR2:real) - (nF1:real / nF2:real)`
DISJ_CASES_TAC)
\\ fs[realTheory.abs]
(* 0 ≤ nR1 / nR2 - nF1 / nF2 *)
>- (
fs[real_sub, real_div]
\\ qspecl_then [`nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* nF1 < 0 *)
>- (
irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 + err2 * inv ((e2hi + err2) pow 2))`
\\ conj_tac
>- (
fs[REAL_LE_LADD] \\ rewrite_tac [REAL_NEG_LMUL]
\\ irule REAL_LE_LMUL_IMP \\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ qabbrev_tac `err_inv = err2 * inv ((e2hi + err2) pow 2)`
\\ qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* inv nR2 + err_inv < 0 *)
>- (irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD])
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
\\ rewrite_tac [real_sub, GSYM REAL_ADD_ASSOC]
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (unabbrev_all_tac
\\ fs[REAL_POW_INV]
\\ once_rewrite_tac [REAL_NEG_LMUL]
\\ once_rewrite_tac [REAL_NEG_RMUL]
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_LMUL_IMP
\\ conj_tac \\ fs[]
\\ irule REAL_LE_RMUL_IMP \\ fs[])
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (once_rewrite_tac [REAL_NEG_RMUL]
\\ real_rewrite `- (inv nR2) * err1 = err1 * - (inv nR2)`
\\ irule REAL_LE_LMUL_IMP
\\ conj_tac \\ fs[])
\\ unabbrev_all_tac
\\ fs[REAL_POW_INV]
\\ irule REAL_LE_TRANS \\ qexists_tac `0`
\\ conj_tac \\ fs[REAL_NEG_LE0, REAL_LE_MUL])
(* 0 ≤ inv nR2 + err_inv *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ irule REAL_LE_RMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
\\ unabbrev_all_tac
\\ rewrite_tac [real_sub, GSYM REAL_ADD_ASSOC]
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ real_rewrite `-nR1 * (err2 * (inv ((e2hi + err2) pow 2))) =
err2 * (-nR1 * (inv ((e2hi + err2) pow 2)))`
\\ irule REAL_LE_LMUL_IMP
\\ conj_tac \\ fs[REAL_POW_INV]
\\ irule REAL_LE_RMUL_IMP \\ fs[])
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (real_rewrite `inv nR2 * err1 = err1 * inv nR2`
\\ irule REAL_LE_LMUL_IMP
\\ conj_tac \\ fs[])
\\ fs[REAL_POW_INV])
(* 0 ≤ nF1 *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 - err2 * inv ((e2hi + err2) pow 2))`
\\ conj_tac
>- (fs[REAL_LE_LADD]
\\ once_rewrite_tac [REAL_NEG_LMUL]
\\ irule REAL_MUL_LE_COMPAT_NEG_L
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ qabbrev_tac `err_inv = (err2 * inv ((e2hi + err2) pow 2))`
\\ qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* inv nR2 - err_inv < 0 *)
>- (irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ once_rewrite_tac [REAL_MUL_COMM]
\\ irule REAL_MUL_LE_COMPAT_NEG_L
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ fs [REAL_LE_NEG])
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv) =
nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 ≤ inv nR2 - err_inv *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 - err1) * (inv nR2 - err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ irule REAL_LE_RMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ real_rewrite `nR1 * inv nR2 + - (nR1 - err1) * (inv nR2 - err_inv) =
nR1 * err_inv + inv nR2 * err1 + - err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (
irule REAL_LE_ADD2 \\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
\\ irule REAL_LE_TRANS \\ qexists_tac `0`
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* nR1 / nR2 - nF1 / nF2 < 0; Absolute value negative *)
\\ fs[real_sub, real_div]
\\ qspecl_then [`nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* nF1 < 0 *)
>- (
irule REAL_LE_TRANS
\\ qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 - err2 * inv ((e2hi + err2) pow 2))`
\\ conj_tac
>- (once_rewrite_tac [REAL_NEG_ADD, GSYM REAL_NEG_LMUL]
\\ fs[REAL_LE_LADD, real_sub])
\\ qabbrev_tac `err_inv = err2 * inv ((e2hi + err2) pow 2)`
\\ qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* inv nR2 - err_inv < 0 *)
>- (irule REAL_LE_TRANS
\\ qexists_tac `- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD, real_sub])
\\ real_rewrite `- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 ≤ inv nR2 - err_inv *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ irule REAL_LE_RMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ real_rewrite `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
\\ qunabbrev_tac `err_inv` \\ rewrite_tac [real_sub]
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2, REAL_POW_INV]
\\ once_rewrite_tac [REAL_NEG_LMUL]
\\ once_rewrite_tac [REAL_NEG_RMUL]
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_LMUL_IMP
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
\\ irule REAL_LE_TRANS \\ qexists_tac `0`
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 <= nF1 *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 + err2 * inv ((e2hi + err2) pow 2))`
\\ conj_tac
>- (fs[REAL_LE_LADD, REAL_NEG_ADD]
\\ irule REAL_LE_LMUL_IMP \\ REAL_ASM_ARITH_TAC)
\\ qabbrev_tac `err_inv = (err2 * inv ((e2hi + err2) pow 2))`
\\ qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* inv nR2 + err_inv < 0 *)
>- (irule REAL_LE_TRANS
\\ qexists_tac `-nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD, real_sub])
\\ real_rewrite `- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv) =
nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ simp[real_sub]
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_LMUL_IMP \\ fs[REAL_POW_INV]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
\\ irule REAL_LE_TRANS \\ qexists_tac `0`
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 ≤ inv nR2 + err_inv *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 + err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ irule REAL_LE_RMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ real_rewrite `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 + err_inv) =
nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* - (nR2 - nF2) ≤ err2; nR2 - nF2 < 0;
Absolute value negative *)
\\ fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2]
\\ `nR2 <= nF2` by REAL_ASM_ARITH_TAC
\\ qpat_x_assum `nR2 <= nF2` (fn thm => assume_tac (ONCE_REWRITE_RULE [GSYM REAL_LE_NEG] thm))
\\ `inv (- nR2) <= inv (- nF2)` by (irule REAL_INV_LE_ANTIMONO_IMPR \\ REAL_ASM_ARITH_TAC)
\\ `inv (- nR2) = - (inv nR2)` by (irule (GSYM REAL_NEG_INV) \\ REAL_ASM_ARITH_TAC)
\\ `inv (- nF2) = - (inv nF2)` by (irule (GSYM REAL_NEG_INV) \\ REAL_ASM_ARITH_TAC)
\\ rpt (
qpat_x_assum `inv (- _) = - (inv _)`
(fn thm => rule_assum_tac (fn hyp => REWRITE_RULE [thm] hyp)))
\\ `inv nF2 <= inv nR2` by REAL_ASM_ARITH_TAC
\\ qpat_x_assum `- _ <= - _` kall_tac
\\ `0 <= inv nR2 - inv nF2` by REAL_ASM_ARITH_TAC
\\ `(inv nR2 − inv nF2) ≤ err2 * inv ((e2hi + err2) pow 2)` by REAL_ASM_ARITH_TAC
\\ `inv nF2 <= inv nR2 + err2 * inv ((e2hi + err2) pow 2)` by REAL_ASM_ARITH_TAC
\\ `inv nR2 - err2 * inv ((e2hi + err2) pow 2) <= inv nF2` by REAL_ASM_ARITH_TAC
(* Next do a case distinction for the absolute value *)
\\ `! (x:real). ((abs x = x) /\ 0 <= x) \/ ((abs x = - x) /\ x < 0)` by REAL_ASM_ARITH_TAC
\\ qpat_x_assum `!x. A /\ B \/ C`
(qspec_then `(nR1:real / nR2:real) - (nF1:real / nF2:real)` DISJ_CASES_TAC)
\\ fs[real_sub, real_div, REAL_NEG_ADD, realTheory.abs]
(* Combine with a case split for nF1 *)
\\ qspecl_then [`nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* 0 ≤ nR1 * inv nR2 + - (nF1 * inv nF2); nF1 < 0*)
>- (
\\ irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 + err2 * inv ((e2hi + err2) * (e2hi + err2)))`
\\ conj_tac
>- (fs[REAL_LE_LADD] \\ once_rewrite_tac [REAL_NEG_LMUL]
\\ irule REAL_LE_LMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ qabbrev_tac `err_inv = (err2 * inv ((e2hi + err2) pow 2))`
\\ qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* inv nR2 + err_inv < 0 *)
>- (
irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD])
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ rewrite_tac[real_sub]
\\ irule REAL_LE_ADD2
\\ conj_tac
>- (irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
\\ irule REAL_LE_TRANS \\ qexists_tac `0`
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 ≤ inv nR2 + err_inv *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ irule REAL_LE_RMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 ≤ nR1 * inv nR2 + - (nF1 * inv nF2); 0 ≤ nF1 *)
>- (
irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 - err2 * inv ((e2hi + err2) pow 2))`
\\ conj_tac
>- (fs[REAL_LE_LADD] \\ once_rewrite_tac [REAL_NEG_LMUL]
\\ irule REAL_MUL_LE_COMPAT_NEG_L
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ qabbrev_tac `err_inv = (err2 * inv ((e2hi + err2) pow 2))` \\
\\ qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* inv nR2 - err_inv < 0 *)
>- (irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD])
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv) =
nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
\\ qunabbrev_tac `err_inv`
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* 0 <= inv nR2 - err_inv *)
\\ irule REAL_LE_TRANS
\\ qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 - err_inv)`
\\ conj_tac
>- (fs [REAL_LE_ADD]
\\ irule REAL_LE_RMUL_IMP
\\ conj_tac \\ REAL_ASM_ARITH_TAC)
\\ real_rewrite `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 - err_inv) =
nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
\\ qunabbrev_tac `err_inv` \\ rewrite_tac[real_sub]
\\ irule REAL_LE_ADD2 \\ conj_tac
>- (irule REAL_LE_ADD2
\\ conj_tac \\ TRY REAL_ASM_ARITH_TAC
\\ real_rewrite `err2 * maxAbs (e1lo, e1hi) = maxAbs (e1lo, e1hi) * err2`
\\ rewrite_tac [GSYM REAL_MUL_ASSOC]
\\ irule REAL_LE_RMUL_IMP
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
\\ rewrite_tac [REAL_NEG_LMUL]
\\ irule REAL_LE_TRANS \\ qexists_tac `0`
\\ fs[REAL_LE_POW2, REAL_LE_MUL])
(* nR1 * inv nR2 + - (nF1 * inv nF2) < 0; nF1 < 0 *)
>- (
)
\\
)
(* 0 < e2lo, 0 < e2lo - err2;
The range of the divisor is positive *)
\\
(* Case 2: Absolute value negative; nF1 < 0 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 - err2 * inv ((e2hi + err2) * (e2hi + err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_MUL_LE_COMPAT_NEG_L \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2hi + err2) * (e2hi + err2))⁻¹)` \\
qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2hi + err2) * (e2hi + err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC))
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2hi + err2) * (e2hi + err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp [real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2, REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* 0 <= - nF1 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 + err2 * inv ((e2hi + err2) * (e2hi + err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_LE_LMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2hi + err2) * (e2hi + err2))⁻¹)` \\
qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `-nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv) =
nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2hi + err2) * (e2hi + err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp[real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (fs [REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 + err_inv) =
nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2hi + err2) * (e2hi + err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* The range of the divisor lies in the range from 0 to infinity *)
>- (rule_assum_tac
(fn thm =>
REWRITE_RULE[IVlo_def, IVhi_def, widenInterval_def, contained_def, invertInterval_def] thm) \\
`abs (inv nR2 - inv nF2) <= err2 * (inv ((e2lo - err2) * (e2lo -err2)))`
by (irule err_prop_inversion_pos \\
qexists_tac `e2hi` \\ rpt(conj_tac) \\
fs[contained_def, IVlo_def, IVhi_def]) \\
fs [widenInterval_def, IVlo_def, IVhi_def, invertInterval_def] \\
`minAbsFun (e2lo - err2, e2hi + err2) = (e2lo - err2)`
by (irule minAbs_positive_iv_is_lo \\ REAL_ASM_ARITH_TAC) \\
simp[] \\
qpat_x_assum `minAbsFun _ = _ ` kall_tac \\
`nF1 <= err1 + nR1` by REAL_ASM_ARITH_TAC \\
`nR1 - err1 <= nF1` by REAL_ASM_ARITH_TAC \\
`(nR2 - nF2 > 0 /\ nR2 - nF2 <= err2) \/ (nR2 - nF2 <= 0 /\ - (nR2 - nF2) <= err2)`
by REAL_ASM_ARITH_TAC
(* Positive case for abs (nR2 - nF2) <= err2 *)
>- (`nF2 < nR2` by REAL_ASM_ARITH_TAC \\
`inv nR2 < inv nF2` by (irule REAL_LT_INV \\ TRY REAL_ASM_ARITH_TAC) \\
`inv nR2 - inv nF2 < 0` by REAL_ASM_ARITH_TAC \\
`nR2⁻¹ − nF2⁻¹ ≤ err2 * ((e2lo - err2) * (e2lo - err2))⁻¹` by REAL_ASM_ARITH_TAC \\
`inv nF2 <= inv nR2 + err2 * inv ((e2lo - err2) * (e2lo - err2))` by REAL_ASM_ARITH_TAC \\
`inv nR2 - err2 * inv ((e2lo - err2) * (e2lo - err2)) <= inv nF2` by REAL_ASM_ARITH_TAC \\
(* Next do a case distinction for the absolute value *)
`! (x:real). ((abs x = x) /\ 0 <= x) \/ ((abs x = - x) /\ x < 0)` by REAL_ASM_ARITH_TAC \\
qpat_x_assum `!x. A /\ B \/ C`
(fn thm => qspec_then `(nR1:real / nR2:real) - (nF1:real / nF2:real)` DISJ_CASES_TAC thm)
\\ fs[realTheory.abs]
(* Case 1: Absolute value positive *)
>- (fs[real_sub, real_div, REAL_NEG_LMUL] \\
qspecl_then [`-nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* -nF1 < 0 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 - err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_MUL_LE_COMPAT_NEG_L \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv) =
nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + - err2) * (e2lo + - err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp[GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC))
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 - err_inv) =
nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp [real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (simp [REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* 0 <= - nF1 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 + err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_LE_LMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp[real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (simp [REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp [GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* Case 2: Absolute value negative *)
>- (fs[real_sub, real_div, REAL_NEG_LMUL, REAL_NEG_ADD] \\
qspecl_then [`nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* nF1 < 0 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 - err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_MUL_LE_COMPAT_NEG_L \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp [GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC))
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp [real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2, REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* 0 <= - nF1 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 + err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_LE_LMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `-nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv) =
nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp[real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (fs [REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 + err_inv) =
nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp [GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC))))))
(* Negative case for abs (nR2 - nF2) <= err2 *)
>- (fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2, REAL_NEG_LMUL] \\
`nR2 <= nF2` by REAL_ASM_ARITH_TAC \\
`inv nF2 <= inv nR2` by (irule REAL_INV_LE_ANTIMONO_IMPR \\ REAL_ASM_ARITH_TAC) \\
`0 <= inv nR2 - inv nF2` by REAL_ASM_ARITH_TAC \\
`(nR2⁻¹ − nF2⁻¹) ≤ err2 * ((e2lo - err2) * (e2lo - err2))⁻¹` by REAL_ASM_ARITH_TAC \\
`inv nF2 <= inv nR2 + err2 * inv ((e2lo - err2) * (e2lo - err2))` by REAL_ASM_ARITH_TAC \\
`inv nR2 - err2 * inv ((e2lo - err2) * (e2lo - err2)) <= inv nF2` by REAL_ASM_ARITH_TAC \\
(* Next do a case distinction for the absolute value *)
`! (x:real). ((abs x = x) /\ 0 <= x) \/ ((abs x = - x) /\ x < 0)` by REAL_ASM_ARITH_TAC \\
qpat_x_assum `!x. A /\ B \/ C`
(fn thm => qspec_then `(nR1:real / nR2:real) - (nF1:real / nF2:real)` DISJ_CASES_TAC thm) \\
fs[real_sub, real_div, REAL_NEG_LMUL, REAL_NEG_ADD, realTheory.abs]
(* Case 1: Absolute value positive *)
>- (qspecl_then [`-nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* -nF1 < 0 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 - err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_MUL_LE_COMPAT_NEG_L \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 - err_inv) =
nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp [GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC))
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 - err_inv) =
nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp [real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (simp [REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* 0 <= - nF1 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - nF1 * (inv nR2 + err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_LE_LMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`nR1 * inv nR2 + - (nR1 + err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp[real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (simp [REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))
>- (irule REAL_LE_TRANS \\
qexists_tac `nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`nR1 * inv nR2 + - (nR1 + - err1) * (inv nR2 + err_inv) =
- nR1 * err_inv + inv nR2 * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp [GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* Case 2: Absolute value negative *)
>- (qspecl_then [`nF1`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
(* nF1 < 0 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 - err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_MUL_LE_COMPAT_NEG_L \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 - err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + - (inv nR2) * err1 + err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
simp [GSYM real_sub] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC))
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (`- nR1 * inv nR2 + (nR1 + err1) * (inv nR2 - err_inv) =
- nR1 * err_inv + inv nR2 * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\
qspecl_then [`inv ((e2lo + -err2) * (e2lo + -err2))`,`err2`]
(fn thm => once_rewrite_tac [thm]) REAL_MUL_COMM \\
qunabbrev_tac `err_inv` \\
simp [real_sub] \\
irule REAL_LE_ADD2 \\
conj_tac
>- (irule REAL_LE_ADD2 \\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (fs [GSYM REAL_NEG_ADD, REAL_NEG_MUL2, REAL_NEG_LMUL] \\
irule REAL_LE_RMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)))))
(* 0 <= - nF1 *)
>- (irule REAL_LE_TRANS \\
qexists_tac `- nR1 * inv nR2 + nF1 * (inv nR2 + err2 * inv ((e2lo - err2) * (e2lo - err2)))` \\
conj_tac
>- (fs[REAL_LE_LADD] \\
irule REAL_LE_LMUL_IMP \\
conj_tac \\ REAL_ASM_ARITH_TAC)
>- (qabbrev_tac `err_inv = (err2 * ((e2lo - err2) * (e2lo - err2))⁻¹)` \\
qspecl_then [`inv nR2 + err_inv`, `0`] DISJ_CASES_TAC REAL_LTE_TOTAL
>- (irule REAL_LE_TRANS \\
qexists_tac `-nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv)` \\
conj_tac
>- (fs [REAL_LE_ADD] \\
once_rewrite_tac [REAL_MUL_COMM] \\
irule REAL_MUL_LE_COMPAT_NEG_L\\
conj_tac \\ TRY REAL_ASM_ARITH_TAC \\
fs [REAL_LE_NEG])
>- (`- nR1 * inv nR2 + (nR1 - err1) * (inv nR2 + err_inv) =
nR1 * err_inv + - (inv nR2) * err1 - err1 * err_inv`
by REAL_ASM_ARITH_TAC \\
simp[REAL_NEG_MUL2] \\