napply (λx.x).
nqed.
-nlemma symmetric_eqbit : symmetricT bitrigesim bool eq_bit.
- #t1;
- nelim t1;
- ##[ ##1: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##2: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##3: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##4: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##5: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##6: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##7: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##8: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##9: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##10: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##11: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##12: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##13: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##14: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##15: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##16: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##17: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##18: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##19: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##20: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##21: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##22: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##23: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##24: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##25: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##26: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##27: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##28: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##29: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##30: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##31: #t2; nelim t2; nnormalize; napply refl_eq
- ##| ##32: #t2; nelim t2; nnormalize; napply refl_eq
- ##]
-nqed.
-
-nlemma eqbit_to_eq1 : ∀t2.eq_bit t00 t2 = true → t00 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##1: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq2 : ∀t2.eq_bit t01 t2 = true → t01 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##2: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq3 : ∀t2.eq_bit t02 t2 = true → t02 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##3: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq4 : ∀t2.eq_bit t03 t2 = true → t03 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##4: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq5 : ∀t2.eq_bit t04 t2 = true → t04 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##5: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq6 : ∀t2.eq_bit t05 t2 = true → t05 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##6: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq7 : ∀t2.eq_bit t06 t2 = true → t06 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##7: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq8 : ∀t2.eq_bit t07 t2 = true → t07 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##8: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq9 : ∀t2.eq_bit t08 t2 = true → t08 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##9: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq10 : ∀t2.eq_bit t09 t2 = true → t09 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##10: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq11 : ∀t2.eq_bit t0A t2 = true → t0A = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##11: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq12 : ∀t2.eq_bit t0B t2 = true → t0B = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##12: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq13 : ∀t2.eq_bit t0C t2 = true → t0C = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##13: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq14 : ∀t2.eq_bit t0D t2 = true → t0D = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##14: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq15 : ∀t2.eq_bit t0E t2 = true → t0E = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##15: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq16 : ∀t2.eq_bit t0F t2 = true → t0F = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##16: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq17 : ∀t2.eq_bit t10 t2 = true → t10 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##17: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq18 : ∀t2.eq_bit t11 t2 = true → t11 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##18: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq19 : ∀t2.eq_bit t12 t2 = true → t12 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##19: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq20 : ∀t2.eq_bit t13 t2 = true → t13 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##20: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq21 : ∀t2.eq_bit t14 t2 = true → t14 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##21: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq22 : ∀t2.eq_bit t15 t2 = true → t15 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##22: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq23 : ∀t2.eq_bit t16 t2 = true → t16 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##23: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq24 : ∀t2.eq_bit t17 t2 = true → t17 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##24: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq25 : ∀t2.eq_bit t18 t2 = true → t18 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##25: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq26 : ∀t2.eq_bit t19 t2 = true → t19 = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##26: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq27 : ∀t2.eq_bit t1A t2 = true → t1A = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##27: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq28 : ∀t2.eq_bit t1B t2 = true → t1B = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##28: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq29 : ∀t2.eq_bit t1C t2 = true → t1C = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##29: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq30 : ∀t2.eq_bit t1D t2 = true → t1D = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##30: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq31 : ∀t2.eq_bit t1E t2 = true → t1E = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##31: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
-
-nlemma eqbit_to_eq32 : ∀t2.eq_bit t1F t2 = true → t1F = t2.
- #t2; ncases t2; nnormalize; #H; ##[ ##32: napply refl_eq ##| ##*: napply (bool_destruct … H) ##]
-nqed.
+nlemma eq_to_eqbit : ∀n1,n2.n1 = n2 → eq_bit n1 n2 = true.
+ #n1; #n2; #H;
+ nrewrite > H;
+ nelim n2;
+ nnormalize;
+ napply refl_eq.
+nqed.
+
+nlemma neqbit_to_neq : ∀n1,n2.eq_bit n1 n2 = false → n1 ≠ n2.
+ #n1; #n2; #H;
+ napply (not_to_not (n1 = n2) (eq_bit n1 n2 = true) …);
+ ##[ ##1: napply (eq_to_eqbit n1 n2)
+ ##| ##2: napply (eqfalse_to_neqtrue … H)
+ ##]
+nqed.
+
+nlemma eqbit_to_eq1 : ∀t2.eq_bit t00 t2 = true → t00 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##1: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq2 : ∀t2.eq_bit t01 t2 = true → t01 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##2: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq3 : ∀t2.eq_bit t02 t2 = true → t02 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##3: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq4 : ∀t2.eq_bit t03 t2 = true → t03 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##4: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq5 : ∀t2.eq_bit t04 t2 = true → t04 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##5: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq6 : ∀t2.eq_bit t05 t2 = true → t05 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##6: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq7 : ∀t2.eq_bit t06 t2 = true → t06 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##7: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq8 : ∀t2.eq_bit t07 t2 = true → t07 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##8: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq9 : ∀t2.eq_bit t08 t2 = true → t08 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##9: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq10 : ∀t2.eq_bit t09 t2 = true → t09 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##10: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq11 : ∀t2.eq_bit t0A t2 = true → t0A = t2. #t2; ncases t2; nnormalize; #H; ##[ ##11: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq12 : ∀t2.eq_bit t0B t2 = true → t0B = t2. #t2; ncases t2; nnormalize; #H; ##[ ##12: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq13 : ∀t2.eq_bit t0C t2 = true → t0C = t2. #t2; ncases t2; nnormalize; #H; ##[ ##13: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq14 : ∀t2.eq_bit t0D t2 = true → t0D = t2. #t2; ncases t2; nnormalize; #H; ##[ ##14: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq15 : ∀t2.eq_bit t0E t2 = true → t0E = t2. #t2; ncases t2; nnormalize; #H; ##[ ##15: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq16 : ∀t2.eq_bit t0F t2 = true → t0F = t2. #t2; ncases t2; nnormalize; #H; ##[ ##16: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq17 : ∀t2.eq_bit t10 t2 = true → t10 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##17: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq18 : ∀t2.eq_bit t11 t2 = true → t11 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##18: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq19 : ∀t2.eq_bit t12 t2 = true → t12 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##19: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq20 : ∀t2.eq_bit t13 t2 = true → t13 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##20: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq21 : ∀t2.eq_bit t14 t2 = true → t14 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##21: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq22 : ∀t2.eq_bit t15 t2 = true → t15 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##22: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq23 : ∀t2.eq_bit t16 t2 = true → t16 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##23: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq24 : ∀t2.eq_bit t17 t2 = true → t17 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##24: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq25 : ∀t2.eq_bit t18 t2 = true → t18 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##25: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq26 : ∀t2.eq_bit t19 t2 = true → t19 = t2. #t2; ncases t2; nnormalize; #H; ##[ ##26: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq27 : ∀t2.eq_bit t1A t2 = true → t1A = t2. #t2; ncases t2; nnormalize; #H; ##[ ##27: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq28 : ∀t2.eq_bit t1B t2 = true → t1B = t2. #t2; ncases t2; nnormalize; #H; ##[ ##28: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq29 : ∀t2.eq_bit t1C t2 = true → t1C = t2. #t2; ncases t2; nnormalize; #H; ##[ ##29: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq30 : ∀t2.eq_bit t1D t2 = true → t1D = t2. #t2; ncases t2; nnormalize; #H; ##[ ##30: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq31 : ∀t2.eq_bit t1E t2 = true → t1E = t2. #t2; ncases t2; nnormalize; #H; ##[ ##31: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
+nlemma eqbit_to_eq32 : ∀t2.eq_bit t1F t2 = true → t1F = t2. #t2; ncases t2; nnormalize; #H; ##[ ##32: napply refl_eq ##| ##*: napply (bool_destruct … H) ##] nqed.
nlemma eqbit_to_eq : ∀t1,t2.eq_bit t1 t2 = true → t1 = t2.
#t1; ncases t1;
##]
nqed.
-nlemma eq_to_eqbit1 : ∀t2.t00 = t2 → eq_bit t00 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##1: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit2 : ∀t2.t01 = t2 → eq_bit t01 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##2: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit3 : ∀t2.t02 = t2 → eq_bit t02 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##3: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit4 : ∀t2.t03 = t2 → eq_bit t03 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##4: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit5 : ∀t2.t04 = t2 → eq_bit t04 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##5: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit6 : ∀t2.t05 = t2 → eq_bit t05 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##6: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit7 : ∀t2.t06 = t2 → eq_bit t06 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##7: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit8 : ∀t2.t07 = t2 → eq_bit t07 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##8: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit9 : ∀t2.t08 = t2 → eq_bit t08 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##9: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit10 : ∀t2.t09 = t2 → eq_bit t09 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##10: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit11 : ∀t2.t0A = t2 → eq_bit t0A t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##11: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit12 : ∀t2.t0B = t2 → eq_bit t0B t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##12: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit13 : ∀t2.t0C = t2 → eq_bit t0C t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##13: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit14 : ∀t2.t0D = t2 → eq_bit t0D t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##14: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit15 : ∀t2.t0E = t2 → eq_bit t0E t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##15: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit16 : ∀t2.t0F = t2 → eq_bit t0F t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##16: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit17 : ∀t2.t10 = t2 → eq_bit t10 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##17: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit18 : ∀t2.t11 = t2 → eq_bit t11 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##18: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit19 : ∀t2.t12 = t2 → eq_bit t12 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##19: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit20 : ∀t2.t13 = t2 → eq_bit t13 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##20: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit21 : ∀t2.t14 = t2 → eq_bit t14 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##21: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit22 : ∀t2.t15 = t2 → eq_bit t15 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##22: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit23 : ∀t2.t16 = t2 → eq_bit t16 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##23: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit24 : ∀t2.t17 = t2 → eq_bit t17 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##24: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit25 : ∀t2.t18 = t2 → eq_bit t18 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##25: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit26 : ∀t2.t19 = t2 → eq_bit t19 t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##26: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit27 : ∀t2.t1A = t2 → eq_bit t1A t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##27: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit28 : ∀t2.t1B = t2 → eq_bit t1B t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##28: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit29 : ∀t2.t1C = t2 → eq_bit t1C t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##29: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit30 : ∀t2.t1D = t2 → eq_bit t1D t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##30: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit31 : ∀t2.t1E = t2 → eq_bit t1E t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##31: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit32 : ∀t2.t1F = t2 → eq_bit t1F t2 = true.
- #t2; ncases t2; nnormalize; #H; ##[ ##32: napply refl_eq ##| ##*: napply (bitrigesim_destruct … H) ##]
-nqed.
-
-nlemma eq_to_eqbit : ∀t1,t2.t1 = t2 → eq_bit t1 t2 = true.
- #t1; ncases t1;
- ##[ ##1: napply eq_to_eqbit1 ##| ##2: napply eq_to_eqbit2
- ##| ##3: napply eq_to_eqbit3 ##| ##4: napply eq_to_eqbit4
- ##| ##5: napply eq_to_eqbit5 ##| ##6: napply eq_to_eqbit6
- ##| ##7: napply eq_to_eqbit7 ##| ##8: napply eq_to_eqbit8
- ##| ##9: napply eq_to_eqbit9 ##| ##10: napply eq_to_eqbit10
- ##| ##11: napply eq_to_eqbit11 ##| ##12: napply eq_to_eqbit12
- ##| ##13: napply eq_to_eqbit13 ##| ##14: napply eq_to_eqbit14
- ##| ##15: napply eq_to_eqbit15 ##| ##16: napply eq_to_eqbit16
- ##| ##17: napply eq_to_eqbit17 ##| ##18: napply eq_to_eqbit18
- ##| ##19: napply eq_to_eqbit19 ##| ##20: napply eq_to_eqbit20
- ##| ##21: napply eq_to_eqbit21 ##| ##22: napply eq_to_eqbit22
- ##| ##23: napply eq_to_eqbit23 ##| ##24: napply eq_to_eqbit24
- ##| ##25: napply eq_to_eqbit25 ##| ##26: napply eq_to_eqbit26
- ##| ##27: napply eq_to_eqbit27 ##| ##28: napply eq_to_eqbit28
- ##| ##29: napply eq_to_eqbit29 ##| ##30: napply eq_to_eqbit30
- ##| ##31: napply eq_to_eqbit31 ##| ##32: napply eq_to_eqbit32
- ##]
-nqed.
-
-nlemma decidable_bit1 : ∀x:bitrigesim.decidable (t00 = x).
- #x; nnormalize; nelim x;
- ##[ ##1: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit2 : ∀x:bitrigesim.decidable (t01 = x).
- #x; nnormalize; nelim x;
- ##[ ##2: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit3 : ∀x:bitrigesim.decidable (t02 = x).
- #x; nnormalize; nelim x;
- ##[ ##3: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit4 : ∀x:bitrigesim.decidable (t03 = x).
- #x; nnormalize; nelim x;
- ##[ ##4: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit5 : ∀x:bitrigesim.decidable (t04 = x).
- #x; nnormalize; nelim x;
- ##[ ##5: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit6 : ∀x:bitrigesim.decidable (t05 = x).
- #x; nnormalize; nelim x;
- ##[ ##6: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit7 : ∀x:bitrigesim.decidable (t06 = x).
- #x; nnormalize; nelim x;
- ##[ ##7: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit8 : ∀x:bitrigesim.decidable (t07 = x).
- #x; nnormalize; nelim x;
- ##[ ##8: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit9 : ∀x:bitrigesim.decidable (t08 = x).
- #x; nnormalize; nelim x;
- ##[ ##9: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit10 : ∀x:bitrigesim.decidable (t09 = x).
- #x; nnormalize; nelim x;
- ##[ ##10: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit11 : ∀x:bitrigesim.decidable (t0A = x).
- #x; nnormalize; nelim x;
- ##[ ##11: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit12 : ∀x:bitrigesim.decidable (t0B = x).
- #x; nnormalize; nelim x;
- ##[ ##12: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit13 : ∀x:bitrigesim.decidable (t0C = x).
- #x; nnormalize; nelim x;
- ##[ ##13: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit14 : ∀x:bitrigesim.decidable (t0D = x).
- #x; nnormalize; nelim x;
- ##[ ##14: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit15 : ∀x:bitrigesim.decidable (t0E = x).
- #x; nnormalize; nelim x;
- ##[ ##15: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit16 : ∀x:bitrigesim.decidable (t0F = x).
- #x; nnormalize; nelim x;
- ##[ ##16: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit17 : ∀x:bitrigesim.decidable (t10 = x).
- #x; nnormalize; nelim x;
- ##[ ##17: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit18 : ∀x:bitrigesim.decidable (t11 = x).
- #x; nnormalize; nelim x;
- ##[ ##18: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit19 : ∀x:bitrigesim.decidable (t12 = x).
- #x; nnormalize; nelim x;
- ##[ ##19: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit20 : ∀x:bitrigesim.decidable (t13 = x).
- #x; nnormalize; nelim x;
- ##[ ##20: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit21 : ∀x:bitrigesim.decidable (t14 = x).
- #x; nnormalize; nelim x;
- ##[ ##21: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit22 : ∀x:bitrigesim.decidable (t15 = x).
- #x; nnormalize; nelim x;
- ##[ ##22: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit23 : ∀x:bitrigesim.decidable (t16 = x).
- #x; nnormalize; nelim x;
- ##[ ##23: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit24 : ∀x:bitrigesim.decidable (t17 = x).
- #x; nnormalize; nelim x;
- ##[ ##24: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit25 : ∀x:bitrigesim.decidable (t18 = x).
- #x; nnormalize; nelim x;
- ##[ ##25: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit26 : ∀x:bitrigesim.decidable (t19 = x).
- #x; nnormalize; nelim x;
- ##[ ##26: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit27 : ∀x:bitrigesim.decidable (t1A = x).
- #x; nnormalize; nelim x;
- ##[ ##27: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit28 : ∀x:bitrigesim.decidable (t1B = x).
- #x; nnormalize; nelim x;
- ##[ ##28: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit29 : ∀x:bitrigesim.decidable (t1C = x).
- #x; nnormalize; nelim x;
- ##[ ##29: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit30 : ∀x:bitrigesim.decidable (t1D = x).
- #x; nnormalize; nelim x;
- ##[ ##30: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit31 : ∀x:bitrigesim.decidable (t1E = x).
- #x; nnormalize; nelim x;
- ##[ ##31: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
-nqed.
-
-nlemma decidable_bit32 : ∀x:bitrigesim.decidable (t1F = x).
- #x; nnormalize; nelim x;
- ##[ ##32: napply (or2_intro1 (? = ?) (? ≠ ?) …); napply refl_eq
- ##| ##*: napply (or2_intro2 (? = ?) (? ≠ ?) …); nnormalize; #H; napply False_ind; napply (bitrigesim_destruct … H)
- ##]
+nlemma neq_to_neqbit : ∀n1,n2.n1 ≠ n2 → eq_bit n1 n2 = false.
+ #n1; #n2; #H;
+ napply (neqtrue_to_eqfalse (eq_bit n1 n2));
+ napply (not_to_not (eq_bit n1 n2 = true) (n1 = n2) ? H);
+ napply (eqbit_to_eq n1 n2).
nqed.
nlemma decidable_bit : ∀x,y:bitrigesim.decidable (x = y).
- #x; nnormalize; nelim x;
- ##[ ##1: napply decidable_bit1 ##| ##2: napply decidable_bit2
- ##| ##3: napply decidable_bit3 ##| ##4: napply decidable_bit4
- ##| ##5: napply decidable_bit5 ##| ##6: napply decidable_bit6
- ##| ##7: napply decidable_bit7 ##| ##8: napply decidable_bit8
- ##| ##9: napply decidable_bit9 ##| ##10: napply decidable_bit10
- ##| ##11: napply decidable_bit11 ##| ##12: napply decidable_bit12
- ##| ##13: napply decidable_bit13 ##| ##14: napply decidable_bit14
- ##| ##15: napply decidable_bit15 ##| ##16: napply decidable_bit16
- ##| ##17: napply decidable_bit17 ##| ##18: napply decidable_bit18
- ##| ##19: napply decidable_bit19 ##| ##20: napply decidable_bit20
- ##| ##21: napply decidable_bit21 ##| ##22: napply decidable_bit22
- ##| ##23: napply decidable_bit23 ##| ##24: napply decidable_bit24
- ##| ##25: napply decidable_bit25 ##| ##26: napply decidable_bit26
- ##| ##27: napply decidable_bit27 ##| ##28: napply decidable_bit28
- ##| ##29: napply decidable_bit29 ##| ##30: napply decidable_bit30
- ##| ##31: napply decidable_bit31 ##| ##32: napply decidable_bit32
- ##]
-nqed.
-
-nlemma neqbit_to_neq1 : ∀t2.eq_bit t00 t2 = false → t00 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##1: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq2 : ∀t2.eq_bit t01 t2 = false → t01 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##2: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq3 : ∀t2.eq_bit t02 t2 = false → t02 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##3: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq4 : ∀t2.eq_bit t03 t2 = false → t03 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##4: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq5 : ∀t2.eq_bit t04 t2 = false → t04 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##5: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq6 : ∀t2.eq_bit t05 t2 = false → t05 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##6: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq7 : ∀t2.eq_bit t06 t2 = false → t06 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##7: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq8 : ∀t2.eq_bit t07 t2 = false → t07 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##8: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq9 : ∀t2.eq_bit t08 t2 = false → t08 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##9: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq10 : ∀t2.eq_bit t09 t2 = false → t09 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##10: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq11 : ∀t2.eq_bit t0A t2 = false → t0A ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##11: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq12 : ∀t2.eq_bit t0B t2 = false → t0B ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##12: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq13 : ∀t2.eq_bit t0C t2 = false → t0C ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##13: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq14 : ∀t2.eq_bit t0D t2 = false → t0D ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##14: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq15 : ∀t2.eq_bit t0E t2 = false → t0E ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##15: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq16 : ∀t2.eq_bit t0F t2 = false → t0F ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##16: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq17 : ∀t2.eq_bit t10 t2 = false → t10 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##17: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq18 : ∀t2.eq_bit t11 t2 = false → t11 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##18: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq19 : ∀t2.eq_bit t12 t2 = false → t12 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##19: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq20 : ∀t2.eq_bit t13 t2 = false → t13 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##20: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq21 : ∀t2.eq_bit t14 t2 = false → t14 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##21: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq22 : ∀t2.eq_bit t15 t2 = false → t15 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##22: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
+ #x; #y; nnormalize;
+ napply (or2_elim (eq_bit x y = true) (eq_bit x y = false) ? (decidable_bexpr ?));
+ ##[ ##1: #H; napply (or2_intro1 (x = y) (x ≠ y) (eqbit_to_eq … H))
+ ##| ##2: #H; napply (or2_intro2 (x = y) (x ≠ y) (neqbit_to_neq … H))
##]
nqed.
-nlemma neqbit_to_neq23 : ∀t2.eq_bit t16 t2 = false → t16 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##23: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq24 : ∀t2.eq_bit t17 t2 = false → t17 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##24: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq25 : ∀t2.eq_bit t18 t2 = false → t18 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##25: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq26 : ∀t2.eq_bit t19 t2 = false → t19 ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##26: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq27 : ∀t2.eq_bit t1A t2 = false → t1A ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##27: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq28 : ∀t2.eq_bit t1B t2 = false → t1B ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##28: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq29 : ∀t2.eq_bit t1C t2 = false → t1C ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##29: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq30 : ∀t2.eq_bit t1D t2 = false → t1D ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##30: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq31 : ∀t2.eq_bit t1E t2 = false → t1E ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##31: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq32 : ∀t2.eq_bit t1F t2 = false → t1F ≠ t2.
- #t2; ncases t2; nnormalize; #H;
- ##[ ##32: napply (bool_destruct … H)
- ##| ##*: #H1; napply False_ind; napply (bitrigesim_destruct … H1)
- ##]
-nqed.
-
-nlemma neqbit_to_neq : ∀t1,t2.eq_bit t1 t2 = false → t1 ≠ t2.
- #t1; nelim t1;
- ##[ ##1: napply neqbit_to_neq1 ##| ##2: napply neqbit_to_neq2
- ##| ##3: napply neqbit_to_neq3 ##| ##4: napply neqbit_to_neq4
- ##| ##5: napply neqbit_to_neq5 ##| ##6: napply neqbit_to_neq6
- ##| ##7: napply neqbit_to_neq7 ##| ##8: napply neqbit_to_neq8
- ##| ##9: napply neqbit_to_neq9 ##| ##10: napply neqbit_to_neq10
- ##| ##11: napply neqbit_to_neq11 ##| ##12: napply neqbit_to_neq12
- ##| ##13: napply neqbit_to_neq13 ##| ##14: napply neqbit_to_neq14
- ##| ##15: napply neqbit_to_neq15 ##| ##16: napply neqbit_to_neq16
- ##| ##17: napply neqbit_to_neq17 ##| ##18: napply neqbit_to_neq18
- ##| ##19: napply neqbit_to_neq19 ##| ##20: napply neqbit_to_neq20
- ##| ##21: napply neqbit_to_neq21 ##| ##22: napply neqbit_to_neq22
- ##| ##23: napply neqbit_to_neq23 ##| ##24: napply neqbit_to_neq24
- ##| ##25: napply neqbit_to_neq25 ##| ##26: napply neqbit_to_neq26
- ##| ##27: napply neqbit_to_neq27 ##| ##28: napply neqbit_to_neq28
- ##| ##29: napply neqbit_to_neq29 ##| ##30: napply neqbit_to_neq30
- ##| ##31: napply neqbit_to_neq31 ##| ##32: napply neqbit_to_neq32
- ##]
-nqed.
-
-nlemma neq_to_neqbit1 : ∀t2.t00 ≠ t2 → eq_bit t00 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##1: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit2 : ∀t2.t01 ≠ t2 → eq_bit t01 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##2: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit3 : ∀t2.t02 ≠ t2 → eq_bit t02 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##3: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit4 : ∀t2.t03 ≠ t2 → eq_bit t03 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##4: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit5 : ∀t2.t04 ≠ t2 → eq_bit t04 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##5: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit6 : ∀t2.t05 ≠ t2 → eq_bit t05 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##6: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit7 : ∀t2.t06 ≠ t2 → eq_bit t06 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##7: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit8 : ∀t2.t07 ≠ t2 → eq_bit t07 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##8: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit9 : ∀t2.t08 ≠ t2 → eq_bit t08 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##9: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit10 : ∀t2.t09 ≠ t2 → eq_bit t09 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##10: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit11 : ∀t2.t0A ≠ t2 → eq_bit t0A t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##11: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit12 : ∀t2.t0B ≠ t2 → eq_bit t0B t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##12: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit13 : ∀t2.t0C ≠ t2 → eq_bit t0C t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##13: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit14 : ∀t2.t0D ≠ t2 → eq_bit t0D t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##14: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit15 : ∀t2.t0E ≠ t2 → eq_bit t0E t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##15: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit16 : ∀t2.t0F ≠ t2 → eq_bit t0F t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##16: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit17 : ∀t2.t10 ≠ t2 → eq_bit t10 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##17: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit18 : ∀t2.t11 ≠ t2 → eq_bit t11 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##18: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit19 : ∀t2.t12 ≠ t2 → eq_bit t12 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##19: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit20 : ∀t2.t13 ≠ t2 → eq_bit t13 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##20: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit21 : ∀t2.t14 ≠ t2 → eq_bit t14 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##21: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit22 : ∀t2.t15 ≠ t2 → eq_bit t15 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##22: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit23 : ∀t2.t16 ≠ t2 → eq_bit t16 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##23: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit24 : ∀t2.t17 ≠ t2 → eq_bit t17 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##24: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit25 : ∀t2.t18 ≠ t2 → eq_bit t18 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##25: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit26 : ∀t2.t19 ≠ t2 → eq_bit t19 t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##26: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit27 : ∀t2.t1A ≠ t2 → eq_bit t1A t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##27: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit28 : ∀t2.t1B ≠ t2 → eq_bit t1B t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##28: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit29 : ∀t2.t1C ≠ t2 → eq_bit t1C t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##29: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit30 : ∀t2.t1D ≠ t2 → eq_bit t1D t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##30: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit31 : ∀t2.t1E ≠ t2 → eq_bit t1E t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##31: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit32 : ∀t2.t1F ≠ t2 → eq_bit t1F t2 = false.
- #t2; ncases t2; nnormalize; #H; ##[ ##32: nelim (H (refl_eq …)) ##| ##*: napply refl_eq ##]
-nqed.
-
-nlemma neq_to_neqbit : ∀t1,t2.t1 ≠ t2 → eq_bit t1 t2 = false.
- #t1; nelim t1;
- ##[ ##1: napply neq_to_neqbit1 ##| ##2: napply neq_to_neqbit2
- ##| ##3: napply neq_to_neqbit3 ##| ##4: napply neq_to_neqbit4
- ##| ##5: napply neq_to_neqbit5 ##| ##6: napply neq_to_neqbit6
- ##| ##7: napply neq_to_neqbit7 ##| ##8: napply neq_to_neqbit8
- ##| ##9: napply neq_to_neqbit9 ##| ##10: napply neq_to_neqbit10
- ##| ##11: napply neq_to_neqbit11 ##| ##12: napply neq_to_neqbit12
- ##| ##13: napply neq_to_neqbit13 ##| ##14: napply neq_to_neqbit14
- ##| ##15: napply neq_to_neqbit15 ##| ##16: napply neq_to_neqbit16
- ##| ##17: napply neq_to_neqbit17 ##| ##18: napply neq_to_neqbit18
- ##| ##19: napply neq_to_neqbit19 ##| ##20: napply neq_to_neqbit20
- ##| ##21: napply neq_to_neqbit21 ##| ##22: napply neq_to_neqbit22
- ##| ##23: napply neq_to_neqbit23 ##| ##24: napply neq_to_neqbit24
- ##| ##25: napply neq_to_neqbit25 ##| ##26: napply neq_to_neqbit26
- ##| ##27: napply neq_to_neqbit27 ##| ##28: napply neq_to_neqbit28
- ##| ##29: napply neq_to_neqbit29 ##| ##30: napply neq_to_neqbit30
- ##| ##31: napply neq_to_neqbit31 ##| ##32: napply neq_to_neqbit32
+nlemma symmetric_eqbit : symmetricT bitrigesim bool eq_bit.
+ #n1; #n2;
+ napply (or2_elim (n1 = n2) (n1 ≠ n2) ? (decidable_bit n1 n2));
+ ##[ ##1: #H; nrewrite > H; napply refl_eq
+ ##| ##2: #H; nrewrite > (neq_to_neqbit n1 n2 H);
+ napply (symmetric_eq ? (eq_bit n2 n1) false);
+ napply (neq_to_neqbit n2 n1 (symmetric_neq ? n1 n2 H))
##]
nqed.
projects / helm.git / blobdiff