Let w be given.
Let w' be given.
Apply IHk0 w Hw to the current goal.
Assume H.
Apply H to the current goal.
Apply IHk0 w' Hw' to the current goal.
Assume H.
Apply H to the current goal.
We prove the intermediate
claim Lww':
SNo (w * w').
An
exact proof term for the current goal is
SNo_mul_SNo w w' Hw1 Hw'1.
We prove the intermediate
claim Lxpww':
SNo (x + w * w').
An
exact proof term for the current goal is
SNo_add_SNo x (w * w') Hx Lww'.
We prove the intermediate
claim Lwpw':
SNo (w + w').
An
exact proof term for the current goal is
SNo_add_SNo w w' Hw1 Hw'1.
We prove the intermediate
claim Ly:
SNo y.
rewrite the current goal using Hyww' (from left to right).
An
exact proof term for the current goal is
SNo_div_SNo (x + w * w') (w + w') Lxpww' Lwpw'.
We prove the intermediate
claim Lxpww'nonneg:
0 ≤ x + w * w'.
We will
prove 0 ≤ w * w'.
We prove the intermediate
claim Lynonneg:
0 ≤ y.
rewrite the current goal using Hyww' (from left to right).
We will
prove 0 ≤ (x + w * w') :/: (w + w').
Apply SNoLeE 0 (x + w * w') SNo_0 Lxpww' Lxpww'nonneg to the current goal.
We will
prove 0 < x + w * w'.
An exact proof term for the current goal is H1.
We will
prove 0 < w + w'.
An exact proof term for the current goal is Hww'pos.
rewrite the current goal using H1 (from right to left).
rewrite the current goal using
div_SNo_0_num (w + w') Lwpw' (from left to right).
Apply and3I to the current goal.
An exact proof term for the current goal is Ly.
An exact proof term for the current goal is Lynonneg.
rewrite the current goal using Hyww' (from left to right).
We will
prove x < ((x + w * w') :/: (w + w')) * ((x + w * w') :/: (w + w')).
rewrite the current goal using
mul_div_SNo_both (x + w * w') (w + w') (x + w * w') (w + w') Lxpww' Lwpw' Lxpww' Lwpw' (from left to right).
We will
prove x < ((x + w * w') * (x + w * w')) :/: ((w + w') * (w + w')).
We will
prove 0 < (w + w') * (w + w').
An
exact proof term for the current goal is
mul_SNo_pos_pos (w + w') (w + w') Lwpw' Lwpw' Hww'pos Hww'pos.
We will
prove x * ((w + w') * (w + w')) < ((x + w * w') * (x + w * w')).
rewrite the current goal using
SNo_foil x (w * w') x (w * w') Hx Lww' (from left to right).
rewrite the current goal using
SNo_foil w w' w w' Hw1 Hw'1 Hw1 Hw'1 (from left to right).
We will
prove x * (w * w + w * w' + w' * w + w' * w') < x * x + x * w * w' + (w * w') * x + (w * w') * w * w'.
rewrite the current goal using
mul_SNo_com w' w Hw'1 Hw1 (from left to right).
We will
prove x * (w * w' + w * w' + w' * w' + w * w) < x * x + x * w * w' + (w * w') * x + (w * w') * w * w'.
rewrite the current goal using
mul_SNo_com (w * w') x Lww' Hx (from left to right).
We will
prove x * (w * w' + w * w' + w' * w' + w * w) < x * x + x * w * w' + x * w * w' + (w * w') * w * w'.
We prove the intermediate
claim Lxww':
SNo (x * w * w').
An
exact proof term for the current goal is
SNo_mul_SNo x (w * w') Hx Lww'.
We will
prove x * (w * w' + w * w' + w' * w' + w * w) < x * w * w' + x * w * w' + (w * w') * w * w' + x * x.
We will
prove x * w * w' + x * (w * w' + w' * w' + w * w) < x * w * w' + x * w * w' + (w * w') * w * w' + x * x.
We will
prove x * w * w' + x * w * w' + x * (w' * w' + w * w) < x * w * w' + x * w * w' + (w * w') * w * w' + x * x.
We prove the intermediate
claim Lxww':
SNo (x * w * w').
An
exact proof term for the current goal is
SNo_mul_SNo_3 x w w' Hx Hw1 Hw'1.
We will
prove x * w * w' + x * (w' * w' + w * w) < x * w * w' + (w * w') * w * w' + x * x.
We will
prove x * (w' * w' + w * w) < (w * w') * w * w' + x * x.
We will
prove x * w' * w' + x * w * w < (w * w') * w * w' + x * x.
We will
prove x * w' * w' + (w * w) * x < (w * w') * w * w' + x * x.
We will
prove x * w' * w' + (w * w) * x < 0 + (w * w') * w * w' + x * x.
We will
prove (x * w' * w' + (w * w) * x) + - ((w * w') * w * w' + x * x) < 0.
We will
prove x * w' * w' + (w * w) * x + - (w * w') * w * w' + - x * x < 0.
We will
prove x * w' * w' + - x * x + (w * w) * x + - (w * w') * w * w' < 0.
We will
prove x * w' * w' + - x * x + - (w * w') * w * w' + (w * w) * x < 0.
rewrite the current goal using
mul_SNo_assoc w w' (w * w') Hw1 Hw'1 Lww' (from right to left).
rewrite the current goal using
mul_SNo_com_3_0_1 w' w w' Hw'1 Hw1 Hw'1 (from left to right).
We will
prove (x + - w * w) * (w' * w' + - x) < 0.
We will
prove 0 < x + - w * w.
We will
prove w' * w' + - x < 0.
We will
prove w' * w' < 0 + x.
rewrite the current goal using
add_SNo_0L x Hx (from left to right).
We will
prove w' * w' < x.
An exact proof term for the current goal is Hw'3.
An exact proof term for the current goal is Hx.
We will
prove SNo (w * w').
An exact proof term for the current goal is Lww'.