Search for blocks/addresses/...

Proofgold Proof

pf
Apply nat_inv_impred with λ x0 . (x0 = 0∀ x1 : ο . x1)∀ x1 . nat_p x1∃ x2 . and (nat_p x2) (divides_nat x0 (add_nat x1 x2)) leaving 2 subgoals.
Assume H0: 0 = 0∀ x0 : ο . x0.
Apply FalseE with ∀ x0 . nat_p x0∃ x1 . and (nat_p x1) (divides_nat 0 (add_nat x0 x1)).
Apply H0.
Let x0 of type ιιο be given.
Assume H1: x0 0 0.
The subproof is completed by applying H1.
Let x0 of type ι be given.
Assume H0: nat_p x0.
Assume H1: ordsucc x0 = 0∀ x1 : ο . x1.
Claim L2: ...
...
Apply nat_ind with λ x1 . ∃ x2 . and (nat_p x2) (divides_nat (ordsucc x0) (add_nat x1 x2)) leaving 2 subgoals.
Let x1 of type ο be given.
Assume H3: ∀ x2 . and (nat_p x2) (divides_nat (ordsucc x0) (add_nat 0 x2))x1.
Apply H3 with 0.
Apply andI with nat_p 0, divides_nat (ordsucc x0) (add_nat 0 0) leaving 2 subgoals.
The subproof is completed by applying nat_0.
Apply add_nat_0R with 0, λ x2 x3 . divides_nat (ordsucc x0) x3.
Apply unknownprop_94b9b73b1207350973a964cfd79fac000c8d717e12b3149994867d613d318c69 with ordsucc x0.
The subproof is completed by applying L2.
Let x1 of type ι be given.
Assume H3: nat_p x1.
Assume H4: ∃ x2 . and (nat_p x2) (divides_nat (ordsucc x0) (add_nat x1 x2)).
Apply H4 with ∃ x2 . and (nat_p x2) (divides_nat (ordsucc x0) (add_nat (ordsucc x1) x2)).
Let x2 of type ι be given.
Assume H5: (λ x3 . and (nat_p x3) (divides_nat (ordsucc x0) (add_nat x1 x3))) x2.
Apply H5 with ∃ x3 . and (nat_p x3) (divides_nat (ordsucc x0) (add_nat (ordsucc x1) x3)).
Apply nat_inv_impred with λ x3 . divides_nat (ordsucc x0) (add_nat x1 x3)∃ x4 . and (nat_p x4) (divides_nat (ordsucc x0) (add_nat (ordsucc x1) x4)), x2 leaving 2 subgoals.
Apply add_nat_0R with x1, λ x3 x4 . divides_nat (ordsucc x0) x4∃ x5 . and (nat_p x5) (divides_nat (ordsucc x0) (add_nat (ordsucc x1) x5)).
Assume H6: divides_nat (ordsucc x0) x1.
Let x3 of type ο be given.
Assume H7: ∀ x4 . and (nat_p x4) (divides_nat (ordsucc x0) (add_nat (ordsucc x1) x4))x3.
Apply H7 with x0.
Apply andI with nat_p x0, divides_nat (ordsucc x0) (add_nat (ordsucc x1) x0) leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_nat_SL with x1, x0, λ x4 x5 . divides_nat (ordsucc x0) x5 leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H0.
Apply add_nat_SR with x1, x0, λ x4 x5 . divides_nat (ordsucc x0) x4 leaving 2 subgoals.
The subproof is completed by applying H0.
Apply add_nat_add_SNo with x1, ordsucc x0, λ x4 x5 . divides_nat (ordsucc x0) x5 leaving 3 subgoals.
Apply nat_p_omega with x1.
The subproof is completed by applying H3.
Apply nat_p_omega with ordsucc x0.
The subproof is completed by applying L2.
Apply unknownprop_4f580385494c3bb0b65abf1bcb00277688faff94da8eb184b6015c42d53d3c52 with ordsucc x0, x1, ordsucc x0 leaving 3 subgoals.
The subproof is completed by applying L2.
The subproof is completed by applying H6.
Apply unknownprop_98e7544997c6005aa9c4dc938c620a6df3d89601058380b842e41b24814e6d5b with ordsucc x0.
The subproof is completed by applying L2.
Let x3 of type ι be given.
Assume H6: nat_p x3.
Assume H7: divides_nat (ordsucc x0) (add_nat x1 (ordsucc x3)).
Let x4 of type ο be given.
Assume H8: ∀ x5 . and (nat_p x5) ...x4.
...