Let x0 of type ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι) be given.

Let x1 of type ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι) be given.

Let x2 of type ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι) be given.

Let x3 of type ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → CN (ι → ι) be given.

Assume H0: ChurchNum_3ary_proj_p x0.

Assume H1: ChurchNum_8ary_proj_p x2.

Assume H2: ChurchNum_3ary_proj_p x1.

Assume H3: ChurchNum_8ary_proj_p x3.

Apply H0 with λ x4 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x4 = (λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) x1 ⟶ ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x2 x4) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x2) = ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x3 x1) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x3) ⟶ ∀ x5 : ο . x5 leaving 3 subgoals.

Apply H2 with λ x4 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) = (λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) x4 ⟶ ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x2 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x2) = ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x3 x4) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x3) ⟶ ∀ x5 : ο . x5 leaving 3 subgoals.

Assume H4: (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) = λ x4 x5 x6 : (ι → ι) → ι → ι . x5.

Apply FalseE with ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x2 (λ x4 x5 x6 : (ι → ι) → ι → ι . x4)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x2) = ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x3 (λ x4 x5 x6 : (ι → ι) → ι → ι . x4)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x3) ⟶ ∀ x4 : ο . x4.

Apply neq_0_1.

Apply H4 with λ x4 x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x5 (λ x6 : ι → ι . λ x7 . x7) (λ x6 : ι → ι . x6) (λ x6 : ι → ι . λ x7 . x7) ordsucc 0 = (λ x6 x7 x8 : (ι → ι) → ι → ι . x7) (λ x6 : ι → ι . λ x7 . x7) (λ x6 : ι → ι . x6) (λ x6 : ι → ι . λ x7 . x7) ordsucc 0.

Let x4 of type ι → ι → ο be given.

Assume H5: x4 ((λ x5 x6 x7 : (ι → ι) → ι → ι . x6) (λ x5 : ι → ι . λ x6 . x6) (λ x5 : ι → ι . x5) (λ x5 : ι → ι . λ x6 . x6) ordsucc 0) ((λ x5 x6 x7 : (ι → ι) → ι → ι . x6) (λ x5 : ι → ι . λ x6 . x6) (λ x5 : ι → ι . x5) (λ x5 : ι → ι . λ x6 . x6) ordsucc 0).

The subproof is completed by applying H5.

Assume H4: (λ x4 x5 x6 : (ι → ι) → ι → ι . x4) = λ x4 x5 x6 : (ι → ι) → ι → ι . x6.

Apply FalseE with ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x2 (λ x4 x5 x6 : (ι → ι) → ι → ι . x4)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x2) = ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x3 (λ x4 x5 x6 : (ι → ι) → ι → ι . x5)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x3) ⟶ ∀ x4 : ο . x4.

Apply neq_0_1.

Apply H4 with λ x4 x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . x5 (λ x6 : ι → ι . λ x7 . x7) (λ x6 : ι → ι . λ x7 . x7) (λ x6 : ι → ι . x6) ordsucc 0 = (λ x6 x7 x8 : (ι → ι) → ι → ι . x8) (λ x6 : ι → ι . λ x7 . x7) (λ x6 : ι → ι . λ x7 . x7) (λ x6 : ι → ι . x6) ordsucc 0.

Let x4 of type ι → ι → ο be given.

Assume H5: x4 ((λ x5 x6 x7 : (ι → ι) → ι → ι . x7) (λ x5 : ι → ι . λ x6 . x6) (λ x5 : ι → ι . λ x6 . x6) (λ x5 : ι → ι . x5) ordsucc 0) ((λ x5 x6 x7 : (ι → ι) → ι → ι . x7) (λ x5 : ι → ι . λ x6 . x6) (λ x5 : ι → ι . λ x6 . x6) (λ x5 : ι → ι . x5) ordsucc 0).

The subproof is completed by applying H5.

Apply H1 with λ x4 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . (λ x5 x6 x7 : (ι → ι) → ι → ι . x5) = (λ x5 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . λ x6 x7 x8 : (ι → ι) → ι → ι . x5 x7 x8 x6) (λ x5 x6 x7 : (ι → ι) → ι → ι . x7) ⟶ ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x4 (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x4) = ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 x3 (λ x5 x6 x7 : (ι → ι) → ι → ι . x7)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 x3) ⟶ ∀ x5 : ο . x5 leaving 8 subgoals.

Apply H3 with λ x4 : ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → ((ι → ι) → ι → ι) → (ι → ι) → ι → ι . ... ⟶ ChurchNums_3x8_to_u24 (ChurchNums_8x3_to_3_lt7_id_ge7_rot2 (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . x5) (λ x5 x6 x7 : (ι → ι) → ι → ι . x5)) (ChurchNums_8_perm_1_2_3_4_5_6_7_0 (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι) → ι → ι . ...)) = ... ⟶ ∀ x5 : ο . x5 leaving 8 subgoals.

...

■

Proofgold Proof