Search for blocks/addresses/...

Proofgold Proof

pf
Let x0 of type ((ιι) → ιι) → ((ιι) → ιι) → CN (ιι) be given.
Let x1 of type ((ιι) → ιι) → ((ιι) → ιι) → CN (ιι) be given.
Let x2 of type ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → CN (ιι) be given.
Let x3 of type ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → ((ιι) → ιι) → CN (ιι) be given.
Assume H0: ChurchNum_3ary_proj_p x0.
Assume H1: ChurchNum_3ary_proj_p x1.
Assume H2: ChurchNum_8ary_proj_p x2.
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 . TwoRamseyGraph_4_5_24_ChurchNums_3x8 x4 x2 x1 x3 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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) x2 ((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) x3 leaving 3 subgoals.
Apply H2 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 . TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) x4 x1 x3 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) 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) x3 leaving 8 subgoals.
Apply H1 with λ x4 : ((ι → ι)ι → ι)((ι → ι)ι → ι)((ι → ι)ι → ι)(ι → ι)ι → ι . TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5) x4 x3 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (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) x3 leaving 3 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 . TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 : (ι → ι)ι → ι . x5) x4 = TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (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 . x5)) x4 leaving 8 subgoals.
Let x4 of type (ιιι) → (ιιι) → ο be given.
Assume H4: x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5)).
The subproof is completed by applying H4.
Let x4 of type (ιιι) → (ιιι) → ο be given.
Assume H4: x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x6)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x6)).
The subproof is completed by applying H4.
Let x4 of type (ιιι) → (ιιι) → ο be given.
Assume H4: x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x7)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x7)).
The subproof is completed by applying H4.
Let x4 of type (ιιι) → (ιιι) → ο be given.
Assume H4: x4 (TwoRamseyGraph_4_5_24_ChurchNums_3x8 (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 : (ι → ι)ι → ι . x5) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x8)) (TwoRamseyGraph_4_5_24_ChurchNums_3x8 ((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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (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 . x5)) (λ x5 x6 x7 x8 x9 x10 x11 x12 : (ι → ι)ι → ι . x8)).
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...
...