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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Assume H0: CSNo x0.
Assume H1: CSNo x1.
Apply CSNo_ReIm_split with add_CSNo x0 x1, add_CSNo x1 x0 leaving 4 subgoals.
Apply CSNo_add_CSNo with x0, x1 leaving 2 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
Apply CSNo_add_CSNo with x1, x0 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
set y2 to be CSNo_Re (add_CSNo x1 x0)
Claim L2: ∀ x3 : ι → ο . x3 y2 ⟶ x3 (CSNo_Re (add_CSNo x0 x1))
Let x3 of type ι → ο be given.
Assume H2: x3 (CSNo_Re (add_CSNo y2 x1)).
Apply unknownprop_af0b151803c2dc4bc6691b166645c0a8471b89f2da30fa0948427517708d6da0 with x1, y2, λ x4 . x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
set y4 to be CSNo_Re (add_CSNo y2 x1)
Claim L3: ∀ x5 : ι → ο . x5 y4 ⟶ x5 (add_SNo (CSNo_Re x1) (CSNo_Re y2))
Let x5 of type ι → ο be given.
Assume H3: x5 (CSNo_Re (add_CSNo x3 y2)).
Apply add_SNo_com with CSNo_Re y2, CSNo_Re x3, λ x6 . x5 leaving 3 subgoals.
Apply CSNo_ReR with y2.
The subproof is completed by applying H0.
Apply CSNo_ReR with x3.
The subproof is completed by applying H1.
Apply unknownprop_af0b151803c2dc4bc6691b166645c0a8471b89f2da30fa0948427517708d6da0 with x3, y2, λ x6 x7 . (λ x8 . x5) x7 x6 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
set y5 to be λ x5 . y4
Apply L3 with λ x6 . y5 x6 y4 ⟶ y5 y4 x6 leaving 2 subgoals.
Assume H4: y5 y4 y4.
The subproof is completed by applying H4.
The subproof is completed by applying L3.
Let x3 of type ι → ι → ο be given.
Apply L2 with λ x4 . x3 x4 y2 ⟶ x3 y2 x4.
Assume H3: x3 y2 y2.
The subproof is completed by applying H3.
set y2 to be CSNo_Im (add_CSNo x1 x0)
Claim L2: ∀ x3 : ι → ο . x3 y2 ⟶ x3 (CSNo_Im (add_CSNo x0 x1))
Let x3 of type ι → ο be given.
Assume H2: x3 (CSNo_Im (add_CSNo y2 x1)).
Apply unknownprop_7f97cbea1b316ccd537155d989f2889dd5c3074e8edefbeca1dbc06c62876158 with x1, y2, λ x4 . x3 leaving 3 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
set y4 to be CSNo_Im (add_CSNo y2 x1)
Claim L3: ∀ x5 : ι → ο . x5 y4 ⟶ x5 (add_SNo (CSNo_Im x1) (CSNo_Im y2))
Let x5 of type ι → ο be given.
Assume H3: x5 (CSNo_Im (add_CSNo x3 y2)).
Apply add_SNo_com with CSNo_Im y2, CSNo_Im x3, λ x6 . x5 leaving 3 subgoals.
Apply CSNo_ImR with y2.
The subproof is completed by applying H0.
Apply CSNo_ImR with x3.
The subproof is completed by applying H1.
Apply unknownprop_7f97cbea1b316ccd537155d989f2889dd5c3074e8edefbeca1dbc06c62876158 with x3, y2, λ x6 x7 . (λ x8 . x5) x7 x6 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H0.
The subproof is completed by applying H3.
set y5 to be λ x5 . y4
Apply L3 with λ x6 . y5 x6 y4 ⟶ y5 y4 x6 leaving 2 subgoals.
Assume H4: y5 y4 y4.
The subproof is completed by applying H4.
The subproof is completed by applying L3.
Let x3 of type ι → ι → ο be given.
Apply L2 with λ x4 . x3 x4 y2 ⟶ x3 y2 x4.
Assume H3: x3 y2 y2.
The subproof is completed by applying H3.
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