Theorem wl-df-3mintru2 35166

Description: Alternative definition of wcad 1609. See df-cad 1610 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 15835. If inputs are viewed as binary digits (true is 1, false is 0), the result is whether ordinary binary full addition yields a carry bit. That is what the name df-cad 1610 is derived from: "carry of an addition". Here we stick with this abbreviated form of our notation above, but still use "adder carry" as a shorthand for "at least 2 out of 3" in text.

The core meaning is to check whether at least two of three inputs are true. So, if the first input is true, at least one of the two remaining must be true, else even both. This theorem is the in-between of "at least 1 out of 3", given by triple disjunction df-3or 1086, and "(at least) 3 out of 3", expressed by triple conjunction df-3an 1087.

The notion above can be generalized to other input numbers with other minimum values as follows. Let us introduce informally a logical operation "n-mintru-m" taking n inputs, and requiring at least m of them be true to let the operation itself be true. There now exists a recursive scheme to define it for increasing n, m. We start with the base case n = 0. Here "n-mintru-0" is equivalent to ⊤ (any sequence of inputs contains at least zero true inputs), the other "0-mintru-m" is for any m > 0 equivalent to ⊥, because a sequence of zero inputs never has a positive number of them true. The general case adds a new input 𝜑 to a given sequence of n inputs, and reduces that case for all integers m to that of the smaller sequence by recursion, informally written as:

"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" )

Our definition here matches "3-mintru-2" with inputs 𝜑, 𝜓 and 𝜒. Starting from the base cases we find after simplifications: "2-mintru-2" (𝜓, 𝜒) ↔ (𝜓 ∧ 𝜒) (wl-2mintru2 35173), and "2-mintru-1" (𝜓, 𝜒) ↔ (𝜓 ∨ 𝜒) (wl-2mintru1 35172). Plugging these expressions into the formula above for n = 3, m = 2 yields exactly our definition here. (Contributed by Wolf Lammen, 2-May-2024.)

Assertion

Ref	Expression
wl-df-3mintru2	⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))

Proof of Theorem wl-df-3mintru2

Step	Hyp	Ref	Expression
1		cadrot 1617	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜒, 𝜑))
2		cadifp 1621	. 2 ⊢ (cadd(𝜓, 𝜒, 𝜑) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))
3	1, 2	bitri 278	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∨ wo 845  if-wif 1059  caddwcad 1609

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-ifp 1060  df-3or 1086  df-3an 1087  df-xor 1504  df-cad 1610

This theorem is referenced by:  wl-df2-3mintru2  35167  wl-df3-3mintru2  35168  wl-df3maxtru1  35174