Theorem wl-df-3mintru2 37450

Description: Alternative definition of wcad 1603. See df-cad 1604 to learn how it is currently introduced. The only use case so far is being a binary addition primitive for df-sad 16497. 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 1604 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 1088, and "(at least) 3 out of 3", expressed by triple conjunction df-3an 1089.

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 37457), and "2-mintru-1" (𝜓, 𝜒) ↔ (𝜓 ∨ 𝜒) (wl-2mintru1 37456). 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 1611	. 2 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ cadd(𝜓, 𝜒, 𝜑))
2		cadifp 1617	. 2 ⊢ (cadd(𝜓, 𝜒, 𝜑) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))
3	1, 2	bitri 275	1 ⊢ (cadd(𝜑, 𝜓, 𝜒) ↔ if-(𝜑, (𝜓 ∨ 𝜒), (𝜓 ∧ 𝜒)))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∨ wo 846  if-wif 1063  caddwcad 1603

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 207  df-an 396  df-or 847  df-ifp 1064  df-3or 1088  df-3an 1089  df-xor 1509  df-cad 1604

This theorem is referenced by:  wl-df2-3mintru2  37451  wl-df3-3mintru2  37452  wl-df3maxtru1  37458