Intuitionize section "Definition and basic properties of monoids" #4453
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This is the syntax and df-mnd . Copied without change from set.mm.
Stated as in set.mm. The proof needs a little intuitionizing but is basically the set.mm proof.
Stated as in set.mm. The proof needs some intuitionizing but is basically the set.mm proof.
This is similar to elfv but for any iota expression, not just a function value.
This is analogous to relelfvdm but for any iota expression not just function value.
This is iotan0 from set.mm with non-empty changed to inhabited. The proof is similar to the set.mm proof at least in the sense of being built on iota2 .
This is sgrpidmnd from set.mm with non-empty changed to inhabited. The proof needs some intuitionizing but is based on the set.mm proof. Remove references in comment to theorems not present in iset.mm
Copied from set.mm. The only change is to the comment of mndbn0 , to mention non-empty versus inhabited.
This is hashelne0d from set.mm but for finite sets.
Stated as in set.mm. The proof needs a very small amount of intuitionizing but is other than that the set.mm proof.
Copied without change from set.mm.
Stated as in set.mm. The proof needs a small amount of intuitionizing but is basically the set.mm proof.
Stated as in set.mm. The proof needs a little intuitionizing but is basically the set.mm proof.
This is prdsidlem , prdsmndd , prds0g , pwsmnd , and pws0g .
This is imasmnd2 , imasmnd , and imasmndf1
Stated as in set.mm. The proof needs some intuitionizing but is basically the set.mm proof.
Stated as in set.mm. The proof needs some intuitionizing but is basically the set.mm proof.
avekens
approved these changes
Nov 26, 2024
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Although most proofs here require some intuitionizing, most of this section carries over to iset.mm readily.
The main exceptions are structure restriction (see #3022 ), structure product, and image structure. None of those is well enough developed in iset.mm for the proofs in the monoid section to work, so these are noted in mmil.html as part of this pull request.
There are a few minor theorem additions, perhaps the most interesting of which is
iotam, that if an iota expression is inhabited, then its proposition is true for a unique value of the variable bound in the iota.