Commit 2025-07-11 15:12 c9ce0e86

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refactor(Order/Concept): rename intentClosurelowerPolar (#26976) The former name was being misused; the intent closure of a set s is actually lowerPolar r (upperPolar r s). The name "polar" can be found in the reference. Although the two types of polars don't have separate names, we choose lower and upper in analogy to lowerBounds and upperBounds; in fact, lowerPolar (· ≤ ·) s is def-eq to lowerBounds s, and likewise for upperPolar. While we're at it, we also get rid of the Concept.toProd projection, in favor of giving more informative names to its fields. This PR contains no mathematics, it's just renaming.

Estimated changes

deleted theorem Concept.bot_fst
deleted theorem Concept.bot_snd
modified theorem Concept.ext'
modified theorem Concept.ext
added theorem Concept.extent_bot
added theorem Concept.extent_inf
added theorem Concept.extent_sInf
added theorem Concept.extent_sSup
added theorem Concept.extent_sup
added theorem Concept.extent_top
deleted theorem Concept.fst_injective
deleted theorem Concept.inf_fst
deleted theorem Concept.inf_snd
added theorem Concept.intent_bot
added theorem Concept.intent_inf
added theorem Concept.intent_sInf
added theorem Concept.intent_sSup
added theorem Concept.intent_sup
added theorem Concept.intent_top
deleted theorem Concept.sInf_fst
deleted theorem Concept.sInf_snd
deleted theorem Concept.sSup_fst
deleted theorem Concept.sSup_snd
deleted theorem Concept.snd_injective
deleted theorem Concept.strictAnti_snd
deleted theorem Concept.strictMono_fst
deleted theorem Concept.sup_fst
deleted theorem Concept.sup_snd
deleted theorem Concept.top_fst
deleted theorem Concept.top_snd
modified structure Concept
deleted def extentClosure
deleted theorem extentClosure_anti
deleted theorem extentClosure_empty
deleted theorem extentClosure_iUnion
deleted theorem extentClosure_iUnion₂
deleted theorem extentClosure_swap
deleted theorem extentClosure_union
deleted def intentClosure
deleted theorem intentClosure_anti
deleted theorem intentClosure_empty
deleted theorem intentClosure_iUnion
deleted theorem intentClosure_iUnion₂
deleted theorem intentClosure_swap
deleted theorem intentClosure_union
added def lowerPolar
added theorem lowerPolar_anti
added theorem lowerPolar_empty
added theorem lowerPolar_iUnion
added theorem lowerPolar_iUnion₂
added theorem lowerPolar_swap
added theorem lowerPolar_union
added def upperPolar
added theorem upperPolar_anti
added theorem upperPolar_empty
added theorem upperPolar_iUnion
added theorem upperPolar_iUnion₂
added theorem upperPolar_swap
added theorem upperPolar_union