Datasets:
theorem_name stringclasses 737
values | cleaned_theorem_name stringclasses 737
values | field stringclasses 62
values | subfield stringclasses 254
values | selected_family stringclasses 5
values | theorem_shape stringclasses 12
values | expression_type stringclasses 5
values | short_statement stringclasses 746
values | canonical_formula_or_condition stringclasses 738
values | required_assumptions stringclasses 56
values | wikipedia_title stringclasses 748
values | wikipedia_url stringclasses 737
values | variant_id stringlengths 19 72 | family stringclasses 5
values | original_formula stringclasses 923
values | variant_formula stringlengths 6 475 | latex_math_representation stringlengths 6 475 | transformation_pipeline stringclasses 145
values | difficulty_level stringclasses 4
values | validity_label stringclasses 1
value | validation_type stringclasses 112
values | inverse_mapping_to_original stringlengths 174 421 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_001 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \frac{\exp(min(|A|+|B|-1, p))}{\exp(min(|A|+|B|-1, p))+\exp(|A+B|)}\le\frac{1}{2} | \frac{\exp(min(|A|+|B|-1, p))}{\exp(min(|A|+|B|-1, p))+\exp(|A+B|)}\le\frac{1}{2} | logistic order embedding | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_002 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \operatorname{softplus}((min(|A|+|B|-1, p))-(|A+B|))\le\log 2 | \operatorname{softplus}((min(|A|+|B|-1, p))-(|A+B|))\le\log 2 | softplus threshold | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_003 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \arctan(min(|A|+|B|-1, p))\le\arctan(|A+B|) | \arctan(min(|A|+|B|-1, p))\le\arctan(|A+B|) | bounded monotone transform | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_004 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \tanh(min(|A|+|B|-1, p))\le\tanh(|A+B|) | \tanh(min(|A|+|B|-1, p))\le\tanh(|A+B|) | hyperbolic monotone transform | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_005 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \max\{0,(min(|A|+|B|-1, p))-(|A+B|)\}=0 | \max\{0,(min(|A|+|B|-1, p))-(|A+B|)\}=0 | positive-part zero set | hard | valid_equivalent | positive-part identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_006 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \operatorname{dist}((min(|A|+|B|-1, p))-(|A+B|),(-\infty,0])=0 | \operatorname{dist}((min(|A|+|B|-1, p))-(|A+B|),(-\infty,0])=0 | distance to halfline | hard | valid_equivalent | distance-to-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_007 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | ((min(|A|+|B|-1, p)),(|A+B|))\in\{(u,v)\in\mathbb{R}^2:u\le v\} | ((min(|A|+|B|-1, p)),(|A+B|))\in\{(u,v)\in\mathbb{R}^2:u\le v\} | order-cone membership | hard | valid_equivalent | set-membership equivalence | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_008 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \operatorname{dist}(((min(|A|+|B|-1, p)),(|A+B|)),\{(u,v)\in\mathbb{R}^2:u\le v\})=0 | \operatorname{dist}(((min(|A|+|B|-1, p)),(|A+B|)),\{(u,v)\in\mathbb{R}^2:u\le v\})=0 | distance to order cone | very_hard | valid_equivalent | distance-to-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_009 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \{t\in\mathbb{R}:t\ge (|A+B|)\}\subseteq\{t\in\mathbb{R}:t\ge (min(|A|+|B|-1, p))\} | \{t\in\mathbb{R}:t\ge (|A+B|)\}\subseteq\{t\in\mathbb{R}:t\ge (min(|A|+|B|-1, p))\} | upper-ray inclusion | hard | valid_equivalent | set-inclusion proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_010 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \{t\in\mathbb{R}:t\le (min(|A|+|B|-1, p))\}\subseteq\{t\in\mathbb{R}:t\le (|A+B|)\} | \{t\in\mathbb{R}:t\le (min(|A|+|B|-1, p))\}\subseteq\{t\in\mathbb{R}:t\le (|A+B|)\} | lower-cut inclusion | hard | valid_equivalent | Dedekind-cut proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_011 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \forall t\in\mathbb{R},\;(|A+B|)\le t\Rightarrow (min(|A|+|B|-1, p))\le t | \forall t\in\mathbb{R},\;(|A+B|)\le t\Rightarrow (min(|A|+|B|-1, p))\le t | universal upper test | hard | valid_equivalent | universal quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_012 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \forall t\in\mathbb{R},\;t\le (min(|A|+|B|-1, p))\Rightarrow t\le (|A+B|) | \forall t\in\mathbb{R},\;t\le (min(|A|+|B|-1, p))\Rightarrow t\le (|A+B|) | universal lower test | hard | valid_equivalent | universal quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_013 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \forall q\in\mathbb{Q},\;q<(min(|A|+|B|-1, p))\Rightarrow q<(|A+B|) | \forall q\in\mathbb{Q},\;q<(min(|A|+|B|-1, p))\Rightarrow q<(|A+B|) | rational cut | very_hard | valid_equivalent | density of rationals | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_014 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \forall \varepsilon\in\mathbb{Q}_{>0},\;(min(|A|+|B|-1, p))<(|A+B|)+\varepsilon | \forall \varepsilon\in\mathbb{Q}_{>0},\;(min(|A|+|B|-1, p))<(|A+B|)+\varepsilon | epsilon relaxation | very_hard | valid_equivalent | Archimedean order proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_015 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \exists s\in\mathbb{R}_{\ge0}\;\text{such that}\;(min(|A|+|B|-1, p))+s=(|A+B|) | \exists s\in\mathbb{R}_{\ge0}\;\text{such that}\;(min(|A|+|B|-1, p))+s=(|A+B|) | nonnegative slack | hard | valid_equivalent | slack-variable equivalence | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_016 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \sup_{\lambda\in[0,1]}\lambda((min(|A|+|B|-1, p))-(|A+B|))=0 | \sup_{\lambda\in[0,1]}\lambda((min(|A|+|B|-1, p))-(|A+B|))=0 | interval optimization | very_hard | valid_equivalent | optimization identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_017 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \inf_{\lambda\in[0,1]}\lambda((|A+B|)-(min(|A|+|B|-1, p)))=0 | \inf_{\lambda\in[0,1]}\lambda((|A+B|)-(min(|A|+|B|-1, p)))=0 | dual interval optimization | very_hard | valid_equivalent | optimization identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_018 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \lim_{n\to\infty}\max\{0,(min(|A|+|B|-1, p))-(|A+B|)-1/n\}=0 | \lim_{n\to\infty}\max\{0,(min(|A|+|B|-1, p))-(|A+B|)-1/n\}=0 | limit positive-part test | very_hard | valid_equivalent | limit identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_019 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \int_0^1\mathbf{1}_{\{(min(|A|+|B|-1, p))-(|A+B|)>t\}}\,dt=0 | \int_0^1\mathbf{1}_{\{(min(|A|+|B|-1, p))-(|A+B|)>t\}}\,dt=0 | indicator integral | very_hard | valid_equivalent | indicator integral proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Cauchy–Davenport theorem | Cauchy–Davenport theorem | combinatorics | additive combinatorics | Combinatorics / Graph Theory | scalar_inequality | inequality | For finite subsets A and B of the cyclic group of prime order p, the cardinality of their sumset is at least the minimum of p and the sum of their cardinalities minus one. | |A+B| >= min(|A|+|B|-1, p) | A and B are nonempty finite subsets of Z/pZ; p is prime. | Additive combinatorics | https://en.wikipedia.org/wiki/Additive_combinatorics | CAUCHYDAVENPORT_THEOREM_R0001_020 | Combinatorics / Graph Theory | |A+B| ≥ min(|A|+|B|-1, p) | \inf_{t\in\mathbb{R}}((t-(min(|A|+|B|-1, p)))^2+\max\{0,t-(|A+B|)\}^2)=0 | \inf_{t\in\mathbb{R}}((t-(min(|A|+|B|-1, p)))^2+\max\{0,t-(|A+B|)\}^2)=0 | zero-loss optimization | very_hard | valid_equivalent | optimization feasibility proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_001 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \exists x\in D:x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | \exists x\in D:x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | element witness | hard | valid_equivalent | existence proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_002 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \sup_{x\in D}\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x)=1 | \sup_{x\in D}\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x)=1 | indicator supremum | hard | valid_equivalent | indicator proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_003 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \neg\forall x\in D,x\notin \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | \neg\forall x\in D,x\notin \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | quantifier negation | hard | valid_equivalent | quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_004 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \{x\in D:x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\}\ne\emptyset | \{x\in D:x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\}\ne\emptyset | comprehension nonempty | hard | valid_equivalent | comprehension proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_005 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \operatorname{card}(\{x\in D:x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\})\ge1 | \operatorname{card}(\{x\in D:x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\})\ge1 | cardinality witness | hard | valid_equivalent | cardinality proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_006 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \exists f:\{*\}\to D:f(*)\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | \exists f:\{*\}\to D:f(*)\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | singleton map witness | very_hard | valid_equivalent | singleton proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_007 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \{*\}\preceq \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | \{*\}\preceq \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | singleton injection | very_hard | valid_equivalent | injection proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_008 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \bigcup_{x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}\{x\}\ne\emptyset | \bigcup_{x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}\{x\}\ne\emptyset | singleton union | very_hard | valid_equivalent | union identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_009 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \operatorname{proj}_1(\{(x,0):x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\})\ne\emptyset | \operatorname{proj}_1(\{(x,0):x\in \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\})\ne\emptyset | tagged projection | very_hard | valid_equivalent | projection proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_010 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \inf_{x\in D}(1-\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x))=0 | \inf_{x\in D}(1-\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x))=0 | indicator infimum | hard | valid_equivalent | indicator proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_011 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \operatorname{dist}_{\delta}(a,\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\})=0\text{ for some }a\in D | \operatorname{dist}_{\delta}(a,\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\})=0\text{ for some }a\in D | distance witness | very_hard | valid_equivalent | distance proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_012 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | D\not\subseteq D\setminus \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | D\not\subseteq D\setminus \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\} | complement failure | hard | valid_equivalent | set complement proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_013 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \mathcal{P}(\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\})\ne\{\emptyset\} | \mathcal{P}(\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\})\ne\{\emptyset\} | power-set test | very_hard | valid_equivalent | power-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_014 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \exists A\subseteq \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}:A\ne\emptyset | \exists A\subseteq \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}:A\ne\emptyset | nonempty subset witness | hard | valid_equivalent | subset proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_015 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \operatorname{Id}_D[\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}]\ne\emptyset | \operatorname{Id}_D[\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}]\ne\emptyset | identity image | hard | valid_equivalent | image proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_016 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \exists R\subseteq D\times D:\operatorname{dom}(R)=\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\land\operatorname{dom}(R)\ne\emptyset | \exists R\subseteq D\times D:\operatorname{dom}(R)=\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\land\operatorname{dom}(R)\ne\emptyset | relation domain witness | very_hard | valid_equivalent | domain proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_017 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \lim_{n\to\infty}(1-\sup_{x\in D}\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x))^n=0 | \lim_{n\to\infty}(1-\sup_{x\in D}\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x))^n=0 | asymptotic indicator | very_hard | valid_equivalent | limit proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_018 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \max_{x\in D}\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x)=1 | \max_{x\in D}\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x)=1 | indicator maximum | hard | valid_equivalent | indicator proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_019 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \exists x\in D:\delta_{\{0,1\}}(\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x),1)=0 | \exists x\in D:\delta_{\{0,1\}}(\mathbf{1}_{\{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}}(x),1)=0 | indicator distance | very_hard | valid_equivalent | distance proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Bauer maximum principle | Bauer maximum principle | mathematical analysis | convex analysis | Analysis | existence | none | A convex and continuous function defined on a convex, compact set attains its maximum at an extreme point of that set. | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | K is compact convex; f:K->R is continuous and convex; ext(K) is the set of extreme points. | Bauer maximum principle | https://en.wikipedia.org/wiki/Bauer_maximum_principle | BAUER_MAXIMUM_PRINCIPLE_R0005_020 | Analysis | exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x) | \operatorname{coeq}(D\setminus \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\hookrightarrow D)\ne0 | \operatorname{coeq}(D\setminus \{x_e\in ext(K):f(x_e)=max_{x in K} f(x)\}\hookrightarrow D)\ne0 | quotient residual | very_hard | valid_equivalent | quotient proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as exists x_e in ext(K) such that f(x_e)=max_{x in K} f(x)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_001 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | ((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\in\{(z,z):z\in D\} | ((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\in\{(z,z):z\in D\} | diagonal membership | hard | valid_equivalent | diagonal membership proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_002 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \operatorname{dist}_{\delta_D}(((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))),\{(z,z):z\in D\})=0 | \operatorname{dist}_{\delta_D}(((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))),\{(z,z):z\in D\})=0 | distance to diagonal | very_hard | valid_equivalent | distance-to-diagonal proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_003 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \{(V(p))\}=\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\} | \{(V(p))\}=\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\} | singleton equality | hard | valid_equivalent | singleton proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_004 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | (V(p))\in\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\}\;\land\;(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in\{(V(p))\} | (V(p))\in\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\}\;\land\;(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in\{(V(p))\} | mutual singleton membership | hard | valid_equivalent | singleton membership proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_005 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \{S\subseteq D:(V(p))\in S\}=\{S\subseteq D:(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in S\} | \{S\subseteq D:(V(p))\in S\}=\{S\subseteq D:(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in S\} | containing-subset profile equality | very_hard | valid_equivalent | extensionality proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_006 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \forall S\subseteq D,\;((V(p))\in S\Longleftrightarrow (max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in S) | \forall S\subseteq D,\;((V(p))\in S\Longleftrightarrow (max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in S) | Leibniz subset quantifier | very_hard | valid_equivalent | Leibniz principle | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_007 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \forall z\in D,\;(z=(V(p))\Longleftrightarrow z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | \forall z\in D,\;(z=(V(p))\Longleftrightarrow z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | equality-fiber equivalence | hard | valid_equivalent | fiber equality proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_008 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \forall F:D\to\{0,1\},\;F((V(p)))=F((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | \forall F:D\to\{0,1\},\;F((V(p)))=F((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | Boolean observer equivalence | very_hard | valid_equivalent | separating function proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_009 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \delta_D((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))=0 | \delta_D((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))=0 | discrete metric zero | hard | valid_equivalent | discrete metric identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_010 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \inf_{z\in D}(\delta_D(z,(V(p)))+\delta_D(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))))=0 | \inf_{z\in D}(\delta_D(z,(V(p)))+\delta_D(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))))=0 | zero-loss domain optimization | very_hard | valid_equivalent | optimization proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_011 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \{(V(p))\}\times\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\}\subseteq\{(z,z):z\in D\} | \{(V(p))\}\times\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\}\subseteq\{(z,z):z\in D\} | singleton product diagonal containment | very_hard | valid_equivalent | set-containment proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_012 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | (\{(V(p))\}\times\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\})\cap\{(z,z):z\in D\}=\{((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\} | (\{(V(p))\}\times\{(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\})\cap\{(z,z):z\in D\}=\{((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\} | diagonal intersection | very_hard | valid_equivalent | intersection proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_013 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \operatorname{argmin}_{z\in D}\delta_D(z,(V(p)))=\operatorname{argmin}_{z\in D}\delta_D(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | \operatorname{argmin}_{z\in D}\delta_D(z,(V(p)))=\operatorname{argmin}_{z\in D}\delta_D(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | argmin profile equality | very_hard | valid_equivalent | argmin identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_014 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \bigcap\{S\subseteq D:(V(p))\in S\}=\bigcap\{S\subseteq D:(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in S\} | \bigcap\{S\subseteq D:(V(p))\in S\}=\bigcap\{S\subseteq D:(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))\in S\} | intersection profile equality | very_hard | valid_equivalent | intersection representation proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_015 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \forall z\in D,\;((V(p)),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z) | \forall z\in D,\;((V(p)),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z) | first coordinate pair test | hard | valid_equivalent | ordered-pair extensionality | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_016 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \forall z\in D,\;(z,(V(p)))=(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | \forall z\in D,\;(z,(V(p)))=(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))) | second coordinate pair test | hard | valid_equivalent | ordered-pair extensionality | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_017 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \{((V(p)),z):z\in D\}=\{((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z):z\in D\} | \{((V(p)),z):z\in D\}=\{((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z):z\in D\} | first-coordinate cylinder equality | very_hard | valid_equivalent | cylinder equality proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_018 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \{(z,(V(p))):z\in D\}=\{(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))):z\in D\} | \{(z,(V(p))):z\in D\}=\{(z,(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega))):z\in D\} | second-coordinate cylinder equality | very_hard | valid_equivalent | cylinder equality proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_019 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \exists z\in D\;\text{such that}\;(V(p))=z\;\land\;z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)) | \exists z\in D\;\text{such that}\;(V(p))=z\;\land\;z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)) | common witness factorization | hard | valid_equivalent | witness proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Blackwell's informativeness theorem | Blackwell's informativeness theorem | decision theory | information theory | Probability / Information Theory | equality | equation | The theorem establishes the equivalence between three ways of ranking information structures based on expected utility, informativeness (garbling), and feasibility. It provides conditions under which a decision maker would prefer one information structure over another. | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | Blackwell's informativeness theorem | https://en.wikipedia.org/wiki/Blackwell%27s_informativeness_theorem | BLACKWELL_S_INFORMATIVENESS_THEOREM_R0006_020 | Probability / Information Theory | V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega) | \{T\subseteq D\times D:((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\in T\}=\{T\subseteq D\times D:((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\in T\} | \{T\subseteq D\times D:((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\in T\}=\{T\subseteq D\times D:((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))\in T\} | pair relation profile equality | very_hard | valid_equivalent | power-set extensionality proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_001 | Analysis | 2GM/Rc^2 ≤ 8/9 | \frac{\exp(2GM/Rc^2)}{\exp(2GM/Rc^2)+\exp(8/9)}\le\frac{1}{2} | \frac{\exp(2GM/Rc^2)}{\exp(2GM/Rc^2)+\exp(8/9)}\le\frac{1}{2} | logistic order embedding | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_002 | Analysis | 2GM/Rc^2 ≤ 8/9 | \operatorname{softplus}((2GM/Rc^2)-(8/9))\le\log 2 | \operatorname{softplus}((2GM/Rc^2)-(8/9))\le\log 2 | softplus threshold | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_003 | Analysis | 2GM/Rc^2 ≤ 8/9 | \arctan(2GM/Rc^2)\le\arctan(8/9) | \arctan(2GM/Rc^2)\le\arctan(8/9) | bounded monotone transform | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_004 | Analysis | 2GM/Rc^2 ≤ 8/9 | \tanh(2GM/Rc^2)\le\tanh(8/9) | \tanh(2GM/Rc^2)\le\tanh(8/9) | hyperbolic monotone transform | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_005 | Analysis | 2GM/Rc^2 ≤ 8/9 | \max\{0,(2GM/Rc^2)-(8/9)\}=0 | \max\{0,(2GM/Rc^2)-(8/9)\}=0 | positive-part zero set | hard | valid_equivalent | positive-part identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_006 | Analysis | 2GM/Rc^2 ≤ 8/9 | \operatorname{dist}((2GM/Rc^2)-(8/9),(-\infty,0])=0 | \operatorname{dist}((2GM/Rc^2)-(8/9),(-\infty,0])=0 | distance to halfline | hard | valid_equivalent | distance-to-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_007 | Analysis | 2GM/Rc^2 ≤ 8/9 | ((2GM/Rc^2),(8/9))\in\{(u,v)\in\mathbb{R}^2:u\le v\} | ((2GM/Rc^2),(8/9))\in\{(u,v)\in\mathbb{R}^2:u\le v\} | order-cone membership | hard | valid_equivalent | set-membership equivalence | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_008 | Analysis | 2GM/Rc^2 ≤ 8/9 | \operatorname{dist}(((2GM/Rc^2),(8/9)),\{(u,v)\in\mathbb{R}^2:u\le v\})=0 | \operatorname{dist}(((2GM/Rc^2),(8/9)),\{(u,v)\in\mathbb{R}^2:u\le v\})=0 | distance to order cone | very_hard | valid_equivalent | distance-to-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_009 | Analysis | 2GM/Rc^2 ≤ 8/9 | \{t\in\mathbb{R}:t\ge (8/9)\}\subseteq\{t\in\mathbb{R}:t\ge (2GM/Rc^2)\} | \{t\in\mathbb{R}:t\ge (8/9)\}\subseteq\{t\in\mathbb{R}:t\ge (2GM/Rc^2)\} | upper-ray inclusion | hard | valid_equivalent | set-inclusion proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_010 | Analysis | 2GM/Rc^2 ≤ 8/9 | \{t\in\mathbb{R}:t\le (2GM/Rc^2)\}\subseteq\{t\in\mathbb{R}:t\le (8/9)\} | \{t\in\mathbb{R}:t\le (2GM/Rc^2)\}\subseteq\{t\in\mathbb{R}:t\le (8/9)\} | lower-cut inclusion | hard | valid_equivalent | Dedekind-cut proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_011 | Analysis | 2GM/Rc^2 ≤ 8/9 | \forall t\in\mathbb{R},\;(8/9)\le t\Rightarrow (2GM/Rc^2)\le t | \forall t\in\mathbb{R},\;(8/9)\le t\Rightarrow (2GM/Rc^2)\le t | universal upper test | hard | valid_equivalent | universal quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_012 | Analysis | 2GM/Rc^2 ≤ 8/9 | \forall t\in\mathbb{R},\;t\le (2GM/Rc^2)\Rightarrow t\le (8/9) | \forall t\in\mathbb{R},\;t\le (2GM/Rc^2)\Rightarrow t\le (8/9) | universal lower test | hard | valid_equivalent | universal quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_013 | Analysis | 2GM/Rc^2 ≤ 8/9 | \forall q\in\mathbb{Q},\;q<(2GM/Rc^2)\Rightarrow q<(8/9) | \forall q\in\mathbb{Q},\;q<(2GM/Rc^2)\Rightarrow q<(8/9) | rational cut | very_hard | valid_equivalent | density of rationals | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_014 | Analysis | 2GM/Rc^2 ≤ 8/9 | \forall \varepsilon\in\mathbb{Q}_{>0},\;(2GM/Rc^2)<(8/9)+\varepsilon | \forall \varepsilon\in\mathbb{Q}_{>0},\;(2GM/Rc^2)<(8/9)+\varepsilon | epsilon relaxation | very_hard | valid_equivalent | Archimedean order proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_015 | Analysis | 2GM/Rc^2 ≤ 8/9 | \exists s\in\mathbb{R}_{\ge0}\;\text{such that}\;(2GM/Rc^2)+s=(8/9) | \exists s\in\mathbb{R}_{\ge0}\;\text{such that}\;(2GM/Rc^2)+s=(8/9) | nonnegative slack | hard | valid_equivalent | slack-variable equivalence | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_016 | Analysis | 2GM/Rc^2 ≤ 8/9 | \sup_{\lambda\in[0,1]}\lambda((2GM/Rc^2)-(8/9))=0 | \sup_{\lambda\in[0,1]}\lambda((2GM/Rc^2)-(8/9))=0 | interval optimization | very_hard | valid_equivalent | optimization identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_017 | Analysis | 2GM/Rc^2 ≤ 8/9 | \inf_{\lambda\in[0,1]}\lambda((8/9)-(2GM/Rc^2))=0 | \inf_{\lambda\in[0,1]}\lambda((8/9)-(2GM/Rc^2))=0 | dual interval optimization | very_hard | valid_equivalent | optimization identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_018 | Analysis | 2GM/Rc^2 ≤ 8/9 | \lim_{n\to\infty}\max\{0,(2GM/Rc^2)-(8/9)-1/n\}=0 | \lim_{n\to\infty}\max\{0,(2GM/Rc^2)-(8/9)-1/n\}=0 | limit positive-part test | very_hard | valid_equivalent | limit identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_019 | Analysis | 2GM/Rc^2 ≤ 8/9 | \int_0^1\mathbf{1}_{\{(2GM/Rc^2)-(8/9)>t\}}\,dt=0 | \int_0^1\mathbf{1}_{\{(2GM/Rc^2)-(8/9)>t\}}\,dt=0 | indicator integral | very_hard | valid_equivalent | indicator integral proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
Buchdahl's theorem | Buchdahl's theorem | physics | general relativity | Analysis | scalar_inequality | inequality | Buchdahl's theorem establishes a maximal sustainable density for static, spherically symmetric configurations of perfect fluid matter, expressed as a limit on the ratio of the object's mass to its radius. | 2GM/Rc^2 <= 8/9 | All displayed quantities are finite real-valued under the theorem hypotheses. | Buchdahl's theorem | https://en.wikipedia.org/wiki/Buchdahl%27s_theorem | BUCHDAHL_S_THEOREM_R0007_020 | Analysis | 2GM/Rc^2 ≤ 8/9 | \inf_{t\in\mathbb{R}}((t-(2GM/Rc^2))^2+\max\{0,t-(8/9)\}^2)=0 | \inf_{t\in\mathbb{R}}((t-(2GM/Rc^2))^2+\max\{0,t-(8/9)\}^2)=0 | zero-loss optimization | very_hard | valid_equivalent | optimization feasibility proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as 2GM/Rc^2 ≤ 8/9."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_001 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \frac{\exp(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))}{\exp(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))+\exp(α)}\le\frac{1}{2} | \frac{\exp(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))}{\exp(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))+\exp(α)}\le\frac{1}{2} | logistic order embedding | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_002 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \operatorname{softplus}((P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α))\le\log 2 | \operatorname{softplus}((P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α))\le\log 2 | softplus threshold | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_003 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \arctan(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\le\arctan(α) | \arctan(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\le\arctan(α) | bounded monotone transform | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_004 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \tanh(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\le\tanh(α) | \tanh(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\le\tanh(α) | hyperbolic monotone transform | hard | valid_equivalent | monotonicity proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_005 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \max\{0,(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α)\}=0 | \max\{0,(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α)\}=0 | positive-part zero set | hard | valid_equivalent | positive-part identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_006 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \operatorname{dist}((P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α),(-\infty,0])=0 | \operatorname{dist}((P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α),(-\infty,0])=0 | distance to halfline | hard | valid_equivalent | distance-to-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_007 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | ((P(x ~ X_θ : Bel(A|x) ≥ 1 - α)),(α))\in\{(u,v)\in\mathbb{R}^2:u\le v\} | ((P(x ~ X_θ : Bel(A|x) ≥ 1 - α)),(α))\in\{(u,v)\in\mathbb{R}^2:u\le v\} | order-cone membership | hard | valid_equivalent | set-membership equivalence | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_008 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \operatorname{dist}(((P(x ~ X_θ : Bel(A|x) ≥ 1 - α)),(α)),\{(u,v)\in\mathbb{R}^2:u\le v\})=0 | \operatorname{dist}(((P(x ~ X_θ : Bel(A|x) ≥ 1 - α)),(α)),\{(u,v)\in\mathbb{R}^2:u\le v\})=0 | distance to order cone | very_hard | valid_equivalent | distance-to-set proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_009 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \{t\in\mathbb{R}:t\ge (α)\}\subseteq\{t\in\mathbb{R}:t\ge (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\} | \{t\in\mathbb{R}:t\ge (α)\}\subseteq\{t\in\mathbb{R}:t\ge (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\} | upper-ray inclusion | hard | valid_equivalent | set-inclusion proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_010 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \{t\in\mathbb{R}:t\le (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\}\subseteq\{t\in\mathbb{R}:t\le (α)\} | \{t\in\mathbb{R}:t\le (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\}\subseteq\{t\in\mathbb{R}:t\le (α)\} | lower-cut inclusion | hard | valid_equivalent | Dedekind-cut proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_011 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \forall t\in\mathbb{R},\;(α)\le t\Rightarrow (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\le t | \forall t\in\mathbb{R},\;(α)\le t\Rightarrow (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\le t | universal upper test | hard | valid_equivalent | universal quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_012 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \forall t\in\mathbb{R},\;t\le (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\Rightarrow t\le (α) | \forall t\in\mathbb{R},\;t\le (P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\Rightarrow t\le (α) | universal lower test | hard | valid_equivalent | universal quantifier proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_013 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \forall q\in\mathbb{Q},\;q<(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\Rightarrow q<(α) | \forall q\in\mathbb{Q},\;q<(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))\Rightarrow q<(α) | rational cut | very_hard | valid_equivalent | density of rationals | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_014 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \forall \varepsilon\in\mathbb{Q}_{>0},\;(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))<(α)+\varepsilon | \forall \varepsilon\in\mathbb{Q}_{>0},\;(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))<(α)+\varepsilon | epsilon relaxation | very_hard | valid_equivalent | Archimedean order proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_015 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \exists s\in\mathbb{R}_{\ge0}\;\text{such that}\;(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))+s=(α) | \exists s\in\mathbb{R}_{\ge0}\;\text{such that}\;(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))+s=(α) | nonnegative slack | hard | valid_equivalent | slack-variable equivalence | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_016 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \sup_{\lambda\in[0,1]}\lambda((P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α))=0 | \sup_{\lambda\in[0,1]}\lambda((P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α))=0 | interval optimization | very_hard | valid_equivalent | optimization identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_017 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \inf_{\lambda\in[0,1]}\lambda((α)-(P(x ~ X_θ : Bel(A|x) ≥ 1 - α)))=0 | \inf_{\lambda\in[0,1]}\lambda((α)-(P(x ~ X_θ : Bel(A|x) ≥ 1 - α)))=0 | dual interval optimization | very_hard | valid_equivalent | optimization identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_018 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \lim_{n\to\infty}\max\{0,(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α)-1/n\}=0 | \lim_{n\to\infty}\max\{0,(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α)-1/n\}=0 | limit positive-part test | very_hard | valid_equivalent | limit identity | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_019 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \int_0^1\mathbf{1}_{\{(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α)>t\}}\,dt=0 | \int_0^1\mathbf{1}_{\{(P(x ~ X_θ : Bel(A|x) ≥ 1 - α))-(α)>t\}}\,dt=0 | indicator integral | very_hard | valid_equivalent | indicator integral proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
False confidence theorem | False confidence theorem | statistics | decision theory | Probability / Information Theory | probability_bound | inequality | A result suggesting that analysts should not assign a high degree of belief to an assertion based on low-quality data, as additive measures of belief (like probability) can lead to unjustified confidence. | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed. | False confidence theorem | https://en.wikipedia.org/wiki/False_confidence_theorem | FALSE_CONFIDENCE_THEOREM_R0014_020 | Probability / Information Theory | P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α | \inf_{t\in\mathbb{R}}((t-(P(x ~ X_θ : Bel(A|x) ≥ 1 - α)))^2+\max\{0,t-(α)\}^2)=0 | \inf_{t\in\mathbb{R}}((t-(P(x ~ X_θ : Bel(A|x) ≥ 1 - α)))^2+\max\{0,t-(α)\}^2)=0 | zero-loss optimization | very_hard | valid_equivalent | optimization feasibility proof | ["Step 1: identify the encoding family in the variant.", "Step 2: apply the stated validation identity to recover the direct source condition.", "Step 3: rewrite it as P(x ~ X_θ : Bel(A|x) ≥ 1 - α) ≤ α."] |
End of preview. Expand in Data Studio
TREAT Dataset Release
TREAT stands for Theorem Recognition under Equivalence-preserving mAthematical Transformation. The dataset evaluates whether a model can recover the underlying named theorem from mathematically equivalent but unfamiliar formula-level representations.
This directory contains a public-facing release of the theorem-variant dataset and associated supporting artifacts. It is organized to separate the raw scraped source material, the final benchmark, theorem-level catalogs, a fixed evaluation panel, and machine-readable metadata required for verification and reuse.
Directory Structure
TREAT/
README.md
raw/
raw_theorem_pages.jsonl
benchmark/
theorem_variants.csv
catalog/
benchmark_catalog.csv
canonical_formula_pool_benchmark.csv
evaluation_panels/
shared_known_intersection_all4_960.csv
shared_known_intersection_all4_960_summary.json
reproducibility/
README.md
source_snapshot.md
model_provenance.csv
recognition_prompt.txt
familiarity_prompt.txt
paired_significance.py
metadata/
benchmark_catalog_schema.json
benchmark_schema.json
canonical_formula_pool_benchmark_schema.json
checksums.sha256
dataset_summary.json
raw_schema.json
release_manifest.json
shared_panel_schema.json
Release Components
1. Raw Source Collection
File: raw/raw_theorem_pages.jsonl
This file contains the raw scraped theorem/page collection that served as the initial intake pool for subsequent curation and benchmark construction.
- Format: JSON Lines
- Records: 2,456
- Role: upstream source archive
- Frozen: March 27, 2026
- SHA-256:
6b002898d2487c080ec7f274891eb05d3d6bad601960535a80fc803cb69c4206
Typical record fields include:
titleurlcategory_pathextractfullurlcategories
This is the exact preserved source text used by the project, not a reconstruction from current Wikipedia pages. It contains a mixture of directly useful theorem pages and pages later filtered, normalized, or excluded. Page revision IDs were not retained; see reproducibility/source_snapshot.md for the precise reproducibility boundary.
2. Final Benchmark
File: benchmark/theorem_variants.csv
This is the principal benchmark release file. It contains the final collection of transformed theorem statements together with canonical theorem metadata and validation-related fields.
- Format: CSV
- Rows: 29,900
- Unique theorem identities: 737
This file is intended for experiments in:
- theorem recognition under transformation
- canonicalization and reverse mapping
- route reconstruction
- robustness to non-canonical mathematical surface forms
High-level family counts in the current release are:
Analysis: 13,440Probability / Information Theory: 5,800Number Theory: 4,320Algebra / Linear Algebra: 4,100Combinatorics / Graph Theory: 2,240
Difficulty distribution:
easy: 4,235medium: 4,001hard: 11,680very_hard: 9,984
3. Theorem-Level Catalogs
catalog/benchmark_catalog.csv
This file provides the theorem-level inventory underlying the final benchmark.
- Format: CSV
- Rows: 737
- Granularity: one row per canonical theorem identity represented in the benchmark
It is the recommended entry point for readers who require a compact overview of the canonical theorem coverage without the full variant expansion.
Representative fields include:
canonical_namefieldsubfieldselected_familytheorem_shapeexpression_typeformula_seedshort_statement
catalog/canonical_formula_pool_benchmark.csv
This file contains a benchmark-relevant pool of canonical formulas or defining conditions for theorem-level entries.
- Format: CSV
- Rows: 1,034
It is broader than the strict 737-theorem benchmark catalog because it retains additional formula-backed entries assembled during curation. It is the most useful file for theorem-level inspection when canonical mathematical forms are required.
4. Fixed Evaluation Panel
evaluation_panels/shared_known_intersection_all4_960.csv
This file contains the fixed 960-item shared evaluation panel used for same-item comparison across multiple models.
- Format: CSV
- Rows: 960
This panel is included because it is substantially easier to inspect and share than the full benchmark while still preserving benchmark-style metadata and transformed statements. It is appropriate for:
- manual review
- small-scale replication
- same-item cross-model comparison
The associated summary file is:
evaluation_panels/shared_known_intersection_all4_960_summary.json
Reproducibility and Model Use
The reproducibility/ directory documents the frozen source snapshot, model roles and retained identifiers, construction and evaluation provenance, prompt contracts, six exact-correctness outcome vectors, and dependency-free paired-test code.
The public artifacts support data inspection, integrity verification, panel-level rescoring, and reproduction of the reported paired tests. They do not claim bit-for-bit regeneration of commercial-model outputs; immutable provider snapshots, exact access dates, decoding settings, and complete construction traces were not retained.
Benchmark File Structure
The final benchmark CSV contains several categories of columns:
Theorem Identity and Taxonomy
theorem_namecleaned_theorem_namefieldsubfieldselected_familyfamilytheorem_shapeexpression_type
Canonical Theorem Description
short_statementcanonical_formula_or_conditionrequired_assumptions
Variant-Level Content
original_formulavariant_formulalatex_math_representation
Transformation and Difficulty Metadata
transformation_pipelinedifficulty_level
Validation and Inverse-Mapping Metadata
validity_labelvalidation_typeinverse_mapping_to_original
Exact schema definitions are provided in:
metadata/benchmark_schema.json
Metadata Files
metadata/release_manifest.json
Machine-readable release manifest containing:
- file inventory
- row counts
- byte sizes
- SHA-256 hashes
- benchmark-level summary statistics
metadata/checksums.sha256
Integrity file containing SHA-256 checksums for the release payload files.
metadata/raw_schema.json
Schema summary for records in the raw JSONL source collection.
metadata/benchmark_schema.json
Column-level schema summary for benchmark/theorem_variants.csv.
metadata/benchmark_catalog_schema.json
Column-level schema summary for catalog/benchmark_catalog.csv.
metadata/canonical_formula_pool_benchmark_schema.json
Column-level schema summary for catalog/canonical_formula_pool_benchmark.csv.
metadata/shared_panel_schema.json
Column-level schema summary for evaluation_panels/shared_known_intersection_all4_960.csv.
metadata/dataset_summary.json
Compact machine-readable summary of the release.
Encoding
All files are distributed in UTF-8-compatible formats:
raw/raw_theorem_pages.jsonl: UTF-8 JSON Linesbenchmark/theorem_variants.csv: UTF-8 CSV- supporting CSV files: UTF-8 CSV
- metadata files: UTF-8 JSON or text
If a downstream system has limited Unicode support, explicit UTF-8 import settings are recommended.
Recommended Usage
The files in this release serve different purposes:
- use
benchmark/theorem_variants.csvfor the full transformed-theorem benchmark - use
catalog/benchmark_catalog.csvfor theorem-level benchmark coverage - use
catalog/canonical_formula_pool_benchmark.csvwhen canonical formulas or conditions are needed - use
evaluation_panels/shared_known_intersection_all4_960.csvfor smaller fixed-panel evaluation or manual inspection - use
raw/raw_theorem_pages.jsonlonly when access to the original scrape-level source material is required
License and Attribution
This release is distributed under CC BY-SA 4.0 because the source collection is derived from public Wikipedia theorem pages. Downstream users should preserve source attribution, including page titles and URLs where present in the released files.
For reuse, cite this dataset as described in CITATION.cff.
Known Limitations
TREAT is a benchmark of representation-robust theorem recognition, not a fully formal proof corpus. The transformed rows include validation metadata and inverse-mapping notes, but not every row has a machine-checkable proof certificate. Users who require formally certified theorem transformations should filter by the validation columns and/or perform additional domain-specific review.
Dataset card
| Paper | TREAT: Evaluating Access to Formal Knowledge across Equivalent Mathematical Representations |
| Venue | 28th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2026) |
| Authors | Fateme Mazdarani, Carlos Toxtli |
| License | Creative Commons Attribution-ShareAlike 4.0 International (cc-by-sa-4.0) |
| Dataset DOI | 10.5281/zenodo.24179071 |
| Version | 1.0.0 |
| Code repository | https://github.com/mazdarani/TREAT |
| Maintainer | Human-AI Empowerment Lab, Clemson University |
Dataset structure
| Path | Format | Size | Records |
|---|---|---|---|
benchmark/theorem_variants.csv |
csv | 32.5 MB | 29,900 |
catalog/benchmark_catalog.csv |
csv | 246.6 KB | 737 |
catalog/canonical_formula_pool_benchmark.csv |
csv | 395.2 KB | 1,034 |
evaluation_panels/shared_known_intersection_all4_960.csv |
csv | 979.0 KB | 960 |
evaluation_panels/shared_known_intersection_all4_960_summary.json |
json | 4.0 KB | |
metadata/benchmark_catalog_schema.json |
json | 1.1 KB | |
metadata/benchmark_schema.json |
json | 2.8 KB | |
metadata/canonical_formula_pool_benchmark_schema.json |
json | 1.0 KB | |
metadata/checksums.sha256 |
sha256 | 1.2 KB | |
metadata/dataset_summary.json |
json | 1.8 KB | |
metadata/raw_schema.json |
json | 633 B | |
metadata/release_manifest.json |
json | 3.8 KB | |
metadata/shared_panel_schema.json |
json | 2.7 KB | |
raw/raw_theorem_pages.jsonl |
jsonl | 41.6 MB | 2,456 |
reproducibility/README.md |
md | 5.7 KB | |
reproducibility/familiarity_prompt.txt |
txt | 338 B | |
reproducibility/model_provenance.csv |
csv | 2.3 KB | 10 |
reproducibility/paired_significance.py |
py | 7.5 KB | |
reproducibility/recognition_prompt.txt |
txt | 817 B | |
reproducibility/source_snapshot.md |
md | 1.1 KB |
20 files, 75.7 MB in total (metadata files excluded). Records are rows for CSV/TSV/JSONL files and list length for JSON arrays.
Loading the data
from datasets import load_dataset
ds = load_dataset("ai4he/treat-theorem-recognition", "theorem_variants")
Available subsets: theorem_variants, shared_known_intersection_all4_960, raw_theorem_pages, canonical_formula_pool_benchmark, benchmark_catalog, model_provenance. All other files can be downloaded with huggingface_hub.snapshot_download or from the Zenodo record.
Known limitations
- Wikipedia revision IDs were not recorded; a new crawl will not reproduce the archived source exactly.
Personal and sensitive information
None.
License
The data and documentation are released under the Creative Commons Attribution-ShareAlike 4.0 International. Derived from English Wikipedia theorem pages (CC BY-SA 4.0).
Citation
If you use this dataset, please cite the paper:
@inproceedings{Mazdarani2026TREAT,
title = {TREAT: Evaluating Access to Formal Knowledge across Equivalent Mathematical Representations},
author = {Mazdarani, Fateme and Toxtli, Carlos},
booktitle = {Proceedings of the 28th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2026)},
publisher = {IEEE},
year = {2026},
note = {Forthcoming},
url = {https://synasc.ro/2026/}
}
and the dataset:
@dataset{Mazdarani2026TreatTheoremRecognitionData,
author = {Mazdarani, Fateme and Toxtli, Carlos},
title = {{TREAT Theorem Recognition}: Dataset for ``TREAT: Evaluating Access to Formal Knowledge across Equivalent Mathematical Representations''},
year = {2026},
version = {1.0.0},
publisher = {Zenodo},
doi = {10.5281/zenodo.24179071},
url = {https://doi.org/10.5281/zenodo.24179071}
}
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