## Code Snippet Analysis: Theorem Prover Syntax Comparison
### Overview
The image contains code snippets from multiple formal theorem provers (PFR, SciLean, Coqeter, MiniF2F, Formal Book) demonstrating mathematical proofs. Each section includes syntax-highlighted code with specific terms emphasized in green.
### Components/Axes
- **Structure**: Five distinct code blocks labeled by prover name
- **Syntax Highlighting**:
- Blue: Keywords (theorem, lemma, simp, apply, etc.)
- Brown: Theorem/lemma names
- Green: Highlighted terms (key components)
- Black: Mathematical symbols and variables
### Detailed Analysis
#### PFR Section
```text
lemma condRho_of_translate {Ω S : Type*} [MeasureSpace Ω] (X : Ω → G) (Y : Ω → S) (A : Finset G) (s:G) :
condRho (fun ω ↦ X ω + s) Y A = condRho X Y A := by
simp only [condRho, rho_of_translate]
```
- **Highlighted Terms**: `condRho`, `rho_of_translate`
#### SciLean Section
```text
theorem re_float (a : Float) : RCLike.re a = a := by
exact RCLike.re_eq_self_of_le le_rfl
```
- **Highlighted Terms**: `RCLike.re_eq_self_of_le`, `le_rfl`
#### Coqeter Section
```text
lemma invmaps_of_eq {S:Set G} [CoxeterSystem G S] {s :S} :
invmaps S s = s := by
simp [CoxeterSystem.Presentation.invmaps]
unfold CoxeterSystem.toMatrix
apply CoxeterSystem.monoidLift.mapLift.of
```
- **Highlighted Terms**: `CoxeterSystem.Presentation.invmaps`, `CoxeterSystem.toMatrix`, `CoxeterSystem.monoidLift.mapLift.of`
#### MiniF2F Section
```text
theorem induction_12dvd4expnp1p20 (n : N) :
12 | 4^(n+1) + 20 := by
norm_num
induction' n with n hn
simp
omega
theorem amc12a_2002_p6 (n : N) (h₀ : 0 < n) :
∃ m, (m > n ∧ ∃ p, m * p ≤ m + p) := by
lift n to N+ using h₀
cases' n with n
exact (_, lt_add_of_pos_right _ _ _ zero_lt_one 1, by simp)
```
- **Highlighted Terms**: `lt_add_of_pos_right`, `zero_lt_one`
#### Formal Book Section
```text
theorem wedderburn (h: Fintype R) : IsField R := by
apply Field.toIsField
```
- **Highlighted Terms**: `Field.toIsField`
### Key Observations
1. **Syntax Consistency**: All provers use similar proof structures (`by`, `simp`, `apply`)
2. **Highlighted Patterns**:
- Mathematical operations (`condRho`, `re_float`)
- Type constraints (`Finset G`, `Fintype R`)
- Core proof tactics (`simp`, `apply`, `exact`)
3. **Mathematical Focus**: All proofs involve algebraic structures (groups, fields, Coxeter systems)
### Interpretation
The image demonstrates comparative analysis of formal proof structures across different theorem provers. The highlighted terms represent:
- Core mathematical concepts (`condRho`, `re_float`)
- Type system constraints (`Finset`, `Fintype`)
- Fundamental proof strategies (`simp`, `apply`)
- Algebraic structure properties (`CoxeterSystem`, `Field`)
The consistent use of `simp` and `apply` across provers suggests shared foundational proof techniques, while the specific highlighted terms reveal domain-specific implementations. The MiniF2F section's number theory focus contrasts with the algebraic emphasis in other sections, showing the versatility of these provers in handling different mathematical domains.
