def test_inequality_sequence(self):
natural = r"$\left|a_n - b_n \right| \leq n$"
lean = r"|a n - b n| ≤ n"
context = Context()
context.add("a", "sequence")
context.add("b", "sequence")
test_bijective(self, Inequality, natural, lean, context)
def test_basic(self):
natural = r"$a_n \leq b_n$ by H1 with n"
lean = "have A1 : a n ≤ b n := H1 n"
context = Context()
context.add("a", "sequence")
context.add("b", "sequence")
test_bijective(self, BySentenceWith, natural, lean, context)
def test_sequence(self):
nat = r"$a_n \leq b_n$"
lean = "a n ≤ b n"
context = Context()
context.add("a", "sequence")
context.add("b", "sequence")
test_bijective(self, Inequality, nat, lean, context)
def test_sequence(self):
natural = r"$\left|a_n - b_n \right|$"
lean = "|a n - b n|"
context = Context()
context.add("a", "sequence")
context.add("b", "sequence")
test_bijective(self, AbsoluteDiff, natural, lean, context)
def test_basic(self):
natural = "Let $n$"
lean = "intros n A1"
context = Context()
context.current_goal = ForAllNatIneqThen(
"n", Inequality.from_natural(r"$n \leq N$"),
r"|b_n - l| < \epsilon$")
test_bijective(self, LetNInequality, natural, lean, context)
def test_basic(self):
context = Context()
RealValuedSequences.from_natural("$a_n, b_n, c_n$ are real-valued",
context)
RealDeclaration.from_natural(r"$l \in \mathbb{R}$", context)
SequenceLimit.from_natural(r"$a_n \rightarrow l$", context)
context.current_goal = SequenceLimit.from_natural(
r"$b_n \rightarrow l$", context)
LetGoalLimit.from_natural(r"Let $\epsilon$", context)
natural = r"Let's choose $N_a$ such that H1 uses $\epsilon$"
lean = r"cases H1 ε A1 with N_a A2"
test_bijective(self, ChooseNEpsilonLimit, natural, lean, context)
def test_basic(self):
nat_gt = "$a > b$"
lean_gt = "a > b"
test_bijective(self, Inequality, nat_gt, lean_gt)
nat_ge = r"$a \geq b$"
lean_ge = "a ≥ b"
test_bijective(self, Inequality, nat_ge, lean_ge)
nat_lt = "$a < b$"
lean_lt = "a < b"
test_bijective(self, Inequality, nat_lt, lean_lt)
nat_le = r"$a \leq b$"
lean_le = "a ≤ b"
test_bijective(self, Inequality, nat_le, lean_le)
def test_inequality(self):
natural = r"$\forall x : x < a$"
lean = "∀ x, x < a"
test_bijective(self, ForAll, natural, lean)
def test_basic(self):
natural = "Let's separate H1"
lean = "cases H1"
test_bijective(self, Cases, natural, lean)
def test_basic(self):
natural = r"$a_n \rightarrow l$"
lean = "is_limit a l"
test_bijective(self, SequenceLimit, natural, lean)
def test_basic(self):
natural = r"$l \in \mathbb{R}$"
lean = "l : ℝ"
test_bijective(self, RealDeclaration, natural, lean)
def test_basic(self):
natural = "$a_n, b_n, c_n$ are real-valued sequences"
lean = "a b c : ℕ → ℝ"
test_bijective(self, RealValuedSequences, natural, lean)
def test_basic(self):
natural = "Let's split the goal and do on all subgoals"
lean = "split;"
test_bijective(self, DoAllSubgoals, natural, lean)
def test_inequality(self):
natural = r"$\left|a - b \right| \leq n$"
lean = "|a - b| ≤ n"
test_bijective(self, Inequality, natural, lean)
def test_basic(self):
natural = "We claim $N$ works"
lean = "use N"
test_bijective(self, Use, natural, lean)
def test_basic(self):
natural = "Let $N = \\max\\left(N_a, N_c\\right)$"
lean = "let N := max N_a N_c"
test_bijective(self, LetMax, natural, lean)
def test_basic(self):
context = Context()
natural = r"Let $\epsilon$"
lean = "intros ε A1"
context.current_goal = SequenceLimit("b", "l")
test_bijective(self, LetGoalLimit, natural, lean, context)
def test_basic(self):
natural = "By linear arithmetic"
lean = "linarith"
test_bijective(self, LinearArithmetic, natural, lean)
def test_basic(self):
natural = r"$\left|a - b \right|$"
lean = "|a - b|"
test_bijective(self, AbsoluteDiff, natural, lean)
def test_basic(self):
natural = "Let's use absolute value inequality property on H1 H2 and on goal"
lean = "rw abs_sub_lt_iff at H1 H2 ⊢"
test_bijective(self, AbsoluteValueIneqProperty, natural, lean)
def test_absdiff(self):
natural = r"$\left|a_n - l \right| < \epsilon$ by H1 with H2"
lean = "have A1 : |a n - l| < ε := H1 H2"
context = Context()
context.add("a", "sequence")
test_bijective(self, BySentenceWith, natural, lean, context)
def test_basic(self):
natural = "Let's split the goal"
lean = "split"
test_bijective(self, SplitGoal, natural, lean)
def test_basic(self):
natural = r"By inequality properties, $N_a \leq N$"
lean = r"have A1 : N_a ≤ N := by obvious_ineq"
test_bijective(self, ByInequalityProperties, natural, lean)
def test_basic(self):
natural = "Let's choose n in H1"
lean = "have A1 := H1 n"
test_bijective(self, LetsChooseIn, natural, lean)
