100 lines
2.4 KiB
Text
100 lines
2.4 KiB
Text
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definition double x := x + x
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(*
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function show(x) print(x) end
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local t1 = parse_lean("0 ≠ 1")
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show(simplify(t1))
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local t2 = parse_lean("3 ≥ 2")
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show(simplify(t2))
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local t3 = parse_lean("double (double 2) + 1")
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show(simplify(t3))
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local t4 = parse_lean("0 = 1")
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show(simplify(t4))
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*)
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(*
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local opt = options({"simplifier", "unfold"}, true, {"simplifier", "eval"}, false)
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local t1 = parse_lean("double (double 2) + 1 ≥ 3")
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show(simplify(t1, 'default', get_environment(), context(), opt))
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*)
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set_opaque Nat::ge false
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add_rewrite Nat::add_assoc
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add_rewrite Nat::distributer
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add_rewrite Nat::distributel
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(*
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local opt = options({"simplifier", "unfold"}, true, {"simplifier", "eval"}, false)
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local t1 = parse_lean("2 * double (double 2) + 1 ≥ 3")
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show(simplify(t1, 'default', get_environment(), context(), opt))
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*)
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variables a b c d : Nat
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import if_then_else
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add_rewrite if_true
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add_rewrite if_false
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add_rewrite if_a_a
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add_rewrite Nat::add_zeror
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add_rewrite Nat::add_zerol
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add_rewrite eq_id
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(*
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local t1 = parse_lean("(a + b) * (c + d)")
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local r, pr = simplify(t1)
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print(r)
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print(pr)
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*)
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theorem congr2_congr1 {A B C : TypeU} {f g : A → B} (h : B → C) (Hfg : f = g) (a : A) :
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congr2 h (congr1 a Hfg) = congr2 (λ x, h (x a)) Hfg
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:= proof_irrel (congr2 h (congr1 a Hfg)) (congr2 (λ x, h (x a)) Hfg)
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theorem congr2_congr2 {A B C : TypeU} {a b : A} (f : A → B) (h : B → C) (Hab : a = b) :
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congr2 h (congr2 f Hab) = congr2 (λ x, h (f x)) Hab
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:= proof_irrel (congr2 h (congr2 f Hab)) (congr2 (λ x, h (f x)) Hab)
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theorem congr1_congr2 {A B C : TypeU} {a b : A} (f : A → B → C) (Hab : a = b) (c : B):
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congr1 c (congr2 f Hab) = congr2 (λ x, f x c) Hab
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:= proof_irrel (congr1 c (congr2 f Hab)) (congr2 (λ x, f x c) Hab)
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rewrite_set proofsimp
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add_rewrite congr2_congr1 congr2_congr2 congr1_congr2 : proofsimp
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(*
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local t2 = parse_lean("(if a > 0 then b else b + 0) + 10 = (if a > 0 then b else b) + 10")
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local r, pr = simplify(t2)
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print(r)
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print(pr)
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show(simplify(pr, 'proofsimp'))
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*)
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(*
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local t1 = parse_lean("(a + b) * (a + b)")
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local t2 = simplify(t1)
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print(t2)
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*)
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-- add_rewrite imp_truer imp_truel imp_falsel imp_falser not_true not_false
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-- print rewrite_set
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(*
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local t1 = parse_lean("true → false")
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print(simplify(t1))
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*)
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(*
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local t1 = parse_lean("true → true")
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print(simplify(t1))
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*)
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(*
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local t1 = parse_lean("false → false")
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print(simplify(t1))
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*)
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(*
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local t1 = parse_lean("true ↔ false")
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print(simplify(t1))
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*)
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