8655f7706f
This PR changes how Lean proves the equational theorems for structural recursion. The core idea is to let-bind the `f` argument to `brecOn` and rewriting `.brecOn` with an unfolding theorem. This means no extra case split for the `.rec` in `.brecOn` is needed, and `simp` doesn't change the `f` argument which can break the definitional equality with the defined function. With this, we can prove the unfolding theorem first, and derive the equational theorems from that, like for all other ways of defining recursive functions. Backs out the changes from #10415, the old strategy works well with the new goals. Fixes #5667 Fixes #10431 Fixes #10195 Fixes #2962
44 lines
1,020 B
Text
44 lines
1,020 B
Text
inductive N where | zero | succ (n : N)
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/--
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info: protected theorem N.brecOn.eq.{u} : ∀ {motive : N → Sort u} (t : N) (F_1 : (t : N) → N.below t → motive t),
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N.brecOn t F_1 = F_1 t (N.brecOn.go t F_1).2
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-/
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#guard_msgs in #print sig N.brecOn.eq
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-- set_option trace.Elab.definition.structural.eqns true
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def g (i : Nat) (j : N) : N :=
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if i < 5 then .zero else
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match j with
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| .zero => N.zero.succ
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| .succ j => g i j
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termination_by structural j
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/--
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info: theorem g.eq_def : ∀ (i : Nat) (j : N),
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g i j =
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if i < 5 then N.zero
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else
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match j with
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| N.zero => N.zero.succ
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| j.succ => g i j
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-/
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#guard_msgs(pass trace, all) in #print sig g.eq_def
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def N.beq : (m n : N) → Bool
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| .zero, .zero => true
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| .succ m, .succ n => N.beq m n
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| _, _ => false
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/--
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info: theorem N.beq.eq_def : ∀ (x x_1 : N),
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x.beq x_1 =
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match x, x_1 with
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| N.zero, N.zero => true
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| m.succ, n.succ => m.beq n
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| x, x_2 => false
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-/
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#guard_msgs(pass trace, all) in #print sig N.beq.eq_def
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