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44 lines
1.3 KiB
Text
44 lines
1.3 KiB
Text
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/- Chapter 3 — Propositions as Types: exercises.
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Rules of the game: TERM MODE ONLY — no `by`, no tactics.
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Your toolkit: fun ... => ..., function application, And.intro,
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h.1 / h.2, Or.inl / Or.inr, match ... with. -/
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namespace Ch03
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variable (P Q R : Prop)
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-- Warm-ups from the Try It box ---------------------------------------------
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/- W1: "if P then (Q implies P)". A constant function. -/
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theorem const_imp : P → Q → P :=
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sorry
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/- W2: a conjunction gives a disjunction (pick a side). -/
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theorem and_to_or : P ∧ Q → P ∨ Q :=
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sorry
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/- W3: modus ponens — "calling a function". -/
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theorem modus_ponens : P → (P → Q) → Q :=
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sorry
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-- Numbered exercises --------------------------------------------------------
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/- 3.1: reassociate evidence with h.1, h.2, And.intro. -/
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theorem and_assoc' : (P ∧ Q) ∧ R → P ∧ (Q ∧ R) :=
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sorry
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/- 3.2: case analysis on which side holds.
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Shape: fun h => match h with
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| Or.inl p => ...
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| Or.inr q => ... -/
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theorem or_swap : P ∨ Q → Q ∨ P :=
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sorry
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/- 3.3: Not P is DEFINED as P → False. Unfold it in your head, and this
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becomes a two-argument function. Afterwards, write in a comment the
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one-sentence description of the program you wrote. -/
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theorem not_not_intro : P → ¬¬P :=
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sorry
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end Ch03
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