SLIDE 1 Forward vs. backward proof
Consider the following Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Rule can be read in two ways:
suffjces to prove ”
holds then holds” With apply-style proofs we are reasoning backwards Decomposing complex goals into simpler goals Using a backward reading of rules and theorems
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SLIDE 2 Forward vs. backward proof
Consider the following Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Rule can be read in two ways:
- Backwards: “to prove Γ ⊢ φ → ψ suffjces to prove Γ, φ ⊢ ψ”
- Forwards: “if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds”
With apply-style proofs we are reasoning backwards Decomposing complex goals into simpler goals Using a backward reading of rules and theorems
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SLIDE 3 Forward vs. backward proof
Consider the following Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Rule can be read in two ways:
- Backwards: “to prove Γ ⊢ φ → ψ suffjces to prove Γ, φ ⊢ ψ”
- Forwards: “if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds”
With apply-style proofs we are reasoning backwards Decomposing complex goals into simpler goals Using a backward reading of rules and theorems
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SLIDE 4
Meta vs. object-level
Consider the same Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Recall the forward reading: “if holds then holds” Note there are two different kinds of implication at play here! The implication in the logic we are reasoning about: The implication in our meta-language, informal English: “if-then” Isabelle uses a logic rather than informal English for this purpose The “fat arrow” replaces the English “if-then” in Isabelle
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SLIDE 5
Meta vs. object-level
Consider the same Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Recall the forward reading: “if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds” Note there are two different kinds of implication at play here! The implication in the logic we are reasoning about: The implication in our meta-language, informal English: “if-then” Isabelle uses a logic rather than informal English for this purpose The “fat arrow” replaces the English “if-then” in Isabelle
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SLIDE 6
Meta vs. object-level
Consider the same Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Recall the forward reading: “if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds” Note there are two different kinds of implication at play here! The implication in the logic we are reasoning about: φ → ψ The implication in our meta-language, informal English: “if-then” Isabelle uses a logic rather than informal English for this purpose The “fat arrow” replaces the English “if-then” in Isabelle
2
SLIDE 7
Meta vs. object-level
Consider the same Natural Deduction rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ Recall the forward reading: “if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds” Note there are two different kinds of implication at play here! The implication in the logic we are reasoning about: φ → ψ The implication in our meta-language, informal English: “if-then” Isabelle uses a logic rather than informal English for this purpose The “fat arrow” = ⇒ replaces the English “if-then” in Isabelle
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SLIDE 8
Moreover...
In the rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ ...there’s also hidden universal quantifjcation “For every , , and : if holds then holds” The meta-level universal quantifjer ( ) replaces the English “for-every” in Isabelle Thus the Natural Deduction rule above would be rendered as when embedded in HOL
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SLIDE 9
Moreover...
In the rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ ...there’s also hidden universal quantifjcation “For every Γ, φ, and ψ: if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds” The meta-level universal quantifjer ( ) replaces the English “for-every” in Isabelle Thus the Natural Deduction rule above would be rendered as when embedded in HOL
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SLIDE 10 Moreover...
In the rule: Γ, φ ⊢ ψ (ImpI) Γ ⊢ φ → ψ ...there’s also hidden universal quantifjcation “For every Γ, φ, and ψ: if Γ, φ ⊢ ψ holds then Γ ⊢ φ → ψ holds” The meta-level universal quantifjer () replaces the English “for-every” in Isabelle Thus the Natural Deduction rule above would be rendered as
Γ, φ ⊢ ψ = ⇒ Γ ⊢ φ → ψ when embedded in HOL
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