Logiweb sequent calculus, Chores

Klaus Grue GRD-2006-06-21.UTC:10:43:40.552317

Contents

1 Test cases 1 2 Pyk definitions 3 3 TEX definitions 7

3.1 Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Priority table 17

1 Test cases

[⌈\overline{x}⌉#^{0}⌈\overline{y} = \overline{z} ⇒ ∀_{obj}\overline{x}: \overline{x} = \overline{y}⌉ ’ check]^{·} [⌈\overline{x}⌉#^{0}⌈\overline{x} = \overline{z} ⇒ ∀_{obj}\overline{x}: \overline{x} = \overline{y}⌉ ’ check]^{−} [⌈\overline{x}⌉#^{0}⌈\overline{y} = \overline{x} ⇒ ∀_{obj}\overline{x}: \overline{x} = \overline{y}⌉ ’ check]^{−} [⌈\overline{x}⌉#^{0}⌈\overline{y} = \overline{z} ⇒ ∀_{obj}\overline{y}: \overline{x} = \overline{y}⌉ ’ check]^{−} [⟨⌈\overline{a}⌉≡^{0}⌈\overline{a}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{·} [⟨⌈\overline{b}⌉≡^{0}⌈\overline{a}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{−} [⟨⌈\overline{c}⌉≡^{0}⌈\overline{a}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{−} [⟨⌈\overline{a}⌉≡^{0}⌈\overline{b}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{−} [⟨⌈\overline{b}⌉≡^{0}⌈\overline{b}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{−} [⟨⌈\overline{c}⌉≡^{0}⌈\overline{b}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{·} [⟨⌈∀_{obj}\overline{a}: \overline{a} = \overline{b}⌉≡^{0}⌈∀_{obj}\overline{a}: \overline{a} = \overline{b}⌉|⌈\overline{a}⌉:=⌈\overline{c}⌉⟩ ’ check]^{·} [⟨⌈∀_{obj}\overline{a}: \overline{a} = \overline{c}⌉≡^{0}⌈∀_{obj}\overline{a}: \overline{a} = \overline{b}⌉|⌈\overline{b}⌉:=⌈\overline{c}⌉⟩ ’ check]^{·} [⟨⌈∀_{obj}\overline{a}: \overline{a} = 0 + \overline{a} ⇒ \overline{c} · \overline{d} = 0 + \overline{c} · \overline{d}⌉≡^{0}⌈∀_{obj}\overline{a}: \overline{a} = 0 + \overline{a} ⇒ \overline{b} = 0 + \overline{b}⌉|⌈\overline{b}⌉:=⌈\overline{c} ·

\overline{d}⌉⟩ ’ check]^{·} [⟨⌈∀_{obj}\overline{a}: \overline{a} = 0+\overline{a} ⇒ \overline{b} = 0+\overline{b}⌉≡^{0}⌈∀_{obj}\overline{a}: \overline{a} = 0+\overline{a} ⇒ \overline{b} = 0+\overline{b}⌉|⌈\overline{a}⌉:=⌈\overline{c}⌉⟩’check]^{·}

[λx.Ded_{0}(⌈0⌉, ⌈0⌉) ’ check]^{·}

[λx.Ded_{0}(⌈0⌉, ⌈1⌉) ’ check]^{−}

[Ded_{8}(⌈∀\underline{a}: \underline{a}⌉, T)]^{·}

[Ded_{7}(⌈∀\underline{a}: \underline{a}⌉) =^{t} ⌈\underline{a}⌉]^{·}

[λx.Ded_{0}(⌈∀\underline{a}: \underline{a}⌉, ⌈\underline{a}⌉) ’ check]^{·}

[λx.Ded_{0}(⌈\underline{a}⌉, ⌈\underline{b}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈∀\underline{a}: \underline{a}⌉, ⌈\underline{b}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b}⌉, ⌈\underline{a} ⇒ \underline{b}⌉) ’ check]^{·}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b}⌉, ⌈\underline{a} ⇒ \underline{a}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b}⌉, ⌈\underline{b} ⇒ \underline{b}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b}⌉, ⌈0⌉) ’ check]^{−}

[λx.Ded_{0}(⌈0⌉, ⌈\underline{a} ⇒ \underline{a}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⊢ \underline{b} ⊢ \underline{c}⌉, ⌈\underline{a} ⇒ \underline{b} ⇒ \underline{c}⌉) ’ check]^{·}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: \underline{a} ⊢ \underline{b} ⊢ \underline{a}⌉, ⌈\underline{a} ⇒ \underline{b} ⇒ \underline{c}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈∀\underline{a}: ∀\underline{b}: ∀\underline{c}: \underline{a} ⊢ \underline{b} ⊢ \underline{c}⌉, ⌈\underline{a} ⇒ \underline{b} ⇒ \underline{c}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈0⌉, ⌈\overline{x}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈\overline{x}⌉, ⌈0⌉) ’ check]^{·}

[λx.Ded_{0}(⌈\overline{x}⌉, ⌈\overline{x}⌉) ’ check]^{·}

[λx.Ded_{0}(⌈∀_{obj}\overline{x}: \overline{x}⌉, ⌈\overline{x}⌉) ’ check]^{−}

[λx.Ded_{0}(⌈\overline{x}⌉, ⌈∀_{obj}\overline{y}: \overline{z}⌉) ’ check]^{·}

[λx.Ded_{0}(⌈∀_{obj}\overline{x}: \overline{x}⌉, ⌈∀_{obj}\overline{x}: \overline{x}⌉) ’ check]^{·}

[λx.Ded_{0}(⌈0 ⊢ 0⌉, ⌈0 ⇒ 0⌉) ’ check]^{·}

[λx.Ded_{0}(⌈\overline{x} ⊢ 0⌉, ⌈0 ⇒ 0⌉) ’ check]^{−}

[λx.Ded_{0}(⌈0 ⊢ \overline{x}⌉, ⌈0 ⇒ 0⌉) ’ check]^{·}

[λx.Ded_{0}(⌈\overline{x} ⊢ \overline{x}⌉, ⌈0 ⇒ 0⌉) ’ check]^{−}

[λx.Ded_{0}(⌈0 ⊢ 0⌉, ⌈∀_{obj}\overline{x}: 0 ⇒ 0⌉) ’ check]^{·}

[λx.Ded_{0}(⌈\overline{x} ⊢ 0⌉, ⌈∀_{obj}\overline{x}: \overline{x} ⇒ 0⌉) ’ check]^{·} [λx.Ded_{0}(⌈0 ⊢ \overline{x}⌉, ⌈∀_{obj}\overline{x}: 0 ⇒ \overline{x}⌉) ’ check]^{·} [λx.Ded_{0}(⌈\overline{x} ⊢ \overline{x}⌉, ⌈∀_{obj}\overline{x}: \overline{x} ⇒ \overline{x}⌉) ’ check]^{·} [λx.Ded_{0}(⌈0 ⊢ 0⌉, ⌈0 ⇒ ∀_{obj}\overline{x}: 0⌉) ’ check]^{−} [λx.Ded_{0}(⌈\overline{x} ⊢ 0⌉, ⌈0 ⇒ ∀_{obj}\overline{x}: 0⌉) ’ check]^{−} [λx.Ded_{0}(⌈0 ⊢ \overline{x}⌉, ⌈0 ⇒ ∀_{obj}\overline{y}: \overline{z}⌉) ’ check]^{·} [λx.Ded_{0}(⌈\overline{x} ⊢ \overline{x}⌉, ⌈0 ⇒ ∀_{obj}\overline{x}: \overline{x}⌉) ’ check]^{−} [λx.Ded_{0}(⌈0 ⊢ 0⌉, ⌈∀_{obj}\overline{x}: 0 ⇒ ∀_{obj}\overline{x}: 0⌉) ’ check]^{−} [λx.Ded_{0}(⌈\overline{x} ⊢ 0⌉, ⌈∀_{obj}\overline{x}: \overline{x} ⇒ ∀_{obj}\overline{x}: 0⌉) ’ check]^{−} [λx.Ded_{0}(⌈0 ⊢ \overline{x}⌉, ⌈∀_{obj}\overline{x}: 0 ⇒ 2⌉) ’ check]^{·} [λx.Ded_{0}(⌈\overline{x} ⊢ \overline{x}⌉, ⌈∀_{obj}\overline{x}: \overline{x} ⇒ 3⌉) ’ check]^{·} [λx.Ded_{0}(⌈\overline{x} + \overline{y} = \overline{y} + \overline{x}⌉, ⌈2 + 3 = 3 + 2⌉) ’ check]^{·} [λx.Ded_{0}(⌈\overline{x} + \overline{y} = \overline{y} + \overline{x}⌉, ⌈2 + 3 = 2 + 3⌉) ’ check]^{−} [λx.Ded_{0}(⌈\overline{x} + \overline{y} = \overline{y} + \overline{x}⌉, ⌈2 + 3 = 2 + 2⌉) ’ check]^{−} [λx.Ded_{0}(⌈\overline{x} + \overline{y} = \overline{y} + \overline{x}⌉, ⌈2 + 3 = 3 + 3⌉) ’ check]^{−}

2 Pyk definitions

[[∗ =^{∘} ∗] →^{pyk} “general macro define " as " end define”] [RootVisible(∗) →^{pyk} “make root visible " end visible”] [A →^{pyk} “sequent example axiom”] [R →^{pyk} “sequent example rule”] [C →^{pyk} “sequent example contradiction”] [T →^{pyk} “sequent example theory”] [L →^{pyk} “sequent example lemma”] [{∗} →^{pyk} “set " end set”] [\overline{∗} →^{pyk} “object var " end var”] [a →^{pyk} “object a”] [b →^{pyk} “object b”] [c →^{pyk} “object c”] [d →^{pyk} “object d”] [e →^{pyk} “object e”] [f →^{pyk} “object f”] [g →^{pyk} “object g”] [h →^{pyk} “object h”] [i →^{pyk} “object i”] [j →^{pyk} “object j”] [k →^{pyk} “object k”] [l →^{pyk} “object l”] [m →^{pyk} “object m”] [n →^{pyk} “object n”] [o →^{pyk} “object o”] [p →^{pyk} “object p”] [q →^{pyk} “object q”] [r →^{pyk} “object r”] [s →^{pyk} “object s”] [t →^{pyk} “object t”] [u →^{pyk} “object u”] [v →^{pyk} “object v”] [w →^{pyk} “object w”] [x →^{pyk} “object x”] [y →^{pyk} “object y”] [z →^{pyk} “object z”] [⟨∗≡ ∗ | ∗:=∗⟩ →^{pyk} “sub " is " where " is " end sub”] [⟨∗≡^{0} ∗ | ∗:=∗⟩ →^{pyk} “sub zero " is " where " is " end sub”] [⟨∗≡^{1} ∗ | ∗:=∗⟩ →^{pyk} “sub one " is " where " is " end sub”] [⟨∗≡^{∗} ∗ | ∗:=∗⟩ →^{pyk} “sub star " is " where " is " end sub”] [Ded(∗, ∗) →^{pyk} “deduction " conclude " end deduction”] [Ded_{0}(∗, ∗) →^{pyk} “deduction zero " conclude " end deduction”] [Ded_{1}(∗, ∗, ∗) →^{pyk} “deduction one " conclude " condition " end deduction”] [Ded_{2}(∗, ∗, ∗) →^{pyk} “deduction two " conclude " condition " end deduction”] [Ded_{3}(∗, ∗, ∗, ∗) →^{pyk} “deduction three " conclude " condition " bound " end deduction”] [Ded_{4}(∗, ∗, ∗, ∗) →^{pyk} “deduction four " conclude " condition " bound " end deduction”] [Ded_{4}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “deduction four star " conclude " condition " bound " end deduction”] [Ded_{5}(∗, ∗, ∗) →^{pyk} “deduction five " condition " bound " end deduction”] [Ded_{6}(∗, ∗, ∗, ∗) →^{pyk} “deduction six " conclude " exception " bound " end deduction”] [Ded_{6}^{∗}(∗, ∗, ∗, ∗) →^{pyk} “deduction six star " conclude " exception " bound " end deduction”] [Ded_{7}(∗) →^{pyk} “deduction seven " end deduction”] [Ded_{8}(∗, ∗) →^{pyk} “deduction eight " bound " end deduction”] [Ded_{8}^{∗}(∗, ∗) →^{pyk} “deduction eight star " bound " end deduction”] [S →^{pyk} “system s”] [Neg →^{pyk} “double negation”] [MP →^{pyk} “rule mp”] [Gen →^{pyk} “rule gen”] [Ded →^{pyk} “deduction”] [S1 →^{pyk} “axiom s one”] [S2 →^{pyk} “axiom s two”] [S3 →^{pyk} “axiom s three”] [S4 →^{pyk} “axiom s four”] [S5 →^{pyk} “axiom s five”] [S6 →^{pyk} “axiom s six”] [S7 →^{pyk} “axiom s seven”] [S8 →^{pyk} “axiom s eight”] [S9 →^{pyk} “axiom s nine”] [Repetition →^{pyk} “repetition”] [A1′ →^{pyk} “lemma a one”] [A2′ →^{pyk} “lemma a two”] [A4′ →^{pyk} “lemma a four”] [A5′ →^{pyk} “lemma a five”] [Prop 3.2a →^{pyk} “prop three two a”] [Prop 3.2b →^{pyk} “prop three two b”] [Prop 3.2c →^{pyk} “prop three two c”] [Prop 3.2d →^{pyk} “prop three two d”] [Prop 3.2e_{1} →^{pyk} “prop three two e one”] [Prop 3.2e_{2} →^{pyk} “prop three two e two”] [Prop 3.2e →^{pyk} “prop three two e”] [Prop 3.2f_{1} →^{pyk} “prop three two f one”] [Prop 3.2f_{2} →^{pyk} “prop three two f two”] [Prop 3.2f →^{pyk} “prop three two f”] [Prop 3.2g_{1} →^{pyk} “prop three two g one”] [Prop 3.2g_{2} →^{pyk} “prop three two g two”] [Prop 3.2g →^{pyk} “prop three two g”] [Prop 3.2h_{1} →^{pyk} “prop three two h one”] [Prop 3.2h_{2} →^{pyk} “prop three two h two”] [Prop 3.2h →^{pyk} “prop three two h”] [Prop 3.2i →^{pyk} “prop three two i”] [Prop 3.2l_{1} →^{pyk} “prop three two l one”] [Block_{1}(∗, ∗, ∗) →^{pyk} “block one " state " cache " end block”] [Block_{2}(∗) →^{pyk} “block two " end block”] [∗^{hide} →^{pyk} “" hide”] [MacroIndent(∗) →^{pyk} “macro indent "”]

[∗′ →^{pyk} “" suc”]

[∗ = ∗ →^{pyk} “" equal "”] [∗ ≠ ∗ →^{pyk} “" unequal "”] [∗^{var} →^{pyk} “" is object var”] [∗#^{0}∗ →^{pyk} “" avoid zero "”] [∗#^{1}∗ →^{pyk} “" avoid one "”] [∗#^{∗}∗ →^{pyk} “" avoid star "”] [∃∗: ∗ →^{pyk} “exist " indeed "”] [∀∗: ∗ →^{pyk} “for all " indeed "”] [∀_{obj}∗: ∗ →^{pyk} “for all objects " indeed "”] [∗ ⇒ ∗ →^{pyk} “" imply "”] [∗ ⇔ ∗ →^{pyk} “" if and only if "”] [∗#∗ →^{pyk} “" avoid "”] [∗ ⊵ ∗ →^{pyk} “" object modus ponens "”] [Π∗: ∗ →^{pyk} “for all terms " indeed "”] [Begin ∗; ∗ : End; ∗ →^{pyk} “block " line " end block "”] [Last block line ∗ ≫ ∗; →^{pyk} “because " indeed " end line”] [Arbitrary ≫ ∗; ∗ →^{pyk} “any term " end line "”] [∗ | ∗ →^{pyk} “" alternative "”] [→ →^{pyk} “evaluates to”] [∗\\∗ →^{pyk} “" safe row "”] [check →^{pyk} “check”]

3 TEX definitions

[[x =^{∘} y] →^{tex} “

[#1/tex name/tex. \stackrel {\circ}{=}#2. ]”]

[RootVisible(x) →^{tex} “#1/tex name/tex.”]

[RootVisible(x) →^{name} “ RootVisible(#1. )”]

[x^{hide} →^{tex} “#1.

{}ˆ {hide}”]

[x′ →^{tex} “#1.

{}’”]

[x = y →^{tex} “#1.

= #2.”]

[x ≠ y →^{tex} “#1.

\neq #2.”]

[x ⇒ y →^{tex} “#1.

\Rightarrow #2.”]

[x ⇔ y →^{tex} “#1.

\Leftrightarrow #2.”]

[x | y →^{tex} “#1.

\mathrel{|} #2.”]

[∃x: y →^{tex} “

\exists #1. \colon #2.”]

[∀x: y →^{tex} “

\forall #1. \colon #2.”]

[∀_{obj}x: y →^{tex} “ \forall {obj} #1. \colon #2.”]

[Πx: y →^{tex} “

\Pi #1. \colon #2.”] [Arbitrary ≫ i; p →^{tex} “

\newline \makebox [0.1\textwidth ][l]{$ \if \relax \csname lgwprooflinep\endcsname L ? \else \global \advance \lgwproofline by 1 L\ifnum \lgwproofline <10 0\fi \number \lgwproofline \fi $:}\makebox [0.4\textwidth ][l]{$Arbitrary{}\gg{}$}\quad \parbox [t]{0.4\textwidth }{$#1. $\hfill \makebox [0mm][l]{\quad ; }}#2.”]

[Arbitrary ≫ i; p →^{name} “ Arbitrary \gg #1. ; #2.”]

[x\\y →^{tex} “#1.

\\{}#2.”]

[x\\y →^{name} “#1.

\backslash \backslash #2.”]

[A →^{tex} “

A”]

[R →^{tex} “

R”]

[C →^{tex} “

C”]

[T →^{tex} “

T”]

[L →^{tex} “

L”]

[{x} →^{tex} “

\{#1. \}”]

[S →^{tex} “

S”]

[Neg →^{tex} “

Neg”]

[S1 →^{tex} “

S1”]

[S2 →^{tex} “

S2”]

[S3 →^{tex} “

S3”]

[S4 →^{tex} “

S4”]

[S5 →^{tex} “

S5”]

[S6 →^{tex} “

S6”]

[S7 →^{tex} “

S7”]

[S8 →^{tex} “

S8”]

[S9 →^{tex} “

S9”]

[MP →^{tex} “

MP”]

[Gen →^{tex} “

Gen”]

[Ded →^{tex} “

Ded”]

[Repetition →^{tex} “

Repetition”]

[A1′ →^{tex} “

A1’”]

[A2′ →^{tex} “

A2’”]

[A4′ →^{tex} “

A4’”]

[A5′ →^{tex} “

A5’”]

[x^{var} →^{tex} “#1.

{}ˆ {var}”]

[x#y →^{tex} “#1.

\#.#2.”]

[x#^{0}y →^{tex} “#1.

\#.ˆ 0#2.”]

[x#^{1}y →^{tex} “#1.

\#.ˆ 1#2.”]

[x#^{∗}y →^{tex} “#1.

\#.ˆ ∗#2.”]

[⟨x≡y|z:=u⟩ →^{tex} “

\langle #1. {\equiv} #2. | #3. {:=} #4. \rangle ”]

[⟨x≡^{0}y|z:=u⟩ →^{tex} “

\langle #1. {\equiv}ˆ 0 #2. | #3. {:=} #4. \rangle ”]

[⟨x≡^{1}y|z:=u⟩ →^{tex} “

\langle #1. {\equiv}ˆ 1 #2. | #3. {:=} #4. \rangle ”]

[⟨x≡^{∗}y|z:=u⟩ →^{tex} “

\langle #1. {\equiv}ˆ ∗ #2. | #3. {:=} #4. \rangle ”]

[Ded(x, y) →^{tex} “ Ded(#1. , #2. )”] [Ded_{0}(x, y) →^{tex} “

Ded 0(#1. , #2. )”]

[Ded_{1}(x, y, z) →^{tex} “ Ded 1(#1. , #2. , #3. )”]

[Ded_{2}(x, y, z) →^{tex} “ Ded 2(#1. , #2. , #3. )”]

[Ded_{3}(x, y, z, u) →^{tex} “ Ded 3(#1. , #2. , #3. , #4. )”]

[Ded_{4}(x, y, z, u) →^{tex} “ Ded 4(#1. , #2. , #3. , #4. )”]

[Ded_{4}^{∗}(x, y, z, u) →^{tex} “ Ded 4ˆ ∗(#1. , #2. , #3. , #4. )”]

[Ded_{5}(x, y, z) →^{tex} “ Ded 5(#1. , #2. , #3. )”]

[Ded_{6}(p, c, e, b) →^{tex} “ Ded 6(#1. , #2.

, #3. , #4. )”]

[Ded_{6}^{∗}(p, c, e, b) →^{tex} “ Ded 6ˆ ∗(#1. , #2. , #3. , #4. )”]

[Ded_{7}(p) →^{tex} “

Ded 7(#1. )”]

[Ded_{8}(p, b) →^{tex} “ Ded 8(#1. , #2. )”]

[Ded_{8}^{∗}(p, b) →^{tex} “ Ded 8ˆ ∗(#1. , #2. )”]

[Begin b; l : End; p →^{tex} “ \newline \makebox [0.1\textwidth]{}% \parbox [b]{0.4\textwidth }{\raggedright \setlength {\parindent }{-0.1\textwidth }% \makebox [0.1\textwidth ][l]{$ \if \relax \csname lgwprooflinep\endcsname L ? \else \global \advance \lgwproofline by 1 L\ifnum \lgwproofline <10 0\fi \number \lgwproofline \fi $:}$Block {}\gg {}$}\quad \parbox [t]{0.4\textwidth }{$Begin $\hfill \makebox [0mm][l]{\quad ; }}#1. \newline \makebox [0.1\textwidth]{}% \parbox [b]{0.4\textwidth }{\raggedright \setlength {\parindent }{-0.1\textwidth }% \makebox [0.1\textwidth ][l]{$#2. $:}$Block {}\gg {}$}\quad \parbox [t]{0.4\textwidth }{$End $\hfill \makebox [0mm][l]{\quad ; }}#3.”]

[Begin b; l : End; p →^{name} “ Begin \, #1.

; #2. : End ; #3.”]

[Last block line a ≫ i ; →^{tex} “ \newline \makebox [0.1\textwidth]{}% \parbox [b]{0.4\textwidth }{\raggedright \setlength {\parindent }{-0.1\textwidth }% \makebox [0.1\textwidth ][l]{$ \if \relax \csname lgwprooflinep\endcsname L ? \else \global \advance \lgwproofline by 1 L\ifnum \lgwproofline <10 0\fi \number \lgwproofline \fi $:}$#1. {}\gg {}$}\quad \parbox [t]{0.4\textwidth }{$#2. $\hfill \makebox [0mm][l]{\quad ; }}”]

[Last block line a ≫ i ; →^{name} “ Last\ block\ line \, #1. \gg #2. \,; ”]

[x ⊵ y →^{tex} “#1.

\unrhd #2.”]

[Prop 3.2a →^{tex} “ Prop\ 3.2a”]

[Prop 3.2b →^{tex} “ Prop\ 3.2b”]

[Prop 3.2c →^{tex} “ Prop\ 3.2c”]

[Prop 3.2d →^{tex} “ Prop\ 3.2d”]

[Prop 3.2e_{1} →^{tex} “ Prop\ 3.2e 1”]

[Prop 3.2e_{2} →^{tex} “ Prop\ 3.2e 2”]

[Prop 3.2e →^{tex} “ Prop\ 3.2e”]

[Prop 3.2f_{1} →^{tex} “ Prop\ 3.2f 1”] [Prop 3.2f_{2} →^{tex} “

Prop\ 3.2f 2”]

[Prop 3.2f →^{tex} “ Prop\ 3.2f”]

[Prop 3.2g_{1} →^{tex} “ Prop\ 3.2g 1”]

[Prop 3.2g_{2} →^{tex} “ Prop\ 3.2g 2”]

[Prop 3.2g →^{tex} “ Prop\ 3.2g”]

[Prop 3.2h_{1} →^{tex} “ Prop\ 3.2h 1”]

[Prop 3.2h_{2} →^{tex} “ Prop\ 3.2h 2”]

[Prop 3.2h →^{tex} “ Prop\ 3.2h”]

[Prop 3.2i →^{tex} “ Prop\ 3.2i”]

[Prop 3.2l_{1} →^{tex} “ Prop\ 3.2l 1”]

[MacroIndent(x) →^{tex} “ $% \leftskip=1em% $#1.”]

[MacroIndent(x) →^{name} “ MacroIndent(#1. )”]

[Block_{1}(t, s, c) →^{tex} “ Block 1(#1. , #2. , #3. )”]

[Block_{2}(b) →^{tex} “ Block 2(#1. )”]

[→→^{tex} “

\rightarrow ”]

3.1 Variables

[\overline{x} →^{tex} “\overline\{#1.\}”] [a →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[a ¨= \overline{a}]⌉)] [b →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[b ¨= \overline{b}]⌉)] [c →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[c ¨= \overline{c}]⌉)] [d →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[d ¨= \overline{d}]⌉)] [e →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[e ¨= \overline{e}]⌉)] [f →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[f ¨= \overline{f}]⌉)] [g →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[g ¨= \overline{g}]⌉)] [h →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[h ¨= \overline{h}]⌉)] [i →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[i ¨= \overline{i}]⌉)] [j →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[j ¨= \overline{j}]⌉)] [k →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[k ¨= \overline{k}]⌉)] [l →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[l ¨= \overline{l}]⌉)] [m →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[m ¨= \overline{m}]⌉)] [n →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[n ¨= \overline{n}]⌉)] [o →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[o ¨= \overline{o}]⌉)] [p →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[p ¨= \overline{p}]⌉)] [q →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[q ¨= \overline{q}]⌉)] [r →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[r ¨= \overline{r}]⌉)] [s →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[s ¨= \overline{s}]⌉)] [t →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[t ¨= \overline{t}]⌉)] [u →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[u ¨= \overline{u}]⌉)] [v →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[v ¨= \overline{v}]⌉)] [w →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[w ¨= \overline{w}]⌉)] [x →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[x ¨= \overline{x}]⌉)] [y →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[y ¨= \overline{y}]⌉)] [z →^{macro} λt.λs.λc.M̃_{4}(t, s, c, ⌈[z ¨= \overline{z}]⌉)]

[a →^{tex} “

\mathit{a}”]

[b →^{tex} “

\mathit{b}”]

[c →^{tex} “

\mathit{c}”]

[d →^{tex} “

\mathit{d}”]

[e →^{tex} “

\mathit{e}”]

[f →^{tex} “

\mathit{f}”]

[g →^{tex} “

\mathit{g}”]

[h →^{tex} “

\mathit{h}”]

[i →^{tex} “

\mathit{i}”]

[j →^{tex} “

\mathit{j}”]

[k →^{tex} “

\mathit{k}”]

[l →^{tex} “

\mathit{l}”]

[m →^{tex} “

\mathit{m}”]

[n →^{tex} “

\mathit{n}”]

[o →^{tex} “

\mathit{o}”]

[p →^{tex} “

\mathit{p}”]

[q →^{tex} “

\mathit{q}”]

[r →^{tex} “

\mathit{r}”]

[s →^{tex} “

\mathit{s}”]

[t →^{tex} “

\mathit{t}”]

[u →^{tex} “

\mathit{u}”]

[v →^{tex} “

\mathit{v}”]

[w →^{tex} “

\mathit{w}”]

[x →^{tex} “

\mathit{x}”]

[y →^{tex} “

\mathit{y}”]

[z →^{tex} “

\mathit{z}”]

4 Priority table

[check →^{prio} Preassociative [check], [base], [bracket ∗ end bracket], [big bracket ∗ end bracket], [ $ ∗ $ ], [flush left [∗]], [x], [y], [z], [[∗ ▷◁ ∗]], [[∗ →^{∗} ∗]], [pyk], [tex], [name], [prio], [∗], [T], [if(∗, ∗, ∗)], [[∗ ⇒^{∗} ∗]], [val], [claim], [⊥], [f(∗)], [(∗)^{I}], [F], [\underline{0}], [\underline{1}], [\underline{2}], [\underline{3}], [\underline{4}], [\underline{5}], [\underline{6}], [\underline{7}], [\underline{8}], [\underline{9}], [0], [1], [2], [3], [4], [5], [6], [7], [8], [9], [a], [b], [c], [d], [e], [f], [g], [h], [i], [j], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [(∗)^{M}], [If(∗, ∗, ∗)], [array{∗} ∗ end array], [l], [c], [r], [empty], [⟨∗ | ∗:= ∗⟩], [M(∗)], [Ũ(∗)], [U(∗)], [U^{M}(∗)], [apply(∗, ∗)], [apply_{1}(∗, ∗)], [identifier(∗)], [identifier_{1}(∗, ∗)], [arrayplus(∗, ∗)], [array-remove(∗, ∗, ∗)], [array-put(∗, ∗, ∗, ∗)], [array-add(∗, ∗, ∗, ∗, ∗)], [bit(∗, ∗)], [bit_{1}(∗, ∗)], [rack], ["vector"], ["bibliography"], ["dictionary"], ["body"], ["codex"], ["expansion"], ["code"], ["cache"], ["diagnose"], ["pyk"], ["tex"], ["texname"], ["value"], ["message"], ["macro"], ["definition"], ["unpack"], ["claim"], ["priority"], ["lambda"], ["apply"], ["true"], ["if"], ["quote"], ["proclaim"], ["define"], ["introduce"], ["hide"], ["pre"], ["post"], [E(∗, ∗, ∗)], [E_{2}(∗, ∗, ∗, ∗, ∗)], [E_{3}(∗, ∗, ∗, ∗)], [E_{4}(∗, ∗, ∗, ∗)], [lookup(∗, ∗, ∗)], [abstract(∗, ∗, ∗, ∗)], [⌈∗⌉], [M(∗, ∗, ∗)], [M_{2}(∗, ∗, ∗, ∗)], [M^{∗}(∗, ∗, ∗)], [macro], [s_{0}], [zip(∗, ∗)], [assoc_{1}(∗, ∗, ∗)], [(∗)^{p}], [self], [[∗ ¨= ∗]], [[∗ ˙= ∗]], [[∗ ´= ∗]], [[∗ =^{pyk} ∗]], [[∗ =^{tex} ∗]], [[∗ =^{name} ∗]], [Priority table[∗]], [M̃_{1}], [M̃_{2}(∗)], [M̃_{3}(∗)], [M̃_{4}(∗, ∗, ∗, ∗)], [M̃(∗, ∗, ∗)], [Q̃(∗, ∗, ∗)], [Q̃_{2}(∗, ∗, ∗)], [Q̃_{3}(∗, ∗, ∗, ∗)], [Q̃^{∗}(∗, ∗, ∗)], [(∗)], [(∗)], [display(∗)], [statement(∗)], [[∗]^{·}], [[∗]^{−}], [aspect(∗, ∗)], [aspect(∗, ∗, ∗)], [⟨∗⟩], [tuple_{1}(∗)], [tuple_{2}(∗)], [let_{2}(∗, ∗)], [let_{1}(∗, ∗)], [[∗ =^{claim} ∗]], [checker], [check(∗, ∗)], [check_{2}(∗, ∗, ∗)], [check_{3}(∗, ∗, ∗)], [check^{∗}(∗, ∗)], [check_{2}^{∗}(∗, ∗, ∗)], [[∗]^{·}], [[∗]^{−}], [[∗]^{∘}], [msg], [[∗ =^{msg} ∗]], [<stmt>], [stmt], [[∗ =^{stmt} ∗]], [HeadNil′], [HeadPair′], [Transitivity′], [⊥ ⊥], [Contra′], [T′_{E}], [L_{1}], [\underline{∗}], [A], [B], [C], [D], [E], [F], [G], [H], [I], [J], [K], [L], [M], [N], [O], [P], [Q], [R], [S], [T ], [U], [V], [W], [X], [Y], [Z], [⟨∗ | ∗:= ∗⟩], [⟨^{∗}∗ | ∗:= ∗⟩], [∅], [Remainder], [(∗)^{v}], [intro(∗, ∗, ∗, ∗)], [intro(∗, ∗, ∗)], [error(∗, ∗)], [error_{2}(∗, ∗)], [proof(∗, ∗, ∗)], [proof_{2}(∗, ∗)], [S(∗, ∗)], [S^{I}(∗, ∗)], [S^{⊳} (∗, ∗)], [S_{1}^{⊳} (∗, ∗, ∗)], [S^{E}(∗, ∗)], [S_{1}^{E}(∗, ∗, ∗)], [S^{+}(∗, ∗)], [S_{1}^{+}(∗, ∗, ∗)], [S^{−}(∗, ∗)], [S_{1}^{−}(∗, ∗, ∗)], [S^{∗}(∗, ∗)], [S_{1}^{∗}(∗, ∗, ∗)], [S_{2}^{∗}(∗, ∗, ∗, ∗)], [S^{@}(∗, ∗)], [S_{1}^{@}(∗, ∗, ∗)], [S^{⊢}(∗, ∗)], [S_{1}^{⊢}(∗, ∗, ∗, ∗)], [S⊢ ⊢(∗, ∗)],

[S⊢ ⊢ ; 1(∗, ∗, ∗, ∗)], [S^{i.e.}(∗, ∗)], [S_{1}^{i.e.}(∗, ∗, ∗, ∗)], [S_{2}^{i.e.}(∗, ∗, ∗, ∗, ∗)], [S^{∀}(∗, ∗)],

[S_{1}^{∀}(∗, ∗, ∗, ∗)], [S^{;}(∗, ∗)], [S_{1}^{;}(∗, ∗, ∗)], [S_{2}^{;}(∗, ∗, ∗, ∗)], [T (∗)], [claims(∗, ∗, ∗)], [claims_{2}(∗, ∗, ∗)], [<proof>], [proof], [[Lemma ∗: ∗]], [[Proof of ∗: ∗]], [[∗ lemma ∗: ∗]], [[∗ antilemma ∗: ∗]], [[∗ rule ∗: ∗]], [[∗ antirule ∗: ∗]], [verifier], [V_{1}(∗)], [V_{2}(∗, ∗)], [V_{3}(∗, ∗, ∗, ∗)], [V_{4}(∗, ∗)], [V_{5}(∗, ∗, ∗, ∗)], [V_{6}(∗, ∗, ∗, ∗)], [V_{7}(∗, ∗, ∗, ∗)], [Cut(∗, ∗)], [Head_{⊕}(∗)], [Tail_{⊕}(∗)], [rule_{1}(∗, ∗)], [rule(∗, ∗)], [Rule tactic], [Plus(∗, ∗)], [[Theory ∗]], [theory_{2}(∗, ∗)], [theory_{3}(∗, ∗)], [theory_{4}(∗, ∗, ∗)], [HeadNil′′], [HeadPair′′], [Transitivity′′], [Contra′′], [HeadNil], [HeadPair], [Transitivity], [Contra], [T_{E}], [ragged right], [ragged right expansion ], [parm(∗, ∗, ∗)], [parm^{∗}(∗, ∗, ∗)], [inst(∗, ∗)], [inst^{∗}(∗, ∗)], [occur(∗, ∗, ∗)], [occur^{∗}(∗, ∗, ∗)], [unify(∗ = ∗, ∗)], [unify^{∗}(∗ = ∗, ∗)], [unify_{2}(∗ = ∗, ∗)], [L_{a}], [L_{b}], [L_{c}], [L_{d}], [L_{e}], [L_{f}], [L_{g}], [L_{h}], [L_{i}], [L_{j}], [L_{k}], [L_{l}], [L_{m}], [L_{n}], [L_{o}], [L_{p}], [L_{q}], [L_{r}], [L_{s}], [L_{t}], [L_{u}], [L_{v}], [L_{w}], [L_{x}], [L_{y}], [L_{z}], [L_{A}], [L_{B}], [L_{C}], [L_{D}], [L_{E}], [L_{F}], [L_{G}], [L_{H}], [L_{I}], [L_{J}], [L_{K}], [L_{L}], [L_{M}], [L_{N}], [L_{O}], [L_{P}], [L_{Q}], [L_{R}], [L_{S}], [L_{T}], [L_{U}], [L_{V}], [L_{W}], [L_{X}], [L_{Y}], [L_{Z}], [L_{?}], [Reflexivity], [Reflexivity_{1}], [Commutativity], [Commutativity_{1}], [<tactic>], [tactic], [[∗ =^{tactic} ∗]], [P(∗, ∗, ∗)], [P^{∗}(∗, ∗, ∗)], [p_{0}], [conclude_{1}(∗, ∗)], [conclude_{2}(∗, ∗, ∗)], [conclude_{3}(∗, ∗, ∗, ∗)], [conclude_{4}(∗, ∗)], [[∗ =^{∘} ∗]], [RootVisible(∗)], [A], [R], [C], [T], [L], [\{∗\}], [\overline{∗}], [a], [b], [c], [d], [e], [f ], [g], [h], [i], [j ], [k], [l], [m], [n], [o], [p], [q], [r], [s], [t], [u], [v], [w], [x], [y], [z], [⟨∗≡ ∗ | ∗:=∗⟩], [⟨∗≡^{0} ∗ | ∗:=∗⟩], [⟨∗≡^{1} ∗ | ∗:=∗⟩], [⟨∗≡^{∗} ∗ | ∗:=∗⟩], [Ded(∗, ∗)], [Ded_{0}(∗, ∗)], [Ded_{1}(∗, ∗, ∗)], [Ded_{2}(∗, ∗, ∗)], [Ded_{3}(∗, ∗, ∗, ∗)], [Ded_{4}(∗, ∗, ∗, ∗)], [Ded_{4}^{∗}(∗, ∗, ∗, ∗)], [Ded_{5}(∗, ∗, ∗)], [Ded_{6}(∗, ∗, ∗, ∗)], [Ded_{6}^{∗}(∗, ∗, ∗, ∗)], [Ded_{7}(∗)], [Ded_{8}(∗, ∗)], [Ded_{8}^{∗}(∗, ∗)], [S], [Neg], [MP], [Gen], [Ded], [S1], [S2], [S3], [S4], [S5], [S6], [S7], [S8], [S9], [Repetition], [A1′], [A2′], [A4′], [A5′], [Prop 3.2a], [Prop 3.2b], [Prop 3.2c], [Prop 3.2d], [Prop 3.2e_{1}], [Prop 3.2e_{2}], [Prop 3.2e], [Prop 3.2f_{1}], [Prop 3.2f_{2}], [Prop 3.2f], [Prop 3.2g_{1}], [Prop 3.2g_{2}], [Prop 3.2g], [Prop 3.2h_{1}], [Prop 3.2h_{2}], [Prop 3.2h], [Prop 3.2i], [Prop 3.2l_{1}], [Block_{1}(∗, ∗, ∗)], [Block_{2}(∗)]; Preassociative [∗ {∗}], [∗/indexintro(∗, ∗, ∗, ∗)], [∗/intro(∗, ∗, ∗)], [∗/bothintro(∗, ∗, ∗, ∗, ∗)], [∗/nameintro(∗, ∗, ∗, ∗)], [∗′], [∗[ ∗]], [∗[∗→∗]], [∗[∗⇒∗]], [∗0], [∗1], [0b], [∗-color(∗)], [∗-color^{∗}(∗)], [∗^{H}], [∗^{T}], [∗^{U}], [∗^{h}], [∗^{t}], [∗^{s}], [∗^{c}], [∗^{d}], [∗^{a}], [∗^{C}], [∗^{M}], [∗^{B}], [∗^{r}], [∗^{i}], [∗^{d}], [∗^{R}], [∗^{0}], [∗^{1}], [∗^{2}], [∗^{3}], [∗^{4}], [∗^{5}], [∗^{6}], [∗^{7}], [∗^{8}], [∗^{9}], [∗^{E}], [∗^{V}], [∗^{C}], [∗^{C^{∗}}], [∗^{hide}]; Preassociative [“ ∗ ”], [], [(∗)^{t}], [string(∗) + ∗], [string(∗) ++ ∗], [ ∗], [ ∗], [!∗], ["∗], [#∗], [$∗], [%∗], [&∗], [’∗], [(∗], [)∗], [∗∗], [+∗], [, ∗], [-∗], [.∗], [/∗], [0∗], [1∗], [2∗], [3∗], [4∗], [5∗], [6∗], [7∗], [8∗], [9∗], [:∗], [; ∗], [<∗], [=∗], [>∗], [?∗], [@∗], [A∗], [B∗], [C∗], [D∗], [E∗], [F∗], [G∗], [H∗], [I∗], [J∗], [K∗], [L∗], [M∗], [N∗], [O∗], [P∗], [Q∗], [R∗], [S∗], [T∗], [U∗], [V∗], [W∗], [X∗], [Y∗], [Z∗], [[∗], [\∗], []∗], [ˆ ∗], [ ∗], [‘∗], [a∗], [b∗], [c∗], [d∗], [e∗], [f∗], [g∗], [h∗], [i∗], [j∗], [k∗], [l∗], [m∗], [n∗], [o∗], [p∗], [q∗], [r∗], [s∗], [t∗], [u∗], [v∗], [w∗], [x∗], [y∗], [z∗], [{∗], [|∗], [}∗], [˜ ∗], [Preassociative ∗; ∗], [Postassociative ∗; ∗], [[∗], ∗], [priority ∗ end], [newline ∗], [macro newline ∗], [MacroIndent(∗)]; Preassociative [∗ ’ ∗], [∗ ‘ ∗]; Preassociative [∗′]; Preassociative [∗ · ∗], [∗ ·_{0} ∗]; Preassociative [∗ + ∗], [∗ +_{0} ∗], [∗ +_{1} ∗], [∗ − ∗], [∗ −_{0} ∗], [∗ −_{1} ∗]; Preassociative [∗ ∪ {∗}], [∗ ∪ ∗], [∗\{∗}]; Postassociative [∗ ˙. . ∗], [∗ \underline{˙. .} ∗], [∗ \underline{: :} ∗], [∗ \underline{+2∗} ∗], [∗ : : ∗], [∗ +2∗ ∗]; Postassociative [∗, ∗]; Preassociative

[∗ ≈^{B} ∗], [∗ ≈^{D} ∗], [∗ ≈^{C} ∗], [∗ ≈^{P} ∗], [∗ ≈ ∗], [∗ = ∗], [∗ →^{+} ∗], [∗ =^{t} ∗], [∗ =^{t^{∗}} ∗], [∗ =^{r} ∗],

[∗ ∈_{t} ∗], [∗ ⊆_{T} ∗], [∗ =^{T} ∗], [∗ =^{s} ∗], [∗ free in ∗], [∗ free in^{∗} ∗], [∗ free for ∗ in ∗], [∗ free for^{∗} ∗ in ∗], [∗ ∈_{c} ∗], [∗ < ∗], [∗ <′ ∗], [∗ ≤′ ∗], [∗ = ∗], [∗ ≠ ∗], [∗^{var}], [∗#^{0}∗], [∗#^{1}∗], [∗#^{∗}∗]; Preassociative [¬∗]; Preassociative [∗ ∧ ∗], [∗ ¨∧ ∗], [∗ ˜∧ ∗], [∗ ∧_{c} ∗]; Preassociative [∗ ∨ ∗], [∗ ∥ ∗], [∗ ¨∨ ∗]; Preassociative [∃∗: ∗], [∀∗: ∗], [∀_{obj}∗: ∗]; Postassociative [∗ ¨⇒ ∗], [∗ ⇒ ∗], [∗ ⇔ ∗]; Postassociative [∗ : ∗], [∗ spy ∗], [∗!∗]; Preassociative

[∗ \{ \begin{matrix}∗ \\ ∗ \end{matrix} ];

Preassociative [λ ∗ .∗], [Λ ∗ .∗], [Λ∗], [if ∗ then ∗ else ∗], [let ∗ = ∗ in ∗], [let ∗ ¨= ∗ in ∗]; Preassociative [∗#∗]; Preassociative [∗^{I}], [∗^{⊳} ], [∗^{V}], [∗^{+}], [∗^{−}], [∗^{∗}]; Preassociative [∗ @ ∗], [∗ ⊳ ∗], [∗ ⊳ ⊳ ∗], [∗ ≫ ∗], [∗ ⊵ ∗]; Postassociative [∗ ⊢ ∗], [∗ ⊢⊢ ∗], [∗ i.e. ∗]; Preassociative [∀∗: ∗], [Π∗: ∗]; Postassociative [∗ ⊕ ∗]; Postassociative [∗; ∗]; Preassociative [∗ proves ∗]; Preassociative [∗ proof of ∗ : ∗], [Line ∗ : ∗ ≫ ∗; ∗], [Last line ∗ ≫ ∗ □], [Line ∗ : Premise ≫ ∗; ∗], [Line ∗ : Side-condition ≫ ∗; ∗], [Arbitrary ≫ ∗; ∗], [Local ≫ ∗ = ∗; ∗], [Begin ∗; ∗ : End; ∗], [Last block line ∗ ≫ ∗; ], [Arbitrary ≫ ∗; ∗]; Postassociative [∗ | ∗]; Postassociative [∗ , ∗], [∗[ ∗]∗]; Preassociative [∗&∗], [→]; Preassociative [∗\\∗], [∗ linebreak[4] ∗], [∗\\∗]; ]