[ BeginEnv "definition"
, Label "subseteq"
, BeginEnv "math"
, Variable "A"
, Command "subseteq"
, Variable "B"
, EndEnv "math"
, Word "iff"
, BeginEnv "math"
, Variable "A"
, Symbol "="
, Variable "B"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "definition"
, Label "pow"
, BeginEnv "math"
, Command "pow"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, Symbol "="
, Command "emptyset"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "definition"
, Label "cons"
, BeginEnv "math"
, Command "cons"
, InvisibleBraceL
, Variable "a"
, InvisibleBraceR
, InvisibleBraceL
, Variable "B"
, InvisibleBraceR
, Symbol "="
, Command "emptyset"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "definition"
, Label "times"
, BeginEnv "math"
, Variable "A"
, Command "times"
, Variable "B"
, Symbol "="
, Command "emptyset"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "definition"
, Label "fld"
, BeginEnv "math"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Symbol "="
, Command "emptyset"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "definition"
, Label "preimg"
, BeginEnv "math"
, Command "preimg"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, Symbol "="
, Command "emptyset"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "axiom"
, Label "lmao"
, BeginEnv "math"
, Variable "x"
, Command "in"
, Command "emptyset"
, EndEnv "math"
, Symbol "."
, EndEnv "axiom"
, BeginEnv "inductive"
, Label "fin"
, Word "define"
, BeginEnv "math"
, Command "fin"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, Command "subseteq"
, Command "pow"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, EndEnv "math"
, Word "inductively"
, Word "as"
, Word "follows"
, Symbol "."
, BeginEnv "enumerate"
, Command "item"
, BeginEnv "math"
, Command "emptyset"
, Command "in"
, Command "fin"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, Command "item"
, Word "if"
, BeginEnv "math"
, Variable "a"
, Command "in"
, Variable "A"
, EndEnv "math"
, Word "and"
, BeginEnv "math"
, Variable "B"
, Command "in"
, Command "fin"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, Command "cons"
, InvisibleBraceL
, Variable "a"
, InvisibleBraceR
, InvisibleBraceL
, Variable "B"
, InvisibleBraceR
, Command "in"
, Command "fin"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, EndEnv "enumerate"
, EndEnv "inductive"
, BeginEnv "lemma"
, Label "fin_mono"
, Word "let"
, BeginEnv "math"
, Variable "A"
, Symbol ","
, Variable "B"
, EndEnv "math"
, Word "be"
, Word "sets"
, Symbol "."
, Word "suppose"
, BeginEnv "math"
, Variable "A"
, Command "subseteq"
, Variable "B"
, EndEnv "math"
, Symbol "."
, Word "then"
, BeginEnv "math"
, Command "fin"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, Command "subseteq"
, Command "fin"
, InvisibleBraceL
, Variable "B"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, EndEnv "lemma"
, BeginEnv "inductive"
, Label "tracl"
, Word "define"
, BeginEnv "math"
, Command "tracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Command "subseteq"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Command "times"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "inductively"
, Word "as"
, Word "follows"
, Symbol "."
, BeginEnv "enumerate"
, Command "item"
, Word "if"
, BeginEnv "math"
, Variable "w"
, Command "in"
, Variable "R"
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, Variable "w"
, Command "in"
, Command "tracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, Command "item"
, Word "if"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "b"
, ParenR
, Command "in"
, Command "tracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "and"
, BeginEnv "math"
, ParenL
, Variable "b"
, Symbol ","
, Variable "c"
, ParenR
, Command "in"
, Variable "R"
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "c"
, ParenR
, Command "in"
, Command "tracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, EndEnv "enumerate"
, EndEnv "inductive"
, BeginEnv "inductive"
, Label "quasirefltracl"
, Word "define"
, BeginEnv "math"
, Command "qrefltracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Command "subseteq"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Command "times"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "inductively"
, Word "as"
, Word "follows"
, Symbol "."
, BeginEnv "enumerate"
, Command "item"
, Word "if"
, BeginEnv "math"
, Variable "a"
, Command "in"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "a"
, ParenR
, Command "in"
, Command "qrefltracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, Command "item"
, Word "if"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "b"
, ParenR
, Command "in"
, Command "qrefltracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "and"
, BeginEnv "math"
, ParenL
, Variable "b"
, Symbol ","
, Variable "c"
, ParenR
, Command "in"
, Variable "R"
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "c"
, ParenR
, Command "in"
, Command "qrefltracl"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, EndEnv "enumerate"
, EndEnv "inductive"
, BeginEnv "inductive"
, Label "refltracl"
, Word "define"
, BeginEnv "math"
, Command "refltracl"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Command "subseteq"
, Variable "A"
, Command "times"
, Variable "A"
, EndEnv "math"
, Word "inductively"
, Word "as"
, Word "follows"
, Symbol "."
, BeginEnv "enumerate"
, Command "item"
, Word "if"
, BeginEnv "math"
, Variable "a"
, Command "in"
, Variable "A"
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "a"
, ParenR
, Command "in"
, Command "refltracl"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, Command "item"
, Word "if"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "b"
, ParenR
, Command "in"
, Command "refltracl"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "and"
, BeginEnv "math"
, ParenL
, Variable "b"
, Symbol ","
, Variable "c"
, ParenR
, Command "in"
, Variable "R"
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, ParenL
, Variable "a"
, Symbol ","
, Variable "c"
, ParenR
, Command "in"
, Command "refltracl"
, InvisibleBraceL
, Variable "A"
, InvisibleBraceR
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, EndEnv "enumerate"
, EndEnv "inductive"
, BeginEnv "inductive"
, Label "acc"
, Word "define"
, BeginEnv "math"
, Command "acc"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, Command "subseteq"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "inductively"
, Word "as"
, Word "follows"
, Symbol "."
, BeginEnv "enumerate"
, Command "item"
, Word "if"
, BeginEnv "math"
, Variable "a"
, Command "in"
, Command "fld"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Word "and"
, BeginEnv "math"
, Command "preimg"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, InvisibleBraceL
, VisibleBraceL
, Variable "a"
, VisibleBraceR
, InvisibleBraceR
, Command "in"
, Command "pow"
, InvisibleBraceL
, Command "acc"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, InvisibleBraceR
, EndEnv "math"
, Symbol ","
, Word "then"
, BeginEnv "math"
, Variable "a"
, Command "in"
, Command "acc"
, InvisibleBraceL
, Variable "R"
, InvisibleBraceR
, EndEnv "math"
, Symbol "."
, EndEnv "enumerate"
, EndEnv "inductive"
]