ICL Website Index Table of Contents
  • Base Library for ICL
    • De Morgan laws
    • Size recursion
    • Iteration
    • Decidability
    • Lists
    • Decidability laws for lists
    • Membership
    • Disjointness
    • Inclusion
    • Setoid rewriting with list inclusion and list equivalence
    • Equivalence
    • Filter
    • Element removal
    • Cardinality
    • Duplicate-free lists
    • Power lists
    • Finite closure iteration
    • Deprecated names, defined for backward compatibilitly
  • Chapter 1 - Types and Functions
    • 1.1 Booleans
    • 1.2 Cascaded Functions
    • 1.3 Natural Numbers
    • 1.4 Structural Induction and Rewriting
    • 1.5 More on Rewriting
    • 1.6 Recursive Abstractions
    • 1.7 Defined Notations
    • 1.8 Standard Library.
    • 1.9 Pairs and Implicit Arguments
    • 1.10 Lists
    • 1.11 Quantified Inductive Hypotheses
    • 1.12 Iteration as Polymorphic Higher-Order Function
    • 1.13 Options and Finite Types
    • 1.14 More about Functions
  • Chapter 2 - Propositions and Proofs
    • 2.2 Implication and Universal Quantification
    • 2.3 Predicates
    • 2.4 The Apply Tactic
    • Leibniz Characterization of Equality
    • 2.6 Propositions are Types
    • 2.7 Falsity and Negation
    • 2.8 Conjunction and Disjunction
    • 2.9 Equivalence and Rewriting
    • 2.10 Automation Tactics
    • 2.11 Existential Quantification
    • 2.13 Proof Rules as Lemmas
    • 2.14 Inductive Propositions
    • 2.15 An Observation
    • 2.16 Excluded Middle
  • Chapter 3 - Definitional Equality and Propositional Equality
    • 3.1 Conversion Principle
    • 3.2 Disjointness and Injectivity of Constructors
    • 3.3 Leibniz Equality
    • 3.4 By Name Specification of Implicit Arguments
    • 3.5 Local Definitions
    • 3.6 Proof of nat <> bool
    • 3.7 Cantor's Theorem
    • 3.8 Kaminski's Equation
    • 3.9 Boolean Equality Tests
  • Chapter 4 - Inductive Predicates
    • 4.1 Induction Lemmas
    • 4.2 Primitive Recursion
    • 4.3 Size Induction
    • 4.4 Equational Specifcation of Functions
  • Chapter 5 - Truth Value Semantics and Elim Restriction
    • 5.1 Truth Value Semantics
    • 5.2 Elim Restriction
    • 5.3 Propositional Extensionality Entails Proof Irrelevance
    • 5.4 A Simpler Proof
  • Chapter 6 - Sum and Sigma Types
    • 6.1 Boolean Sums and Certifying Test
    • 6.2 - Inhabitation and Decidability
    • 6.3 Writing Certifying Test
    • 6.5 Decidable Predicates
    • 6.6 Sigma Types
    • 6.7 Strong Truth Value Semantics
  • Chapter 7 - Inductive Predicates
    • 7.1 Nonparametric Arguments and Linerization
    • 7.2 Even
    • 7.3 Less or Equal
    • 7.4 Equality
    • 7.5 Exceptions from the Elim Restriction
    • 7.6 Safe and Nonuniform Parameters
    • 7.7 Constructive Choice for Nat
    • 7.8 Technical Summary
    • 7.9 Induction Lemmas
  • Chapter 8 - Lists
    • 8.1 Constructors and Notation
    • 8.2 Recursion and Induction
    • 8.3 Membership
    • 8.4 Inclusion and Equivalence
    • 8.5 Disjointness
    • 8.6 Decidability
    • 8.7 Filtering
    • 8.8 Element Removal
    • 8.9 Cardinality
    • 8.10 Duplicate-Freeness
    • 8.11 Power Lists
  • Chapter 10 - Propositional Entailment
    • 10.1 Propositional Formulas
    • 10.2 Structural Properties of Entailment Relations
    • 10.3 Logical Properties of Entailment Relations
    • 10.4 Variables and Substitutions
    • 10.5 Natural Deduction System
    • 10.6 Classical Natural Deduction
    • 10.7 Glivenko's Theorem
    • 10.8 Hilbert System
  • Chapter 11 - Classical Tableau Method
    • 11.1 Boolean Evaluation and Satisfiability
    • 11.2 Validity and Boolean Entailment
    • 11.3 Signed Formulas and Clauses
    • 11.4 Solved Clauses
    • 11.6 DNF Procedure
    • 11.7 Recursion Trees
    • 11.8 Assisted Decider for Satisfiability
    • 11.9 Main Results
    • 11.10 Refutation Lemma
  • Chapter 12 - Intuitionistic Gentzen System
    • 12.1 - Gentzen System GS
    • 12.2 Completeness of GS
    • 12.3 Decidability
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