Mathematics / research
Constructive mathematics and type theory
Connect proofs and programs through constructive logic, dependent types, identity, normalization, and formal mathematical structures.
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Module 01 research
Constructive mathematics and type theory
Connect proofs and programs through constructive logic, dependent types, identity, normalization, and formal mathematical structures.
- 01
Constructive logic and witnesses
Track the evidence supplied by a proof and distinguish an explicit construction from double-negated existence.
35 min - 02
Curry–Howard and typed lambda calculus
Build typed proof terms, perform capture-avoiding substitution, and interpret reduction as simplifying a derivation.
35 min - 03
Dependent products and dependent sums
Let result types depend on inputs, count finite families, and distinguish data-level choice from truncated choice.
36 min - 04
Inductive types and elimination principles
Read constructors as a specification of recursion and induction, including indexed outputs and positivity restrictions.
36 min - 05
Identity types and path induction
Separate computation from equality evidence and derive transport coherence using the identity eliminator.
37 min - 06
Mere propositions and propositional truncation
Erase witness distinctions deliberately, respect the truncation eliminator, and recover uniquely characterized data safely.
37 min - 07
Normalization, canonicity, and decidable checking
State metatheorems for an explicit total fragment and separate termination, canonical results, checking, and theorem discovery.
38 min - 08
Constructive reals and choice principles
Carry quantitative approximation data explicitly and state the extra principles used to compare real-number constructions.
38 min - 09
Universes, univalence, and homotopy type theory
Control universe levels, interpret type equality by equivalence, and separate axiomatic transport from cubical computation.
39 min - 10
Formalizing structures and trusted kernels
Package algebraic data with laws, transport a finite structure, and audit what a checked proof actually assumes.
39 min