The method can be extended to prove statements about more general well-founded structures, such as trees ; this generalization, known as structural induction, is used in mathematical logic and computer science.

Structural recursion is a recursion method bearing the same relationship to structural induction as ordinary recursion bears to ordinary mathematical induction.

Structural recursion is usually proved correct by structural induction ; in particularly easy cases, the inductive step is often left out.

Thus, the induction principle should be respected to avoid overfitted hypotheses and deriving overfitted and useless interpretations on structural / molecular data.

Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.

Some chemists, notably Adolph Wilhelm Hermann Kolbe, heavily criticized the use of structural formulas that were offered, as he thought, without proof.

Proof theory is important in philosophical logic, where the primary interest is in the idea of a proof-theoretic semantics, an idea which depends upon technical ideas in structural proof theory to be feasible.

In parallel, the foundations of structural proof theory were being founded.

Structural proof theory is connected to type theory by means of the Curry-Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus.

In linguistics, type-logical grammar, categorial grammar and Montague grammar apply formalisms based on structural proof theory to give a formal natural language semantics.

Analytic tableaux apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics.

Several important logics have come from insights into logical structure arising in structural proof theory.

* A structural rule in proof theory

The barrel and action are then inspected and, if they have maintained structural integrity, they will be stamped with proof marks indicating the test pressure, bore diameter in millimetres at 9 " from the breech face, chamber length, suitability for use with steel shot in the case of a smoothbore shotgun, a date stamp or code and the mark of the house ( London or Birmingham ).

*** Analytic proof, in structural proof theory, a proof whose structure is simple in a special way

The perfect graph theorem has a short proof, but the proof of the strong perfect graph theorem is long and technical, based on a deep structural decomposition of Berge graphs.

Affine logic is a substructural logic whose proof theory rejects the structural rule of contraction.

In proof theory, a structural rule is an inference rule that does not refer to any logical connective, but instead operates on the judgements or sequents directly.

In mathematical logic, structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof.

The term structure in structural proof theory comes from a technical notion introduced in the sequent calculus: the sequent calculus represents the judgement made at any stage of an inference using special, extra-logical operators which we call structural operators: in, the commas to the left of the turnstile are operators normally interpreted as conjunctions, those to the right as disjunctions, whilst the turnstile symbol itself is interpreted as an implication.

Although the play does show a certain structural amateurishness ( there are eleven acts varying in length from twenty-five seconds to an hour and a half ), the statement it makes concerning the ceaseless yearning and searching of youth is profound and worthy of our attention.

Af appeared to be well suited for the study of these matters, since it is a normal paramagnet, with three unpaired electrons on the chromium, its crystal structure is very simple, and the unknown position of the hydrogen in the strong Af bond provides structural interest.

The first stage of translation after glossary lookup is structural analysis of the input text.

The paper has a certain value as a comparatively easy introduction to this approach, particularly since it treats a fairly simple and straightforward phenomenon where it is possible to compare it with a more traditional ( though not structural ) statement.

The aim is to collect a very broad range of physical, chemical, morphological, and structural data for crystals on an encyclopedic scale and to seek all possible useful and revealing correlations of properties with internal structure.

The overall chemical and structural composition is used to divide ambers into five classes.

A critical structural and tonal component of an acoustic guitar is the bracing, a systems of struts glued to the inside of the back and top.

In 1941 it was rumored to have been inspected by the federal Works Progress Administration to determine its structural strength, but there is no evidence of this.

Compared with most other classes of natural compounds, alkaloids are characterized by a great structural diversity and there is no uniform classification of alkaloids.

It is the most polyphetic known enzyme: six distinct classes have been described, all catalyzing the same reaction but representing unrelated gene families with no known sequence or structural homology.

As in Classical architecture, in Gothic architecture, too, an aedicule or tabernacle frame is a structural framing device that gives importance to its contents, whether an inscribed plaque, a cult object, a bust or the like, by assuming the tectonic vocabulary of a little building that sets it apart from the wall against which it is placed.

Since the inefficiency is related to government tax policy, and hence is structural in nature, it has not been arbitraged away.

Agar or agar-agar is a gelatinous substance derived by boiling from a polysaccharide in red algae, where it accumulates in the cell walls of agarophyte and serves as the primary structural support for the algae's cell walls.

A bridge's structural efficiency is the ratio of load carried to bridge weight, given a specific set of material types.

The IMF loan, approved on September 26, is designed to support Bosnia's economic programme for the next two years, ease the effects of the “ external environment and address domestic structural weaknesses ”.

* Cellulose is made by plants and is an important structural component of their cell walls.

Structural biology is the study of the structural properties of the biopolymers.

The hull is the main structural component of the boat which actually provides buoyancy for the boat.

The " floor " of a cabin is properly known as the sole, but is more likely to be called the floor ( a floor is properly, a structural member which ties a frame to the keelson and keel ).

The case holds in position all of the important structural members: pinblock, soundboard, hitchpins, keyboard, and the jack action.

Physical gills are a type of structural adaptation common among some types of aquatic insects, which holds atmospheric oxygen in an area with small openings called spiracles.

They signal the structural relationships that words have to one another and are the glue that holds sentences together.

* Structural engineer – Typically holds a bachelors or master's degree in structural engineering, new university graduates must hold a master's degree to gain chartered status from the Engineering Council, mainly through the Institution of Structural Engineers.

* Frame ( beehive ), a structural element that holds honeycomb

Standing rigging is a structural element that holds up the masts, and loss of standing rigging puts them at risk of being sprung ( cracked ) or simply snapped off.

MEP, she holds globalization, intergovernmental organizations, ' euro-mondialism ', free trade and ultra-liberalism responsible for the decline of agriculture and the fishing industry, deindustrialization, offshoring and structural unemployment.

WSA holds that struggles against gender inequality, structural racism, and oppression of LGBTQ people are also part of the larger fight for social liberation and self-management.

A frame in a beehive is the structural element that holds the honeycomb or brood comb within the hive body ( or " super ").