Dimitri Della Faille

A Sociological Understanding of Neoliberal Discourses of Development

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A Sociological Understanding of Neoliberal Discourses of Development

Dimitri della Faille

When understanding development issues, language and discourse are subjects viewed with much conventional wisdom. According to a commonly held idea, words are different than deeds, or language must be followed by actions. Some years ago, the World Bankan international financial institution supporting "developing" countriesdeclared that, in terms of sustainable development, "it is necessary that each relevant stakeholdergovernment, industry, technology developer and financial institutiondoes whatever it takes to unlock the value of wasted gas because actions will always speak louder than words." (World Bank 2009) Around the same time, another multilateral institution, the World Health Organization (WHO)an institution from the United Nations system specializing in coordinating and implementing international efforts related to health policies and programs -declared that "governments, including members of the G8 and other development partners, international organizations, civil society, the private sector, academia and others must maintain momentum and work together to ensure that the exceptional work of 2008 moves from words to deeds, resulting in concrete progress on the ground in 2009." (World Health Organization 2009) Undeniably, and these are just two of countless examples, it seems commonly held that words and actions are two distinct domains, implying that development occurs on two separate levels. First, development planning and policymaking begin with discussions, usually around a table. Ideas are exchanged and concerns are expressed by academics, activists, public intellectuals, and policymakers. Next, these debates must be followed by actions. Once they are discussed, ideas must be implemented by the "real" actors of development, those who are in the field, dealing with "real" issues. According to another common idea, discourse, especially when uttered by politicians, is nothing but smoke and mirrors, a linguistic spin used by persons of influence to misrepresent their objectives, cover for their inaction, or more deceptively, to lie. This paper will attempt to show that social scientists studying development issues must consider these common ideas with considerable caution. We argue that words are, in fact, actions. And as such, they must be investigated. We contend that an examination of underdevelopment and "developing" societies must go beyond an artificial divide between discourse and action. But also, that it must not limit its definition of discourse to an act of deception. Otherwise, we run the risk of misunderstanding social problems, which is the basis for much social action and collective mobilization in the "developing" world. We will also propose in this paper a number of ways to examine language and discourse that go beyond received ideas. We will attempt to show that they are integral parts of actionwhether scholarly, activist, administrative or otherwiseagainst underdevelopment. In the first place, we will focus most of our explanation on how neoliberal governance and policymaking use language, social representation and discourse to achieve their goals. Using example of neoliberal discourses, we will attempt to show how the main ideologies of the various contemporary development discourses transforms our perception and understanding of development problems. This transformation, we argue, exists both in imposing the use of specific words and in successfully controlling means of communication.

We will begin with a quick presentation of discourse and a definition of neoliberal ideologies. Then, we will demonstrate how discourse analysis could study neoliberal discourses by applying to documents about a natural disaster in the Philippines. After this demonstration, we present other various examples of discourse analysis as it applies to development discourses. Then, we present some of the major approaches and methodologies of discourse analysis. Before concluding, we will present some ethical considerations for the analysis of development discourses.

Words of Caution

A paper about language and discourse would fall short of its goal to draw attention to the use of language if it did not contain at least some form of criticism of usages of the word "development". We argue that calling societies "developing" is actually making a normative statement about the past trajectory, current status and expected future of these societies. Social scientists may contend that political, scientific, ethical or lay statements about development and underdevelopment are in fact "problematizations" of human societies. A problematization is a process by which social relations, practices, rules, institutions, and habits previously established are suddenly viewed as doubtful and problematic [START_REF] Foucault | Dits et écrits[END_REF]). The word "development" itself may carry different meanings around the world [START_REF] Thornton | Knowledge and beliefs about national development and developmental hierarchies: the viewpoints of ordinary people in thirteen countries[END_REF]. The understanding and expectations of actions in the name of "development" are conditioned by social representations and interpretations. However, we contend that development discourses are problematizations of the "developing" world because they transform the history of societies of Latin America, Asia, Africa and some parts of Europe into a long story of troubles and failures. They do that in order to justify social transformations and interventions [START_REF] Escobar | Encountering Development: The Making and Unmaking of the Third World[END_REF]. We also contend that they are problematizations because they produce cultural discourses that apply specifically to "developing" countries, and therefore reinforce ideas about the perceived superiority of "developed" countries over the rest of the world [START_REF] Mohanty | Under Western Eyes: Feminist Scholarship and Colonial Discourses[END_REF].

This paper refuses to hierarchize societies based on perceptions of their economic achievement, their form of political governance or the global recognition of their cultural products. We recognize that discourses about "development" are problematizations, and that perceptions of any social, political or cultural inferiority of these regions, countries or populations must be criticized. We therefore use the term "developing" for some societies, not as a normative statement on regions, countries, and populations viewed as economically, socially, politically or culturally inferior to the "developed world", but rather as an unfortunate shortcut to describe regions and countries in which actors desire to act in the name of "development". There is a wealth of scholarly literature on criticism of the use of the word "development", some of which is evoked further in this paper.

We will give further explanations that might help you better understand why we must be cautious when comparing societies in terms of their perceived "development". Now that we explained why we, in this paper, are cautious of talking about "development" and "underdevelopment", let us very briefly present some aspects of discourse and its analysis.

Understanding discourse and its analysis

If discourse analysis is getting more recognition in development studies, before we further embark in this paper it must be noted that if you chose to study discourse, you might encounter disapproval [START_REF] Ziai | The Contribution of Discourse Analysis to Development Studies[END_REF]. As we have argued elsewhere, discourse analysis is often viewed with reservations or criticized in the context of the study of "development" and "underdevelopment" [START_REF] Faille | Discourse Analysis in International Development Studies: Mapping Some Contemporary Contributions[END_REF][START_REF] Bello | Neoliberalism as Hegemonic Ideology in the Philippines[END_REF]. But very often, the criticism comes from misunderstanding of what discourse actually is. Discourse analysts face many commonly held ideas, as per the examples we have provided in the introduction of this paper. We believe that the best way for social scientists to justify the analysis of words, language and communication is to approach it with a clear definition of discourse that relates to the study of social relations and also to present convincing analysis. This section attempts to clarify our definition of discourse analysis and the following sections will attempt to illustrate how this analysis relates to the study of social relations and "development".

Social scientists studying discourses are examining the social and institutional constraints of language. At the conceptual level, language can be apprehended either as a social fact determined by material conditions and social domination, or as a field of social activity with specific rules and a social environment where meaning, social relations, and society are produced. Most discourse analysts adopt the latter conception. They attempt to reveal the strategies that aim to convey cultural values and ideologies, whether implicitly or explicitly. They define language as the production of meaning and the results of acts of communication that are conditioned by collective rules and social codes. Through the use of language, social groups and individuals come to build their identity, describe themselves, interact, and share ideas. Language is thus more than the use of specific vocabularies and grammars. It is an organized sequence of social acts that is not limited to speech or utterance. Some analysts study images and material artefacts as sequences of social acts and social strategies to convey ideologies.

In the 1960s French and British philosophers, sociologists and political scientists began to understand the production of language in terms of communication strategies. This new direction was dubbed the "linguistic turn" of humanities and social sciences (Rorty 1967). Based on several decades of debate in literary study, linguistics and anthropology, discourse analysis emerged as a new discipline. It proposed a way to see language as a field of social confrontation and struggles. Discourse is therefore understood as the social usage of language and studied as a social practice and a materialization of social relations. It means that discourse analysts are interested in the social practice of using language to put forward agendas, to express dissent, to defend a position, or to transmit values. They also study acts of silencing and censoringsuch as prohibiting other worldviews from circulating and being heard. Therefore, discourse analysts see language as a series of social processes and they acknowledge that language is not limited to otherwise unrelated individual acts. Discourse analysis could be described as a political understanding of the use of language in the context of unequal access to platforms of decision making, economic resources, and social recognition. As we will attempt to demonstrate throughout this paper, the study of discourse is not limited to looking for hidden agendas, lies or the uttering of meaningless and empty words. Deception is only one of the strategies used to convey worldviews, and it is not necessarily the most effective or even the most interesting for discourse analysts. Some schools of discourse analysis criticize social reproduction of gender inequality, racism and social class. Critical Discourse Analysis is an example of this field. For this school of thought, discourse analysis is the social study of language, its social constraints and its effects [START_REF] Fairclough | Critical Discourse Analysis[END_REF]. Through language, social groups come to represent society in a way that perpetuates domination, positive or negative discrimination, and social repression. Critical discourse analysts look at the perpetuation of social conflicts and unequal relations of power. They examine issues related to gender, sexuality, social class, and ethnicity.

While our presentation of neoliberal discourses and its analysis does not fall totally under the umbrella of the school of Critical Discourse Analysis, this paper demonstrates how to analyse discourse in the context of the study of global inequalities, social discrimination and repression. We are critical of the current state of global politics, economy and society as it reproduces and reinforces inequalities. Therefore, the next section presents a critical analysis of neoliberalism understood as an ideology whose aim is to impose its worldviews and the interest of the actors it attempts to defend and whose interests this ideology is putting forward in the context of development discourses.

Neoliberal ideologies

In October 2014, a press release of the World Trade Organization (WTO)an international institution whose aim is to protect free global trade -declared that "trade has allowed many developing countries to benefit from the opportunities created by emerging new markets, to integrate into the world market through global value chains at lower costs" but added that "developing countries need flexibilities because their economic circumstances can hamper their ability to implement obligations." (World Trade Organization 2014) With declarations such as these, the World Trade Organization attempts to circulate, and ultimately to impose, its worldview. The organization is using language as a political act of transforming society by making ideas circulate to reinforce its ideology and limit the capacity of other worldviews to be heard. Since the organization is powerful, it is very successful at imposing its own terms and reality that give shape to its legal and policymaking actions. Declarations, such as the previous ones, are part of a larger plan to transform societies according to ideologies of neoliberalism. This paper argues that current development thinking and practice are mainly shaped by neoliberal ideologies and that these ideologies are successful at achieving most of their goals through the use of words, language and communication. Neoliberal discourses, understood widely as social usages of language and acts of communication, shape our understanding of development and the desirability of social transformation. We argue that neoliberalism was able to impose its own worldviews through the shaping of development vocabulary and means of communication. For instance, as illustrated in the previous quote, the WTO is representing the world in economic terms. In its language, societies are defined as "economies" and their achievements are measured according to the strength of their markets. This neoliberal ideology is, of course, only but one way to represent the "developing world". Other ideologies, market-focused or not, define countries and achievements with other terms.

For critical social scientists, an ideology is a set of beliefs and doctrines used by a social group or institution in an effort to achieve or to maintain domination over other groups by means other than force and coercion. More generally, ideologies are worldviews or conflicting views about how the world works and how it should be working. Ideologies are being circulated through language and discourse. Development is, in fact, a multifaceted term and many ideologies have shaped how we come to understand it [START_REF] Peet | Theories of Development[END_REF]. It has been influenced by several of the ideologies that marked the second part of twentieth century including modernization, welfare capitalism, socialism and communism.

Neoliberalism is a diverse, yet relatively well-organized, set of ideas that emerged in the 1970s and particularly visible starting in the 1980s with the beginning of the mandates President Ronald Reagan and Prime Minister Margaret Thatcher, respectively in the United States and in Great Britain [START_REF] Harvey | A Brief History of Neoliberalism[END_REF]. It is based on interpretations of the free market philosophies of the 1800s. It should be noted here that there is not a single neoliberal ideology; it should be more appropriate to talk about neoliberal ideologies. Indeed, each institutional actor interprets and advocate for neoliberalism in different ways that relate to the history of the organization and the social, political, economic and cultural context.

As with every ideology, critical social scientists contend that neoliberalism envisages development according to a worldview whose assumptions are difficult to prove (Rist 2002a). At the core of this free market ideology resides the ideas that markets create wealth and that they help the accumulation of capital in forms of money and goods [START_REF] Munck | Neoliberalism and Politics, and the Politics of Neoliberalism[END_REF]. According to these ideologies, markets and accumulation of capital are the most efficient ways to redistribute wealth through all layers of society. Therefore, countries are compared according to their economic growth, which measure that potential. Neoliberal ideologies believe that the state and its bureaucracy is a major obstacle to market efficiency. Consequently, David Harvey describes neoliberalism as a project to "disembed capitalism from its constrains" (Harvey 2005, 11).

Neoliberalism believes that markets and borders should be open to anyone, including and especially foreign companies. Tariffs and protectionists barriers have been taken down and foreign investments have been given priority. It also aggressively advocates for the protection of private property and individual choices, as neoliberalism sees is it as the key motivation for engaging in markets (Clark 2005). Under neoliberal ideologies, engagement in market economies is advertised as a norm, as an expected normal behaviour. In fact, Arturo Escobar describes neoliberalism as a successful "entrenchment of individualism and consumption as cultural norms" (Escobar 2010, 41).

Since the end of the 1980s, free market capitalism has been promoted as the main vehicle of development policies. As demonstrated by Daniel Stedman Jones, in the United States, series of lobbies and think thanks have coordinated their influence to shape consent for neoliberal policies [START_REF] Jones | Masters of the Universe: Hayek, Friedman, and the Birth of Neoliberal Politics[END_REF]. Starting in the 1970-1980s, the "education" of judges, law professors and congressional aides has been supported by lobbies. In major media outlets, influential journalists pertaining to neoliberal think thanks have shaped the acceptability for such policies.

Starting around the same period, important reforms and structural adjustments have been carried in order to transform most "developing" countries into free market economies as their best way on the path to development [START_REF] Gore | The Rise and Fall of the Washington Consensus as a Paradigm for Developing Countries[END_REF]. Under the ideologies of neoliberalism, state bureaucracies have been downsized, sometimes to the bare minimum, healthcare has been privatized and social security significantly reduced when not abolished. In fact, neoliberal ideologies believe that private, for-profit initiatives, and civil society organizations, often faithbased ideological groups, are much better equipped to serve needs of gender equality, health, education, and poverty alleviation [START_REF] Clark | Faith Matters: Faith-Based Organisations, Civil Society and International[END_REF].

Even though neoliberal ideologies have been shaping governance practices and policymaking since the 1980s, it is important to understand that these ideologies have never been fully achieved. In its realization, policymaking made in the name of neoliberalism often contradicts the credos of the ideologies. For instance, countries that are strong advocates of neoliberal global policies are very often subsidizing and protecting their national industrial productions [START_REF] Prasad | The Politics of Free Markets: The Rise of Neoliberal Economic Policies in Britain, France, Germany, and the United States[END_REF]). These policies refute many of the assumptions of neoliberal ideologies.

If we are seeing today a reintroduction of the state as a key player of development policies, most core elements of neoliberalism still remain among the main motivations in development policymaking. And neoliberal ideologies have been quite successful at reintegrating state in their policies without much affecting their beliefs in free market and development through capitalism, industrialisation and global finance.

Neoliberal tales of a super typhoon, a brown woman and the Filipino public service

Let us now to illustrate how neoliberal ideologies are successful at controlling language, at shaping our understanding of development and, ultimately, how they are successful at transforming social relations. This section presents the cases of a discourse about a natural disaster, a super typhoon. The players of this tale are the World Bank, a victimized Filipino woman and the government of the Philippines. This tale reveals why we contend that words are not so different than deeds. We chose to quickly examine some texts, chosen without much organizing principles other than their common topic.

On November 8, 2013, super typhoon Yolanda, also known as typhoon Haiyan, hit the coast of the Philippines. This was no ordinary natural disaster. Since the 1880s, no other typhoon had left such traces of devastation and death. Towns and villages have literally been wiped out, written off the map. It left more than 6000 dead and hundreds of thousands homeless. Local response teams were ill equipped for this disaster and were overwhelmed by the gravity of the situation. In an instant, images of the destruction circulated around the world. International response, to what quickly became a humanitarian crisis, was both immediate and significant. Governments, international organizations, non-governmental organizations and aid agencies quickly channelled money and material support to the Visayas region of the Philippines where the death toll was the highest.

On October 24, 2014, one year after super typhoon Yolanda, the World Bank, some foreign aid agencies and the government of the Philippines released a joint statement to the press. This press release argued that, in the aftermath of the typhoon, to prevent such disasters from happening again more transparency in government spending was needed, especially in climate change public expenditures (World Bank 2014a). In that press release, the World Bank calls for stronger institutional review processes, as they will help the Philippines to strengthen its resilience against climate change impacts.

Social science questions pertaining to the cases of super typhoon Yolanda are as numerous and rich as is the imagination of the social scientist studying it. In this paper, we are calling development social scientists to open their imagination to discourse beyond the commonly-held ideas. As we will attempt to demonstrate with this example, acts of communication and language mold our perceptions of development problems and, consequently affect how we perceive the available solutions to these problems. Neoliberalism, as an ideology, is framing the way we see "reality" and is consequently successful at imposing its diagnosis about social problems and the adequate response needed.

In the case of quoted press release about super typhoon Yolanda, the World Bank is using halftruths in regards to an event that impacted the Philippines to move the focus of attention. For instance, the World Bank is calling for more transparency in government spending, not something for which there are many detractors. It is very well possible that more transparency might strengthen the resilience of the government of the Philippines against forthcoming natural disasters. However, the World Bank fails to tell us many things. To name only but a few, the World Bank neglects to tell us that several years of neoliberal policies in the Philippines have destroyed the capacity of the government to provide the people with adequate public services [START_REF] Bello | Neoliberalism as Hegemonic Ideology in the Philippines[END_REF]). In the Philippines, public service and local government services have been purposely shattered according to a neoliberal ideology carried, in part, by the World Bank. One could argue in fact that a well working government with enough local public servants would be able to respond very quickly to disasters and help evacuation. It could also be argued that local public service might be more efficient at organizing response where there are very few customers to pay for a similar private service. But according to a neoliberal ideology such an adequate government would need to inflate the number of public servants. In its press releases, the World Bank is saying the government is not efficient enough, but it is in fact very careful at not requesting for more public service. It is calling for more efficiency of an already very reduced public service.

The World Bank also fails to tell us that, as many scientists have warned, the neoliberal capitalist policies of industrialization and deregulation might have, in part, accelerated climate change [START_REF] Andrew | Carbon tax: Challenging neoliberal solutions to climate change[END_REF]. To many, the link between neoliberal ideologies and a super typhoon hitting the Philippines might appear as a stretch. However, it could be argued that there seems to be clear evidence that the increased intensity of typhoons is related to climate change for which there is enough to link with capitalism and industrialization. So we can claim that, as in the case of super typhoon Yolanda, the World Bank is carefully choosing what and how to say it. The World Bank is successfully moving our attention away from several important questions to another aspect that concurs perfectly with core elements of its ideology. But discourse analysis is not only about revealing "smoke and mirrors" it is also about understanding how text, speech, and image mold the perception of problems and their available solutions. As such, the World Bank is using the language of organizational efficiency, inherited from financial capitalism, to describe goals of development. If there are not many people to argue against efficiency, the development of infrastructure of public service in the Philippines is not limited to that dimension. It also needs proper investments and to be discussed in terms other than profits and clientele. By using an economic vocabulary, the World Bank is effective at transforming our representation of the world problems in its own terms. As we have argued earlier, an ideological discourse does not necessary lie, it just needs to attract attention to elements favouring its worldviews and in its own words. And very often, it does not do it to conspire. Neoliberalism is so strongly convinced it is the only possible way that it does not even contemplate there can be alternatives.

In another document entitled "Philippines: A Lesson that Helped Save Lives" (World Bank 2014b), the World Bank uses a different rhetoric. In this case, it is using a discursive strategy appealing to emotions in order to persuade called pathos. In such a discursive strategy, the emitter of the discourse will attempt to personalize a tragedy by giving it a face, an embodiment, a name. More often than not, the name or the embodiment of the tragedy is presented in sufficiently general terms that the reader sees beyond the individual's social specifics. Here, in this case, the World Bank is using the image of a "helpless victimized brown woman" that needs international intervention. With plenty of details, the World Bank paints a picturesque portrait of the town of Daanbantayan in the Cebu province said to a bucolic village full of colourful flowers in bloom. A housewife named "Heidi" is testifying of the help the World Bank provided her. According to Heidi, social programs helped her buying new pairs of slippers for her children. The organization seems to have helped the housewife better her life condition after the super typhoon in two ways: by implementing Conditional Cash Transfers (CCT) and by providing education in the form of family development sessions. But the document reads more than simply a tale about "how lives were saved", it tells about the World Bank discursive strategy to reinforce its larger designs. It is using Yolanda and a tearjerker or "feel good" story about a victimized brown woman as opportunities to promote a controversial program and claim supposed success of cash transfer programs. And therefore, it attempts to reinforce the legitimacy of its actions.

The CCT, that Heidi declares having benefited from, is Pantawid Pamilyang Pilipino Program. It is one of the largest programs in the world after Brazil and Mexico [START_REF] Albert | The costs and benefits of Pantawid Pamilya. There is growing evidence showing positive outcomes for the government's conditional cash transfer[END_REF]). Its goal is to have an incidence on poverty and child labor. In various ways, those who receive aid have earned it or who meet certain criteria. Criteria include actively engaging in providing education and health services to children. This falls under many of the assumptions of neoliberal ideologes such as efficiency and individualism. However, studies have contended that Pantawid Pamilyang Pilipino Program did not affect the incidence of child labor while only reducing the amount of paid work (Philippines Institute for Development Studies 2014). Discourse analysts might say that indeed, the World Bank is using opportunities of the agenda to produce textual weight to the supposed success by abstracting facts. While it is doing it, it is not telling lies. Discourses should therefore not only being studied in their relation to a perceived material empirical reality. Studying how the emitter supports its ideology is at least as interesting for the discourse scholar.

In fact, discourses are also about what the emitter abstracts, what he or she choose not to say or not to present. The quoted document does not in fact really talk about what was announced in its title. It does not really talk about how lives would have been saved thanks to policies that should have been implemented before Yolanda. Rather, the document claiming to tell about how lives were saved talks more about post-disaster recovery. Strangely enough, reading the document quoting the story of Heidi, one wonders why the World Bank would produce it. The World Bank is hardly citing itself and its actions. Would it be possible that the organization does it only to keep occupying means of communication with tales supporting its ideology? The organization must constantly struggle to have its worldviews circulate. Because of its prestige and resources, the World Bank is more successful than others.

But many other questions come to mind. Such as, why would an organization spend resources to promote a program it almost does not take credit for? The development scholar Emma Lynn Dadap [START_REF] Dadap | The Politics of Conditional Cash Transfers of the Philippines[END_REF] helps us understand better. She tells us that:

The World Bank and the ADB narrative has always been that Pantawid Pamilya is governmentowned as explained by the fact that both institutions' participation in the program is limited to providing partial financing (i.e. through loans) and technical assistance. The term -governmentowned-, however, is a tricky one. It is possible that the government assumes overall control of design and implementation, but was led by some external pressures to decide for the adoption of the program.

To that aspect, the document is very successful. The World Bank appears to be only an actor in the shadow. While celebrating the Government of the Philippines, it uses Yolanda to tell about its ideology, including its goal to disseminate "Social protection programs" as both a basic right and a social investment [START_REF] Merrien | Social Protection as Development Policy: A New International Agenda for Action[END_REF]. It artificially inflates the number of supporters for its ideology, making it appear as if it is universally supporter.

Discourses are not just about uttering meaningless words. They are about choosing vocabulary and facts, about using discursive strategies in order to reinforce their difficult to prove credos. Therefore, this document, and others are elements of an ongoing struggle to circulate neoliberal ideologies and actively transform the world. They are social acts.

The neoliberal institutions of development need to justify their radical transformation of societies and economies in their own terms. One way to achieve that is to ensure that social groups desire that transformation. It is less costly in political and economic terms than the use of force and physical coercion. Through discursive strategies, such institutions represent the "developing" world in terms of constant failures and crisis. A crisis commands an immediate response and responses are shaped by the initial diagnosis. Very often, "developing" societies come to see themselves according to such ideological discourses and are willingly enabling neoliberal solutions change their societies and economies.

The ideologies of neoliberalism, as presented in declarations of international organizations such as the World Bank or the World Health Organization, are very successful at shaping the way responses to "underdevelopment" are perceived. For instance, when the World Bank declares that the response to natural disasters is more effectiveness in public service and aid allocation, it is framing "development" problems and is actively transforming societies according to a neoliberal worldview. Its vocabulary, a limited and ideological representation of the world, a specific "problematization" of the Philippines provides the basis in which policymaking is designed. We could therefore argue that its words are in fact deeds. Neoliberal discourse is a social action part of a larger plan to frame the perceived failures of the "developing" world in terms of free markets, competitive economy and individualism.

Further examinations of development discourses

Now that we have illustrated some elements of the analysis of neoliberal discourses by examining a few aspects of the discourses of the World Bank, let us expand our understanding of development discourses further to other institutions. The World Bank is one of many, and perhaps, one of the most visible institution supporting neoliberal ideologies. In this section we will also expand our understanding to ideologies other than simply those of neoliberalism. As we have argued earlier, neoliberalism provides the main ideology of development policymaking and practice. But contemporary development discourses are not strictly limited to neoliberal ideologies. This section attempts to exemplify how discourse analysts could study "development" by the examination of words, language and acts of communication. This section takes some examples from the literature; some others are derived from our own reflections. These examples might identify an ideology, others not. But all examples demonstrate how critical social scientists should study institutional discourses and ideologies in the context of international development. This section is certainly not exhaustive; it aims at giving multiple varied examples to sparkle the imagination of the aspiring development social scientist.

Development discourses can be examined as carriers of ethnical and cultural biases. Scholars have examined biased or exaggerated representations of the "developing" world as regions stricken with poverty and conflict. Called "poverty porn", these representations of the "developing" world are mostly used to raise funds or concern [START_REF] Plewes | The Pornography of Poverty: A Cautionary Fundraising Tale[END_REF]. "Poverty porn" uses striking images, sometimes of distressed and dirty adults and sometimes of crying children. Around Christmas time, non-governmental organizations (NGO) such as World Vision diffuse representations of poverty in the mass media in order to sell their products to concerned customers in the "North" who look for an easy way to help foster good education for children in need and then buy "peace of mind" from the comfort of their homes [START_REF] Jefferess | For sale-peace of mind: (neo)-colonial discourse and the commodification of Third World poverty in World Vision's 'Telethons[END_REF]. But beyond the highly questionable objectification of populations in "developing" countries, the omnipresent images of poverty frame one's perception and reinforce clichés that strengthen colonial discourses and interventions. After the global success of the 2008 movie Slumdog Millionaire, which featured the implausibly rapid social ascent of a young man born in conditions of extreme vulnerability, some scholars criticized the perpetuation of clichés about poverty in the Indian Subcontinent [START_REF] Sengupta | A Million Dollar Exit from the Anarchic Slum-world: Slumdog Millionaire's Hollow Idioms of Social Justice[END_REF]. For many people outside India, the movie was one of the very few, if not the only, sources of information about the cast system. It has a potential for advocacy, but its representation of India reinforced biases toward Western social advancement, and therefore toward a Western-oriented definition of development.

Development discourses can be examined as elements of story telling. Swiss sociologist Gilbert Rist contends that many of the policymaking in development is based on an idea of development that is specific to a cultural and historical context (Rist 2002b). He contends that the definition we have adopted for "development" is in fact specific to Western society. The social model development policymaking is attempting to reproduce cannot be achieved outside of the context that led it to appear in the first time. Rist goes further, declaring that development policies and practices try to reproduce facts have never actually existed in the first place. According to him, development discourses are therefore often based on historically non-factual elements. Consequently, development discourses must be considered as fables whose objectives are to transmit messages about the values and ideologies of international organizations.

Development discourses can be examined as specific stylistic productions. For instance, development policymaking is said to have a specific language that perpetuates relations of power (Athorpe 1997). U.S. anthropologist James Ferguson showed that development institutions produce their own literary genre and create a vision of the world that allows them to justify the spending of budgets received from governments in the name of development [START_REF] Ferguson | The Anti-politics Machine: Development, Depoliticization and Bureaucratic Power in Lesotho[END_REF]. But, language can also be used to help social mobilization. The stylistic productions of international organizations are very different than of social activism. If words can help push a neoliberal agenda, they can also help to advance social causes and empower social groups. But U.K. and U.S. feminist social anthropologists Rosalind Eyben and Rebecca Napier-Moore have advised us to use vocabulary with great care in the context of pushing development agendas [START_REF] Eyben | Choosing Words with Care? Shifting Meanings of Women's Empowerment in International Development[END_REF]. Language of social activism should be careful not to fall into the stylistic clichés of the genre. As this paper argues, discourses frame policymaking and practices. Social action made in the name of gender inequality is different than that of free markets. Both are making claims about the urgency of the social problems they have identified. Arguments used will give shape to the response. Different organizations have different styles in their argumentations. While some are using figures to convince, others are using pictures. Using figures gives the impression that statistics and numbers presented are a factual reality. Such strategy is usually very effective at appealing government actors. That is what the World Health Organization is doing in the examples shown earlier. On the other hand, using pictures of poverty is usually very effective at appealing individual donors and raise the concerns of social activists.

That is what many NGOs do for the reason that, most of the time, the audience they are addressing to is different.

Development discourses can be examined as elements of how organizations socialize or interact with each other. One way to understand the annual reports of the World Bank or the United Nations Development Programme (UNDP) is by studying the internal meaning that can be accessed by reading and analyzing the documents. An analyst could state that the text of these reports is a method used by these organizations to describe their achievements, influence policymaking or criticize the actions of others. But another way to understand these documents is to give them a meaning beyond words. These annual reports are part of institutional relationships and of positive or negative behavior reinforcement. These institutions tell governments that they did well in implementing the right policies. But these organizations are part of a larger endeavor of producing text, and they occupy a global communicative space. These reports, almost regardless of their content, can be understood as ways for organizations to socialize, for them to keep the communication channels open and constantly reinforce their positions.

Development discourses can be examined as the production of concerted action. Through the use of language, international organizations produce concerted action and reaction. The production of documents helps various organizations to work together. Sometimes, the words laid in the document do not make much sense, but these words have to be understood as the result of collective actions. Intergovernmental summits and "high-level" fora gather hundreds of civil servants, members of civil society organizations and scholars [START_REF] Pianta | UN World Summits and Civil Society: The State of the Art[END_REF]. A public statement is usually produced at the end of the summits and fora. Various actors spend the entire time of the forum lobbying heavily to ensure that their interpretations of past or current events, as well as their particular ways to frame future actions, are included in the final declaration. Governments inevitably react to these declarations in one way or another. They publicly declare their rejection, state their doubts or express their praise. These reactions will frame possible changes in organizations, policies and practices. Regardless of the actual will of the organizations and the governments to be faithful to their words, the final declaration is the result of acts of language, and this social act in turn conditions other forthcoming social acts.

Development discourses can be examined as fluxes of ideas. Examinations of how discourses change over time might study the trajectories of buzzwords. Such examination have, for instance, focused on how trendy words have moved from one environment to another and changed their meaning [START_REF] Cornwall | What do Buzzwords do for Development Policy? A critical Look at 'Participation', 'Empowerment' and 'Poverty Reduction[END_REF]. The study of buzzwords considers discourse as a social practice, since these words dictate the terms of the debate, frame the discussion and limit the range of possible solutions. Throughout the years, if the core arguments of neoliberalism remain the same, on the surface, using buzzwords allows them to appear to be changing very often. This allows neoliberal ideologies to appear to be constantly changing and adapting. Additionally, these buzzwords condition money allocation, they dictate institutional and social changes. When the World Bank declares that the Philippines' problems are government efficiency which must now be understood in terms of review processes it is setting a new agenda. Consequently, the government of the Philippines might have to allocate resources. It is likely that a position or an office of "review processes" will be created. If the core argument of government efficiency remains, on the surface, it appears that the World Bank is constantly reacting and adjusting. These constantly renewed buzzwords could also testify of the capacity of neoliberal ideologies to integrate and neutralize criticism. Neoliberal organizations such as the World Bank, which are critical of the ecological discourses, are known to have successfully neutralized environmental concerns by adopting a "green" vocabulary while rejecting its more larger critical implications of the industrial capitalist "development" model [START_REF] Goodman | The 'Green Economy': Class Hegemony and Counter-Hegemony[END_REF].

Development discourses can be examined as containers of ideas as they are reactions to other people's ideas. If the fluidity of the language can be examined as demonstrated in the previous paragraph, discourse can also be examined as interactions containing many traces of other texts and speeches by others. This presence of other elements of communicationsometimes explicit, sometimes implicit -is called "intertextuality". Development discourses very often contain reactions of the authors to other texts and discourses. Sometimes this reaction is done in the form of a direct quotation; sometimes it just comes as a hint. Intertextuality is not limited to international organizations. It is also often used by social activists in their reaction to neoliberal ideologies. For instance, in 2011, the international non-governmental organization Oxfam launched the "Grow" campaign. Using a word similar to "growth", which is promoted by development banks and economic development institutions, the NGO proposed another understanding of development by the appropriation of a word that is popular in the field of economic development. The discourse of Oxfam therefore contains ideas of others, in the form of acknowledgment, or in this case, in the form of criticism.

Development discourses can be examined as elements of the creation and justification of social hierarchy. Through the use of words, groups come to claim their superiority and justify their domination. Once well established, the domination is accepted as being natural, as being the only possible way. Development discourses contain many forms of domination. A look at documents produced by international organizations may show biases related to gender, sexuality, social class, and ethnicity. For instance, in photographs of annual reports, men can constantly be shown to be holders of scientific knowledge and mastering machinery while women are shown to take care of children and food. In the same photographs, white persons can be shown standing or driving cars while other ethnicities are shown sitting on the floor or riding old bicycles. Other biases exist in development discourses such as representations of human superiority over other biological species. For instance, discourse analysts can contend that development discourses use language to justify transforming nature into a "resource" that can be exploited for the exclusive benefit of humans. Indeed, regarding non-human species, animals or otherwise, as well as rocks and minerals as part of the "environment" creates social and cognitive conditioninga divide between humans and naturejustifying human ownership of nature and its exploitation for human benefit. Development discourses can be examined as strategies of reinforcing domination. Discourses showing important cultures and development differences between "developed" and "developing" or "underdeveloped" societies are very common. This precedes neoliberal ideology. In fact, these strategies appear in documentation produced by colonial administrators, in the accounts of explorers-adventurers, in literature and science in Europe and elsewhere. For instance, according to US-Palestinian literary scholar Edward Said, representations of the Orient in 19 th century French literature as exotic and fundamentally different from civilized France contributed to the general public's acceptance and support of large-scale economic and military interventions [START_REF] Said | Orientalism[END_REF]. But these strategies also appear elsewhere, within other imperialist endeavors. According to U.S. anthropologist James C. Scott, the Chinese Qing dynasty used cartography, literature and art to justify the civilizing of "barbaric" hill tribes of South East Asia [START_REF] Scott | The Art of Not Being Governed: An Anarchist History of Upland Southeast Asia[END_REF]. Scott shows that these development discourses constructed images of indigenous peoples in such ways that it would be desired to intervene and transform societies and ultimately to control them. U.S.-Colombian anthropologist Arturo Escobar makes a similar argument when he states that the poverty of the "developing" countries is actually an invention [START_REF] Escobar | Encountering Development: The Making and Unmaking of the Third World[END_REF]. Development discourses, in particular those of modernization and neoliberalism, are representing "developing" societies in terms specific to capitalism, as global economy needs their participation. Societies only become "poor" when compared and dominated by Western economies. Poverty can be understood as a capitalist-centered representation. Escobar argues that once self-represented as poor, those countries wilfully participated in global economy where they accepted to be dominated.

Development discourses can be examined as they are carried through figures and statistics. In this paper, we have focused mainly on the analysis of words and texts, but we could also see figures and statistics as discourses. Discourse analysis is helping opening fields of study into the interpretation of the causes of underdevelopment [START_REF] Faille | Discourse Analysis in International Development Studies: Mapping Some Contemporary Contributions[END_REF]. Discourse analysts need to question concepts, ideas, vocabularies and narratives dealing with the "developing" world. Development statistics, for instance, are a product of communication and can be examined as an ideological representation of society. They are a narrative or an act of communication that carries a meaning and a purpose. They are produced in an environment of interaction between institutions and of disagreements on methodology and on what to measure (Della Faille 2013). In that sense, they are never neutral. Measures such as the Gross National Product, the Human Development Index, and even the World Values Survey are contentious. As a consequence, discourse analysts should criticize the working definitions of statistics. Education statistics, for example, might require schoolmasters to document the number of pupils in each class. But the definition of a pupil is problematic. How do you define a pupil ? Is it somebody who attends school every once in a while or regularly ? Is a pupil somebody who passes exams ? How do you define levels of education ? By age category ? And are female and male students included in the same categories ? Without necessarily challenging the material conditions of education in "developing" societies, discourse analysts may say that what we think we know is wrongly formulated and ill conceived. One could say that problems of education must be solved by more and better infrastructure. But there may also be a need to reconsider our formulation of the problems, and to look closely at our biases and ideology. We must be aware of the limitations of our measurement tools. Solving a "development" problem is not only a question of perception; it also requires a critical understanding of how we investigate problems and how we know about reality.

Development discourses can be examined as they are carried through artefacts and objects. If we were to push the definition of language further than words, texts, images and figures, we could also see in cultural products, objects, art, and artefacts elements of a communication system, and therefore conceive them as they are part of a social act aimed at creating meaning. For instance, in the "developing" world, the omnipresent Land Cruiser 4X4 used by many international organizations must also be understood as elements of social interaction embedded in an effort to carry meaning [START_REF] Mills | The Postmodern Tank of the Humanitarian International[END_REF]. Those vehicles are a status symbol for those driving or being driven, but they are also a symbol of the supposed technical and financial superiority of organizations using them in an environment of scarce imported goods transiting through a global market economy. This section has exemplified various focuses of the analysis of discourse as it applies to the study of "development". There are many other focuses and levels of discourse analysis. Fortunately, discourse analysts must not necessarily investigate all of these levels. The goal of this section was to illustrate some of the most common perspectives of discourse analysis. Now, before concluding, let us briefly introduce some ethical considerations.

Overview of methodological approaches to discourses

The methodological approaches to the analysis of discourses are multiple. In the context of the sociological study of underdevelopment and so-called developing societies, a number of methodologies will grant the analyst access to language as a social fact. This section attempts to demonstrate that there are many methodologies in discourse analysis; some are more formal and more encompassing than what we have inelegantly demonstrated with the example of the analysis of documents about super typhoon Yolanda.

Discourse analysis is characterized by the same dynamics and methodological reasoning as other fields of social research. Before presenting some of the major methodological tools of discourse analyses, it is important to stress that discourse analysis is not a unified field of research and analysts examining discourses about underdevelopment have adopted various approaches and methodologies to study speech, text and images (Della Faille 2011).

The three major epistemological approaches to research found in social sciences are also found in discourse analysis. First, some analysts apply a systematic reading based on preconceived ideas and knowledge. They will then attempt to verify general intuitions. This is the case of deductive reasoning. Second, others will attempt to test a specific formal hypothesis. Throughout the research, this analyst will attempt to prove or disprove her or his hypothesis built, for instance, from the results of previous research or from a review of scientific literature. This is the method of hypothetico-deductive reasoning. Positivist research usually falls within this approach. Positivist researchers believe that their experience is independent from scientific observation and consider societies a series of neutrally observable facts. Third, still other analysts will let text, speech, and image reveal themselves and unfold. This analyst then examines them and hopes to slowly come to understand social dynamics. This is the method of inductive reasoning. One of the examples of inductive reasoning most familiar to qualitative sociologists is the Grounded Theory, which allows explanatory models to emerge from analysis [START_REF] Glaser | Basics of Grounded Theory Analysis[END_REF]. As these approaches testify, researching language and discourse is truly synchronized with major methodological developments in social science research.

The objectives and level of methodological formality also vary. Qualitative, quantitative or hybrid analyses can be applied to the study of language. This section does not focus on data collection, but rather, on the methodologies chosen once most of the data has already been collected in the form of a corpus. A corpus is an ensemble of texts, speech, and image gathered under an organizing principle or a research hypothesis. The corpus should not be seen as a fixed series of data. There are several instances where data is incorporated continually throughout the analysis. Many language and discourse analysts will thus recognize that data collection is not totally distinct from analysis.

The methodologies of language and discourse analyses as they apply to the study of underdevelopment and so-called developing societies are diversified. Among the most popular are the analyses of texts, conversations, and speech, more often than not "transcribed speech" rather than in its audible form. Texts and transcribed speech can originate from various sources. To name but a few, analysts have examined texts produced by governments, politicians, international organizations, non-governmental organizations, activists, scholars, scientists, writers, poets, song makers, and others. Analysts study corpora, some with the assistance of computer software that helps classify text or code, or attributes a sociological or semantic category to text segments, sentences or words that are then compared against an analytical grid. Computer-assisted qualitative data analysis software (CAQDAS) has been popular for many years among content and discourse analysts. Some CAQDAS are open-source or freely available software developed by a research community. They allow for the color-coding of texts and transcribed speech and the creation of categories that can be hierarchized. Other types of software that have created interest since the 2000's are data mining software, which allow for rapid analysis of textual corpora, but with less control over the parameters and less freedom for interpretation than CAQDAS. Additionally, the quantitative study of texts thanks to specialized software packages, often called "statistical analysis of textual data" is a vibrant academic community. Other analysts, less prone to the use of computers, will take notes directly in their word processing software. Although access to text has been greatly improved by the widespread use of the Internet, there is a great wealth of documents yet to be analyzed outside of electronic databases and the Internet. Notwithstanding conversations gathered by analysts and texts produced during interactions as part of the research, the sources of second-hand texts are almost unlimited. This easy access is tempting and, very unfortunately, it is not uncommon for analysts to be too focused on easily accessible text or transcribed speech. Some of the other methodologies analysts are using include visual sociology, studying images, photography, and the analysis of feature films. Photographs can be taken during fieldwork to emphasize the analyst's visual observations. For instance, in their analyses, some scholars will include comments on photographs made by research subjects. Photos, scanned magazines and images can be analyzed thanks to computer software enabling organization of data and annotations. Some sociologists also consider the communication of their results through the production of images, complex interactive graphics or video blogs (vlogs) as an element of their methodology. The communication of early results through these channels allows sociologists to gather reactions from research subjects. In a feedback or iterative process, these reactions are then reinserted into the research material. Other sociologists analyze recorded conversations directly within audio software. Annotations are then applied to the computer files, often directly in relation to wave forms.

Some ethical considerations

After this too brief overview of methodological approaches to discourse analysis and before concluding, we would like to raise some of the many ethical questions that social research on development issues raises. Some concerns are directly related to the specificities of the study of language and discourse, while others are more general to social science. Some analysts in the "developing" world and elsewhere, have expressed concerns about imperialist tendencies and the ethnic, class, and gender biases of some of the methodologies of social science [START_REF] Connell | Southern Theory. The Global Dynamics of Knowledge in Social Science[END_REF][START_REF] Denzin | Handbook of Critical and Indigenous Methodologies[END_REF]. In essence, social science methodologies are said to create artificial divides, to disregard continuities, to overlook hybridity, and to be blind to the complexity of social relations. Some contend that the methodologies of discourse analysis extract social facts from their context of production and consider society as a series of disjointed facts to be measured, quantified and manipulated. This is a very reasonable criticism, as it pushes us to reflect on the goal of research. Do we want a society guided by the impersonal, analytical approach of science? Or do we envisage another role for social science? Some of these concerns emerge from academia, but indigenous communities and some marginalized or subaltern groups have also voiced reservations. Major research councils now have specific research guidelines on free, prior and informed consent [START_REF] Rosenthal | Politics, Culture, and Governance in the Development of Prior Informed Consent in Indigenous Communities[END_REF]. Some years ago already, calls for the decolonization of methodologies have been voiced [START_REF] Smith | Decolonizing Methodologies, Research and Indigenous Peoples[END_REF]. Participative approaches involving studied groups and communities in the definition of the tools, the research objectives and the means of communicating results have been increasingly adopted, especially in indigenous studies. Projects that integrate mythologies and dreams as jump boards for research are emerging. But some contend research is still highly biased towards a Western or Euro-specific comprehensive approach. Decolonizing language and discourse analysis means more than involving communities. It means that the researcher must consider the overall colonial character of the research relationship and favor indigenous ways of accessing knowledge [START_REF] Kovach | Indigenous Methodologies: Characteristics, Conversations, and Context[END_REF].

How this can decolonizing of language and research can be reconciled with the analysis of discourse is still to be explored in a convincing way. But we have to consider that critical discourse analysis is only but one explanation of social relations and one among many competing representations of the world.

Conclusion

This paper has focused on neoliberal discourses of development. Underlying in our understanding of discourse, we have argued that every aspect of "development" is based on interpretation and the framework provided by language. Every social actor, whether individual or institutional, interacts by using language, attempting to produce meaning, to transmit a message through text, speech, image and objects. We have also argued that development discourses are limited representation of the world. They are ever neutral. Groups tend to use them in ways specific to their own history, ideology, values, political and economic systems. We have attempted to illustrate that language is as much a social act as are riding a bicycle, posting a letter, applying for a position or patting somebody on the back. Language is a tool to organize social facts, and is embedded in social action.

Social problems are nearly unlimited in number, but only a few are actually brought to light and generate concern. And, as U.S. sociologist Joel Best has stated, social problems compete for attention [START_REF] Best | Social Problems[END_REF]. The fact that a problem is successfully brought forward for public discussion is the result of persuasion in strategies, power dynamics, and even circumstantial chance (Author). Natural disasters are also worthy of investigation for discourse analysts. Although natural catastrophes do exist outside of human language, their effects and our reaction to them are not independent of interpretation. Through language, we assess a situation, analyze the causes, debate the importance of acting, and suggest ways to implement social response and organize social action. It is through language that we come to understand our experience, describe natural disasters and learn about what we have not experienced first-hand. Once they are experienced, represented and interpreted through language, droughts, tsunamis, typhoons, and earthquakes become social facts.

In its attempt to operate a radical transformation of "developing" societies and economies, the ideologies of neoliberalism are using language to establish a diagnosis, to justify action and to make sure the solutions are implemented. For these reasons, words are part of a contentious action. As we have seen throughout this paper, language and discourse analyses offer subtle ways to interpret development and underdevelopment.

As a discipline of the social sciences, discourse analysis conceptualizes social problems through the examination of language as a social act, the production of meaning, and the perpetuation of worldviews and ideologies. When placed in the context of social domination and limited access to resources, it helps the researcher to challenge representations of underdevelopment at home or abroad. Discourse analysis grants the scholar privileged access to the production of meaning, its materiality and its social consequences. It also offers the activist and the development practitioner grounds for the critical understanding of action and its limits, and provides lines for future interventions in social transformation.


Colin Riba
email: colin.riba@sophia.inria.fr

Union of Reducibility Candidates for Orthogonal Constructor Rewriting

We revisit Girard's reducibility candidates by proposing a general of the notion of neutral terms. They are the terms which do not interact with some contexts called elimination contexts. We apply this framework to constructor rewriting, and show that for orthogonal constructor rewriting, Girard's reducibility candidates are stable by union.

Introduction

The most flexible termination proof methods for various extensions of typed λ-calculi use type interpretations [2,5,[START_REF] Blanqui | Inductive-Data-Types Systems[END_REF][START_REF] Blanqui | Combining Typing and Size Constraints for Checking the Termination of Higher-Order Conditional Rewrite Systems[END_REF]16]. Among them we distinguish three families: Girard's reducibility candidates [START_REF] Girard | Interprétation Fonctionnelle et Élimination des Coupures de l'Arithmétique d[END_REF], Tait's saturated sets [START_REF] Tait | A Realizability Interpretation of the Theory of Species[END_REF], and interpretations based on biorthogonality [START_REF] Girard | Linear Logic[END_REF]16]. An interesting way to compare different type interpretations is to study their stability by union. This is even a necessary property in some cases [2, [START_REF] Blanqui | Combining Typing and Size Constraints for Checking the Termination of Higher-Order Conditional Rewrite Systems[END_REF][START_REF] Tatsuta | Simple Saturated Sets for Disjunction and Second-Order Existential Quantification[END_REF].

This paper concerns the extension of the simply-typed λ-calculus with constructor rewriting. We are not interested in termination criteria by themselves, but by the investigation of the closure properties of types interpretations that allow to formulate different termination criteria. We focus on Girard's reducibility candidates and their stability by union.

We give a generalization of the notion of neutral terms that allows to define Girard's reducibility candidates in a generic way. Neutral terms are the terms that do not interact with some contexts called elimination contexts. Terms which are not neutral are observable since they interact with some elimination contexts. We call them values. We instantiate this framework with constructor rewriting. In order to get interesting values, we use elimination contexts with destructors to eliminate the constructors.

Next, we study the question of stability by union. By instantiating the condition of [START_REF] Riba | On the Stability by Union of Reducibility Candidates[END_REF], we show that reducibility candidates are stable by union for orthogonal constructor rewriting. The proof uses a result of [START_REF] Khasidashvili | Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems[END_REF] on the existence of external redexes for orthogonal CCERSs.

The paper is organized as follows. We expose our notations in Sec. 2.

⋆ 2004 route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France.

Section 3 presents a general definition of reducibility families and type interpretations. We apply it by briefly discussing Tait's saturated sets for the pure λ-calculus and one possible extension to deal with rewriting.

Section 4 is devoted to Girard's reducibility candidates, of which we suggest a generalization in Sec. 4.1. We instantiate it in Sec. 4.2 to the framework of λ-calculus plus constructor rewriting.

We then discuss stability by union in Sec. 5. We first briefly present the key problems and known results. Section 5.1 recalls a necessary and sufficient condition for the stability by union of Girard's candidates. In Sec. 5.2, we show that this condition is met with orthogonal constructor rewriting.

We assume familiarity with typed λ-calculus [4], reducibility [9, 14] and rewriting [START_REF]Term Rewriting Systems[END_REF]. Concerning CCERSs, we refer to [START_REF] Glauert | Expression Reduction Systems and Extensions: An Overview. In Processes, Terms and Cycles: Steps to the Road of Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday[END_REF]. The paper (except Sec. 5.2) is based on parts of the Phd thesis of the author [START_REF] Riba | Definitions par Réécriture dans le λ-Calcul: Confluence, Réductibilité et Typage[END_REF] (in French).

Simply Typed λ-Calculus with Constructor Rewriting

Given a set A, a denotes a finite sequence of elements of A of length | a|.

Terms and Types. We consider λ-terms with uncurried symbols f in a signature Σ = (Σ n ) n∈N and variables x ∈ X :

t, u ∈ Λ(Σ) ::= x | λx.t | t u | f(t 1 , . . . , t n ) ,
where f ∈ Σ n . Let Λ be the set of pure λ-terms Λ(∅). A substitution is a function σ : X → Λ(Σ) of finite domain. The capture avoiding application of σ to the term t is written tσ or t[σ(x 1 )/x 1 , . . . , σ(x n )/x n ] if Dom(σ) = {x 1 , . . . , x n }.

Given base types B ∈ B, simple types are defined as usual:

T, U ∈ T (B) ::= B | U ⇒ T .
Typing contexts are functions Γ of finite domain from variables to types, written x 1 : T 1 , . . . , x n : T n . Given a type assignment τ : Π n∈N .Σ n → T (B) n+1 , the typing relation Γ ⊢ τ t : T is inductively defined by the following rules:

(Ax) Γ, x : T ⊢ τ x : T (Symb) Γ ⊢ τ t 1 : T 1 . . . Γ ⊢ τ t n : T n Γ ⊢ τ f(t 1 , . . . , t n ) : T τ(f) = (T 1 , . . . , T n , T ) (⇒ I) Γ, x : U ⊢ τ t : T Γ ⊢ τ λx.t : U ⇒ T (⇒ E) Γ ⊢ τ t : U ⇒ T Γ ⊢ τ u : U Γ ⊢ τ t u : T Constructor Rewriting. Assume given a set C ⊆ Σ of constructor symbols c of type ( T , B) with B ∈ B.
For normalization, B must occur only at positive positions in T [START_REF] Mendler | Recursive Types and Type Constraints in Second Order Lambda-Calculus[END_REF]. As we are not interested in strong normalization conditions, we do not care of this restriction here. A constructor rewrite system (or rewrite system) on C is a set R of rewrite rules f( l) → R r such that r ∈ Λ(Σ), FV(r) ⊆ FV(f( l)), f ∈ Σ\C (defined symbols are not constructors) and l are terms of the grammar

p ::= x | c(p 1 , . . . , p n ) ,
where c ∈ C (hence, l are patterns).

A rewrite system R is typed if for each rewrite rule f( l) → R r with τ(f) = ( T , T ), there exists a (necessarily unique) context Γ with Dom(Γ ) = FV(f( l)) such that Γ ⊢ τ f( l) : T and Γ ⊢ τ r : T .

Example 2.1. We consider the type Nat of Peano's numbers, with constructors 0 : Nat and S : (Nat, Nat). The following system, defining addition, is a constructor rewrite system: plus(x, 0) → x plus(x, S(y)) → plus(S(x), y) .

Reductions. A rewrite relation is a binary relation

→ R ⊆(Λ(Σ) \ X ) × Λ(Σ) which is stable by contexts and substitutions. Let (t) R = def {v | t → R v} and write (t 1 , . . . , t n ) → R (u 1 , . . . , u n ) for the product extension of → R . A term t is R-reducible (or reducible) if (t) R = ∅.
We denote by SN R the set of strongly normalizing terms for → R . Note that it is the smallest set of terms such that ∀t.

(∀u. t → R u =⇒ u ∈ SN R ) =⇒ t ∈ SN R .
Given a constructor rewrite system R, we let → βR be the smallest rewrite relation on Λ(Σ) containing → R and β-reduction:

(λx.t)u → β t[u/x].

Reducibility Families

In this section, we present a general notion of reducibility family and of type interpretation. We then briefly take a look at their instantiation to deal with the pure λ-calculus and with the combination of λ-calculus with rewriting.

Definition 3.1. Let → R be a rewrite relation on Λ(Σ).

(i) The function space is the function ⇒ : P(Λ(Σ)) 2 → P(Λ(Σ)) defined as

A ⇒ B = def {t | ∀u. u ∈ A =⇒ t u ∈ B} . (ii) A reducibility family for → R is a set of sets Red ⊆ {A | X ⊆ A ⊆ SN R }
which is stable by intersections and by the function space. (iii) A type interpretation in Red is a map : T (B) → Red such that for all T, U ∈ T (B) we have U ⇒ T = U ⇒ T . (iv) A type interpretation is adequate if for all Γ , t, T and σ we have

Γ ⊢ τ t : T ∧ σ | = Γ =⇒ tσ ∈ T , where σ | = Γ iff σ(x) ∈ Γ (x) for all x ∈ Dom(Γ ).
Pure λ-Calculus. Let Red be a reducibility family and : T (B) → Red be a type interpretation. Let us see, in the case of the pure λ-calculus, some sufficient conditions to ensure that is adequate. As usual, we reason by induction on Γ ⊢ t : T and by cases on the last applied typing rule.

We only have to check the rules (Ax), (⇒ E) and (⇒ I). The rule (Ax) is trivial while (⇒ E) is dealt with by definition of the function space ⇒ . Concerning the rule (⇒ I), it is sufficient that for all A ∈ Red,

∀t, u ∈ Λ. (t[u/x] ∈ A ∧ u ∈ SN β ) =⇒ (λx.t)u ∈ A . (1)
As for X ⊆ A ⊆ SN β , condition (1) has to be preserved by ⇒ : Red 2 → Red.

We can conveniently formulate this by using elimination contexts [START_REF] Abel | Termination Checking with Types[END_REF]. For the pure λ-calculus, they are defined by the grammar

E[ ] ∈ E ⇒ ::= [ ] | E[ ] t .
Then, we get the following clauses: for all E[ ] ∈ E, all x ∈ X and all t, u ∈ Λ,

E[ ] ∈ SN β =⇒ E[x] ∈ A , (2)
(E[t[u/x]] ∈ A ∧ u ∈ SN β ) =⇒ E[(λx.t)u] ∈ A . (3)
The sets A ⊆ SN β satisfying (2) and ( 3) are Tait's saturated sets [START_REF] Tait | A Realizability Interpretation of the Theory of Species[END_REF].

∀v. E[x] → β v =⇒ (v = E ′ [x] with E[ ] → β E ′ [ ]) (4) ∀v. E[(λx.t)u] → β v =⇒ (v = E ′ [s] with (E[ ], (λx.t)u) → β (E ′ [ ], s)) (5)
Second, property (3) follows from (5) and the fact that (λx.t)u ∈ SN β as soon as t[u/x] ∈ SN β and u ∈ SN β . This property holds in turn thanks to the Weak Standardization Lemma, which was used in [START_REF] Altenkirch | Constructions, Inductive Types and Strong Normalization[END_REF] for extensions of the Calculus of Constructions. This is obvious for the pure λ-calculus.

Lemma 3.2 (Weak Standardization). A reduct of a β-redex (λx.t)u is either t[u/x] or a β-redex (λx.t ′ )u ′ with (t, u) → β (t ′ , u ′ ).
λ-Calculus with Rewriting. To deal with rewriting, we must consider the rule (Symb). For constructor, we have to use specific interpretations of base types (eg. using inductive types, as in [START_REF] Blanqui | Inductive-Data-Types Systems[END_REF]). We concentrate on symbols f ∈ Σ \ C. Given f of type ( T , T ) and t ∈ T , we have to make sure that f( t) ∈ T . Sufficient conditions for this are given by termination criteria, a subject that we do not treat in this paper (see for instance [START_REF] Blanqui | Inductive-Data-Types Systems[END_REF][START_REF] Abel | Termination Checking with Types[END_REF]2,5,[START_REF] Blanqui | Combining Typing and Size Constraints for Checking the Termination of Higher-Order Conditional Rewrite Systems[END_REF]).

Here, we are interested in the exploration of reducibility families that allow to formulate termination criteria. As for the λ-calculus, we can use a non-interaction property similar to (5):

∀v. E[f( t)] → βR v =⇒ v = E ′ [s] with (E[ ], f( t)) → βR (E ′ [ ], s) . ( 6
)
But rewrite systems do not satisfy in general the weak standardization lemma. Therefore, in order to get f( t) ∈ SN βR , we need v ∈ SN βR for all v such that f( t) → βR v. This is subsumed by the clause

∀v. E[f( t)] → βR v =⇒ v ∈ A =⇒ E[f( t)] ∈ A . (7)
In this case, we also need saturated sets to be stable by reduction: if t ∈ A and t → βR u then u ∈ A.

Neutral Terms and Reducibility Candidates

We now turn to Girard's reducibility candidates [START_REF] Girard | Interprétation Fonctionnelle et Élimination des Coupures de l'Arithmétique d[END_REF]. They form a reducibility family in which properties ( 3) and ( 7) can be formulated in a uniform and elegant way. This is due to neutral terms, that enjoy non-interaction properties such as (4), ( 5) and [START_REF] Blanqui | Inductive-Data-Types Systems[END_REF].

We first give a general formulation, and then apply it to constructor rewriting.

A General Formulation

We give a generalization of the original notion of neutral terms that allows to define reducibility candidates in a generic way. The key idea is that neutral terms are the terms that do not interact with elimination contexts. In the whole section, we assume given a rewrite relation → R . Elimination contexts will be defined as a special case of evaluation contexts.

Definition 4.1 (Evaluation Contexts). Let [ ] ∈ X be a distinguished vari- able. A set of evaluation contexts for → R is a set E of terms E[ ] linear in [ ], which is (i) stable by reduction: if E[ ] ∈ E and E[ ] → R t then t = F[ ] ∈ E; (ii) stable by composition: if E[ ] ∈ E and F[ ] ∈ E then E[F[ ]] ∈ E, where E[t] = def E[ ](t/[ ]).
We now assume given a set E of evaluation contexts for → R .

Definition 4.2 (Neutral Terms

). A neutral term for → R in E is a term t such that for all E[ ] ∈ E, ∀v. E[t] → R v =⇒ (v = E ′ [s] with (E[ ], t) → R (E ′ [ ], s)) .
We denote by N RE the set of neutral terms for → R in E.

The terms that are not neutral interact with evaluation contexts. They are therefore observable, and we think of them as being values.

Definition 4.3 (Values). A value for → R in E is a term which is not neutral. We denote by V RE the set of values for → R in E.

Example 4.4. In the pure λ-calculus, taking E ⇒ as evaluation contexts, the values are exactly the terms of the form λx.t.

To build reducibility candidates, we are interested in neutral terms and evaluation contexts that enjoy some properties. This leads to the notion of elimination contexts.

Definition 4.5 (Elimination Contexts). Let E be a set of evaluation contexts for → R . Then E is a set of elimination contexts for → R if all variables are neutral (X ⊆ N RE ), and if for all t ∈ N RE and all E[ ] ∈ E we have E[t] ∈ N RE .

Example 4.6. For the pure λ-calculus, E ⇒ is a set of elimination contexts.

We now define reducibility candidates in the usual way: our generalization regards neutral terms and their definition using elimination contexts. Assume that E is a set of elimination contexts for → R .

Definition 4.7 (Reducibility Candidates). The set CR

RE of reducibility can- didates for → R in E is the set of all C ⊆ SN R such that (CR0) if t ∈ C and t → R u then u ∈ C, (CR1) if t ∈ N RE and ∀u. t → R u =⇒ u ∈ C then t ∈ C.
Note that CR RE is a complete lattice for ⊆ whose top element is SN R . In order to verify that X is contained in any candidate, it is interesting to look at the least reducibility candidate. This is HN RE , the set of hereditary neutral terms, defined as the smallest set of terms such that

∀t ∈ N RE . (∀u. t → R u =⇒ u ∈ HN RE ) =⇒ t ∈ HN RE .
Since variables are neutral terms in normal form, we have X ⊆ HN RE ⊆ C for every C ∈ CR RE .

When E contains contexts of the form [ ] t with t ∈ SN R , reducibility candidates form a reducibility family. To show this, it remains to see that ⇒ is a function from CR 2 RE to CR RE . This follows from a simple but fundamental consequence of the non-interaction between neutral terms and evaluation contexts: Let

Lemma 4.8. Let t ∈ N RE and E[ ] ∈ E ∩ SN R . Then, for all C ∈ CR RE , (∀u. t → R u =⇒ E[u] ∈ C) =⇒ E[t] ∈ C . Proof. Since E[t] ∈ N RE , we have to show that (E[t]) R ⊆ C. We have t ∈ SN R because (t) R ⊆ C ⊆ SN R ,
v such that E[t] → R v. Since t is neutral, we have v = E ′ [t ′ ] with (E[ ], t) → R (E ′ [ ], t ′ )
, and there are two cases.

Case of E[

] → R E ′ [ ]. Note that E ′ [ ] ∈ E. For all u ∈ (t) R , since E[u] → R E ′ [u] and E[u] ∈ C, we have E ′ [u] ∈ C by (CR0

Application to Constructor Rewriting

Let R be a constructor rewrite system on C. If we use elimination contexts of the form E ⇒ , then the values are the terms of the form λx.t. However, we would like to build values from constructors. This is particularly useful with inductive types [START_REF] Blanqui | Inductive-Data-Types Systems[END_REF]. According to Def. 4.3, we have to make them observable. To this end, we introduce appropriate destructors in elimination contexts. To each c ∈ C of type ( T , B) with | T | > 0 and each i ∈ {1, . . . , | T |}, we associate a new unary destructor symbol d c,i defined by the rewrite rule

d c,i (c(x 1 , . . . , x n )) → D x i .
Let ℧ be a new nullary symbol. For the elimination of a nullary constructor c, we use a new unary destructor c defined by the rewrite rule

d c (c) → D ℧ .
Lemma 4.11. Let E ⇒ C be the set of terms defined by the grammar (iii) E ⇒ C is a set of elimination contexts for → βRD .

E[ ] ∈ E ⇒ C ::= [ ] | E[ ] t | d(E[ ]) ,
(iv) Each C ∈ CR βRDE ⇒ C satisfies properties (3) and [START_REF] Blanqui | Combining Typing and Size Constraints for Checking the Termination of Higher-Order Conditional Rewrite Systems[END_REF]. Indeed, we have

Proof. (i) It is clear that E βRDE ⇒ C is
(λx.t) u → β t[u/x] d S,1 (S(t)) → D t d 0 (0) → D ℧ .

Stability by Union

A reducibility family Red is stable by union if

∀R. R ⊆ Red =⇒ R ∈ Red .
The main question on stability by union is the following: given a rewrite relation → R , does there exists a reducibility family Red for → R which is stable by union and leads to an adequate type interpretation ? For the pure λ-calculus, it is well-known that the answer is positive: Tait's saturated sets (presented in Sec. 3) are stable by union and lead to an adequate type interpretation. This has been exploited for instance in [2,[START_REF] Tatsuta | Simple Saturated Sets for Disjunction and Second-Order Existential Quantification[END_REF].

The question becomes more difficult with rewriting. We have seen in Sec. 3 that rewrite systems do not satisfy in general the weak standardization lemma (Lem. 3.2), and that we need a reducibility family satisfying a clause like [START_REF] Blanqui | Combining Typing and Size Constraints for Checking the Termination of Higher-Order Conditional Rewrite Systems[END_REF]. But this is precisely what makes stability by union difficult. Assume given R ⊆ Red such that for all v ∈ (E[f( t)]) βR , we have v ∈ R. Then, unless we find some

A ∈ R such that (E[f( t)]) βR ⊆ A, there is no reason to have E[f( t)] ∈ R.
Besides, using intersection and union types, we have shown in [START_REF] Riba | Strong Normalization as Safe Interaction[END_REF] that there are confluent typed rewrite systems for which no reducibility family that is stable by union leads to an adequate type interpretation.

However, we can in some cases obtain a reducibility family which is stable by union. In [START_REF] Riba | On the Stability by Union of Reducibility Candidates[END_REF], we have given a necessary and sufficient condition for reducibility candidates to be stable by union; and in [START_REF] Riba | Strong Normalization as Safe Interaction[END_REF], we have given a necessary and sufficient condition for the closure by union of Biorthogonals [START_REF] Girard | Linear Logic[END_REF]16,[START_REF] Danos | Disjunctive Tautologies as Synchronisation Schemes[END_REF] to be reducibility candidates. The second condition is strictly more general than the first one.

We now recall the condition established in [START_REF] Riba | On the Stability by Union of Reducibility Candidates[END_REF] for the stability by union of reducibility candidates, and then show that it is met with orthogonal constructor rewriting.

Reducibility Candidates

The study of stability by union of reducibility candidates of [START_REF] Riba | On the Stability by Union of Reducibility Candidates[END_REF] carries on with our generalization of neutral terms and elimination contexts. The key observation is a characterization of the membership of a term to a candidate using a weak observational preorder.

Definition 5.1. Let t N u if and only if t, u ∈ SN R and ∀v ∈ V RE . t → * R v =⇒ u → * R v .
Every candidate C ∈ CR RE is a non-empty subset of SN R downward-closed wrt. N . Reducibility candidates are stable by union exactly when the converse is also true. Theorem 5.2 ( [START_REF] Riba | On the Stability by Union of Reducibility Candidates[END_REF]). The following are equivalent: (i) CR RE is stable by union; (ii) CR RE is the set of non-empty C ⊆ SN R that are downward-closed wrt. N ; (iii) every strongly normalizable neutral term t which is reducible has a reduct u such that t N u. Such u is a strong principal reduct1 of t.

Example 5. 3 ([18, 21]). For the pure λ-calculus, thanks to the weak standardization lemma (Lem. 3.2), CR βE⇒ is stable by union.

Application to Orthogonal Constructor Rewriting

Recall that a rewrite system R is orthogonal if it is left linear and has no critical pairs [START_REF]Term Rewriting Systems[END_REF]. For instance, the system of Ex. 2.1 is orthogonal.

In this section, we show that if R is an orthogonal constructor rewrite system, then CR βRDE ⇒ C is stable by union. According to Thm. 5.2, this amounts to show that every strongly normalizing reducible neutral term has a strong principal reduct. We prove it by using a general theorem on external redexes [START_REF] Khasidashvili | Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems[END_REF] (see also [START_REF] Glauert | Expression Reduction Systems and Extensions: An Overview. In Processes, Terms and Cycles: Steps to the Road of Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday[END_REF]). This result applies to the framework of orthogonal CCERSs, of which our higher-order rewrite systems β ∪ R ∪ D are an instance. t = f( t) and t is a RD-redex. Since φ is external in t, it is the roof RD-redex of t. Hence there is a rule l → RD r and a substitution σ such that t = lσ and u = rσ. Because l is linear, there is a substitution σ ′ such that σ → βRD σ ′ and v = lσ ′ . We are done by taking w = def rσ ′ since lσ ′ is a descendant of lσ and rσ → * βRD rσ ′ . ⊓ ⊔ Lemma 5.5. Assume that R is orthogonal and let t ∈ N βRDE ⇒ C . If t → * βRD v with v ∈ V βRDE ⇒ C , then for all u such that t → βRD u by contracting an external redex of t, we have u → * βRD v. Proof. Let u such that t → βRD u by contracting an external redex of t and let v ∈ V βRDE ⇒ C such that t → * βRD v. Since t is neutral, we have t → + βRD v. We reason by induction on the length of the derivation.

Base Case. Since t is neutral, it follows from Lem. 4.11.(ii) that t → βRD v by contracting a top redex of t. Since every external redex is outermost, it follows that every external redex of t is a root redex of t. By orthogonality, this is the redex contracted in t → βRD v, and we have u = v. Induction Case. Assume that t → βRD s → * βRD v. If s is a value, then reasoning as in the base case we have s = u, hence u → * βRD v. Otherwise s is neutral. If s is obtained from t by contracting the same redex as in t → βRD u, then s = u and u → * βRD v. Otherwise, by weak standardization (Lem. 5.4), there is w such that u → * βRD w and s → βRD w by contracting an external redex of s. We then have w → * βRD v by induction hypothesis, hence u → * βRD v.

⊓ ⊔

If t is neutral and t → βRD u by contracting an external redex of t, then we have t N u. Hence u is a strong principal reduct of t. Then, by Thm. 5.2 CR βRDE ⇒ C is stable by union if all neutral terms have external redexes. It reto show this last property, which follows from the next theorem, proved in [START_REF] Khasidashvili | Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems[END_REF]. We can apply it because the CCERS β ∪ R ∪ D is orthogonal as soon as the rewrite system R is orthogonal.

Theorem 5.6 ( [START_REF] Khasidashvili | Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems[END_REF][START_REF] Glauert | Expression Reduction Systems and Extensions: An Overview. In Processes, Terms and Cycles: Steps to the Road of Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday[END_REF]). If R is orthogonal then every reducible term has an external redex.

Theorem 5.7. If R is orthogonal then CR βRDE ⇒ C is stable by union.

  and moreover we have E[ ] ∈ SN R by assumption. We can therefore reason by induction on pairs (E[ ], t) ordered by → R .

  where d is a destructor of a constructor of C. Then, (i) E ⇒ C is a set of evaluation contexts for → βRD . (ii) The values in E ⇒ C for → βRD are exactly the terms of the form λx.t; or c( t) with c ∈ C.

  ). Hence, we can apply the induction hypothesis on (E ′ [ ], t) and we conclude thatE ′ [t] ∈ C. Case of t → R t ′ . In this case, we have E[t ′ ] ∈ C by assumption. If E contains [ ] t for all t ∈ Λ(Σ), then CR RE is a reducibility family. Proof. It remains to show that ⇒ : CR 2 RE → CR RE . Let A, B ∈ CR RE . First, we have A ⇒ B ⊆ SN R : for all t ∈ A ⇒ B, since X ⊆ A we have tx ∈ B ⊆ SN R , hence t ∈ SN R .Let us now check the clauses (CR0) and (CR1).(CR0) Let t ∈ A ⇒ B and u ∈ (t) R . For all a ∈ A, we have ta ∈ B, hence ua ∈ B by (CR0) applied to B. It follows that u ∈ A ⇒ B. (CR1) Let t ∈ N RE such that (t) R ⊆ A ⇒ B and let a ∈ A. For all u ∈ (t)

	⊓ ⊔
	Proposition 4.9.

R , we have ua ∈ B. Since [ ] a ∈ SN R , it follows from Lem. 4.8 that ta ∈ B. We conclude that t ∈ A ⇒ B. ⊓ ⊔ Example 4.10. For the pure λ-calculus, CR βE⇒ is the usual set of reducibility candidates. In particular, each C ∈ CR βE⇒ satisfies property (3).

  a set of evaluation contexts for → βRD . (ii) It is clear that the terms of (ii) are values. We check that if t is a value, then it is of one of these forms. Assume now that E[ ] ∈ E ⇒ C is a minimal context that interact with t. Note that the top symbol of E[ ] is either an application or a destructor d. We consider these two cases: The top symbol of E[ ] is an application. In this case, E[ ] is of the form F[ ] u. By minimality, F[ ] = [ ] and t is an abstraction. The top symbol of E[ ] is a destructor. In this case, E[ ] is of the form d(F[ ]). By minimality, F[ ] = [ ] and t is a constructor. (iii) The fact that E ⇒ C is a set elimination contexts is a direct consequence of the shape of values (ii). (iv) From (ii) we know that terms of the form (λx.t)u and f( t) with f ∈ Σ \ C are neutral. Properties (3) and (7) then follows from Lem. 4.8.

			⊓ ⊔
	Example 4.12. Consider the system presented at example 2.1. Its values are
	the terms of the form
	λx.t	S(t)	0 .

Called "principal reduct" in[START_REF] Riba | On the Stability by Union of Reducibility Candidates[END_REF].

We use the notions of descendants and of redex-arguments of [START_REF] Khasidashvili | Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems[END_REF][START_REF] Glauert | Expression Reduction Systems and Extensions: An Overview. In Processes, Terms and Cycles: Steps to the Road of Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday[END_REF]. The key notion is that of external redex [START_REF] Khasidashvili | Perpetuality ans Uniform Normalization in Orthogonal Rewrite Systems[END_REF][START_REF] Glauert | Expression Reduction Systems and Extensions: An Overview. In Processes, Terms and Cycles: Steps to the Road of Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday[END_REF]: a redex u in a term t is external if for any derivation P : t → * βRD v, no descendant of u along P appears inside redex-arguments. Hence every descendant of an external redex is external. Note that every external redex is outermost, but the converse is false.

Using external redexes, we have the following weak standardization lemma. We apply it in Lem. 5.5 to show that every neutral term that has an external redex has a strong reduct.

Lemma 5.4 (Weak Standardization). Assume that R is orthogonal. Let t → βRD u by contracting an external redex of t. If t → βRD v by contracting a different redex, then there is w such that u → * βRD w and v → βRD w by contracting a descendant of t → βRD u (which is therefore external in v).

Proof. We use the following standard notations: a position in a term t is a finite word on N \ {0}, φ • ψ denotes the concatenations of the words φ and ψ, and t[u] φ is the term t in which the subterm at position φ is textually replaced by u.

Let φ and ψ be the respective positions of the redexes contracted in t → βRD u and t → βRD v. We show that there exists w such that the redex contracted in v → βRD w is a descendant of φ. It is an external redex of v because φ is external in t.

Note that since φ is external in t, for every